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This book contains a valuable discussion of renormalization through the addition of counterterms to the Lagrangian, giving the first complete proof of the cancellation of all divergences in an arbitrary interaction. The author also introduces a new method of renormalizing an arbitrary Feynman amplitude, a method that is simpler than previous approaches and can be used to study the renormalized perturbation series in quantum field theory.

Mathematical physics. --- Quantum field theory. --- Addition. --- Adjoint. --- Amplitude. --- Analytic continuation. --- Analytic function. --- Antiparticle. --- C-number. --- Calculation. --- Change of variables. --- Classical electromagnetism. --- Coefficient. --- Commutative property. --- Compact space. --- Complex analysis. --- Complex number. --- Connectivity (graph theory). --- Constant term. --- Convolution. --- Derivative. --- Diagram (category theory). --- Differentiable function. --- Distribution (mathematics). --- Equation. --- Estimation. --- Explicit formulae (L-function). --- Fermion. --- Fock space. --- Formal power series. --- Fourier transform. --- Free field. --- Gauge theory. --- Graph theory. --- Hilbert space. --- Incidence matrix. --- Interaction picture. --- Invertible matrix. --- Irreducibility (mathematics). --- Isolated singularity. --- Lagrangian (field theory). --- Laurent series. --- Mathematical induction. --- Mathematics. --- Momentum. --- Monomial. --- Multiple integral. --- National Science Foundation. --- Notation. --- Parameter. --- Path integral formulation. --- Permutation. --- Polynomial. --- Power series. --- Probability. --- Propagator. --- Quadratic form. --- Quantity. --- Remainder. --- Renormalization. --- Requirement. --- S-matrix. --- Scattering amplitude. --- Scientific notation. --- Second quantization. --- Several complex variables. --- Simple extension. --- Special case. --- Subset. --- Subtraction. --- Suggestion. --- Summation. --- Taylor series. --- Tensor product. --- Theorem. --- Theory. --- Topological space. --- Translational symmetry. --- Tree (data structure). --- Uniform convergence. --- Vacuum expectation value. --- Vacuum state. --- Vacuum. --- Variable (mathematics). --- Vector field. --- Vector potential. --- Wick's theorem. --- Z0.

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Reciprocity laws of various kinds play a central role in number theory. In the easiest case, one obtains a transparent formulation by means of roots of unity, which are special values of exponential functions. A similar theory can be developed for special values of elliptic or elliptic modular functions, and is called complex multiplication of such functions. In 1900 Hilbert proposed the generalization of these as the twelfth of his famous problems. In this book, Goro Shimura provides the most comprehensive generalizations of this type by stating several reciprocity laws in terms of abelian varieties, theta functions, and modular functions of several variables, including Siegel modular functions. This subject is closely connected with the zeta function of an abelian variety, which is also covered as a main theme in the book. The third topic explored by Shimura is the various algebraic relations among the periods of abelian integrals. The investigation of such algebraicity is relatively new, but has attracted the interest of increasingly many researchers. Many of the topics discussed in this book have not been covered before. In particular, this is the first book in which the topics of various algebraic relations among the periods of abelian integrals, as well as the special values of theta and Siegel modular functions, are treated extensively.

