Positive Polynomials
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Positive Polynomials
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Author : Alexander Prestel
language : en
Publisher: Springer Science & Business Media
Release Date : 2001-04-24
Positive Polynomials written by Alexander Prestel 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 2001-04-24 with Mathematics categories.
Positivity is one of the most basic mathematical concepts, involved in many areas of mathematics (analysis, real algebraic geometry, functional analysis, etc.). The main objective of the book is to give useful characterizations of polynomials. Beyond basic knowledge in algebra, only valuation theory as explained in the appendix is needed.
Positive Polynomials And Sums Of Squares
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Author : Murray Marshall
language : en
Publisher: American Mathematical Soc.
Release Date : 2008
Positive Polynomials And Sums Of Squares written by Murray Marshall 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 with Mathematics categories.
The study of positive polynomials brings together algebra, geometry and analysis. The subject is of fundamental importance in real algebraic geometry when studying the properties of objects defined by polynomial inequalities. Hilbert's 17th problem and its solution in the first half of the 20th century were landmarks in the early days of the subject. More recently, new connections to the moment problem and to polynomial optimization have been discovered. The moment problem relates linear maps on the multidimensional polynomial ring to positive Borel measures. This book provides an elementary introduction to positive polynomials and sums of squares, the relationship to the moment problem, and the application to polynomial optimization. The focus is on the exciting new developments that have taken place in the last 15 years, arising out of Schmudgen's solution to the moment problem in the compact case in 1991. The book is accessible to a well-motivated student at the beginning graduate level. The objects being dealt with are concrete and down-to-earth, namely polynomials in $n$ variables with real coefficients, and many examples are included. Proofs are presented as clearly and as simply as possible. Various new, simpler proofs appear in the book for the first time. Abstraction is employed only when it serves a useful purpose, but, at the same time, enough abstraction is included to allow the reader easy access to the literature. The book should be essential reading for any beginning student in the area.
Positive Trigonometric Polynomials And Signal Processing Applications
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Author : Bogdan Dumitrescu
language : en
Publisher: Springer
Release Date : 2018-07-21
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 2018-07-21 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.
Moments Positive Polynomials And Their Applications
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Author : Jean-Bernard Lasserre
language : en
Publisher: World Scientific
Release Date : 2010
Moments Positive Polynomials And Their Applications written by Jean-Bernard Lasserre and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010 with Mathematics categories.
Many important problems in global optimization, algebra, probability and statistics, applied mathematics, control theory, financial mathematics, inverse problems, etc. can be modeled as a particular instance of the Generalized Moment Problem (GMP). This book introduces, in a unified manual, a new general methodology to solve the GMP when its data are polynomials and basic semi-algebraic sets. This methodology combines semidefinite programming with recent results from real algebraic geometry to provide a hierarchy of semidefinite relaxations converging to the desired optimal value. Applied on appropriate cones, standard duality in convex optimization nicely expresses the duality between moments and positive polynomials. In the second part of this invaluable volume, the methodology is particularized and described in detail for various applications, including global optimization, probability, optimal context, mathematical finance, multivariate integration, etc., and examples are provided for each particular application.
Moments Positive Polynomials And Their Applications
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Author : Jean Bernard Lasserre
language : en
Publisher: World Scientific
Release Date : 2009-10-02
Moments Positive Polynomials And Their Applications written by Jean Bernard Lasserre and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-10-02 with Mathematics categories.
Many important applications in global optimization, algebra, probability and statistics, applied mathematics, control theory, financial mathematics, inverse problems, etc. can be modeled as a particular instance of the Generalized Moment Problem (GMP).This book introduces a new general methodology to solve the GMP when its data are polynomials and basic semi-algebraic sets. This methodology combines semidefinite programming with recent results from real algebraic geometry to provide a hierarchy of semidefinite relaxations converging to the desired optimal value. Applied on appropriate cones, standard duality in convex optimization nicely expresses the duality between moments and positive polynomials.In the second part, the methodology is particularized and described in detail for various applications, including global optimization, probability, optimal control, mathematical finance, multivariate integration, etc., and examples are provided for each particular application.
