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In a noncommutative ring R, a left ideal is a subset I which is an additive subgroup of R and such that for all r in R and all a in I, ra in I. A left ideal of R can be ...

An ideal I of a partial order P is a subset of the elements of P which satisfy the property that if y in I and x<y, then x in I. For k disjoint chains in which the ith chain ...

A positive matrix is a real or integer matrix (a)_(ij) for which each matrix element is a positive number, i.e., a_(ij)>0 for all i, j. Positive matrices are therefore a ...

Evans et al. (2000, p. 6) use the unfortunate term "probability domain" to refer to the range of the distribution function of a probability density function. For a continuous ...

The set of all Schwartz functions is called a Schwartz space and is denoted S(R^n). If C_0^infty(R^n) denotes the set of smooth functions of compact support on R^n, then this ...

A subset X of R^n is star convex if there exists an x_0 in X such that the line segment from x_0 to any point in X is contained in X. A star-shaped figure is star convex but ...

A submonoid is a subset of the elements of a monoid that are themselves a monoid under the same monoid operation. For example, consider the monoid formed by the nonnegative ...

If V and W are Banach spaces and T:V->W is a bounded linear operator, the T is said to be a compact operator if it maps the unit ball of V into a relatively compact subset of ...

A free Abelian group is a group G with a subset which generates the group G with the only relation being ab=ba. That is, it has no group torsion. All such groups are a direct ...

A set of generators (g_1,...,g_n) is a set of group elements such that possibly repeated application of the generators on themselves and each other is capable of producing ...