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Five papers by distinguished American and European mathematicians describe some current trends in mathematics in the perspective of the recent past and in terms of expectations for the future. Among the subjects discussed are algebraic groups, quadratic forms, topological aspects of global analysis, variants of the index theorem, and partial differential equations.

Mathematics --- Mathématiques --- Congresses --- Congrès --- 51 --- -Math --- Science --- Congresses. --- -Mathematics --- 51 Mathematics --- -51 Mathematics --- Math --- Mathématiques --- Congrès --- A priori estimate. --- Addition. --- Additive group. --- Affine space. --- Algebraic geometry. --- Algebraic group. --- Atiyah–Singer index theorem. --- Bernoulli number. --- Boundary value problem. --- Bounded operator. --- C*-algebra. --- Canonical transformation. --- Cauchy problem. --- Characteristic class. --- Clifford algebra. --- Coefficient. --- Cohomology. --- Commutative property. --- Commutative ring. --- Complex manifold. --- Complex number. --- Complex vector bundle. --- Dedekind sum. --- Degenerate bilinear form. --- Diagram (category theory). --- Diffeomorphism. --- Differentiable manifold. --- Differential operator. --- Dimension (vector space). --- Ellipse. --- Elliptic operator. --- Equation. --- Euler characteristic. --- Euler number. --- Existence theorem. --- Exotic sphere. --- Finite difference. --- Finite group. --- Fourier integral operator. --- Fourier transform. --- Fourier. --- Fredholm operator. --- Hardy space. --- Hilbert space. --- Holomorphic vector bundle. --- Homogeneous coordinates. --- Homomorphism. --- Homotopy. --- Hyperbolic partial differential equation. --- Identity component. --- Integer. --- Integral transform. --- Isomorphism class. --- John Milnor. --- K-theory. --- Lebesgue measure. --- Line bundle. --- Local ring. --- Mathematics. --- Maximal ideal. --- Modular form. --- Module (mathematics). --- Monoid. --- Normal bundle. --- Number theory. --- Open set. --- Parametrix. --- Parity (mathematics). --- Partial differential equation. --- Piecewise linear manifold. --- Poisson bracket. --- Polynomial ring. --- Polynomial. --- Prime number. --- Principal part. --- Projective space. --- Pseudo-differential operator. --- Quadratic form. --- Rational variety. --- Real number. --- Reciprocity law. --- Resolution of singularities. --- Riemann–Roch theorem. --- Shift operator. --- Simply connected space. --- Special case. --- Square-integrable function. --- Subalgebra. --- Submanifold. --- Support (mathematics). --- Surjective function. --- Symmetric bilinear form. --- Symplectic vector space. --- Tangent space. --- Theorem. --- Topology. --- Variable (mathematics). --- Vector bundle. --- Vector space. --- Winding number. --- Mathematics - Congresses

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Rabinowitz's classical global bifurcation theory, which concerns the study in-the-large of parameter-dependent families of nonlinear equations, uses topological methods that address the problem of continuous parameter dependence of solutions by showing that there are connected sets of solutions of global extent. Even when the operators are infinitely differentiable in all the variables and parameters, connectedness here cannot in general be replaced by path-connectedness. However, in the context of real-analyticity there is an alternative theory of global bifurcation due to Dancer, which offers a much stronger notion of parameter dependence. This book aims to develop from first principles Dancer's global bifurcation theory for one-parameter families of real-analytic operators in Banach spaces. It shows that there are globally defined continuous and locally real-analytic curves of solutions. In particular, in the real-analytic setting, local analysis can lead to global consequences--for example, as explained in detail here, those resulting from bifurcation from a simple eigenvalue. Included are accounts of analyticity and implicit function theorems in Banach spaces, classical results from the theory of finite-dimensional analytic varieties, and the links between these two and global existence theory. Laying the foundations for more extensive studies of real-analyticity in infinite-dimensional problems and illustrating the theory with examples, Analytic Theory of Global Bifurcation is intended for graduate students and researchers in pure and applied analysis.

