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Pluripotential theory is a very powerful tool in geometry, complex analysis and dynamics. This volume brings together the lectures held at the 2011 CIME session on "pluripotential theory" in Cetraro, Italy. This CIME course focused on complex Monge-Ampére equations, applications of pluripotential theory to Kahler geometry and algebraic geometry and to holomorphic dynamics. The contributions provide an extensive description of the theory and its very recent developments, starting from basic introductory materials and concluding with open questions in current research.

Mathematics --- Physical Sciences & Mathematics --- Geometry --- Mathematics. --- Algebraic geometry. --- Potential theory (Mathematics). --- Functions of complex variables. --- Geometry. --- Algebraic Geometry. --- Several Complex Variables and Analytic Spaces. --- Potential Theory. --- Monge-Ampère equations --- Pluripotential theory --- Nonlinear theories --- Potential theory (Mathematics) --- Equations, Monge-Ampère --- Differential equations, Partial --- Geometry, algebraic. --- Differential equations, partial. --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Partial differential equations --- Algebraic geometry --- Euclid's Elements --- Complex variables --- Elliptic functions --- Functions of real variables

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This monograph presents the geoscientific context arising in decorrelative gravitational exploration to determine the mass density distribution inside the Earth. First, an insight into the current state of research is given by reducing gravimetry to mathematically accessible, and thus calculable, decorrelated models. In this way, the various unresolved questions and problems of gravimetry are made available to a broad scientific audience and the exploration industry. New theoretical developments will be given, and innovative ways of modeling geologic layers and faults by mollifier regularization techniques are shown. This book is dedicated to surface as well as volume geology with potential data primarily of terrestrial origin. For deep geology, the geomathematical decorrelation methods are to be designed in such a way that depth information (e.g., in boreholes) may be canonically entered. Bridging several different geo-disciplines, this book leads in a cycle from the potential measurements made by geoengineers, to the cleansing of data by geophysicists and geoengineers, to the subsequent theory and model formation, computer-based implementation, and numerical calculation and simulations made by geomathematicians, to interpretation by geologists, and, if necessary, back. It therefore spans the spectrum from geoengineering, especially geodesy, via geophysics to geomathematics and geology, and back. Using the German Saarland area for methodological tests, important new fields of application are opened, particularly for regions with mining-related cavities or dense development in today's geo-exploration. .

Potential theory (Mathematics). --- Numerical analysis. --- Physical geography. --- Gravitation. --- Potential Theory. --- Numerical Analysis. --- Earth System Sciences. --- Classical and Quantum Gravitation, Relativity Theory. --- Geography --- Field theory (Physics) --- Matter --- Physics --- Antigravity --- Centrifugal force --- Relativity (Physics) --- Mathematical analysis --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mechanics --- Properties --- Potential theory (Mathematics) --- Teoria del potencial (Matemàtica) --- Fórmula de Green --- Funcions potencials --- Operadors de Green --- Teorema de Green --- Teoria del potencial --- Anàlisi matemàtica --- Mecànica --- Efecte túnel --- Teoria del potencial (Física) --- Varietats de Riemann

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Potential Theory presents a clear path from calculus to classical potential theory and beyond, with the aim of moving the reader into the area of mathematical research as quickly as possible. The subject matter is developed from first principles using only calculus. Commencing with the inverse square law for gravitational and electromagnetic forces and the divergence theorem, the author develops methods for constructing solutions of Laplace's equation on a region with prescribed values on the boundary of the region. The latter half of the book addresses more advanced material aimed at those with the background of a senior undergraduate or beginning graduate course in real analysis. Starting with solutions of the Dirichlet problem subject to mixed boundary conditions on the simplest of regions, methods of morphing such solutions onto solutions of Poisson's equation on more general regions are developed using diffeomorphisms and the Perron-Wiener-Brelot method, culminating in application to Brownian motion. In this new edition, many exercises have been added to reconnect the subject matter to the physical sciences. This book will undoubtedly be useful to graduate students and researchers in mathematics, physics, and engineering.

