Szegő's Theorem and Its Descendants
Szegő's Theorem and Its Descendants

Author: Barry Simon

Publisher: Princeton University Press

Published: 2010-11-08

Total Pages: 663

ISBN-13: 1400837057

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This book presents a comprehensive overview of the sum rule approach to spectral analysis of orthogonal polynomials, which derives from Gábor Szego's classic 1915 theorem and its 1920 extension. Barry Simon emphasizes necessary and sufficient conditions, and provides mathematical background that until now has been available only in journals. Topics include background from the theory of meromorphic functions on hyperelliptic surfaces and the study of covering maps of the Riemann sphere with a finite number of slits removed. This allows for the first book-length treatment of orthogonal polynomials for measures supported on a finite number of intervals on the real line. In addition to the Szego and Killip-Simon theorems for orthogonal polynomials on the unit circle (OPUC) and orthogonal polynomials on the real line (OPRL), Simon covers Toda lattices, the moment problem, and Jacobi operators on the Bethe lattice. Recent work on applications of universality of the CD kernel to obtain detailed asymptotics on the fine structure of the zeros is also included. The book places special emphasis on OPRL, which makes it the essential companion volume to the author's earlier books on OPUC.

Geometric Science of Information
Geometric Science of Information

Author: Frank Nielsen

Publisher: Springer

Published: 2019-08-19

Total Pages: 764

ISBN-13: 3030269809

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This book constitutes the proceedings of the 4th International Conference on Geometric Science of Information, GSI 2019, held in Toulouse, France, in August 2019. The 79 full papers presented in this volume were carefully reviewed and selected from 105 submissions. They cover all the main topics and highlights in the domain of geometric science of information, including information geometry manifolds of structured data/information and their advanced applications.

Orthogonal Polynomials
Orthogonal Polynomials

Author: Gabor Szegš

Publisher: American Mathematical Soc.

Published: 1939-12-31

Total Pages: 448

ISBN-13: 0821810235

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The general theory of orthogonal polynomials was developed in the late 19th century from a study of continued fractions by P. L. Chebyshev, even though special cases were introduced earlier by Legendre, Hermite, Jacobi, Laguerre, and Chebyshev himself. It was further developed by A. A. Markov, T. J. Stieltjes, and many other mathematicians. The book by Szego, originally published in 1939, is the first monograph devoted to the theory of orthogonal polynomials and its applications in many areas, including analysis, differential equations, probability and mathematical physics. Even after all the years that have passed since the book first appeared, and with many other books on the subject published since then, this classic monograph by Szego remains an indispensable resource both as a textbook and as a reference book. It can be recommended to anyone who wants to be acquainted with this central topic of mathematical analysis.

Orthogonal Polynomials on the Unit Circle
Orthogonal Polynomials on the Unit Circle

Author: Barry Simon

Publisher: American Mathematical Soc.

Published: 2009-08-05

Total Pages: 498

ISBN-13: 0821848631

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This two-part book is a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analog of the spectral theory of one-dimensional Schrodinger operators. Among the topics discussed along the way are the asymptotics of Toeplitz determinants (Szego's theorems), limit theorems for the density of the zeros of orthogonal polynomials, matrix representations for multiplication by $z$ (CMV matrices), periodic Verblunsky coefficients from the point of view of meromorphic functions on hyperelliptic surfaces, and connections between the theories of orthogonal polynomials on the unit circle and on the real line.

The Andrews Festschrift
The Andrews Festschrift

Author: Dominique Foata

Publisher: Springer Science & Business Media

Published: 2011-06-28

Total Pages: 430

ISBN-13: 3642565131

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This book contains seventeen contributions made to George Andrews on the occasion of his sixtieth birthday, ranging from classical number theory (the theory of partitions) to classical and algebraic combinatorics. Most of the papers were read at the 42nd session of the Sminaire Lotharingien de Combinatoire that took place at Maratea, Basilicata, in August 1998. This volume contains a long memoir on Ramanujan's Unpublished Manuscript and the Tau functions studied with a contemporary eye, together with several papers dealing with the theory of partitions. There is also a description of a maple package to deal with general q-calculus. More subjects on algebraic combinatorics are developed, especially the theory of Kostka polynomials, the ice square model, the combinatorial theory of classical numbers, a new approach to determinant calculus.

Mathematical Writing
Mathematical Writing

Author: Donald E. Knuth

Publisher: Cambridge University Press

Published: 1989

Total Pages: 132

ISBN-13: 9780883850633

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This book will help those wishing to teach a course in technical writing, or who wish to write themselves.

Mathematical Encounters of the Second Kind
Mathematical Encounters of the Second Kind

Author: Philip J. Davis

Publisher: Springer Science & Business Media

Published: 1997

Total Pages: 316

ISBN-13: 9780817639396

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A memoir of mathematician Philip Davis's life and encounters, some actual and some imaginary, with a number of mathematicians and historical figures. His message focuses on the idea that mathematics can bring people into contacts with each other across centuries, oceans, and cultural difference. Annotation copyrighted by Book News, Inc., Portland, OR

Quantum Mechanics for Hamiltonians Defined as Quadratic Forms
Quantum Mechanics for Hamiltonians Defined as Quadratic Forms

Author: Barry Simon

Publisher: Princeton University Press

Published: 2015-03-08

Total Pages: 261

ISBN-13: 1400868831

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This monograph combines a thorough introduction to the mathematical foundations of n-body Schrodinger mechanics with numerous new results. Originally published in 1971. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Emergent Results of Artificial Economics
Emergent Results of Artificial Economics

Author: Sjoukje Osinga

Publisher: Springer Science & Business Media

Published: 2011-06-22

Total Pages: 226

ISBN-13: 3642211089

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Artificial economics is a computational approach that aims to explain economic systems by modeling them as societies of intelligent software agents. The individual agents make autonomous decisions, but their actual behaviors are constrained by available resources, other individuals' behaviors, and institutions. Intelligent software agents have communicative skills that enable simulation of negotiation, trade, reputation, and other forms of knowledge transfer that are at the basis of economic life. Incorporated learning mechanisms may adapt the agents' behaviors. In artificial economics, all system behavior is generated from the individual agents' simulated decisions; no system level laws are a priori imposed. For instance, price convergence and market clearing may emerge, but not necessarily. Thus, artificial economics facilitates the study of the mechanisms that make the economy function. This book presents a selection of peer-reviewed papers addressing recent developments in this field between economics and computer science.