A general group-theoretic procedure is indicated for representing rational integers as products of other integers. A detailed example is given.

Let R be an associative ring which is not necessarily commutative. For any torsion theory τ on the category of left R-modules and for any nonnegative integer n we define and study the notion of the nth local cohomology functor with respect to τ. For suitably nice rings a bound for the nonvanishing of these functors is given in terms of the τ-dimension of the modules.

Let S be a regular semigroup and A a D(S)-module. We proved in a previous paper that the set Ext(S, A) of equivalence classes of extensions of A by S admits an abelian group structure and studied its functorial properties. The main aim of this paper is to describe Ext(S, A) as a second cohomology group of certain chain complex.

It is well-known that in any near-ring, any intersection of prime ideals is a semi-prime ideal. The aim of this note is to prove that any ideal is a prime ideal if and only if it is equal to its prime radical. As a consequence of this we have any semi-prime ideal I in a near-ring N is the intersection of minimal prime ideals of I in N and that I is the intersection of all prime ideals containing I.

In this paper, the questions about bitopological spaces proposed by S. Lal are solved and one of his counterexamples rectified.

Given an ordered Banach space ℬ equipped with an order-norm we construct a larger space ℬ¯ with an order-norm and order-identity such that B is isometrically order-isomorphic to a Banach subspace of B. We also discuss the extension of positive operators from ℬto ℬ¯.

An undirected simple graph G is called chordal if every circle of G of length greater than 3 has a chord. For a chordal graph G, we prove the following: (i) If m is an odd positive integer, Gm is chordal. (ii) If m is an even positive integer and if Gm is not chordal, then none of the edges of any chordless cycle of Gm is an edge of Gr, r < m.

We prove a conjecture of Lennox and Wiegold that a finitely generated soluble group, in which every infinite subset contains two elements generating a supersoluble group, is finite-by-supersoluble.

Stampfli and Embry have shown that a point of the numerical range of an operator is extreme if and only if a set of vectors corresponding to it is linear. This is generalized here to show that a point of the closure of the numerical range is extreme if and only if a corresponding set of sequences forms a linear space. A more geometric alternative proof is given for a theorem of Das and Garske concerning weak convergence to zero at the unattained extreme points of the closure of the numerical range.The result is shown to hold also for lone extreme points of the numerical range which lie on line segments on its boundary. Further, a bound is obtained on the norm of the weak limit of the weakly convergent sequences corresponding to points on a line segment on the boundary of numerical range.

Several morphisms of this lattice V(CR) are found, leading to decompostions of it, and various sublattices, into subdirect products of interval sublattices. For example the map V → V ∪ G (where G is the variety of groups) is shown to be a retraction of V(CR); from modularity of the lattice V(BG) of varieties of bands of groups it follows that the map V → (V ∪ V V G) is an isomorphism of V(BG).

This paper defines descriptive, Riemann, and constructive integrals equivalent to the approximately continuous integral of Burkill.

This paper is concerned with a characterization of the optimal order of convergence of Tikhonov regularization for first kind operator equations in terms of the “smoothness” of the data.

The structure and geometry of Banach spaces with the property that E(4) = Ê** + E⊥ ⊥ are investigated: such spaces are called quotient reflexive spaces here. For these spaces, if E is very smooth, Ê is also very smooth, and if E* is weakly locally uniformly rotund (WLUR), E(4) is smooth on a certain (relatively) norm dense subset of Ê**. Consequently, for quotient reflexive spaces, WLUR and very-WLUR are equivalent in E*.

We introduce the class of p-adic F-functions which contains both the p-adic E-function and p-adic G-functions, as well as other functions. In this paper we obtain lower bounds for polynomials in the values at algebraic points of a class of p-adic F-functions defined over the completion of the algebraic closure of a p-adic field.

For a cardinal m with ω ≤ m ≤ 2≤m is embeddable in it as a closed subspace. We derive this result from the corresponding statement for metric spaces.