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The topics in this research monograph are at the interface of several areas of mathematics such as harmonic analysis, functional analysis, analysis on spaces of homogeneous type, topology, and quasi-metric geometry. The presentation is self-contained with complete, detailed proofs, and a large number of examples and counterexamples are provided. Unique features of Metrization Theory for Groupoids: With Applications to Analysis on Quasi-Metric Spaces and Functional Analysis include: * treatment of metrization from a wide, interdisciplinary perspective, with accompanying applications ranging across diverse fields; * coverage of topics applicable to a variety of scientific areas within pure mathematics; * useful techniques and extensive reference material; * includes sharp results in the field of metrization. Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties.

Functional analysis. --- Nonsymmetric matrices. --- Probabilities. --- Quasi-metric spaces. --- Groupoids --- Harmonic analysis --- Functional analysis --- Mathematics --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Algebra --- Operations Research --- Groupoids. --- Harmonic analysis. --- Functional calculus --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Mathematics. --- Algebraic geometry. --- Mathematical analysis. --- Analysis (Mathematics). --- Measure theory. --- Topology. --- Abstract Harmonic Analysis. --- Functional Analysis. --- Analysis. --- Measure and Integration. --- Algebraic Geometry. --- Calculus of variations --- Functional equations --- Integral equations --- Banach algebras --- Calculus --- Mathematical analysis --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Group theory --- Global analysis (Mathematics). --- Geometry, algebraic. --- Algebraic geometry --- Geometry --- Math --- Science --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- 517.1 Mathematical analysis

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The aim of this book is to offer, in a concise, rigorous, and largely self-contained manner, a rapid introduction to the theory of distributions and its applications to partial differential equations and harmonic analysis. The book is written in a format suitable for a graduate course spanning either over one-semester,  when the focus is primarily on the foundational aspects, or over a two-semester period that allows for the proper amount of time to cover all intended applications as well. It presents a balanced treatment of the topics involved, and contains a large number of exercises (upwards of two hundred, more than half of which are accompanied by solutions), which have been carefully chosen to amplify the effect, and substantiate the power and scope, of the theory of distributions. Graduate students, professional mathematicians, and scientifically trained people with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. Throughout, a special effort has been made to develop the theory of distributions not as an abstract edifice but rather give the reader a chance to see the rationale behind various seemingly technical definitions, as well as the opportunity to apply the newly developed tools (in the natural build-up of the theory) to concrete problems in partial differential equations and harmonic analysis, at the earliest opportunity.

Functional analysis --- Harmonic analysis. Fourier analysis --- Partial differential equations --- Differential equations --- Mathematics --- Fourieranalyse --- differentiaalvergelijkingen --- Laplacetransformatie --- functies (wiskunde) --- wiskunde

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Mathematical analysis --- Hardy spaces. --- Harmonic analysis. --- Interpolation spaces. --- Pseudodifferential operators. --- Hardy, Espaces de --- Analyse harmonique --- Espaces d'interpolation --- Opérateurs pseudo-différentiels --- 51 <082.1> --- Mathematics--Series --- Opérateurs pseudo-différentiels --- Hardy spaces --- Harmonic analysis --- Interpolation spaces --- Pseudodifferential operators --- Operators, Pseudodifferential --- Pseudo-differential operators --- Operator theory --- Spaces, Interpolation --- Function spaces --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Spaces, Hardy --- Functional analysis --- Functions of complex variables --- Analyse harmonique (mathématiques) --- Linear operators --- Opérateurs linéaires --- Selfadjoint operators --- Opérateurs auto-adjoints

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Divergence theorem. --- Functional analysis. --- Anàlisi funcional --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Gauss-Ostrogradsky theorem --- Gauss's theorem --- Vector algebra --- Vector analysis --- Càlcul funcional --- Càlcul de variacions --- Àlgebres de Hilbert --- Àlgebres topològiques --- Anàlisi funcional no lineal --- Anàlisi microlocal --- Espais analítics --- Espais de Hardy --- Espais d'Orlicz --- Espais funcionals --- Espais vectorials normats --- Espais vectorials --- Filtres digitals (Matemàtica) --- Funcionals --- Funcions vectorials --- Multiplicadors (Anàlisi matemàtica) --- Pertorbació (Matemàtica) --- Teoria d'operadors --- Teoria de distribucions (Anàlisi funcional) --- Teoria de functors --- Teoria de l'aproximació --- Teoria del funcional de densitat --- Teoria espectral (Matemàtica) --- Equacions funcionals --- Equacions integrals

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This monograph presents a comprehensive, self-contained, and novel approach to the Divergence Theorem through five progressive volumes. Its ultimate aim is to develop tools in Real and Harmonic Analysis, of geometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problems formulated in rather general geometric and analytic settings. The text is intended for researchers, graduate students, and industry professionals interested in applications of harmonic analysis and geometric measure theory to complex analysis, scattering, and partial differential equations. Traditionally, the label “Calderón-Zygmund theory” has been applied to a distinguished body of works primarily pertaining to the mapping properties of singular integral operators on Lebesgue spaces, in various geometric settings. Volume IV amounts to a versatile Calderón-Zygmund theory for singular integral operators of layer potential type in open sets with uniformly rectifiable boundaries, considered on a diverse range of function spaces. Novel applications to complex analysis in several variables are also explored here.

Mathematical analysis. --- Integral Transforms and Operational Calculus. --- 517.1 Mathematical analysis --- Mathematical analysis --- Teoria de la mesura geomètrica --- Àlgebra vectorial --- Capa límit