Polynomial Resolution Theory
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Polynomial Resolution Theory
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Author : William A. Hardy
language : en
Publisher: Trafford Publishing
Release Date : 2005
Polynomial Resolution Theory written by William A. Hardy and has been published by Trafford Publishing this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with Education categories.
This book is the definitive work on polynomial solution theory. Starting with the simplest linear equations with complex coefficients, this book proceeds in a step by step logical manner to outline the method for solving equations of arbitrarily high degree. Polynomial Resolution Theory is an invaluable book because of its unique perspective on the age old problem of solving polynomial equations of arbitrarily high degree. First of all Hardy insists upon pursuing the subject by using general complex coefficients rather than restricting himself to real coefficients. Complex numbers are used in ordered pair (x,y) form rather than the more traditional x + iy (or x + jy) notation. As Hardy comments, "The Fundamental Theorem of Algebra makes the treatments of polynomials with complex coefficients mandatory. We must not allow applications to direct the way mathematics is presented, but must permit the mathematical results themselves determine how to present the subject. Although practical, real-world applications are important, they must not be allowed to dictate the way in which a subject is treated. Thus, although there are at present no practical applications which employ polynomials with complex coefficients, we must present this subject with complex rather than restrictive real coefficients." This book then proceeds to recast familiar results in a more consistent notation for later progress. Two methods of solution to the general cubic equation with complex coefficients are presented. Then Ferrari's solution to the general complex bicubic (fourth degree) polynomial equation is presented. After this Hardy seamlessly presents the first extension of Ferrari's work to resolving the general bicubic (sixth degree) equation with complex coefficients into two component cubic equations. Eight special cases of this equation which are solvable in closed form are developed with detailed examples. Next the resolution of the octal (eighth degree) polynomial equation is developed along with twelve special cases which are solvable in closed form. This book is appropriate for students at the advanced college algebra level who have an understanding of the basic arithmetic of the complex numbers and know how to use a calculator which handles complex numbers directly. Hardy continues to develop the theory of polynomial resolution to equations of degree forty-eight. An extensive set of appendices is useful for verifying derived results and for rigging various special case equations. This is the 3rd edition of Hardy's book.
Noncommutative Polynomial Algebras Of Solvable Type And Their Modules
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Author : Huishi Li
language : en
Publisher: CRC Press
Release Date : 2021-11-08
Noncommutative Polynomial Algebras Of Solvable Type And Their Modules written by Huishi Li and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-11-08 with Mathematics categories.
Noncommutative Polynomial Algebras of Solvable Type and Their Modules is the first book to systematically introduce the basic constructive-computational theory and methods developed for investigating solvable polynomial algebras and their modules. In doing so, this book covers: A constructive introduction to solvable polynomial algebras and Gröbner basis theory for left ideals of solvable polynomial algebras and submodules of free modules The new filtered-graded techniques combined with the determination of the existence of graded monomial orderings The elimination theory and methods (for left ideals and submodules of free modules) combining the Gröbner basis techniques with the use of Gelfand-Kirillov dimension, and the construction of different kinds of elimination orderings The computational construction of finite free resolutions (including computation of syzygies, construction of different kinds of finite minimal free resolutions based on computation of different kinds of minimal generating sets), etc. This book is perfectly suited to researchers and postgraduates researching noncommutative computational algebra and would also be an ideal resource for teaching an advanced lecture course.
Introduction To The Theory Of Weighted Polynomial Approximation
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Author : H N Mhaskar
language : en
Publisher: World Scientific
Release Date : 1997-01-04
Introduction To The Theory Of Weighted Polynomial Approximation written by H N Mhaskar and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-01-04 with Mathematics categories.
In this book, we have attempted to explain a variety of different techniques and ideas which have contributed to this subject in its course of successive refinements during the last 25 years. There are other books and surveys reviewing the ideas from the perspective of either potential theory or orthogonal polynomials. The main thrust of this book is to introduce the subject from an approximation theory point of view. Thus, the main motivation is to study analogues of results from classical trigonometric approximation theory, introducing other ideas as needed. It is not our objective to survey the most recent results, but merely to introduce to the readers the thought processes and ideas as they are developed.This book is intended to be self-contained, although the reader is expected to be familiar with rudimentary real and complex analysis. It will also help to have studied elementary trigonometric approximation theory, and have some exposure to orthogonal polynomials.
Resolution Of Curve And Surface Singularities In Characteristic Zero
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Author : K. Kiyek
language : en
Publisher: Springer Science & Business Media
Release Date : 2004-10
Resolution Of Curve And Surface Singularities In Characteristic Zero written by K. Kiyek 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 2004-10 with Mathematics categories.
