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Bilinear Algebra

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Bilinear Algebra

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Author : Kazimierz Szymiczek
language : en
Publisher: Routledge
Release Date : 2017-11-22

Bilinear Algebra written by Kazimierz Szymiczek and has been published by Routledge this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-11-22 with Mathematics categories.

Giving an easily accessible elementary introduction to the algebraic theory of quadratic forms, this book covers both Witt's theory and Pfister's theory of quadratic forms. Leading topics include the geometry of bilinear spaces, classification of bilinear spaces up to isometry depending on the ground field, formally real fields, Pfister forms, the Witt ring of an arbitrary field (characteristic two included), prime ideals of the Witt ring, Brauer group of a field, Hasse and Witt invariants of quadratic forms, and equivalence of fields with respect to quadratic forms. Problem sections are included at the end of each chapter. There are two appendices: the first gives a treatment of Hasse and Witt invariants in the language of Steinberg symbols, and the second contains some more advanced problems in 10 groups, including the u-invariant, reduced and stable Witt rings, and Witt equivalence of fields.

Symmetric Bilinear Forms

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Author : John Milnor
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-17

Symmetric Bilinear Forms written by John Milnor and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-17 with Mathematics categories.

The theory cf quadratic forms and the intimately related theory of sym metrie bilinear forms have a lang and rich his tory, highlighted by the work of Legendre, Gauss, Minkowski, and Hasse. (Compare [Dickson] and [Bourbaki, 24, p. 185].) Our exposition will concentrate on the rela tively recent developments which begin with and are inspired by Witt's 1937 paper "Theorie der quadratischen Formen in beliebigen Körpern." We will be particularly interested in the work of A. Pfister and M. Knebusch. However, some older material will be described, particularly in Chapter II. The presentation is based on lectures by Milnor at the Institute for Ad vanced Study, and at Haverford College under the Phillips Lecture Pro gram, during the Fall of 1970, as weIl as Iectures at Princeton University il1 1966. We want to thank J. Cunningham, M. Knebusch, M. Kneser, A. Rosenberg, W. Scharlau and J.-P. Serre for helpful suggestions and corrections. Prerequisites. The reader should be familiar with the rudiments of algebra., incJuding for example the concept of tensor product for mo dules over a commutative ring. A few individual sections will require quite a bit more. The logical relationship between the various chapters can be roughly described by the diagram below. There are also five appendices, largely self-contained, which treat special topics. I. Arbitrary commutative rings I H. The ring of V. Miscellaneous IIl. Fields integers examples IV. Dedekind domains Contents Chapter r. Basie Coneepts ...

Specialization Of Quadratic And Symmetric Bilinear Forms

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Author : Manfred Knebusch
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-01-22

Specialization Of Quadratic And Symmetric Bilinear Forms written by Manfred Knebusch and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-01-22 with Mathematics categories.

A Mathematician Said Who Can Quote Me a Theorem that’s True? For the ones that I Know Are Simply not So, When the Characteristic is Two! This pretty limerick ?rst came to my ears in May 1998 during a talk by T.Y. Lam 1 on ?eld invariants from the theory of quadratic forms. It is—poetic exaggeration allowed—a suitable motto for this monograph. What is it about? At the beginning of the seventies I drew up a specialization theoryofquadraticandsymmetricbilinear formsover ?elds[32].Let? : K? L?? be a place. Then one can assign a form? (?)toaform? over K in a meaningful way ? if? has “good reduction” with respect to? (see§1.1). The basic idea is to simply apply the place? to the coe?cients of?, which must therefore be in the valuation ring of?. The specialization theory of that time was satisfactory as long as the ?eld L, and therefore also K, had characteristic 2. It served me in the ?rst place as the foundation for a theory of generic splitting of quadratic forms [33], [34]. After a very modest beginning, this theory is now in full bloom. It became important for the understanding of quadratic forms over ?elds, as can be seen from the book [26]of Izhboldin–Kahn–Karpenko–Vishik for instance. One should note that there exists a theoryof(partial)genericsplittingofcentralsimplealgebrasandreductivealgebraic groups, parallel to the theory of generic splitting of quadratic forms (see [29] and the literature cited there).

Lectures On The Complexity Of Bilinear Problems

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Author : Hans F. de Groote
language : en
Publisher: Springer Science & Business Media
Release Date : 1987-02-23

Lectures On The Complexity Of Bilinear Problems written by Hans F. de Groote and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1987-02-23 with Computers categories.