Differential equations --- Probability theory --- wiskunde --- Mathematics --- differentiaalvergelijkingen --- kansrekening --- waarschijnlijkheidstheorie --- Laplacetransformatie --- Operational research. Game theory --- Partial differential equations --- stochastische analyse --- Differential equations, partial --- Potential theory (Mathematics) --- Distribution (Probability theory) --- Mathématiques --- Potentiel, Théorie du --- Distribution (Théorie des probabilités) --- EPUB-LIV-FT LIVMATHE LIVSTATI SPRINGER-B --- Differential equations, partial. --- Potential theory (Mathematics). --- Distribution (Probability theory. --- Partial Differential Equations. --- Potential Theory. --- Probability Theory and Stochastic Processes. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Partial differential equations. --- Probabilities. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk

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Random walks, Markov chains and electrical networks serve as an introduction to the study of real-valued functions on finite or infinite graphs, with appropriate interpretations using probability theory and current-voltage laws. The relation between this type of function theory and the (Newton) potential theory on the Euclidean spaces is well-established. The latter theory has been variously generalized, one example being the axiomatic potential theory on locally compact spaces developed by Brelot, with later ramifications from Bauer, Constantinescu and Cornea. A network is a graph with edge-weights that need not be symmetric. This book presents an autonomous theory of harmonic functions and potentials defined on a finite or infinite network, on the lines of axiomatic potential theory. Random walks and electrical networks are important sources for the advancement of the theory.

Harmonic functions. --- Mathematics. --- Operator theory. --- Harmonic functions --- Potential theory (Mathematics) --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Operations Research --- Functions, Harmonic --- Laplace's equations --- Functions of complex variables. --- Partial differential equations. --- Potential theory (Mathematics). --- Potential Theory. --- Functions of a Complex Variable. --- Partial Differential Equations. --- Bessel functions --- Differential equations, Partial --- Fourier series --- Harmonic analysis --- Lamé's functions --- Spherical harmonics --- Toroidal harmonics --- Differential equations, partial. --- Partial differential equations --- Complex variables --- Elliptic functions --- Functions of real variables --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics

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Aimed at graduate students and researchers in mathematics, physics, and engineering, this book presents a clear path from calculus to classical potential theory and beyond, moving the reader into a fertile area of mathematical research as quickly as possible. The author revises and updates material from his classic work, Introduction to Potential Theory (1969), to provide a modern text that introduces all the important concepts of classical potential theory. In the first half of the book, the subject matter is developed meticulously from first principles using only calculus. Commencing with the inverse square law for gravitational and electromagnetic forces and the divergence theorem of the calculus, the author develops methods for constructing solutions of Laplace’s equation on a region with prescribed values on the boundary of the region. The second half addresses more advanced material aimed at those with a background of a senior undergraduate or beginning graduate course in real analysis. For specialized regions, namely spherical chips, solutions of Laplace’s equation are constructed having prescribed normal derivatives on the flat portion of the boundary and prescribed values on the remaining portion of the boundary. By means of transformations known as diffeomorphisms, these solutions are morphed into local solutions on regions with curved boundaries. The Perron-Weiner-Brelot method is then used to construct global solutions for elliptic partial differential equations involving a mixture of prescribed values of a boundary differential operator on part of the boundary and prescribed values on the remainder of the boundary.

Electronic books. -- local. --- Mathematics. --- Potential theory (Mathematics). --- Potential theory (Mathematics) --- Operations Research --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Math --- Partial differential equations. --- Applied mathematics. --- Engineering mathematics. --- Physics. --- Engineering. --- Potential Theory. --- Mathematical Methods in Physics. --- Engineering, general. --- Partial Differential Equations. --- Applications of Mathematics. --- Science --- Construction --- Industrial arts --- Technology --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Engineering --- Engineering analysis --- Mathematical analysis --- Mechanics --- Partial differential equations --- Mathematics --- Mathematical physics. --- Differential equations, partial. --- Physical mathematics --- Physics