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This book presents the first “How To” guide to the use of radial basis functions (RBF). It provides a clear vision of their potential, an overview of ready-for-use computational tools and precise guidelines to implement new engineering applications of RBF. Radial basis functions (RBF) are a mathematical tool mature enough for useful engineering applications. Their mathematical foundation is well established and the tool has proven to be effective in many fields, as the mathematical framework can be adapted in several ways. A candidate application can be faced considering the features of RBF: multidimensional space (including 2D and 3D), numerous radial functions available, global and compact support, interpolation/regression. This great flexibility makes RBF attractive – and their great potential has only been partially discovered. This is because of the difficulty in taking a first step toward RBF as they are not commonly part of engineers’ cultural background, but also due to the numerical complexity of RBF problems that scales up very quickly with the number of RBF centers. Fast RBF algorithms are available to alleviate this and high-performance computing (HPC) can provide further aid. Nevertheless, a consolidated tradition in using RBF in engineering applications is still missing and the beginner can be confused by the literature, which in many cases is presented with language and symbolisms familiar to mathematicians but which can be cryptic for engineers. The book is divided in two main sections. The first covers the foundations of RBF, the tools available for their quick implementation and guidelines for facing new challenges; the second part is a collection of practical RBF applications in engineering, covering several topics, including response surface interpolation in n-dimensional spaces, mapping of magnetic loads, mapping of pressure loads, up-scaling of flow fields, stress/strain analysis by experimental displacement fields, implicit surfaces, mesh to cad deformation, mesh morphing for crack propagation in 3D, ice and snow accretion using computational fluid dynamics (CFD) data, shape optimization for external aerodynamics, and use of adjoint data for surface sculpting. For each application, the complete path is clearly and consistently exposed using the systematic approach defined in the first section.

Radial basis functions. --- Mathematics. --- Algorithms. --- Computer mathematics. --- Structural mechanics. --- Computational Science and Engineering. --- Algorithm Analysis and Problem Complexity. --- Structural Mechanics. --- Basis functions, Radial --- Functions, Radial basis --- Radial basis function method --- Approximation theory --- Computer science. --- Computer software. --- Mechanics. --- Mechanics, Applied. --- Solid Mechanics. --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Software, Computer --- Computer systems --- Informatics --- Science --- Algorism --- Algebra --- Arithmetic --- Computer mathematics --- Electronic data processing --- Mathematics --- Foundations

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In many areas of mathematics, science and engineering, from computer graphics to inverse methods to signal processing, it is necessary to estimate parameters, usually multidimensional, by approximation and interpolation. Radial basis functions are a powerful tool which work well in very general circumstances and so are becoming of widespread use as the limitations of other methods, such as least squares, polynomial interpolation or wavelet-based, become apparent. The author's aim is to give a thorough treatment from both the theoretical and practical implementation viewpoints. For example, he emphasises the many positive features of radial basis functions such as the unique solvability of the interpolation problem, the computation of interpolants, their smoothness and convergence and provides a careful classification of the radial basis functions into types that have different convergence. A comprehensive bibliography rounds off what will prove a very valuable work.

Radial basis functions. --- Basis functions, Radial --- Functions, Radial basis --- Radial basis function method --- Approximation theory --- Radial basis functions --- 517.518.8 --- 681.3*G12 --- 681.3*G12 Approximation: chebyshev; elementary function; least squares; linear approximation; minimax approximation and algorithms; nonlinear and rational approximation; spline and piecewise polynomial approximation (Numerical analysis) --- Approximation: chebyshev; elementary function; least squares; linear approximation; minimax approximation and algorithms; nonlinear and rational approximation; spline and piecewise polynomial approximation (Numerical analysis) --- 517.518.8 Approximation of functions by polynomials and their generalizations --- Approximation of functions by polynomials and their generalizations

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The purpose of the Special Issue “Quantitative Methods in Economics and Finance” of the journal Risks was to provide a collection of papers that reflect the latest research and problems of pricing complex derivates, simulation pricing, analysis of financial markets, and volatility of exchange rates in the international context. This book can be used as a reference for academicians and researchers who would like to discuss and introduce new developments in the field of quantitative methods in economics and finance and explore applications of quantitative methods in other business areas.