Bilinear Forms And Zonal Polynomials

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Author : Arak M. Mathai
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Bilinear Forms And Zonal Polynomials written by Arak M. Mathai and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

The book deals with bilinear forms in real random vectors and their generalizations as well as zonal polynomials and their applications in handling generalized quadratic and bilinear forms. The book is mostly self-contained. It starts from basic principles and brings the readers to the current research level in these areas. It is developed with detailed proofs and illustrative examples for easy readability and self-study. Several exercises are proposed at the end of the chapters. The complicated topic of zonal polynomials is explained in detail in this book. The book concentrates on the theoretical developments in all the topics covered. Some applications are pointed out but no detailed application to any particular field is attempted. This book can be used as a textbook for a one-semester graduate course on quadratic and bilinear forms and/or on zonal polynomials. It is hoped that this book will be a valuable reference source for graduate students and research workers in the areas of mathematical statistics, quadratic and bilinear forms and their generalizations, zonal polynomials, invariant polynomials and related topics, and will benefit statisticians, mathematicians and other theoretical and applied scientists who use any of the above topics in their areas. Chapter 1 gives the preliminaries needed in later chapters, including some Jacobians of matrix transformations. Chapter 2 is devoted to bilinear forms in Gaussian real ran dom vectors, their properties, and techniques specially developed to deal with bilinear forms where the standard methods for handling quadratic forms become complicated.

Neutrosophic Bilinear Algebras And Their Generalizations

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Author : W. B. Vasantha Kandasamy
language : en
Publisher: Infinite Study
Release Date : 2010

Neutrosophic Bilinear Algebras And Their Generalizations written by W. B. Vasantha Kandasamy and has been published by Infinite Study this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010 with Mathematics categories.

This book introduces over one hundred new concepts related to neutrosophic bilinear algebras and their generalizations. Illustrated by more than 225 examples, these innovative new notions find applications in various fields.

Bilinear Algebra

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Author : Kazimierz Szymiczek
language : en
Publisher: Routledge
Release Date : 2017-11-22

Bilinear Control Systems

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Author : David Elliott
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-09-01

Bilinear Control Systems written by David Elliott and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-09-01 with Science categories.

The mathematical theory of control became a ?eld of study half a century ago in attempts to clarify and organize some challenging practical problems and the methods used to solve them. It is known for the breadth of the mathematics it uses and its cross-disciplinary vigor. Its literature, which can befoundinSection93ofMathematicalReviews,wasatonetimedominatedby the theory of linear control systems, which mathematically are described by linear di?erential equations forced by additive control inputs. That theory led to well-regarded numerical and symbolic computational packages for control analysis and design. Nonlinear control problems are also important; in these either the - derlying dynamical system is nonlinear or the controls are applied in a n- additiveway.Thelastfourdecadeshaveseenthedevelopmentoftheoretical work on nonlinear control problems based on di?erential manifold theory, nonlinear analysis, and several other mathematical disciplines. Many of the problems that had been solved in linear control theory, plus others that are new and distinctly nonlinear, have been addressed; some resulting general de?nitions and theorems are adapted in this book to the bilinear case.

Computational Complexity Of Bilinear Forms

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Author : Hari Krishna
language : en
Publisher: Springer
Release Date : 1987-04-07

Computational Complexity Of Bilinear Forms written by Hari Krishna and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 1987-04-07 with Language Arts & Disciplines categories.

The Algebraic And Geometric Theory Of Quadratic Forms

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Author : Richard S. Elman
language : en
Publisher: American Mathematical Soc.
Release Date : 2008-07-15

The Algebraic And Geometric Theory Of Quadratic Forms written by Richard S. Elman and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-07-15 with Mathematics categories.

This book is a comprehensive study of the algebraic theory of quadratic forms, from classical theory to recent developments, including results and proofs that have never been published. The book is written from the viewpoint of algebraic geometry and includes the theory of quadratic forms over fields of characteristic two, with proofs that are characteristic independent whenever possible. For some results both classical and geometric proofs are given. Part I includes classical algebraic theory of quadratic and bilinear forms and answers many questions that have been raised in the early stages of the development of the theory. Assuming only a basic course in algebraic geometry, Part II presents the necessary additional topics from algebraic geometry including the theory of Chow groups, Chow motives, and Steenrod operations. These topics are used in Part III to develop a modern geometric theory of quadratic forms.

Bilinear Algebra

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