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The pair group of a group G is the group that acts on the 2-subsets of {1,...,p} whose permutations are induced by G. Pair groups can be calculated using PairGroup[g] in the ...

Given a smooth manifold M with an open cover U_i, a partition of unity subject to the cover U_i is a collection of smooth, nonnegative functions psi_i, such that the support ...

A point group is a group of symmetry operations which all leave at least one point unmoved. Although an isolated object may have an arbitrary Schönflies symbol, the ...

The highest order power in a univariate polynomial is known as its order (or, more properly, its polynomial degree). For example, the polynomial ...

A set S of integers is said to be recursive if there is a total recursive function f(x) such that f(x)=1 for x in S and f(x)=0 for x not in S. Any recursive set is also ...

A regular number, also called a finite decimal (Havil 2003, p. 25), is a positive number that has a finite decimal expansion. A number such as 1/3=0.33333... which is not ...

The Robertson-Seymour theorem, also called the graph minor theorem, is a generalization of the Kuratowski reduction theorem by Robertson and Seymour, which states that the ...

In n dimensions for n>=5 the arrangement of hyperspheres whose convex hull has minimal content is always a "sausage" (a set of hyperspheres arranged with centers along a ...

A commutative Noetherian unit ring having only finitely many maximal ideals. A ring having the same properties except Noetherianity is called quasilocal. If K is a field, the ...

A Lie algebra over a field of characteristic zero is called semisimple if its Killing form is nondegenerate. The following properties can be proved equivalent for a ...