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A necessary and sufficient condition for a measure which is quasi-invariant under a transformation to be equivalent to an invariant probability measure is that the ...

A measure algebra which has many properties associated with the convolution measure algebra of a group, but no algebraic structure is assumed for the underlying space.

Let a plane figure have area A and perimeter p. Then Q=(4piA)/(p^2)<=1, where Q is known as the isoperimetric quotient. The equation becomes an equality only for a circle.

A roll of 4 on a pair of six-sided dice in the game of craps (Gardner 1978, p. 256). The probability of rolling "a little Joe" is 1/12, or 8.333...%. A roll of "4 the hard ...

A measure space is a measurable space possessing a nonnegative measure. Examples of measure spaces include n-dimensional Euclidean space with Lebesgue measure and the unit ...

Find two numbers such that x^2=y^2 (mod n). If you know the greatest common divisor of n and x-y, there exists a high probability of determining a prime factor. Taking small ...

A Radon measure is a Borel measure that is finite on compact sets.

Rényi entropy is defined as: H_alpha(p_1,p_2,...,p_n)=1/(1-alpha)ln(sum_(i=1)^np_i^alpha), where alpha>0, alpha!=1. As alpha->1, H_alpha(p_1,p_2,...,p_n) converges to ...

A set of methods that are generally superior to ANOVA for small data sets or where sample distributions are non-normal.

phi(A)+phi(B)-phi(A union B)>=phi(A intersection B).