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Given a matrix A, let |A| denote its determinant. Then |A||A_(rs,pq)|=|A_(r,p)||A_(s,q)|-|A_(r,q)||A_(s,p)|, (1) where A_(u,w) is the submatrix of A formed by the ...

Informally, a symplectic map is a map which preserves the sum of areas projected onto the set of (p_i,q_i) planes. It is the generalization of an area-preserving map. ...

The Lorentz group is the group L of time-preserving linear isometries of Minkowski space R^((3,1)) with the Minkowski metric dtau^2=-(dx^0)^2+(dx^1)^2+(dx^2)^2+(dx^3)^2 ...

The oriented matroid of a finite configuration of points extracts relative position and orientation information from the configuration. An oriented matroid can be described ...

When two cycles have a transversal intersection X_1 intersection X_2=Y on a smooth manifold M, then Y is a cycle. Moreover, the homology class that Y represents depends only ...

An oriented knot is an oriented link of one component, or equivalently, it is a knot which has been given an orientation. Given an oriented knot K, reversing the orientation ...

Let L denote the n×n triangular lattice with wraparound. Call an orientation of L an assignment of a direction to each edge of L, and denote the number of orientations of L ...

A circle-preserving transformation composed of an odd number of inversions.

An area-preserving affine transformation. Equiaffinities include the crossed hyperbolic rotation, elliptic rotation, hyperbolic rotation, and parabolic rotation.

A circle-preserving transformation composed of an even number of inversions.