The Riemann boundary problem on Riemann surfaces:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
D. Reidel Publishing Company
1988
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| Series: | Mathematics and its Applications / Soviet series
16 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 199 Seiten graph. Darst. |
| ISBN: | 9027726531 |
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MARC
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| 245 | 1 | 0 | |a The Riemann boundary problem on Riemann surfaces |c Yu. L. Rodin, Institute of Solid State Physics, Academy of Sciences of the U.S.S.R., Moscow, U.S.S.R. |
| 264 | 1 | |a Dordrecht [u.a.] |b D. Reidel Publishing Company |c 1988 | |
| 300 | |a XIII, 199 Seiten |b graph. Darst. | ||
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Record in the Search Index
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CONTENTS
Preface xj
Chapter 1. The Riemann Boundary Problem on
Closed Riemann Surfaces 1
§1. Riemann Surfaces 1
§2. Functions and Differential Forms. Abelian Integrals and
Differentials 5
§3. Riemann Bilinear Relations. The Riemann Roch Theorem 11
A. Bilinear Relations 11
B. A Differential Order 16
C. The Riemann Roch Theorem 17
§4. Cauchy type Integrals 19
§5. The Riemann Problem. Number of Solutions 26
§6. Inversion of Abelian Integrals and Abel's Theorem. Solvability
of the Riemann Problem 35
A. Abel's Theorem 35
B. Inversion of Abelian Integrals. The Boundary Problem
Solvability 38
C. Jacobi Variety 41
§7. Riemann Theta Functions. Solvability of the Riemann
Boundary Problem 42
A. Zeros of the Riemann Thet a Function 42
B. The Problem of Inversion of Abelian Integrals 44
C. Divisor Classes 46
D. The Solvability of the Riemann Problem 47
§8. Explicit Formulae for Solutions of the Riemann Problem 48
Chapter 2. Complex Vector Bundles over Compact
Riemann Surfaces 54
§9. De Rham and Dolbeault Theorems 54
§10. Divisors. Complex Vector Bundles. Serre and
Riemann Theorems 60
A. Divisors and Complex Line Bundles 60
B. The Serre Duality Theorem 62
C. The Riemann Theorem 63
§11. The Riemann Roch Theorem. The Riemann Problem 66
A. The Riemann Roch Theorem 66
viii Contents
B. Some Corollaries 67
C. The Riemann Boundary Problem 69
§12. The Second Cousin Problem. Solvability of the
Riemann Problem 71
A. Characteristic Classes. Abel's Theorem 71
B. The Second Cousin Problem 73
C. Classification of Complex Line Bundles 77
D. Solvability of the Riemann Problem, k = 0 78
E. Solvability of the Riemann Problem, 0 k g 80
F. The Nonhomogeneous Riemann Problem 82
Chapter 3. The Riemann Boundary Problem for Vectors
on Compact Riemann Surfaces 83
§13. The Riemann Boundary Problem for Vector Functions 83
A. The Riemann Problem and Complex Vector Bundles 83
B. The General Solution of the Riemann Problem 88
C. The Conjugate Problem. The Riemann Roch Theorem
for Vector Bundles 91
Chapter 4. The Riemann Boundary Problem on Open
Riemann Surfaces 94
§14. Open Riemann Surfaces 94
A. Finite Surfaces 94
B. Triviality of Cohomologies on Open Riemann Surfaces 96
C. The Riemann Bilinear Relations 99
D. The Hodge Royden Theorem 102
§15. D Cohomologies 102
A. £ Cohomology Groups. The Singular Group 102
B. Serre Duality 105
§16. P Divisors. The Second Cousin Problem 106
A. Divisor Degree 106
B. Infinite Divisors 108
C. 5 Divisors 111
D. The Second Cousin Problem 113
§17. The Riemann Problem. Solvability 116
A. The Problem Statement. The Bundle Ba 116
B. The Existence of a Solution 118
C. The Cauchy Index. The Solvability Conditions 120
D. The Case /c = 0. S Problems 123
§18. The Solving of the Riemann Problem in the Explicit Form 127
A. Cauchy type Integrals 127
B. Construction of a Solution 129
Chapter 5. Generalized Analytic Functions 132
§19. Bers Vekua Integral Representations 132
A. Generalized Analytic Functions on a Plane 132
B. Generalized Analytic Functions on a Compact Riemann
