Download or read book From Polynomials to Sums of Squares written by T.H Jackson and published by CRC Press. This book was released on 2023-05-09 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: From Polynomials to Sums of Squares describes a journey through the foothills of algebra and number theory based around the central theme of factorization. The book begins by providing basic knowledge of rational polynomials, then gradually introduces other integral domains, and eventually arrives at sums of squares of integers. The text is complemented with illustrations that feature specific examples. Other than familiarity with complex numbers and some elementary number theory, very little mathematical prerequisites are needed. The accompanying disk enables readers to explore the subject further by removing the tedium of doing calculations by hand. Throughout the text there are practical activities involving the computer.
Download or read book From Polynomials to Sums of Squares written by Terence Hugh Jackson and published by . This book was released on 2020 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: From Polynomials to Sums of Squares describes a journey through the foothills of algebra and number theory based around the central theme of factorization. The book begins by providing basic knowledge of rational polynomials, then gradually introduces other integral domains, and eventually arrives at sums of squares of integers. The text is complemented with illustrations that feature specific examples. Other than familiarity with complex numbers and some elementary number theory, very little mathematical prerequisites are needed. The accompanying disk enables readers to explore the subject further by removing the tedium of doing calculations by hand. Throughout the text there are practical activities involving the computer.
Download or read book Positive Polynomials and Sums of Squares written by Murray Marshall and published by American Mathematical Soc.. This book was released on 2008 with total page 201 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book From Polynomials to Sums of Squares written by T.H Jackson and published by CRC Press. This book was released on 2023-05-09 with total page 195 pages. Available in PDF, EPUB and Kindle. Book excerpt: From Polynomials to Sums of Squares describes a journey through the foothills of algebra and number theory based around the central theme of factorization. The book begins by providing basic knowledge of rational polynomials, then gradually introduces other integral domains, and eventually arrives at sums of squares of integers. The text is complemented with illustrations that feature specific examples. Other than familiarity with complex numbers and some elementary number theory, very little mathematical prerequisites are needed. The accompanying disk enables readers to explore the subject further by removing the tedium of doing calculations by hand. Throughout the text there are practical activities involving the computer.
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.
Download or read book Semidefinite Optimization and Convex Algebraic Geometry written by Grigoriy Blekherman and published by SIAM. This book was released on 2013-03-21 with total page 487 pages. Available in PDF, EPUB and Kindle. Book excerpt: An accessible introduction to convex algebraic geometry and semidefinite optimization. For graduate students and researchers in mathematics and computer science.
Download or read book Positive Polynomials in Control written by Didier Henrion and published by Springer Science & Business Media. This book was released on 2005-01-14 with total page 332 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Emerging Applications of Algebraic Geometry written by Mihai Putinar and published by Springer Science & Business Media. This book was released on 2008-12-10 with total page 382 pages. Available in PDF, EPUB and Kindle. Book excerpt: Recent advances in both the theory and implementation of computational algebraic geometry have led to new, striking applications to a variety of fields of research. The articles in this volume highlight a range of these applications and provide introductory material for topics covered in the IMA workshops on "Optimization and Control" and "Applications in Biology, Dynamics, and Statistics" held during the IMA year on Applications of Algebraic Geometry. The articles related to optimization and control focus on burgeoning use of semidefinite programming and moment matrix techniques in computational real algebraic geometry. The new direction towards a systematic study of non-commutative real algebraic geometry is well represented in the volume. Other articles provide an overview of the way computational algebra is useful for analysis of contingency tables, reconstruction of phylogenetic trees, and in systems biology. The contributions collected in this volume are accessible to non-experts, self-contained and informative; they quickly move towards cutting edge research in these areas, and provide a wealth of open problems for future research.
Download or read book Positive Trigonometric Polynomials and Signal Processing Applications written by Bogdan Alexandru Dumitrescu and published by Springer Science & Business Media. This book was released on 2007-04-27 with total page 253 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Positive Trigonometric Polynomials and Signal Processing Applications written by Bogdan Dumitrescu and published by Springer. This book was released on 2017-03-20 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Moments Positive Polynomials and Their Applications written by Jean-Bernard Lasserre and published by World Scientific. This book was released on 2010 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Download or read book Polynomials written by and published by Springer Science & Business Media. This book was released on 2009 with total page 311 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Sum of Squares Theory and Applications written by Pablo A. Parrilo and published by American Mathematical Soc.. This book was released on with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is based on lectures delivered at the 2019 AMS Short Course “Sum of Squares: Theory and Applications”, held January 14–15, 2019, in Baltimore, Maryland. This book provides a concise state-of-the-art overview of the theory and applications of polynomials that are sums of squares. This is an exciting and timely topic, with rich connections to many areas of mathematics, including polynomial and semidefinite optimization, real and convex algebraic geometry, and theoretical computer science. The six chapters introduce and survey recent developments in this area; specific topics include the algebraic and geometric aspects of sums of squares and spectrahedra, lifted representations of convex sets, and the algorithmic and computational implications of viewing sums of squares as a meta algorithm. The book also showcases practical applications of the techniques across a variety of areas, including control theory, statistics, finance and machine learning.
Download or read book Polynomials written by Victor V. Prasolov and published by Springer Science & Business Media. This book was released on 2009-09-23 with total page 311 pages. Available in PDF, EPUB and Kindle. Book excerpt: Covers its topic in greater depth than the typical standard books on polynomial algebra
Download or read book Real Algebraic Geometry written by Michel Coste and published by Springer. This book was released on 2006-11-15 with total page 425 pages. Available in PDF, EPUB and Kindle. Book excerpt: Ten years after the first Rennes international meeting on real algebraic geometry, the second one looked at the developments in the subject during the intervening decade - see the 6 survey papers listed below. Further contributions from the participants on recent research covered real algebra and geometry, topology of real algebraic varieties and 16thHilbert problem, classical algebraic geometry, techniques in real algebraic geometry, algorithms in real algebraic geometry, semialgebraic geometry, real analytic geometry. CONTENTS: Survey papers: M. Knebusch: Semialgebraic topology in the last ten years.- R. Parimala: Algebraic and topological invariants of real algebraic varieties.- Polotovskii, G.M.: On the classification of decomposing plane algebraic curves.- Scheiderer, C.: Real algebra and its applications to geometry in the last ten years: some major developments and results.- Shustin, E.L.: Topology of real plane algebraic curves.- Silhol, R.: Moduli problems in real algebraic geometry. Further contributions by: S. Akbulut and H. King; C. Andradas and J. Ruiz; A. Borobia; L. Br|cker; G.W. Brumfield; A. Castilla; Z. Charzynski and P. Skibinski; M. Coste and M. Reguiat; A. Degtyarev; Z. Denkowska; J.-P. Francoise and F. Ronga; J.M. Gamboa and C. Ueno; D. Gondard- Cozette; I.V. Itenberg; P. Jaworski; A. Korchagin; T. Krasinksi and S. Spodzieja; K. Kurdyka; H. Lombardi; M. Marshall and L. Walter; V.F. Mazurovskii; G. Mikhalkin; T. Mostowski and E. Rannou; E.I. Shustin; N. Vorobjov.
Download or read book System Structure and Control 2004 written by Sabine Mondie and published by Elsevier. This book was released on 2005-05-11 with total page 780 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Intermediate Algebra 2e written by Lynn Marecek and published by . This book was released on 2020-05-06 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
