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The main focus of this textbook, in two parts, is on showing how self-similar solutions are useful in studying the behavior of solutions of nonlinear partial differential equations, especially those of parabolic type. The exposition moves systematically from the basic to more sophisticated concepts with recent developments and several open problems. With challenging exercises, examples, and illustrations to help explain the rigorous analytic basis for the Navier--Stokes equations, mean curvature flow equations, and other important equations describing real phenomena, this book is written for graduate students and researchers, not only in mathematics but also in other disciplines. Nonlinear Partial Differential Equations will serve as an excellent textbook for a first course in modern analysis or as a useful self-study guide. Key topics in nonlinear partial differential equations as well as several fundamental tools and methods are presented. The only prerequisite required is a basic course in calculus.

Functional analysis --- Partial differential equations --- Mathematical analysis --- Numerical approximation theory --- Mathematics --- Computer science --- differentiaalvergelijkingen --- analyse (wiskunde) --- functies (wiskunde) --- informatica --- wiskunde

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Several mathematical areas that have been developed independently over the last 30 years are brought together revolving around the computation of the number of integral points in suitable families of polytopes. The problem is formulated here in terms of partition functions and multivariate splines. In its simplest form, the problem is to compute the number of ways a given nonnegative integer can be expressed as the sum of h fixed positive integers. This goes back to ancient times and was investigated by Euler, Sylvester among others; in more recent times also in the higher dimensional case of vectors. The book treats several topics in a non-systematic way to show and compare a variety of approaches to the subject. No book on the material is available in the existing literature. Key topics and features include: - Numerical analysis treatments relating this problem to the theory of box splines - Study of regular functions on hyperplane and toric arrangements via D-modules - Residue formulae for partition functions and multivariate splines - Wonderful completion of the complement of hyperplane arrangements - Theory and properties of the Tutte polynomial of a matroid and of zonotopes Graduate students as well as researchers in algebra, combinatorics and numerical analysis, will benefit from Topics in Hyperplane Arrangements, Polytopes, and Box Splines.

Algebra --- Geometry --- Functional analysis --- Differential equations --- Mathematical analysis --- Numerical approximation theory --- Mathematics --- Computer science --- differentiaalvergelijkingen --- algebra --- analyse (wiskunde) --- matrices --- informatica --- wiskunde --- geometrie

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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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Recent results in local convergence and semi-local convergence analysis constitute a natural framework for the theoretical study of iterative methods. This monograph provides a comprehensive study of both basic theory and new results in the area. Each chapter contains new theoretical results and important applications in engineering, modeling dynamic economic systems, input-output systems, optimization problems, and nonlinear and linear differential equations. Several classes of operators are considered, including operators without Lipschitz continuous derivatives, operators with high order derivatives, and analytic operators. Each section is self-contained. Examples are used to illustrate the theory and exercises are included at the end of each chapter. The book assumes a basic background in linear algebra and numerical functional analysis. Graduate students and researchers will find this book useful. It may be used as a self-study reference or as a supplementary text for an advanced course in numerical functional analysis.

Convergence --- Iterative methods (Mathematics) --- Newton-Raphson method --- Convergence (Mathématiques) --- Itération (Mathématiques) --- Convergence. --- Iterative methods (Mathematics). --- Newton-Raphson method. --- Engineering & Applied Sciences --- Applied Mathematics --- Method, Newton-Raphson --- Method of tangents --- Newton approximation method --- Newton iterative process --- Newton method --- Newton-Raphson algorithm --- Newton-Raphson formula --- Newton-Raphson process --- Newton's approximation method --- Newton's method --- Quadratically convergent Newton-Raphson process --- Raphson method, Newton --- -Second-order Newton-Raphson process --- Iteration (Mathematics) --- Mathematics. --- Functional analysis. --- Computer mathematics. --- Numerical analysis. --- Numerical Analysis. --- Computational Mathematics and Numerical Analysis. --- Functional Analysis. --- Numerical analysis --- Functions --- Computer science --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Mathematical analysis --- Mathematics

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Many nonlinear problems in physics, engineering, biology, and social sciences can be reduced to finding critical points of functionals. While minimax and Morse theories provide answers to many situations and problems on the existence of multiple critical points of a functional, they often cannot provide much-needed additional properties of these critical points. Sign-changing critical point theory has emerged as a new area of rich research on critical points of a differentiable functional with important applications to nonlinear elliptic PDEs. Key features of this book: * Self-contained in-depth treatment of sign-changing critical point theory * Further explorations in minimax and Morse theories * Topics devoted to linking and nodal solutions, the sign-changing saddle point theory, the generalized Brezis–Nirenberg critical point theorem, the parameter dependence of sign-changing critical points * Applications of sign-changing critical point theory studied within the classical symmetric mountain pass theorem *Applies sign-changing concepts to Schrödinger equations and boundary value problems This book is intended for advanced graduate students and researchers involved in sign-changing critical point theory, PDEs, global analysis, and nonlinear functional analysis. Also by the author: (with Martin Schechter) Critical Point Theory and Its Applications, ©2006, Springer, ISBN: 978-0-387-32965-9.

Mathematics. --- Approximations and Expansions. --- Topology. --- Functional Analysis. --- Partial Differential Equations. --- Calculus of Variations and Optimal Control; Optimization. --- Global Analysis and Analysis on Manifolds. --- Functional analysis. --- Global analysis. --- Differential equations, partial. --- Mathematical optimization. --- Mathématiques --- Analyse fonctionnelle --- Optimisation mathématique --- Topologie --- Critical point theory (Mathematical analysis). --- Mathematical analysis. --- Critical point theory (Mathematical analysis) --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Global analysis (Mathematics) --- Analysis, Global (Mathematics) --- Approximation theory. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Partial differential equations. --- Calculus of variations. --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Calculus of variations --- Math --- Science --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Functional calculus --- Functional equations --- Integral equations --- Partial differential equations --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Isoperimetrical problems --- Variations, Calculus of --- Geometry, Differential --- Topology