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This Special Edition contains new results on Differential and Integral Equations and Systems, covering higher-order Initial and Boundary Value Problems, fractional differential and integral equations and applications, non-local optimal control, inverse, and higher-order nonlinear boundary value problems, distributional solutions in the form of a finite series of the Dirac delta function and its derivatives, asymptotic properties’ oscillatory theory for neutral nonlinear differential equations, the existence of extremal solutions via monotone iterative techniques, predator–prey interaction via fractional-order models, among others. Our main goal is not only to show new trends in this field but also to showcase and provide new methods and techniques that can lead to future research.

Research & information: general --- Mathematics & science --- fourth-order differential equations --- neutral delay --- oscillation --- ψ-Caputo fractional derivative --- Cauchy problem extremal solutions --- monotone iterative technique --- upper and lower solutions --- third-order differential equation --- boundary value problem --- existence --- sign conditions --- mixed type nonlinear equation --- hilfer operator --- mittag–leffler function --- spectral parameter --- solvability --- equations of the pseudo-elliptic type of third order --- energy estimate --- analog of the Saint-Venant principle --- even-order differential equations --- Dirac delta function --- distributional solution --- Laplace transform --- power series solution --- integro-differential equation --- mixed type equation --- small parameter --- spectral parameters --- Caputo operators of different fractional orders --- inverse problem --- one value solvability --- Rosenzweig–MacArthur model --- fractional derivatives --- threshold harvesting --- distributed-order fractional calculus --- basic optimal control problem --- Pontryagin extremals --- fourth-order differential equations --- neutral delay --- oscillation --- ψ-Caputo fractional derivative --- Cauchy problem extremal solutions --- monotone iterative technique --- upper and lower solutions --- third-order differential equation --- boundary value problem --- existence --- sign conditions --- mixed type nonlinear equation --- hilfer operator --- mittag–leffler function --- spectral parameter --- solvability --- equations of the pseudo-elliptic type of third order --- energy estimate --- analog of the Saint-Venant principle --- even-order differential equations --- Dirac delta function --- distributional solution --- Laplace transform --- power series solution --- integro-differential equation --- mixed type equation --- small parameter --- spectral parameters --- Caputo operators of different fractional orders --- inverse problem --- one value solvability --- Rosenzweig–MacArthur model --- fractional derivatives --- threshold harvesting --- distributed-order fractional calculus --- basic optimal control problem --- Pontryagin extremals

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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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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.

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In Hypo-Analytic Structures Franois Treves provides a systematic approach to the study of the differential structures on manifolds defined by systems of complex vector fields. Serving as his main examples are the elliptic complexes, among which the De Rham and Dolbeault are the best known, and the tangential Cauchy-Riemann operators. Basic geometric entities attached to those structures are isolated, such as maximally real submanifolds and orbits of the system. Treves discusses the existence, uniqueness, and approximation of local solutions to homogeneous and inhomogeneous equations and delimits their supports. The contents of this book consist of many results accumulated in the last decade by the author and his collaborators, but also include classical results, such as the Newlander-Nirenberg theorem. The reader will find an elementary description of the FBI transform, as well as examples of its use. Treves extends the main approximation and uniqueness results to first-order nonlinear equations by means of the Hamiltonian lift.Originally published in 1993.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.

Differential equations, Partial. --- Manifolds (Mathematics) --- Vector fields. --- Direction fields (Mathematics) --- Fields, Direction (Mathematics) --- Fields, Slope (Mathematics) --- Fields, Vector --- Slope fields (Mathematics) --- Vector analysis --- Geometry, Differential --- Topology --- Partial differential equations --- Algebra homomorphism. --- Analytic function. --- Automorphism. --- Basis (linear algebra). --- Bijection. --- Bounded operator. --- C0. --- CR manifold. --- Cauchy problem. --- Cauchy sequence. --- Cauchy–Riemann equations. --- Characterization (mathematics). --- Coefficient. --- Cohomology. --- Commutative property. --- Commutator. --- Complex dimension. --- Complex manifold. --- Complex number. --- Complex space. --- Complex-analytic variety. --- Continuous function (set theory). --- Corollary. --- Coset. --- De Rham cohomology. --- Diagram (category theory). --- Diffeomorphism. --- Differential form. --- Differential operator. --- Dimension (vector space). --- Dirac delta function. --- Dirac measure. --- Eigenvalues and eigenvectors. --- Embedding. --- Equation. --- Exact differential. --- Existential quantification. --- Exterior algebra. --- F-space. --- Formal power series. --- Frobenius theorem (differential topology). --- Frobenius theorem (real division algebras). --- H-vector. --- Hadamard three-circle theorem. --- Hahn–Banach theorem. --- Holomorphic function. --- Hypersurface. --- Hölder condition. --- Identity matrix. --- Infimum and supremum. --- Integer. --- Integral equation. --- Integral transform. --- Intersection (set theory). --- Jacobian matrix and determinant. --- Linear differential equation. --- Linear equation. --- Linear map. --- Lipschitz continuity. --- Manifold. --- Mean value theorem. --- Method of characteristics. --- Monomial. --- Multi-index notation. --- Neighbourhood (mathematics). --- Norm (mathematics). --- One-form. --- Open mapping theorem (complex analysis). --- Open mapping theorem. --- Open set. --- Ordinary differential equation. --- Partial differential equation. --- Poisson bracket. --- Polynomial. --- Power series. --- Projection (linear algebra). --- Pullback (category theory). --- Pullback (differential geometry). --- Pullback. --- Riemann mapping theorem. --- Riemann surface. --- Ring homomorphism. --- Sesquilinear form. --- Sobolev space. --- Special case. --- Stokes' theorem. --- Stone–Weierstrass theorem. --- Submanifold. --- Subset. --- Support (mathematics). --- Surjective function. --- Symplectic geometry. --- Symplectic vector space. --- Taylor series. --- Theorem. --- Unit disk. --- Upper half-plane. --- Vector bundle. --- Vector field. --- Volume form.

