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This book presents an overview of recent developments in the area of localization for quasi-periodic lattice Schrödinger operators and the theory of quasi-periodicity in Hamiltonian evolution equations. The physical motivation of these models extends back to the works of Rudolph Peierls and Douglas R. Hofstadter, and the models themselves have been a focus of mathematical research for two decades. Jean Bourgain here sets forth the results and techniques that have been discovered in the last few years. He puts special emphasis on so-called "non-perturbative" methods and the important role of subharmonic function theory and semi-algebraic set methods. He describes various applications to the theory of differential equations and dynamical systems, in particular to the quantum kicked rotor and KAM theory for nonlinear Hamiltonian evolution equations. Intended primarily for graduate students and researchers in the general area of dynamical systems and mathematical physics, the book provides a coherent account of a large body of work that is presently scattered in the literature. It does so in a refreshingly contained manner that seeks to convey the present technological "state of the art."

Schrödinger operator. --- Green's functions. --- Hamiltonian systems. --- Evolution equations. --- Evolutionary equations --- Equations, Evolution --- Equations of evolution --- Hamiltonian dynamical systems --- Systems, Hamiltonian --- Functions, Green's --- Functions, Induction --- Functions, Source --- Green functions --- Induction functions --- Source functions --- Operator, Schrödinger --- Differential equations --- Differentiable dynamical systems --- Potential theory (Mathematics) --- Differential operators --- Quantum theory --- Schrödinger equation --- Almost Mathieu operator. --- Analytic function. --- Anderson localization. --- Betti number. --- Cartan's theorem. --- Chaos theory. --- Density of states. --- Dimension (vector space). --- Diophantine equation. --- Dynamical system. --- Equation. --- Existential quantification. --- Fundamental matrix (linear differential equation). --- Green's function. --- Hamiltonian system. --- Hermitian adjoint. --- Infimum and supremum. --- Iterative method. --- Jacobi operator. --- Linear equation. --- Linear map. --- Linearization. --- Monodromy matrix. --- Non-perturbative. --- Nonlinear system. --- Normal mode. --- Parameter space. --- Parameter. --- Parametrization. --- Partial differential equation. --- Periodic boundary conditions. --- Phase space. --- Phase transition. --- Polynomial. --- Renormalization. --- Self-adjoint. --- Semialgebraic set. --- Special case. --- Statistical significance. --- Subharmonic function. --- Summation. --- Theorem. --- Theory. --- Transfer matrix. --- Transversality (mathematics). --- Trigonometric functions. --- Trigonometric polynomial. --- Uniformization theorem.

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Polynomials play a crucial role in many areas of mathematics including algebra, analysis, number theory, and probability theory. They also appear in physics, chemistry, and economics. Especially extensively studied are certain infinite families of polynomials. Here, we only mention some examples: Bernoulli, Euler, Gegenbauer, trigonometric, and orthogonal polynomials and their generalizations. There are several approaches to these classical mathematical objects. This Special Issue presents nine high quality research papers by leading researchers in this field. I hope the reading of this work will be useful for the new generation of mathematicians and for experienced researchers as well

Research & information: general --- Mathematics & science --- Shivley’s matrix polynomials --- Generating matrix functions --- Matrix recurrence relations --- summation formula --- Operational representations --- Euler polynomials --- higher degree equations --- degenerate Euler numbers and polynomials --- degenerate q-Euler numbers and polynomials --- degenerate Carlitz-type (p, q)-Euler numbers and polynomials --- 2D q-Appell polynomials --- twice-iterated 2D q-Appell polynomials --- determinant expressions --- recurrence relations --- 2D q-Bernoulli polynomials --- 2D q-Euler polynomials --- 2D q-Genocchi polynomials --- Apostol type Bernoulli --- Euler and Genocchi polynomials --- Euler numbers and polynomials --- Carlitz-type degenerate (p,q)-Euler numbers and polynomials --- Carlitz-type higher-order degenerate (p,q)-Euler numbers and polynomials --- symmetric identities --- (p, q)-cosine Bernoulli polynomials --- (p, q)-sine Bernoulli polynomials --- (p, q)-numbers --- (p, q)-trigonometric functions --- Bernstein operators --- rate of approximation --- Voronovskaja type asymptotic formula --- q-cosine Euler polynomials --- q-sine Euler polynomials --- q-trigonometric function --- q-exponential function --- multiquadric --- radial basis function --- radial polynomials --- the shape parameter --- meshless --- Kansa method

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The description for this book, Elementary Differential Topology. (AM-54), Volume 54, will be forthcoming.

