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Written for advanced undergraduate and first-year graduate students, this book aims to introduce students to a serious level of p-adic analysis with important implications for number theory. The main object is the study of G-series, that is, power series y=aij=0 Ajxj with coefficients in an algebraic number field K. These series satisfy a linear differential equation Ly=0 with LIK(x) [d/dx] and have non-zero radii of convergence for each imbedding of K into the complex numbers. They have the further property that the common denominators of the first s coefficients go to infinity geometrically with the index s. After presenting a review of valuation theory and elementary p-adic analysis together with an application to the congruence zeta function, this book offers a detailed study of the p-adic properties of formal power series solutions of linear differential equations. In particular, the p-adic radii of convergence and the p-adic growth of coefficients are studied. Recent work of Christol, Bombieri, André, and Dwork is treated and augmented. The book concludes with Chudnovsky's theorem: the analytic continuation of a G -series is again a G -series. This book will be indispensable for those wishing to study the work of Bombieri and André on global relations and for the study of the arithmetic properties of solutions of ordinary differential equations.

Analyse p-adique --- H-fonction --- H-functie --- H-function --- p-adic analyse --- p-adic analysis --- H-functions --- H-functions. --- p-adic analysis. --- Analysis, p-adic --- Algebra --- Calculus --- Geometry, Algebraic --- Fox's H-function --- G-functions, Generalized --- Generalized G-functions --- Generalized Mellin-Barnes functions --- Mellin-Barnes functions, Generalized --- Hypergeometric functions --- Adjoint. --- Algebraic Method. --- Algebraic closure. --- Algebraic number field. --- Algebraic number theory. --- Algebraic variety. --- Algebraically closed field. --- Analytic continuation. --- Analytic function. --- Argument principle. --- Arithmetic. --- Automorphism. --- Bearing (navigation). --- Binomial series. --- Calculation. --- Cardinality. --- Cartesian coordinate system. --- Cauchy sequence. --- Cauchy's theorem (geometry). --- Coefficient. --- Cohomology. --- Commutative ring. --- Complete intersection. --- Complex analysis. --- Conjecture. --- Density theorem. --- Differential equation. --- Dimension (vector space). --- Direct sum. --- Discrete valuation. --- Eigenvalues and eigenvectors. --- Elliptic curve. --- Equation. --- Equivalence class. --- Estimation. --- Existential quantification. --- Exponential function. --- Exterior algebra. --- Field of fractions. --- Finite field. --- Formal power series. --- Fuchs' theorem. --- G-module. --- Galois extension. --- Galois group. --- General linear group. --- Generic point. --- Geometry. --- Hypergeometric function. --- Identity matrix. --- Inequality (mathematics). --- Intercept method. --- Irreducible element. --- Irreducible polynomial. --- Laurent series. --- Limit of a sequence. --- Linear differential equation. --- Lowest common denominator. --- Mathematical induction. --- Meromorphic function. --- Modular arithmetic. --- Module (mathematics). --- Monodromy. --- Monotonic function. --- Multiplicative group. --- Natural number. --- Newton polygon. --- Number theory. --- P-adic number. --- Parameter. --- Permutation. --- Polygon. --- Polynomial. --- Projective line. --- Q.E.D. --- Quadratic residue. --- Radius of convergence. --- Rational function. --- Rational number. --- Residue field. --- Riemann hypothesis. --- Ring of integers. --- Root of unity. --- Separable polynomial. --- Sequence. --- Siegel's lemma. --- Special case. --- Square root. --- Subring. --- Subset. --- Summation. --- Theorem. --- Topology of uniform convergence. --- Transpose. --- Triangle inequality. --- Unipotent. --- Valuation ring. --- Weil conjecture. --- Wronskian. --- Y-intercept.

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This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part. Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization. This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.