Positive Polynomials
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Author : Alexander Prestel
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-17
Positive Polynomials written by Alexander Prestel 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.
Positivity is one of the most basic mathematical concepts. In many areas of mathematics (like analysis, real algebraic geometry, functional analysis, etc.) it shows up as positivity of a polynomial on a certain subset of R^n which itself is often given by polynomial inequalities. The main objective of the book is to give useful characterizations of such polynomials. It takes as starting point Hilbert's 17th Problem from 1900 and explains how E. Artin's solution of that problem eventually led to the development of real algebra towards the end of the 20th century. Beyond basic knowledge in algebra, only valuation theory as explained in the appendix is needed. Thus the monograph can also serve as the basis for a 2-semester course in real algebra.
Positivity In Algebraic Geometry 2
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Author : R.K. Lazarsfeld
language : en
Publisher: Springer Science & Business Media
Release Date : 2004-08-24
Positivity In Algebraic Geometry 2 written by R.K. Lazarsfeld 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 2004-08-24 with Mathematics categories.
This two volume work on "Positivity in Algebraic Geometry" contains a contemporary account of a body of work in complex algebraic geometry loosely centered around the theme of positivity. Topics in Volume I include ample line bundles and linear series on a projective variety, the classical theorems of Lefschetz and Bertini and their modern outgrowths, vanishing theorems, and local positivity. Volume II begins with a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. A good deal of this material has not previously appeared in book form, and substantial parts are worked out here in detail for the first time. At least a third of the book is devoted to concrete examples, applications, and pointers to further developments. Whereas Volume I is more elementary, the present Volume II is more at the research level and somewhat more specialized. Both volumes are also available as hardcover edition as Vols. 48 and 49 in the series "Ergebnisse der Mathematik und ihrer Grenzgebiete".
Certificates Of Positivity For Real Polynomials
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Author : Victoria Powers
language : en
Publisher: Springer Nature
Release Date : 2021-11-26
Certificates Of Positivity For Real Polynomials written by Victoria Powers and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-11-26 with Mathematics categories.
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.
Handbook On Semidefinite Conic And Polynomial Optimization
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Author : Miguel F. Anjos
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-11-19
Handbook On Semidefinite Conic And Polynomial Optimization written by Miguel F. Anjos 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 2011-11-19 with Business & Economics categories.
Semidefinite and conic optimization is a major and thriving research area within the optimization community. Although semidefinite optimization has been studied (under different names) since at least the 1940s, its importance grew immensely during the 1990s after polynomial-time interior-point methods for linear optimization were extended to solve semidefinite optimization problems. Since the beginning of the 21st century, not only has research into semidefinite and conic optimization continued unabated, but also a fruitful interaction has developed with algebraic geometry through the close connections between semidefinite matrices and polynomial optimization. This has brought about important new results and led to an even higher level of research activity. This Handbook on Semidefinite, Conic and Polynomial Optimization provides the reader with a snapshot of the state-of-the-art in the growing and mutually enriching areas of semidefinite optimization, conic optimization, and polynomial optimization. It contains a compendium of the recent research activity that has taken place in these thrilling areas, and will appeal to doctoral students, young graduates, and experienced researchers alike. The Handbook’s thirty-one chapters are organized into four parts: Theory, covering significant theoretical developments as well as the interactions between conic optimization and polynomial optimization; Algorithms, documenting the directions of current algorithmic development; Software, providing an overview of the state-of-the-art; Applications, dealing with the application areas where semidefinite and conic optimization has made a significant impact in recent years.
An Introduction To Polynomial And Semi Algebraic Optimization
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Author : Jean Bernard Lasserre
language : en
Publisher: Cambridge University Press
Release Date : 2015-02-19
An Introduction To Polynomial And Semi Algebraic Optimization written by Jean Bernard Lasserre 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 2015-02-19 with Mathematics categories.
The first comprehensive introduction to the powerful moment approach for solving global optimization problems.