Let ψ be a non-decreasing continuous subadditive function defined on [0, ∞) and satisfy ψ(x) = 0 if and only if x = 0. The space H(ψ) is defined as the set of analytic functions in the unit disk which satisfy

and the space H+ (ψ) is the space of a f ∊ H(ψ) for which

where almost everywhere.

In this paper we study the H(ψ) spaces and characterize the continuous linear functionals on H+ (ψ).

An asymptotic approximation is obtained, as k → ∞, for the integral

where Φ is the cumulative distribution function for a standard normal random variable, and L is a positive constant. The problem is motivated by a question in statistics, and an outline of'the application is given. Similar methods may be used to approximate other integrals involving the normal distribution.

The concept of the M-radical of a submodule B of an R-module A is discussed (R is a commutative ring with identity and A is a unitary fl-module). The M-radical of B is defined as the intersection of all prime submodules of A containing B. The main result of the paper is that if denotes the ideal radical of (B:A), then M-rad B = provided that A is a finitely generated multiplication module. Additionally, it is shown that if A is an arbitrary module, where for some

This paper gives a sharper explicit representation of the set of optimal solutions for a class of linear programs than those obtained by A. Ben-Israel, A. Charnes and S. Zlobec since 1968. The representation is used to determine bounds on the coefficients of the objective function that produce the same set of optimal solutions (sensitivity analysis).

Let G be the group of integers of a p-series field. A class {E(θ)} of perfect null subsets of G is introduced and classified into M-sets and U-sets according to the arithmetical nature of the field element θ. This is analogous to the well-known classification, involving Pisot numbers, of certain Cantor sets on the circle.

The (F, dn) method is investigated with respect to perfectness, strong regularity, and summing bounded divergent sequences. In the process the columns of the inverse matrix are characterized in terms of {dn}.

We show that a topological space X admits a unique quasiuniformity if and only if every interior-preserving open collection of X is finite.

The surjectivity of epimorphisms in the category of planar graphs and edge-preserving maps follows from and is implied by the Four Colour Theorem. The argument that establishes the equivalence is not combinatorially complex.

Let R be a prime algebra over a field F and let d be a non-zero derivation in R such that for every x ∊ R, d(x) is algebraic over F of bounded degree. Then R is a primitive ring with a minimal right ideal eR, where e2 = e and eRe is a finite dimensional central division algebra.

Greene and Wu have shown that any continuous strongly convex function on a Riemannian manifold can be uniformly approximated by infinitely differentiable strongly convex functions. This result is not true if the word “strongly” is omitted; in this paper, we give examples of manifolds on which convex functions cannot be approximated by convex functions (k = 0, 1,2,...).

It is a question of Arhangel'skiĭ [1] (Problem 2) whether the identity ψ(G) = X(G) holds for every minimal Hausdorff topological group G = 〈G,u〉). (Here, as usual, ψ(G), the pseudocharacter of G, is the least cardinal number K for which there is such that and and x(G), the character of G,is the least cardinality of a local base at e for (〈G,u〉.) That 〈G, u〉 is minimal means that, if v is a Hausdorff topological group topology for G and v ⊂ u, then v = u.

In this paper, we give some conditions on G sufficient to ensure a positive response to Arhangel'skiï's question, and we offer an example which responds negatively to a question on minimal groups posed some years ago (cf. [6] (p. 107) and [4] (p. 259)).

Let Tk = T1#...#T1 T1 = Sl x Sl, Uk = ℝP2#... #ℝP2, and G is a finite group. We prove (1) Every free action of G on Ul + 2 lifts to a free action of G on the orientable two fold cover Tl+1 → Ul+1 and (2) The minimum k such that can act freely on Tk is ml((l - 2)/2) + 1 if m = 2 or l is even and ml((l - 1)/2) + 1 otherwise.

E. Pirtle has conjectured that if D is an almost Krull domain in which the height one prime ideals are divisorial then D is a Krull domain. An example is given to show that this is not the case. Further, let U = and let denote the set of prime ideals of D which are minimal over some ideal (a):(b), where a, b ∈ D. If Dp is a valuation ring for each let then Huckaba and Papick have asked whether D[x]U must be a Prufer domain. The given example shows that it need not be.

The oscillatory behavior of the solutions of the neutral delay differential equation

where p, τ, and a are positive constants and Q ∊ C([t0, ∞), ℝ+), are studied.

Let L be a finitely generated field extension of a field K. The order of inseparability of L/K is the minimum of {n|[L:S] = pn where S is a separable extension of K}. If V is a subfield of L/K, then its order of inseparability is less than or equal to that of L/K. This paper examines the question of when there are unique minimal subfields of order of inseparability n — j, 0 ≤ j ≤ n.

Let L be a lattice and q a convergence structure (or a topology) finer than the interval topology of L. In case of compact maximal chains and continuous lattice translations, the connected components of the space (L,q) are characterized using lattice conditions only. Moreover, lattice conditions of L are related to connectedness conditions of the order convergence space (L, o). Throughout this note, maximal chain conditions and maximal chain techniques play an important role.

The abelian groups which are the additive groups of only finitely many non-isomorphic (associative) nilpotent rings are studied. Progress is made toward a complete classification of these groups. In the torsion free case, the actual number of non-isomorphic nilpotent rings these groups support is obtained.

Let G be a group, let RG be the group ring of the group G over the unital commutative ring R and let U(RG) be its group of units. Conditions which imply that U(RG) contains no free noncyclic group are studied, when R is a field of characteristic p ≠ 0, not algebraic over its prime field, and G is a solvable-by-finite group without p-elements. We also consider the case R = ℤp, the ring of p-adic integers and G torsionby- nilpotent torsion free group. Finally, the residual nilpotence of U(ℤpG) is investigated.

We follow P. Fletcher and W. F. Lindgren's work in the study of topological spaces with a unique quasi-uniformity by generalizing some of their results and constructing larger classes of uqu spaces which contain some of their examples as a particular case.

Let A and A denote the classes of ordinal spaces with the order topology and Σ-product spaces of the two point discrete space respectively. Characterizations are given in terms of ultrafiIters of clopen sets of those O-dimensional Hausdorff topological spaces that can be embedded homeomorphically as a closed subspace of a topological product of either spaces from the class Λ or the class Δ. Both classes consist of spaces that are ω0-bounded. An example is given of a O-dimensional Hausdorff ω0-bounded space that cannot be homeomorphically embedded as a closed subset of a product of spaces from either Λ or Δ, answering a question of R. G. Woods.