Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This book discusses some aspects of the theory of partial differential equations from the viewpoint of probability theory. It is intended not only for specialists in partial differential equations or probability theory but also for specialists in asymptotic methods and in functional analysis. It is also of interest to physicists who use functional integrals in their research. The work contains results that have not previously appeared in book form, including research contributions of the author.

Partial differential equations --- Differential equations, Partial. --- Probabilities. --- Integration, Functional. --- Functional integration --- Functional analysis --- Integrals, Generalized --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- A priori estimate. --- Absolute continuity. --- Almost surely. --- Analytic continuation. --- Axiom. --- Big O notation. --- Boundary (topology). --- Boundary value problem. --- Bounded function. --- Calculation. --- Cauchy problem. --- Central limit theorem. --- Characteristic function (probability theory). --- Chebyshev's inequality. --- Coefficient. --- Comparison theorem. --- Continuous function (set theory). --- Continuous function. --- Convergence of random variables. --- Cylinder set. --- Degeneracy (mathematics). --- Derivative. --- Differential equation. --- Differential operator. --- Diffusion equation. --- Diffusion process. --- Dimension (vector space). --- Direct method in the calculus of variations. --- Dirichlet boundary condition. --- Dirichlet problem. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Elliptic operator. --- Elliptic partial differential equation. --- Equation. --- Existence theorem. --- Exponential function. --- Feynman–Kac formula. --- Fokker–Planck equation. --- Function space. --- Functional analysis. --- Fundamental solution. --- Gaussian measure. --- Girsanov theorem. --- Hessian matrix. --- Hölder condition. --- Independence (probability theory). --- Integral curve. --- Integral equation. --- Invariant measure. --- Iterated logarithm. --- Itô's lemma. --- Joint probability distribution. --- Laplace operator. --- Laplace's equation. --- Lebesgue measure. --- Limit (mathematics). --- Limit cycle. --- Limit point. --- Linear differential equation. --- Linear map. --- Lipschitz continuity. --- Markov chain. --- Markov process. --- Markov property. --- Maximum principle. --- Mean value theorem. --- Measure (mathematics). --- Modulus of continuity. --- Moment (mathematics). --- Monotonic function. --- Navier–Stokes equations. --- Nonlinear system. --- Ordinary differential equation. --- Parameter. --- Partial differential equation. --- Periodic function. --- Poisson kernel. --- Probabilistic method. --- Probability space. --- Probability theory. --- Probability. --- Random function. --- Regularization (mathematics). --- Schrödinger equation. --- Self-adjoint operator. --- Sign (mathematics). --- Simultaneous equations. --- Smoothness. --- State-space representation. --- Stochastic calculus. --- Stochastic differential equation. --- Stochastic. --- Support (mathematics). --- Theorem. --- Theory. --- Uniqueness theorem. --- Variable (mathematics). --- Weak convergence (Hilbert space). --- Wiener process.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

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

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Graduate students in the natural sciences-including not only geophysics and space physics but also atmospheric and planetary physics, ocean sciences, and astronomy-need a broad-based mathematical toolbox to facilitate their research. In addition, they need to survey a wider array of mathematical methods that, while outside their particular areas of expertise, are important in related ones. While it is unrealistic to expect them to develop an encyclopedic knowledge of all the methods that are out there, they need to know how and where to obtain reliable and effective insights into these broader areas. Here at last is a graduate textbook that provides these students with the mathematical skills they need to succeed in today's highly interdisciplinary research environment. This authoritative and accessible book covers everything from the elements of vector and tensor analysis to ordinary differential equations, special functions, and chaos and fractals. Other topics include integral transforms, complex analysis, and inverse theory; partial differential equations of mathematical geophysics; probability, statistics, and computational methods; and much more. Proven in the classroom, Mathematical Methods for Geophysics and Space Physics features numerous exercises throughout as well as suggestions for further reading. Provides an authoritative and accessible introduction to the subject Covers vector and tensor analysis, ordinary differential equations, integrals and approximations, Fourier transforms, diffusion and dispersion, sound waves and perturbation theory, randomness in data, and a host of other topics Features numerous exercises throughout Ideal for students and researchers alike an online illustration package is available to professors

