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1: 21.7 Riemann Surfaces
§21.7 Riemann Surfaces
§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces
For example, Figure 21.7.1 depicts a genus 2 surface. …
§21.7(iii) Frobenius’ Identity
The genus of this surface is g . …
2: 21.10 Methods of Computation
§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. …
  • Belokolos et al. (1994, Chapter 5) and references therein. Here the Riemann surface is represented by the action of a Schottky group on a region of the complex plane. The same representation is used in Gianni et al. (1998).

  • 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: 1.6 Vectors and Vector-Valued Functions
    §1.6(v) Surfaces and Integrals over Surfaces
    A parametrized surface S is defined by … The area A ( S ) of a parametrized smooth surface is given by …
    4: 21.9 Integrable Equations
    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)): …Here x and y are spatial variables, t is time, and u ( x , y , t ) is the elevation of the surface wave. …These parameters, including 𝛀 , are not free: they are determined by a compact, connected Riemann surface (Krichever (1976)), or alternatively by an appropriate initial condition u ( x , y , 0 ) (Deconinck and Segur (1998)). … 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). …
    5: Publications
  • B. V. Saunders and Q. Wang (2000) From 2D to 3D: Numerical Grid Generation and the Visualization of Complex Surfaces, Proceedings of the 7th International Conference on Numerical Grid Generation in Computational Field Simulations, Whistler, British Columbia, Canada, September 25-28, 2000. PDF
  • B. V. Saunders and Q. Wang (2006) From B-Spline Mesh Generation to Effective Visualizations for the NIST Digital Library of Mathematical Functions, in Curve and Surface Design, Proceedings of the Sixth International Conference on Curves and Surfaces, Avignon, France June 29–July 5, 2006, pp. 235–243. PDF
  • B. Saunders and Q. Wang (2010) Tensor Product B-Spline Mesh Generation for Accurate Surface Visualizations in the NIST Digital Library of Mathematical Functions, in Mathematical Methods for Curves and Surfaces, Proceedings of the 2008 International Conference on Mathematical Methods for Curves and Surfaces (MMCS 2008), Lecture Notes in Computer Science, Vol. 5862, (M. Dæhlen, M. Floater., T. Lyche, J. L. Merrien, K. Mørken, L. L. Schumaker, eds), Springer, Berlin, Heidelberg (2010) pp. 385–393. PDF
  • B. I. Schneider, B. R. Miller and B. V. Saunders (2018) NIST’s Digital Library of Mathematial Functions, Physics Today 71, 2, 48 (2018), pp. 48–53. PDF
  • 6: 19.33 Triaxial Ellipsoids
    §19.33(i) Surface Area
    The surface area of an ellipsoid with semiaxes a , b , c , and volume V = 4 π a b c / 3 is given by
    19.33.1 S = 3 V R G ( a 2 , b 2 , c 2 ) ,
    If a conducting ellipsoid with semiaxes a , b , c bears an electric charge Q , then the equipotential surfaces in the exterior region are confocal ellipsoids: …
    7: Sidebar 21.SB1: Periodic Surface Waves
    Sidebar 21.SB1: Periodic Surface Waves
    The caption reads “Mosaic of two overhead photographs, showing surface patterns of waves in shallow water”. …
    8: 5.19 Mathematical Applications
    The volume V and surface area S of the n -dimensional sphere of radius r are given by …
    9: 36.5 Stokes Sets
    Stokes sets are surfaces (codimension one) in 𝐱 space, across which Ψ K ( 𝐱 ; k ) or Ψ ( U ) ( 𝐱 ; k ) acquires an exponentially-small asymptotic contribution (in k ), associated with a complex critical point of Φ K or Φ ( U ) . … In Figure 36.5.4 the part of the Stokes surface inside the bifurcation set connects two complex saddles. …
    See accompanying text
    Figure 36.5.7: Sheets of the Stokes surface for the swallowtail catastrophe (colored and with mesh) and the bifurcation set (gray). … Magnify
    See accompanying text
    Figure 36.5.8: Sheets of the Stokes surface for the elliptic umbilic catastrophe (colored and with mesh) and the bifurcation set (gray). Magnify
    See accompanying text
    Figure 36.5.9: Sheets of the Stokes surface for the hyperbolic umbilic catastrophe (colored and with mesh) and the bifurcation set (gray). Magnify
    10: William P. Reinhardt
    This is closely connected with his interests in classical dynamical “chaos,” an area where he coauthored a book, Chaos in atomic physics with Reinhold Blümel. …
  • In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.