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Riemann integral. --- Integral, Riemann --- Definite integrals --- Integral de Riemann --- Integrals definides

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This textbook provides a thorough introduction to measure and integration theory, fundamental topics of advanced mathematical analysis. Proceeding at a leisurely, student-friendly pace, the authors begin by recalling elementary notions of real analysis before proceeding to measure theory and Lebesgue integration. Further chapters cover Fourier series, differentiation, modes of convergence, and product measures. Noteworthy topics discussed in the text include Lp spaces, the Radon–Nikodým Theorem, signed measures, the Riesz Representation Theorem, and the Tonelli and Fubini Theorems. This textbook, based on extensive teaching experience, is written for senior undergraduate and beginning graduate students in mathematics. With each topic carefully motivated and hints to more than 300 exercises, it is the ideal companion for self-study or use alongside lecture courses.

Mathematics. --- Fourier analysis. --- Functional analysis. --- Measure and Integration. --- Real Functions. --- Fourier Analysis. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Analysis, Fourier --- Mathematical analysis --- Math --- Science --- Riemann integral. --- Integral, Riemann --- Definite integrals --- Measure theory. --- Functions of real variables. --- Real variables --- Functions of complex variables --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra)

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This book deals with the numerical analysis and efficient numerical treatment of high-dimensional integrals using sparse grids and other dimension-wise integration techniques with applications to finance and insurance. The book focuses on providing insights into the interplay between coordinate transformations, effective dimensions and the convergence behaviour of sparse grid methods. The techniques, derivations and algorithms are illustrated by many examples, figures and code segments. Numerical experiments with applications from finance and insurance show that the approaches presented in this book can be faster and more accurate than (quasi-) Monte Carlo methods, even for integrands with hundreds of dimensions.

Calculus, Integral. --- Numerical grid generation (Numerical analysis). --- Numerical analysis --- Finance --- Risk --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Applied Mathematics --- Mathematics - General --- Mathematical models --- Numerical integration. --- Insurance --- Mathematical models. --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Mathematics. --- Economics, Mathematical. --- Computer mathematics. --- Computational Mathematics and Numerical Analysis. --- Quantitative Finance. --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Economics --- Mathematical economics --- Econometrics --- Math --- Science --- Methodology --- Definite integrals --- Interpolation --- Computer science --- Finance. --- Funding --- Funds --- Currency question --- Economics, Mathematical .

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The book addresses the problem of calculation of d-dimensional integrals (conditional expectations) in filter problems. It develops new methods of deterministic numerical integration, which can be used to speed up and stabilize filter algorithms. With the help of these methods, better estimates and predictions of latent variables are made possible in the fields of economics, engineering and physics. The resulting procedures are tested within four detailed simulation studies.

Statistics. --- Computer mathematics. --- Vibration. --- Dynamical systems. --- Dynamics. --- Econometrics. --- Statistics for Business/Economics/Mathematical Finance/Insurance. --- Computational Mathematics and Numerical Analysis. --- Vibration, Dynamical Systems, Control. --- Statistical Physics and Dynamical Systems. --- Cubature formulas. --- Numerical integration. --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Definite integrals --- Interpolation --- Numerical analysis --- Formulas, Cubature --- Numerical integration --- Computer science --- Statistical physics. --- Statistics for Business, Management, Economics, Finance, Insurance. --- Mathematics. --- Physics --- Mathematical statistics --- Cycles --- Mechanics --- Sound --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Economics, Mathematical --- Statistics --- Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Mathematics --- Econometrics --- Statistics . --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Statics

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Numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions are the subject of this book. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by many figures, it treats applications from physics and astronomy and contains many numerical experiments and comparisons of different approaches. The second edition is substantially revised and enlarged, with many improvements in the presentation and additions concerning in particular non-canonical Hamiltonian systems, highly oscillatory mechanical systems, and the dynamics of multistep methods.

Numerical integration. --- Hamiltonian systems. --- Differential equations --- Numerical solutions. --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Numerical analysis. --- Biomathematics. --- Physics. --- Numerical Analysis. --- Analysis. --- Theoretical, Mathematical and Computational Physics. --- Mathematical Methods in Physics. --- Numerical and Computational Physics. --- Mathematical and Computational Biology. --- 517.91 Differential equations --- Hamiltonian dynamical systems --- Systems, Hamiltonian --- Differentiable dynamical systems --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Definite integrals --- Interpolation --- Numerical analysis --- Global analysis (Mathematics). --- Mathematical physics. --- Numerical and Computational Physics, Simulation. --- Physical mathematics --- Physics --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematical analysis --- Mathematics --- Biology --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- 517.1 Mathematical analysis --- Hamiltonian systems --- Differential equations - Numerical solutions --- 517.91 --- Numerical integration --- 519.62 --- 681.3*G17 --- 681.3*G17 Ordinary differential equations: boundary value problems; convergence and stability; error analysis; initial value problems; multistep methods; single step methods; stiff equations (Numerical analysis) --- Ordinary differential equations: boundary value problems; convergence and stability; error analysis; initial value problems; multistep methods; single step methods; stiff equations (Numerical analysis) --- 519.62 Numerical methods for solution of ordinary differential equations --- Numerical methods for solution of ordinary differential equations --- Numerical solutions