Let G be a nondiscrete locally compact abelian group with dual group Γ. For 1 ≤ p ≤ ∞, denote by Ap(G) the space of integrable functions on G whose Fourier transforms belong to Lp(Γ). We investigate multipliers from Ap(G) to Aq(G). If G is compact and 2 < p1, p2 < ∞, we show that multipliers of and multipliers of are different, provided Pl ≠ P2. For compact G, we also exhibit a relationship between lr (Γ) and the multipliers from Ap(G) to Aq(G). If G is a compact nonabelian group we observe that the spaces Ap(G) behave in the same way as in the abelian case as far as the multiplier problems are concerned.

A Stieltjes-Volterra integral equation system

is firstly considered. Pointwise estimates and boundedness of its solutions are obtained under various conditions on the function K. To do this, the well-known Gronwall-Bellman integral inequality is generalized. For a particular choice of u, it is shown that the integral equation reduces to a difference equation. The problem of existence (and non-existence), uniqueness (and non-uniqueness) of the difference equation is discussed. Gronwall-Bellman inequality is further generalized to n linear terms and is subsequently applied to obtain sufficient conditions in order that a certain stability of the unperturbed Volterra system

implies the corresponding local stability of the (discontinuously) perturbed system

Given an n × n matrix A, an n-dimensional vector q, and a closed, convex cone S of Rn, the generalized linear complementarity problem considered here is the following: find a z ∈ Rn such that

where s* is the polar cone of S. The existence of a solution to this problem for arbitrary vector q has been established both analytically and constructively for several classes of matrices A. In this note, a new class of matrices, denoted by J, is introduced. A is a J-matrix if

The new class can be seen to be broader than previously studied classes. We analytically show that for any A in this class, a solution to the above problem exists for arbitrary vector q. This is achieved by using a result on variational inequalities.

We obtain sufficient conditions for the convergence of martingale triangular arrays to infinitely divisible laws with finite variances, without making the usual assumptions of uniform asymptotic negligibility. Our results generalise known results for both the martingale case under a negligibility assumption and the classical (independence) case without such assumptions.

The natural completion functor on the category of cλ-embedded Cauchy spaces induces a completion functor on a category of sequential Cauchy spaces which preserves sequential regularity.

A characterization has previously been given for linear transformations in Hilbert space whose first N + 1 powers are partial isometries. An analogous characterization is now obtained for transformations whose first N+ 1 powers have closed ranges. A hypothesis (that transformations have no isometric part) is found to be unnecessary in previous work.

Explicit expressions for the Green's functions due to a point force in one of two half space fluids are presented for the case when inertial effects of the fluid are negligible (Stokes flow) and the interface between the two fluids is considered to be flat due to the action of surface tension. The analytic expressions are discussed in terms of singularity diagrams. For the case of a force parallel to the interface a first approximation to the interface displacement is made.

A new graph product is introduced, and the characteristic polynomial of a graph so–formed is given as a function of the characteristic polynomials of the factor graphs. A class of trees produced using this product is shown to be characterized by spectral properties.

It is shown that in a variety of (not necessarily associative) algebras which satisfies a variant of Andrunakievich's Lemma, a class C containing no solvable algebras is the semi-simple class corresponding to some supernilpotent radical class if and only ifCis hereditary and is closed under extensions and sub-direct products. Semi-simple classes in general are not characterized by these properties. If the variety satisfies the further condition that some proper power of every ideal is an ideal, then analogous results hold for the semi-simple classes corresponding to radical classes containing no solvable algebras. In particular, for algebras over a field in the latter situation, all semi-simple classes are characterized by the three closure properties mentioned.

In this note we give external characterizations of absolutely topologicai functors U: A →X (where A may be a large category) as functors which are injective with respect to all functors based on the category X, or equivalently, which do not admit essential extensions over X. From this we deduce several characterizations and properties of the MacNeille completion of faithful functors. These results generalize the work of Horst Herriich [Math. Z. 150 (1976), 101–110] in case of small categories, where the crucial point of this generalization lies in the fact that we cannot make use of the existence of MacNeille completions and we therefore have to introduce a local-initial-completion-construction.

This article contains one method of fully embedding a symmetric closed category into a symmetric monoidal closed category. Such an embedding is very useful in the study of coherence problems. Also we give an example of a non-symmetric closed category for which, under the embedding discussed in this article, the resultant monoidal closed structure has associativity not an isomorphism.

We announce results of Landesman-Lazer type for boundary-value problems for ordinary differential equations. Details will appear elsewhere.

Finite O-simple semigroups with a Rees representation over an elementary abelian group are considered. The isomorphism classes of such semigroups with class graph γ and coefficient group of order pd are shown to correspond to the orbits under the automorphism group of γ of those subspaces of the circuit space of γ over Zp which have codimension at most d. A semigroup interpretation of the inclusion relation between subspaces is given and results are obtained on the structure of the automorphism group and the enumeration of the semigroups under consideration.

A lemma is presented which is a weak version of the inverse function theorem, in that differentiability is assumed instead of continuous differentiability. The result holds only for finite dimensional spaces; a counter-example is given for the infinite dimensional analogue. The lemma is used to answer a question posed by Nadler concerning differentiable retracts.

Let X be a set and G a group which acts on X and is generated by two elements α and b. Motivated by a geometric problem of L. Fejes Tóth, we define a subset S ⊂ X to have [α, b]-symmetry if its images under α and b satisfy Sα ∩ Sb = S. The problem of finding all sets with [α, b]- symmetry when an arbitrary 2-generator group G acts on an arbitrary space X is shown to be equivalent to the same problem in the special case when the 2-generator free group acts on itself by right translation. This action is modelled in the hyperbolic plane in a way that helps to reveal the [α, b]- symmetric subsets of the free group.

Let N denote the set of natural numbers and let φ be a mapping from N into itself. Then the composition transformmation Cφ, on the weighted l2 space with weights a2n, where n ∈ N and 0 < a < 1 is defined by Cφf = f ∘ φ. If Cφ is a bounded operator, then it is called a composition operator. The adjoint of the composition operator Cφ, is computed, and it is used to characterise normal, unitary, isometric, and co-isometric composition operators. Not every invertible φ induces an invertible composition operator, as is shown by examples. At the end of this note all invertible composition operators are characterised.

An oscillation criterion is established for a class of functional differential equations including the generalized Emden-Fowler equation

as a special case. The deviating arguments involved may be retarded or advanced or otherwise. The result extends and improves known fundamental oscillation criteria for superlinear differential equations with retarded arguments.

This paper, in conjunction with its algebraic sequel, aims to provide a foundation, long outstanding, to the theory of quasigroups obeying the law x(xy) = yx , otherwise known as Stein quasigroups.

The oscillation property of the semilinear hyperbolic or ultra-hyperbolic operator L defined by

is studied. Sufficient conditions are provided for all solutions of uL[u] ≤ 0 satisfying certain boundary conditions to be oscillatory. The basis of our results is the non-existence of positive solutions of the associated differential inequalities.

Let K be a closed convex set in the plane containing no nonzero point of the integral lattice. We show that if the area A(K) of K is equally distributed amongst the four principal quadrants of the plane, then A(K) < 4.