Differential topology. --- Addition. --- Affine transformation. --- Algebraic topology. --- Analytic manifold. --- Approximation. --- Barycentric coordinate system. --- Barycentric subdivision. --- Basis (linear algebra). --- Brouwer fixed-point theorem. --- CR manifold. --- Centroid. --- Chain rule. --- Closed set. --- Combinatorics. --- Compact space. --- Conjecture. --- Continuous function. --- Convex set. --- Coordinate system. --- Corollary. --- Degeneracy (mathematics). --- Diameter. --- Diffeomorphism. --- Differentiable function. --- Differentiable manifold. --- Dimension (vector space). --- Dimension theory (algebra). --- Dimension theory. --- Disjoint sets. --- Elementary proof. --- Empty set. --- Equation. --- Euclidean space. --- Existential quantification. --- Function composition. --- Fundamental theorem. --- General topology. --- Geometry. --- Grassmannian. --- Homeomorphism. --- Homotopy. --- Hyperplane. --- Identity matrix. --- Inclusion map. --- Integer. --- Intersection (set theory). --- Invariance of domain. --- Jacobian matrix and determinant. --- Line segment. --- Linear algebra. --- Linear equation. --- Linear map. --- Locally compact space. --- Manifold. --- Mathematical induction. --- Matrix multiplication. --- Metrization theorem. --- Natural number. --- Number theory. --- Open set. --- Partial derivative. --- Partition of unity. --- Polyhedron. --- Polytope. --- Regular homotopy. --- Remainder. --- Scientific notation. --- Secant. --- Similarity (geometry). --- Simplex. --- Simplicial complex. --- Smoothness. --- Special case. --- Submanifold. --- Subset. --- Tangent bundle. --- Tangent vector. --- Theorem. --- Thickness (graph theory). --- Topological manifold. --- Topology. --- Trigonometric functions. --- Unit cube. --- Word problem (mathematics).

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This book describes the theory and applications of discrete orthogonal polynomials--polynomials that are orthogonal on a finite set. Unlike other books, Discrete Orthogonal Polynomials addresses completely general weight functions and presents a new methodology for handling the discrete weights case. J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin & P. D. Miller focus on asymptotic aspects of general, nonclassical discrete orthogonal polynomials and set out applications of current interest. Topics covered include the probability theory of discrete orthogonal polynomial ensembles and the continuum limit of the Toda lattice. The primary concern throughout is the asymptotic behavior of discrete orthogonal polynomials for general, nonclassical measures, in the joint limit where the degree increases as some fraction of the total number of points of collocation. The book formulates the orthogonality conditions defining these polynomials as a kind of Riemann-Hilbert problem and then generalizes the steepest descent method for such a problem to carry out the necessary asymptotic analysis.

Orthogonal polynomials --- Asymptotic theory --- Orthogonal polynomials -- Asymptotic theory. --- Polynomials. --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Operations Research --- Asymptotic theory. --- Asymptotic theory of orthogonal polynomials --- Algebra --- Airy function. --- Analytic continuation. --- Analytic function. --- Ansatz. --- Approximation error. --- Approximation theory. --- Asymptote. --- Asymptotic analysis. --- Asymptotic expansion. --- Asymptotic formula. --- Beta function. --- Boundary value problem. --- Calculation. --- Cauchy's integral formula. --- Cauchy–Riemann equations. --- Change of variables. --- Complex number. --- Complex plane. --- Correlation function. --- Degeneracy (mathematics). --- Determinant. --- Diagram (category theory). --- Discrete measure. --- Distribution function. --- Eigenvalues and eigenvectors. --- Equation. --- Estimation. --- Existential quantification. --- Explicit formulae (L-function). --- Factorization. --- Fredholm determinant. --- Functional derivative. --- Gamma function. --- Gradient descent. --- Harmonic analysis. --- Hermitian matrix. --- Homotopy. --- Hypergeometric function. --- I0. --- Identity matrix. --- Inequality (mathematics). --- Integrable system. --- Invariant measure. --- Inverse scattering transform. --- Invertible matrix. --- Jacobi matrix. --- Joint probability distribution. --- Lagrange multiplier. --- Lax equivalence theorem. --- Limit (mathematics). --- Linear programming. --- Lipschitz continuity. --- Matrix function. --- Maxima and minima. --- Monic polynomial. --- Monotonic function. --- Morera's theorem. --- Neumann series. --- Number line. --- Orthogonal polynomials. --- Orthogonality. --- Orthogonalization. --- Parameter. --- Parametrix. --- Pauli matrices. --- Pointwise convergence. --- Pointwise. --- Polynomial. --- Potential theory. --- Probability distribution. --- Probability measure. --- Probability theory. --- Probability. --- Proportionality (mathematics). --- Quantity. --- Random matrix. --- Random variable. --- Rate of convergence. --- Rectangle. --- Rhombus. --- Riemann surface. --- Special case. --- Spectral theory. --- Statistic. --- Subset. --- Theorem. --- Toda lattice. --- Trace (linear algebra). --- Trace class. --- Transition point. --- Triangular matrix. --- Trigonometric functions. --- Uniform continuity. --- Unit vector. --- Upper and lower bounds. --- Upper half-plane. --- Variational inequality. --- Weak solution. --- Weight function. --- Wishart distribution. --- Orthogonal polynomials - Asymptotic theory

