First, we give a more general extension theorem about continuous mappings. It is shown that the Taimanov-Eilenberg-Steenrod extension theorem and the Engelking extension theorem are special cases and that this theorem implies the Dugunji extension theorem. The localic version of this theorem also generalises Joyal's extension theorem for locales. Then, using the same technique, we obtain another more interesting extension theorem and its applications. In particular, we sharpen a well-known result due to Wallman.

Let B be a Banach space. consider the convex proper weakly complete cones X contained in B′ with σ(B′, B) such that X ∩ B′, is conic in the sense of Asimow: that is, there exists α ≥ 0 and f ∈ B″ such that ‖ ‖B ≤ f ≤ α·‖ ‖B on X. This class arises in the theory of integral representations.

If B is reflexive, such a cone has a weakly-compact basis. This paper considers the converse problem:- if one requires that X ∩ B′1 be σ(B′, B) metrisable, the existence of X (without a compact σ(B′, B) basis) is equivalent to the statement that B is not a Grothendieck space.

However, in every space C(K) with infinitely compact K, one can find such a cone X. If two such cones in B′ are not too far apart, their sum belongs to this class.

A necessary and sufficient condition is obtained for a first order linear delay differential inequality to be oscillatory. Our main result improves and extends some known results.

Sullivan's problem of describing rings, all of whose additive endomorphisms are multiplicative, is generalised to the study of rings R satisfying ϕ(a1 … an) = ϕ(a1)…ϕ(an) for every additive endomorphism ϕ of R, and all a1,…,an ∈ R, with n > 1 a fixed positive integer. It is shown that such rings possess a bounded (finite) ideal A such that [R/A]n = 0 ([R/A]2n−1 = 0). More generally, if f(X1, …, Xt) is a homogeneous polynomial with integer coefficients, of degree > 1, and if a ring R satisfies ϕ[f(a1, …, at)] = f[ϕ(a1), …, ϕ(at)] for all additive endomorphisms ϕ, and all a1, …, at ∈ R, then R possesses a bounded ideal A such that R/A satisfies the polynomial identity f.

Integral mean and coefficient bounds for some Bazilevic functions are determined.

Λ is a lattice and K a bounded, open, convex set in En. An axial diameter of K is the maximal length Xi, of chords of K parallel to the ith lattice basis vector (1 ≤ i ≤ n). A number of properties of the axial diameters are developed. For sets K containing just one lattice point, an inequality is established; when Λ is the integral lattice, this inequality takes the form .

In this article we show that, under suitable conditions a quadratic functional attains its minimum on a closed convex cone (in a finite dimensional real Hilbert space) whenever it is bounded below on the cone. As an application, we solve Generalised Linear Complementarity Problems over closed convex cones.

Let C denote the class of all near-rings which have the property that the subnear-ring of constants forms an ideal. Prominent examples are abstract affine near-rings and a generalisation of these by Feigelstock [1]. In this note we show C forms a variety and construct a proper sub-class such that every N ε C can be embedded into some . It turns out that near-rings have an ideal structure which is similar to the ideal structure of abstract affine near-rings, in contrast to the situation for arbitrary elements of C.

The main object of this paper is to extend the theorem of W.H. Young on convergence of the conjugate Fourier series to Wiener's class. We prove that the conjugate Fourier series of a function of Wiener's class converges to for every x for which exists.

Let G be a torsion-free abelian group of finite rank n and let F be a full free subgroup of G. Then G/F is isomorphic to T1 ⊕ … ⊕ Tn, where T1 ⊆ T2 ⊆ … ⊆ Tn ⊆ ℚ/ℤ. It is well known that type T1 = inner type G and type Tn = outer type G. In this note we give two characterisations of type Ti for 1 < i < n.

We characterise artinian left duo rings via Morita duality, and show that there is a large class of such rings that possess a duality.

Using a weighted Poincaré inequality, we study (ω1,…,ωn)-elliptic operators. This method is applied to solve singular elliptic equations with boundary conditions in W1,2. We also obtain a result about the regularity of solutions of singular elliptic equations. An application to (ω1,…,ωn)-parabolic equations is given.

