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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.

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This book is a spectacular introduction to the modern mathematical discipline known as the Theory of Games. Harold Kuhn first presented these lectures at Princeton University in 1952. They succinctly convey the essence of the theory, in part through the prism of the most exciting developments at its frontiers half a century ago. Kuhn devotes considerable space to topics that, while not strictly the subject matter of game theory, are firmly bound to it. These are taken mainly from the geometry of convex sets and the theory of probability distributions. The book opens by addressing "matrix games," a name first introduced in these lectures as an abbreviation for two-person, zero-sum games in normal form with a finite number of pure strategies. It continues with a treatment of games in extensive form, using a model introduced by the author in 1950 that quickly supplanted von Neumann and Morgenstern's cumbersome approach. A final section deals with games that have an infinite number of pure strategies for the two players. Throughout, the theory is generously illustrated with examples, and exercises test the reader's understanding. A historical note caps off each chapter. For readers familiar with the calculus and with elementary matrix theory or vector analysis, this book offers an indispensable store of vital insights on a subject whose importance has only grown with the years.

Operational research. Game theory --- Game theory --- 519.83 --- Theory of games --- 519.83 Theory of games --- Game theory. --- Games, Theory of --- Mathematical models --- Mathematics --- Abstract algebra. --- Addition. --- Algorithm. --- Almost surely. --- Analytic geometry. --- Axiom. --- Basic solution (linear programming). --- Big O notation. --- Bijection. --- Binary relation. --- Boundary (topology). --- Bounded set (topological vector space). --- Branch point. --- Calculation. --- Cardinality of the continuum. --- Cardinality. --- Cartesian coordinate system. --- Characteristic function (probability theory). --- Combination. --- Computation. --- Connectivity (graph theory). --- Constructive proof. --- Convex combination. --- Convex function. --- Convex hull. --- Convex set. --- Coordinate system. --- David Gale. --- Diagram (category theory). --- Differential equation. --- Dimension (vector space). --- Dimensional analysis. --- Disjoint sets. --- Distribution function. --- Embedding. --- Empty set. --- Enumeration. --- Equation. --- Equilibrium point. --- Equivalence relation. --- Estimation. --- Euclidean space. --- Existential quantification. --- Expected loss. --- Extreme point. --- Formal scheme. --- Fundamental theorem. --- Galois theory. --- Geometry. --- Hyperplane. --- Inequality (mathematics). --- Infimum and supremum. --- Integer. --- Iterative method. --- Line segment. --- Linear equation. --- Linear inequality. --- Matching Pennies. --- Mathematical induction. --- Mathematical optimization. --- Mathematical theory. --- Mathematician. --- Mathematics. --- Matrix (mathematics). --- Measure (mathematics). --- Min-max theorem. --- Minimum distance. --- Mutual exclusivity. --- Prediction. --- Probability distribution. --- Probability interpretations. --- Probability measure. --- Probability theory. --- Probability. --- Proof by contradiction. --- Quantity. --- Rank (linear algebra). --- Rational number. --- Real number. --- Requirement. --- Scientific notation. --- Sign (mathematics). --- Solution set. --- Special case. --- Statistics. --- Strategist. --- Strategy (game theory). --- Subset. --- Theorem. --- Theory of Games and Economic Behavior. --- Theory. --- Three-dimensional space (mathematics). --- Total order. --- Two-dimensional space. --- Union (set theory). --- Unit interval. --- Unit square. --- Vector Analysis. --- Vector calculus. --- Vector space.

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Based on seven lecture series given by leading experts at a summer school at Peking University, in Beijing, in 1984. this book surveys recent developments in the areas of harmonic analysis most closely related to the theory of singular integrals, real-variable methods, and applications to several complex variables and partial differential equations. The different lecture series are closely interrelated; each contains a substantial amount of background material, as well as new results not previously published. The contributors to the volume are R. R. Coifman and Yves Meyer, Robert Fcfferman,Carlos K. Kenig, Steven G. Krantz, Alexander Nagel, E. M. Stein, and Stephen Wainger.

