A space is said to be subparacompact if every open covering of it has a σ-discrete closed refinement. Subparacompactness is equivalent to Fσ -screenability of McAuley and also to σ-paracompactness of Arhangel'skiĭ. Some properties of these spaces have been obtained in this note. Countably subparacompact spaces, which can be defined in an analogous manner, have also been studied.

By constructing a special set of A-orthonormal functions, it is shown that, under certain smoothness assumptions, the variational and Fourier series representations for the solution of first initial boundary value problems for the simple parabolic differential equation coincide. This result is then extended in order to construct a variational representation for the solution of a very general first initial boundary value problem for this equation.

Strongly differentiable solutions of the minimal surface equation are shown to be classical solutions and consequently locally analytic. A global regularity result is also proved.

Since Paul J. Cohen's 1963 result, it has been possible to investigate the consequences of the axioms of Zermelo-Fraenkel set theory without the Axiom of Choice. In this paper we examine the relative strengths of a number of finiteness criteria against the background of ZF set theory without AC.

We show that each archimedean lattice-ordered group is contained in a unique (up to isomorphism) minimal archimedean vector lattice. This improves a result of Paul F. Conrad appearing previously in this Bulletin. Moreover, we show that this relationship between archimedean lattice-ordered groups and archimedean vector lattices is functorial.

Let H be a subgroup of a finite group G. In this paper the Brauer correspondence between blocks of H and blocks of G is characterized in terms of a relationship among the block idempotents.

In Problem 1 of her book Varieties of groups, Hanna Neumann asked whether a fully invariant subsemigroup of a free group of infinite rank is necessarily a subgroup. This note presents an example which shows that the answer is negative.

In this paper we establish (c, 1) summability of the sequence {nBn (x)} and by using Tauber's Second Theorem, we deduce the convergence criterion of the conjugate series of a Fourier series.

Let for t ∈ [a, b] ⊂ [0, ∞)

where Ws is an n-dimensional Wiener process, f(s) an n-vector process and G(s) an n × m matrix process. f and G are nonanticipating and sample continuous. Then the set of limit points of the net

in Rn is equal, almost surely, to the random ellipsoid Et = G(t)Sm, Sm = {x ∈ Rm: |x| ≤ 1}. The analogue of Lévy's law is also given. The results apply to n-dimensional diffusion processes which are solutions of stochastic differential equations, thus extending the versions of Hinčin's and Lévy's laws proved by H.P. McKean, Jr, and W.J. Anderson.

A power-associative ring A is called a p-ring provided there exists a prime p so that for every x in A, xp = x and px = 0. It is shown that if A is such a ring with p ≠ 2, then A is isomorphic to a subdirect sum of copies of GF(p), the Galois field with p elements.

We prove that if C is an abelian category and M is the class of all coretractions, then the class of M-fibrations is the class of all retractions. As a corollary we prove that the class of all retractions is contained in the class of M-fibrations for any M.

Let p denote a prime, G a. finite soluble p'-group and A an elementary abelian p-group of operators on G. Suppose that A has order p4 and that if ω ∈ A# then CG(ω) has nilpotent derived group. Then G has nilpotent derived group.

There exists a non-solvable group which is third Engel. More generally, the existence of a non-solvable group in which every n-generator subgroup is nilpotent of class at most 2n - 1 is confirmed.

Let X be a commutative locally convex Hausdorff topological algebra with identity over a non-trivially valued field F. Let Mc denote the continuous nontrivial homomorphisms of X into F and M the set of all maximal ideals of X. If the spectrum of each element x in X is the set of scalars {f(x) | f ∈ Mc}, it is shown that the singular elements of X are weakly dense in X if and only if M is an infinite set.

Let denote the variety of all abelian groups and, for each prime p, let p be the variety of all elementary abelian p-groups. Let be a subvariety of a product of (finitely many) varieties each of which is either soluble or Cross.

Let N be a subgroup of the torsion-free abelian group G. Then a partial order for N is contained in one, two or uncountably many full orders for G, and a full order for nonzero N is contained in one or uncountably many full orders for G.

It is proved that, for any m given distinct real numbers a1, …, am, there exist transcendental entire functions f(z) at most of order m for which all the values

are rational integers.