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Moduli theory --- Smoothness of functions --- Variétés topologiques à 4 dimensions --- Modules, théorie des --- 517.5 --- Smooth functions --- Functions --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Theory of functions --- Moduli theory. --- Smoothness of functions. --- 517.5 Theory of functions --- Variétés topologiques à 4 dimensions --- Modules, théorie des --- Fonctions d'une variable reelle --- Approximation

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These notes, based on lectures delivered in Saint Flour, provide an easy introduction to the authors’ 2007 Springer monograph “Random Fields and Geometry.” While not as exhaustive as the full monograph, they are also less exhausting, while still covering the basic material, typically at a more intuitive and less technical level. They also cover some more recent material relating to random algebraic topology and statistical applications. The notes include an introduction to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness. This is followed by a quick review of geometry, both integral and Riemannian, with an emphasis on tube formulae, to provide the reader with the material needed to understand and use the Gaussian kinematic formula, the main result of the notes. This is followed by chapters on topological inference and random algebraic topology, both of which provide applications of the main results.

Gaussian processes --- Random fields --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Mathematical Statistics --- Geometry --- Smoothness of functions. --- Stochastic processes. --- Functions. --- Analysis (Mathematics) --- Random processes --- Smooth functions --- Mathematics. --- Geometry. --- Statistics. --- Statistical Theory and Methods. --- Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Econometrics --- Euclid's Elements --- Math --- Science --- Differential equations --- Mathematical analysis --- Numbers, Complex --- Set theory --- Probabilities --- Functions --- Mathematical statistics. --- Statistical inference --- Statistics, Mathematical --- Statistics --- Sampling (Statistics) --- Statistics .

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Mathematical analysis --- 517.54 --- 51 <082.1> --- Conformal mapping and geometric problems in the theory of functions of a complex variable. Analytic functions and their generalizations --- Mathematics--Series --- 517.54 Conformal mapping and geometric problems in the theory of functions of a complex variable. Analytic functions and their generalizations --- Harmonic functions. --- Green's functions. --- Inequalities (Mathematics) --- Smoothness of functions. --- Fonctions harmoniques. --- Green, Fonctions de. --- Inégalités (mathématiques) --- Fonctions de lissage. --- Green's functions --- Harmonic functions --- Smoothness of functions --- Smooth functions --- Functions --- Processes, Infinite --- Functions, Harmonic --- Laplace's equations --- Bessel functions --- Differential equations, Partial --- Fourier series --- Harmonic analysis --- Lamé's functions --- Spherical harmonics --- Toroidal harmonics --- Functions, Green's --- Functions, Induction --- Functions, Source --- Green functions --- Induction functions --- Source functions --- Differential equations --- Potential theory (Mathematics)

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Many phenomena in engineering and mathematical physics can be modeled by means of boundary value problems for a certain elliptic differential operator in a given domain. When the differential operator under discussion is of second order a variety of tools are available for dealing with such problems, including boundary integral methods, variational methods, harmonic measure techniques, and methods based on classical harmonic analysis. When the differential operator is of higher-order (as is the case, e.g., with anisotropic plate bending when one deals with a fourth order operator) only a few options could be successfully implemented. In the 1970s Alberto Calderón, one of the founders of the modern theory of Singular Integral Operators, advocated the use of layer potentials for the treatment of higher-order elliptic boundary value problems. The present monograph represents the first systematic treatment based on this approach. This research monograph lays, for the first time, the mathematical foundation aimed at solving boundary value problems for higher-order elliptic operators in non-smooth domains using the layer potential method and addresses a comprehensive range of topics, dealing with elliptic boundary value problems in non-smooth domains including layer potentials, jump relations, non-tangential maximal function estimates, multi-traces and extensions, boundary value problems with data in Whitney–Lebesque spaces, Whitney–Besov spaces, Whitney–Sobolev- based Lebesgue spaces, Whitney–Triebel–Lizorkin spaces,Whitney–Sobolev-based Hardy spaces, Whitney–BMO and Whitney–VMO spaces.

Boundary value problems --- Differential equations, Elliptic --- Lipschitz spaces --- Smoothness of functions --- Calderâon-Zygmund operator --- Mathematics --- Civil & Environmental Engineering --- Physical Sciences & Mathematics --- Engineering & Applied Sciences --- Calculus --- Operations Research --- Mathematical Theory --- Smooth functions --- Hölder spaces --- Elliptic differential equations --- Elliptic partial differential equations --- Linear elliptic differential equations --- Boundary conditions (Differential equations) --- Mathematics. --- Fourier analysis. --- Integral equations. --- Partial differential equations. --- Potential theory (Mathematics). --- Potential Theory. --- Partial Differential Equations. --- Integral Equations. --- Fourier Analysis. --- Boundary value problems. --- Differential equations, Elliptic. --- Lipschitz spaces. --- Smoothness of functions. --- Calderón-Zygmund operator. --- Calderón-Zygmund singular integral operator --- Mikhlin-Calderon-Zygmund operator --- Operator, Calderón-Zygmund --- Singular integral operator, Calderón-Zygmund --- Zygmund-Calderón operator --- Linear operators --- Functions --- Function spaces --- Differential equations, Linear --- Differential equations, Partial --- Differential equations --- Functions of complex variables --- Mathematical physics --- Initial value problems --- Differential equations, partial. --- Analysis, Fourier --- Mathematical analysis --- Equations, Integral --- Functional equations --- Functional analysis --- Partial differential equations --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mechanics