omnichannel (omni-channel) sales --- sales funnel --- cost of sales --- customer relationship management (CRM), Big Data --- robo-advisor --- financial innovations --- diffusion --- exchange traded funds --- stock index futures --- stock index options --- stock market indexes --- business finance --- earnings management --- EBIT --- financial modelling --- homogeneity --- stationarity --- time series methods --- unit root --- loan pricing --- RAROC --- loan origination --- exchange-rate risk --- long-range dependency --- wavelets --- multi-frequency analysis --- AUD–USD exchange rate --- π-option --- American-type option --- optimal stopping --- Monte Carlo simulation --- economic security of companies --- valuation of intangible assets and intellectual property --- International Valuation Standards (IVS) --- legal disputes over intellectual rights --- time series --- prediction --- exchange rate --- artificial neural networks --- radial basis function --- multi-layer perceptron --- seasonal fluctuations --- global economy

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This book focuses on empirical analyses of the Environmental Kuznets Curve. Most papers apply their research on CO2 emissions. The papers in this Special Issue seek to improve modeling techniques to prevent econometrical flaws (e.g., by adding additional explanatory variables and/or moving away from linear regression models) and apply their models to specific countries or groups of countries.

Technology: general issues --- History of engineering & technology --- productivity changes --- technical efficiency --- energy industry --- DEA-based Malmquist productivity index --- European Union --- Environmental Kuznets Curve --- carbon dioxide emissions --- environmental degradation --- financial development --- energy use --- institutional quality --- institutional development --- human capital --- CO2 emissions --- co-integration analysis --- pollution-income --- Environmental Kunzets Curve --- education --- income-inequality --- Europe --- panel data --- clustering --- carbon tax --- price elasticity --- translog cost function --- energy and carbon performance --- environmental kuznets curve --- kink regression model --- G7 countries --- EKC estimation --- CO2 emissions prediction --- neural networks --- radial basis function neural network --- renewable energy consumption

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This book is the first to be devoted to the theory and applications of spherical (radial) basis functions (SBFs), which is rapidly emerging as one of the most promising techniques for solving problems where approximations are needed on the surface of a sphere. The aim of the book is to provide enough theoretical and practical details for the reader to be able to implement the SBF methods to solve real world problems. The authors stress the close connection between the theory of SBFs and that of the more well-known family of radial basis functions (RBFs), which are well-established tools for solving approximation theory problems on more general domains. The unique solvability of the SBF interpolation method for data fitting problems is established and an in-depth investigation of its accuracy is provided. Two chapters are devoted to partial differential equations (PDEs). One deals with the practical implementation of an SBF-based solution to an elliptic PDE and another which describes an SBF approach for solving a parabolic time-dependent PDE, complete with error analysis. The theory developed is illuminated with numerical experiments throughout. Spherical Radial Basis Functions, Theory and Applications will be of interest to graduate students and researchers in mathematics and related fields such as the geophysical sciences and statistics.

Mathematics. --- Approximations and Expansions. --- Partial Differential Equations. --- Numerical Analysis. --- Global Analysis and Analysis on Manifolds. --- Geophysics/Geodesy. --- Physical geography. --- Global analysis. --- Differential equations, partial. --- Numerical analysis. --- Mathématiques --- Géographie physique --- Analyse numérique --- Radial basis functions. --- Spherical functions. --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Operations Research --- Basis functions, Radial --- Functions, Radial basis --- Radial basis function method --- Functions, Spherical --- Geophysics. --- Approximation theory. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Partial differential equations. --- Approximation theory --- Spherical harmonics --- Transcendental functions --- Spheroidal functions --- Geography --- Mathematical analysis --- Partial differential equations --- Math --- Science --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Geological physics --- Terrestrial physics --- Earth sciences --- Physics --- Geometry, Differential --- Topology --- Differential equations, Partial. --- Global analysis (Mathematics) --- Manifolds (Mathematics)