Matrices. --- Numerical analysis. --- Mathematical analysis --- Algebra, Matrix --- Cracovians (Mathematics) --- Matrix algebra --- Matrixes (Algebra) --- Algebra, Abstract --- Algebra, Universal --- Matrices --- Numerical analysis --- Algorithm. --- Analysis of algorithms. --- Analytic function. --- Asymptotic analysis. --- Basis (linear algebra). --- Basis function. --- Biconjugate gradient method. --- Bidiagonal matrix. --- Bilinear form. --- Calculation. --- Characteristic polynomial. --- Chebyshev polynomials. --- Coefficient. --- Complex number. --- Computation. --- Condition number. --- Conjugate gradient method. --- Conjugate transpose. --- Cross-validation (statistics). --- Curve fitting. --- Degeneracy (mathematics). --- Determinant. --- Diagonal matrix. --- Dimension (vector space). --- Eigenvalues and eigenvectors. --- Equation. --- Estimation. --- Estimator. --- Exponential function. --- Factorization. --- Function (mathematics). --- Function of a real variable. --- Functional analysis. --- Gaussian quadrature. --- Hankel matrix. --- Hermite interpolation. --- Hessenberg matrix. --- Hilbert matrix. --- Holomorphic function. --- Identity matrix. --- Interlacing (bitmaps). --- Inverse iteration. --- Inverse problem. --- Invertible matrix. --- Iteration. --- Iterative method. --- Jacobi matrix. --- Krylov subspace. --- Laguerre polynomials. --- Lanczos algorithm. --- Linear differential equation. --- Linear regression. --- Linear subspace. --- Logarithm. --- Machine epsilon. --- Matrix function. --- Matrix polynomial. --- Maxima and minima. --- Mean value theorem. --- Meromorphic function. --- Moment (mathematics). --- Moment matrix. --- Moment problem. --- Monic polynomial. --- Monomial. --- Monotonic function. --- Newton's method. --- Numerical integration. --- Numerical linear algebra. --- Orthogonal basis. --- Orthogonal matrix. --- Orthogonal polynomials. --- Orthogonal transformation. --- Orthogonality. --- Orthogonalization. --- Orthonormal basis. --- Partial fraction decomposition. --- Polynomial. --- Preconditioner. --- QR algorithm. --- QR decomposition. --- Quadratic form. --- Rate of convergence. --- Recurrence relation. --- Regularization (mathematics). --- Rotation matrix. --- Singular value. --- Square (algebra). --- Summation. --- Symmetric matrix. --- Theorem. --- Tikhonov regularization. --- Trace (linear algebra). --- Triangular matrix. --- Tridiagonal matrix. --- Upper and lower bounds. --- Variable (mathematics). --- Vector space. --- Weight function.

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The theory of pseudo-differential operators (which originated as singular integral operators) was largely influenced by its application to function theory in one complex variable and regularity properties of solutions of elliptic partial differential equations. Given here is an exposition of some new classes of pseudo-differential operators relevant to several complex variables and certain non-elliptic problems.Originally published in 1979.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.

517.982.4 --- Pseudodifferential operators --- Operators, Pseudodifferential --- Pseudo-differential operators --- Theory of generalized functions (distributions) --- Pseudodifferential operators. --- 517.982.4 Theory of generalized functions (distributions) --- Operator theory --- Differential equations, Partial --- Équations aux dérivées partielles --- Opérateurs pseudo-différentiels --- Addition. --- Adjoint. --- Approximation. --- Asymptotic expansion. --- Banach space. --- Bounded operator. --- Boundedness. --- Calculation. --- Change of variables. --- Coefficient. --- Compact space. --- Complex analysis. --- Computation. --- Corollary. --- Cotangent bundle. --- Derivative. --- Differential operator. --- Disjoint union. --- Elliptic partial differential equation. --- Estimation. --- Euclidean distance. --- Euclidean vector. --- Existential quantification. --- Fourier integral operator. --- Fourier transform. --- Geometric series. --- Heat equation. --- Heisenberg group. --- Homogeneous distribution. --- Infimum and supremum. --- Integer. --- Integration by parts. --- Intermediate value theorem. --- Jacobian matrix and determinant. --- Left inverse. --- Linear combination. --- Linear map. --- Mean value theorem. --- Monograph. --- Monomial. --- Nilpotent group. --- Operator (physics). --- Operator norm. --- Order of magnitude. --- Orthogonal complement. --- Parametrix. --- Parity (mathematics). --- Partition of unity. --- Polynomial. --- Projection (linear algebra). --- Pseudo-differential operator. --- Quadratic function. --- Regularity theorem. --- Remainder. --- Requirement. --- Right inverse. --- Scientific notation. --- Self-reference. --- Several complex variables. --- Singular integral. --- Smoothness. --- Sobolev space. --- Special case. --- Submanifold. --- Subset. --- Sum of squares. --- Summation. --- Support (mathematics). --- Tangent space. --- Taylor's theorem. --- Theorem. --- Theory. --- Transpose. --- Triangle inequality. --- Uniform boundedness. --- Upper and lower bounds. --- Variable (mathematics). --- Without loss of generality. --- Zero set. --- Équations aux dérivées partielles --- Opérateurs pseudo-différentiels

