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This work offers a contribution in the geometric form of the theory of several complex variables. Since complex Grassmann manifolds serve as classifying spaces of complex vector bundles, the cohomology structure of a complex Grassmann manifold is of importance for the construction of Chern classes of complex vector bundles. The cohomology ring of a Grassmannian is therefore of interest in topology, differential geometry, algebraic geometry, and complex analysis. Wilhelm Stoll treats certain aspects of the complex analysis point of view.This work originated with questions in value distribution theory. Here analytic sets and differential forms rather than the corresponding homology and cohomology classes are considered. On the Grassmann manifold, the cohomology ring is isomorphic to the ring of differential forms invariant under the unitary group, and each cohomology class is determined by a family of analytic sets.

Algebraic geometry --- Differential geometry. Global analysis --- Grassmann manifolds --- Differential forms. --- Grassmann manifolds. --- Invariants. --- Geometry, Differential. --- Géométrie différentielle. --- Differential invariants. --- Invariants différentiels. --- Forms, Differential --- Continuous groups --- Geometry, Differential --- Grassmannians --- Differential topology --- Manifolds (Mathematics) --- Calculation. --- Cohomology ring. --- Cohomology. --- Complex space. --- Cotangent bundle. --- Diagram (category theory). --- Exterior algebra. --- Grassmannian. --- Holomorphic vector bundle. --- Manifold. --- Regular map (graph theory). --- Remainder. --- Representation theorem. --- Schubert variety. --- Sesquilinear form. --- Theorem. --- Vector bundle. --- Vector space.

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The application by Fadeev and Pavlov of the Lax-Phillips scattering theory to the automorphic wave equation led Professors Lax and Phillips to reexamine this development within the framework of their theory. This volume sets forth the results of that work in the form of new or more straightforward treatments of the spectral theory of the Laplace-Beltrami operator over fundamental domains of finite area; the meromorphic character over the whole complex plane of the Eisenstein series; and the Selberg trace formula.CONTENTS: 1. Introduction. 2. An abstract scattering theory. 3. A modified theory for second order equations with an indefinite energy form. 4. The Laplace-Beltrami operator for the modular group. 5. The automorphic wave equation. 6. Incoming and outgoing subspaces for the automorphic wave equations. 7. The scattering matrix for the automorphic wave equation. 8. The general case. 9. The Selberg trace formula.

Harmonic analysis. Fourier analysis --- Automorphic functions --- Scattering (Mathematics) --- Fonctions automorphes --- Dispersion (Mathématiques) --- Automorphic functions. --- Scattering (Mathematics). --- Dispersion (Mathématiques) --- Selberg, Formule de trace de --- Selberg trace formula --- Eisenstein series --- Eisenstein, Séries d' --- Scattering theory (Mathematics) --- Boundary value problems --- Differential equations, Partial --- Scattering operator --- Fuchsian functions --- Functions, Automorphic --- Functions, Fuchsian --- Functions of several complex variables --- Absolute continuity. --- Algebra. --- Analytic continuation. --- Analytic function. --- Annulus (mathematics). --- Asymptotic distribution. --- Automorphic function. --- Bilinear form. --- Boundary (topology). --- Boundary value problem. --- Bounded operator. --- Calculation. --- Cauchy sequence. --- Change of variables. --- Complex plane. --- Conjugacy class. --- Convolution. --- Cusp neighborhood. --- Cyclic group. --- Derivative. --- Differential equation. --- Differential operator. --- Dimension (vector space). --- Dimensional analysis. --- Dirichlet integral. --- Dirichlet series. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Elliptic operator. --- Elliptic partial differential equation. --- Equation. --- Equivalence class. --- Even and odd functions. --- Existential quantification. --- Explicit formula. --- Explicit formulae (L-function). --- Exponential function. --- Fourier transform. --- Function space. --- Functional analysis. --- Functional calculus. --- Fundamental domain. --- Harmonic analysis. --- Hilbert space. --- Hyperbolic partial differential equation. --- Infinitesimal generator (stochastic processes). --- Integral equation. --- Integration by parts. --- Invariant subspace. --- Laplace operator. --- Laplace transform. --- Lebesgue measure. --- Linear differential equation. --- Linear space (geometry). --- Matrix (mathematics). --- Maximum principle. --- Meromorphic function. --- Modular group. --- Neumann boundary condition. --- Norm (mathematics). --- Null vector. --- Number theory. --- Operator theory. --- Orthogonal complement. --- Orthonormal basis. --- Paley–Wiener theorem. --- Partial differential equation. --- Perturbation theory (quantum mechanics). --- Perturbation theory. --- Primitive element (finite field). --- Principal component analysis. --- Projection (linear algebra). --- Quadratic form. --- Removable singularity. --- Representation theorem. --- Resolvent set. --- Riemann hypothesis. --- Riemann surface. --- Riemann zeta function. --- Riesz representation theorem. --- Scatter matrix. --- Scattering theory. --- Schwarz reflection principle. --- Selberg trace formula. --- Self-adjoint. --- Semigroup. --- Sign (mathematics). --- Spectral theory. --- Subgroup. --- Subsequence. --- Summation. --- Support (mathematics). --- Theorem. --- Trace class. --- Trace formula. --- Unitary operator. --- Wave equation. --- Weighted arithmetic mean. --- Winding number. --- Eisenstein, Séries d'. --- Analyse harmonique

