isomonodromy problems

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31: 31.16 Mathematical Applications

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§31.16(i) Uniformization Problem for Heun’s Equation

… ► thesis “Inversion problem for a second-order linear differential equation with four singular points”. …

32: 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)). …

33: William P. Reinhardt

… ►Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions. …

34: 7.21 Physical Applications

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35: 16.24 Physical Applications

… ►They are also potentially useful for the solution of more complicated restricted lattice walk problems, and the 3D Ising model; see Barber and Ninham (1970, pp. 147–148). …

36: 20.12 Mathematical Applications

… ►For applications of ${\theta }_3(0,q)$ to problems involving sums of squares of integers see §27.13(iv), and for extensions see Estermann (1959), Serre (1973, pp. 106–109), Koblitz (1993, pp. 176–177), and McKean and Moll (1999, pp. 142–143). …

37: 26.15 Permutations: Matrix Notation

… ►For the problem of derangements, $r_j(B)=\left(\genfrac{}{}{0.0pt}{}{n}{j}\right)$. … ►

Example 1

►The problème des ménages asks for the number of ways of seating $n$ married couples around a circular table with labeled seats so that no men are adjacent, no women are adjacent, and no husband and wife are adjacent. …

38: 27.16 Cryptography

… ►But if $p$ and $q$ are not known, the problem of recovering $x$ from $y$ seems insurmountable. …

39: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$, or $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$, to determine the scattering $S$-matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951). ►For bound-state problems only the exponentially decaying solution is required, usually taken to be the Whittaker function $W_{-\eta ,\mathrm{\ell }+\frac{1}{2}}\left(2\rho \right)$. … ►

40: 18.40 Methods of Computation

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§18.40(ii) The Classical Moment Problem

►The problem of moments is simply stated and the early work of Stieltjes, Markov, and Chebyshev on this problem was the origin of the understanding of the importance of both continued fractions and OP’s in many areas of analysis. Given the power moments, ${\mu }_n={\int }_a^b{x}^{n}d\mu (x)$, $n=0,1,2,\mathrm{\dots }$, can these be used to find a unique $\mu (x)$, a non-decreasing, real, function of $x$, in the case that the moment problem is determined? Should a unique solution not exist the moment problem is then indeterminant. …