Kashiwara / Kawai Algebraic Analysis

Papers Dedicated to Professor Mikio Sato on the Occasion of His Sixtieth Birthday
1. Auflage 2014
ISBN: 978-1-4832-6794-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 500 Seiten, Web PDF

Algebraic Analysis: Papers Dedicated to Professor Mikio Sato on the Occasion of his 60th Birthday, Volume II is a collection of research papers on algebraic analysis and related topics in honor to Professor Mikio Sato's 60th birthday. This volume is divided into 29 chapters and starts with research works concerning the fundamentals of KP equations, strings, Schottky problem, and the applications of transformation theory for nonlinear integrable systems to linear prediction problems and isospectral deformations,. The subsequent chapters contain papers on the approach to nonlinear integrable systems, the Hodge numbers, the stochastic different equation for the multi-dimensional weakly stationary process, and a method of harmonic analysis on semisimple symmetric spaces. These topics are followed by studies on the quantization of extended vortices, moduli space for Fuchsian groups, microfunctions for boundary value problems, and the issues of multi-dimensional integrable systems. The remaining chapters explore the practical aspects of pseudodifferential operators in hyperfunction theory, the elliptic solitons, and Carlson's theorem for holomorphic functions. 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 I;12
8;Contributors;16
9;Chapter
1. KP Equations, Strings, and the Schottky Problem;24
9.1;0. Historical Origin;24
9.2;1. Motivations;27
9.3;2. Schottky Problem;29
9.4;3. Algebraic Curves;30
9.5;4. KP System and Jacobian Varieties;33
9.6;5. Virasoro Action;37
9.7;6. Supersymmetrization;40
9.8;References;42
10;Chapter
2. A Note on the Holonomic System of Invariant Hyperfunctions on a Certain Prehomogeneous Vector Space;44
10.1;Introduction;44
10.2;1. An Example of Prehomogeneous Vector Space;45
10.3;2. Holonomy Diagrams of the Holonomic System of Relatively Invariant Hyperfunctions;47
10.4;3. Fourier Transforms of the Local Zeta Functions;51
10.5;References;53
11;Chapter
3. Applications of Transformation Theory for Nonlinear Integrable Systems to Linear Prediction Problems and Isospectral Deformations;56
11.1;1. Introduction;56
11.2;2. Application to Linear Prediction Problems;57
11.3;3. Application to Isospectral Deformations;62
11.4;References;65
12;Chapter
4. New Local Supersymmetry in the Framework of Einstein Gravity;68
12.1;1. Quantum Field Theory;68
12.2;2. Symmetry Principles;69
12.3;3. Einstein Gravity and Supergravity;71
12.4;4. New Local Supersymmetry;74
12.5;References;77
13;Chapter
5. K3 Surfaces Related to Root Systems in E8;78
13.1;Introduction;78
13.2;1. Del Pezzo Surfaces;79
13.3;2. Construction of Universal Family;80
13.4;3. Simultaneous Resolution and Period Mapping;83
13.5;4. Section-Module, Fiber-Module and Main Property of the Family;85
13.6;5. Monodromy Module;87
13.7;6. Double Coverings Branched over Singular Fibers;88
13.8;7. Computation of Monodromy;90
13.9;8. Example;93
13.10;Acknowledgements;98
13.11;References;98
14;Chapter
6. Wronskian Determinants and the Gröbner Representation of a Linear Differential Equation: An Approach to Nonlinear Integrable Systems;100
14.1;Introduction;100
14.2;1. Wronskian Determinants;101
14.3;2. Grassmannian Formalism;105
14.4;3, Gröbner Representations;109
