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The mean of a distribution with probability density function P(x) is the first raw moment mu_1^', defined by mu=<x>, (1) where <f> is the expectation value. For a continuous ...

Consider solutions to the equation x^y=y^x. (1) Real solutions are given by x=y for x,y>0, together with the solution of (lny)/y=(lnx)/x, (2) which is given by ...

A sieving procedure that can be used in conjunction with Dixon's factorization method to factor large numbers n. Pick values of r given by r=|_sqrt(n)_|+k, (1) where k=1, 2, ...

A public-key cryptography algorithm which uses prime factorization as the trapdoor one-way function. Define n=pq (1) for p and q primes. Also define a private key d and a ...

Let the two-dimensional cylinder function be defined by f(x,y)={1 for r<R; 0 for r>R. (1) Then the Radon transform is given by ...

A random variable is a measurable function from a probability space (S,S,P) into a measurable space (S^',S^') known as the state space (Doob 1996). Papoulis (1984, p. 88) ...

Closed forms are known for the sums of reciprocals of even-indexed Lucas numbers P_L^((e)) = sum_(n=1)^(infty)1/(L_(2n)) (1) = sum_(n=1)^(infty)1/(phi^(2n)+phi^(-2n)) (2) = ...

The consideration of the result of a numerical calculation as a function of an adjustable parameter (usually the step size). The function can then be fitted and evaluated at ...

The second Steiner circle (a term coined here for the first time) is the circumcircle of the Steiner triangle DeltaS_AS_BS_C. Its center has center function ...

The sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ... (OEIS A002024) consisting of 1 copy of 1, 2 copies of 2, 3 copies of 3, and so on. Surprisingly, there exist simple formulas ...