Flat Extensions of Positive Moment Matrices: Recursively Generated Relations

Flat Extensions of Positive Moment Matrices: Recursively Generated Relations PDF Author: Raúl E. Curto
Publisher: American Mathematical Soc.
ISBN: 0821808699
Category : Mathematics
Languages : en
Pages : 73

Get Book Here

Book Description
In this book, the authors develop new computational tests for existence and uniqueness of representing measures $\mu$ in the Truncated Complex Moment Problem: $\gamma {ij}=\int \bar zizj\, d\mu$ $(0\le i+j\le 2n)$. Conditions for the existence of finitely atomic representing measures are expressed in terms of positivity and extension properties of the moment matrix $M(n)(\gamma )$ associated with $\gamma \equiv \gamma {(2n)}$: $\gamma {00}, \dots ,\gamma {0,2n},\dots ,\gamma {2n,0}$, $\gamma {00}>0$. This study includes new conditions for flat (i.e., rank-preserving) extensions $M(n+1)$ of $M(n)\ge 0$; each such extension corresponds to a distinct rank $M(n)$-atomic representing measure, and each such measure is minimal among representing measures in terms of the cardinality of its support. For a natural class of moment matrices satisfying the tests of recursive generation, recursive consistency, and normal consistency, the existence problem for minimal representing measures is reduced to the solubility of small systems of multivariable algebraic equations. In a variety of applications, including cases of the quartic moment problem ($n=2$), the text includes explicit contructions of minimal representing measures via the theory of flat extensions. Additional computational texts are used to prove non-existence of representing measures or the non-existence of minimal representing measures. These tests are used to illustrate, in very concrete terms, new phenomena, associated with higher-dimensional moment problems that do not appear in the classical one-dimensional moment problem.

Flat Extensions of Positive Moment Matrices: Recursively Generated Relations

Flat Extensions of Positive Moment Matrices: Recursively Generated Relations PDF Author: Raúl E. Curto
Publisher: American Mathematical Soc.
ISBN: 0821808699
Category : Mathematics
Languages : en
Pages : 73

Get Book Here

Book Description
In this book, the authors develop new computational tests for existence and uniqueness of representing measures $\mu$ in the Truncated Complex Moment Problem: $\gamma {ij}=\int \bar zizj\, d\mu$ $(0\le i+j\le 2n)$. Conditions for the existence of finitely atomic representing measures are expressed in terms of positivity and extension properties of the moment matrix $M(n)(\gamma )$ associated with $\gamma \equiv \gamma {(2n)}$: $\gamma {00}, \dots ,\gamma {0,2n},\dots ,\gamma {2n,0}$, $\gamma {00}>0$. This study includes new conditions for flat (i.e., rank-preserving) extensions $M(n+1)$ of $M(n)\ge 0$; each such extension corresponds to a distinct rank $M(n)$-atomic representing measure, and each such measure is minimal among representing measures in terms of the cardinality of its support. For a natural class of moment matrices satisfying the tests of recursive generation, recursive consistency, and normal consistency, the existence problem for minimal representing measures is reduced to the solubility of small systems of multivariable algebraic equations. In a variety of applications, including cases of the quartic moment problem ($n=2$), the text includes explicit contructions of minimal representing measures via the theory of flat extensions. Additional computational texts are used to prove non-existence of representing measures or the non-existence of minimal representing measures. These tests are used to illustrate, in very concrete terms, new phenomena, associated with higher-dimensional moment problems that do not appear in the classical one-dimensional moment problem.

Analysis of Operators on Function Spaces

Analysis of Operators on Function Spaces PDF Author: Alexandru Aleman
Publisher: Springer
ISBN: 3030146405
Category : Mathematics
Languages : en
Pages : 283

Get Book Here

Book Description
This book contains both expository articles and original research in the areas of function theory and operator theory. The contributions include extended versions of some of the lectures by invited speakers at the conference in honor of the memory of Serguei Shimorin at the Mittag-Leffler Institute in the summer of 2018. The book is intended for all researchers in the fields of function theory, operator theory and complex analysis in one or several variables. The expository articles reflecting the current status of several well-established and very dynamical areas of research will be accessible and useful to advanced graduate students and young researchers in pure and applied mathematics, and also to engineers and physicists using complex analysis methods in their investigations.

Nonselfadjoint Operator Algebras, Operator Theory, and Related Topics

Nonselfadjoint Operator Algebras, Operator Theory, and Related Topics PDF Author: H. Bercovicii
Publisher: Birkhäuser
ISBN: 3034887795
Category : Mathematics
Languages : en
Pages : 213

Get Book Here

Book Description
This volume, dedicated to Carl Pearcy on the occasion of his 60th birthday, presents recent results in operator theory, nonselfadjoint operator algebras, measure theory and the theory of moments. The articles on these subjects have been contributed by leading area experts, many of whom were associated with Carl Pearcy as students or collaborators. The book testifies to his multifaceted interests and includes a biographical sketch and a list of publications.

