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Guillemin V., Sternberg S. — Geometric Asymptotics

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Название: Geometric Asymptotics

Авторы: Guillemin V., Sternberg S.

Аннотация:

Book Description
Symplectic geometry and the theory of Fourier integral operators are modern manifestations of themes that have occupied a central position in mathematical thought for the past three hundred years — the relations between the wave and the corpuscular theories of light. The purpose of this book is to develop these themes, and present some of the recent advances, using the language of differential geometry as a unifying influence.

Chapters included in this book are: Chapter I, Introduction. The method of stationary phase; Appendix I, Morse's lemma and some generalizations; Chapter II, Differential operators and asymptotic solutions; Chapter III, Geometrical optics; Chapter IV, Symplectic geometry; Chapter V, Geometric quantization; Chapter VI, Geometric aspects of distribution; Appendix to Chapter VI, The Plancherel formula for the complex semisimple Lie groups; Chapter VII, Compound Asymptotics; Appendix II, Various functorial constructions; Index.

Язык:

Рубрика: Математика/Анализ/Асимптотические методы, Теория возмущений/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: revised

Год издания: 1990

Количество страниц: 480

Добавлена в каталог: 23.11.2004

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Предметный указатель

      198
singularities, elliptic      449
singularities, hyperbolic      449
singularities, parabolic      449
Abbe sine conditions      83
Aberrations      96
Aberrations spherical      78
Admissible filtration      430
Airy function, elliptic      466
Airy function, generalized      462
Airy function, parabolic      466
Angle characteristic      88
Astigmatism      99
Asymptotic differential equation      29
Asymptotic differential operator      29
Asymptotic expansion      28
Asymptotic number      28
Asymptotic operator      29
Asymptotic sections      27
Asymptotic vector      28
Asymptotics, compound      400
Asymptotics, simple      30
Atiyah — Bott fixed point formula      324
Backward wave, nonexistence of      13
Barrel distortion      100
Basic formula of differential calculus      110
Bergmann kernel      ix
Bicharacteristic field      43
Blattner, Bob      xiii
Boundary layer      441 467
Bundle isotropy      316
Bundle line      219
Bundle, Maslov      145
Canonical relations      149
Canonical relations, composition of      149
Cartan decomposition      267
Cartan involution      267
Character formula      321
Characteristic angle      88
Characteristic equation      31
Characteristic mixed      88 96
Characteristic point      87
Characteristic variety      35
Charge density      102
Coadjoint representation      176
Complex polarization      233
Compound asymptotic      400
Compound asymptotic, associated to $\Lambda$       408
Compound asymptotic, associated with $(\Lambda,\varphi)$       422
Compound asymptotic, order of a      419
Compound asymptotic, symbol of a      416
Conformal symplectic algebra      116
Conformal symplectic group      115
Conformal transformation      104
Current density      103
Curvature of the field      99
Darboux — Weinstein theorem      109
Decomposition Cartan      267
Decomposition Iwasawa      321 392
Decomposition Plancherel      390
Decomposition, polar      266 269
Delta section      309 310
Delta section, pullback of a      310
Delta section, pushforward of a      311
density      53
Density, charge      102
Density, current      103
Density, generalized      306
dielectric constant      105
Dielectric displacement      103
Differential calculus basic, formula of      110
Differential operator      22
Differential operator, asymptotic      29
Differential operator, symbol of a      22
Discontinuities, propagation of      33
Distortion      99
Distortion, barrel      100
distribution      306
Distribution, support of a      306
Double fibration      340
Double mirror experiment      1
Eikonal equation      34 72
electric field      103
Elementary particles      183 195
Elementary symplectic action      183
Elliptic singularities      449
Elliptic Airy function      466
Envelope      150
Exponential map      37
Fermat’s principle      72
Feyman integrals      72
Fibration, double      341
Field, bicharacteristic      43
Field, curvature of the      99
Field, electric      103
Filtration, admissible      430
First fundamental form      10
First order optics      84
Fixed point formula      314 320
Fixed point formula, Atiyah — Bott      324
Fixed point formula, Lefschetz      324
Flat connection      217
Focal points      37
Focus, perfect      79
Focus, strong      79
Frequency set      399
Fresnel      1
Frobenius reciprocity formula      323
Fundamental forms, first      10
Fundamental forms, of a surface      7
Fundamental forms, second      10
Gaussian optics      84 90
Gelfand — Naimark formula      322
Generalized Airy function      462
Generalized density      306
Generalized section      307
Generalized trace      318
Golubitsky, Marthy      xiii
Gouy      1 15
Guckenheimer, John      xiii
Half form      251 258
Hamilton — Cartan formalism      206