Ordered algebraic structures --- 512.74 --- Abelian varieties --- Modular functions --- Functions, Modular --- Elliptic functions --- Group theory --- Number theory --- Varieties, Abelian --- Geometry, Algebraic --- Algebraic groups. Abelian varieties --- 512.74 Algebraic groups. Abelian varieties --- Abelian varieties. --- Modular functions. --- Abelian extension. --- Abelian group. --- Abelian variety. --- Absolute value. --- Adele ring. --- Affine space. --- Affine variety. --- Algebraic closure. --- Algebraic equation. --- Algebraic extension. --- Algebraic number field. --- Algebraic structure. --- Algebraic variety. --- Analytic manifold. --- Automorphic function. --- Automorphism. --- Big O notation. --- CM-field. --- Characteristic polynomial. --- Class field theory. --- Coefficient. --- Complete variety. --- Complex conjugate. --- Complex multiplication. --- Complex number. --- Complex torus. --- Corollary. --- Degenerate bilinear form. --- Differential form. --- Direct product. --- Direct proof. --- Discrete valuation ring. --- Divisor. --- Eigenvalues and eigenvectors. --- Embedding. --- Endomorphism. --- Existential quantification. --- Field of fractions. --- Finite field. --- Fractional ideal. --- Function (mathematics). --- Fundamental theorem. --- Galois extension. --- Galois group. --- Galois theory. --- Generic point. --- Ground field. --- Group theory. --- Groupoid. --- Hecke character. --- Homology (mathematics). --- Homomorphism. --- Identity element. --- Integer. --- Irreducibility (mathematics). --- Irreducible representation. --- Lie group. --- Linear combination. --- Linear subspace. --- Local ring. --- Modular form. --- Natural number. --- Number theory. --- Polynomial. --- Prime factor. --- Prime ideal. --- Projective space. --- Projective variety. --- Rational function. --- Rational mapping. --- Rational number. --- Real number. --- Residue field. --- Riemann hypothesis. --- Root of unity. --- Scientific notation. --- Semisimple algebra. --- Simple algebra. --- Singular value. --- Special case. --- Subgroup. --- Subring. --- Subset. --- Summation. --- Theorem. --- Vector space. --- Zero element.

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The theory of D-modules deals with the algebraic aspects of differential equations. These are particularly interesting on homogeneous manifolds, since the infinitesimal action of a Lie algebra consists of differential operators. Hence, it is possible to attach geometric invariants, like the support and the characteristic variety, to representations of Lie groups. By considering D-modules on flag varieties, one obtains a simple classification of all irreducible admissible representations of reductive Lie groups. On the other hand, it is natural to study the representations realized by functions on pseudo-Riemannian symmetric spaces, i.e., spherical representations. The problem is then to describe the spherical representations among all irreducible ones, and to compute their multiplicities. This is the goal of this work, achieved fairly completely at least for the discrete series representations of reductive symmetric spaces. The book provides a general introduction to the theory of D-modules on flag varieties, and it describes spherical D-modules in terms of a cohomological formula. Using microlocalization of representations, the author derives a criterion for irreducibility. The relation between multiplicities and singularities is also discussed at length.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.

Differentiable manifolds. --- D-modules. --- Representations of groups. --- Lie groups. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Modules (Algebra) --- Differential manifolds --- Manifolds (Mathematics) --- Affine space. --- Algebraic cycle. --- Algebraic element. --- Analytic function. --- Annihilator (ring theory). --- Automorphism. --- Banach space. --- Base change. --- Big O notation. --- Bijection. --- Bilinear form. --- Borel subgroup. --- Cartan subalgebra. --- Cofibration. --- Cohomology. --- Commutative diagram. --- Commutative property. --- Commutator subgroup. --- Complexification (Lie group). --- Conjugacy class. --- Coproduct. --- Coset. --- Cotangent space. --- D-module. --- Derived category. --- Diagram (category theory). --- Differential operator. --- Dimension (vector space). --- Direct image functor. --- Discrete series representation. --- Disk (mathematics). --- Dot product. --- Double coset. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Endomorphism. --- Euler operator. --- Existential quantification. --- Fibration. --- Function space. --- Functor. --- G-module. --- Gelfand pair. --- Generic point. --- Hilbert space. --- Holomorphic function. --- Homomorphism. --- Hyperfunction. --- Ideal (ring theory). --- Infinitesimal character. --- Inner automorphism. --- Invertible sheaf. --- Irreducibility (mathematics). --- Irreducible representation. --- Levi decomposition. --- Lie algebra. --- Line bundle. --- Linear algebraic group. --- Linear space (geometry). --- Manifold. --- Maximal compact subgroup. --- Maximal torus. --- Metric space. --- Module (mathematics). --- Moment map. --- Morphism. --- Noetherian ring. --- Open set. --- Presheaf (category theory). --- Principal series representation. --- Projective line. --- Projective object. --- Projective space. --- Projective variety. --- Reductive group. --- Riemannian geometry. --- Riemann–Hilbert correspondence. --- Right inverse. --- Ring (mathematics). --- Root system. --- Satake diagram. --- Sheaf (mathematics). --- Sheaf of modules. --- Special case. --- Sphere. --- Square-integrable function. --- Sub"ient. --- Subalgebra. --- Subcategory. --- Subgroup. --- Summation. --- Surjective function. --- Symmetric space. --- Symplectic geometry. --- Tensor product. --- Theorem. --- Triangular matrix. --- Vector bundle. --- Volume form. --- Weyl group.