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The description for this book, Linear Inequalities and Related Systems. (AM-38), Volume 38, will be forthcoming.

Operational research. Game theory --- Linear programming. --- Matrices. --- Game theory. --- Games, Theory of --- Theory of games --- Mathematical models --- Mathematics --- Algebra, Matrix --- Cracovians (Mathematics) --- Matrix algebra --- Matrixes (Algebra) --- Algebra, Abstract --- Algebra, Universal --- Production scheduling --- Programming (Mathematics) --- Banach space. --- Basic solution (linear programming). --- Big O notation. --- Bilinear form. --- Boundary (topology). --- Brouwer fixed-point theorem. --- Characterization (mathematics). --- Coefficient. --- Combination. --- Computation. --- Computational problem. --- Convex combination. --- Convex cone. --- Convex hull. --- Convex set. --- Corollary. --- Correlation and dependence. --- Cramer's rule. --- Cyclic permutation. --- Dedekind cut. --- Degeneracy (mathematics). --- Determinant. --- Diagram (category theory). --- Dilworth's theorem. --- Dimension (vector space). --- Directional derivative. --- Disjoint sets. --- Doubly stochastic matrix. --- Dual space. --- Duality (mathematics). --- Duality (optimization). --- Eigenvalues and eigenvectors. --- Elementary proof. --- Equation solving. --- Equation. --- Equivalence class. --- Euclidean space. --- Existence theorem. --- Existential quantification. --- Extreme point. --- Fixed-point theorem. --- Functional analysis. --- Fundamental theorem. --- General equilibrium theory. --- Hall's theorem. --- Hilbert space. --- Incidence matrix. --- Inequality (mathematics). --- Infimum and supremum. --- Invertible matrix. --- Kakutani fixed-point theorem. --- Lagrange multiplier. --- Linear equation. --- Linear inequality. --- Linear map. --- Linear space (geometry). --- Linear subspace. --- Loss function. --- Main diagonal. --- Mathematical induction. --- Mathematical optimization. --- Mathematical problem. --- Max-flow min-cut theorem. --- Maxima and minima. --- Maximal set. --- Maximum flow problem. --- Menger's theorem. --- Minor (linear algebra). --- Monotonic function. --- N-vector. --- Nonlinear programming. --- Nonnegative matrix. --- Parity (mathematics). --- Partially ordered set. --- Permutation matrix. --- Permutation. --- Polyhedron. --- Quantity. --- Representation theorem. --- Row and column vectors. --- Scientific notation. --- Sensitivity analysis. --- Set notation. --- Sign (mathematics). --- Simplex algorithm. --- Simultaneous equations. --- Solution set. --- Special case. --- Subset. --- Summation. --- System of linear equations. --- Theorem. --- Transpose. --- Unit sphere. --- Unit vector. --- Upper and lower bounds. --- Variable (mathematics). --- Vector space. --- Von Neumann's theorem.

