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Polynomial Resolution Theory
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Book Synopsis Polynomial Resolution Theory by : William A. Hardy
Download or read book Polynomial Resolution Theory written by William A. Hardy and published by Trafford Publishing. This book was released on 2005 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Polynomials written by Cheon Seoung Ryoo and published by BoD – Books on Demand. This book was released on 2019-05-02 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt: Polynomials are well known for their ability to improve their properties and for their applicability in the interdisciplinary fields of engineering and science. Many problems arising in engineering and physics are mathematically constructed by differential equations. Most of these problems can only be solved using special polynomials. Special polynomials and orthonormal polynomials provide a new way to analyze solutions of various equations often encountered in engineering and physical problems. In particular, special polynomials play a fundamental and important role in mathematics and applied mathematics. Until now, research on polynomials has been done in mathematics and applied mathematics only. This book is based on recent results in all areas related to polynomials. Divided into sections on theory and application, this book provides an overview of the current research in the field of polynomials. Topics include cyclotomic and Littlewood polynomials; Descartes' rule of signs; obtaining explicit formulas and identities for polynomials defined by generating functions; polynomials with symmetric zeros; numerical investigation on the structure of the zeros of the q-tangent polynomials; investigation and synthesis of robust polynomials in uncertainty on the basis of the root locus theory; pricing basket options by polynomial approximations; and orthogonal expansion in time domain method for solving Maxwell's equations using paralleling-in-order scheme.
Book Synopsis Introduction To The Theory Of Weighted Polynomial Approximation by : H N Mhaskar
Download or read book Introduction To The Theory Of Weighted Polynomial Approximation written by H N Mhaskar and published by World Scientific. This book was released on 1997-01-04 with total page 398 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Polynomial Methods and Incidence Theory by : Adam Sheffer
Download or read book Polynomial Methods and Incidence Theory written by Adam Sheffer and published by Cambridge University Press. This book was released on 2022-03-24 with total page 263 pages. Available in PDF, EPUB and Kindle. Book excerpt: A thorough yet accessible introduction to the mathematical breakthroughs achieved by using new polynomial methods in the past decade.
Book Synopsis Noncommutative Polynomial Algebras of Solvable Type and Their Modules by : Huishi Li
Download or read book Noncommutative Polynomial Algebras of Solvable Type and Their Modules written by Huishi Li and published by CRC Press. This book was released on 2021-11-08 with total page 177 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Analytic Theory of Polynomials by : Qazi Ibadur Rahman
Download or read book Analytic Theory of Polynomials written by Qazi Ibadur Rahman and published by Oxford University Press. This book was released on 2002 with total page 760 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents easy to understand proofs of same of the most difficult results about polynomials demonstrated by means of applications
Book Synopsis Minimal Free Resolutions over Complete Intersections by : David Eisenbud
Download or read book Minimal Free Resolutions over Complete Intersections written by David Eisenbud and published by Springer. This book was released on 2016-03-08 with total page 113 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces a theory of higher matrix factorizations for regular sequences and uses it to describe the minimal free resolutions of high syzygy modules over complete intersections. Such resolutions have attracted attention ever since the elegant construction of the minimal free resolution of the residue field by Tate in 1957. The theory extends the theory of matrix factorizations of a non-zero divisor, initiated by Eisenbud in 1980, which yields a description of the eventual structure of minimal free resolutions over a hypersurface ring. Matrix factorizations have had many other uses in a wide range of mathematical fields, from singularity theory to mathematical physics.
Book Synopsis Theory of Uniform Approximation of Functions by Polynomials by : Vladislav K. Dzyadyk
Download or read book Theory of Uniform Approximation of Functions by Polynomials written by Vladislav K. Dzyadyk and published by Walter de Gruyter. This book was released on 2008-09-25 with total page 497 pages. Available in PDF, EPUB and Kindle. Book excerpt: A thorough, self-contained and easily accessible treatment of the theory on the polynomial best approximation of functions with respect to maximum norms. The topics include Chebychev theory, Weierstraß theorems, smoothness of functions, and continuation of functions.
Download or read book Polynomials written by Cheon Seoung Ryoo and published by . This book was released on 2020-10-12 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: Polynomial and its applications are well known for their proven properties and excellent applicability in interdisciplinary fields of science. Until now, research on polynomial and its applications has been done in mathematics, applied mathematics, and sciences. This book is based on recent results in all areas related to polynomial and its applications. This book provides an overview of the current research in the field of polynomials and its applications. The following papers have been published in this volume: 'A Parametric Kind of the Degenerate Fubini Numbers and Polynomials'; 'On 2-Variables Konhauser Matrix Polynomials and Their Fractional Integrals'; 'Fractional Supersymmetric Hermite Polynomials'; 'Rational Approximation for Solving an Implicitly Given Colebrook Flow Friction Equation'; 'Iterating the Sum of Möbius Divisor Function and Euler Totient Function'; 'Differential Equations Arising from the Generating Function of the (r, β)-Bell Polynomials and Distribution of Zeros of Equations'; 'Truncated Fubini Polynomials'; 'On Positive Quadratic Hyponormality of a Unilateral Weighted Shift with Recursively Generated by Five Weights'; 'Ground State Solutions for Fractional Choquard Equations with Potential Vanishing at Infinity'; 'Some Identities on Degenerate Bernstein and Degenerate Euler Polynomials'; 'Some Identities Involving Hermite Kampé de Fériet Polynomials Arising from Differential Equations and Location of Their Zeros.'
