LEADER 04530nam 2200493 450 001 996466561803316 005 20230427132327.0 010 $a9783030855475$b(electronic bk.) 010 $z9783030855468$b(print) 035 $a(MiAaPQ)EBC6812200 035 $a(Au-PeEL)EBL6812200 035 $a(CKB)19919399100041 035 $a(OCoLC)1287133630 035 $a(PPN)258838981 035 $a(EXLCZ)9919919399100041 100 $a20220818d2021 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aCertificates of positivity for real polynomials $etheory, practice, and applications /$fVictoria Powers 210 1$aCham, Switzerland :$cSpringer,$d[2021] 210 4$d©2021 215 $a1 online resource (161 pages) $cillustrations 225 1 $aDevelopments in mathematics ;$vVolume 69 311 08$aPrint version: Powers, Victoria Certificates of Positivity for Real Polynomials Cham : Springer International Publishing AG,c2022 9783030855468 320 $aIncludes bibliographical references and index. 327 $aIntro -- 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. 327 $a9.1.1 Pólya's Theorem with Zeros -- 9.2 Certificates in Preprimes and Quadratic Modules -- 9.3 Applications -- 9.3.1 Approximating the Stability Number of a Graph -- 9.3.2 Minimization of Polynomials on Polytopes -- References -- 10 The Noncompact Case -- 10.1 Saturation in the Noncompact Case -- 10.2 Stability and Open Cones -- 10.3 Cylinders with Compact Cross-Section -- References -- 11 Sums of Squares of Rational Polynomials -- 11.1 Weighted Sums of Squares Representations -- 11.2 The Univariate Case -- 11.3 Sums of Rational Squares -- 11.4 Algorithmic Approaches -- References -- 12 Positive Polynomials with Special Structure -- 12.1 Symmetric Polynomials -- 12.2 Diagonal-Tail Forms -- 12.3 Agiforms -- 12.4 Circuit Polynomials -- 12.5 Application: Global Optimization Using Geometric Programming -- References -- Appendix A Real Algebra and Algebraic Geometry -- A.1 Algebraic Sets -- A.2 Real Fields -- A.3 Tarski-Seidenberg Theorem -- A.4 The Real Spectrum -- References -- Appendix Index of Notation -- Index. 410 0$aDevelopments in mathematics ;$vVolume 69. 606 $aPolinomis$2thub 606 $aPolynomials 608 $aLlibres electrònics$2thub 615 7$aPolinomis 615 0$aPolynomials. 676 $a512.942 700 $aPowers$b Victoria$01068607 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 912 $a996466561803316 996 $aCertificates of Positivity for Real Polynomials$92553541 997 $aUNISA LEADER 03449nam 2200637 450 001 9910789388503321 005 20230617013019.0 010 $a0-8132-2060-2 035 $a(CKB)3710000000095179 035 $a(EBL)3135140 035 $a(SSID)ssj0001339259 035 $a(PQKBManifestationID)11832885 035 $a(PQKBTitleCode)TC0001339259 035 $a(PQKBWorkID)11366766 035 $a(PQKB)10396495 035 $a(MiAaPQ)EBC3135140 035 $a(Au-PeEL)EBL3135140 035 $a(CaPaEBR)ebr10842029 035 $a(OCoLC)922997127 035 $a(EXLCZ)993710000000095179 100 $a20140310h20032003 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt 182 $cc 183 $acr 200 10$aDialogue with Trypho /$fSt. Justin Martyr ; translated by Thomas B. Falls ; revised and with a new introduction by Thomas P. Halton ; edited by Michael Slusser 210 1$aWashington, District of Columbia :$cCatholic University of America Press,$d2003. 210 4$d©2003 215 $a1 online resource (256 p.) 225 1 $aSelections from the Fathers of the Church ;$vVolume 3 300 $aDescription based upon print version of record. 311 $a0-8132-1342-8 320 $aIncludes bibliographical references and indexes. 327 $a""Contents ""; ""Acknowledgments""; ""Abbreviations""; ""Introduction to This Edition""; ""Introduction to the First Edition""; ""Dialogue with Trypho""; ""Justin meets Trypho""; ""How Justin found philosophy""; ""Justin offers to defend Christian belief""; ""A new covenant, not a new God""; ""Jewish ritual laws as a remedy for idolatry""; ""A Christian interpretation of ritual laws""; ""The Messiah: already here or still to come?""; ""Divisions among Christians""; ""The divinity of the Messiah""; ""Scriptural types of Jesus""; ""Those who live by the Law of Moses"" 327 $a""The Messiah must be Jesus""""The unity of God and the divinity of Christ""; ""The virginal birth""; ""Corrections to the text of Scripture""; ""[The Second Day's Discussion]""; ""More on the birth of Jesus""; ""Millennium at the end of the world""; ""Interpreting Scripture about Christ""; ""The problem of the crucifixion""; ""How Christians read Psalm 22""; ""The sign of Jonah""; ""Reprise of points from the first day""; ""Jesus as high priest""; ""The Gentiles as God's new people""; ""The names of Christ in Scripture""; ""The unity of God and the divinity of Christ: reprise "" 327 $a""The election of the Jews and Gentiles""""A final appeal""; ""Conclusion""; ""Select Bibliography""; ""General Index""; ""Index of Holy Scripture"" 410 0$aFathers of the church.$kSelections ;$vv. 3. 606 $aChristianity and other religions$xJudaism$vEarly works to 1800 606 $aJudaism$xRelations$xChristianity$vEarly works to 1800 606 $aApologetics$vEarly works to 1800 615 0$aChristianity and other religions$xJudaism 615 0$aJudaism$xRelations$xChristianity 615 0$aApologetics 676 $a239.2 700 $aJustin$cMartyr, Saint.$0742176 701 $aFalls$b Thomas B$01574949 701 $aHalton$b Thomas P$g(Thomas Patrick)$01555183 701 $aSlusser$b Michael$f1940-$01574950 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910789388503321 996 $aDialogue with Trypho$93851553 997 $aUNINA