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This book exploits the classification of a class of linear bounded operators with rank-one self-commutators in terms of their spectral parameter, known as the principal function. The resulting dictionary between two dimensional planar shapes with a degree of shade and Hilbert space operators turns out to be illuminating and beneficial for both sides. An exponential transform, essentially a Riesz potential at critical exponent, is at the heart of this novel framework; its best rational approximants unveil a new class of complex orthogonal polynomials whose asymptotic distribution of zeros is thoroughly studied in the text. Connections with areas of potential theory, approximation theory in the complex domain and fluid mechanics are established. The text is addressed, with specific aims, at experts and beginners in a wide range of areas of current interest: potential theory, numerical linear algebra, operator theory, inverse problems, image and signal processing, approximation theory, mathematical physics.

Mathematics. --- Functions of complex variables. --- Operator theory. --- Potential theory (Mathematics). --- Numerical analysis. --- Functions of a Complex Variable. --- Operator Theory. --- Potential Theory. --- Numerical Analysis. --- Mathematical analysis --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mechanics --- Functional analysis --- Complex variables --- Elliptic functions --- Functions of real variables --- Math --- Science --- Hyponormal operators. --- Operators, Hyponormal --- Linear operators

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The book consists of solicited articles from a select group of mathematicians and physicists working at the interface between positivity and the geometry, combinatorics or analysis of polynomials of one or several variables. It is dedicated to the memory of Julius Borcea (1968-2009), a distinguished mathematician, Professor at the University of Stockholm. With his extremely original contributions and broad vision, his impact on the topics of the planned volume cannot be underestimated. All contributors knew or have exchanged ideas with Dr. Borcea, and their articles reflect, at least partially, his heritage.

Geometry. --- Numerical analysis -- Congresses. --- Numerical analysis. --- Polynomials --- Polynomial operators --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Applied Mathematics --- Algebra --- Polynomials. --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Analysis. --- Global analysis (Mathematics). --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- 517.1 Mathematical analysis --- Mathematical analysis

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Recent advances in both the theory and implementation of computational algebraic geometry have led to new, striking applications to a variety of fields of research. The articles in this volume highlight a range of these applications and provide introductory material for topics covered in the IMA workshops on "Optimization and Control" and "Applications in Biology, Dynamics, and Statistics" held during the IMA year on Applications of Algebraic Geometry. The articles related to optimization and control focus on burgeoning use of semidefinite programming and moment matrix techniques in computational real algebraic geometry. The new direction towards a systematic study of non-commutative real algebraic geometry is well represented in the volume. Other articles provide an overview of the way computational algebra is useful for analysis of contingency tables, reconstruction of phylogenetic trees, and in systems biology. The contributions collected in this volume are accessible to non-experts, self-contained and informative; they quickly move towards cutting edge research in these areas, and provide a wealth of open problems for future research.

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Algebra --- Operator theory --- Functional analysis --- Engineering sciences. Technology --- Artificial intelligence. Robotics. Simulation. Graphics --- algebra --- functies (wiskunde) --- controleleer

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This volume is dedicated to Bill Helton on the occasion of his sixty-fifth birthday. It contains biographical material, a list of Bill's publications, a detailed survey of Bill's contributions to operator theory, optimization and control, and 19 technical articles. Most of the technical articles are expository and should serve as useful introductions to many of the areas which Bill's highly original contributions have helped to shape over the last forty-odd years. These include interpolation, Szegö limit theorems, Nehari problems, trace formulas, systems and control theory, convexity, matrix completion problems, linear matrix inequalities and optimization. The book should be useful to graduate students in mathematics and engineering, as well as to faculty and individuals seeking entry level introductions and references to the indicated topics. It can also serve as a supplementary text to numerous courses in pure and applied  mathematics and engineering, and as a source book for seminars.

Computer programming. --- Computer software -- Development -- Mathematical methods. --- Operator theory --- System theory --- Mathematical optimization --- Control theory --- Civil & Environmental Engineering --- Mathematics --- Physical Sciences & Mathematics --- Engineering & Applied Sciences --- Algebra --- Calculus --- Operations Research --- Operator theory. --- Nonlinear control theory. --- System analysis. --- Helton, J. William, --- Network theory --- Systems analysis --- Helton, John W., --- Helton, Bill, --- Mathematics. --- Algebra. --- Functions of complex variables. --- Operations research. --- Management science. --- Control engineering. --- General Algebraic Systems. --- Control. --- Functions of a Complex Variable. --- Operator Theory. --- Operations Research, Management Science. --- Functional analysis --- Network analysis --- Network science --- Nonlinear theories --- Control and Systems Theory. --- Complex variables --- Elliptic functions --- Functions of real variables --- Mathematical analysis --- Quantitative business analysis --- Management --- Problem solving --- Operations research --- Statistical decision --- Operational analysis --- Operational research --- Industrial engineering --- Management science --- Research --- Control engineering --- Control equipment --- Engineering instruments --- Automation --- Programmable controllers