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Given a unit disk, find the smallest radius r(n) required for n equal disks to completely cover the unit disk. The first few such values are r(1) = 1 (1) r(2) = 1 (2) r(3) = ...

The distribution function D(x), also called the cumulative distribution function (CDF) or cumulative frequency function, describes the probability that a variate X takes on a ...

A formula for the Bell polynomial and Bell numbers. The general formula states that B_n(x)=e^(-x)sum_(k=0)^infty(k^n)/(k!)x^k, (1) where B_n(x) is a Bell polynomial (Roman ...

The most general forced form of the Duffing equation is x^..+deltax^.+(betax^3+/-omega_0^2x)=gammacos(omegat+phi). (1) Depending on the parameters chosen, the equation can ...

The Dyson mod 27 identities are a set of four Rogers-Ramanujan-like identities given by A(q) = 1+sum_(n=1)^(infty)(q^(n^2)(q^3;q^3)_(n-1))/((q;q)_n(q;q)_(2n-1)) (1) = ...

The Egyptian Mathematical Leather Roll (EMLR), dates to the Middle Kingdom, and was purchased in Egypt in 1858 by Henry Rhind, near the time when the Rhind papyrus was ...

Let the elliptic modulus k satisfy 0<k^2<1, and the Jacobi amplitude be given by phi=amu with -pi/2<phi<pi/2. The incomplete elliptic integral of the first kind is then ...

Let the elliptic modulus k satisfy 0<k^2<1. (This may also be written in terms of the parameter m=k^2 or modular angle alpha=sin^(-1)k.) The incomplete elliptic integral of ...

A self-intersecting minimal surface which can be generated using the Enneper-Weierstrass parameterization with f(z) = 1 (1) g(z) = zeta. (2) Letting z=re^(iphi) and taking ...

The path traced out by a point P on the edge of a circle of radius b rolling on the outside of a circle of radius a. An epicycloid is therefore an epitrochoid with h=b. ...