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Four circles c_1, c_2, c_3, and c_4 are tangent to a fifth circle or a straight line iff T_(12)T_(34)+/-T_(13)T_(42)+/-T_(14)T_(23)=0. (1) where T_(ij) is the length of a ...

Also called Macaulay ring, a Cohen Macaulay ring is a Noetherian commutative unit ring R in which any proper ideal I of height n contains a sequence x_1, ..., x_n of elements ...

A sieving procedure that can be used in conjunction with Dixon's factorization method to factor large numbers n. Pick values of r given by r=|_sqrt(n)_|+k, (1) where k=1, 2, ...

A simple directed graph is a directed graph having no multiple edges or graph loops (corresponding to a binary adjacency matrix with 0s on the diagonal). The number of simple ...

The vertex of an isosceles triangle having angle different from the two equal angles is called the apex of the isosceles triangle. The common polygon vertex at the top of a ...

A two-sided (doubly infinite) Laplace transform, L_t[f(t)](s)=int_(-infty)^inftyf(t)e^(-st)dt. While some authors use this as the primary definition of "the" Laplace ...

A section of a fiber bundle gives an element of the fiber over every point in B. Usually it is described as a map s:B->E such that pi degreess is the identity on B. A ...

The first corona of a tile is the set of all tiles that have a common boundary point with that tile (including the original tile itself). The second corona is the set of ...

Attractive compounds can be constructed from the cube 3-compounds and their octahedron 3-compound duals. One is illustrated above. This compound will be implemented in a ...

The disorder number of a simple connected graph on n vertices is defined as the maximum length of a walk along the edges of the graph taken over all ordering of its vertices ...