14.5;4. Applications to Nonlinear Integrable Systems;115
14.6;References;120
15;Chapter
7. Higher-Codimensional Boundary Value Problem and F-Mild Hyperfunctions;122
15.1;Introduction;122
15.2;1. F-Mild Hyperfunctions for Higher-Codimensional Boundary;123
15.3;2. Higher-Codimensional Boundary Value Problem;134
15.4;References;137
16;Chapter 8. Hodge Numbers of a Kummer Covering of P2 Ramified along a Line Configuration;138
16.1;0. Introduction;138
16.2;1. Basic Notation;139
16.3;2. Formulation of Theorem 1;140
16.4;3. Proof of Theorem 1;141
16.5;4. Hodge Type of H2(a) for Generic a;145
16.6;References;150
17;Chapter
9. On a Stochastic Difference Equation for the Multi-Dimensional Weakly Stationary Process with Discrete Time;152
17.1;1. Introduction;152
17.2;2. KM2O–Langevin Equations;155
17.3;3. Relations Among .+(. , *), .-( · , * ), d+( . ) and d_ ( · );158
17.4;4. Relations Among V+( · ), V_( · ), d+( · ) and d_(·);159
17.5;5. The Prediction Formula and Prediction Error;177
17.6;6. A Construction Theorem;178
17.7;References;195
18;Chapter
10. Bäcklund Transformations of Classical Orthogonal Polynomials;198
18.1;1. Truncated Toda Equation;199
18.2;2. Bäcklund Transformation;200
18.3;3. Hermite Polynomials;201
18.4;4. Tchebichef Polynomials;202
18.5;5. Laguerre Polynomials;204
18.6;6. Gegenbauer Polynomials;205
18.7;7. Hypergeometric Functions;205
18.8;8. Jacobi Polynomials;206
18.9;References;208
19;Chapter
11. A Deformation of Dirichlet's Class Number Formula;210
19.1;1. Dirichlet's Formula;210
19.2;2. A Recurrence Relation;211
19.3;3. A Deformation of Dirichlet's Formula;212
19.4;4. An Upper Bound of eh;215
19.5;5. Comparison with Hua's Upper Bound;216
19.6;6. A Parting Remark;217
20;Chapter
12. A Method of Harmonic Analysis on Semisimple Symmetric Spaces;218
20.1;0. Introduction;218
20.2;1. Notation;219
20.3;2. Smooth Imbedding;220
20.4;3. Asymptotic Expansion;221
20.5;4. Asymptotic Behavior of Spherical Functions;224
20.6;5. Discrete Series;227
20.7;References;230
21;Chapter
13. A Note on Ehrenpreis' Fundamental Principle on a Symmetric Space;232
21.1;0. Introduction;232
21.2;1. A Paley–Wiener Theorem;234
21.3;2. Ehrenpreis' Fundamental Principle;237
21.4;3. Crucial Point in the Proof of Theorem 2;241
21.5;References;247
22;Chapter
14. Resurgence, Quantized Canonical Transformations, and Multi-lnstanton Expansions;250
22.1;1. Resurgence (A Radar Approach);252
22.2;2. Quantized Canonical Transformations and Multi-lnstanton Expansions;259
22.3;References;276
23;Chapter 15. Quantization of Extended Vortices and sDiff R;278
23.1;1. Introduction;278
23.2;2. sDiff R3 as Phase Space for Vortices;279
23.3;3. Vortices and Knots;282
23.4;4. Conclusions;284
23.5;Acknowledgement;284
23.6;References;284
24;Chapter
16. Moduli Space for Fuchsian Groups;286
24.1;1. Introduction;287
24.2;2. SL(2, R) and PSL(2, R);291
24.3;3. Fuchsian Groups;295
24.4;4. Representation Space Ro(G, G);296
24.5;5. The Spaces S1(x(G)), x(G) and I (G);297
24.6;6. The Real Tangent Spaces;300
24.7;7. The Complex Structures on the Tangent Spaces;305
24.8;8. Local Parametrization;309
24.9;9. Integrability of the Complex Structure on I(G);315
24.10;10. Integrability of the Complex Structure on x(G);320
24.11;11. Gauge Transformation I;325
24.12;12. Gauge Transformation II;328
24.13;13. The Unit Circle Bundle;331
24.14;References;336
25;Chapter
17. The Hamburger Theorem for the Epstein Zeta Functions;340
