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A group G is nilpotent if the upper central sequence 1=Z_0<=Z_1<=Z_2<=...<=Z_n<=... of the group terminates with Z_n=G for some n. Nilpotent groups have the property that ...

A pile of counters in a game of nim. The nim-heap illustrated above corresponds to the game of Marienbad.

A conic section on which the midpoints of the sides of any complete quadrangle lie. The three diagonal points P, Q, and R also lie on this conic.

Any cubic curve that passes through eight of the nine intersections of two given cubic curves automatically passes through the ninth.

Let S_N(s)=sum_(n=1)^infty[(n^(1/N))]^(-s), (1) where [x] denotes nearest integer function, i.e., the integer closest to x. For s>3, S_2(s) = 2zeta(s-1) (2) S_3(s) = ...

Let M be a finitely generated module over a commutative Noetherian ring R. Then there exists a finite set {N_i|1<=i<=l} of submodules of M such that 1. intersection ...

If two curves phi and psi of multiplicities r_i!=0 and s_i!=0 have only ordinary points or ordinary singular points and cusps in common, then every curve which has at least ...

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 ...

Any irreducible curve may be carried by a factorable Cremona transformation into one with none but ordinary singular points.

An assemblage of faces forming a polyhedron of zero volume (Holden 1991, p. 124).