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The four-color theorem states that any map in a plane can be colored using four-colors in such a way that regions sharing a common boundary (other than a single point) do not ...

A set of real numbers x_1, ..., x_n is said to possess an integer relation if there exist integers a_i such that a_1x_1+a_2x_2+...+a_nx_n=0, with not all a_i=0. For ...

Let x=[a_0;a_1,...]=a_0+1/(a_1+1/(a_2+1/(a_3+...))) (1) be the simple continued fraction of a "generic" real number x, where the numbers a_i are the partial denominator. ...

A matrix is a concise and useful way of uniquely representing and working with linear transformations. In particular, every linear transformation can be represented by a ...

Minimal surfaces are defined as surfaces with zero mean curvature. A minimal surface parametrized as x=(u,v,h(u,v)) therefore satisfies Lagrange's equation, ...

A normal distribution in a variate X with mean mu and variance sigma^2 is a statistic distribution with probability density function ...

A number is said to be simply normal to base b if its base-b expansion has each digit appearing with average frequency tending to b^(-1). A normal number is an irrational ...

The quaternions are members of a noncommutative division algebra first invented by William Rowan Hamilton. The idea for quaternions occurred to him while he was walking along ...

A right triangle is triangle with an angle of 90 degrees (pi/2 radians). The sides a, b, and c of such a triangle satisfy the Pythagorean theorem a^2+b^2=c^2, (1) where the ...

In his Meditationes algebraicae, Waring (1770, 1782) proposed a generalization of Lagrange's four-square theorem, stating that every rational integer is the sum of a fixed ...