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Interest in the spin-c Dirac operator originally came about from the study of complex analytic manifolds, where in the non-Kähler case the Dolbeault operator is no longer suitable for getting local formulas for the Riemann–Roch number or the holomorphic Lefschetz number. However, every symplectic manifold (phase space in classical mechanics) also carries an almost complex structure and hence a corresponding spin-c Dirac operator. Using the heat kernels theory of Berline, Getzler, and Vergne, this work revisits some fundamental concepts of the theory, and presents the application to symplectic geometry. J.J. Duistermaat was well known for his beautiful and concise expositions of seemingly familiar concepts, and this classic study is certainly no exception. Reprinted as it was originally published, this work is as an affordable text that will be of interest to a range of researchers in geometric analysis and mathematical physics. Overall this is a carefully written, highly readable book on a very beautiful subject. —Mathematical Reviews The book of J.J. Duistermaat is a nice introduction to analysis related [to the] spin-c Dirac operator. ... The book is almost self contained, [is] readable, and will be useful for anybody who is interested in the topic. —EMS Newsletter The author's book is a marvelous introduction to [these] objects and theories. —Zentralblatt MATH.

Almost complex manifolds. --- Differential topology. --- Dirac equation. --- Mathematical physics. --- Operator theory. --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Physical mathematics --- Physics --- Manifolds, Almost complex --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Partial differential equations. --- Differential geometry. --- Global Analysis and Analysis on Manifolds. --- Partial Differential Equations. --- Differential Geometry. --- Analysis. --- Operator Theory. --- Mathematical Physics. --- Functional analysis --- Differential geometry --- Partial differential equations --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- 517.1 Mathematical analysis --- Mathematical analysis --- Math --- Science --- Differential equations, Partial --- Quantum field theory --- Wave equation --- Complex manifolds --- Global analysis. --- Differential equations, partial. --- Global differential geometry.

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Differential geometry. Global analysis --- Operator theory --- Partial differential equations --- Mathematical analysis --- differentiaalvergelijkingen --- analyse (wiskunde) --- differentiaal geometrie

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Singularities (Mathematics) --- Fourier integral operators. --- Fourier series. --- Pseudodifferential operators. --- Singularités (mathématiques) --- Fourier, Opérateurs intégraux de. --- Fourier, Séries de. --- Opérateurs pseudo-différentiels. --- Singularités (mathématiques) --- Fourier, Opérateurs intégraux de. --- Fourier, Séries de. --- Opérateurs pseudo-différentiels.

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This textbook is an application-oriented introduction to the theory of distributions, a powerful tool used in mathematical analysis. The treatment emphasizes applications that relate distributions to linear partial differential equations and Fourier analysis problems found in mechanics, optics, quantum mechanics, quantum field theory, and signal analysis. Throughout the book, methods are developed to deal with formal calculations involving functions, series, and integrals that cannot be mathematically justified within the classical framework. Key features: • Many examples, exercises, hints, and solutions guide the reader throughout the text. • Includes an introduction to distributions, differentiation, convergence, convolution, the Fourier transform, and spaces of distributions having special properties. • Original proofs, which may be difficult to locate elsewhere, are given for many well-known results. • The Fourier transform is transparently treated and applied to provide a new proof of the Kernel Theorem, which in turn is used to efficiently derive numerous important results. • The systematic use of pullback and pushforward introduces concise notation. • Emphasizes the role of symmetry in obtaining short arguments and investigates distributions that are invariant under the actions of various groups of transformations. Distributions: Theory and Applications is aimed at advanced undergraduates and graduate students in mathematics, theoretical physics, and engineering, who will find this textbook a welcome introduction to the subject, requiring only a minimal mathematical background. The work may also serve as an excellent self-study guide for researchers who use distributions in various fields.

Electronic books. -- local. --- Theory of distributions (Functional analysis). --- Theory of distributions (Functional analysis) --- Fourier transformations --- Civil & Environmental Engineering --- Mathematics --- Physical Sciences & Mathematics --- Engineering & Applied Sciences --- Operations Research --- Calculus --- Distribution (Functional analysis) --- Distributions, Theory of (Functional analysis) --- Functions, Generalized --- Generalized functions --- Mathematics. --- Approximation theory. --- Fourier analysis. --- Functional analysis. --- Differential equations. --- Partial differential equations. --- Applied mathematics. --- Engineering mathematics. --- Functional Analysis. --- Approximations and Expansions. --- Applications of Mathematics. --- Partial Differential Equations. --- Fourier Analysis. --- Ordinary Differential Equations. --- Engineering --- Engineering analysis --- Mathematical analysis --- Partial differential equations --- 517.91 Differential equations --- Differential equations --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Analysis, Fourier --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Math --- Science --- Differential equations, partial. --- Differential Equations. --- Differential equations, Partial.