Differential geometry. Global analysis --- Bifurcation theory. --- Differential equations, Nonlinear --- Stability --- Numerical solutions --- Addition. --- Algebraic equation. --- Analytic function. --- Analytic manifold. --- Atmospheric pressure. --- Banach space. --- Bernhard Riemann. --- Bifurcation diagram. --- Boundary value problem. --- Bounded operator. --- Bounded set (topological vector space). --- Boundedness. --- Canonical form. --- Cartesian coordinate system. --- Codimension. --- Compact operator. --- Complex analysis. --- Complex conjugate. --- Complex number. --- Connected space. --- Coordinate system. --- Corollary. --- Curvature. --- Derivative. --- Diagram (category theory). --- Differentiable function. --- Differentiable manifold. --- Dimension (vector space). --- Dimension. --- Direct sum. --- Eigenvalues and eigenvectors. --- Elliptic integral. --- Embedding. --- Equation. --- Euclidean division. --- Euler equations (fluid dynamics). --- Existential quantification. --- First principle. --- Fredholm operator. --- Froude number. --- Functional analysis. --- Hilbert space. --- Homeomorphism. --- Implicit function theorem. --- Integer. --- Linear algebra. --- Linear function. --- Linear independence. --- Linear map. --- Linear programming. --- Linear space (geometry). --- Linear subspace. --- Linearity. --- Linearization. --- Metric space. --- Morse theory. --- Multilinear form. --- N0. --- Natural number. --- Neumann series. --- Nonlinear functional analysis. --- Nonlinear system. --- Numerical analysis. --- Open mapping theorem (complex analysis). --- Operator (physics). --- Ordinary differential equation. --- Parameter. --- Parametrization. --- Partial differential equation. --- Permutation group. --- Permutation. --- Polynomial. --- Power series. --- Prime number. --- Proportionality (mathematics). --- Pseudo-differential operator. --- Puiseux series. --- Quantity. --- Real number. --- Resultant. --- Singularity theory. --- Skew-symmetric matrix. --- Smoothness. --- Solution set. --- Special case. --- Standard basis. --- Sturm–Liouville theory. --- Subset. --- Symmetric bilinear form. --- Symmetric group. --- Taylor series. --- Taylor's theorem. --- Theorem. --- Total derivative. --- Two-dimensional space. --- Union (set theory). --- Variable (mathematics). --- Vector space. --- Zero of a function.

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This book contains an exposition of some of the main developments of the last twenty years in the following areas of harmonic analysis: singular integral and pseudo-differential operators, the theory of Hardy spaces, Lsup estimates involving oscillatory integrals and Fourier integral operators, relations of curvature to maximal inequalities, and connections with analysis on the Heisenberg group.

