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Serving both as an introduction to the subject and as a reference, this book presents the theory in elegant form and with modern concepts and notation. It covers the general theory and emphasizes the classical types of orthogonal polynomials whose weight functions are supported on standard domains. The approach is a blend of classical analysis and symmetry group theoretic methods. Finite reflection groups are used to motivate and classify symmetries of weight functions and the associated polynomials. This revised edition has been updated throughout to reflect recent developments in the field. It contains 25% new material, including two brand new chapters on orthogonal polynomials in two variables, which will be especially useful for applications, and orthogonal polynomials on the unit sphere. The most modern and complete treatment of the subject available, it will be useful to a wide audience of mathematicians and applied scientists, including physicists, chemists and engineers.
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Molecular orbitals. --- Density functionals. --- Nonmonotonic reasoning. --- Polynomials.
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Molecular orbitals. --- Density functionals. --- Nonmonotonic reasoning. --- Polynomials.
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Polynomials. --- Inequalities. --- Theorem proving. --- Roots of equations. --- Mathematical programming.
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The current book makes several useful topics from the theory of special functions, in particular the theory of spherical harmonics and Legendre polynomials in arbitrary dimensions, available to undergraduates studying physics or mathematics. With this audience in mind, nearly all details of the calculations and proofs are written out, and extensive background material is covered before exploring the main subject matter. Contents: Introduction and Motivation; Working in p Dimensions; Orthogonal Polynomials; Spherical Harmonics in p Dimensions; Solutions to Problems. Readership: Undergraduate an
Spherical harmonics. --- Spherical functions. --- Legendre's polynomials. --- Mathematical physics. --- Physical mathematics --- Physics --- Functions, Spherical --- Spherical harmonics --- Transcendental functions --- Spheroidal functions --- Functions, Potential --- Potential functions --- Harmonic analysis --- Harmonic functions --- Polynomials, Legendre's --- Orthogonal polynomials --- Mathematics
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This volume developed from papers presented at the international conference Approximation Theory XIV, held April 7–10, 2013 in San Antonio, Texas. The proceedings contains surveys by invited speakers, covering topics such as splines on non-tensor-product meshes, Wachspress and mean value coordinates, curvelets and shearlets, barycentric interpolation, and polynomial approximation on spheres and balls. Other contributed papers address a variety of current topics in approximation theory, including eigenvalue sequences of positive integral operators, image registration, and support vector machines. This book will be of interest to mathematicians, engineers, and computer scientists working in approximation theory, computer-aided geometric design, numerical analysis, and related approximation areas.
Approximation theory --- Mathematics. --- Approximations and Expansions. --- Math --- Science --- Approximation theory. --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems
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Orthogonal polynomials --- Functions of several real variables --- 517.5 --- Theory of functions --- 517.5 Theory of functions
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Differential geometry. Global analysis --- Polytopes. --- Orthogonal polynomials. --- Geometry, Riemannian. --- Polytopes --- Polynômes orthogonaux --- Riemann, Géométrie de --- Polynômes orthogonaux --- Riemann, Géométrie de
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This book is aimed at the approximation of set-valued functions with compact sets in an Euclidean space as values. The interest in set-valued functions is rather new. Such functions arise in various modern areas such as control theory, dynamical systems and optimization. The authors' motivation also comes from the newer field of geometric modeling, in particular from the problem of reconstruction of 3D objects from 2D cross-sections. This is reflected in the focus of this book, which is the approximation of set-valued functions with general (not necessarily convex) sets as values, while previo
Approximation theory. --- Linear operators. --- Function spaces. --- Spaces, Function --- Functional analysis --- Linear maps --- Maps, Linear --- Operators, Linear --- Operator theory --- Theory of approximation --- Functions --- Polynomials --- Chebyshev systems
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John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing. This collection will be useful to students and researchers for decades to come. The contributors are Marco Abate, Marco Arizzi, Alexander Blokh, Thierry Bousch, Xavier Buff, Serge Cantat, Tao Chen, Robert Devaney, Alexandre Dezotti, Tien-Cuong Dinh, Romain Dujardin, Hugo García-Compeán, William Goldman, Rotislav Grigorchuk, John Hubbard, Yunping Jiang, Linda Keen, Jan Kiwi, Genadi Levin, Daniel Meyer, John Milnor, Carlos Moreira, Vincente Muñoz, Viet-Anh Nguyên, Lex Oversteegen, Ricardo Pérez-Marco, Ross Ptacek, Jasmin Raissy, Pascale Roesch, Roberto Santos-Silva, Dierk Schleicher, Nessim Sibony, Daniel Smania, Tan Lei, William Thurston, Vladlen Timorin, Sebastian van Strien, and Alberto Verjovsky.
Functions of complex variables. --- Differentiable dynamical systems. --- Milnor, John W. --- John Milnor. --- Jordan curves. --- Julia sets. --- K-theory. --- M. Hakim. --- Maldelbrot set. --- Milnor's conjecture. --- Sierpinski carpets. --- Sierpinski gaskets. --- Thurston Rigidity Theorem. --- William Thurston. --- antiholomorphic quadratic polynomials. --- antiholomorphic unicritical polynomials. --- arithmetics. --- asymptotic behavior. --- automorphisms. --- biholomorphisms. --- combinatorial group theory. --- complex manifolds. --- complex polynomials. --- conformal distortion. --- conjugacy. --- connectedness loci. --- critical objects. --- critical point. --- critically finite case. --- decomposition. --- differential geometry. --- differential topology. --- dynamical cores. --- dynamical derivatives. --- dynamical scales. --- dynamical systems. --- dynamics. --- eigenvalues. --- entropy theory. --- entropy. --- escape regions. --- expanding critical orbits. --- external rays. --- geometrically finite rational maps. --- global meromorphic maps. --- holomorphic dynamics studies. --- holomorphic dynamics. --- holomorphic germs. --- holomorphic maps. --- hyperbolic distortion. --- hyperbolic geometry. --- hyperbolicity. --- identity germ. --- implosions. --- index theorems. --- infinitely renormalizable quadratic polynomials. --- integral closure. --- interval dynamics. --- irreducibility. --- kneading sequences. --- laminations. --- leading monomial vector. --- leading monomials. --- limiting behavior. --- local connectivity. --- local dynamics. --- mathematics. --- mating. --- metric stability. --- monotonicity. --- multicorn. --- non-locally connected Julia sets. --- one-dimensional maps. --- periodic critical orbit. --- periodic objects. --- periodic orbits. --- perturbations. --- polynomials. --- postcritically finite maps. --- projective space. --- pushforwards. --- quadratic differentials. --- quadratic dynatomic curves. --- quadratic polynomials. --- random walks. --- rational maps. --- renormalization theory. --- resurgence theory. --- rigidity. --- robustness. --- singularly perturbed rational maps. --- smoothness. --- spherical geometry. --- summability condition. --- summable critical points. --- topological entropy. --- topological polynomials. --- topological space. --- transversality theory. --- tricorn. --- unbounded hyperbolic components. --- unicritical polynomial maps. --- unimodal interval map. --- unmating.
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