Noncommutative Plurisubharmonic Polynomials PDF Download
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Author: Jeremy Michael Greene
Publisher:
ISBN: 9781124670843
Category :
Languages : en
Pages : 67
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Book Description
Many optimization problems and engineering problems connected with linear systems lead to matrix inequalities. Matrix inequalities are constraints in which a polynomial or a matrix of polynomials with matrix variables is required to take a positive semidefinite value. Many of these problems have the property that they are "dimension free" and, in this case, the form of the polynomials remains the same for matrices of all size. In other words, we have noncommutative polynomials. One very much desires these polynomials to be "convex" in the unknown matrix variables, since if they are, then numerical calculations are reliable and local optima are global optima. In this dissertation, we classify all changes of variables (not containing transposes) from noncommutative non-convex polynomials to noncommutative convex polynomials. This introduces notions of noncommutative complex Hessians and plurisubharmonicity, classical notions from several complex variables. In addition, we present a theory of noncommutative integration and we prove a "local implies global" result in that we show noncommutative plurisubharmonicity on a noncommutative open set implies noncommutative plurisubharmonicity everywhere.
Author: Jeremy Michael Greene
Publisher:
ISBN: 9781124670843
Category :
Languages : en
Pages : 67
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Book Description
Many optimization problems and engineering problems connected with linear systems lead to matrix inequalities. Matrix inequalities are constraints in which a polynomial or a matrix of polynomials with matrix variables is required to take a positive semidefinite value. Many of these problems have the property that they are "dimension free" and, in this case, the form of the polynomials remains the same for matrices of all size. In other words, we have noncommutative polynomials. One very much desires these polynomials to be "convex" in the unknown matrix variables, since if they are, then numerical calculations are reliable and local optima are global optima. In this dissertation, we classify all changes of variables (not containing transposes) from noncommutative non-convex polynomials to noncommutative convex polynomials. This introduces notions of noncommutative complex Hessians and plurisubharmonicity, classical notions from several complex variables. In addition, we present a theory of noncommutative integration and we prove a "local implies global" result in that we show noncommutative plurisubharmonicity on a noncommutative open set implies noncommutative plurisubharmonicity everywhere.
Author: Christopher Scott Nelson
Publisher:
ISBN: 9781267607898
Category :
Languages : en
Pages : 88
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Book Description
Author: Sabine Burgdorf
Publisher: Springer
ISBN: 9783319333366
Category : Mathematics
Languages : en
Pages : 0
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Book Description
This 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.
Author: HUISHI. LI
Publisher: Chapman & Hall/CRC Monographs and Research Notes in Mathematics
ISBN: 9781032079882
Category :
Languages : en
Pages : 232
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Book Description
This is the first book to systematically introduce the basic constructive-computational theory and methods developed for investigating solvable polynomial algebras and their modules. This book is perfectly suited to researchers and postgrads researching noncommutative computational algebra.
Author: Jeremy Judah Stone
Publisher:
ISBN:
Category : Algebraic functions
Languages : en
Pages : 90
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Book Description
Author: Victor V. Prasolov
Publisher: Springer Science & Business Media
ISBN: 9783540407140
Category : Mathematics
Languages : en
Pages : 324
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Book Description
Covers its topic in greater depth than the typical standard books on polynomial algebra
Author: Francisco Marcellàn
Publisher: Springer
ISBN: 3540367160
Category : Mathematics
Languages : en
Pages : 432
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Book Description
Special functions and orthogonal polynomials in particular have been around for centuries. Can you imagine mathematics without trigonometric functions, the exponential function or polynomials? The present set of lecture notes contains seven chapters about the current state of orthogonal polynomials and special functions and gives a view on open problems and future directions.
Author: G. V. Milovanovi?
Publisher: World Scientific
ISBN: 9789810204990
Category : Science
Languages : en
Pages : 842
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Book Description
The book contains some of the most important results on the analysis of polynomials and their derivatives. Besides the fundamental results which are treated with their proofs, the book also provides an account of the most recent developments concerning extremal properties of polynomials and their derivatives in various metrics with an extensive analysis of inequalities for trigonometric sums and algebraic polynomials, as well as their zeros. The final chapter provides some selected applications of polynomials in approximation theory and computer aided geometric design (CAGD). One can also find in this book several new research problems and conjectures with sufficient information concerning the results obtained to date towards the investigation of their solution.
Author:
Publisher: Elsevier
ISBN: 0080533507
Category : Mathematics
Languages : en
Pages : 873
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Book Description
Handbook of the Geometry of Banach Spaces
Author: Alex Barnett
Publisher: American Mathematical Soc.
ISBN: 0821853198
Category : Mathematics
Languages : en
Pages : 354
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Book Description
This volume contains the proceedings of the International Conference on Spectral Geometry, held July 19-23, 2010, at Dartmouth College, Dartmouth, New Hampshire. Eigenvalue problems involving the Laplace operator on manifolds have proven to be a consistently fertile area of geometric analysis with deep connections to number theory, physics, and applied mathematics. Key questions include the measures to which eigenfunctions of the Laplacian on a Riemannian manifold condense in the limit of large eigenvalue, and the extent to which the eigenvalues and eigenfunctions of a manifold encode its geometry. In this volume, research and expository articles, including those of the plenary speakers Peter Sarnak and Victor Guillemin, address the flurry of recent progress in such areas as quantum unique ergodicity, isospectrality, semiclassical measures, the geometry of nodal lines of eigenfunctions, methods of numerical computation, and spectra of quantum graphs. This volume also contains mini-courses on spectral theory for hyperbolic surfaces, semiclassical analysis, and orbifold spectral geometry that prepared the participants, especially graduate students and young researchers, for conference lectures.
