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Given two distributions Y and X with joint probability density function f(x,y), let U=Y/X be the ratio distribution. Then the distribution function of u is D(u) = P(U<=u) (1) ...

Let u_k be a series with positive terms and suppose rho=lim_(k->infty)(u_(k+1))/(u_k). Then 1. If rho<1, the series converges. 2. If rho>1 or rho=infty, the series diverges. ...

A rational amicable pair consists of two integers a and b for which the divisor functions are equal and are of the form sigma(a)=sigma(b)=(P(a,b))/(Q(a,b))=R(a,b), (1) where ...

If alpha is any number and m and n are integers, then there is a rational number m/n for which |alpha-m/n|<=1/n. (1) If alpha is irrational and k is any whole number, there ...

Any square matrix T has a canonical form without any need to extend the field of its coefficients. For instance, if the entries of T are rational numbers, then so are the ...

The rational distance problem asks to find a geometric configuration satisfying given properties such that all distances along specific edges are rational numbers. (This is ...

It is possible to find six points in the plane, no three on a line and no four on a circle (i.e., none of which are collinear or concyclic), such that all the mutual ...

There are nine possible types of isolated singularities on a cubic surface, eight of them rational double points. Each type of isolated singularity has an associated normal ...

A quotient of two polynomials P(z) and Q(z), R(z)=(P(z))/(Q(z)), is called a rational function, or sometimes a rational polynomial function. More generally, if P and Q are ...

A synonym for integer. The word "rational" is sometimes used for emphasis to distinguish it from other types of "integers" such as cyclotomic integers, Eisenstein integers, ...