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Author: Jeremy Michael Greene Publisher: ISBN: 9781124670843 Category : Languages : en Pages : 67
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
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: Sabine Burgdorf Publisher: Springer ISBN: 9783319333366 Category : Mathematics Languages : en Pages : 0
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
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: Francisco Marcellàn Publisher: Springer ISBN: 3540367160 Category : Mathematics Languages : en Pages : 432
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: Renato Alvarez-Nodarse Publisher: Nova Publishers ISBN: 9781594540097 Category : Mathematics Languages : en Pages : 222
Book Description
This new book presents research in orthogonal polynomials and special functions. Recent developments in the theory and accomplishments of the last decade are pointed out and directions for research in the future are identified. The topics covered include matrix orthogonal polynomials, spectral theory and special functions, Asymptotics for orthogonal polynomials via Riemann-Hilbert methods, Polynomial wavelets and Koornwinder polynomials.
Author: A. Schinzel Publisher: Cambridge University Press ISBN: 9781139426718 Category : Mathematics Languages : en Pages : 590
Book Description
This book covers most of the known results on reducibility of polynomials over arbitrary fields, algebraically closed fields and finitely generated fields. Results valid only over finite fields, local fields or the rational field are not covered here, but several theorems on reducibility of polynomials over number fields that are either totally real or complex multiplication fields are included. Some of these results are based on recent work of E. Bombieri and U. Zannier (presented here by Zannier in an appendix). The book also treats other subjects like Ritt's theory of composition of polynomials, and properties of the Mahler measure, and it concludes with a bibliography of over 300 items. This unique work will be a necessary resource for all number theorists and researchers in related fields.
Author: G. V. Milovanovi? Publisher: World Scientific ISBN: 9789810204990 Category : Science Languages : en Pages : 842
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: Jeremy Michael Greene
Author: Jeremy Michael Greene
Author: Christopher Scott Nelson
Author: Sabine Burgdorf
Author: Victor V. Prasolov
Author: HUISHI. LI
Author: Jeremy Judah Stone
Author: Francisco Marcellàn
Author: Renato Alvarez-Nodarse
Author: A. Schinzel
Author: G. V. Milovanovi?