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Anisotropy --- Orthotropism --- Singular integral equations

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This book develops a new theory of multi-parameter singular integrals associated with Carnot-Carathéodory balls. Brian Street first details the classical theory of Calderón-Zygmund singular integrals and applications to linear partial differential equations. He then outlines the theory of multi-parameter Carnot-Carathéodory geometry, where the main tool is a quantitative version of the classical theorem of Frobenius. Street then gives several examples of multi-parameter singular integrals arising naturally in various problems. The final chapter of the book develops a general theory of singular integrals that generalizes and unifies these examples. This is one of the first general theories of multi-parameter singular integrals that goes beyond the product theory of singular integrals and their analogs. Multi-parameter Singular Integrals will interest graduate students and researchers working in singular integrals and related fields.

Singular integrals. --- Transformations (Mathematics) --- CaldernКygmund singular integrals. --- CaldernКygmund. --- CarnotЃarathodory balls. --- CarnotЃarathodory geometry. --- CarnotЃarathodory metric. --- Euclidean singular integral operators. --- Frobenius theorem. --- Frobenius. --- LittlewoodАaley theory. --- Schwartz space. --- Sobolev spaces. --- convolution. --- elliptic partial differential equations. --- elliptic partial differential operators. --- flag kernels. --- invariant operators. --- linear partial differential equation. --- non-homogeneous kernels. --- pseudodifferential operators. --- singular integral operator. --- singular integral operators. --- singular integrals. --- strengthened cancellation.

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Now in paperback, this remains one of the classic expositions of the theory of wavelets from two of the subject's leading experts. In this volume the theory of paradifferential operators and the Cauchy kernel on Lipschitz curves are discussed with the emphasis firmly on their connection with wavelet bases. Sparse matrix representations of these operators can be given in terms of wavelet bases which have important applications in image processing and numerical analysis. This method is now widely studied and can be used to tackle a wide variety of problems arising in science and engineering. Put simply, this is an essential purchase for anyone researching the theory of wavelets.

517.98 --- Calderon-Zygmund operator --- Wavelets (Mathematics) --- 517.518.8 --- Wavelet analysis --- Harmonic analysis --- Calderón-Zygmund singular integral operator --- Mikhlin-Calderon-Zygmund operator --- Operator, Calderón-Zygmund --- Singular integral operator, Calderón-Zygmund --- Zygmund-Calderón operator --- Linear operators --- Functional analysis and operator theory --- Approximation of functions by polynomials and their generalizations --- Caldéron-Zygmund operator. --- Wavelets (Mathematics). --- 517.518.8 Approximation of functions by polynomials and their generalizations --- 517.98 Functional analysis and operator theory --- Caldéron-Zygmund operator --- Caldéron-Zygmund operator

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Many phenomena in engineering and mathematical physics can be modeled by means of boundary value problems for a certain elliptic differential operator in a given domain. When the differential operator under discussion is of second order a variety of tools are available for dealing with such problems, including boundary integral methods, variational methods, harmonic measure techniques, and methods based on classical harmonic analysis. When the differential operator is of higher-order (as is the case, e.g., with anisotropic plate bending when one deals with a fourth order operator) only a few options could be successfully implemented. In the 1970s Alberto Calderón, one of the founders of the modern theory of Singular Integral Operators, advocated the use of layer potentials for the treatment of higher-order elliptic boundary value problems. The present monograph represents the first systematic treatment based on this approach. This research monograph lays, for the first time, the mathematical foundation aimed at solving boundary value problems for higher-order elliptic operators in non-smooth domains using the layer potential method and addresses a comprehensive range of topics, dealing with elliptic boundary value problems in non-smooth domains including layer potentials, jump relations, non-tangential maximal function estimates, multi-traces and extensions, boundary value problems with data in Whitney–Lebesque spaces, Whitney–Besov spaces, Whitney–Sobolev- based Lebesgue spaces, Whitney–Triebel–Lizorkin spaces,Whitney–Sobolev-based Hardy spaces, Whitney–BMO and Whitney–VMO spaces.