This book covers the beautiful theory of resolutions of surface singularities in characteristic zero. The primary goal is to present in detail, and for the first time in one volume, two proofs for the existence of such resolutions. One construction was introduced by H.W.E. Jung, and another is due to O. Zariski. Jung's approach uses quasi-ordinary singularities and an explicit study of specific surfaces in affine three-space. In particular, a new proof of the Jung-Abhyankar theorem is given via ramification theory. Zariski's method, as presented, involves repeated normalisation and blowing up points. It also uses the uniformization of zero-dimensional valuations of function fields in two variables, for which a complete proof is given. Despite the intention to serve graduate students and researchers of Commutative Algebra and Algebraic Geometry, a basic knowledge on these topics is necessary only. This is obtained by a thorough introduction of the needed algebraic tools in the two appendices.
Positive Trigonometric Polynomials And Signal Processing Applications
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Author : Bogdan Dumitrescu
language : en
Publisher: Springer
Release Date : 2017-03-20
Positive Trigonometric Polynomials And Signal Processing Applications written by Bogdan Dumitrescu and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-03-20 with Technology & Engineering categories.
This book gathers the main recent results on positive trigonometric polynomials within a unitary framework. The book has two parts: theory and applications. The theory of sum-of-squares trigonometric polynomials is presented unitarily based on the concept of Gram matrix (extended to Gram pair or Gram set). The applications part is organized as a collection of related problems that use systematically the theoretical results.
Fields And Galois Theory
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Author : John M. Howie
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-09-05
Fields And Galois Theory written by John M. Howie 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 2007-09-05 with Mathematics categories.
A modern and student-friendly introduction to this popular subject: it takes a more "natural" approach and develops the theory at a gentle pace with an emphasis on clear explanations Features plenty of worked examples and exercises, complete with full solutions, to encourage independent study Previous books by Howie in the SUMS series have attracted excellent reviews
Sparse Polynomial Optimization Theory And Practice
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Author : Victor Magron
language : en
Publisher: World Scientific
Release Date : 2023-04-25
Sparse Polynomial Optimization Theory And Practice written by Victor Magron and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-04-25 with Mathematics categories.
Many applications, including computer vision, computer arithmetic, deep learning, entanglement in quantum information, graph theory and energy networks, can be successfully tackled within the framework of polynomial optimization, an emerging field with growing research efforts in the last two decades. One key advantage of these techniques is their ability to model a wide range of problems using optimization formulations. Polynomial optimization heavily relies on the moment-sums of squares (moment-SOS) approach proposed by Lasserre, which provides certificates for positive polynomials. On the practical side, however, there is 'no free lunch' and such optimization methods usually encompass severe scalability issues. Fortunately, for many applications, including the ones formerly mentioned, we can look at the problem in the eyes and exploit the inherent data structure arising from the cost and constraints describing the problem.This book presents several research efforts to resolve this scientific challenge with important computational implications. It provides the development of alternative optimization schemes that scale well in terms of computational complexity, at least in some identified class of problems. It also features a unified modeling framework to handle a wide range of applications involving both commutative and noncommutative variables, and to solve concretely large-scale instances. Readers will find a practical section dedicated to the use of available open-source software libraries.This interdisciplinary monograph is essential reading for students, researchers and professionals interested in solving optimization problems with polynomial input data.
Orthogonal Polynomials On The Unit Circle Spectral Theory
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Author : Barry Simon
language : en
Publisher: American Mathematical Soc.
Release Date : 2005
Orthogonal Polynomials On The Unit Circle Spectral Theory written by Barry Simon 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 2005 with Mathematics categories.
Presents an overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. This book discusses topics such as asymptotics of Toeplitz determinants (Szego's theorems), and limit theorems for the density of the zeros of orthogonal polynomials.
Solving Polynomial Equation Systems Iv Volume 4 Buchberger Theory And Beyond
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Author : Teo Mora
language : en
Publisher: Cambridge University Press
Release Date : 2016-04-01
Solving Polynomial Equation Systems Iv Volume 4 Buchberger Theory And Beyond written by Teo Mora and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-04-01 with Mathematics categories.
In this fourth and final volume the author extends Buchberger's Algorithm in three different directions. First, he extends the theory to group rings and other Ore-like extensions, and provides an operative scheme that allows one to set a Buchberger theory over any effective associative ring. Second, he covers similar extensions as tools for discussing parametric polynomial systems, the notion of SAGBI-bases, Gröbner bases over invariant rings and Hironaka's theory. Finally, Mora shows how Hilbert's followers - notably Janet, Gunther and Macaulay - anticipated Buchberger's ideas and discusses the most promising recent alternatives by Gerdt (involutive bases) and Faugère (F4 and F5). This comprehensive treatment in four volumes is a significant contribution to algorithmic commutative algebra that will be essential reading for algebraists and algebraic geometers.
Polynomial Methods And Incidence Theory
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Author : Adam Sheffer
language : en
Publisher: Cambridge University Press
Release Date : 2022-03-24
Polynomial Methods And Incidence Theory written by Adam Sheffer and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-03-24 with Mathematics categories.
A thorough yet accessible introduction to the mathematical breakthroughs achieved by using new polynomial methods in the past decade.