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Kashiwara M., Kawai T., Kimura T. — Foundations of Algebraic Analysis
Kashiwara M., Kawai T., Kimura T. — Foundations of Algebraic Analysis

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Название: Foundations of Algebraic Analysis

Авторы: Kashiwara M., Kawai T., Kimura T.


Prior to its founding in 1963, the Research Institute for Mathematical Sciences was the focus of divers discussions concerning goals. One of the more modest goals was to set up an institution that would create a "Courant-Hilbert" for a new age.1 Indeed, our intention here—even though this book is small in scale and only the opening chapter of our Utopian "Treatise of Analysis"—is to write just such a "Courant-Hilbert" for the new generation. Each researcher in this field may have his own definition of "algebraic analysis," a term included in the title of this book. On the other hand, algebraic analysts may well share a common attitude toward the study of analysis: the essential use of algebraic methods such as cohomology theory. This characterization is, of course, too vague: one can observe such common trends whenever analysis has made serious reformations. Professor K. Oka, for example, once spoke of the "victory of abstract algebra" in regard to his theory of ideals of undetermined domains.2 Furthermore, even Leibniz's main interest, in the early days of analysis, seems to have been in the algebraization of infinitesimal calculus. As used in the title of our book, however, "algebraic analysis" has a more special meaning, after Professor M. Sato: it is that analysis which holds onto substance and survives the shifts of fashion in the field of analysis, as Euler's mathematics, for example, has done. In this book, as the most fruitful result of our philosophy, we pay particular attention to the microlocal theory of linear partial differential equations, i.e. the new thinking on the local analysis on contangent bundles. We hope that the fundamental ideas that appear in this book will in the near future become the conventional wisdom among analysts and theoretical physicists, just as the Courant-Hilbert treatise did.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1986

Количество страниц: 254

Добавлена в каталог: 05.12.2009

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Предметный указатель
$\delta$-function      85
Analytic polyhedron      53
Antipodal mapping      40
Bicharacteristic curve      159
Bicharacteristic manifold      233
Bicharacteristic strip      159
Boundary value      80
Canonical coordinates      217
Canonical form      216
Cauchy problem      146
Cauchy — Kovalevsky’s theorem      139
Cauchy — Riemann differential equation      184
Cech cohomology group      22
Characteristic conoid      164
Characteristic variety      145
Coboundary      9
Cochain complex      9
Cocycle      9
Cohomology group      9
Cohomology group, Cech      22
Cohomology group, relative      11
Complex conjugate      182
Complexification      18
Conjugate operator      143
Conoidal neighborhood      80
Conormal bundle      35 105
Conormal sphere bundle      36
Contact structure      216
Contact transformation      218
Contact transformation having a generating function      219
Contractible      30
Convex (a subset of $\sqrt{-1}SM$)      78
Convex (a subset of $\sqrt{-1}SM$), properly convex      79
Convex hull      79
Cotangent bundle      35
Cotangential sphere bundle      35
Darboux theorem      217
De Rham system      229
Derived sheaf      13
Direct image      4
Domain of dependence      158
Domain of influence      158
Duhamel principle      140
d’Alambertian      150
Elementary solution      144
Elliptic operator      144
Exact sequence      4
Feynman diagram      131
Feynman integral      132
Fibre product      35
Finite propagation      164
Five lemma      11
Flabby dimension      16
Flabby resolution      8
Flabby resolution, canonical      8
Flabby sheaf      5
Formal norm (of a microdifferential operator)      212
Fundamental solution      144
Fundamental solution, for Cauchy problem      158
Fundamental theorem of Sato      140
Generalized Levi form      246
Generalized relative cohomology      61
Generating function      219
Hamiltonian vector field      217
Heaviside function      83
Holmgren’s uniqueness theorem      147
Holomorphic parameter      184
Holonomic system      220
Homotopic      30
Homotopy      30
Hyperbolic operator      161
Hyperfunction      19
Hyperfunction containing real holomorphic parameter      147
Hyperfunction, real-valued      183
Inductive limit      3
Initial value problem      146
Inverse image      4
Involutory      220
Lagrangian (manifold)      220
Landau — Nakanishi manifold      134
Leray covering      24
Lewy — Mizohata system      246
Lewy — Mizohata type      243
Micro-analytic      81
Microdifferential operator      195
Microdifferential operator of finite order      195
Microdifferential operator of infinite order      195
Microfunction      40
Microlocal operator      139
Morphism of complexes      9
Morphism of presheaves      4
Nine lemma      10
Non-characteristic      139
Normal bundle      35
Normal sphere bundle      36
Order, of microdifferential operator      195
Orientation      18
Orientation, sheaf      18
Poisson bracket      150 217
Poisson’s summation formula      94
Polar set      79
Positive type      91
Presheaf      3
Principal symbol      139
Principal symbol of microdifferential operator      208
Proper map      45
Properly convex      79
Purely r-codimensional map      64
Purely r-codimensional submanifold      18
Purely r-dimensional, map      40
Quantized contact transformation      221
Real monoidal transform      36
Regular (submanifold of a contact manifold)      220
Relative cohomology      11
Relative cohomology, generalized      61
Restriction (of a sheaf)      5
Section      5
Sheaf      3
Sheaf space      41
Sheafication      4
Singularity spectrum      61 71
Spath theorem (for a microdifferential operator)      214 215
Spectrum      71
Spectrum map      50
Stalk      3
Stein manifold      25
Stein neighborhood      26
Support      5
Symplectic manifold      216
Tangent bundle      35
Tangent sphere bundle      35
Weierstrass preparation theorem      91
Weierstrass preparation theorem for a microdifferential operator      215
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