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A method for predicting the onset of widespread chaos. It is based on the hypothesis that the dissolution of an invariant torus can be associated with the sudden change from ...

The kernel of a group homomorphism f:G-->G^' is the set of all elements of G which are mapped to the identity element of G^'. The kernel is a normal subgroup of G, and always ...

The second-order ordinary differential equation (1+x^2)^2y^('')+lambday=0 (Hille 1969, p. 357; Zwillinger 1997, p. 122).

Let D=D(z_0,R) be an open disk, and let u be a harmonic function on D such that u(z)>=0 for all z in D. Then for all z in D, we have 0<=u(z)<=(R/(R-|z-z_0|))^2u(z_0).

A similarity transformation which preserves orientation, also called a homothety.

Two similar figures with parallel homologous lines and connectors of homologous points concurrent at the homothetic center are said to be in homothetic position. If two ...

The roulette of the pole of a hyperbolic spiral rolling on a straight line is a tractrix.

A number of attractive 11-compounds of the regular icosahedron can be constructed. The compound illustrated above will be implemented in a future version of the Wolfram ...

The ideal quotient (a:b) is an analog of division for ideals in a commutative ring R, (a:b)={x in R:xb subset a}. The ideal quotient is always another ideal. However, this ...

The identity element I (also denoted E, e, or 1) of a group or related mathematical structure S is the unique element such that Ia=aI=a for every element a in S. The symbol ...