The Riemann Boundary Problem on Riemann Surfaces

ISBN-13: 9789027726537 / Angielski / Twarda / 1988 / 199 str.

Iurii Leonidovich Rodin; Y. Rodin

Opis

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van GuIik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Recenzje

`This valuable book characterized by an original point of view in both choice and organization of the material, will be much appropriated by mathematicians and physicists and will favourably influence the future researchers.'
Zentralblatt für Mathematik, 695

Spis treści

1. The Riemann Boundary Problem on Closed Riemann Surfaces.- §1. Riemann Surfaces.- §2. Functions and Differential Forms. Abelian Integrals and Differentials.- §3. Riemann Bilinear Relations. The Riemann — Roch Theorem.- A. Bilinear Relations.- B. A Differential Order.- C. The Riemann — Roch Theorem.- §4. Cauchy-type Integrals.- §5. The Riemann Problem. Number of Solutions.- §6. Inversion of Abelian Integrals and Abel’s Theorem. Solvability of the Riemann Problem.- A. Abel’s Theorem.- B. Inversion of Abelian Integrals. The Boundary Problem Solvability.- C. Jacobi Variety.- §7. Riemann Theta-Functions. Solvability of the Riemann Boundary Problem.- A. Zeros of the Riemann Theta-Function.- B. The Problem of Inversion of Abelian Integrals.- C. Divisor Classes.- D. The Solvability of the Riemann Problem.- §8. Explicit Formulae for Solutions of the Riemann Problem.- 2. Complex Vector Bundles over Compact Riemann Surfaces.- §9. De Rham and Dolbeault Theorems.- §10. Divisors. Complex Vector Bundles. Serre and Riemann Theorems.- A. Divisors and Complex Line Bundles.- B. The Serre Duality Theorem.- C. The Riemann Theorem.- §11. The Riemann — Roch Theorem. The Riemann Problem.- A. The Riemann — Roch Theorem.- B. Some Corollaries.- C. The Riemann Boundary Problem.- §12. The Second Cousin Problem. Solvability of the Riemann Problem.- A. Characteristic Classes. Abel’s Theorem.- B. The Second Cousin Problem.- C. Classification of Complex Line Bundles.- D. Solvability of the Riemann Problem, ? = 0.- E. Solvability of the Riemann Problem, 0 < ? < g.- F. The Nonhomogeneous Riemann Problem.- 3. The Riemann Boundary Problem for Vectors on Compact Riemann Surfaces.- §13. The Riemann Boundary Problem for Vector Functions.- A. The Riemann Problem and Complex Vector Bundles.- B. The General Solution of the Riemann Problem.- C. The Conjugate Problem. The Riemann — Roch Theorem for Vector Bundles.- 4. The Riemann Boundary Problem on Open Riemann Surfaces.- §14. Open Riemann Surfaces.- A. Finite Surfaces.- B. Triviality of Cohomologies on Open Riemann Surfaces.- C. The Riemann Bilinear Relations.- D. The Hodge — Royden Theorem.- §15. D-Cohomologies.- A. D-Cohomology Groups. The Singular Group.- B. Serre Duality.- §16. D-Divisors. The Second Cousin Problem.- A. Divisor Degree.- B. Infinite Divisors.- C. S-Divisors.- D. The Second Cousin Problem.- §17. The Riemann Problem. Solvability.- A. The Problem Statement. The Bundle BG.- B. The Existence of a Solution.- C. The Cauchy Index. The Solvability Conditions.- D. The Case ? = 0. S-Problems.- §18. The Solving of the Riemann Problem in the Explicit Form.- A. Cauchy-type Integrals.- B. Construction of a Solution.- 5. Generalized Analytic Functions.- §19. Bers — Vekua Integral Representations.- A. Generalized Analytic Functions on a Plane.- B. Generalized Analytic Functions on a Compact Riemann Surface. Basic Definitions.- C. The First Bers — Vekua Equation.- D. Equation ??u = Au.- §20. The Riemann — Roch Theorem.- A. Generalized Constants.- B. The Riemann — Roch Theorem.- §21. Nonlinear Aspects of the Generalized Analytic Function Theory.- A. Multiplicative Multivalued Solution. Existence.- B. Multiplicative Constants. Uniqueness.- C. Abel’s Theorem.- 6. Integrable Systems.- §22. The Schrödinger Equation.- A. Fast-Decreasing Potentials.- B. Reflection Finite-Zone Potentials.- §23. The Landau — Lifschitz Equation.- A. Fast-Decreasing Potentials.- B. Reflection Finite-Zone Potentials.- §24. Riemann — Hilbert and Related Problems.- A. D-Bar Problem.- B. The Dressing Method.- C. The Riemann-Hilbert Problem.- Appendix 1 Hyperelliptic Surfaces.- Appendix 2 The Matrix Riemann Problem on the Plane.- Appendix 3 One Approximate Method of Solving the Matrix Riemann Problem.- Appendix 4 The Riemann — Hilbert Boundary Problem.- Notations.- References.