25.1;0. Introduction;340
25.2;1. Statement of the Results;341
25.3;2. Proof of Theorem 1;348
25.4;References;357
26;Chapter
18. Microfunctions for Boundary Value Problems;360
26.1;1. The Functor µhom (cf. [K-S 2]);360
26.2;2. Wave Front Sets at the Boundary (cf. [S 4]);361
26.3;3. Boundary Values (cf. [S 4]);364
26.4;4. Application to Diffraction;367
26.5;References;369
27;Chapter
19. Regularization of the Product of Complex Powers of Polynomials and Its Application;372
27.1;Introduction;372
27.2;1. Meromorphic Continuation of Distributions Defined by the Product of Complex Powers of Polynomials;374
27.3;2. An Application of Theorem 1;376
27.4;3. Properties of Functions Related with the Distribution .a,ß;381
27.5;References;386
28;Chapter
20. On the Local Solvability of Fuchsian Type Partial Differential Equations;388
28.1;1. Main Result;389
28.2;2. Proof of Theorem 1;390
28.3;3. Proof of (2.8);393
28.4;4. Generalization;397
28.5;References;399
29;Chapter 21. .b-Cohomology and the Bochner-Martinelli Kernel;400
29.1;References;403
30;Chapter
22. Issues of Multi-Dimensional Integrable Systems;404
30.1;0. Introduction;404
30.2;1. Self-Dual Connections, Linear System, and Twistors;405
30.3;2. Riemann–Hilbert Transformations;406
30.4;3. Cauchy Problem;407
30.5;4. Grassmann Manifold;409
30.6;5. Enlarged Groups and Lie Algebras;410
30.7;6. Issue of Multi-Dimensional Spectral Parameters;411
30.8;7. Self-Dual Metrics, or Deformation of Integrable G- Structures;413
30.9;References;415
31;Chapter
23. Second Microlocalization and Conical Refraction (II);418
31.1;1. Introduction;418
31.2;2. Preliminary;419
31.3;3. Statement of the Main Theorem;424
31.4;4. Proof of the Main Theorem;426
31.5;References;432
32;Chapter
24. Pseudodifferential Operators in Hyperfunction Theory;434
32.1;0 Introduction;434
32.2;1. Symbol Functions of Microlocal Operators;435
32.3;2. Symbol Formulae for Adjoint Operators and for Composite Operators;438
32.4;3. Examples;441
32.5;References;443
33;Chapter 25. Some Observations on Geometric Representations of the Superconformal Algebras and a Super Analogue of the Mumford Sheaves;444
33.1;Introduction;444
33.2;1.;446
33.3;2.;448
33.4;References;451
34;Chapter
26. New Elliptic Solitons;452
34.1;1. Solitons;452
34.2;2. Real Periodic Solitons;454
34.3;3. Special Complex and Algebraic Solitons;455
34.4;4. Elliptic Solitons;456
34.5;5. Tangent Covers;457
34.6;6. Hyperelliptic Tangent Covers;458
34.7;7. Solitons Versus Diffusons;460
34.8;References;461
35;Chapter
27. On the Microlocal Smoothing Effect of Dispersive Partial Differential Equations, I: Second-Order Linear Equations;462
35.1;0. Introduction;462
35.2;1. Notations and the Statement of the Main Results;463
35.3;2. Proof of Theorem 1;465
35.4;3. A Proposition;467
35.5;4. Proof of Theorem 2;470
35.6;Acknowledgements;477
35.7;References;477
36;Chapter
28. Locally Prehomogeneous Spaces and Their Transverse Localizations;478
36.1;1. Introduction;478
36.2;2. Locally Prehomogeneous Space;480
36.3;3. Localization;483
36.4;4.;488
36.5;5. Examples;491
36.6;References;493
37;Chapter
29. On Carlson's Theorem for Holomorphic Functions;494
37.1;1. Brief History of Carlson's Theorem;494
37.2;2. Some Applications of Carlson's Theorems;496
37.3;3. Existence of Interpolating Function;497
37.4;4. Carlson's Theorem for Holomorphic Functions with Proximate Order 1+log(log r)/log r;499
37.5;References;501