Boundary value problems --- Differential equations, Elliptic --- Lipschitz spaces --- Smoothness of functions --- Calderâon-Zygmund operator --- Mathematics --- Civil & Environmental Engineering --- Physical Sciences & Mathematics --- Engineering & Applied Sciences --- Calculus --- Operations Research --- Mathematical Theory --- Smooth functions --- Hölder spaces --- Elliptic differential equations --- Elliptic partial differential equations --- Linear elliptic differential equations --- Boundary conditions (Differential equations) --- Mathematics. --- Fourier analysis. --- Integral equations. --- Partial differential equations. --- Potential theory (Mathematics). --- Potential Theory. --- Partial Differential Equations. --- Integral Equations. --- Fourier Analysis. --- Boundary value problems. --- Differential equations, Elliptic. --- Lipschitz spaces. --- Smoothness of functions. --- Calderón-Zygmund operator. --- Calderón-Zygmund singular integral operator --- Mikhlin-Calderon-Zygmund operator --- Operator, Calderón-Zygmund --- Singular integral operator, Calderón-Zygmund --- Zygmund-Calderón operator --- Linear operators --- Functions --- Function spaces --- Differential equations, Linear --- Differential equations, Partial --- Differential equations --- Functions of complex variables --- Mathematical physics --- Initial value problems --- Differential equations, partial. --- Analysis, Fourier --- Mathematical analysis --- Equations, Integral --- Functional equations --- Functional analysis --- Partial differential equations --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mechanics

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Numerical methods are a specific form of mathematics that involve creating and use of algorithms to map out the mathematical core of a practical problem. Numerical methods naturally find application in all fields of engineering, physical sciences, life sciences, social sciences, medicine, business, and even arts. The common uses of numerical methods include approximation, simulation, and estimation, and there is almost no scientific field in which numerical methods do not find a use. Results communicated here include topics ranging from statistics (Detecting Extreme Values with Order Statistics in Samples from Continuous Distributions) and Statistical software packages (dCATCH—A Numerical Package for d-Variate near G-Optimal Tchakaloff Regression via Fast NNLS) to new approaches for numerical solutions (Exact Solutions to the Maxmin Problem max‖Ax‖ Subject to ‖Bx‖≤1; On q-Quasi-Newton’s Method for Unconstrained Multiobjective Optimization Problems; Convergence Analysis and Complex Geometry of an Efficient Derivative-Free Iterative Method; On Derivative Free Multiple-Root Finders with Optimal Fourth Order Convergence; Finite Integration Method with Shifted Chebyshev Polynomials for Solving Time-Fractional Burgers’ Equations) to the use of wavelets (Orhonormal Wavelet Bases on The 3D Ball Via Volume Preserving Map from the Regular Octahedron) and methods for visualization (A Simple Method for Network Visualization).

Research & information: general --- Mathematics & science --- Clenshaw–Curtis–Filon --- high oscillation --- singular integral equations --- boundary singularities --- local convergence --- nonlinear equations --- Banach space --- Fréchet-derivative --- finite integration method --- shifted Chebyshev polynomial --- Caputo fractional derivative --- Burgers’ equation --- coupled Burgers’ equation --- maxmin --- supporting vector --- matrix norm --- TMS coil --- optimal geolocation --- probability computing --- Monte Carlo simulation --- order statistics --- extreme values --- outliers --- multiobjective programming --- methods of quasi-Newton type --- Pareto optimality --- q-calculus --- rate of convergence --- wavelets on 3D ball --- uniform 3D grid --- volume preserving map --- Network --- graph drawing --- planar visualizations --- multiple root solvers --- composite method --- weight-function --- derivative-free method --- optimal convergence --- multivariate polynomial regression designs --- G-optimality --- D-optimality --- multiplicative algorithms --- G-efficiency --- Caratheodory-Tchakaloff discrete measure compression --- Non-Negative Least Squares --- accelerated Lawson-Hanson solver