Lax pairs
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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: 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. …4: 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. …5: 24.19 Methods of Computation
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►For number-theoretic applications it is important to compute for ; in particular to find the irregular pairs
for which .
We list here three methods, arranged in increasing order of efficiency.
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6: 10.25 Definitions
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§10.25(iii) Numerically Satisfactory Pairs of Solutions
►Table 10.25.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.25.1). … ►7: 27.5 Inversion Formulas
8: 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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9: 27.19 Methods of Computation: Factorization
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►Type II probabilistic algorithms for factoring rely on finding a pseudo-random pair of integers that satisfy .
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