This paper discusses the rate of approximation of functions of bounded variation using the Modified Lupas operator. We obtain an approximation theorem and our estimate is essentially the best possible.

The aim of this paper is to give a metric regularity theorem for multifunctions between metric spaces involving some known results for multifunctions by using the notion of strict (G, δ)-differentiability of multifunctions and a simple convergence procedure.

An associative ring is called strongly right (left) bounded if every nonzero right (left) ideal contains a nonzero ideal. We prove that if R is a strongly right bounded finite ring with unity and the order |R| of R has no factors of the form p5, then R is strongly left bounded. This answers a question of Birkenmeier and Tucci.

The aim of this note is to obtain an intrinsic product formula for the centraliser of the injective tensor product of a couple of Banach spaces, Z(). More precisely, we are going to prove that

Here, the spaces and depend only on X and Y, respectively, and Xk denotes the topological k-product.

A Counterexaple used to demonstrate that the k-product cannot beavoided serves as an answer to a question posed by W. Rueß and D. Werner concerning the behaviour of M-ideals on Y.

Given a torsion-free abelian group G, a subgroup A of G is said to be a quasi-summand of G if nG ≤ A ⊕ B ≤ G for some subgroup B of G and some positive integer n. If the intersection of any two quasi-summands of G is a quasi-summand, then G is said to have the quasi-summand intersection property. This is a generalisation of the summand intersection property of L. Fuchs. In this note, we give a complete characterisation of the torsion-free abelian groups (in fact, torsion-free modules over torsion-free rings) with the quasi-summand intersection property. It is shown that such a characterisation cannot be given via endomorphism rings alone but must involve the way in which the endomorphism ring acts on the underlying group. For torsion-free groups G of finite rank without proper fully invariant quasi-summands however, the structure of its quasi-endomorphism ring QE(G) suffices: G has the quasi-summand intersection property if and only if the ring QE(G) is simple or else G is strongly indecomposable.

Let X and Y be metric spaces and C(X, Y) be the space of all continuous functions from X to Y. If X is a locally connected space, the compact-open topology on C(X, Y) is weaker than the Attouch-Wets topology on C(X, Y). The result is applied on the space of continuous linear functions. Let X be a locally convex topological linear space metrisable with an invariant metric and X* be a continuous dual. X is normable if and only if the strong topology on X* and the Attouch-Wets topology coincide.

This note is to indicate a nonsemiprime ring R such that the smallest supernilpotent (respectively special) radical containing the ring R is an atom of the lattice of all supernilpotent (respectively special) radicals. This gives a positive answer to Puczylowski's and Roszkowska's question.

We prove that, in a finite soluble group, all of whose Sylow normalisers are super-soluble, the Fitting length is at most 2m + 2, where pm is the highest power of the smallest prime p dividing |G/Gs| here Gs is the supersoluble residual of G. The bound 2m + 2 is best possible. However under certain structural constraints on G/GS, typical of the small examples one makes by way of experimentation, the bound is sharply reduced. More precisely let p be the smallest, and r the largest, prime dividing the order of a group G in the class under consideration. If a Sylow p–subgroup of G/GS acts faithfully on every r-chief factor of G/GS, then G has Fitting length at most 3.

We present a simple one-parameter family of interval maps which has a direct bifurcation from order to chaos and then a direct (reverse) bifurcation from chaos back to order.

The topological entropy of a continuous map of the interval is the supremum of the topological entropies of the piecewise linear maps associated to its finite invariant sets. We show that for transitive maps, this supremum is attained at some finite invariant set if and only if the map is piecewise monotone and the set contains the endpoints of the interval and the turning points of the map.

This paper connects the literature on weaving with that on coloured tilings of the plane. It sets the many published designs of woven cloth in the context of (2, 2, 2)-fabrics, that is, those in which strands of two colours run in two perpendicular directions to cover the plane doubly almost everywhere. At the same time, it connects with the interesting topological question of when a weave hangs together. The main result is that isonemal (2, 2, 2)-fabrics can be made from isonemal un-coloured fabrics either by colouring them in the conventional way (black warp, white weft), or by colouring warp and weft in thin or thick stripes to give the visual appearance of conventionally coloured isonemal pre-fabrics that fall apart, and in no other way. Such visual appearance can always be realised by the striping of an isonemal fabric, but not uniquely. A figure showing the fifty isonemal (2, 2, 2)-fabrics of exact orders up to sixteen coloured by striping is included.

By characterising CR-hypersurfaces of dimension m having m strongly independent CR-functions, we find the necessary condition for a CR-hypersurface to be locally integrable and show that the same condition is also sufficient if the CR-hypersurface is strongly pseudoconvex and has m strongly independent CR-functions.

We prove the following result: let R be an arbitrary ring with centre Z such that for every x, y ∈ R, there exists a positive integer n = n(x, y) ≥ 1 such that (xy)n − ynxn ∈ Z and (yx)n − xnyn ∈ Z; then, if R has no non-zero nil ideals, R is commutative. We also prove a result on commutativity of general rings: if R is r!-torsion free and for all x, y ∈ R, [xr, ys] = 0 for fixed integers r ≥ s ≥ 1, then R is commutative. As a corollary we obtain that if R is (n + 1)!-torsion free and there exists a fixed n ≥ 1 such that (xy)n − ynxn = (yx)n − xnyn ∈ Z for all x, y ∈ R, then R is commutative.

We give an upper bound for the number of zeros of recurrence sequences defined over an algebraic number field in terms of their order, the degree of their field of definition and the number of prime ideal divisors of the characteristic roots of the sequence.

Running a computer search for new, cyclic, 2-perfect 6-cycle systems of order 13 and constructing the quasigroups which arise from such systems enabled the author to establish that there are at most two such non-isomorphic systems. Then by using two-variable laws of the quasigroups it is shown that there are exactly two non-isomorphic 2-perfect 6-cycle systems of order 13 which are cyclic.

We prove that if a nonflat complete regular minimal surface immersed in Rn is of finite total curvature, then its Gauss map can omit at most (n – 1)(n + 2)/2 hyperplanes in general position in Pn–1 (ℂ).

A class of second-order asymptotically stationary random fields is shown to contain the class of almost harmonisable random fields. A continuity theorem which leads to the spectral representation for the covariance function of asymptotically stationary random fields is established. A mean ergodic theorem for the fields is also given. When stationarity is assumed, the results reduce to the well-known corresponding theorems for stationary random fields.

We prove the following theorem: Let R be a ring, l a positive integer, and n a non-negative integer. If for each x, y ∈ R, either xy = yx or xy = xn f(y)x1 for some f(X) ∈ X2Z[X], then R is commutative.

The notion of finite type submanifolds was introduced by B.Y. Chen. In this article we study 2- and 3-type pseudo-umbilical submanifolds in a hypersphere. Two theorems in this respect are obtained.

It is shown that an n-torsion-free ring R with identity such that, for all x, y in R, xnyn = ynyn and (xy)n+1 − xn+1yn+1 is central, must be commutative. It is also shown that a periodic n–torsion-free ring (not necessarily with identity) for which (xy)n − (yx)n is always in the centre is commutative provided that the nilpotents of R form a commutative set. Further, examples are given which show that all the hypotheses of both theorems are essential.