Lax%20pairs
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1: 32.4 Isomonodromy Problems
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– can be expressed as the compatibility condition of a linear system, called an isomonodromy problem or Lax pair.
Suppose
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2: 18.38 Mathematical Applications
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Integrable Systems
… ►It has elegant structures, including -soliton solutions, Lax pairs, and Bäcklund transformations. …3: 27.15 Chinese Remainder Theorem
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►The Chinese remainder theorem states that a system of congruences , always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod ), where is the product of the moduli.
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►Their product has 20 digits, twice the number of digits in the data.
…These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result , which is correct to 20 digits.
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4: 20 Theta Functions
Chapter 20 Theta Functions
…5: 17.12 Bailey Pairs
§17.12 Bailey Pairs
… ►Bailey Pairs
… ►When (17.12.5) is iterated the resulting infinite sequence of Bailey pairs is called a Bailey Chain. … ►The Bailey pair that implies the Rogers–Ramanujan identities §17.2(vi) is: … ►The Bailey pair and Bailey chain concepts have been extended considerably. …6: 28.17 Stability as
§28.17 Stability as
►If all solutions of (28.2.1) are bounded when along the real axis, then the corresponding pair of parameters is called stable. All other pairs are unstable. ►For example, positive real values of with comprise stable pairs, as do values of and that correspond to real, but noninteger, values of . ►However, if , then always comprises an unstable pair. …7: 10.75 Tables
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Achenbach (1986) tabulates , , , , , 20D or 18–20S.
Kerimov and Skorokhodov (1985a) tabulates 5 (nonreal) complex conjugate pairs of zeros of the principal branches of and for , 8D.
Bickley et al. (1952) tabulates or , or , , (.01 or .1) 10(.1) 20, 8S; , , , or , 10S.
Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of and , for , 9S.
Zhang and Jin (1996, p. 322) tabulates , , , , , , , , , 7S.
8: 3.8 Nonlinear Equations
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►Let be an approximation to the real quadratic factor of that corresponds to a pair of conjugate complex zeros or to a pair of real zeros.
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3.8.15
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►Consider and .
We have and .
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3.8.16
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9: 36.5 Stokes Sets
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36.5.4
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36.5.7
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►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4).
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