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This book presents a systematic approach to a solution theory for linear partial differential equations developed in a Hilbert space setting based on a Sobolev lattice structure, a simple extension of the well-established notion of a chain (or scale) of Hilbert spaces. The focus on a Hilbert space setting (rather than on an apparently more general Banach space) is not a severe constraint, but rather a highly adaptable and suitable approach providing a more transparent framework for presenting the main issues in the development of a solution theory for partial differential equations. In contrast to other texts on partial differential equations, which consider either specific equation types or apply a collection of tools for solving a variety of equations, this book takes a more global point of view by focusing on the issues involved in determining the appropriate functional analytic setting in which a solution theory can be naturally developed. Applications to many areas of mathematical physics are also presented. The book aims to be largely self-contained. Full proofs to all but the most straightforward results are provided, keeping to a minimum references to other literature for essential material. It is therefore highly suitable as a resource for graduate courses and also for researchers, who will find new results for particular evolutionary systems from mathematical physics.

Hilbert space. --- Differential equations, Partial. --- Partial differential equations --- Banach spaces --- Hyperspace --- Inner product spaces --- Evolution Equation. --- Hilbert Space. --- Mathematics. --- Partial Differential Equations. --- Sobolev.

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This book presents a largely self-contained account of the main results of convex analysis, monotone operator theory, and the theory of nonexpansive operators in the context of Hilbert spaces. Unlike existing literature, the novelty of this book, and indeed its central theme, is the tight interplay among the key notions of convexity, monotonicity, and nonexpansiveness. The presentation is accessible to a broad audience and attempts to reach out in particular to the applied sciences and engineering communities, where these tools have become indispensable. Graduate students and researchers in pure and applied mathematics will benefit from this book. It is also directed to researchers in engineering, decision sciences, economics, and inverse problems, and can serve as a reference book. Author Information: Heinz H. Bauschke is a Professor of Mathematics at the University of British Columbia, Okanagan campus (UBCO) and currently a Canada Research Chair in Convex Analysis and Optimization. He was born in Frankfurt where he received his "Diplom-Mathematiker (mit Auszeichnung)" from Goethe Universität in 1990. He defended his Ph.D. thesis in Mathematics at Simon Fraser University in 1996 and was awarded the Governor General's Gold Medal for his graduate work. After a NSERC Postdoctoral Fellowship spent at the University of Waterloo, at the Pennsylvania State University, and at the University of California at Santa Barbara, Dr. Bauschke became College Professor at Okanagan University College in 1998. He joined the University of Guelph in 2001, and he returned to Kelowna in 2005, when Okanagan University College turned into UBCO. In 2009, he became UBCO's first "Researcher of the Year". Patrick L. Combettes received the Brevet d'Études du Premier Cycle from Académie de Versailles in 1977 and the Ph.D. degree from North Carolina State University in 1989. In 1990, he joined the City College and the Graduate Center of the City University of New York where he became a Full Professor in 1999. Since 1999, he has been with the Faculty of Mathematics of Université Pierre et Marie Curie -- Paris 6, laboratoire Jacques-Louis Lions, where he is presently a Professeur de Classe Exceptionnelle. He was elected Fellow of the IEEE in 2005.

Numerical methods of optimisation --- Computer science --- Computer architecture. Operating systems --- visualisatie --- algoritmen --- kansrekening --- optimalisatie --- Hilbert space --- Nonlinear functional analysis --- Monotone operators --- Espace de Hilbert --- Analyse fonctionnelle non linéaire --- Opérateurs monotones --- EPUB-LIV-FT LIVMATHE LIVSTATI SPRINGER-B --- Hilbert space. --- Approximation theory. --- Monotone operators. --- Nonlinear functional analysis. --- Functional analysis --- Nonlinear theories --- Operator theory --- Theory of approximation --- Functions --- Polynomials --- Chebyshev systems --- Banach spaces --- Hyperspace --- Inner product spaces --- Approximation theory --- Calculus of variations. --- Algorithms. --- Mathematics. --- Visualization. --- Calculus of Variations and Optimal Control; Optimization. --- Math --- Science --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Visualisation --- Imagination --- Visual perception --- Imagery (Psychology) --- Algorism --- Algebra --- Arithmetic --- Foundations

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Approximation theory. --- Stochastic partial differential equations. --- Approximation theory --- Stochastic partial differential equations --- 517.95 --- 519.63 --- Banach spaces, Stochastic differential equations in --- Hilbert spaces, Stochastic differential equations in --- SPDE (Differential equations) --- Stochastic differential equations in Banach spaces --- Stochastic differential equations in Hilbert spaces --- Differential equations, Partial --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- 519.63 Numerical methods for solution of partial differential equations --- Numerical methods for solution of partial differential equations --- 517.95 Partial differential equations --- Partial differential equations

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Along with finite differences and finite elements, spectral methods are one of the three main methodologies for solving partial differential equations on computers. This book provides a detailed presentation of basic spectral algorithms, as well as a systematical presentation of  basic convergence theory and error analysis for spectral methods. Readers of this book will be exposed to a unified framework for designing and analyzing spectral algorithms for a variety of problems, including in particular high-order differential equations and problems in unbounded domains. The book contains a large number of figures which are designed to illustrate various concepts stressed in the book. A set of basic matlab codes has been made available online to help the readers to develop their own spectral codes for their specific applications.

Computer. Automation --- informatica --- differentiaalvergelijkingen --- wiskunde --- Computer science --- Partial differential equations --- Spectral theory (Mathematics) --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Differential equations. --- 517.91 Differential equations --- Differential equations --- Differential equations, partial. --- Computer science. --- Computational Mathematics and Numerical Analysis. --- Partial Differential Equations. --- Mathematics of Computing. --- Mathematics. --- Informatics --- Science --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Mathematics --- Computer mathematics. --- Partial differential equations. --- Computer science—Mathematics.

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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.

Matrices. --- Algebra, Matrix --- Cracovians (Mathematics) --- Matrix algebra --- Matrixes (Algebra) --- Algebra, Abstract --- Algebra, Universal --- Algebras, Linear --- Hermitian forms --- Matrices --- Forms, Hermitian --- Forms (Mathematics) --- Linear algebra --- Generalized spaces --- Mathematical analysis --- Calculus of operations --- Line geometry --- Topology --- Bernstein–Szeg ő measures. --- Carathéodory problem. --- Christoel–Darboux formulas. --- Corona problem. --- Fejéer–Riesz factorization. --- Fejér–Riesz factorization. --- Hamburger problem. --- Hermitian matrices. --- Hermitian matrix expressions. --- Hermitian squares problems. --- Hilbert spaces. --- Hilbert–Schmidt norm control. --- MATLAB codes. --- Nehari problem. --- Nevanlinna–Pick problem. --- Schur complement. --- Toeplitz case. --- Toeplitz matrices. --- banded case. --- chordal case. --- completion problems. --- complex function theory. --- cones. --- contractive completion. --- contractive completions. --- control theory. --- electrical engineering. --- linear algebra. --- mathematics. --- measure theory. --- minimal rank completions. --- multivariables. --- operator theory. --- partial operator matrices. --- positive Carathéodory interpolation. --- positive definite completions. --- positive semidefinite completion. --- quantum information. --- semidefinite completions. --- semidefinite matrices. --- semidefinite operator matrices. --- semidefinite programming. --- separability problem. --- signal processing. --- trigonometric polynomials.