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The second-order ordinary differential equation (d^2y)/(dx^2)+[theta_0+2sum_(n=1)^inftytheta_ncos(2nx)]y=0, (1) where theta_n are fixed constants. A general solution can be ...

The inversive distance is the natural logarithm of the ratio of two concentric circles into which the given circles can be inverted. Let c be the distance between the centers ...

A number which is simultaneously octagonal and square. Let O_n denote the nth octagonal number and S_m the mth square number, then a number which is both octagonal and square ...

The distribution with probability density function and distribution function P(r) = (re^(-r^2/(2s^2)))/(s^2) (1) D(r) = 1-e^(-r^2/(2s^2)) (2) for r in [0,infty) and parameter ...

Let f:R×R->R be a one-parameter family of C^2 maps satisfying f(0,mu)=0 (1) [(partialf)/(partialx)]_(mu=0,x=0)=0 (2) [(partial^2f)/(partialxpartialmu)]_(0,0)>0 (3) ...

WireWorld is a two-dimensional four-color cellular automaton introduced by Brian Silverman in 1987. The rule for the automaton uses the cell's old value a together with the ...

A leaf of an unrooted tree is a node of vertex degree 1. Note that for a rooted or planted tree, the root vertex is generally not considered a leaf node, whereas all other ...

The associated Legendre polynomials P_l^m(x) and P_l^(-m)(x) generalize the Legendre polynomials P_l(x) and are solutions to the associated Legendre differential equation, ...

The value for zeta(2)=sum_(k=1)^infty1/(k^2) (1) can be found using a number of different techniques (Apostol 1983, Choe 1987, Giesy 1972, Holme 1970, Kimble 1987, Knopp and ...

There are two definitions of the Carmichael function. One is the reduced totient function (also called the least universal exponent function), defined as the smallest integer ...