In this note we consider commutative rings with identity over which every unitary module is a zero-divisor module. We call such rings Universally Zero-divisor (UZD) rings. We show (1) a Noetherian ring R is a UZD if and only if R is semilocal and the Krull dimension of R is at most one, (2) a Prüfer domain R is a UZD if and only if R has only a finite number of maximal ideals, and (3) if a ring R has Noetherian spectrum and descending chain condition on prime ideals then R is a UZD if and only if Spec (R) is a finite set. The question of ascent and descent of the property of a ring being a UZD with respect to integral extension of rings has also been answered.

In this paper we study the asymptotic behaviour, as |x| → ∞, of solutions to the initial value problem in nonlinear three-dimensional thermoelasticity in some weighted Sobolev spaces. We show that under some conditions, solutions decrease fast for each t as x tends to infinity. We also consider the possible extension of the method presented in this paper to the initial boundary value problem in exterior domains.

Let X be a complete regular space. We denote by Cb(X) the Banach space of all real-valued bounded continuous functions on X endowed with the supremumnorm.

In this paper we give a characterisation of weakly compact operators defined from Cb(X) into a Banach space E which are β∞-continuous, where β∞ is a locally convex topology on Cb(X) introduced by Wheeler. We also prove that (Cb(X), β∞) has the strict Dunford-Pettis property and, if X is a σ-compact space, (Cb(X), β∞), has the Dunford-Pettis property.

It is shown that a bounded linear operator T from Lρ(Rn) to itself which commutes both with translations and dilatations is a finite linear combination of Hilbert-type transforms. Using this we show that the ρ-norm of the Hilbert transform is the same as the ρ-norm of its truncation to any Lebesgue measurable subset of Rn with non-zero measure.

At the Second Oklahoma Conference on Convexity and Related Combinatorial Geometry (1980) G.Sierksma posed several problems dealing with the generalised Helly and Radon numbers of a convexity space. The aim of this note is to give answers to and comment on some of Sierksma's questions.

We consider C∞ manifolds endowed with torsionfree affine connections and C∞ projective submersions between them which, by definition, map geodesics into geodesics up to parametrisation. After giving a differential characterisation of these mappings, we deal with the case when one of the given connections is projectively flat or satisfies certain conditions concerning its Ricci tensor; under these hypotheses we prove that the projective submersion is actually a covering.

The main result of this paper is that a continuous convex function with domain in a locally convex space factors through a normed space. In a recent paper by Sharp, topological linear spaces are categorised according to the differentiability properties of their continuous convex functions; we show that under suitable conditions the classification is preserved by linear maps. A technique for deducing results for locally convex spaces from Banach space theory is an immediate consequence. Examples are given and Asplund C(S) spaces are characterised.

The obvious necessary conditions for a triple system of index two having a given automorphism group to underly a directed triple system having the same group are shown to be sufficient. An efficient algorithm for directing the triple system in a way that preserves its group is given. Applications to the existence of directed triple systems with various automorphisms are outlined.

In this paper we determine the left ideals in the near-ring Aff(V) of all affine transformations of a vector space V. It is shown that there is a Galois correspondence between the filters of affine subspaces of V and those left ideals of Aff(V) which are not left invariant. In particular, the not left invariant finitely generated left ideals of Aff(V) are precisely the annihilators of the affine subspaces of V. A similar correspondence exists between the filters of linear subspaces of V and the left invariant left ideals of Aff (V). If V is finite-dimensional, then all left ideals of Aff(V) are finitely generated.

I shall consider the following problem: given a stable, conservative, single-exit q-matrix, Q, over an irreducible state-space S and a μ-subinvariant measure, m, for Q, determine all Q-processes for which m is a μ-invariant measure. I shall provide necessary and sufficient conditions for the existence and uniqueness of such a process.

We prove several commutativity theorems for unital rings with polynomial constraints on certain subsets, which improve and generalise the recent results of Grosen, and Ashraf and Quadri.

For a ring R with identity we show that the existence of certain nilpotent elements forces R to be a matrix ring of size ≥ 2.

In this paper the result of Sobczyk about complemented copies of c0 is extended to a class of Banach spaces X such that the unit ball of their dual endowed with the weak* topology has a certain topological property satisfied by every Corson-compact space. By means of a simple example it is shown that if Corson-compact is replaced by Rosenthal-compact, this extension does not hold. This example gives an easy proof of a result of Phillips and an easy solution to a question of Sobczyk about the existence of a Banach space E, c0 ⊂ E ⊂ l∞, such that E is not complemented in l∞ and c0 is not complemented in E. Assuming the continuum hypothesis, it is proved that there exists a Rosenthal-compact space K such that C(K) has no projectional resolution of the identity.

Let F be a free group on a1,…, ap (p ≥ 1), and X a finitely generated subgroup in F. Suppose either that X contains some non-trivial power of , or that p is even and X contains some nontrivial power of [a1, a2]…[ap−1, ap]. We discuss some properties of X which we can derive from this assumption.

Shortly connected and shortly linked inverse semigroups arise in the study of inverse semigroups determined by the lattices of inverse subsemigroups and by the partial automorphism semigroups. It has been shown that every shortly linked inverse semigroup is shortly connected, but the question of whether the converse is true has not been addressed. Here we construct two examples of combinatorial shortly connected inverse semigroups which are not shortly linked. One of them is completely semisimple while the other is not.

A uniform space is trans-separable if every uniform cover has a countable subcover. We show that a uniform space is trans-separable if it contains a suitable family of precompact sets. Applying this result to locally convex spaces, we are able to deduce that the precompact subsets of a wide class of spaces are metrisable. The proof of our main Theorem is based on a cardinality argument, and is reminiscent of the classical Bolzano-Weierstrass Theorem.

This paper establishes new criteria for meromorphic starlike univalent functions of the form

Further property preserving integrals are considered.

In this paper we give some new types of inequalities which can be regarded as discrete forms of the Wirtinger inequality and several new isoperimetric-type inequalities.

Let {Sn, n ≥ 1} denote the partial sum of sequence (Xn) of identically distributed martingale differences. It is shown that E|X1|q (lg |X1|)r < ∞ implies E(sup((lg n)pr/q/npr/q)|Sn|p) < ∞, where 1 < p < 2, p < q, r ∈ R and lg x = max{1, log+x} For the independent identically distributed case, the converse of the above statement holds.

Some general theorems concerning the relative behaviour of moduli of smoothness are established. In particular an open problem raised by Dickmeis, Nessel and van Wickeren is negated.