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This volume offers a systematic treatment of certain basic parts of algebraic geometry, presented from the analytic and algebraic points of view. The notes focus on comparison theorems between the algebraic, analytic, and continuous categories.Contents include: 1.1 sheaf theory, ringed spaces; 1.2 local structure of analytic and algebraic sets; 1.3 Pn 2.1 sheaves of modules; 2.2 vector bundles; 2.3 sheaf cohomology and computations on Pn; 3.1 maximum principle and Schwarz lemma on analytic spaces; 3.2 Siegel's theorem; 3.3 Chow's theorem; 4.1 GAGA; 5.1 line bundles, divisors, and maps to Pn; 5.2 Grassmanians and vector bundles; 5.3 Chern classes and curvature; 5.4 analytic cocycles; 6.1 K-theory and Bott periodicity; 6.2 K-theory as a generalized cohomology theory; 7.1 the Chern character and obstruction theory; 7.2 the Atiyah-Hirzebruch spectral sequence; 7.3 K-theory on algebraic varieties; 8.1 Stein manifold theory; 8.2 holomorphic vector bundles on polydisks; 9.1 concluding remarks; bibliography.Originally published in 1974.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.

Geometry, Algebraic --- Geometry, Analytic --- Additive map. --- Affine variety. --- Algebraic curve. --- Algebraic geometry. --- Algebraic operation. --- Algebraic surface. --- Algebraic variety. --- Analytic continuation. --- Analytic set. --- Analytic space. --- Atiyah–Hirzebruch spectral sequence. --- Automorphism. --- Bott periodicity theorem. --- Cauchy's integral formula. --- Chern class. --- Chow's theorem. --- Codimension. --- Coherence condition. --- Cohomology operation. --- Cohomology ring. --- Cohomology. --- Cokernel. --- Commutative diagram. --- Comparison theorem. --- Complex manifold. --- Complex vector bundle. --- De Rham cohomology. --- Diagram (category theory). --- Differentiable manifold. --- Differential form. --- Differential geometry. --- Differential topology. --- Dimension (vector space). --- Divisor (algebraic geometry). --- Divisor. --- Duality (mathematics). --- Epimorphism. --- Equation. --- Exact sequence. --- Fiber bundle. --- Fundamental class. --- General linear group. --- Grassmannian. --- Group homomorphism. --- Hilbert's syzygy theorem. --- Holomorphic function. --- Holomorphic sheaf. --- Holomorphic vector bundle. --- Homomorphism. --- Homotopy group. --- Homotopy. --- Irreducibility (mathematics). --- Isomorphism class. --- Jacobian matrix and determinant. --- K-theory. --- Laurent series. --- Lie derivative. --- Line bundle. --- Linear map. --- Manifold. --- Mathematical induction. --- Measure (mathematics). --- Meromorphic function. --- Module (mathematics). --- Morphism. --- Neighbourhood (mathematics). --- Obstruction theory. --- Polynomial. --- Power series. --- Presheaf (category theory). --- Principal bundle. --- Product topology. --- Projective space. --- Projective variety. --- Resolution of singularities. --- Riemann surface. --- Ring (mathematics). --- Ring homomorphism. --- Schwarz lemma. --- Sheaf (mathematics). --- Sheaf cohomology. --- Sheaf of modules. --- Simplicial complex. --- Special case. --- Spectral sequence. --- Stein manifold. --- Submanifold. --- Subset. --- Surjective function. --- Tangent bundle. --- Tangent space. --- Tensor algebra. --- Theorem. --- Topological space. --- Topology. --- Transcendence degree. --- Variable (mathematics). --- Vector bundle. --- Weierstrass preparation theorem. --- Zariski topology.

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This work deals with an extension of the classical Littlewood-Paley theory in the context of symmetric diffusion semigroups. In this general setting there are applications to a variety of problems, such as those arising in the study of the expansions coming from second order elliptic operators. A review of background material in Lie groups and martingale theory is included to make the monograph more accessible to the student.

Harmonic analysis. Fourier analysis --- Harmonic analysis --- Semigroups --- 517.986.6 --- Lie groups --- Littlewood-Paley theory --- #WWIS:d.d. Prof. L. Bouckaert/BOUC --- Fourier analysis --- Functions of several real variables --- Group theory --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Harmonic analysis of functions of groups and homogeneous spaces --- Harmonic analysis. --- Littlewood-Paley theory. --- Lie groups. --- Semigroups. --- 517.986.6 Harmonic analysis of functions of groups and homogeneous spaces --- Addition. --- Analytic function. --- Axiom. --- Boundary value problem. --- Central limit theorem. --- Change of variables. --- Circle group. --- Classification theorem. --- Commutative property. --- Compact group. --- Complex analysis. --- Convex set. --- Coset. --- Covering space. --- Derivative. --- Differentiable manifold. --- Differential geometry. --- Differential operator. --- Dimension (vector space). --- Dimension. --- Direct sum. --- E6 (mathematics). --- E7 (mathematics). --- E8 (mathematics). --- Elementary proof. --- Equation. --- Equivalence class. --- Existence theorem. --- Existential quantification. --- Fourier analysis. --- Fourier series. --- Fourier transform. --- Function space. --- General linear group. --- Haar measure. --- Harmonic function. --- Hermite polynomials. --- Hilbert transform. --- Homogeneous space. --- Homomorphism. --- Ideal (ring theory). --- Identity matrix. --- Indecomposability. --- Integral transform. --- Invariant measure. --- Invariant subspace. --- Irreducibility (mathematics). --- Irreducible representation. --- Lebesgue measure. --- Legendre polynomials. --- Lie algebra. --- Lie group. --- Linear combination. --- Linear map. --- Local diffeomorphism. --- Markov process. --- Martingale (probability theory). --- Matrix group. --- Measurable function. --- Measure (mathematics). --- Multiple integral. --- Normal subgroup. --- One-dimensional space. --- Open set. --- Ordinary differential equation. --- Orthogonality. --- Orthonormality. --- Parseval's theorem. --- Partial differential equation. --- Probability space. --- Quadratic form. --- Rank of a group. --- Regular representation. --- Riemannian manifold. --- Riesz transform. --- Schur orthogonality relations. --- Scientific notation. --- Semigroup. --- Sequence. --- Special case. --- Stone–Weierstrass theorem. --- Sturm–Liouville theory. --- Subgroup. --- Subset. --- Summation. --- Tensor algebra. --- Tensor product. --- Theorem. --- Theory. --- Topological group. --- Topological space. --- Torus. --- Trigonometric polynomial. --- Trivial representation. --- Uniform convergence. --- Unitary operator. --- Unitary representation. --- Vector field. --- Vector space. --- Lie, Groupes de --- Analyse harmonique