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For a normed space (X,||·||), define X^~ to be the set of all equivalent classes of Cauchy sequences obtained by the relation {x_n}∼{y_n} if and only if lim_(n)||x_n-y_n||=0. ...

A number n is called a barrier of a number-theoretic function f(m) if, for all m<n, m+f(m)<=n. Neither the totient function phi(n) nor the divisor function sigma(n) has a ...

Given a complex Hilbert space H with associated space L(H) of continuous linear operators from H to itself, the bicommutant M^('') of an arbitrary subset M subset= L(H) is ...

If {a_j} subset= D(0,1) (with possible repetitions) satisfies sum_(j=1)^infty(1-|a_j|)<=infty, where D(0,1) is the unit open disk, and no a_j=0, then there is a bounded ...

Church proved several important theorems that now go by the name Church's theorem. One of Church's theorems states that there is no consistent decidable extension of Peano ...

The circumsphere of given set of points, commonly the vertices of a solid, is a sphere that passes through all the points. A circumsphere does not always exist, but when it ...

Let m(G) be the cycle rank of a graph G, m^*(G) be the cocycle rank, and the relative complement G-H of a subgraph H of G be defined as that subgraph obtained by deleting the ...

Given a complex Hilbert space H with associated space L(H) of continuous linear operators from H to itself, the commutant M^' of an arbitrary subset M subset= L(H) is the ...

The compact-open topology is a common topology used on function spaces. Suppose X and Y are topological spaces and C(X,Y) is the set of continuous maps from f:X->Y. The ...

In general, a cross is a figure formed by two intersecting line segments. In linear algebra, a cross is defined as a set of n mutually perpendicular pairs of vectors of equal ...