Kashiwara / Kawai Algebraic Analysis

Papers Dedicated to Professor Mikio Sato on the Occasion of His Sixtieth Birthday
1. Auflage 2014
ISBN: 978-1-4832-6802-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 494 Seiten, Web PDF

Algebraic Analysis: Papers Dedicated to Professor Mikio Sato on the Occasion of his 60th Birthday, Volume I is a collection of research papers on algebraic analysis and related topics in honor to Professor Mikio Sato's 60th birthday. This volume is composed of 35 chapters and begins with papers concerning Sato's early career in algebraic analysis. The succeeding chapters deal with research works on the existence of local holomorphic solutions, the holonomic q-difference systems, partial differential equations, and the properties of solvable models. Other chapters explore the fundamentals of hypergeometric functions, the Toda lattice in the complex domain, the Lie algebras, b-functions, p-adic integrals, analytic parameters of hyperfunctions, and some applicatioins of microlocal energy methods to analytic hypoeellipticity. This volume also presents studies on the complex powers of p-adic fields, operational calculus, extensions of microfunction sheaves up to the boundary, and the irregularity of holonomic modules. The last chapters feature research works on error analysis of quadrature formulas obtained by variable transformation and the analytic functional on the complex light cone, as well as their Fourier-Borel transformations. This book will prove useful to mathematicians and advance mathematics students.

Autoren/Hrsg.

