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Integraalvergelijkingen --- Theorie van Lebesgue --- Integrals, Generalized

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This is a graduate level textbook on measure theory and probability theory. The book can be used as a text for a two semester sequence of courses in measure theory and probability theory, with an option to include supplemental material on stochastic processes and special topics. It is intended primarily for first year Ph.D. students in mathematics and statistics although mathematically advanced students from engineering and economics would also find the book useful. Prerequisites are kept to the minimal level of an understanding of basic real analysis concepts such as limits, continuity, differentiability, Riemann integration, and convergence of sequences and series. A review of this material is included in the appendix. The book starts with an informal introduction that provides some heuristics into the abstract concepts of measure and integration theory, which are then rigorously developed. The first part of the book can be used for a standard real analysis course for both mathematics and statistics Ph.D. students as it provides full coverage of topics such as the construction of Lebesgue-Stieltjes measures on real line and Euclidean spaces, the basic convergence theorems, Lp spaces, signed measures, Radon-Nikodym theorem, Lebesgue's decomposition theorem and the fundamental theorem of Lebesgue integration on R, product spaces and product measures, and Fubini-Tonelli theorems. It also provides an elementary introduction to Banach and Hilbert spaces, convolutions, Fourier series and Fourier and Plancherel transforms. Thus part I would be particularly useful for students in a typical Statistics Ph.D. program if a separate course on real analysis is not a standard requirement. Part II (chapters 6-13) provides full coverage of standard graduate level probability theory. It starts with Kolmogorov's probability model and Kolmogorov's existence theorem. It then treats thoroughly the laws of large numbers including renewal theory and ergodic theorems with applications and then weak convergence of probability distributions, characteristic functions, the Levy-Cramer continuity theorem and the central limit theorem as well as stable laws. It ends with conditional expectations and conditional probability, and an introduction to the theory of discrete time martingales. Part III (chapters 14-18) provides a modest coverage of discrete time Markov chains with countable and general state spaces, MCMC, continuous time discrete space jump Markov processes, Brownian motion, mixing sequences, bootstrap methods, and branching processes. It could be used for a topics/seminar course or as an introduction to stochastic processes. Krishna B. Athreya is a professor at the departments of mathematics and statistics and a Distinguished Professor in the College of Liberal Arts and Sciences at the Iowa State University. He has been a faculty member at University of Wisconsin, Madison; Indian Institute of Science, Bangalore; Cornell University; and has held visiting appointments in Scandinavia and Australia. He is a fellow of the Institute of Mathematical Statistics USA; a fellow of the Indian Academy of Sciences, Bangalore; an elected member of the International Statistical Institute; and serves on the editorial board of several journals in probability and statistics. Soumendra N. Lahiri is a professor at the department of statistics at the Iowa State University. He is a fellow of the Institute of Mathematical Statistics, a fellow of the American Statistical Association, and an elected member of the International Statistical Institute.

Measure theory --- Probabilities --- Integrals, Generalized --- Stochastic processes --- Measure theory. Mathematical integration --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Measure algebras --- Rings (Algebra) --- Calculus, Integral --- 303.3 --- AA / International- internationaal --- Waarschijnlijkheid. Probabiliteit. Nauwkeurigheid. Residuals: measurement and specification (wiskundige statistiek)

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This monograph presents the first comprehensive treatment in book form of shape-preserving approximation by real or complex polynomials in one or several variables. Such approximation methods are useful in many problems that arise in science and engineering and require an optimal mathematical representation of physical reality. The main topics are structured in four chapters, followed by an appendix: shape-preserving approximation and interpolation of real functions of one real variable by real polynomials; shape-preserving approximation of real functions of several real variables by multivariate real polynomials; shape-preserving approximation of analytic functions of one complex variable by complex polynomials in the unit disk; and shape-preserving approximation of analytic functions of several complex variables on the unit ball or the unit polydisk by polynomials of several complex variables. The appendix treats related results of non-polynomial and non-spline approximations preserving shape including those by complexified operators with applications to complex partial differential equations. Shape-Preserving Approximation by Real and Complex Polynomials contains many open problems at the end of each chapter to stimulate future research along with a rich and updated bibliography surveying the vast literature. The text will be useful to graduate students and researchers interested in approximation theory, mathematical analysis, numerical analysis, computer aided geometric design, robotics, data fitting, chemistry, fluid mechanics, and engineering.

Mathematics. --- Approximations and Expansions. --- Real Functions. --- Functions of a Complex Variable. --- Computational Mathematics and Numerical Analysis. --- Appl.Mathematics/Computational Methods of Engineering. --- Math Applications in Computer Science. --- Computer science. --- Functions of complex variables. --- Computer science --- Engineering mathematics. --- Mathématiques --- Informatique --- Fonctions d'une variable complexe --- Mathématiques de l'ingénieur --- Approximation theory. --- Bernstein polynomials. --- Mathematical optimization. --- Multivariate analysis. --- Approximation theory --- Bernstein polynomials --- Multivariate analysis --- Mathematics --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Operations Research --- Algebra --- Multivariate distributions --- Multivariate statistical analysis --- Statistical analysis, Multivariate --- Polynomials, Bernstein --- Theory of approximation --- Algebra. --- Functions of real variables. --- Computer mathematics. --- Applied mathematics. --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Analysis of variance --- Mathematical statistics --- Matrices --- Convergence --- Integrals --- Probabilities --- Series --- Mathematical and Computational Engineering. --- Engineering --- Engineering analysis --- Mathematical analysis --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Complex variables --- Elliptic functions --- Functions of real variables --- Math --- Science --- Real variables --- Functions of complex variables

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Devoted to a theory of gradient flows in spaces which are not necessarily endowed with a natural linear or differentiable structure, this book focuses on gradient flows in metric spaces. It covers gradient flows in the space of probability measures on a separable Hilbert space, endowed with the Kantorovich-Rubinstein-Wasserstein distance.

Differential geometry. Global analysis --- Mathematical physics --- Operational research. Game theory --- differentiaalvergelijkingen --- kansrekening --- differentiaal geometrie --- stochastische analyse --- Measure theory --- Metric spaces --- Differential equations, Partial --- Monotone operators --- Evolution equations, Nonlinear --- Mesure, Théorie de la --- Espaces métriques --- Equations aux dérivées partielles --- Opérateurs monotones --- Equations d'évolution non linéaires --- EPUB-LIV-FT LIVMATHE LIVSTATI SPRINGER-B --- Global analysis (Mathematics). --- Mathematics. --- Global differential geometry. --- Distribution (Probability theory. --- Analysis. --- Measure and Integration. --- Differential Geometry. --- Probability Theory and Stochastic Processes. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Geometry, Differential --- Math --- Science --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematical analysis. --- Analysis (Mathematics). --- Measure theory. --- Differential geometry. --- Probabilities. --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Differential geometry --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- 517.1 Mathematical analysis --- Mathematical analysis --- Metric spaces. --- Differential equations, Parabolic. --- Monotone operators. --- Evolution equations, Nonlinear. --- Operator theory --- Parabolic differential equations --- Parabolic partial differential equations --- Spaces, Metric --- Generalized spaces --- Set theory --- Topology --- Nonlinear equations of evolution --- Nonlinear evolution equations --- Differential equations, Nonlinear