A metrizable topological space has a metric taking values in a closed subset of the real numbers having Čech dimension zero if and only if the space itself has Čech dimension zero. We call a development D = {Dn} for a topological space (X, T) a sieve for X if the sets in each Dn are pairwise disjoint. Then a Hausdorff topological space (X, T) has a compatible metric taking values in a closed subset of the real numbers having Čech dimension zero if and only if there exists a sieve for X.

An α-distributive (respectively α-implicative) semilattice S is a lower semilattice (with greatest lower bound denoted by juxtaposition) in which the annihilator 〈x, a〉, that is {y ∈ S: xy ≤ α}, is an ideal (respectively a principal ideal) for the fixed element α and any x of S. These semilattices appear as natural links between general and distributive semi-lattices on the one hand, and between pseudo-complemented and implicative semilattices on the other hand. Prime and dense elements, as well as maximal and prime filters, are essential. Mandelker's result, a lattice L is distributive if and only if 〈x, y〉 is an ideal for any x, y ∈ L is extended to semi-lattices.

This paper studies the connection between the best possible value of a constant in the compact abelian case of a known inequality due to Helgason and the Λ2-constants of sets of characters. Various estimates of and expressions for the best possible value are given.

Let H be a fixed Hilbert space and B(H, H) be the Banach space of bounded linear operators from H to H with the uniform operator topology. Oscillation criteria are obtained for the operator differential equation

where the coefficients A, C are linear operators from B(H, H) to B(H, H), for each t ≤ 0. A solution Y: R+ → B(H, H) is said to be oscillatory if there exists a sequence of points ti ∈ R+, so that ti → ∞ as i → ∞, and Y(ti) fails to have a bounded inverse. The main theorem states that a solution Y is oscillatory if an associated scalar differential equation is oscillatory.

The object of this present paper is to study a nonlinear perturbed stochastic integral equation of the form

where ω ∈ Ω, the supporting set of the complete probability measure space (Ω A, μ). We are concerned with the existence and uniqueness of a random solution to the above equation. A random solution, x(t; ω), of the above equation is defined to be a vector random variable which satisfies the equation μ almost everywhere.

It is shown that a saturated formation F has the property that in each group G each F-projector of G is complemented if and only if F is the formation of finite soluble π-groups for some set π of primes.

The complex linear complementarity problem considered here is the following: Find z such that

where S is a polyhedral convex cone in Cp, S* the polar cone, M ∈ Cp×p and q ∈ Cp.

Generalizing earlier results in real and complex space, it is shown that if M satisfies RezHMz ≥ 0 for all z ∈ Cp and if the set satisfying Mz + q ∈ S*, z ∈ S is not empty, then a solution to the complex linear complementarity problem exists. If RezHMz > 0 unless z = 0, then a solution to this problem always exists.

This note presents a characterization of connected simple expansions of topologies in terms of conditions on subspaces, and gives corollaries improving and unifying known sufficient conditions. These results are applied to obtain information about maximally connected spaces, and it is shown that maximal connectedness is inherited by connected subspaces.

Suppose G = AB where G is a finite group and A and B are nilpotent subgroups. It is proved that the derived length of G modulo its Frattini subgroup is at most the sum of the classes of A and B. An upper bound for the derived length of G in terms of the derived lengths of A and B also is obtained.

In this paper, we exhibit the necessary and sufficient conditions for a ring R to have only the trivial torsion parts with respect to any (hereditary) radical on the category of left R-modules.

Let G be a Hausdorff locally compact abelian group, Γ its character group. We shall prove that, if S is a translation-invariant subspace of Lp (G) (p ∈ [1, ∞]),

for each a ∈ G and , then is relatively compact (where Σ(f) denotes the spectrum of f). We also obtain a similar result when G is a Hausdorff compact (not necessarily abelian) group. These results can be considered as a converse of Bernstein's inequality for locally compact groups.