The Defect Relation of Meromorphic Maps on Parabolic Manifolds

The Defect Relation of Meromorphic Maps on Parabolic Manifolds PDF Author: George Lawrence Ashline
Publisher: American Mathematical Soc.
ISBN: 0821810693
Category : Mathematics
Languages : en
Pages : 95

Get Book Here

Book Description
This book is intended for graduate students and research mathematicians working in several complex variables and analytic spaces.

Complexes Associated to Two Vectors and a Rectangular Matrix

Complexes Associated to Two Vectors and a Rectangular Matrix PDF Author: Andrew R. Kustin
Publisher: American Mathematical Soc.
ISBN: 0821820737
Category : Mathematics
Languages : en
Pages : 97

Get Book Here

Book Description
This book is intended for graduate student and research mathematicians interested in commutative rings and algebras.

Optimization of Polynomials in Non-Commuting Variables

Optimization of Polynomials in Non-Commuting Variables PDF Author: Sabine Burgdorf
Publisher: Springer
ISBN: 3319333380
Category : Mathematics
Languages : en
Pages : 118

Get Book Here

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.

Algebraic and Strong Splittings of Extensions of Banach Algebras

Algebraic and Strong Splittings of Extensions of Banach Algebras PDF Author: William G. Bade
Publisher: American Mathematical Soc.
ISBN: 0821810588
Category : Mathematics
Languages : en
Pages : 129

Get Book Here

Book Description
In this volume, the authors address the following: Let $A$ be a Banach algebra, and let $\sum\:\ 0\rightarrow I\rightarrow\frak A\overset\pi\to\longrightarrow A\rightarrow 0$ be an extension of $A$, where $\frak A$ is a Banach algebra and $I$ is a closed ideal in $\frak A$. The extension splits algebraically (respectively, splits strongly) if there is a homomorphism (respectively, continuous homomorphism) $\theta\: A\rightarrow\frak A$ such that $\pi\circ\theta$ is the identity on $A$. Consider first for which Banach algebras $A$ it is true that every extension of $A$ in a particular class of extensions splits, either algebraically or strongly, and second for which Banach algebras it is true that every extension of $A$ in a particular class which splits algebraically also splits strongly. These questions are closely related to the question when the algebra $\frak A$ has a (strong) Wedderburn decomposition. The main technique for resolving these questions involves the Banach cohomology group $\cal H2(A,E)$ for a Banach $A$-bimodule $E$, and related cohomology groups. Later chapters are particularly concerned with the case where the ideal $I$ is finite-dimensional. Results are obtained for many of the standard Banach algebras $A$.

Matrix Completions, Moments, and Sums of Hermitian Squares

Matrix Completions, Moments, and Sums of Hermitian Squares PDF Author: Mihály Bakonyi
Publisher: Princeton University Press
ISBN: 1400840597
Category : Mathematics
Languages : en
Pages : 533

Get Book Here

Book Description
Intensive research in matrix completions, moments, and sums of Hermitian squares has yielded a multitude of results in recent decades. This book provides a comprehensive account of this quickly developing area of mathematics and applications and gives complete proofs of many recently solved problems. With MATLAB codes and more than 200 exercises, the book is ideal for a special topics course for graduate or advanced undergraduate students in mathematics or engineering, and will also be a valuable resource for researchers. Often driven by questions from signal processing, control theory, and quantum information, the subject of this book has inspired mathematicians from many subdisciplines, including linear algebra, operator theory, measure theory, and complex function theory. In turn, the applications are being pursued by researchers in areas such as electrical engineering, computer science, and physics. The book is self-contained, has many examples, and for the most part requires only a basic background in undergraduate mathematics, primarily linear algebra and some complex analysis. The book also includes an extensive discussion of the literature, with close to 600 references from books and journals from a wide variety of disciplines.

Realization of Vector Fields and Dynamics of Spatially Homogeneous Parabolic Equations

Realization of Vector Fields and Dynamics of Spatially Homogeneous Parabolic Equations PDF Author: Edward Norman Dancer
Publisher: American Mathematical Soc.
ISBN: 0821811827
Category : Mathematics
Languages : en
Pages : 97

Get Book Here

Book Description
This book is intended for graduate students and research mathematicians working in partial differential equations.

Morava $K$-Theories and Localisation

Morava $K$-Theories and Localisation PDF Author: Mark Hovey
Publisher: American Mathematical Soc.
ISBN: 0821810790
Category : Mathematics
Languages : en
Pages : 114

Get Book Here

Book Description
This book is intended for graduate students and research mathematicians working in group theory and generalizations.