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A function f(x) decreases on an interval I if f(b)<=f(a) for all b>a, where a,b in I. If f(b)<f(a) for all b>a, the function is said to be strictly decreasing. Conversely, a ...

A set partition of the rational numbers into two nonempty subsets S_1 and S_2 such that all members of S_1 are less than those of S_2 and such that S_1 has no greatest ...

A metatheorem in mathematical logic also known under the name "conditional proof." It states that if the sentential formula B can be derived from the set of sentential ...

A subspace A of X is called a deformation retract of X if there is a homotopy F:X×I->X (called a retract) such that for all x in X and a in A, 1. F(x,0)=x, 2. F(x,1) in A, ...

A limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class. For example, the point is a degenerate case of the circle as ...

Degen's eight-square identity is the incredible polynomial identity (1) found around 1818 by the Danish mathematician Ferdinand Degen (1766-1825). It was subsequently ...

A (Delta,D)-graph is a graph with maximum vertex degree Delta and diameter at most D. The order of a graph with degree Delta of diameter D is bounded by ...

A diagonal matrix D=diag(d_1,...,d_n) sometimes also called the valency matrix corresponding to a graph that has the vertex degree of d_i in the ith position (Skiena 1990, p. ...

An embedding of a 1-sphere in a 3-manifold which exists continuously over the 2-disk also extends over the disk as an embedding. An alternate phrasing is that if a knot group ...

The operator partial^_ is defined on a complex manifold, and is called the 'del bar operator.' The exterior derivative d takes a function and yields a one-form. It decomposes ...