Hamiltonian action      183
Hannabus, Keith      56
Helmholtz formula      11
Hilbert transform      337
Homogeneous generalized functions      345
Homogeneous lagrangian, submanifold      36
Hormander cross index      128
Huygens principle      153 332 335
Hyperbolic singularities      449
Integrable submanifold of a symplectic manifolde      39
Intrinsic derivative      41
Invariant polarization      235
Isotropy bundle      316
Iwasawa decomposition      321 392
Kac, Mark      xi
Kahler manifold      233
Kahler type, polarization of      234
Kepler problem      171
Kostant, Bertram      xiii
Lagrangian pair      415
Lagrangian subspace      116
Langrangian submanifold      35
Langrangian submanifold, homogeneous      36
Langrangian submanifold, pullback of a      150
Langrangian submanifold, pushforward of a      152
Lefschetz fixed point theorem      324
Legendre relation      404
Legendre transformation      403
Leray class      130
Line bundle      219
Linear connection      215
Locally transitive action      316
Luneberg’s algorithm      78
Magnetic excitation      103
Magnetic induction      103
Magnetic permeability      105
Malgrange preparation theorem      456
Manifolds, Kahler      233
Manifolds, metalinear      253 258
Maslov bundle      145
Maslov canonical operator      64
Maslov class      63 122 129 130
Maslov cycle      129
Maslov index      125 126
Metalinear manifolds      253 258
Metaplectic structure      263
Milnor condition      19
Mixed characteristic      88 96
Morse’s Lemma      16
Optical length      72 79
Optics, first order      84
Optics, Gaussian      84 90
Order of a compound asymptotic      419
Parabolic singularities      449
Parabolic Airy function      466
Parallel translation      21
Perfect focus      79
Peter — Weyl theorem      391
Phase function      156
Phase function, reduced      158
Phase shift      1
Plancherel decomposition of       390
Plancherel formula      389 396
Plancherel measure      389
Point characteristic      87
Poisson bracket      38
Polar decomposition      266 269
Polarization, complex      233
Polarization, invariant      235
Polarization, Kahler type      234
Principal symbol      374
Projective wave front set      326
Propagation of discontinuities      33
Pullback of a delta section      311
Pullback of a Lagrangian submanifold      148
Pullback of a wave front set      331
Pushforward of a delta section      311
Pushforward of a wave front set      331
Pushforward of Lagrangian manifolds      151
Radon transform      340 370
Reduced phase function      158
Schaeffer, Dave      xiii
Scheffe, Molly      xiii
Schrodinger equation      244
Schuster, Sir Arthur      xi
Second fundamental form      10
Second order transport equation      33
Simple asymptotics      30
Singular support      325
Snell’s law      73
Sommerfeld radiation conditions      12
Specific conductivity      105
Spherical aberation      78
Stationary phase formula      6
Stationary phase, method of      1
Strong focus      79
Subprincipal symbol      57
Support of a distribution      306
Surface, fundamental forms of a      7
Symbol of a compound asymptotic      416
Symbol of a differential operator      22
Symbol, principal      374
Symbol, subprincipal      57
Symplectic action      183
Symplectic action, elementary      183
Symplectic algebra      115
Symplectic algebra, conformal      116
Symplectic group      115
Symplectic group, conformal      115
Symplectic manifolds      37
Theta functions      ix
Time ordered exponential integral      281
Transform, Hilbert      337
Transform, Radon      340 370
Transport equation      32
Transport equation, second order      33
Transversality of a map to a subset of $PT^{\ast}$       332
Unit planes      92
Wave front set      327
Wave front set, projecgtive      326
Wave front set, pullback of a      333
Wave front set, pushforward of a      332

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