Positive Polynomials
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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 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, 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 Product Type Actions Of Compact Groups
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Author : David Handelman
language : en
Publisher: American Mathematical Soc.
Release Date : 1985
Positive Polynomials And Product Type Actions Of Compact Groups written by David Handelman 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 1985 with Mathematics categories.
This paper is largely devoted to the behaviour of operators which are polynomials (in several variables) in full and one-sided shifts, together with their corresponding Toeplitz operators, with consequences for product type actions of compact connected Lie groups. Problems are of the type, for which vectors will some power of the given operator change the vector into one with only non-negative coefficients.
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.
1. The generalized moment problem. 1.1. Formulations. 1.2. Duality theory. 1.3. Computational complexity. 1.4. Summary. 1.5. Exercises. 1.6. Notes and sources -- 2. Positive polynomials. 2.1. Sum of squares representations and semi-definite optimization. 2.2. Nonnegative versus s.o.s. polynomials. 2.3. Representation theorems : univariate case. 2.4. Representation theorems : mutivariate case. 2.5. Polynomials positive on a compact basic semi-algebraic set. 2.6. Polynomials nonnegative on real varieties. 2.7. Representations with sparsity properties. 2.8. Representation of convex polynomials. 2.9. Summary. 2.10. Exercises. 2.11. Notes and sources -- 3. Moments. 3.1. The one-dimensional moment problem. 3.2. The multi-dimensional moment problem. 3.3. The K-moment problem. 3.4. Moment conditions for bounded density. 3.5. Summary. 3.6. Exercises. 3.7. Notes and sources -- 4. Algorithms for moment problems. 4.1. The overall approach. 4.2. Semidefinite relaxations. 4.3. Extraction of solutions. 4.4. Linear relaxations. 4.5. Extensions. 4.6. Exploiting sparsity. 4.7. Summary. 4.8. Exercises. 4.9. Notes and sources. 4.10. Proofs -- 5. Global optimization over polynomials. 5.1. The primal and dual perspectives. 5.2. Unconstrained polynomial optimization. 5.3. Constrained polynomial optimization : semidefinite relaxations. 5.4. Linear programming relaxations. 5.5. Global optimality conditions. 5.6. Convex polynomial programs. 5.7. Discrete optimization. 5.8. Global minimization of a rational function. 5.9. Exploiting symmetry. 5.10. Summary. 5.11. Exercises. 5.12. Notes and sources -- 6. Systems of polynomial equations. 6.1. Introduction. 6.2. Finding a real solution to systems of polynomial equations. 6.3. Finding all complex and/or all real solutions : a unified treatment. 6.4. Summary. 6.5. Exercises. 6.6. Notes and sources -- 7. Applications in probability. 7.1. Upper bounds on measures with moment conditions. 7.2. Measuring basic semi-algebraic sets. 7.3. Measures with given marginals. 7.4. Summary. 7.5. Exercises. 7.6. Notes and sources -- 8. Markov chains applications. 8.1. Bounds on invariant measures. 8.2. Evaluation of ergodic criteria. 8.3. Summary. 8.4. Exercises. 8.5. Notes and sources -- 9. Application in mathematical finance. 9.1. Option pricing with moment information. 9.2. Option pricing with a dynamic model. 9.3. Summary. 9.4. Notes and sources -- 10. Application in control. 10.1. Introduction. 10.2. Weak formulation of optimal control problems. 10.3. Semidefinite relaxations for the OCP. 10.4. Summary. 10.5. Notes and sources -- 11. Convex envelope and representation of convex sets. 11.1. The convex envelope of a rational function. 11.2. Semidefinite representation of convex sets. 11.3. Algebraic certificates of convexity. 11.4. Summary. 11.5. Exercises. 11.6. Notes and sources -- 12. Multivariate integration 12.1. Integration of a rational function. 12.2. Integration of exponentials of polynomials. 12.3. Maximum entropy estimation. 12.4. Summary. 12.5. Exercises. 12.6. Notes and sources -- 13. Min-max problems and Nash equilibria. 13.1. Robust polynomial optimization. 13.2. Minimizing the sup of finitely many rational cunctions. 13.3. Application to Nash equilibria. 13.4. Exercises. 13.5. Notes and sources -- 14. Bounds on linear PDE. 14.1. Linear partial differential equations. 14.2. Notes and sources
Positive Polynomials In Control
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Author : Didier Henrion
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-01-14
Positive Polynomials In Control written by Didier Henrion 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 2005-01-14 with Technology & Engineering categories.
Positive Polynomials in Control originates from an invited session presented at the IEEE CDC 2003 and gives a comprehensive overview of existing results in this quickly emerging area. This carefully edited book collects important contributions from several fields of control, optimization, and mathematics, in order to show different views and approaches of polynomial positivity. The book is organized in three parts, reflecting the current trends in the area: 1. applications of positive polynomials and LMI optimization to solve various control problems, 2. a mathematical overview of different algebraic techniques used to cope with polynomial positivity, 3. numerical aspects of positivity of polynomials, and recently developed software tools which can be employed to solve the problems discussed in the book.
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.
Positive Polynomials Convex Integral Polytopes And A Random Walk Problem
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Author : David E. Handelman
language : en
Publisher: Springer
Release Date : 2006-11-15
Positive Polynomials Convex Integral Polytopes And A Random Walk Problem written by David E. Handelman and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-15 with Mathematics categories.
Emanating from the theory of C*-algebras and actions of tori theoren, the problems discussed here are outgrowths of random walk problems on lattices. An AGL (d,Z)-invariant (which is a partially ordered commutative algebra) is obtained for lattice polytopes (compact convex polytopes in Euclidean space whose vertices lie in Zd), and certain algebraic properties of the algebra are related to geometric properties of the polytope. There are also strong connections with convex analysis, Choquet theory, and reflection groups. This book serves as both an introduction to and a research monograph on the many interconnections between these topics, that arise out of questions of the following type: Let f be a (Laurent) polynomial in several real variables, and let P be a (Laurent) polynomial with only positive coefficients; decide under what circumstances there exists an integer n such that Pnf itself also has only positive coefficients. It is intended to reach and be of interest to a general mathematical audience as well as specialists in the areas mentioned.
Positive Polynomials
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Author : Alexander Prestel
language : en
Publisher:
Release Date : 2014-01-15
Positive Polynomials written by Alexander Prestel and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-15 with categories.
Positive Polynomials In Control
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Author : Didier Henrion
language : en
Publisher: Springer
Release Date : 2009-09-02
Positive Polynomials In Control written by Didier Henrion and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-09-02 with Technology & Engineering categories.
Positive Polynomials in Control originates from an invited session presented at the IEEE CDC 2003 and gives a comprehensive overview of existing results in this quickly emerging area. This carefully edited book collects important contributions from several fields of control, optimization, and mathematics, in order to show different views and approaches of polynomial positivity. The book is organized in three parts, reflecting the current trends in the area: 1. applications of positive polynomials and LMI optimization to solve various control problems, 2. a mathematical overview of different algebraic techniques used to cope with polynomial positivity, 3. numerical aspects of positivity of polynomials, and recently developed software tools which can be employed to solve the problems discussed in the book.
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.