Book Synopsis Solving Systems of Polynomial Equations by : Bernd Sturmfels
Download or read book Solving Systems of Polynomial Equations written by Bernd Sturmfels and published by American Mathematical Soc.. This book was released on 2002 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt: Bridging a number of mathematical disciplines, and exposing many facets of systems of polynomial equations, Bernd Sturmfels's study covers a wide spectrum of mathematical techniques and algorithms, both symbolic and numerical.
Book Synopsis Sparse Polynomial Optimization: Theory And Practice by : Victor Magron
Download or read book Sparse Polynomial Optimization: Theory And Practice written by Victor Magron and published by World Scientific. This book was released on 2023-04-25 with total page 223 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Polynomials written by Edward Barbeau and published by New York : Springer-Verlag. This book was released on 1989 with total page 472 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a backdrop for study in calculus, modern algebra, numerical analysis and complex variable theory, through examples. Includes some 300 problems drawn from journals, contests, and examinations to test understanding, ingenuity, and skill.
Book Synopsis Resolution of Curve and Surface Singularities in Characteristic Zero by : K. Kiyek
Download or read book Resolution of Curve and Surface Singularities in Characteristic Zero written by K. Kiyek and published by Springer Science & Business Media. This book was released on 2004-10 with total page 522 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Polynomial Root-finding and Polynomiography by : Bahman Kalantari
Download or read book Polynomial Root-finding and Polynomiography written by Bahman Kalantari and published by World Scientific. This book was released on 2009 with total page 492 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers fascinating and modern perspectives into the theory and practice of the historical subject of polynomial root-finding, rejuvenating the field via polynomiography, a creative and novel computer visualization that renders spectacular images of a polynomial equation. Polynomiography will not only pave the way for new applications of polynomials in science and mathematics, but also in art and education. The book presents a thorough development of the basic family, arguably the most fundamental family of iteration functions, deriving many surprising and novel theoretical and practical applications such as: algorithms for approximation of roots of polynomials and analytic functions, polynomiography, bounds on zeros of polynomials, formulas for the approximation of Pi, and characterizations or visualizations associated with a homogeneous linear recurrence relation. These discoveries and a set of beautiful images that provide new visions, even of the well-known polynomials and recurrences, are the makeup of a very desirable book. This book is a must for mathematicians, scientists, advanced undergraduates and graduates, but is also for anyone with an appreciation for the connections between a fantastically creative art form and its ancient mathematical foundations.
Book Synopsis Fields and Galois Theory by : John M. Howie
Download or read book Fields and Galois Theory written by John M. Howie and published by Springer Science & Business Media. This book was released on 2007-09-05 with total page 230 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Book Synopsis Solving Polynomial Equation Systems by : Teo Mora
Download or read book Solving Polynomial Equation Systems written by Teo Mora and published by Cambridge University Press. This book was released on 2003 with total page 833 pages. Available in PDF, EPUB and Kindle. Book excerpt: Covers extensions of Buchberger's Theory and Algorithm, and promising recent alternatives to Gröbner bases.
Book Synopsis Certificates of Positivity for Real Polynomials by : Victoria Powers
Download or read book Certificates of Positivity for Real Polynomials written by Victoria Powers and published by Springer Nature. This book was released on 2021-11-26 with total page 161 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book collects and explains the many theorems concerning the existence of certificates of positivity for polynomials that are positive globally or on semialgebraic sets. A certificate of positivity for a real polynomial is an algebraic identity that gives an immediate proof of a positivity condition for the polynomial. Certificates of positivity have their roots in fundamental work of David Hilbert from the late 19th century on positive polynomials and sums of squares. Because of the numerous applications of certificates of positivity in mathematics, applied mathematics, engineering, and other fields, it is desirable to have methods for finding, describing, and characterizing them. For many of the topics covered in this book, appropriate algorithms, computational methods, and applications are discussed. This volume contains a comprehensive, accessible, up-to-date treatment of certificates of positivity, written by an expert in the field. It provides an overview of both the theory and computational aspects of the subject, and includes many of the recent and exciting developments in the area. Background information is given so that beginning graduate students and researchers who are not specialists can learn about this fascinating subject. Furthermore, researchers who work on certificates of positivity or use them in applications will find this a useful reference for their work.