Harmonic analysis. Fourier analysis --- Harmonic analysis --- Analyse harmonique --- Harmonic analysis. --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Groupe de Heisenberg. --- Addition. --- Analytic function. --- Asymptote. --- Asymptotic analysis. --- Asymptotic expansion. --- Asymptotic formula. --- Automorphism. --- Axiom. --- Banach space. --- Bessel function. --- Big O notation. --- Bilinear form. --- Borel measure. --- Boundary value problem. --- Bounded function. --- Bounded mean oscillation. --- Bounded operator. --- Boundedness. --- Cancellation property. --- Cauchy's integral theorem. --- Cauchy–Riemann equations. --- Characteristic polynomial. --- Characterization (mathematics). --- Commutative property. --- Commutator. --- Complex analysis. --- Convolution. --- Differential equation. --- Differential operator. --- Dimension (vector space). --- Dimension. --- Dirac delta function. --- Dirichlet problem. --- Elliptic operator. --- Existential quantification. --- Fatou's theorem. --- Fourier analysis. --- Fourier integral operator. --- Fourier inversion theorem. --- Fourier series. --- Fourier transform. --- Fubini's theorem. --- Function (mathematics). --- Fundamental solution. --- Gaussian curvature. --- Hardy space. --- Harmonic function. --- Heisenberg group. --- Hilbert space. --- Hilbert transform. --- Holomorphic function. --- Hölder's inequality. --- Infimum and supremum. --- Integral transform. --- Interpolation theorem. --- Lagrangian (field theory). --- Laplace's equation. --- Lebesgue measure. --- Lie algebra. --- Line segment. --- Linear map. --- Lipschitz continuity. --- Locally integrable function. --- Marcinkiewicz interpolation theorem. --- Martingale (probability theory). --- Mathematical induction. --- Maximal function. --- Meromorphic function. --- Multiplication operator. --- Nilpotent Lie algebra. --- Norm (mathematics). --- Number theory. --- Operator theory. --- Order of integration (calculus). --- Orthogonality. --- Oscillatory integral. --- Poisson summation formula. --- Projection (linear algebra). --- Pseudo-differential operator. --- Pseudoconvexity. --- Rectangle. --- Riesz transform. --- Several complex variables. --- Sign (mathematics). --- Singular integral. --- Sobolev space. --- Special case. --- Spectral theory. --- Square (algebra). --- Stochastic differential equation. --- Subharmonic function. --- Submanifold. --- Summation. --- Support (mathematics). --- Theorem. --- Translational symmetry. --- Uniqueness theorem. --- Variable (mathematics). --- Vector field. --- Fourier, Analyse de --- Fourier, Opérateurs intégraux de

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Encompassing both introductory and more advanced research material, these notes deal with the author's contributions to stochastic processes and focus on Brownian motion processes and its derivative white noise.Originally published in 1970.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Stationary processes --- Stationary processes. --- Stochastic processes --- 519.216 --- 519.216 Stochastic processes in general. Prediction theory. Stopping times. Martingales --- Stochastic processes in general. Prediction theory. Stopping times. Martingales --- Bochner integral. --- Bochner's theorem. --- Bounded operator. --- Bounded variation. --- Brownian motion. --- Characteristic exponent. --- Characteristic function (probability theory). --- Complexification. --- Compound Poisson process. --- Computation. --- Conditional expectation. --- Continuous function (set theory). --- Continuous function. --- Continuous linear operator. --- Convergence of random variables. --- Coset. --- Covariance function. --- Cyclic subspace. --- Cylinder set. --- Degrees of freedom (statistics). --- Derivative. --- Differential equation. --- Dimension (vector space). --- Dirac delta function. --- Discrete spectrum. --- Distribution function. --- Dual space. --- Eigenfunction. --- Equation. --- Existential quantification. --- Exponential distribution. --- Exponential function. --- Finite difference. --- Fourier series. --- Fourier transform. --- Function (mathematics). --- Function space. --- Gaussian measure. --- Gaussian process. --- Harmonic analysis. --- Hermite polynomials. --- Hilbert space. --- Homeomorphism. --- Independence (probability theory). --- Independent and identically distributed random variables. --- Indicator function. --- Infinitesimal generator (stochastic processes). --- Integral equation. --- Isometry. --- Joint probability distribution. --- Langevin equation. --- Lebesgue measure. --- Lie algebra. --- Limit superior and limit inferior. --- Linear combination. --- Linear function. --- Linear interpolation. --- Linear subspace. --- Mean squared error. --- Measure (mathematics). --- Monotonic function. --- Normal distribution. --- Normal subgroup. --- Nuclear space. --- One-parameter group. --- Orthogonality. --- Orthogonalization. --- Parameter. --- Poisson point process. --- Polynomial. --- Probability distribution. --- Probability measure. --- Probability space. --- Probability. --- Projective linear group. --- Radon–Nikodym theorem. --- Random function. --- Random variable. --- Reproducing kernel Hilbert space. --- Self-adjoint operator. --- Self-adjoint. --- Semigroup. --- Shift operator. --- Special case. --- Stable process. --- Stationary process. --- Stochastic differential equation. --- Stochastic process. --- Stochastic. --- Subgroup. --- Summation. --- Symmetrization. --- Theorem. --- Transformation semigroup. --- Unitary operator. --- Unitary representation. --- Unitary transformation. --- Variance. --- White noise. --- Zero element.