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A mathematical journey through the most fascinating problems of extremes and how to solve them. What is the best way to photograph a speeding bullet? How can lost hikers find their way out of a forest? Why does light move through glass in the least amount of time possible? When Least Is Best combines the mathematical history of extrema with contemporary examples to answer these intriguing questions and more. Paul Nahin shows how life often works at the extremes--with values becoming as small (or as large) as possible--and he considers how mathematicians over the centuries, including Descartes, Fermat, and Kepler, have grappled with these problems of minima and maxima. Throughout, Nahin examines entertaining conundrums, such as how to build the shortest bridge possible between two towns, how to vary speed during a race, and how to make the perfect basketball shot. Moving from medieval writings and modern calculus to the field of optimization, the engaging and witty explorations of When Least Is Best will delight math enthusiasts everywhere.

Mathematics --- Maxima and minima. --- MATHEMATICS / History & Philosophy. --- History. --- Minima --- Math --- Science --- AP Calculus. --- Addition. --- Almost surely. --- American Mathematical Monthly. --- Arc (geometry). --- Calculation. --- Cambridge University Press. --- Cartesian coordinate system. --- Catenary. --- Central angle. --- Chain rule. --- Change of variables. --- Circumference. --- Clockwise. --- Convex function. --- Coordinate system. --- Curve. --- Cycloid. --- Cylinder (geometry). --- Derivative. --- Diameter. --- Differential calculus. --- Differential equation. --- Dimension. --- Dynamic programming. --- Elementary function. --- Equation. --- Equilateral triangle. --- Euler–Lagrange equation. --- Fermat's principle. --- Fluxion. --- Geometry. --- Honeycomb conjecture. --- Hyperbolic function. --- Hypotenuse. --- Illustration. --- Inequality of arithmetic and geometric means. --- Instant. --- Integer. --- Isoperimetric problem. --- Iteration. --- Jensen's inequality. --- Johann Bernoulli. --- Kinetic energy. --- Length. --- Line (geometry). --- Line segment. --- Linear programming. --- Logarithm. --- Mathematical maturity. --- Mathematical problem. --- Mathematician. --- Mathematics. --- Newton's method. --- Notation. --- Parabola. --- Parametric equation. --- Partial derivative. --- Perimeter. --- Philosopher. --- Physicist. --- Pierre de Fermat. --- Polygon. --- Polynomial. --- Potential energy. --- Princeton University Press. --- Projectile. --- Pumping station. --- Pythagorean theorem. --- Quadratic equation. --- Quadratic formula. --- Quantity. --- Ray (optics). --- Real number. --- Rectangle. --- Refraction. --- Refractive index. --- Regiomontanus. --- Requirement. --- Result. --- Right angle. --- Right triangle. --- Science. --- Scientific notation. --- Second derivative. --- Semicircle. --- Sign (mathematics). --- Simple algebra. --- Simplex algorithm. --- Snell's law. --- Special case. --- Square root. --- Summation. --- Surface area. --- Tangent. --- Trigonometric functions. --- Variable (mathematics). --- Vertex angle. --- Writing.