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The monograph is devoted to the use of the moduli method in mapping theory, in particular, the meaning of direct and inverse modulus inequalities and their possible applications. The main goal is the development of a modulus technique in the Euclidean space and some metric spaces (manifolds, surfaces, quotient spaces, etc.). Particular attention is paid to the local and boundary behavior of mappings, as well as to obtaining modulus inequalities for some classes. The reader is invited to familiarize himself with all the main achievements of the author, synthesized in this book. The results presented here are of a high scientific level, are new and have no analogues in the world with such a degree of generality.

Functions of complex variables. --- Potential theory (Mathematics). --- Functions of a Complex Variable. --- Potential Theory.

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Multivariate polysplines are a new mathematical technique that has arisen from a synthesis of approximation theory and the theory of partial differential equations. It is an invaluable means to interpolate practical data with smooth functions.Multivariate polysplines have applications in the design of surfaces and ""smoothing"" that are essential in computer aided geometric design (CAGD and CAD/CAM systems), geophysics, magnetism, geodesy, geography, wavelet analysis and signal and image processing. In many cases involving practical data in these areas, polysplines are proving more effecti

Spline theory. --- Polyharmonic functions. --- Differential equations, Elliptic --- Numerical solutions. --- Functions, Polyharmonic --- Harmonic functions --- Potential theory (Mathematics) --- Spline functions --- Approximation theory --- Interpolation --- Polyharmonic functions --- Spline theory --- Numerical solutions

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This graduate textbook provides a detailed introduction to the probabilistic interpretation of nonlinear potential theory, relying on the recently introduced notion of tug-of-war games with noise. The book explores both basic and more advanced constructions, carefully explaining the parallel between linear and nonlinear cases. The presentation is self-contained with many exercises, making the book suitable as a textbook for a graduate course, as well as for self-study. Extensive background and auxiliary material allow the tailoring of courses to individual student levels.

Potential theory (Mathematics) --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Potential theory (Mathematics). --- Probabilities. --- Mathematical analysis. --- Analysis (Mathematics). --- Potential Theory. --- Probability Theory and Stochastic Processes. --- Analysis. --- 517.1 Mathematical analysis --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk

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"Potential Theory in Applied Geophysics" introduces the principles of gravitational, magnetic, electrostatic, direct current electrical and electromagnetic fields, with detailed solutions of Laplace and electromagnetic wave equations by the method of separation of variables. Behaviour of the scalar and vector potential and the nature of the solutions of these boundary value problems are shown along with the use of complex variables and conformal transformation, Green's theorem, Green's functions and its use in integral equation. Finite element and finite difference methods for two-dimensional potential problems are discussed in considerable detail. The analytical continuation of the potential field and inverse theory, used for the interpretation of potential field data, are also demonstrated.

Geophysics. --- Potential theory (Mathematics) --- Geological physics --- Terrestrial physics --- Earth sciences --- Physics --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Physical geography. --- Potential theory (Mathematics). --- Geography. --- Mathematics. --- Geophysics/Geodesy. --- Theoretical, Mathematical and Computational Physics. --- Potential Theory. --- Earth Sciences, general. --- Applications of Mathematics. --- Math --- Science --- Geography --- Cosmography --- World history --- Mathematical physics. --- Earth sciences. --- Applied mathematics. --- Engineering mathematics. --- Engineering --- Engineering analysis --- Geosciences --- Environmental sciences --- Physical sciences --- Physical mathematics --- Mathematics

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The study of M-matrices, their inverses and discrete potential theory is now a well-established part of linear algebra and the theory of Markov chains. The main focus of this monograph is the so-called inverse M-matrix problem, which asks for a characterization of nonnegative matrices whose inverses are M-matrices. We present an answer in terms of discrete potential theory based on the Choquet-Deny Theorem. A distinguished subclass of inverse M-matrices is ultrametric matrices, which are important in applications such as taxonomy. Ultrametricity is revealed to be a relevant concept in linear algebra and discrete potential theory because of its relation with trees in graph theory and mean expected value matrices in probability theory. Remarkable properties of Hadamard functions and products for the class of inverse M-matrices are developed and probabilistic insights are provided throughout the monograph.

Mathematics. --- Potential theory (Mathematics). --- Game theory. --- Probabilities. --- Potential Theory. --- Probability Theory and Stochastic Processes. --- Game Theory, Economics, Social and Behav. Sciences. --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Math --- Science --- Potential theory (Mathematics) --- Games, Theory of --- Theory of games --- Mathematical models --- Mathematics --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk