Principles of advanced mathematical physics

著者

• Richtmyer, Robert D

書誌事項

Principles of advanced mathematical physics

Robert D. Richtmyer

（Texts and monographs in physics）

Springer Verlag, c1978-1981

• v. 1 : gw
• v. 1 : us
• v. 2 : gw
• v. 2 : us

注記

Includes bibliographies and indexes

内容説明・目次

巻冊次

v. 1 : gw ISBN 9783540088738

内容説明

A first consequence of this difference in texture concerns the attitude we must take toward some (or perhaps most) investigations in "applied mathe- matics," at least when the mathematics is applied to physics. Namely, those investigations have to be regarded as pure mathematics and evaluated as such. For example, some of my mathematical colleagues have worked in recent years on the Hartree-Fock approximate method for determining the structures of many-electron atoms and ions. When the method was intro- duced, nearly fifty years ago, physicists did the best they could to justify it, using variational principles, intuition, and other techniques within the texture of physical reasoning. By now the method has long since become part of the established structure of physics. The mathematical theorems that can be proved now (mostly for two- and three-electron systems, hence of limited interest for physics), have to be regarded as mathematics. If they are good mathematics (and I believe they are), that is justification enough. If they are not, there is no basis for saying that the work is being done to help the physicists. In that sense, applied mathematics plays no role in today's physics. In today's division of labor, the task of the mathematician is to create mathematics, in whatever area, without being much concerned about how the mathematics is used; that should be decided in the future and by physics.

目次

• 1 Hilbert Spaces.- 1.1 Review of pertinent facts about matrices and finite-dimensional spaces.- 1.2 Linear spaces
• normed linear spaces.- 1.3 Hilbert space: axioms and elementary consequences.- 1.4 Examples of Hilbert spaces.- 1.5 Cardinal numbers
• separability
• dimension.- 1.6 Orthonormal sequences.- 1.7 Subspaces
• the projection theorem.- 1.8 Linear functional
• the Riesz-Frechet representation theorem.- 1.9 Strong and weak convergence.- 1.10 Hilbert spaces of analytic functions.- 1.11 Polarization.- 2 Distributions
• General Properties.- 2.1 Origin of the distribution concept.- 2.2 Classes of test functions
• functions of type (Math).- 2.3 Notations for distributions
• the bilinear form.- 2.4 The formal definition
• the continuity of the functionals.- 2.5 Examples of distributions.- 2.6 Distributions as limits of sequences of functions
• convergence of distributions.- 2.7 Differentiation and integration.- 2.8 Change of independent variable
• symmetries.- 2.9 Restrictions, limitations, and warnings.- 2.10 Regularization.- Appendix: A discontinuous linear functional.- 3 Local Properties of Distributions.- 3.1 Quick review of open and closed sets in ?n.- 3.2 Local properties defined.- 3.3 A theorem on open coverings.- 3.4 Theorems on test functions
• partitions of unity.- 3.5 The main theorems on local properties.- 3.6 The support of a distribution.- 4 Tempered Distributions and Fourier Transforms.- 4.1 The space S.- 4.2 Tempered distributions.- 4.3 Growth at infinity.- 4.4 Fourier transformation in S.- 4.5 Fourier transforms of tempered distributions.- 4.6 The power spectrum.- 5 L2 Spaces.- 5.1 Mean convergence
• completeness of function systems.- 5.2 A physical example of approximation in the mean.- 5.3 The spaces L2(?n) and L2(?).- 5.4 Multiplication in L2 spaces.- 5.5 Integration in L2 spaces
• definite integrals.- 5.6 On vanishing at infinity I.- 5.7 Spaces of type L1, Lp, L?.- 5.8 Fourier transforms in L1
• Riemann-Lebesgue Lemma
• Luzin's theorem.- 5.9 Spaces of type (Math).- 5.10 Fourier transforms and mollifiers in L2 spaces.- 5.11 The Sobolev spaces
• the space W1.- 5.12 Boundary values in W1
• the subspace W01.- 5.13 On vanishing at infinity II.- 6 Some Problems Connected with the Laplacian.- 6.1 The potential
• Poisson's equation.- 6.2 Convolutions.- 6.3 Proof of Poisson's equation.- 6.4 The classical potential-theory problems of Poisson, Dirichlet, Green, and Neumann 1..- 6.5 Schwartz's nuclear theorem
• the direct product f(x)g(y).- 6.6 The variational method for the eigenfunctions of the Laplacian.- 6.7 A compactness theorem for the Sobolev space W1.- 6.8 Existence of the eigenfunctions.- 6.9 A problem from hydrodynamical stability
• irrotational and solenoidal vector fields.- 6.10 The Cauchy-Riemann equations
• harmonic distributions.- 7 Linear Operators in a Hilbert Space.- 7.1 Linear operators.- 7.2 Adjoints
• self-adjoint and unitary operators.- 7.3 Examples in l2.- 7.4 Integral operators in L2 (a, b).- 7.5 Differential operators via distribution theory.- 7.6 Closed operators.- 7.7 The graph of an operator
• range and nullspace.- 7.8 The radial momentum operators.- 7.9 Positive operators
• numerical range.- 8 Spectrum and Resolvent.- 8.1 Definitions.- 8.2 Examples and exercises.- 8.3 Spectra of symmetric, self-adjoint, and unitary operators.- 8.4 Modification of the spectrum when an operator is extended.- 8.5 Analytic properties of the resolvent.- 8.6 Extension of a symmetric operator
• deficiency indices
• the Cayley transform
• second definition of self-adjointness.- 9 Spectral Decomposition of Self-Adjoint and Unitary Operators.- 9.1 Spectral decompositions of a Hermitian matrix.- 9.2 Projectors in a Hilbert space ?.- 9.3 Construction of the spectral projectors for a matrix.- 9.4 Connection with analytic functions.- 9.5 Functions and distributions as boundary values of analytic functions.- 9.6 The resolution of the identity for a self-adjoint operator.- 9.7 The properties of the operators Et.- 9.8 The canonical representation of a self-adjoint operator.- 9.9 Modes of convergence of bounded operators
• connection between the continuity properties of Et and the spectrum of A.- 9.10 Unitary operators
• functions of operators
• bounded observables
• polar decomposition.- Appendix A: The properties of the operators Et.- Appendix B: The canonical representations of a self-adjoint operator.- 10 Ordinary Differential Operators.- 10.1 Resolvent and spectral family for the operator -id/dx.- 10.2 Resolvent and spectral family for the operator -(d/dx)2.- 10.3 The Fourier transform method.- 10.4 Regular Sturm-Liouville operator.- 10.5 Existence and uniqueness of the solution
• the integral equation
• the eigenfunctions.- 10.6 The resolvent
• the Green's function
• completeness of the eigenfunctions.- 10.7 More general boundary conditions.- 10.8 Sturm-Liouville operator with one singular endpoint.- 10.9 The boundary condition at a singular endpoint.- 10.10 Regular singular point
• method of Frobenius.- 10.11 Self-adjoint extension of T in the limit-point case.- 10.12 The eigenfunction expansion.- 10.13 The limit-circle case.- 10.14 Case of two singular endpoints.- 10.15 Bessel's equation.- 10.16 The nonrelativistic hydrogen-like atom.- 10.17 The relativistic hydrogen-like atom.- 11 Some Partial Differential Operators of Quantum Mechanics.- 11.1 Self-adjoint Laplacian in ?n.- 11.2 Resolvent, spectrum, and spectral projectors.- 11.3 Schrodinger operators.- 11.4 Perturbation of the spectrum
• essential spectrum
• absolutely continuous spectrum.- 11.5 Continuous spectrum in the sense of Hilbert
• continuous and absolutely continuous subspaces.- 11.6 Dirac Hamiltonians.- 11.7 The Laplacian in a bounded region.- 12 Compact, Hilbert-Schmidt, and Trace-Class Operators.- 12.1 Some properties of matrices.- 12.2 Compact operators.- 12.3 Hilbert-Schmidt and trace-class operators.- 12.4 Hilbert-Schmidt integral operators.- 12.5 Operators with compact resolvent.- 13 Probability
• Measure.- 13.1 Univariate or one-dimensional probability distributions: cumulative probability
• density.- 13.2 Means and expectations.- 13.3 Bivariate and multivariate distributions
• nondecreasing functions of several variables.- 13.4 The normal distributions.- 13.5 The central limit theorem.- 13.6 Sampling.- 13.7 Marginal and conditional probabilities.- 13.8 Simulation
• the Monte Carlo Method.- 13.9 Measures.- 13.10 Measures as set functions.- 13.11 Probability in Hilbert space
• cylinder sets
• Gaussian measures.- Appendix: Functions of Bounded Variation.- 14 Probability and Operators in Quantum Mechanics.- 14.1 States of a system
• observables.- 14.2 Probabilities-a finite model.- 14.3 Probabilities-the general case (? infinite-dimensional).- 14.4 Expectations
• the domain of A.- 14.5 The density matrix.- 14.6 Algebras of bounded operators
• canonical commutation relations.- 14.7 Self-adjoint operator with a simple spectrum.- 14.8 Spectral representation of ? for a self-adjoint operator with a simple spectrum.- 14.9 Complete set of commuting observables.- 15 Problems of Evolution
• Banach Spaces.- 15.1 Initial-value problems in mechanics.- 15.2 Initial-value problems of heat flow.- 15.3 Well- and ill-posed problems.- 15.4 The initial-value problem of wave motion.- 15.5 The function space (state space) of an initial-value problem.- 15.6 Completeness of the state space
• Banach space.- 15.7 Examples of Banach spaces.- 15.8 Inequivalence of various Banach spaces.- 15.9 Linear operators.- 15.10 Linear functionals
• the dual space.- 15.11 Convergence of vectors and operators.- 15.12 Inner product
• Hilbert space.- 15.13 Relativistic problems.- 15.14 Seminorms.- 16 Well-Posed Initial-Value Problems
• Semigroups.- 16.1 Banach-space formulation of an initial-value problem.- 16.2 Well-posed problem
• generalized solutions.- 16.3 Wave motion.- 16.4 The Schrodinger equation.- 16.5 Maxwell's equations in empty space.- 16.6 Semigroups.- 16.7 The infinitesimal generator of a semigroup.- 16.8 The Hille-Yosida theorem.- 16.9 Neutron transport in a slab
• an application of the Hille-Yosida theorem.- 16.10 Inhomogeneous problems.- 16.11 Problems in which the operator is time-dependent.- 17 Nonlinear Problems
• Fluid Dynamics.- 17.1 Wave propagation.- 17.2 Fluid-dynamical conservation laws.- 17.3 Weak solutions.- 17.4 The jump conditions.- 17.5 Shocks and slip surfaces.- 17.6 Instability of negative shocks.- 17.7 Sound waves and characteristics in one dimension.- 17.8 Hyperbolic systems.- 17.9 Fluid-dynamical equations in characteristic form.- 17.10 Remarks on the initial-value problem.- 17.11 Flow of information along the characteristics in one dimension.- 17.12 Characteristics in several dimensions
• the Cauchy-Kovalevski theorem.- 17.13 The Riemann problem and its generalizations.- 17.14 The spontaneous generation of shocks.- 17.15 Helmholtz and Taylor instabilities.- 17.16 A conjecture on piecewise analytic initial-value problems of fluid dynamics.- 17.17 Singularities of flows.- Appendix: The detached shock problem: 17.A The Problem.- 17.B Ill-posedness of the problem.- 17.C The power series method.- 17.D Significance arithmetic.- 17.E Analytic continuation.- References.

巻冊次

v. 2 : gw ISBN 9783540107729

目次

• 18 Elementary Group Theory.- 18.1 The group axioms
• examples.- 18.2 Elementary consequences of the axioms
• further definitions.- 18.3 Isomorphism.- 18.4 Permutation groups.- 18.5 Homomorphisms
• normal subgroups.- 18.6 Cosets.- 18.7 Factor groups.- 18.8 The Law of Homomorphism.- 18.9 The structure of cyclic groups.- 18.10 Translations, inner automorphisms.- 18.11 The subgroups of ?4.- 18.12 Generators and relations
• free groups.- 18.13 Multiply periodic functions and crystals.- 18.14 The space and point groups.- 18.15 Direct and semidirect products of groups
• symmorphic space groups.- 19 Continuous Groups.- 19.1 Orthogonal and rotation groups.- 19.2 The rotation group SO(3)
• Euler's theorem.- 19.3 Unitary groups.- 19.4 The Lorentz groups.- 19.5 Group manifolds.- 19.6 Intrinsic coordinates in the manifold of the rotation group.- 19.7 The homomorphism of SU(2) onto SO(3).- 19.8 The homomorphism of SL(2, ?) onto the proper Lorentz group ? p.- 19.9 Simplicity of the rotation and Lorentz groups.- 20 Group Representations I: Rotations and Spherical Harmonics.- 20.1 Finite-dimensional representations of a group.- 20.2 Vector and tensor transformation laws.- 20.3 Other group representations in physics.- 20.4 Infinite-dimensional representations.- 20.5 A simple case: SO(2).- 20.6 Representations of matrix groups on X?.- 20.7 Homogeneous spaces.- 20.8 Regular representations.- 20.9 Representations of the rotation group SO(3).- 20.10 Tesseral harmonics
• Legendre functions.- 20.11 Associated Legendre functions.- 20.12 Matrices of the irreducible representations of SO(3)
• the Euler angles.- 20.13 The addition theorem for tesseral harmonics.- 20.14 Completeness of the tesseral harmonics.- 21 Group Representations II: General
• Rigid Motions
• Bessel Functions.- 21.1 Equivalence
• unitary representations.- 21.2 The reduction of representations.- 21.3 Schur's Lemma and its corollaries.- 21.4 Compact and noncompact groups.- 21.5 Invariant integration
• Haar measure.- 21.6 Complete system of representations of a compact group.- 21.7 Homogeneous spaces as configuration spaces in physics.- 21.8 M2 and related groups.- 21.9 Representations of M2.- 21.10 Some irreducible representations.- 21.11 Bessel functions.- 21.12 Matrices of the representations.- 21.13 Characters.- 22 Group Representations and Quantum Mechanics.- 22.1 Representations in quantum mechanics.- 22.2 Rotations of the axes.- 22.3 Ray representations.- 22.4 A finite-dimensional case.- 22.5 Local representations.- 22.6 Origin of the two-valued representations.- 22.7 Representations of SU(2) and SL(2, ?).- 22.8 Irreducible representations of SU(2).- 22.9 The characters of SU(2).- 22.10 Functions of z and z?.- 22.11 The finite-dimensional representations of SL(2, ?).- 22.12 The irreducible invariant subspaces of X? for SL(2, ?).- 22.13 Spinors.- 23 Elementary Theory of Manifolds.- 23.1 Examples of manifolds
• method of identification.- 23.2 Coordinate systems or charts
• compatibility
• smoothness.- 23.3 Induced topology.- 23.4 Definition of manifold
• Hausdorff separation axiom.- 23.5 Curves and functions in a manifold.- 23.6 Connectedness
• components of a manifold.- 23.7 Global topology
• homotopic curves
• fundamental group.- 23.8 Mechanical linkages: Cartesian products.- 24 Covering Manifolds.- 24.1 Definition and examples.- 24.2 Principles of lifting.- 24.3 Universal covering manifold.- 24.4 Comments on the construction of mathematical models.- 24.5 Construction of the universal covering.- 24.6 Manifolds covered by a given manifold.- 25 Lie Groups.- 25.1 Definitions and statement of objectives.- 25.2 The expansions of m(*, *) and l(*, *).- 25.3 The Lie algebra of a Lie group.- 25.4 Abstract Lie algebras.- 25.5 The Lie algebras of linear groups.- 25.6 The exponential mapping
• logarithmic coordinates.- 25.7 An auxiliary lemma on inner automorphisms
• the mappings Ad?.- 25.8 Auxiliary lemmas on formal derivatives.- 25.9 An auxiliary lemma on the differentiation of exponentials.- 25.10 The Campbell-Baker-Hausdorf (CBH) formula.- 25.11 Translation of charts
• compatibility
• G as an analytic manifold.- 25.12 Lie algebra homomorphisms.- 25.13 Lie group homomorphisms.- 25.14 Law of homomorphism for Lie groups.- 25.15 Direct and semidirect sums of Lie algebras.- 25.16 Classification of the simple complex Lie algebras.- 25.17 Models of the simple complex Lie algebras.- 25.18 Note on Lie groups and Lie algebras in physics.- Appendix to Chapter 25-s theorem.- 19.3 Unitary groups.- 19.4 The Lorentz groups.- 19.5 Group manifolds.- 19.6 Intrinsic coordinates in the manifold of the rotation group.- 19.7 The homomorphism of SU(2) onto SO(3).- 19.8 The homomorphism of SL(2, ?) onto the proper Lorentz group ? p.- 19.9 Simplicity of the rotation and Lorentz groups.- 20 Group Representations I: Rotations and Spherical Harmonics.- 20.1 Finite-dimensional representations of a group.- 20.2 Vector and tensor transformation laws.- 20.3 Other group representations in physics.- 20.4 Infinite-dimensional representations.- 20.5 A simple case: SO(2).- 20.6 Representations of matrix groups on X?.- 20.7 Homogeneous spaces.- 20.8 Regular representations.- 20.9 Representations of the rotation group SO(3).- 20.10 Tesseral harmonics
• Legendre functions.- 20.11 Associated Legendre functions.- 20.12 Matrices of the irreducible representations of SO(3)
• the Euler angles.- 20.13 The addition theorem for tesseral harmonics.- 20.14 Completeness of the tesseral harmonics.- 21 Group Representations II: General
• Rigid Motions
• Bessel Functions.- 21.1 Equivalence
• unitary representations.- 21.2 The reduction of representations.- 21.3 Schur's Lemma and its corollaries.- 21.4 Compact and noncompact groups.- 21.5 Invariant integration
• Haar measure.- 21.6 Complete system of representations of a compact group.- 21.7 Homogeneous spaces as configuration spaces in physics.- 21.8 M2 and related groups.- 21.9 Representations of M2.- 21.10 Some irreducible representations.- 21.11 Bessel functions.- 21.12 Matrices of the representations.- 21.13 Characters.- 22 Group Representations and Quantum Mechanics.- 22.1 Representations in quantum mechanics.- 22.2 Rotations of the axes.- 22.3 Ray representations.- 22.4 A finite-dimensional case.- 22.5 Local representations.- 22.6 Origin of the two-valued representations.- 22.7 Representations of SU(2) and SL(2, ?).- 22.8 Irreducible representations of SU(2).- 22.9 The characters of SU(2).- 22.10 Functions of z and z?.- 22.11 The finite-dimensional representations of SL(2, ?).- 22.12 The irreducible invariant subspaces of X? for SL(2, ?).- 22.13 Spinors.- 23 Elementary Theory of Manifolds.- 23.1 Examples of manifolds
• method of identification.- 23.2 Coordinate systems or charts
• compatibility
• smoothness.- 23.3 Induced topology.- 23.4 Definition of manifold
• Hausdorff separation axiom.- 23.5 Curves and functions in a manifold.- 23.6 Connectedness
• components of a manifold.- 23.7 Global topology
• homotopic curves
• fundamental group.- 23.8 Mechanical linkages: Cartesian products.- 24 Covering Manifolds.- 24.1 Definition and examples.- 24.2 Principles of lifting.- 24.3 Universal covering manifold.- 24.4 Comments on the construction of mathematical models.- 24.5 Construction of the universal covering.- 24.6 Manifolds covered by a given manifold.- 25 Lie Groups.- 25.1 Definitions and statement of objectives.- 25.2 The expansions of m(*, *) and l(*, *).- 25.3 The Lie algebra of a Lie group.- 25.4 Abstract Lie algebras.- 25.5 The Lie algebras of linear groups.- 25.6 The exponential mapping
• logarithmic coordinates.- 25.7 An auxiliary lemma on inner automorphisms
• the mappings Ad?.- 25.8 Auxiliary lemmas on formal derivatives.- 25.9 An auxiliary lemma on the differentiation of exponentials.- 25.10 The Campbell-Baker-Hausdorf (CBH) formula.- 25.11 Translation of charts
• compatibility
• G as an analytic manifold.- 25.12 Lie algebra homomorphisms.- 25.13 Lie group homomorphisms.- 25.14 Law of homomorphism for Lie groups.- 25.15 Direct and semidirect sums of Lie algebras.- 25.16 Classification of the simple complex Lie algebras.- 25.17 Models of the simple complex Lie algebras.- 25.18 Note on Lie groups and Lie algebras in physics.- Appendix to Chapter 25-Two nonlinear Lie groups.- 26 Metric and Geodesics on a Manifold.- 26.1 Scalar and vector fields on a manifold.- 26.2 Tensor fields.- 26.3 Metric in Euclidean space.- 26.4 Riemannian and pseudo-Riemannian manifolds.- 26.5 Raising and lowering of indices.- 26.6 Geodesies in a Riemannian manifold.- 26.7 Geodesies in a pseudo-Riamannian manifold M.- 26.8 Geodesies
• the initial-value problem
• the Lipschitz condition.- 26.9 The integral equation
• Picard iterations.- 26.10 Geodesies
• the two-point problem.- 26.11 Continuation of geodesies.- 26.12 Affmely connected manifolds.- 26.13 Riemannian and pseudo-Riemannian covering manifolds.- 27 Riemannian, Pseudo-Riemannian, and Affinely Connected Manifolds.- 27.1 Topology and metric.- 27.2 Geodesic or Riemannian coordinates.- 27.3 Normal coordinates in Riemannian and pseudo-Riemannian manifolds.- 27.4 Geometric concepts
• principle of equivalence.- 27.5 Covariant differentiation.- 27.6 Absolute differentiation along a curve.- 27.7 Parallel transport.- 27.8 Orientability.- 27.9 The Riemann tensor, general
• Laplacian and d'Alembertian.- 27.10 The Riemann tensor in a Riemannian or pseudo-Riemannian manifold.- 27.11 The Riemann tensor and the intrinsic curvature of a manifold.- 27.12 Flatness and the vanishing of the Riemann tensor.- 27.13 Eisenhart's analysis of the Stackel systems.- 28 The Extension of Einstein Manifolds.- 28.1 Special relativity.- 28.2 The Einstein gravitational field equations.- 28.3 The Schwarzschild charts.- 28.4 The Finkelstein extensions of the Schwarzschild charts.- 28.5 The Kruskal extension.- 28.6 Maximal extensions
• geodesic completeness.- 28.7 Other extensions of the Schwarzschild manifolds.- 28.8 The Kerr manifolds.- 28.9 The Cauchy problem.- 28.10 Concluding remarks.- 29 Bifurcations in Hydrodynamic Stability Problems.- 29.1 The classical problems of hydrodynamic stability.- 29.2 Examples of bifurcations in hydrodynamics.- 29.3 The Navier-Stokes equations.- 29.4 Hilbert space formulation.- 29.5 The initial-value problem
• the semiflow in ?.- 29.6 The normal modes.- 29.7 Reduction to a finite-dimensional dynamical system.- 29.8 Bifurcation to a new steady state.- 29.9 Bifurcation to a periodic orbit.- 29.10 Bifurcation from a periodic orbit to an invariant torus.- 29.11 Subharmonic bifurcation.- Appendix to Chapter 29-Computational details for the invariant torus.- 30 Invariant Manifolds in the Taylor Problem.- 30.1 Survey of the Taylor problem to 1968.- 30.2 Calculation of invariant manifolds.- 30.3 Cylindrical coordinates.- 30.4 The Hilbert space.- 30.5 Separation of variables in cylindrical coordinates.- 30.6 Results to date for the Taylor problem.- Appendix to Chapter 30-The matrices in Eagles' formulation.- 31 The Early Onset of Turbulence.- 31.1 The Landau-Hopf model.- 31.2 The Hopf example.- 31.3 The Ruelle-Takens model.- 31.4 The co-limit set of a motion.- 31.5 Attractors.- 31.6 The power spectrum for motions in ?n.- 31.7 Almost periodic and aperiodic motions.- 31.8 Lyapounov stability.- 31.9 The Lorenz system
• the bifurcations.- 31.10 The Lorenz attractor
• general description.- 31.11 The Lorenz attractor
• aperiodic motions.- 31.12 Statistics of the mapping f and g.- 31.13 The Lorenz attractor
• detailed structure I.- 31.14 The symbols [i, j] of Williams.- 31.15 Prehistories.- 31.16 The Lorenz attractor
• detailed structure II.- 31.17 Existence of 1-cells in F.- 31.18 Bifurcation to a strange attractor.- 31.19 The Feigenbaum model.- Appendix to Chapter 31 (Parts A-H)-Generic properties of systems:.- 31.A Spaces of systems.- 31.B Absence of Lebesgue measure in a Hilbert space.- 31.C Generic properties of systems.- 31.D Strongly generic
• physical interpretation.- 31.E Peixoto's theorem.- 31.F Other examples of generic and nongeneric properties.- 31.G Lack of correspondence between genericity and Lebesgue measure 308 31.H Probability and physics.- References.

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詳細情報

• NII書誌ID(NCID): BA03452727
• ISBN:
  • 3540088733
  • 0387088733
  • 354010772X
  • 038710772X
• LCCN: 78016494
• 出版国コード: us
• タイトル言語コード: eng
• 本文言語コード: eng
• 出版地: New York ; Tokyo
• ページ数/冊数: 2 v.
• 大きさ: 25 cm
• 分類:
  • LCC : QC20
  • DC : 530.1/5
• 件名:
  • LCSH : Mathematical physics
• 親書誌ID:
  • BA00216852