It is shown that if V is an element of the lattice of the title then the map given by U → (V ∧ U, V ∨ U) is a complete lattice embedding of into (V] × [V) if and only if V is a join-infinitely distributive element. In this case the image of the map is a subdirect product of the principal ideal (V] by the principal filter [V) generated by V. Some important varieties in are shown to be join-infinitely distributive.

In some situations weak convergence in L1, implies strong convergence. Let P, L: T → C∘(ℝd) be measurable multifunctions (C∘(ℝd) being the set of closed, convex subsets of ℝd) the values L(t) affine sets and W(t) = P(t) ∩ L(t) extremal faces of P(t). Let pk be integrable selections of P, the projection of pk,(t) on L(t) and pk(t) on W(t). We prove that if converges weakly to zero then pk − k converges to zero in measure. We give also some extensions of this theorem. As applications to differential inclusions we investigate convergence of derivatives of convergent sequences of solutions and we describe solutions which are in some sense isolated. Finally we discuss what can be said about control functions u when the corresponding trajectories of ẋ = f(t, x, u) are convergent to some trajectory.

We prove that the classical sequence James space has the fixed point property. This gives an example of Banach space with a non-unconditional basis where the Maurey-Lin's method applies.

The weak global dimension of a right coherent ring with left Krull dimension α ≥ 1 is found to be the supremum of the weak dimensions of the β-critical cyclic modules, where β < α. If, in addition, the mapping I → assl gives a bijection between isomorphism classes on injective left R-modules and prime ideals of R, then the weak global dimension of R is the supremum of the weak dimensions of the simple left R-modules. These results are used to compute the left homological dimension of a right coherent, left noetherian ring. Some analogues of our results are also given for rings with Gabriel dimension.

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

In this paper we prove some characterisations of weakly compact operators defined from Cb(X) into a Banach space E which are continuous with respect to fit, βt, βr and βσ introduced by Sentilles.

We also prove that (Cb,(X), βi), i = t, τσ , has the Dunford-Pettis property.

In this paper neighbourhood lattices are developed as a generalisation of topological spaces in order to examine to what extent the concepts of “openness”, “closedness”, and “continuity” defined in topological spaces depend on the lattice structure of P(X), the power set of X.

A general pre-neighbourhood system, which satisfies the poset analogues of the neighbourhood system of points in a topological space, is defined on an ∧-semi-lattice, and is used to define open elements. Neighbourhood systems, which satisfy the poset analogues of the neighbourhood system of sets in a topological space, are introduced and it is shown that it is the conditionally complete atomistic structure of P(X) which determines the extension of pre-neighbourhoods of points to the neighbourhoods of sets.

The duals of pre-neighbourhood systems are used to generate closed elements in an arbitrary lattice, independently of closure operators or complementation. These dual systems then form the backdrop for a brief discussion of the relationship between preneighbourhood systems, topological closure operators, algebraic closure operators, and Čech closure operators.

Continuity is defined for functions between neighbourhood lattices, and it is proved that a function f: X → Y between topological spaces is continuous if and only if corresponding direct image function between the neighbourhood lattices P(X) and P(Y) is continuous in the neighbourhood sense. Further, it is shown that the algebraic character of continuity, that is, the non-convergence aspects, depends only on the properites of pre-neighbourhood systems. This observation leads to a discussion of the continuity properties of residuated mappings. Finally, the topological properties of normality and regularity are characterised in terms of the continuity properties of the closure operator on a topological space.

In this paper we shall define a generalised A-quadratic form and prove that in some way this form and an A-bilinear form are equivalent to each other. Our result characterises that of Vukman in the sense that we use any n vectors for a fixed n ≥ 2, instead of any two vectors. Consequently, a new generalisation of an inner product space among vector spaces is obtained. This also leads to a new relationship between a 2-inner product space and a 2-normed space.

Given a closed and convex set K in Rn and two continuous maps F: K → Rn and η: K × K → Rn, the problem considered here is to find ε K such that

.

We call it a variational-like inequality problem, and develop a theory for the existence of a solution. We also show the relationship between the variational-like inequality problem and some mathematical programming problems.