Let A be an abelian group, let ∧ = End (A), and assume that A is a flat left ∧-module. Then σ = { right ideals I ⊂ ∧ | IA = A} generates a linear topology oil ∧. We prove that Hom(A,·) is an equivalence from the category of those groups B ⊂ An satisfying B = Hom(A, B)A, onto the category of σ-closed submodules of finitely generated free right ∧-modules. Applications classify the right ideal structure of A, and classify torsion-free groups A of finite rank which are (nearly) isomorphic to each A-generated subgroup of finite index in A.

A uniform space can be given a boundedly compact compatible psuedometric if and only if it is uniformly locally compact and second countable and has a countable base for its entourages.

In this paper, we give a new characterisation of pseudo-compact spaces, namely a space X is pseudocompact if and only if each σ-point finite open cover of X has a finite subfamily whose union is dense. As a corollary, we show that every pseudocompact σ-metacompact (or screenable) space is compact, which sharpens some known results.

A geometrically aberrant Banach space with uniformly normal structure, a modification of Brown's example, is given.

Let Ω denote the closed interval [0, 1] and let bA denote the set of all bounded, approximately continuous functions on Ω. Let Q denote the Banach space (sup norm) of quasi-continuous functions on Ω. Let M denote the closed convex cone in Q comprised of non-decreasing functions. Let hp, 1 < p < ∞, denote the best Lp-simultaneaous approximation to the bounded measurable functions f and g by elements of M. It is shown that if f and g are elements of Q, then hp converges unifornily to a best L1-simultaneous approximation of f and g. We also show that if f and g are in bA, then hp is continuous.

This paper shows that amongst the p-groups of order p5, where p denotes an odd prime, there is only one group whose automorphism group is again a p-group. This automorphism group has order p6 and it is shown that this is the smallest order a p-group may have when it occurs as an automorphism group. The paper also shows that all groups of order p5 have an automorphism of order 2 apart from the group above and three other related groups.

Using the Iwasawa structure theorem for connected locally compact Hausdorff groups we show that every locally compact Hausdorff group G is homeomorphic to Rn × K × D, where n is a non-negative integer, K is a compact group and D is a discrete group. This makes recent results on cardinal numbers associated with the topology of locally compact groups more transparent. For abelian G, we note that the dual group, Ĝ, is homeomorphic to This leads us to the relationship card G = ω0(Ĝ) + 2ω0(G), where ω (respectively, ω0) denotes the weight (respectively local weight) of the topological group. From this classical results such as card G = 2 card Ĝ for compact Hausdorff abelian groups, and ω(G) = ω(Ĝ) for general locally compact Hausdorff abelian groups are easily derived.

The p-norm of the Hilbert transform is the same as the p-norm of its truncation to any Lebesgue measurable set with strictly positive measure. This fact follows from two symmetry properties, the joint presence of which is essentially unique to the Hilbert transform. Our result applies, in particular, to the finite Hilbert transform taken over (−1, 1), and to the one-sided Hilbert transform taken over (0, ∞). A related weaker property holds for integral operators with Hardy kernels.

In the present paper the asymptotic and globally asymptotic stability of the zero solution of systems with impulses are investigated. For this purpose piecewise continuous auxiliary functions are used which are analogous to Lyapunov's functions. The theorem of Marachkov on the asymptotic stability of systems without impulses is generalised. The results obtained are formulated in four theorems.

Inserting new knots into B-spline curves is a well-known technique in CAGD to gain extra flexibility for design purposes. However, from a user's point of view, the insertion of knots is somewhat unsatisfactory since the newly generated control points sometimes show up in unexpected locations. The aim of this note is to show that these problems can be circumvented by inserting the control vertices directly, thus also providing a more natural user interface.

The “type II conjecture”, proposed by J. Rhodes, gives an algorithm to compute the kernel of a given finite semigroup. We show that this conjecture is a consequence of another conjecture, of a topological nature. This new conjecture gives a simple and effective characterisation of the recognisable subsets of a free monoid that are closed in the finite group topology for the free monoid.

We show that while farthest points always exist in w* -compact sets in duals to Radon-Nikodym spaces, this is generally not the case in dual Radon-Nikodym spaces. We also show how to characterise weak compactness in terms of farthest points.

Let I = [0, 1] and let f be an unimodal expanding map in C0(I, I). If f has an expanding constant for some integers m ≧ 0 and n ≧ 1, where λn is the unique positive zero of the polynomial x2n+1 − 2x2n−1 −1, then we show that f has a periodic point of period 2m(2n + 1). The converse of the above result is trivially false. The condition in the above result is the best possible in the sense that we cannot have the same conclusion if the number λn is replaced by any smaller positive number and the generalisation of the above result to arbitrary piecewise monotonic expanding maps in C0(I, I) is not possible.

Each ring contains a unique smallest ideal which when factored out yields a ring with zero middle annihilator. Various results concerning this ideal are obtained including theorems about how it behaves in connection with normalising extensions and smash products.

The method of nondiscrete mathematical induction is used to find sharp error bounds for Newton's method. We assume only that the operator has Hölder continuous derivatives. In the case when the Fréchet-derivative of the operator satisfies a Lipschitz condition, our results reduce to the ones obtained by Ptak and Potra in 1972.

Let f be a function meromorphic in D = {|z| < 1} and let X be the chordal distance on the Riemann sphere. Then f satisfies the Lipschitz condition

in D if and only if |f′(z)|/(1 + |f(z)|2) = O((1 – |z|)α−1) and |z| → 1.

Bear's characterisation of central-by-finite groups as groups possessing a finite covering by abelian subgroups if the starting point for this invertigation. We characterise groups with a finite covering by 2-Engel subgroups as groups for which the subgroup of right 2-Engel elements has finite index; and the groups having a finite covering by normal 2-Engel subgroups are exactly the 3-Engel groups among those having a finite covering by 2-Engel subgroups. The second centre of a group having a finite covering by class two subgroups does not necessarily have finite index. However, a group has a finite covering by subgroups in a variety containing all cyclic groups if the margin of thes variety in the group has finite index.