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Let B, A, and e be square matrices with e small, and define B=A(I+e), (1) where I is the identity matrix. Then the inverse of B is approximately B^(-1)=(I-e)A^(-1). (2) This ...

Numbers 1, alpha_1, ..., alpha_L are rationally independent iff under the action of rotation rho_(alpha_1)×...×rho_(alpha_L) on the L-dimensional torus, every orbit is ...

A one-person game played on a rectangular lattice of lamps which can be turned on and off. A move consists of flipping a "switch" inside one of the squares, thereby toggling ...

The surface of revolution generated by the external catenary between a fixed point a and its conjugate on the envelope of the catenary through the fixed point is equal in ...

A matrix with 0 determinant whose determinant becomes nonzero when any element on or below the diagonal is changed from 0 to 1. An example is M=[1 -1 0 0; 0 0 -1 0; 1 1 1 -1; ...

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 ...

The north pole is the point on a sphere with maximum z-coordinate for a given coordinate system. For a rotating sphere like the Earth, the natural coordinate system is ...

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

The pentagonal gyrocupolarotunda is a convex equilateral solid that is Johnson solid J_(33). The unit pentagonal gyrocupolarotunda has volume V=5/(12)(11+5sqrt(5)) (1) and ...

The class m, curve order n, number of ordinary double points delta, number of cusps kappa, number of inflection points (inflection points) iota, number of bitangents tau, and ...