The purpose of this paper is to present three somewhat disparate results on free objects in three different classes of λ-groups. The first is that no proper ideal of a finitely generated free vector lattice can itself be a free vector lattice. Second, each free abelian lgroup is characteristically simple. The third result is that each disjoint subset of a free (non-abelian) lgroup is countable.

A formula is established for differentiating the composite F(ξ)○G(ξ) with respect to the parameter ξ where F and G are maps which assumetheir values in function spaces. This composite is denoted by . Use of this notation, together with the tangent functor T, enables the formula to be written in the variable-free form where π1 is a projection map and TF(ξ) = T(F(ξ)). Formal verificationof this formula is straightforward. It has application to many small divisor problems such as those which occur in celestial mechanics.

A polygon is said to be rational if all its sides and diagonals are rational, and I. J. Schoenberg has posed the difficult question, ‘Can any given polygon be approximated as closely as we like by a rational polygon?’ Many of the known results concerning this question are contained as special cases in theorem 1 below which was proved by one of us (cf. the references).

We provide sufficient conditions to factorise an equivariant spectral triple as a Kasparov product of unbounded classes constructed from the group action on the algebra and from the fixed point spectral triple. We show that if factorisation occurs, then the equivariant index of the spectral triple vanishes. Our results are for the action of compact abelian Lie groups, and we demonstrate them with examples from manifolds and $\unicode[STIX]{x1D703}$-deformations. In particular, we show that equivariant Dirac-type spectral triples on the total space of a torus principal bundle always factorise. Combining this with our index result yields a special case of the Atiyah–Hirzebruch theorem. We also present an example that shows what goes wrong in the absence of our sufficient conditions (and how we get around it for this example).

We show that the complete symmetric digraph DKn, n≧5, can be decomposed into each of the four oriented pentagons if and only if n ≡ 0 or 1 (mod 5).

We study the coefficients An in the expansion of the infinite product We first derive a recurrence relation for An and from it we obtain an explicit expression of An. We then prove a convolution identity involving An.

We study the problem of representing a permutation C as a product of a minimum number, fk(C), of cycles of length k. Upper and lower bounds on fk(C) are obtained and exact results are derived for k = 2, 3, 4.

We define and investigate H-prefrattini subgroups for Schunck classes H of finite soluble groups, and solve a problem of Gaschütz concerning the structure of H-prefrattini groups for H = {1}.

Using the comparsion results for positive compact operators by Aliprantis and Burkinshow, Mokhtar Kharroubi investigated cimpactness properties of positive semigroups on Banach latttices. The aim of this paper is to study these properties in general Banach spaces (without positivity). Our results generalize a part fo those obtained by Mokhtar-Kharroubi to general Banach spaces context. More specifically, we derive conditions which ensure the compactness of the remainder term Rn(t) for some inteter n. The improvement here is that it can applied directly to the neutron transport equation for a wide class of collision operators.

Let ℱ be a family of meromorphic functions defined in D, all of whose zeros have multiplicity at least k+1. Let a and b be distinct finite complex numbers, and let k be a positive integer. If, for each pair of functions f and g in ℱ, f(k) and g(k) share the set S={a,b}, then ℱ is normal in D. The condition that the zeros of functions in ℱ have multiplicity at least k+1 cannot be weakened.

A semigroup is eventually regular if each of its elements has some power that is regular. Let 𝓚 be one of Green's relations and let ρ be a congruence on an eventually regular semigroup S. It is shown for 𝓚 = 𝓛, 𝓡 and 𝓓 that if A and B are regular elements of S/ρ that are 𝓚-related in S/ρ then there exist elements a ∈ A, b ∈ B such that a and b are 𝓚-related in S. The result is not true for 𝓗 or 𝓙.

