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Let there be N_i observations of the ith phenomenon, where i=1, ..., p and N = sumN_i (1) y^__i = 1/(N_i)sum_(alpha)y_(ialpha) (2) y^_ = 1/Nsum_(i)sum_(alpha)y_(ialpha). (3) ...

Let f*g denote the cross-correlation of functions f(t) and g(t). Then f*g = int_(-infty)^inftyf^_(tau)g(t+tau)dtau (1) = ...

The function f(x)=1-2|x|^(1/2) for x in [-1,1]. The natural invariant is rho(y)=1/2(1-y).

The n-cyclohedron, also known as the Bott-Taubes polytope, is defined as the compactification of the configuration space of n points on the circle. The cyclohedron can be ...

The radial curve of the deltoid x = 1/3a[2cost+cos(2t)] (1) y = 1/3a[2sint-sin(2t)] (2) with pedal point (x_0,y_0) is x_p = 1/6[3x+cost+3xcost-cos(2t)-3ysint] (3) y_p = ...

Two cones placed apex to apex. The double cone is given by algebraic equation (z^2)/(c^2)=(x^2+y^2)/(a^2).

The system of partial differential equations u_t = 3ww_x (1) w_t = 2w_(xxx)+2uw_x+u_xw. (2)

For any two ergodic measure-preserving transformations on nonatomic probability spaces, there is an isomorphism between the two probability spaces carrying orbits onto orbits.

A roll of eight on a pair of six-sided dice in the game of craps. The probability of rolling "eighter from Decatur" is 5/36, or 13.888...%.

Consider the family of ellipses (x^2)/(c^2)+(y^2)/((1-c)^2)-1=0 (1) for c in [0,1]. The partial derivative with respect to c is -(2x^2)/(c^3)+(2y^2)/((1-c)^3)=0 (2) ...