Geophysics --- Cosmic physics --- Physics --- Space sciences --- Mathematics. --- Analytical mechanics. --- Applied mathematics. --- Atmospheric physics. --- Bessel function. --- Bifurcation theory. --- Calculation. --- Calculus of variations. --- Cartesian coordinate system. --- Cauchy's theorem (geometry). --- Celestial mechanics. --- Central limit theorem. --- Chaos theory. --- Classical electromagnetism. --- Classical mechanics. --- Classical physics. --- Convolution theorem. --- Deformation (mechanics). --- Degeneracy (mathematics). --- Diagram (category theory). --- Differential equation. --- Drag (physics). --- Earth science. --- Eigenvalues and eigenvectors. --- Einstein notation. --- Elliptic integral. --- Elliptic orbit. --- Equation. --- Expectation value (quantum mechanics). --- Figure of the Earth. --- Forcing function (differential equations). --- Fourier series. --- Fourier transform. --- Fractal dimension. --- Function (mathematics). --- Gaussian function. --- Geochemistry. --- Geochronology. --- Geodesics in general relativity. --- Geometry. --- Geophysics. --- Gravitational acceleration. --- Gravitational constant. --- Gravitational potential. --- Gravitational two-body problem. --- Hamiltonian mechanics. --- Handbook of mathematical functions. --- Harmonic oscillator. --- Helmholtz equation. --- Hilbert transform. --- Hyperbolic partial differential equation. --- Integral equation. --- Isotope geochemistry. --- Lagrangian (field theory). --- Laplace transform. --- Laplace's equation. --- Laws of thermodynamics. --- Limit (mathematics). --- Line (geometry). --- Lorenz system. --- Mathematical analysis. --- Mathematical geophysics. --- Mathematical physics. --- Newton's law of universal gravitation. --- Newton's laws of motion. --- Newton's method. --- Newtonian dynamics. --- Numerical analysis. --- Numerical integration. --- Operator (physics). --- Orbit. --- Orbital resonance. --- Parseval's theorem. --- Partial differential equation. --- Perturbation theory (quantum mechanics). --- Perturbation theory. --- Planetary body. --- Planetary science. --- Poisson's equation. --- Pole (complex analysis). --- Proportionality (mathematics). --- Quantum mechanics. --- Rotation (mathematics). --- Satellite geodesy. --- Scalar (physics). --- Scientific notation. --- Separatrix (mathematics). --- Sign (mathematics). --- Space physics. --- Statistical mechanics. --- Stokes' theorem. --- Three-dimensional space (mathematics). --- Transformation geometry. --- Trapezoidal rule. --- Truncation error (numerical integration). --- Two-dimensional space. --- Van der Pol oscillator. --- Variable (mathematics). --- Vector space. --- Wave equation.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Many parallels between complex dynamics and hyperbolic geometry have emerged in the past decade. Building on work of Sullivan and Thurston, this book gives a unified treatment of the construction of fixed-points for renormalization and the construction of hyperbolic 3- manifolds fibering over the circle. Both subjects are studied via geometric limits and rigidity. This approach shows open hyperbolic manifolds are inflexible, and yields quantitative counterparts to Mostow rigidity. In complex dynamics, it motivates the construction of towers of quadratic-like maps, and leads to a quantitative proof of convergence of renormalization.

Differential dynamical systems --- Drie-menigvuldigheden (Topologie) --- Three-manifolds (Topology) --- Trois-variétés (Topologie) --- Differentiable dynamical systems. --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- 3-manifolds (Topology) --- Manifolds, Three dimensional (Topology) --- Three-dimensional manifolds (Topology) --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Low-dimensional topology --- Topological manifolds --- Algebraic topology. --- Analytic continuation. --- Automorphism. --- Beltrami equation. --- Bifurcation theory. --- Boundary (topology). --- Cantor set. --- Circular symmetry. --- Combinatorics. --- Compact space. --- Complex conjugate. --- Complex manifold. --- Complex number. --- Complex plane. --- Conformal geometry. --- Conformal map. --- Conjugacy class. --- Convex hull. --- Covering space. --- Deformation theory. --- Degeneracy (mathematics). --- Dimension (vector space). --- Disk (mathematics). --- Dynamical system. --- Eigenvalues and eigenvectors. --- Factorization. --- Fiber bundle. --- Fuchsian group. --- Fundamental domain. --- Fundamental group. --- Fundamental solution. --- G-module. --- Geodesic. --- Geometry. --- Harmonic analysis. --- Hausdorff dimension. --- Homeomorphism. --- Homotopy. --- Hyperbolic 3-manifold. --- Hyperbolic geometry. --- Hyperbolic manifold. --- Hyperbolic space. --- Hypersurface. --- Infimum and supremum. --- Injective function. --- Intersection (set theory). --- Invariant subspace. --- Isometry. --- Julia set. --- Kleinian group. --- Laplace's equation. --- Lebesgue measure. --- Lie algebra. --- Limit point. --- Limit set. --- Linear map. --- Mandelbrot set. --- Manifold. --- Mapping class group. --- Measure (mathematics). --- Moduli (physics). --- Moduli space. --- Modulus of continuity. --- Möbius transformation. --- N-sphere. --- Newton's method. --- Permutation. --- Point at infinity. --- Polynomial. --- Quadratic function. --- Quasi-isometry. --- Quasiconformal mapping. --- Quasisymmetric function. --- Quotient space (topology). --- Radon–Nikodym theorem. --- Renormalization. --- Representation of a Lie group. --- Representation theory. --- Riemann sphere. --- Riemann surface. --- Riemannian manifold. --- Schwarz lemma. --- Simply connected space. --- Special case. --- Submanifold. --- Subsequence. --- Support (mathematics). --- Tangent space. --- Teichmüller space. --- Theorem. --- Topology of uniform convergence. --- Topology. --- Trace (linear algebra). --- Transversal (geometry). --- Transversality (mathematics). --- Triangle inequality. --- Unit disk. --- Unit sphere. --- Upper and lower bounds. --- Vector field. --- Differentiable dynamical systems --- 515.16 --- 515.16 Topology of manifolds --- Topology of manifolds

