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A map u:R^n->R^n from a domain G is called a map of class C^r if each component of u(x)=(u_1(x_1,...,x_n),...,u_m(x_1,...,x_n)) is of class C^r (0<=r<=infty or r=omega) in G, ...

An extremely powerful theorem in physics which states that each symmetry of a system leads to a physically conserved quantity. Symmetry under translation corresponds to ...

A nonsingular linear map A:R^n->R^n is orientation-preserving if det(A)>0.

Orthogonal involution, also called absolute involution, is the involution on the line at infinity that maps orthogonal directions to each other.

The Rabinovich-Fabrikant equation is the set of coupled linear ordinary differential equations given by x^. = y(z-1+x^2)+gammax (1) y^. = x(3z+1-x^2)+gammay (2) z^. = ...

An automorphism which interchanges the two vertices of a regular map at each edge without interchanging the vertices.

A transformation consisting of rotations and translations which leaves a given arrangement unchanged.

A translation along a straight line L and a rotation about L such that the angle of rotation is proportional to the translation at each instant. Also known as a twist.

Any motion of a rigid body in space at every instant is a screw motion. This theorem was proved by Mozzi and Cauchy.

A mapping of a domain F:U->U to itself.