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This book offers the first comprehensive introduction to wave scattering in nonstationary materials. G. F. Roach's aim is to provide an accessible, self-contained resource for newcomers to this important field of research that has applications across a broad range of areas, including radar, sonar, diagnostics in engineering and manufacturing, geophysical prospecting, and ultrasonic medicine such as sonograms. New methods in recent years have been developed to assess the structure and properties of materials and surfaces. When light, sound, or some other wave energy is directed at the material in question, "imperfections" in the resulting echo can reveal a tremendous amount of valuable diagnostic information. The mathematics behind such analysis is sophisticated and complex. However, while problems involving stationary materials are quite well understood, there is still much to learn about those in which the material is moving or changes over time. These so-called non-autonomous problems are the subject of this fascinating book. Roach develops practical strategies, techniques, and solutions for mathematicians and applied scientists working in or seeking entry into the field of modern scattering theory and its applications. Wave Scattering by Time-Dependent Perturbations is destined to become a classic in this rapidly evolving area of inquiry.

Waves --- Scattering (Physics) --- Perturbation (Mathematics) --- Perturbation equations --- Perturbation theory --- Approximation theory --- Dynamics --- Functional analysis --- Mathematical physics --- Atomic scattering --- Atoms --- Nuclear scattering --- Particles (Nuclear physics) --- Scattering of particles --- Wave scattering --- Collisions (Nuclear physics) --- Particles --- Collisions (Physics) --- Cycles --- Hydrodynamics --- Benjamin-Feir instability --- Mathematics. --- Scattering --- Acoustic wave equation. --- Acoustic wave. --- Affine space. --- Angular frequency. --- Approximation. --- Asymptotic analysis. --- Asymptotic expansion. --- Banach space. --- Basis (linear algebra). --- Bessel's inequality. --- Boundary value problem. --- Bounded operator. --- C0-semigroup. --- Calculation. --- Characteristic function (probability theory). --- Classical physics. --- Codimension. --- Coefficient. --- Continuous function (set theory). --- Continuous function. --- Continuous spectrum. --- Convolution. --- Differentiable function. --- Differential equation. --- Dimension (vector space). --- Dimension. --- Dimensional analysis. --- Dirac delta function. --- Dirichlet problem. --- Distribution (mathematics). --- Duhamel's principle. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Electromagnetism. --- Equation. --- Existential quantification. --- Exponential function. --- Floquet theory. --- Fourier inversion theorem. --- Fourier series. --- Fourier transform. --- Fredholm integral equation. --- Frequency domain. --- Helmholtz equation. --- Hilbert space. --- Initial value problem. --- Integral equation. --- Integral transform. --- Integration by parts. --- Inverse problem. --- Inverse scattering problem. --- Lebesgue measure. --- Linear differential equation. --- Linear map. --- Linear space (geometry). --- Locally integrable function. --- Longitudinal wave. --- Mathematical analysis. --- Mathematical physics. --- Metric space. --- Operator theory. --- Ordinary differential equation. --- Orthonormal basis. --- Orthonormality. --- Parseval's theorem. --- Partial derivative. --- Partial differential equation. --- Phase velocity. --- Plane wave. --- Projection (linear algebra). --- Propagator. --- Quantity. --- Quantum mechanics. --- Reflection coefficient. --- Requirement. --- Riesz representation theorem. --- Scalar (physics). --- Scattering theory. --- Scattering. --- Scientific notation. --- Self-adjoint operator. --- Self-adjoint. --- Series expansion. --- Sine wave. --- Spectral method. --- Spectral theorem. --- Spectral theory. --- Square-integrable function. --- Subset. --- Theorem. --- Theory. --- Time domain. --- Time evolution. --- Unbounded operator. --- Unitarity (physics). --- Vector space. --- Volterra integral equation. --- Wave function. --- Wave packet. --- Wave propagation.