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

The aim of this work is to provide a proof of the nonlinear gravitational stability of the Minkowski space-time. More precisely, the book offers a constructive proof of global, smooth solutions to the Einstein Vacuum Equations, which look, in the large, like the Minkowski space-time. In particular, these solutions are free of black holes and singularities. The work contains a detailed description of the sense in which these solutions are close to the Minkowski space-time, in all directions. It thus provides the mathematical framework in which we can give a rigorous derivation of the laws of gravitation proposed by Bondi. Moreover, it establishes other important conclusions concerning the nonlinear character of gravitational radiation. The authors obtain their solutions as dynamic developments of all initial data sets, which are close, in a precise manner, to the flat initial data set corresponding to the Minkowski space-time. They thus establish the global dynamic stability of the latter.Originally published in 1994.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.

Space and time --- Generalized spaces --- Nonlinear theories --- Physics --- Physical Sciences & Mathematics --- Atomic Physics --- Nonlinear problems --- Nonlinearity (Mathematics) --- Calculus --- Mathematical analysis --- Mathematical physics --- Geometry of paths --- Minkowski space --- Spaces, Generalized --- Weyl space --- Calculus of tensors --- Geometry, Differential --- Geometry, Non-Euclidean --- Hyperspace --- Relativity (Physics) --- Space of more than three dimensions --- Space-time --- Space-time continuum --- Space-times --- Spacetime --- Time and space --- Fourth dimension --- Infinite --- Metaphysics --- Philosophy --- Space sciences --- Time --- Beginning --- Mathematics --- Angular momentum operator. --- Asymptotic analysis. --- Asymptotic expansion. --- Big O notation. --- Boundary value problem. --- Cauchy–Riemann equations. --- Coarea formula. --- Coefficient. --- Compactification (mathematics). --- Comparison theorem. --- Corollary. --- Covariant derivative. --- Curvature tensor. --- Curvature. --- Cut locus (Riemannian manifold). --- Degeneracy (mathematics). --- Degrees of freedom (statistics). --- Derivative. --- Diffeomorphism. --- Differentiable function. --- Eigenvalues and eigenvectors. --- Eikonal equation. --- Einstein field equations. --- Equation. --- Error term. --- Estimation. --- Euclidean space. --- Existence theorem. --- Existential quantification. --- Exponential map (Lie theory). --- Exponential map (Riemannian geometry). --- Exterior (topology). --- Foliation. --- Fréchet derivative. --- Geodesic curvature. --- Geodesic. --- Geodesics in general relativity. --- Geometry. --- Hodge dual. --- Homotopy. --- Hyperbolic partial differential equation. --- Hypersurface. --- Hölder's inequality. --- Identity (mathematics). --- Infinitesimal generator (stochastic processes). --- Integral curve. --- Intersection (set theory). --- Isoperimetric inequality. --- Laplace's equation. --- Lie algebra. --- Lie derivative. --- Linear equation. --- Linear map. --- Logarithm. --- Lorentz group. --- Lp space. --- Mass formula. --- Mean curvature. --- Metric tensor. --- Minkowski space. --- Nonlinear system. --- Normal (geometry). --- Null hypersurface. --- Orthonormal basis. --- Partial derivative. --- Poisson's equation. --- Projection (linear algebra). --- Quantity. --- Radial function. --- Ricci curvature. --- Riemann curvature tensor. --- Riemann surface. --- Riemannian geometry. --- Riemannian manifold. --- Sard's theorem. --- Scalar (physics). --- Scalar curvature. --- Scale invariance. --- Schwarzschild metric. --- Second derivative. --- Second fundamental form. --- Sobolev inequality. --- Sobolev space. --- Stokes formula. --- Stokes' theorem. --- Stress–energy tensor. --- Symmetric tensor. --- Symmetrization. --- Tangent space. --- Tensor product. --- Theorem. --- Trace (linear algebra). --- Transversal (geometry). --- Triangle inequality. --- Uniformization theorem. --- Unit sphere. --- Vector field. --- Volume element. --- Wave equation. --- Weyl tensor.