Surface. Basic Definitions 134
Contents jx
C. The First Bers Vekua Equation 136
D. Equation Bu = Au 137
§20. The Riemann Roch Theorem 139
A. Generalized Constants 139
B. The Riemann Roch Theorem 142
§21. Nonlinear Aspects of the Generalized Analytic Function Theory 146
A. Multiplicative Multivalued Solution. Existence 146
B. Multiplicative Constants. Uniqueness 147
C. Abel's Theorem 150
Chapter 6. Integrable Systems 151
§22. The Schrodinger Equation 151
A. Fast Decreasing Potentials 151
B. Reflection Finite Zone Potentials 158
§23. The Landau Lifschitz Equation 176
A. Fast Decreasing Potentials 167
B. Reflection Finite Zone Potentials 171
§24. Riemann Hilbert and Related Problems 172
A. Z? Bar Problem 172
B. The Dressing Method 174
C. The Riemann Hilbert Problem 175
Appendix 1 Hyperelliptic Surfaces 179
Appendix 2 The Matrix Riemann Problem on the Plane 181
Appendix 3 One Approximate Method of Solving the
Matrix Riemann Problem 183
Appendix 4 The Riemann Hilbert Boundary Problem 186
Notations 188
References 191
Subject Index 197 |
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| id | DE-604.BV001315843 |
| illustrated | Illustrated |
| indexdate | 2024-08-20T00:34:35Z |
| institution | BVB |
| isbn | 9027726531 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-000794839 |
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| physical | XIII, 199 Seiten graph. Darst. |
| publishDate | 1988 |
| publishDateSearch | 1988 |
| publishDateSort | 1988 |
| publisher | D. Reidel Publishing Company |
| record_format | marc |
| series2 | Mathematics and its Applications / Soviet series |
| spelling | Rodin, Jurij L. 1936- Verfasser (DE-588)172334624 aut The Riemann boundary problem on Riemann surfaces Yu. L. Rodin, Institute of Solid State Physics, Academy of Sciences of the U.S.S.R., Moscow, U.S.S.R. Dordrecht [u.a.] D. Reidel Publishing Company 1988 XIII, 199 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematics and its Applications / Soviet series 16 Randwertproblem - Riemannsche Fläche Riemannsche Fläche (DE-588)4049991-1 gnd rswk-swf Riemannsches Randwertproblem (DE-588)4317184-9 gnd rswk-swf Riemannsches Randwertproblem (DE-588)4317184-9 s Riemannsche Fläche (DE-588)4049991-1 s DE-604 Soviet series Mathematics and its Applications 16 (DE-604)BV004708148 16 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000794839&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rodin, Jurij L. 1936- The Riemann boundary problem on Riemann surfaces Randwertproblem - Riemannsche Fläche Riemannsche Fläche (DE-588)4049991-1 gnd Riemannsches Randwertproblem (DE-588)4317184-9 gnd |
| subject_GND | (DE-588)4049991-1 (DE-588)4317184-9 |
| title | The Riemann boundary problem on Riemann surfaces |
| title_auth | The Riemann boundary problem on Riemann surfaces |
| title_exact_search | The Riemann boundary problem on Riemann surfaces |
| title_full | The Riemann boundary problem on Riemann surfaces Yu. L. Rodin, Institute of Solid State Physics, Academy of Sciences of the U.S.S.R., Moscow, U.S.S.R. |
| title_fullStr | The Riemann boundary problem on Riemann surfaces Yu. L. Rodin, Institute of Solid State Physics, Academy of Sciences of the U.S.S.R., Moscow, U.S.S.R. |
| title_full_unstemmed | The Riemann boundary problem on Riemann surfaces Yu. L. Rodin, Institute of Solid State Physics, Academy of Sciences of the U.S.S.R., Moscow, U.S.S.R. |
| title_short | The Riemann boundary problem on Riemann surfaces |
| title_sort | the riemann boundary problem on riemann surfaces |
| topic | Randwertproblem - Riemannsche Fläche Riemannsche Fläche (DE-588)4049991-1 gnd Riemannsches Randwertproblem (DE-588)4317184-9 gnd |
| topic_facet | Randwertproblem - Riemannsche Fläche Riemannsche Fläche Riemannsches Randwertproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000794839&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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