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The ratio X_1/X_2 of uniform variates X_1 and X_2 on the interval [0,1] can be found directly as P_(X_1/X_2)(u) = int_0^1int_0^1delta((x_1)/(x_2)-u)dx_1dx_2 (1) = ...

Modular arithmetic is the arithmetic of congruences, sometimes known informally as "clock arithmetic." In modular arithmetic, numbers "wrap around" upon reaching a given ...

A prime ideal is an ideal I such that if ab in I, then either a in I or b in I. For example, in the integers, the ideal a=<p> (i.e., the multiples of p) is prime whenever p ...

In a 1631 edition of Academiae Algebrae, J. Faulhaber published the general formula for the power sum of the first n positive integers, sum_(k=1)^(n)k^p = H_(n,-p) (1) = ...

Amazingly, the distribution of a difference of two normally distributed variates X and Y with means and variances (mu_x,sigma_x^2) and (mu_y,sigma_y^2), respectively, is ...

The distribution of a product of two normally distributed variates X and Y with zero means and variances sigma_x^2 and sigma_y^2 is given by P_(XY)(u) = ...

A linear transformation x_1^' = a_(11)x_1+a_(12)x_2+a_(13)x_3 (1) x_2^' = a_(21)x_1+a_(22)x_2+a_(23)x_3 (2) x_3^' = a_(31)x_1+a_(32)x_2+a_(33)x_3, (3) is said to be an ...

Let H=l^2, (alpha_n) be a bounded sequence of complex numbers, and (xi_n) be the (usual) standard orthonormal basis of H, that is, (xi_n)(m)=delta_(nm), n,m in N, where ...

A ring torus constructed out of a square of side length c can be dissected into two squares of arbitrary side lengths a and b (as long as they are consistent with the size of ...

An operator defined on a set S which takes two elements from S as inputs and returns a single element of S. Binary operators are called compositions by Rosenfeld (1968). Sets ...