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Let S be a semigroup and alpha a positive real-valued function on S such that alpha(st)<=alpha(s)alpha(t) (s,t in S). If l^1(S,alpha) is the set of all complex-valued ...

Division by zero is the operation of taking the quotient of any number x and 0, i.e., x/0. The uniqueness of division breaks down when dividing by zero, since the product ...

A univariate function f(x) is said to be even provided that f(x)=f(-x). Geometrically, such functions are symmetric about the y-axis. Examples of even functions include 1 ...

A generalized Vandermonde matrix of two sequences a and b where a is an increasing sequence of positive integers and b is an increasing sequence of nonnegative integers of ...

A group homomorphism is a map f:G->H between two groups such that the group operation is preserved:f(g_1g_2)=f(g_1)f(g_2) for all g_1,g_2 in G, where the product on the ...

Given two modules M and N over a unit ring R, Hom_R(M,N) denotes the set of all module homomorphisms from M to N. It is an R-module with respect to the addition of maps, ...

A differential ideal is an ideal I in the ring of smooth forms on a manifold M. That is, it is closed under addition, scalar multiplication, and wedge product with an ...

The l^2-norm (also written "l^2-norm") |x| is a vector norm defined for a complex vector x=[x_1; x_2; |; x_n] (1) by |x|=sqrt(sum_(k=1)^n|x_k|^2), (2) where |x_k| on the ...

The orthogonal complement of a subspace V of the vector space R^n is the set of vectors which are orthogonal to all elements of V. For example, the orthogonal complement of ...

The prescription that a trigonometry identity can be converted to an analogous identity for hyperbolic functions by expanding, exchanging trigonometric functions with their ...