Harmonic analysis. --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Analytic function. --- Asymptotic formula. --- Bergman metric. --- Bernhard Riemann. --- Bessel function. --- Biholomorphism. --- Boundary value problem. --- Bounded mean oscillation. --- Bounded operator. --- Boundedness. --- Cauchy's integral formula. --- Characteristic function (probability theory). --- Characterization (mathematics). --- Coefficient. --- Commutator. --- Complexification (Lie group). --- Continuous function. --- Convolution. --- Degeneracy (mathematics). --- Differential equation. --- Differential operator. --- Dirac delta function. --- Dirichlet problem. --- Equation. --- Estimation. --- Existence theorem. --- Existential quantification. --- Explicit formula. --- Explicit formulae (L-function). --- Fatou's theorem. --- Fourier analysis. --- Fourier integral operator. --- Fourier transform. --- Fredholm theory. --- Fubini's theorem. --- Function (mathematics). --- Functional calculus. --- Fundamental solution. --- Gaussian curvature. --- Hardy space. --- Harmonic function. --- Harmonic measure. --- Heisenberg group. --- Hilbert space. --- Hilbert transform. --- Hodge theory. --- Holomorphic function. --- Hyperbolic partial differential equation. --- Hölder's inequality. --- Infimum and supremum. --- Integration by parts. --- Interpolation theorem. --- Intersection (set theory). --- Invertible matrix. --- Isometry group. --- Laplace operator. --- Laplace's equation. --- Lebesgue measure. --- Linear map. --- Lipschitz continuity. --- Lipschitz domain. --- Lp space. --- Mathematical induction. --- Mathematical physics. --- Maximal function. --- Maximum principle. --- Measure (mathematics). --- Newtonian potential. --- Non-Euclidean geometry. --- Number theory. --- Operator theory. --- Oscillatory integral. --- Parameter. --- Partial derivative. --- Partial differential equation. --- Polynomial. --- Power series. --- Product metric. --- Radon–Nikodym theorem. --- Riemannian manifold. --- Riesz representation theorem. --- Scientific notation. --- Several complex variables. --- Sign (mathematics). --- Simultaneous equations. --- Singular function. --- Singular integral. --- Sobolev space. --- Square (algebra). --- Statistical hypothesis testing. --- Stokes' theorem. --- Support (mathematics). --- Tangent space. --- Tensor product. --- Theorem. --- Trigonometric series. --- Uniformization theorem. --- Variable (mathematics). --- Vector field.

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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 has as its subject the boundary value theory of holomorphic functions in several complex variables, a topic that is just now coming to the forefront of mathematical analysis. For one variable, the topic is classical and rather well understood. In several variables, the necessary understanding of holomorphic functions via partial differential equations has a recent origin, and Professor Stein's book, which emphasizes the potential-theoretic aspects of the boundary value problem, should become the standard work in the field.Originally published in 1972.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.

Mathematical potential theory --- Holomorphic functions --- Harmonic functions --- Holomorphic functions. --- Harmonic functions. --- Fonctions de plusieurs variables complexes. --- Functions of several complex variables --- Functions, Harmonic --- Laplace's equations --- Bessel functions --- Differential equations, Partial --- Fourier series --- Harmonic analysis --- Lamé's functions --- Spherical harmonics --- Toroidal harmonics --- Functions, Holomorphic --- Absolute continuity. --- Absolute value. --- Addition. --- Ambient space. --- Analytic function. --- Arbitrarily large. --- Bergman metric. --- Borel measure. --- Boundary (topology). --- Boundary value problem. --- Bounded set (topological vector space). --- Boundedness. --- Brownian motion. --- Calculation. --- Change of variables. --- Characteristic function (probability theory). --- Combination. --- Compact space. --- Complex analysis. --- Complex conjugate. --- Computation. --- Conformal map. --- Constant term. --- Continuous function. --- Coordinate system. --- Corollary. --- Cramer's rule. --- Determinant. --- Diameter. --- Dimension. --- Elliptic operator. --- Estimation. --- Existential quantification. --- Explicit formulae (L-function). --- Exterior (topology). --- Fatou's theorem. --- Function space. --- Green's function. --- Green's theorem. --- Haar measure. --- Half-space (geometry). --- Harmonic function. --- Hilbert space. --- Holomorphic function. --- Hyperbolic space. --- Hypersurface. --- Hölder's inequality. --- Invariant measure. --- Invertible matrix. --- Jacobian matrix and determinant. --- Line segment. --- Linear map. --- Lipschitz continuity. --- Local coordinates. --- Logarithm. --- Majorization. --- Matrix (mathematics). --- Maximal function. --- Measure (mathematics). --- Minimum distance. --- Natural number. --- Normal (geometry). --- Open set. --- Order of magnitude. --- Orthogonal complement. --- Orthonormal basis. --- Parameter. --- Poisson kernel. --- Positive-definite matrix. --- Potential theory. --- Projection (linear algebra). --- Quadratic form. --- Quantity. --- Real structure. --- Requirement. --- Scientific notation. --- Sesquilinear form. --- Several complex variables. --- Sign (mathematics). --- Smoothness. --- Subgroup. --- Subharmonic function. --- Subsequence. --- Subset. --- Summation. --- Tangent space. --- Theorem. --- Theory. --- Total variation. --- Transitive relation. --- Transitivity. --- Transpose. --- Two-form. --- Unit sphere. --- Unitary matrix. --- Vector field. --- Vector space. --- Volume element. --- Weak topology.