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000755833 019__ $$a951977463
000755833 020__ $$a9783319333380$$q(electronic book)
000755833 020__ $$a3319333380$$q(electronic book)
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000755833 08204 $$a519.6$$223
000755833 1001_ $$aBurgdorf, Sabine,$$eauthor.
000755833 24510 $$aOptimization of polynomials in non-commuting variables$$h[electronic resource] /$$cSabine Burgdorf, Igor Klep, Janez Povh.
000755833 264_1 $$aSwitzerland :$$bSpringer,$$c2016.
000755833 300__ $$a1 online resource (xv, 104 pages) :$$bcolor illustrations.
000755833 336__ $$atext$$btxt$$2rdacontent
000755833 337__ $$acomputer$$bc$$2rdamedia
000755833 338__ $$aonline resource$$bcr$$2rdacarrier
000755833 4901_ $$aSpringerBriefs in mathematics,$$x2191-8198
000755833 504__ $$aIncludes bibliographical references and index.
000755833 5050_ $$aIntroduction; Organization of the Book; References; Contents; List of Figures; List of Tables; 1 Selected Results from Algebra and Mathematical Optimization ; 1.1 Positive Semidefinite Matrices; 1.2 Words and Polynomials in Non-commuting Variables; 1.3 Sums of Hermitian Squares and Gram Matrices; 1.4 Quadratic Modules and Semialgebraic Sets; 1.5 Gelfand-Naimark-Segal's Construction; 1.6 Sums of Hermitian Squares and Positivity; 1.7 Vanishing Nc Polynomials; 1.8 Hankel Matrices and Flatness; 1.9 Commutators, Cyclic Equivalence, and Trace Zero Polynomials
000755833 5058_ $$a1.10 Cyclic Quadratic Modules and Trace-Positivity1.11 Wedderburn Theorem; 1.12 Curto-Fialkow's Theorems; Implementation; 1.13 Semidefinite Programming; References; 2 Detecting Sums of Hermitian Squares; 2.1 Introduction; 2.2 The Gram Matrix Method; 2.3 Newton Chip Method; 2.4 Augmented Newton Chip Method; 2.5 Implementation; 2.5.1 On the Gram Matrix Method; 2.5.2 Software Package NCSOStools; References; 3 Cyclic Equivalence to Sums of Hermitian Squares; 3.1 Introduction; 3.2 The Cyclic Degree; 3.3 The Tracial Newton Polytope; 3.4 The Tracial Gram Matrix Method; 3.5 Implementation
000755833 5058_ $$a3.5.1 Detecting Members of Theta3.5.2 BMV Polynomials; References; 4 Eigenvalue Optimization of Polynomials in Non-commuting Variables; 4.1 Introduction; 4.2 Unconstrained Optimization; 4.2.1 Unconstrained Optimization as a Single SDP; 4.2.2 Extracting Optimizers for the Unconstrained Case; 4.3 Constrained Eigenvalue Optimization of Non-commutative Polynomials; 4.3.1 Approximation Hierarchy; 4.3.2 Extracting Optimizers; 4.4 Constrained Optimization over the Nc Ball and the Nc Polydisc; 4.4.1 Approximation Hierarchies Contain Only One Member; 4.4.2 Extracting Optimizers; 4.5 Implementation
000755833 5058_ $$a4.5.1 Application to Quantum MechanicsReferences; 5 Trace Optimization of Polynomials in Non-commuting Variables; 5.1 Introduction; 5.2 Unconstrained Trace Optimization; 5.3 Constrained Trace Optimization; 5.4 Flatness and Extracting Optimizers; 5.5 Implementation; References; List of Symbols; Index
000755833 506__ $$aAccess limited to authorized users.
000755833 520__ $$aThis book presents recent results on positivity and optimization of polynomials in non-commuting variables. Researchers in non-commutative algebraic geometry, control theory, system engineering, optimization, quantum physics and information science will find the unified notation and mixture of algebraic geometry and mathematical programming useful. Theoretical results are matched with algorithmic considerations; several examples and information on how to use NCSOStools open source package to obtain the results provided. Results are presented on detecting the eigenvalue and trace positivity of polynomials in non-commuting variables using Newton chip method and Newton cyclic chip method, relaxations for constrained and unconstrained optimization problems, semidefinite programming formulations of the relaxations and finite convergence of the hierarchies of these relaxations, and the practical efficiency of algorithms.
000755833 588__ $$aOnline resource; title from PDF title page (SpringerLink, viewed June 20, 2016).
000755833 650_0 $$aMathematical optimization.
000755833 650_0 $$aPolynomials.
000755833 650_0 $$aVariables (Mathematics)
000755833 7001_ $$aKlep, Igor,$$eauthor.
000755833 7001_ $$aPovh, Janez,$$d1973-$$eauthor.
000755833 77608 $$iPrint version:$$aBurgdorf, Sabine$$tOptimization of Polynomials in Non-Commuting Variables$$dCham : Springer International Publishing,c2016$$z9783319333366
000755833 830_0 $$aSpringerBriefs in mathematics.
000755833 852__ $$bebk
000755833 85640 $$3SpringerLink$$uhttps://univsouthin.idm.oclc.org/login?url=http://link.springer.com/10.1007/978-3-319-33338-0$$zOnline Access$$91397441.1
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