Schottky problem

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1: 21.10 Methods of Computation

… ►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).

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2: 21.9 Integrable Equations

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Figure 21.9.2: Contour plot of a two-phase solution of Equation (21.9.3). … Magnify

►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)). …

3: 29.19 Physical Applications

… ►Simply-periodic Lamé functions ($\nu$ noninteger) can be used to solve boundary-value problems for Laplace’s equation in elliptical cones. … ►

§29.19(ii) Lamé Polynomials

… ►Shail (1978) treats applications to solutions of elliptic crack and punch problems. …

4: 11.12 Physical Applications

§11.12 Physical Applications

►Applications of Struve functions occur in water-wave and surface-wave problems (Hirata (1975) and Ahmadi and Widnall (1985)), unsteady aerodynamics (Shaw (1985) and Wehausen and Laitone (1960)), distribution of fluid pressure over a vibrating disk (McLachlan (1934)), resistive MHD instability theory (Paris and Sy (1983)), and optical diffraction (Levine and Schwinger (1948)). …

5: Bibliography S

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Über die Moduln der Thetafunctionen. Acta Math. 27 (1), pp. 235–288.

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External Links:

ISSN 0001-5962, Document, MathReview Entry, ZentralBlatt

Referenced by:

§21.9.

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Lamé polynomial solutions to some elliptic crack and punch problems. Internat. J. Engrg. Sci. 16 (8), pp. 551–563.

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External Links:

ISSN 0020-7225, Document, MathReview, ZentralBlatt

Referenced by:

§29.19(ii).

Encodings:

BibTeX

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The Problem of Moments. 4th edition, American Mathematical Society Mathematical Surveys, Vol. 1, American Mathematical Society, Providence, RI.

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External Links:

MathReview (R. P. Boas, Jr.), ZentralBlatt

Referenced by:

§1.16(ix), §1.4(v), §18.2(viii), (18.40.7), §18.40(ii), §18.40(ii).

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Some Boundary Value Problems Associated with the Heun Equation. Ph.D. Thesis, London University.

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Notes:

Available from the library, University of Surrey, UK

External Links:

LibraryRecord

Referenced by:

§31.9(i), §31.9(ii), Ch.31, Profile Brian D. Sleeman.

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Mixed Boundary Value Problems in Potential Theory. North-Holland Publishing Co., Amsterdam.

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External Links:

MathReview (R. Shail), ZentralBlatt

Referenced by:

§10.22(iv).

Encodings:

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6: 14.31 Other Applications

… ►Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). … ►The conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^m\left(x\right)$ appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). … ►

§14.31(iii) Miscellaneous

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7: 28.33 Physical Applications

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§28.33(ii) Boundary-Value Problems

►Physical problems involving Mathieu functions include vibrational problems in elliptical coordinates; see (28.32.1). …The general solution of the problem is a superposition of the separated solutions. … ►

§28.33(iii) Stability and Initial-Value Problems

… ►References for other initial-value problems include: …

8: Bibliography Q

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A note on an open problem about the first Painlevé equation. Acta Math. Appl. Sin. Engl. Ser. 24 (2), pp. 203–210.

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External Links:

ISSN 0168-9673, Document, MathReview (Dmitrii Petrovich Novikov), ZentralBlatt

Referenced by:

§32.11(i).

Encodings:

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On two problems concerning means. J. Hangzhou Inst. Elec. Engrg. 17, pp. 1–7 (Chinese).

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Notes:

In Chinese

Referenced by:

§19.9(i).

Encodings:

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9: 26.20 Physical Applications

… ►Applications of combinatorics, especially integer and plane partitions, to counting lattice structures and other problems of statistical mechanics, of which the Ising model is the principal example, can be found in Montroll (1964), Godsil et al. (1995), Baxter (1982), and Korepin et al. (1993). … ►Other applications to problems in engineering, crystallography, biology, and computer science can be found in Beckenbach (1981) and Graham et al. (1995).

10: 6.17 Physical Applications

… ►Geller and Ng (1969) cites work with applications from diffusion theory, transport problems, the study of the radiative equilibrium of stellar atmospheres, and the evaluation of exchange integrals occurring in quantum mechanics. …