# Lax pair

(0.002 seconds)

## 1—10 of 76 matching pages

##### 1: 32.4 Isomonodromy Problems
$\mbox{P}_{\mbox{\scriptsize I}}$$\mbox{P}_{\mbox{\scriptsize VI}}$ can be expressed as the compatibility condition of a linear system, called an isomonodromy problem or Lax pair. Suppose …
##### 2: 18.38 Mathematical Applications
###### Integrable Systems
It has elegant structures, including $N$-soliton solutions, Lax pairs, and Bäcklund transformations. …
##### 3: 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 $x\to\pm\infty$
###### §28.17 Stability as $x\to\pm\infty$
If all solutions of (28.2.1) are bounded when $x\to\pm\infty$ along the real axis, then the corresponding pair of parameters $(a,q)$ is called stable. All other pairs are unstable. For example, positive real values of $a$ with $q=0$ comprise stable pairs, as do values of $a$ and $q$ that correspond to real, but noninteger, values of $\nu$. However, if $\Im\nu\neq 0$, then $(a,q)$ always comprises an unstable pair. …
##### 5: 24.19 Methods of Computation
For number-theoretic applications it is important to compute $B_{2n}\pmod{p}$ for $2n\leq p-3$; in particular to find the irregular pairs $(2n,p)$ for which $B_{2n}\equiv 0\pmod{p}$. We list here three methods, arranged in increasing order of efficiency. …
• A method related to “Stickelberger codes” is applied in Buhler et al. (2001); in particular, it allows for an efficient search for the irregular pairs $(2n,p)$. Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).

• ##### 6: 10.25 Definitions
###### §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
27.5.3 $g(n)=\sum_{d\mathbin{|}n}f(d)\Longleftrightarrow f(n)=\sum_{d\mathbin{|}n}g(d)% \mu\left(\frac{n}{d}\right).$
Special cases of Möbius inversion pairs are: …
##### 8: 27.15 Chinese Remainder Theorem
The Chinese remainder theorem states that a system of congruences $x\equiv a_{1}\pmod{m_{1}},\dots,x\equiv a_{k}\pmod{m_{k}}$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. …
##### 9: 27.19 Methods of Computation: Factorization
Type II probabilistic algorithms for factoring $n$ rely on finding a pseudo-random pair of integers $(x,y)$ that satisfy $x^{2}\equiv y^{2}\pmod{n}$. …
##### 10: 9.2 Differential Equation
###### §9.2(iii) Numerically Satisfactory Pairs of Solutions
Table 9.2.1 lists numerically satisfactory pairs of solutions of (9.2.1) for the stated intervals or regions; compare §2.7(iv).