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AuthorRodin, Yu. L. author
TitleThe Riemann Boundary Problem on Riemann Surfaces [electronic resource] / by Yu. L. Rodin
ImprintDordrecht : Springer Netherlands, 1988
Connect tohttp://dx.doi.org/10.1007/978-94-009-2885-5
Descript XIII, 199 p. online resource

SUMMARY

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


CONTENT

1. The Riemann Boundary Problem on Closed Riemann Surfaces -- ยง1. Riemann Surfaces -- ยง2. Functions and Differential Forms. Abelian Integrals and Differentials -- ยง3. Riemann Bilinear Relations. The Riemann โ{128}{148} Roch Theorem -- ยง4. Cauchy-type Integrals -- ยง5. The Riemann Problem. Number of Solutions -- ยง6. Inversion of Abelian Integrals and Abelโ{128}{153}s Theorem. Solvability of the Riemann Problem -- ยง7. Riemann Theta-Functions. Solvability of the Riemann Boundary Problem -- ยง8. Explicit Formulae for Solutions of the Riemann Problem -- 2. Complex Vector Bundles over Compact Riemann Surfaces -- ยง9. De Rham and Dolbeault Theorems -- ยง10. Divisors. Complex Vector Bundles. Serre and Riemann Theorems -- ยง11. The Riemann โ{128}{148} Roch Theorem. The Riemann Problem -- ยง12. The Second Cousin Problem. Solvability of the Riemann Problem -- 3. The Riemann Boundary Problem for Vectors on Compact Riemann Surfaces -- ยง13. The Riemann Boundary Problem for Vector Functions -- 4. The Riemann Boundary Problem on Open Riemann Surfaces -- ยง14. Open Riemann Surfaces -- ยง15. D-Cohomologies -- ยง16. D-Divisors. The Second Cousin Problem -- ยง17. The Riemann Problem. Solvability -- ยง18. The Solving of the Riemann Problem in the Explicit Form -- 5. Generalized Analytic Functions -- ยง19. Bers โ{128}{148} Vekua Integral Representations -- ยง20. The Riemann โ{128}{148} Roch Theorem -- ยง21. Nonlinear Aspects of the Generalized Analytic Function Theory -- 6. Integrable Systems -- ยง22. The Schrรถdinger Equation -- ยง23. The Landau โ{128}{148} Lifschitz Equation -- ยง24. Riemann โ{128}{148} Hilbert and Related Problems -- 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 โ{128}{148} Hilbert Boundary Problem -- Notations -- References


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