Riemann surface
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1: 21.7 Riemann Surfaces
§21.7 Riemann Surfaces
►§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces
… ►Removing the singularities of this curve gives rise to a two-dimensional connected manifold with a complex-analytic structure, that is, a Riemann surface. All compact Riemann surfaces can be obtained this way. … ► … ►§21.7(iii) Frobenius’ Identity
…2: 21.10 Methods of Computation
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§21.10(ii) Riemann Theta Functions Associated with a Riemann Surface
►In addition to evaluating the Fourier series, the main problem here is to compute a Riemann matrix originating from a Riemann surface. … ►Tretkoff and Tretkoff (1984). Here a Hurwitz system is chosen to represent the Riemann surface.
Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.
3: Bernard Deconinck
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►He has worked on integrable systems, algorithms for computations with Riemann surfaces, Bose-Einstein condensates, and methods to investigate the stability of solutions of nonlinear wave equations.
He is the coauthor of several Maple commands to work with Riemann surfaces and the command to compute multidimensional theta functions numerically.
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4: 21.9 Integrable Equations
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►Particularly important for the use of Riemann theta functions is the Kadomtsev–Petviashvili (KP) equation, which describes the propagation of two-dimensional, long-wave length surface waves in shallow water (Ablowitz and Segur (1981, Chapter 4)):
…These parameters, including , are not free: they are determined by a compact, connected Riemann surface (Krichever (1976)), or alternatively by an appropriate initial condition (Deconinck and Segur (1998)).
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►Furthermore, the solutions of the KP equation solve the Schottky
problem: this is the question concerning conditions that a Riemann matrix needs to satisfy in order to be associated with a Riemann surface (Schottky (1903)).
Following the work of Krichever (1976), Novikov conjectured that the Riemann theta function in (21.9.4) gives rise to a solution of the KP equation (21.9.3) if, and only if, the theta function originates from a Riemann surface; see Dubrovin (1981, §IV.4).
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5: 21.1 Special Notation
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positive integers. | |
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intersection index of and , two cycles lying on a closed surface. if and do not intersect. Otherwise gets an additive contribution from every intersection point. This contribution is if the basis of the tangent vectors of the and cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is . | |
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6: 21.4 Graphics
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►Figure 21.4.1 provides surfaces of the scaled Riemann theta function , with
…This Riemann matrix originates from the Riemann surface represented by the algebraic curve ; compare §21.7(i).
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7: Bibliography R
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Elliptic Functions, Theta Functions, and Riemann Surfaces.
The Williams & Wilkins Co., Baltimore, MD.
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8: Bibliography T
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Combinatorial Group Theory, Riemann Surfaces and Differential Equations.
In Contributions to Group Theory,
Contemp. Math., Vol. 33, pp. 467–519.
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9: Bibliography F
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Theta Functions on Riemann Surfaces.
Springer-Verlag, Berlin.
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10: Bibliography J
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Java Tools for Experimental Mathematics
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