Kashiwara, Masaki

Kawai, Takahiro

Inhaltsverzeichnis

1;Front Cover;1
2;Algebraic Analysis: Papers Dedicated to Professor Mikio Sato on the Occasion of His Sixtieth Birthday;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;VOLUME EDITORS:;7
7;Contents of Volume II;12
8;Contributors;16
9;Introduction;24
10;Professor Mikio Sato and his Work;24
11;Publications of Professor Mikio Sato;28
12;Chapter 1. Three Personal Reminiscences;32
13;Chapter 2. Mikio Sato: Impressions;36
14;Chapter 3. Sato, a Perfectionist;40
15;Chapter 4. Existence of Local Holomorphic Solutions of Differential Equations of Infinite Order;42
15.1;References;46
16;Chapter 5. A Note on Holonomic q- Difference Systems;48
16.1;1. Introduction;48
16.2;2. Existence Theorem;49
16.3;References;51
17;Chapter 6. Solvable Models in Statistical Mechanics and Riemann Surfaces of Genus Greater than One;52
17.1;Acknowledgements;61
17.2;References;62
18;Chapter 7. Linearization and Singular Partial Differential Equations;64
18.1;1. Introduction;64
18.2;2. Preliminaries;66
18.3;3. Proof of Theorem 1;67
18.4;References;69
19;Chapter 8. On the Notions of Scattering State,Potential and Wave-Functionin Quantum Field Theory:An Analytic-Functional Viewpoint;72
19.1;1. Particles and Fields;72
19.2;2. Irreducible Interaction Kernelsand the Analytic Background of Two-Particle Scattering;78
19.3;3. Scattering States in Wave-Mechanics;83
19.4;4. Two-particle Scattering Statesin Quantum Field Theory;87
19.5;References;95
20;Chapter 9. Two Remarks on Recent Developments in Solvable Models;98
20.1;1. Introduction;98
20.2;2. Star-Triangle Relation and the Braid Group;99
20.3;3. Modular Covariant Characters;102
20.4;References;106
21;Chapter 10. Hypergeometric Functions;108
21.1;Preface;108
21.2;1. Introduction;108
21.3;2. The Conformai Group;118
21.4;3. Cartari and Weyl Group and Their Applications;127
21.5;4. Beyond the Conformai Group;134
21.6;5. Miscellanea;141
21.7;References;151
22;Chapter 11. Quantization of Lie Groups and Lie Algebras;152
22.1;1. Quantum Formal Groups;153
22.2;2. Finite-Dimensional Example;156
22.3;3. Infinite-Dimensional Example;159
22.4;4. Deformation Theory and Quantum Groups;160
22.5;References;161
23;Chapter 12. The Toda Lattice in the Complex Domain;164
23.1;1. Introduction;164
23.2;2. Toda Lattice Background;167
23.3;3. Schubert Cells for A,;170
23.4;4. The Lowest Balance;172
23.5;5. Other Balances;174
23.6;6. Conclusion;175
23.7;Acknowledgements;176
23.8;References;176
24;Chapter 13. Zoll Phenomena in (2 + 1) Dimensions;178
24.1;1. Introduction;178
24.2;2. Zoll-Einstein Deformations;181
24.3;3. Deformations of M2,1;184
24.4;4. Floquet Theory;187
24.5;References;192
25;Chapter 14. A Proof of the Bott Inequalities;194
25.1;0. Introduction;194
25.2;1. Some Complements to [6];196
25.3;2. Geometric Preparations;198
25.4;3. Proof of the Bott Inequalities;202
25.5;References;205
26;Chapter 15. What is the Notion of a Complex Manifold with a Smooth Boundary?;208
26.1;1. Complex Manifolds with a Concrete Boundary (dimcM = n);210
26.2;2. Complex Manifolds with an Abstract Boundary (dimR M = 2n);210
26.3;3. Induced CR Structure on the Boundary (dimR .M =2n –1 );211
26.4;4. Noncompact Examples;212
26.5;5. Proof of Lemma 1;213
26.6;6. Proof of Theorem 1;215
26.7;7. Proof of Theorem 2;217
26.8;8. Proof of Theorem 3;218
26.9;9. Compact Examples;218
26.10;10. Influence of Real Analyticity;221
26.11;11. Influence of Pseudoconvexity;222
26.12;12. Remark on Local Embeddability;222
26.13;13. A Conjecture;223
26.14;References;223
27;Chapter 16. Toda Molecule Equations;226
27.1;1. Introduction;226
27.2;2. Transformation of Toda Molecule Equations into Bilinear Forms;229
27.3;3. Discrete Two-Dimensional Toda Molecule Equation;233
27.4;4. Cylindrical Toda Molecule Equation;236
27.5;5. (3 + 1 )-Dimensional Toda Molecule Equation;237
27.6;References;239
28;Chapter 17. Microlocal Analysis and Scattering in Quantum Field Theories;240
28.1;1. Introduction;240
28.2;2. S- Matrix Theory;242
28.3;3. Axiomatic Field Theory;245
28.4;4. Constructive Field Theory;249
28.5;References;252
29;Chapter 18. b-Functions and p-adic Integrals;254
29.1;References;264
30;Chapter 19. On the Poles of the Scattering Matrix for Several Convex Bodies;266
30.1;1. Introduction;266
30.2;2. On the Trace Formula and the Reduction of the Problem;268
30.3;3. Proof of Proposition 2.2;270
30.4;References;274