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Kiyosi Itô's greatest contribution to probability theory may be his introduction of stochastic differential equations to explain the Kolmogorov-Feller theory of Markov processes. Starting with the geometric ideas that guided him, this book gives an account of Itô's program. The modern theory of Markov processes was initiated by A. N. Kolmogorov. However, Kolmogorov's approach was too analytic to reveal the probabilistic foundations on which it rests. In particular, it hides the central role played by the simplest Markov processes: those with independent, identically distributed increments. To remedy this defect, Itô interpreted Kolmogorov's famous forward equation as an equation that describes the integral curve of a vector field on the space of probability measures. Thus, in order to show how Itô's thinking leads to his theory of stochastic integral equations, Stroock begins with an account of integral curves on the space of probability measures and then arrives at stochastic integral equations when he moves to a pathspace setting. In the first half of the book, everything is done in the context of general independent increment processes and without explicit use of Itô's stochastic integral calculus. In the second half, the author provides a systematic development of Itô's theory of stochastic integration: first for Brownian motion and then for continuous martingales. The final chapter presents Stratonovich's variation on Itô's theme and ends with an application to the characterization of the paths on which a diffusion is supported. The book should be accessible to readers who have mastered the essentials of modern probability theory and should provide such readers with a reasonably thorough introduction to continuous-time, stochastic processes.

Markov processes. --- Stochastic difference equations. --- Itō, Kiyosi, --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Itō, K. --- Ito, Kiesi, --- Itō, Kiyoshi, --- 伊藤淸, --- 伊藤清, --- Itō, Kiyosi, --- Itō, Kiyosi, 1915-2008. --- Stochastic difference equations --- Difference equations --- Stochastic processes --- Abelian group. --- Addition. --- Analytic function. --- Approximation. --- Bernhard Riemann. --- Bounded variation. --- Brownian motion. --- Central limit theorem. --- Change of variables. --- Coefficient. --- Complete metric space. --- Compound Poisson process. --- Continuous function (set theory). --- Continuous function. --- Convergence of measures. --- Convex function. --- Coordinate system. --- Corollary. --- David Hilbert. --- Decomposition theorem. --- Degeneracy (mathematics). --- Derivative. --- Diffeomorphism. --- Differentiable function. --- Differentiable manifold. --- Differential equation. --- Differential geometry. --- Dimension. --- Directional derivative. --- Doob–Meyer decomposition theorem. --- Duality principle. --- Elliptic operator. --- Equation. --- Euclidean space. --- Existential quantification. --- Fourier transform. --- Function space. --- Functional analysis. --- Fundamental solution. --- Fundamental theorem of calculus. --- Homeomorphism. --- Hölder's inequality. --- Initial condition. --- Integral curve. --- Integral equation. --- Integration by parts. --- Invariant measure. --- Itô calculus. --- Itô's lemma. --- Joint probability distribution. --- Lebesgue measure. --- Linear interpolation. --- Lipschitz continuity. --- Local martingale. --- Logarithm. --- Markov chain. --- Markov process. --- Markov property. --- Martingale (probability theory). --- Normal distribution. --- Ordinary differential equation. --- Ornstein–Uhlenbeck process. --- Polynomial. --- Principal part. --- Probability measure. --- Probability space. --- Probability theory. --- Pseudo-differential operator. --- Radon–Nikodym theorem. --- Representation theorem. --- Riemann integral. --- Riemann sum. --- Riemann–Stieltjes integral. --- Scientific notation. --- Semimartingale. --- Sign (mathematics). --- Special case. --- Spectral sequence. --- Spectral theory. --- State space. --- State-space representation. --- Step function. --- Stochastic calculus. --- Stochastic. --- Stratonovich integral. --- Submanifold. --- Support (mathematics). --- Tangent space. --- Tangent vector. --- Taylor's theorem. --- Theorem. --- Theory. --- Topological space. --- Topology. --- Translational symmetry. --- Uniform convergence. --- Variable (mathematics). --- Vector field. --- Weak convergence (Hilbert space). --- Weak topology.

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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.