Let B, S, and T be subsets of a (left) near-ring R with B and T nonempty. We say B is (S, T)-distributive if s(b1+b2)t = sb1t + sb2t, for each s ∈ S, b1, 2 ∈ B, t ∈ T. Basic properties for this type of ‘localized distributivity’ condition are developed, examples are given, and applications are made in determining the structure of minimal ideals. Theorem. If I is a minimal ideal of R and Ik is (Im, In)-distributive for some k, n ≧ 1, m ≧ 0, then either I2 = 0 or I is a simple, nonnilpotent ring with every element of I distributive in R. Theorem. Let Rk be (Rm, Rn)-distributive, for some k, n ≧ 1, m ≧ 0; if R is semiprime or is a subdirect product of simple near-rings, then R is a ring. Connections are established with near-rings which satisfy a permutation identity and with weakly distributive near-rings. If R → A → 0 is an exact sequence of near-rings, then conditions on A are given which will impose conditions on the minimal ideals of R.

Suppose the sequence of Taylor coefficients of a rational function f consists of kth powers of elements all belonging to some finitely generated extension field F of Q. Then it is a generalisation of a conjecture of Pisot that there is a rational function with Taylor coefficients term-by-term kth roots of those of f. The authors show that it suffices to prove the conjecture in the case that the field of definition is a number field and prove the conjecture in that case subject to the constraint that f has a dominant pole, that is, that there is a valuation with respect to which f has a unique pole either of maximal or of minimal absolute value.

The theory of isogeny estimates for Abelian varieties provides ‘additive bounds’ of the form ‘d is at most B’ for the degrees d of certain isogenies. We investigate whether these can be improved to ‘multiplicative bounds’ of the form ‘d divides B’. We find that in general the answer is no (Theorem 1), but that sometimes the answer is yes (Theorem 2). Further we apply the affirmative result to the study of exceptional primes ℒ in connexion with modular Galois representations coming from elliptic curves: we prove that the additive bounds for ℒ of Masser and Wüstholz (1993) can be improved to multiplicative bounds (Theorem 3).

Motivated by a class of nonlinear nonlocal equations of interest for string theory, we introduce Sobolev spaces on arbitrary locally compact abelian groups and we examine some of their properties. Specifically, we focus on analogs of the Sobolev embedding and Rellich–Kondrachov compactness theorems. As an application, we prove the existence of continuous solutions to a generalized bosonic string equation posed on an arbitrary compact abelian group, and we also remark that our approach allows us to solve very general linear equations in a $p$-adic context.

A classical theorem states that any open set on the real line is a countable union of disjoint open invervals. Here the numerical content of this theorem is investigated with the methods of constructive topology.

An algorithm called UNLABEL is devised to uniquely label an unlabelled transitive digraph. This is used to construct a one-one correspondence between homeomorphism classes of finite nondiscrete To-topologies and certain generalised Young tableaux of shape-type α. For sets of cardinality 2, 3, 4 and 5 these classes are enumerated and classified in several ways. The notion of a generalised descriptor graph is then introduced to enumerate the homeomorphism classes of all topologies on these sets.

J. Gani and G. F. Yeo ([2] and [3]) have recently investigated certain age-distributions associated with a replacement model which serves, in particular, as a model for phage reproduction. In this paper, the characteristic functional for this model will be obtained explicitly.

In a recent paper In a recent paper the authors proved a multiplier theorem for Hardy spaces Hp (G), 0 < p ≤ 1, defined on a locally compact Vilenkin group G. The assumptions on the multiplier were expressed in terms of the “norms” of certain Herz spaces K (1/p − 1/?r, r, p) with r restricted to 1 ≤ r < ∞ and p < r. In the present paper we show how this restriction on r may be weakened to p ≤ r ∞. Furthermore, we present two modifications of our main theorem and compare these with certain results for multipliers on LP (Rn)-spaces, 1 < p < ∞, due to Seeger and to Cowling, Fendler and Foumier. We also discuss the sharpness of some of our results.

If w is a group word in n variables, x1,…,xn, then R. Horowitz has proved that under an arbitrary mapping of these variables into a two-dimensional special linear group, the trace of the image of w can be expressed as a polynomial with integer coefficients in traces of the images of 2n−1 products of the form xσ1xσ2…xσm 1 ≤ σ1 < σ2 <… <σm ≤ n. A refinement of this result is proved which shows that such trace polynomials fall into 2n classes corresponding to a division of n-variable words into 2n classes. There is also a discussion of conditions which two words must satisfy if their images have the same trace for any mapping of their variables into a two-dimensional special linear group over a ring of characteristic zero.