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This book develops a general analysis and synthesis framework for impulsive and hybrid dynamical systems. Such a framework is imperative for modern complex engineering systems that involve interacting continuous-time and discrete-time dynamics with multiple modes of operation that place stringent demands on controller design and require implementation of increasing complexity--whether advanced high-performance tactical fighter aircraft and space vehicles, variable-cycle gas turbine engines, or air and ground transportation systems. Impulsive and Hybrid Dynamical Systems goes beyond similar treatments by developing invariant set stability theorems, partial stability, Lagrange stability, boundedness, ultimate boundedness, dissipativity theory, vector dissipativity theory, energy-based hybrid control, optimal control, disturbance rejection control, and robust control for nonlinear impulsive and hybrid dynamical systems. A major contribution to mathematical system theory and control system theory, this book is written from a system-theoretic point of view with the highest standards of exposition and rigor. It is intended for graduate students, researchers, and practitioners of engineering and applied mathematics as well as computer scientists, physicists, and other scientists who seek a fundamental understanding of the rich dynamical behavior of impulsive and hybrid dynamical systems.

Automatic control. --- Control theory. --- Dynamics. --- Discrete-time systems. --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Dynamics --- Machine theory --- Control engineering --- Control equipment --- Control theory --- Engineering instruments --- Automation --- Programmable controllers --- DES (System analysis) --- Discrete event systems --- Sampled-data systems --- Digital control systems --- Discrete mathematics --- System analysis --- Linear time invariant systems --- Actuator. --- Adaptive control. --- Algorithm. --- Amplitude. --- Analog computer. --- Arbitrarily large. --- Asymptote. --- Asymptotic analysis. --- Axiom. --- Balance equation. --- Bode plot. --- Boundedness. --- Calculation. --- Center of mass (relativistic). --- Coefficient of restitution. --- Continuous function. --- Convex set. --- Differentiable function. --- Differential equation. --- Dissipation. --- Dissipative system. --- Dynamical system. --- Dynamical systems theory. --- Energy. --- Equations of motion. --- Equilibrium point. --- Escapement. --- Euler–Lagrange equation. --- Exponential stability. --- Forms of energy. --- Hamiltonian mechanics. --- Hamiltonian system. --- Hermitian matrix. --- Hooke's law. --- Hybrid system. --- Identity matrix. --- Inequality (mathematics). --- Infimum and supremum. --- Initial condition. --- Instability. --- Interconnection. --- Invariance theorem. --- Isolated system. --- Iterative method. --- Jacobian matrix and determinant. --- Lagrangian (field theory). --- Lagrangian system. --- Lagrangian. --- Likelihood-ratio test. --- Limit cycle. --- Limit set. --- Linear function. --- Linearization. --- Lipschitz continuity. --- Lyapunov function. --- Lyapunov stability. --- Mass balance. --- Mathematical optimization. --- Melting. --- Mixture. --- Moment of inertia. --- Momentum. --- Monotonic function. --- Negative feedback. --- Nonlinear programming. --- Nonlinear system. --- Nonnegative matrix. --- Optimal control. --- Ordinary differential equation. --- Orthant. --- Parameter. --- Partial differential equation. --- Passive dynamics. --- Poincaré conjecture. --- Potential energy. --- Proof mass. --- Quantity. --- Rate function. --- Requirement. --- Robust control. --- Second law of thermodynamics. --- Semi-infinite. --- Small-gain theorem. --- Special case. --- Spectral radius. --- Stability theory. --- State space. --- Stiffness. --- Supply (economics). --- Telecommunication. --- Theorem. --- Transpose. --- Uncertainty. --- Uniform boundedness. --- Uniqueness. --- Vector field. --- Vibration. --- Zeroth (software). --- Zeroth law of thermodynamics.

