04530nam 2200493 450 99646656180331620230427132327.09783030855475(electronic bk.)9783030855468(print)(MiAaPQ)EBC6812200(Au-PeEL)EBL6812200(CKB)19919399100041(OCoLC)1287133630(PPN)258838981(EXLCZ)991991939910004120220818d2021 uy 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierCertificates of positivity for real polynomials theory, practice, and applications /Victoria PowersCham, Switzerland :Springer,[2021]©20211 online resource (161 pages) illustrationsDevelopments in mathematics ;Volume 69Print version: Powers, Victoria Certificates of Positivity for Real Polynomials Cham : Springer International Publishing AG,c2022 9783030855468 Includes bibliographical references and index.Intro -- Preface -- Contents -- 0 Introduction -- 1 Preliminaries -- 1.1 Basic Notation -- 1.2 The Real Numbers -- 1.3 Polynomials -- 1.4 Polynomials Versus Forms -- 1.5 Matrices and Quadratic Forms -- 1.6 Convex Sets and Cones -- Reference -- 2 Sums of Squares and Positive Polynomials -- 2.1 SOS and PSD Polynomials -- 2.2 Hilbert's 17th Problem -- 2.2.1 Uniform Denominators -- 2.2.2 Degree Bounds -- 2.3 The Gap Between PSD and SOS Polynomials -- References -- 3 Global Certificates of Positivity -- 3.1 The Gram Matrix Method -- 3.2 From Gram Matrices to Certificates -- 3.3 Application: Global Optimization of Polynomials -- References -- 4 Positive Semidefinite Ternary Quartics -- 4.1 Proofs of Hilbert's Theorem on Ternary Quartics -- 4.2 Finding and Counting Sum of Squares Representations -- References -- 5 Positivity on Semialgebraic Sets -- 5.1 Semialgebraic Sets -- 5.2 Preorders and Quadratic Modules -- 5.3 Positivity Properties -- 5.4 Saturated Quadratic Modules and Preorders -- 5.5 The Positivstellensatz -- 5.6 Sums of Squares and Positivity on Algebraic Sets -- 5.7 Application: The Moment Problem -- References -- 6 The Archimedean Property -- 6.1 Archimedean T-Modules -- 6.2 Marshall's Representation Theorem -- 6.3 Application: Theorems of Handelman and Pólya -- References -- 7 Theorems of Schmüdgen and Putinar -- 7.1 Schmüdgen's Positivstellensatz -- 7.1.1 Wörmann's Proof -- 7.2 Putinar's Positivstellenstz -- 7.3 Minimal Representations -- 7.4 Constructive Proofs -- 7.4.1 Degree Bounds -- 7.5 Application: Polynomial Optimization on Compact Semialgebraic Sets -- References -- 8 The Dimension One Case -- 8.1 Positivity on a Closed Interval -- 8.1.1 Bernstein Representations -- 8.2 Basic Closed Semialgebraic Subsets of mathbbR -- 8.3 Positivity on Curves in the Plane -- References -- 9 Positivity on Polytopes -- 9.1 Pólya's Theorem.9.1.1 Pólya's Theorem with Zeros -- 9.2 Certificates in Preprimes and Quadratic Modules -- 9.3 Applications -- 9.3.1 Approximating the Stability Number of a Graph -- 9.3.2 Minimization of Polynomials on Polytopes -- References -- 10 The Noncompact Case -- 10.1 Saturation in the Noncompact Case -- 10.2 Stability and Open Cones -- 10.3 Cylinders with Compact Cross-Section -- References -- 11 Sums of Squares of Rational Polynomials -- 11.1 Weighted Sums of Squares Representations -- 11.2 The Univariate Case -- 11.3 Sums of Rational Squares -- 11.4 Algorithmic Approaches -- References -- 12 Positive Polynomials with Special Structure -- 12.1 Symmetric Polynomials -- 12.2 Diagonal-Tail Forms -- 12.3 Agiforms -- 12.4 Circuit Polynomials -- 12.5 Application: Global Optimization Using Geometric Programming -- References -- Appendix A Real Algebra and Algebraic Geometry -- A.1 Algebraic Sets -- A.2 Real Fields -- A.3 Tarski-Seidenberg Theorem -- A.4 The Real Spectrum -- References -- Appendix Index of Notation -- Index.Developments in mathematics ;Volume 69.PolinomisthubPolynomialsLlibres electrònicsthubPolinomisPolynomials.512.942Powers Victoria1068607MiAaPQMiAaPQMiAaPQ996466561803316Certificates of Positivity for Real Polynomials2553541UNISA01627nam 2200517 450 991082913260332120220525000655.01-5129-7772-1607-98003-3-0(CKB)4100000008951038(MiAaPQ)EBC6433810(Au-PeEL)EBL6433810(OCoLC)1228037392(MiAaPQ)EBC6771213(OCoLC)1165368425(FINmELB)ELB121745(EXLCZ)99410000000895103820220525d2018 uy 0spaurcnu||||||||txtrdacontentcrdamediacrrdacarrierInvestigaciones educativas /edited by Joaquín Hernández GonzálezDistrito Federal, México :Bonilla Artigas Editores,[2018]©20181 online resource (448 pages)Publica Educación ;9"La segunda parte de este libro presenta un conjunto de trabajos de varios discípulos de Eduardo Weiss"--Page 441.607-98003-0-6 Includes bibliographical references.Páublica educación ;9.EducationMexicoHistoryEducationResearchMexicoMexicofastEducationHistory.EducationResearch370.972Hernández González JoaquínMiAaPQMiAaPQMiAaPQBOOK9910829132603321Investigaciones educativas3999281UNINA