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Lévy processes form a wide and rich class of random process, and have many applications ranging from physics to finance. Stochastic calculus is the mathematics of systems interacting with random noise. Here, the author ties these two subjects together, beginning with an introduction to the general theory of Lévy processes, then leading on to develop the stochastic calculus for Lévy processes in a direct and accessible way. This fully revised edition now features a number of new topics. These include: regular variation and subexponential distributions; necessary and sufficient conditions for Lévy processes to have finite moments; characterisation of Lévy processes with finite variation; Kunita's estimates for moments of Lévy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Lévy processes; multiple Wiener-Lévy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Lévy-driven SDEs.

Lévy processes --- Stochastic analysis --- Lévy, Processus de --- Analyse stochastique --- Lévy processes. --- Stochastic analysis. --- Lévy processes --- Lévy, Processus de --- Lévy processes. --- Stochastic integral equations. --- Integral equations --- Random walks (Mathematics) --- Analysis, Stochastic --- Mathematical analysis --- Stochastic processes

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The International Conference on Modern Analysis and Applications, which was dedicated to the 100th anniversary of the birth of Mark Krein, one of the greatest mathematicians of the 20th century, was held in Odessa, Ukraine, on April 9-14, 2007. This title contains peer-reviewed research and survey papers based on invited talks at this conference.

Functional analysis -- Congresses. --- Mathematical analysis -- Congresses. --- Applied Mathematics --- Calculus --- Engineering & Applied Sciences --- Mathematics --- Physical Sciences & Mathematics --- Functional analysis --- Operator theory --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Functional analysis. --- Integral equations. --- Operator theory. --- Analysis. --- Operator Theory. --- Functional Analysis. --- Integral Equations. --- 517.1 Mathematical analysis --- Mathematical analysis --- Math --- Science --- Equations, Integral --- Functional equations --- Functional calculus --- Calculus of variations --- Integral equations --- Global analysis (Mathematics). --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic

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This is a unique collection of lectures on integrability, intended for graduate students or anyone who would like to master the subject from scratch, and written by leading experts in the field including Fields Medallist Serge Novikov. Since integrable systems have found a wide range of applications in modern theoretical and mathematical physics, it is important to recognise integrable models and, ideally, to obtain a global picture of the integrable world. The main aims of the book are to present a variety of views on the definition of integrable systems; to develop methods and tests for integrability based on these definitions; and to uncover beautiful hidden structures associated with integrable equations.

Integral equations --- Mathematical physics --- Applied Physics --- Engineering & Applied Sciences --- Mathematical physics. --- Integral equations. --- Equations, Integral --- Physical mathematics --- Physics --- Mathematics --- Physics. --- Mechanics. --- Fluids. --- Theoretical, Mathematical and Computational Physics. --- Fluid- and Aerodynamics. --- Functional equations --- Functional analysis --- Classical Mechanics. --- Classical mechanics --- Newtonian mechanics --- Dynamics --- Quantum theory --- Hydraulics --- Mechanics --- Hydrostatics --- Permeability

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The purpose of this book is to give a comprehensive exposition of the theory of pointwise multipliers acting in pairs of spaces of differentiable functions. The theory was essentially developed by the authors during the last thirty years and the present volume is mainly based on their results. Part I is devoted to the theory of multipliers and encloses the following topics: trace inequalities, analytic characterization of multipliers, relations between spaces of Sobolev multipliers and other function spaces, maximal subalgebras of multiplier spaces, traces and extensions of multipliers, essential norm and compactness of multipliers, and miscellaneous properties of multipliers. Part II concerns several applications of this theory: continuity and compactness of differential operators in pairs of Sobolev spaces, multipliers as solutions to linear and quasilinear elliptic equations, higher regularity in the single and double layer potential theory for Lipschitz domains, regularity of the boundary in $L_p$-theory of elliptic boundary value problems, and singular integral operators in Sobolev spaces.

Differential operators. --- Integral operators. --- Multipliers (Mathematical analysis). --- Multipliers (Mathematical analysis) --- Sobolev spaces --- Differential operators --- Integral operators --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Sobolev spaces. --- Operators, Integral --- Operators, Differential --- Spaces, Sobolev --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Functional analysis. --- Integral equations. --- Partial differential equations. --- Analysis. --- Integral Equations. --- Partial Differential Equations. --- Functional Analysis. --- Integrals --- Operator theory --- Differential equations --- Function spaces --- Functional analysis --- Harmonic analysis --- Global analysis (Mathematics). --- Differential equations, partial. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Partial differential equations --- Equations, Integral --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- 517.1 Mathematical analysis --- Mathematical analysis --- Calculus. --- Analysis (Mathematics) --- Fluxions (Mathematics) --- Infinitesimal calculus --- Limits (Mathematics) --- Functions --- Geometry, Infinitesimal

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Nonlinear functional analysis and applications is an area of study that has provided fascination for many mathematicians across the world. This monograph delves specifically into the topic of the geometric properties of Banach spaces and nonlinear iterations, a subject of extensive research over the past thirty years. Chapters 1 to 5 develop materials on convexity and smoothness of Banach spaces, associated moduli and connections with duality maps. Key results obtained are summarized at the end of each chapter for easy reference. Chapters 6 to 23 deal with an in-depth, comprehensive and up-to-date coverage of the main ideas, concepts and results on iterative algorithms for the approximation of fixed points of nonlinear nonexpansive and pseudo-contractive-type mappings. This includes detailed workings on solutions of variational inequality problems, solutions of Hammerstein integral equations, and common fixed points (and common zeros) of families of nonlinear mappings. Carefully referenced and full of recent, incisive findings and interesting open-questions, this volume will prove useful for graduate students of mathematical analysis and will be a key-read for mathematicians with an interest in applications of geometric properties of Banach spaces, as well as specialists in nonlinear operator theory.

Banach spaces --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Banach spaces. --- Probabilities. --- Probability --- Statistical inference --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Functional analysis. --- Integral equations. --- Operator theory. --- Numerical analysis. --- Calculus of variations. --- Operator Theory. --- Analysis. --- Functional Analysis. --- Calculus of Variations and Optimal Control; Optimization. --- Integral Equations. --- Numerical Analysis. --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Mathematical analysis --- Functional analysis --- Equations, Integral --- Functional equations --- Functional calculus --- Calculus of variations --- Integral equations --- 517.1 Mathematical analysis --- Math --- Science --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Functions of complex variables --- Generalized spaces --- Topology --- Global analysis (Mathematics). --- Mathematical optimization. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Operations research --- Simulation methods --- System analysis --- Analysis, Global (Mathematics) --- Differential topology --- Geometry, Algebraic