Let V be a group generated by elements ν1 and ν2 of finite coprime order, and let N be the near-ring generated by the inner automorphism induced by ν1.It is proved that V is a monogenic N-group. Certain consequences of this result are discussed. There exist finite near-rings N with identity generated by a single distributive element μ, such that μ2 = 1 and where the radical J2(N) (see Günter Pilz, Near-rings. The theory and its applications, 1977, p. 136) is non-nilpotent.

Let C(X) denote the complete boolean algebra of Borel sets modulo first category sets of the space X. Given an isomorphism τ between C(X) and C(Y), where X and Y are complete metric spaces, it is shown that there exists a homeomorphism T, between residual subsets A of X and B of Y, that induces τ. When X = Y one can make A = B. An analogous result is stated when τ is a complete isomorphism onto a subalgebra.

Let p be a prime number. The Lie ring of the largest finite group of exponent p and nilpotency class 3p − 3 is determined under certain assumptions (which are conjectured always to hold).

For the interpolation polynomial of Hermite-Fejér type An[f] of degree less than or equal to 4n − 1 constructed on the nodes , k = 1, 2, …, n, it is shown that for f ∈ CM(Ω) the inequality

holds where CM(Ω) is the class of continuous functions on [−1, 1] satisfying certain conditions, Ω is a certain modulus of continuity, and c3 and M are positive constants.

If A = {am,n} is a regular summability matrix, the sequence s = {sn} is said to be A uniformly distributed (see L. Kuipers, H. Niederreiter, Uniform distribution of sequences, p. 221, John Wiley & Sons, New York, London, Sydney, Toronto, 1974), if

(h = 1, 2, …). In this paper we examine sequences belonging to A*, where t ∈ A* if and only if t is bounded and s + t is A uniformly distributed whenever s is A uniformly distributed. By A′ are denoted those members t of A* such that at ∈ A* for every real a. The members of A′ form a Banach algebra, A* is not connected under the sup norm, but A′ is a component.

Given a closed, convex cone S, in Rn, its polar S* and a mapping g from Rn into itself, the generalized nonlinear complementarity problem is to find a z ∈ Rn such that

Many existence theorems for the problem have been established under varying conditions on g. We introduce new mappings, denoted by J(S)-functions, each of which is used to guarantee the existence of a solution to the generalized problem under certain coercivity conditions on itself. A mapping g:S → Rn is a J(S)-function if

imply that z = 0. It is observed that the new class of functions is a broader class than the previously studied ones.

The natural order of an inverse semigroup defined by a ≤ b ⇔ a′b = a′a has turned out to be of great importance in describing the structure of it. In this paper an order-theoretical point of view is adopted to characterise inverse semigroups. A complete description is given according to the type of partial order an arbitrary inverse semigroup S can possibly admit: a least element of (S, ≤) is shown to be the zero of (S, ·); the existence of a greatest element is equivalent to the fact, that (S, ·) is a semilattice; (S, ≤) is directed downwards, if and only if S admits only the trivial group-homomorphic image; (S, ≤) is totally ordered, if and only if for all a, b ∈ S, either ab = ba = a or ab = ba = b; a finite inverse semigroup is a lattice, if and only if it admits a greatest element. Finally formulas concerning the inverse of a supremum or an infimum, if it exists, are derived, and right-distributivity and left-distributivity of multiplication with respect to union and intersection are shown to be equivalent.

Applying a new and very elegant method of proof of the Schur-Hardy inequality, given by E.R. Love at the Oberwolfach conference on Linear Spaces and Approximation (1977), norm estimates of integral operators with homogeneous kernels are established in the setting of abstract function norms. Applications to Flett's inequality, to integral means, and to fractional integrals are given.

The local connectivity, νk(G), of a graph G is the minimum of the connectivities of neighbourhoods of the vertices of G. G is minimally locally n-connected if νk(G) = n and for every edge x of G, νk(G−x) = n − 1. A necessary and sufficient condition for a locally connected graph to be minimally locally 1-connected is given, and it is shown that for n ≥ 7, is minimally locally 1-connected.

Let Cφ be a composition operator on L2(λ), where λ is a σ-finite measure on a set X. Conditions under which Cφ is invertible, unitary, and normal are investigated in this paper.

Necessary and sufficient conditions on an arbitrary Gabriel filter of left ideals of a ring R are determined in order that the ring of quotients of R with respect to the filter be semi-simple artinian. Special instances include generalizations of earlier work on classical rings of quotients and maximal rings of quotients.

Let p be a prime and let (mk)k<ω be a strictly increasing sequence of positive integers such that m0 = 1 and mk divides mk+1. A field F is said to be of type (p, (mk)k<ω) if it is the union of an increasing sequence (Fk)k<ω of fields such that Fk has pmk elements. A set X is called “finite” if it has n elements for some nonnegative integer n, and “Dedekind-finite” if every injection f: X → X is a bijection. If the Axiom of Choice is rejected, then it is relatively consistent to assume the existence of medial (that is, infinite, Dedekind-finite) sets. In this paper it is shown that given any type (p, (mk)k<ω) as above, it is relatively consistent with the usual axioms of set theory (minus Choice) to assume the existence of a medial field of type (p, (mk)k<ω). Conversely, it is shown that any medial field must be of type (p, (mk)k<ω) for some (p, (mk)k<ω) as above. The paper concludes with a few observations on Dedekind-finite rings. In the first part of the paper, a general knowledge of Fraenkel-Mostowski set theory and of the Jech-Sochor Embedding Theorems is assumed.

In a single-server queueing system, subject to the queue discipline ‘First come first served’, the equilibrium distribution function of the waiting time of a server depends on the distribution of the random variable (u) which is the difference between the service time and the inter-arrival time. If in two queueing systems u's are equivalent in distribution, the waiting times are also equivalent in distribution (known result). It has been shown in this note that equivalence in waiting time distributions does not necessarily imply equivalence in distributions of u's. The proof is heuristic. This result has useful practical implications.

Let K be a bounded, closed, convex set in the euclidean plane having diameter d, width w, inradius r, and circumradius R. We show that

and

where both these inequalities are best possible.