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Algebra. --- Algèbre. --- Fonctions (Mathématiques). --- Functions. --- Mathematics --- Mathématiques --- algebra. --- functions (mathematics). --- Philosophy. --- Philosophie.

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Potential theory is the broad area of mathematical analysis encompassing such topics as harmonic and subharmonic functions, the Dirichlet problem, harmonic measure, Green's functions, potentials and capacity. This is an introduction to the subject suitable for beginning graduate students, concentrating on the important case of two dimensions. This permits a simpler treatment than other books, yet is still sufficient for a wide range of applications to complex analysis; these include Picard's theorem, the Phragmén-Lindelöf principle, the Koebe one-quarter mapping theorem and a sharp quantitative form of Runge's theorem. In addition there is a chapter on connections with functional analysis and dynamical systems, which shows how the theory can be applied to other parts of mathematics, and gives a flavour of some recent research. Exercises are provided throughout, enabling the book to be used with advanced courses on complex analysis or potential theory.

Potential theory (Mathematics) --- Functions of complex variables. --- Complex variables --- Elliptic functions --- Functions of real variables --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Functions of complex variables --- Functions (Mathematics)

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M. Brelot: Historical introduction.- H. Bauer: Harmonic spaces and associated Markov processes.- J.M. Bony: Opérateurs elliptiques dégénérés associés aux axiomatiques de la theorie du potentiel.- J. Deny: Méthodes hilbertiennes en theory du potentiel.- J.L. Doob: Martingale theory – Potential theory.- G. Mokobodzki: Cônes de potentiels et noyaux subordonnés.

Potential theory (Mathematics) --- Mathematical analysis --- 517.1 Mathematical analysis --- Mathematics. --- Potential theory (Mathematics). --- Potential Theory. --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mechanics

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The book presents a research area in geometric function theory concerned with harmonic quasiconformal mappings and hyperbolic type metrics defined on planar and multidimensional domains. The classes of quasiconformal and quasiregular mappings are well established areas of study in this field as these classes are natural and fruitful generalizations of the class of analytic functions in the planar case. The book contains many concrete examples, as well as detailed proofs and explanations of motivations behind given results, gradually bringing the reader to the forefront of current research in the area. This monograph was written for a wide readership from graduate students of mathematical analysis to researchers working in this or related areas of mathematics who want to learn the tools or work on open problems listed in various parts of the book.

Geometric function theory. --- Functions of complex variables. --- Potential theory (Mathematics). --- Functions of a Complex Variable. --- Potential Theory. --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Complex variables --- Elliptic functions --- Functions of real variables

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This third volume of the monograph examines potential theory. The first chapter develops potential theory with respect to a single kernel (or discrete time semigroup). All the essential ideas of the theory are presented: excessive functions, reductions, sweeping, maximum principle. The second chapter begins with a study of the notion of reduction in the most general situation possible - the ``gambling house'' of Dubins and Savage. The beautiful results presented have never been made accessible to a wide public. These are then connected with the theory of sweeping with respect to a cone of cont

Probabilities. --- Potential theory (Mathematics) --- Semigroups. --- Group theory --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk