# moduli

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## 7 matching pages

##### 1: 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. … Choose four relatively prime moduli $m_{1},m_{2},m_{3}$, and $m_{4}$ of five digits each, for example $2^{16}-3$, $2^{16}-1$, $2^{16}+1$, and $2^{16}+3$. …Because each residue has no more than five digits, the arithmetic can be performed efficiently on these residues with respect to each of the moduli, yielding answers $a_{1}\pmod{m_{1}}$, $a_{2}\pmod{m_{2}}$, $a_{3}\pmod{m_{3}}$, and $a_{4}\pmod{m_{4}}$, where each $a_{j}$ has no more than five digits. …
##### 2: 22.17 Moduli Outside the Interval [0,1]
###### §22.17(i) Real or Purely Imaginary Moduli
Jacobian elliptic functions with real moduli in the intervals $(-\infty,0)$ and $(1,\infty)$, or with purely imaginary moduli are related to functions with moduli in the interval $[0,1]$ by the following formulas. …
##### 5: 29.1 Special Notation
 $m,n,p$ nonnegative integers. … complete elliptic integrals of the first kind with moduli $k,k^{\prime}$, respectively (see §19.2(ii)).
##### 6: 29.2 Differential Equations
This equation has regular singularities at the points $2pK+(2q+1)\mathrm{i}{K^{\prime}}$, where $p,q\in\mathbb{Z}$, and $K$, ${K^{\prime}}$ are the complete elliptic integrals of the first kind with moduli $k$, $k^{\prime}(=(1-k^{2})^{1/2})$, respectively; see §19.2(ii). …
##### 7: 2.11 Remainder Terms; Stokes Phenomenon
We now compute the forward differences $\Delta^{j}$, $j=0,1,2,\dots$, of the moduli of the rounded values of the first 6 neglected terms: …