For positive integers n and k, with n≥2k, let , where each prime factor of u is less than k, and each prime factor of v is at least equal to k. It is shown that u<v holds with just 12 exceptions, which are determined. If , where each prime factor of U is at most equal to k, and each prime factor of V is greater than k, then U<V holds with at most finitely many exceptions, 19 of which are determined. It is conjectured that there are no others.

We discuss generalizations of the Lie-Kolchin-Mal'cev theorem. For example we show that if G is a soluble linear group of degree n, then G contains a triangularizable subgroup T whose index in G is bounded by function of n only and such that T is normalized by every automorphism of G normalizing G0, the Zariski connected component of G containing the identity. We also prove that in certain situations at least the index of G0 in G can be bounded in terms of the degree and the ground field.

A cohomology theory for locally trivial, locally compact topological groupoids with coefficients in vector bundles is constructed, generalizing constructions of Hochschild and Mostow (1962) for topological groups and Higgins (1971) for discrete groupoids. It is calculated to be naturally isomorphic to the cohomology of the vertex groups, and is thus independent of the twistedness of the groupoid. The second cohomology space is accordingly realized as those “rigid” extensions which essentially arise from extensions of the vertex group; the cohomological machinery now yields the unexpected result that in fact all extensions, satisfying some natural weak conditions, are rigid.

For normed linear spaces two similar characterizations of strong differentiability of the norm and rotundity of the dual space are established, but it is shown that in general there is no causal relation between these two concepts.

If a weighted Euler transformation is applied to the asymptotic series for ezE1(z) the remainder can be expressed as an integral. Examination of this integral shows that for a transformation of given order the smallest term of the resulting series remains at approximately a constant distance from the start of the series. If, however, there is no restriction on the order of transformation the remainder may be decreased to zero by increasing the number of terms used, but if z is close to the negative real axis the rate of decrease is small. A more general theorem for alternating real series and Taylor's series is also given.

Groups with the property of the title were considered by Chillag (1977); this paper completes his results by showing that, with known exceptions, they are triply transitive.

We show that rings for which every non-constant multiplicative endomorphism is additive are trivial or power rings (that is, rings R such that R = R2 and x2 = 0 = x+x for all x ∈ R) and that if R is a power ring for which every multiplicative endomorphism is additive, then End (R) is a zero semigroup or a semilattice according to how the product is defined.

In the last few years, the oscillatory behavior of functional differential equations has been investigated by many authors. But much less is known about the first-order functional differential equations. Recently, Tomaras (1975b) considered the functional differential equation and gave very interesting results on this problem, namely the sufficient conditions for its solutions to oscillate. The purpose of this paper is to extend and improve them, by examining the more general functional differential equation

As dual to the notion of “finitely injective modules” introduced and studied by Ramamurth and Rangaswamy (1973), we define a right R-module M to be finitely projective if it is projective. with respect to short exact sequences of right R-modules of the form 0 → A → B → C → 0 with C finitely generated. We have completely characterized finitely projective modules over a Dedekind domain. If R is a Dedekind domain, then an R-module M is finitely projective if and only if its reduced part is torsionless and coseparable.

For a Dedekind domain R, finite projectivity, unlike projectivity is not hereditary. But it is proved to be pure hereditary, that is, every pure submodule of a finitely projective R-module is finitely projective.

This paper is a contribution to the problem of characterizing the ℵ0-categorical Stone algebras. If the dense set is either finite or a chain, the problem is solved by reducing it to the ℵ0-categoricity of the skeleton and the dense set, solutions for these being known. If the dense set is a Boolean algebra, we show that this type of reduction works for certain subclasses but not for all such algebras. For generalized Post algebras the characterization problem is solved completely.

The authors investigate max min μF(x) F(y)for two standard distance functions F in R2, where μ denotes the area of {x ∈ R2; F(x)≤ 1}, the maximum is over all (geometric) lattices Λ in R2 and the minimum is over all positive points x ∈ Λ and y ∈ Λ* (the polar lattice of Λ). An application is given to a problem on fractional parts.

Grant (1976) has attempted to establish a relationship between fixing subgraphs and smoothly embeddable subgraphs. Here we give counterexamples to his two main lemmas and two characterizing theorems. We then go on to give our own version of these lemmas and theorems.

An example is constructed of a locally finite variety of non-associative algebras which satisfies the maximal condition on subvarieties but not the minimal condition. Based on this, counterexamples to various conjectures concerning varieties generated by finite algebras are constructed. The possibility of finding a locally finite variety of algebras which satisfies the minimal condition on subvarieties but not the maximal is also investigated.

The proof of Theorem 5 in a paper with the same title is incorrect. In this note weaker versions of that theorem are proved.