31;Chapter 20. Symmetrie Tensors of the .(1)n–1 Family;276
31.1;1. Introduction;276
31.2;2. Elementary Blocks;277
31.3;3. Fusion;280
31.4;4. Restriction;281
31.5;References;288
32;Chapter 21. On Hyperfunctions with Analytic Parameters;290
32.1;1. Introduction;290
32.2;2. Holmgren-Type Theorem and Watermelon Theorem;291
32.3;3. Restriction Data and Uniqueness;294
32.4;4. Analyticity of the Discrete Loci;296
32.5;References;299
33;Chapter 22. The Invariant Holonomic Systemon a Semisimple Lie Group;300
33.1;0. Introduction;300
33.2;1. Proof of Theorems 1 and 3;302
33.3;2. Proof of Theorem 1 (Continued);304
33.4;3. Proof of Theorem 3 (Continued);305
33.5;4. The Proof of Theorem 2;307
33.6;References;309
34;Chapter 23. Some Applications of Microlocal Energy Methods to Analytic Hypoellipticity;310
34.1;Introduction;310
34.2;1. Preliminaries;311
34.3;2. The Quasi-Positivity and Some Inequalities;311
34.4;3. Applications to Analytic Hypoellipticity;320
34.5;References;325
35;Chapter 24. A Proof of the Transformation Formula of the Theta-Functions;328
36;Chapter 25. Microlocal Analysis of Infrared Singularities;332
36.1;0. Introduction;332
36.2;1. The Photon Propagator;337
36.3;2. Definition of the Problem;338
36.4;3. Properties of F(q);340
36.5;4. The Discontinuity around Lo+ (D);341
36.6;5. Polar Coordinates;342
36.7;6. Some Auxiliary Results;343
36.8;7. Derivation of a Discontinuity Formula;346
36.9;8. Explicit Form of Da, Db, Dc and Dd;351
36.10;References;353
37;Chapter 26. On the Global Existence of Real Analytic Solutions of Systems of Linear Differential Equations;354
37.1;0. Introduction;354
37.2;1. Uniform M-Convexity;355
37.3;2. Main Results;358
37.4;3. Examples;360
37.5;4. Proof of the Theorems;361
37.6;References;367
38;Chapter 27. Complex Powers on p-adic Fields and a Resolution of Singularities;368
38.1;Introduction;368
38.2;1. The Integration Formula Corresponding to the Blowing-up at the Origin;370
38.3;2. The Resolution of Singularities Modulo p;371
38.4;References;378
39;Chapter 28.Operational Calculus, Hyperfunctions and Ultradistributions;380
39.1;1. Introduction;380
39.2;2. Laplace Hyperfunctions and their Laplace Transforms;384
39.3;3, Convolution Algebras of Hyperfunctions;388
39.4;4. Ultradistributions as Elements in B.n.M;391
39.5;5, The Vector-Valued Case;393
39.6;References;394
40;Chapter 29. On a Conjectural Equation of Certain Kinds of Surfaces;396
41;Chapter 29. Vanishing Cycles and Second Microlocalization;404
41.1;1. Formal Microdifferential Operators;405
41.2;2. Sheaves of 2-Microdifferential Operators;406
41.3;3. Equivalence Theorem;409
41.4;4. Vanishing Cycles of a D-Module;412
41.5;References;414
42;Chapter 30. Extensions of Microfunction Sheavesup to the Boundary;416
42.1;1. Introduction;416
42.2;2. Tapering Domains and Sheavesover S F;417
42.3;3. Passage from Sphere to Cosphere;419
42.4;4. Microlocalization;421
42.5;5. Some Applications and Remarks;422
42.6;References;424
43;Chapter 31. Extension of Holonomic D-modules;426
43.1;Introduction;426
43.2;1. Specializable Modules;427
43.3;2. Direct and Inverse Images;429
43.4;3. Extensions;431
43.5;References;433
44;Chapter 32. On the Irregularity of the Dx Holonomic Modules;436
44.1;0. Introduction;436
44.2;1. The Semi-Continuity Theorem;437
44.3;2. The Irregular Vanishing Cycles;438
44.4;3. The Irregularity Sheaf of a Dx Holonomic Module and the Comparison Theorem;441
44.5;References;443
45;Chapter 33. An Error Analysis of Quadrature Formulas Obtained by Variable Transformation;446
45.1;1. Introduction;446
45.2;2. Quadrature Formulas Obtained by Variable Transformation and the DE-Rule;450
45.3;3. Characteristic Function of the Error of the DE-Rule;452
45.4;4. A Generalized DE-Rule;455
45.5;References;459
46;Chapter 34. Analytic Functionals on the Complex Light Cone and Their Fourier-Borel Transformations;462
46.1;Introduction;462
46.2;1. Spaces of Complex Harmonic Functions;463
46.3;2. Holomorphic Functions on the Complex Sphere;466
46.4;3. Holomorphic Functions on the Complex Light Cone;470
46.5;4. The Fourier–Borel Transformation;475
46.6;References;478
47;Chapter 35. A p-adic Theory of Hyperfunctions, II;480
47.1;1. Cohomologies of Rigid Analytic Spaces;481
47.2;2. Lemmas on the Grothendieck Topology on kn;483
47.3;3. The Key Lemma;485
47.4;4. Projective Limits and the Duality;489
47.5;5. The Main Results;493
47.6;References;494