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Many of the operators one meets in several complex variables, such as the famous Lewy operator, are not locally solvable. Nevertheless, such an operator L can be thoroughly studied if one can find a suitable relative parametrix--an operator K such that LK is essentially the orthogonal projection onto the range of L. The analysis is by far most decisive if one is able to work in the real analytic, as opposed to the smooth, setting. With this motivation, the author develops an analytic calculus for the Heisenberg group. Features include: simple, explicit formulae for products and adjoints; simple representation-theoretic conditions, analogous to ellipticity, for finding parametrices in the calculus; invariance under analytic contact transformations; regularity with respect to non-isotropic Sobolev and Lipschitz spaces; and preservation of local analyticity. The calculus is suitable for doing analysis on real analytic strictly pseudoconvex CR manifolds. In this context, the main new application is a proof that the Szego projection preserves local analyticity, even in the three-dimensional setting. Relative analytic parametrices are also constructed for the adjoint of the tangential Cauchy-Riemann operator.Originally published in 1990.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.

Pseudodifferential operators. --- Functions of several complex variables. --- Solvable groups. --- Analytic function. --- Analytic set. --- Associative property. --- Asymptotic expansion. --- Atkinson's theorem. --- Banach space. --- Bilinear map. --- Boundary value problem. --- Bounded function. --- Bounded operator. --- Bump function. --- C space. --- CR manifold. --- Cauchy problem. --- Cauchy's integral formula. --- Cauchy–Schwarz inequality. --- Cayley transform. --- Characteristic function (probability theory). --- Characterization (mathematics). --- Coefficient. --- Cokernel. --- Combinatorics. --- Complex conjugate. --- Complex number. --- Complexification (Lie group). --- Contact geometry. --- Convolution. --- Darboux's theorem (analysis). --- Darboux's theorem. --- Diagram (category theory). --- Diffeomorphism. --- Difference "ient. --- Differential operator. --- Dimension (vector space). --- Dirac delta function. --- Eigenvalues and eigenvectors. --- Elliptic operator. --- Equation. --- Existential quantification. --- Explicit formulae (L-function). --- Factorial. --- Fourier inversion theorem. --- Fourier series. --- Fourier transform. --- Fundamental solution. --- Heisenberg group. --- Hermitian adjoint. --- Hilbert space. --- Hodge theory. --- Hypoelliptic operator. --- Hölder's inequality. --- Implicit function theorem. --- Integral transform. --- Invertible matrix. --- Leibniz integral rule. --- Lie algebra. --- Mathematical induction. --- Mathematical proof. --- Mean value theorem. --- Multinomial theorem. --- Neighbourhood (mathematics). --- Neumann series. --- Nilpotent group. --- Orthogonal transformation. --- Orthonormal basis. --- Oscillatory integral. --- Paley–Wiener theorem. --- Parametrix. --- Parity (mathematics). --- Partial differential equation. --- Partition of unity. --- Plancherel theorem. --- Polynomial. --- Power function. --- Power series. --- Product rule. --- Property B. --- Pseudo-differential operator. --- Pullback (category theory). --- Quadratic form. --- Regularity theorem. --- Riesz transform. --- Schwartz space. --- Scientific notation. --- Self-adjoint operator. --- Self-adjoint. --- Sesquilinear form. --- Several complex variables. --- Singular integral. --- Special case. --- Summation. --- Support (mathematics). --- Symmetrization. --- Theorem. --- Topology. --- Triangle inequality. --- Unbounded operator. --- Union (set theory). --- Unitary transformation. --- Variable (mathematics).