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In this monograph the authors redevelop the theory systematically using two different approaches. A general mechanism for the deformation of structures on manifolds was developed by Donald Spencer ten years ago. A new version of that theory, based on the differential calculus in the analytic spaces of Grothendieck, was recently given by B. Malgrange. The first approach adopts Malgrange's idea in defining jet sheaves and linear operators, although the brackets and the non-linear theory arc treated in an essentially different manner. The second approach is based on the theory of derivations, and its relationship to the first is clearly explained. The introduction describes examples of Lie equations and known integrability theorems, and gives applications of the theory to be developed in the following chapters and in the subsequent volume.

Differential geometry. Global analysis --- Lie groups --- Lie algebras --- Differential equations --- Groupes de Lie --- Algèbres de Lie --- Equations différentielles --- 514.76 --- Groups, Lie --- Symmetric spaces --- Topological groups --- Algebras, Lie --- Algebra, Abstract --- Algebras, Linear --- Equations, Differential --- Bessel functions --- Calculus --- Geometry of differentiable manifolds and of their submanifolds --- Differential equations. --- Lie algebras. --- Lie groups. --- 517.91 Differential equations --- 514.76 Geometry of differentiable manifolds and of their submanifolds --- Algèbres de Lie --- Equations différentielles --- 517.91. --- Numerical solutions --- Surfaces, Deformation of --- Surfaces (mathématiques) --- Déformation --- Pseudogroups. --- Pseudogroupes (mathématiques) --- 517.91 --- Adjoint representation. --- Adjoint. --- Affine transformation. --- Alexander Grothendieck. --- Analytic function. --- Associative algebra. --- Atlas (topology). --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Bundle map. --- Category of topological spaces. --- Cauchy–Riemann equations. --- Coefficient. --- Commutative diagram. --- Commutator. --- Complex conjugate. --- Complex group. --- Complex manifold. --- Computation. --- Conformal map. --- Continuous function. --- Coordinate system. --- Corollary. --- Cotangent bundle. --- Curvature tensor. --- Deformation theory. --- Derivative. --- Diagonal. --- Diffeomorphism. --- Differentiable function. --- Differential form. --- Differential operator. --- Differential structure. --- Direct proof. --- Direct sum. --- Ellipse. --- Endomorphism. --- Equation. --- Exact sequence. --- Exactness. --- Existential quantification. --- Exponential function. --- Exponential map (Riemannian geometry). --- Exterior derivative. --- Fiber bundle. --- Fibration. --- Frame bundle. --- Frobenius theorem (differential topology). --- Frobenius theorem (real division algebras). --- Group isomorphism. --- Groupoid. --- Holomorphic function. --- Homeomorphism. --- Integer. --- J-invariant. --- Jacobian matrix and determinant. --- Jet bundle. --- Linear combination. --- Linear map. --- Manifold. --- Maximal ideal. --- Model category. --- Morphism. --- Nonlinear system. --- Open set. --- Parameter. --- Partial derivative. --- Partial differential equation. --- Pointwise. --- Presheaf (category theory). --- Pseudo-differential operator. --- Pseudogroup. --- Quantity. --- Regular map (graph theory). --- Requirement. --- Riemann surface. --- Right inverse. --- Scalar multiplication. --- Sheaf (mathematics). --- Special case. --- Structure tensor. --- Subalgebra. --- Subcategory. --- Subgroup. --- Submanifold. --- Subset. --- Tangent bundle. --- Tangent space. --- Tangent vector. --- Tensor field. --- Tensor product. --- Theorem. --- Torsion tensor. --- Transpose. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Volume element. --- Surfaces (mathématiques) --- Déformation --- Analyse sur une variété