Burgdorf, Sabine
Overview
Works:  8 works in 27 publications in 3 languages and 310 library holdings 

Roles:  Author, Contributor, htt 
Classifications:  QA402.5, 510 
Publication Timeline
.
Most widely held works by
Sabine Burgdorf
Optimization of polynomials in noncommuting variables by
Sabine Burgdorf(
)
14 editions published in 2016 in English and held by 277 WorldCat member libraries worldwide
This book presents recent results on positivity and optimization of polynomials in noncommuting variables. Researchers in noncommutative 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 noncommuting 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
14 editions published in 2016 in English and held by 277 WorldCat member libraries worldwide
This book presents recent results on positivity and optimization of polynomials in noncommuting variables. Researchers in noncommutative 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 noncommuting 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
Tracepositive polynomials, sums of hermitian squares and the tracial moment problem by
Sabine Burgdorf(
Book
)
7 editions published in 2011 in English and French and held by 23 WorldCat member libraries worldwide
A polynomial $f$ in noncommuting variables is tracepositive if the trace of $f(\underline{A})$ is positive for all tuples $\underline{A}$ of symmetric matrices of the same size. The investigation of tracepositive polynomials and of the question of when they can be written as a sum of hermitian squares and commutators of polynomials are motivated by their connection to two famous conjectures: The BMV conjecture from statistical quantum mechanics and the embedding conjecture of Alain Connes concerning von Neumann algebras. First, results on the question of when a tracepositive polynomial in two noncommuting variables can be written as a sum of hermitian squares and commutators are presented. For instance, any bivariate tracepositive polynomial of degree at most four has such a representation, whereas this is false in general if the degree is at least six. This is in perfect analogy to Hilbert's results from the commutative context. Further, a partial answer to the LiebSeiringer formulation of the BMV conjecture is given by presenting some concrete representations of the polynomials $S_{m,4}(X^2; Y^2)$ as a sum of hermitian squares and commutators. The second part of this work deals with the tracial moment problem. That is, how can one describe sequences of real numbers that are given by tracial moments of a probability measure on symmetric matrices of a fixed size. The truncated tracial moment problem, where one considers only finite sequences, as well as the tracial analog of the $K$moment problem are also investigated. Several results from the classical moment problem in Functional Analysis can be transferred to this context. For instance, a tracial analog of Haviland's theorem holds: A traciallinear functional $L$ is given by the tracial moments of a positive Borel measure on symmetric matrices of a fixed size s if and only if $L$ takes only positive values on all polynomials which are tracepositive on all tuples of symmetric $s \times s$matrices. This result uses tracial versions of the results of Fialkow and Nie on positive extensions of truncated sequences. Further, tracial analogs of results of Stochel and of Bayer and Teichmann are given. Defining a tracial Hankel matrix in analogy to the Hankel matrix in the classical moment problem, the results of Curto and Fialkow concerning sequences with Hankel matrices of finite rank or Hankel matrices of finite size which admit a flat extension also hold true in the tracial context. Finally, a relaxation for traceminimization of polynomials using sums of hermitian squares and commutators is proposed. While this relaxation is not always exact, the tracial analogs of the results of Curto and Fialkow give a sufficient condition for the exactness of this relaxation
7 editions published in 2011 in English and French and held by 23 WorldCat member libraries worldwide
A polynomial $f$ in noncommuting variables is tracepositive if the trace of $f(\underline{A})$ is positive for all tuples $\underline{A}$ of symmetric matrices of the same size. The investigation of tracepositive polynomials and of the question of when they can be written as a sum of hermitian squares and commutators of polynomials are motivated by their connection to two famous conjectures: The BMV conjecture from statistical quantum mechanics and the embedding conjecture of Alain Connes concerning von Neumann algebras. First, results on the question of when a tracepositive polynomial in two noncommuting variables can be written as a sum of hermitian squares and commutators are presented. For instance, any bivariate tracepositive polynomial of degree at most four has such a representation, whereas this is false in general if the degree is at least six. This is in perfect analogy to Hilbert's results from the commutative context. Further, a partial answer to the LiebSeiringer formulation of the BMV conjecture is given by presenting some concrete representations of the polynomials $S_{m,4}(X^2; Y^2)$ as a sum of hermitian squares and commutators. The second part of this work deals with the tracial moment problem. That is, how can one describe sequences of real numbers that are given by tracial moments of a probability measure on symmetric matrices of a fixed size. The truncated tracial moment problem, where one considers only finite sequences, as well as the tracial analog of the $K$moment problem are also investigated. Several results from the classical moment problem in Functional Analysis can be transferred to this context. For instance, a tracial analog of Haviland's theorem holds: A traciallinear functional $L$ is given by the tracial moments of a positive Borel measure on symmetric matrices of a fixed size s if and only if $L$ takes only positive values on all polynomials which are tracepositive on all tuples of symmetric $s \times s$matrices. This result uses tracial versions of the results of Fialkow and Nie on positive extensions of truncated sequences. Further, tracial analogs of results of Stochel and of Bayer and Teichmann are given. Defining a tracial Hankel matrix in analogy to the Hankel matrix in the classical moment problem, the results of Curto and Fialkow concerning sequences with Hankel matrices of finite rank or Hankel matrices of finite size which admit a flat extension also hold true in the tracial context. Finally, a relaxation for traceminimization of polynomials using sums of hermitian squares and commutators is proposed. While this relaxation is not always exact, the tracial analogs of the results of Curto and Fialkow give a sufficient condition for the exactness of this relaxation
Pure states, nonnegative polynomials, and sums of squares by
Sabine Burgdorf(
)
1 edition published in 2011 in English and held by 2 WorldCat member libraries worldwide
1 edition published in 2011 in English and held by 2 WorldCat member libraries worldwide
Algorithmic aspects of sums of hermitian squares by
Sabine Burgdorf(
)
1 edition published in 2012 in English and held by 2 WorldCat member libraries worldwide
1 edition published in 2012 in English and held by 2 WorldCat member libraries worldwide
Tracepositive polynomials and the quartic tracial moment problem by
Sabine Burgdorf(
)
1 edition published in 2010 in English and held by 2 WorldCat member libraries worldwide
1 edition published in 2010 in English and held by 2 WorldCat member libraries worldwide
21. Mainzer AllergieWorkshop by S Rennert(
)
1 edition published in 2009 in German and held by 2 WorldCat member libraries worldwide
1 edition published in 2009 in German and held by 2 WorldCat member libraries worldwide
The tracial moment problem and traceoptimization of polynomials by
Sabine Burgdorf(
)
1 edition published in 2011 in English and held by 2 WorldCat member libraries worldwide
1 edition published in 2011 in English and held by 2 WorldCat member libraries worldwide
Optimization of Polynomials in NonCommuting Variables by
Sabine Burgdorf(
)
1 edition published in 2016 in English and held by 0 WorldCat member libraries worldwide
This book presents recent results on positivity and optimization of polynomials in noncommuting variables. Researchers in noncommutative 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 noncommuting 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
1 edition published in 2016 in English and held by 0 WorldCat member libraries worldwide
This book presents recent results on positivity and optimization of polynomials in noncommuting variables. Researchers in noncommutative 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 noncommuting 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
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Related Identities
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