The order structure of the space of all continuous linear operators on an ordered Banach space is studied. The main topic is the Robinson property, that is, the norm of a positive linear operator is attained on the positive unit cone.

We obtain uniqueness theorems for L-functions in the extended Selberg class when the functions share values in a finite set and share values weighted by multiplicities.

We introduce a property formally weaker than weak uniform rotundity, which we call equatorial weak uniform rotundity. We show that an equatorially weakly uniformly rotund norm need not be weakly locally uniformly rotund. Nevertheless, we show that an equatorially weakly uniformly rotund Banach space is an Asplund space.

For a field k of characteristic not two, it is known that k is algebraically closed in the function field of any (non-degenerate) quadratic form in three or more variables. In this note we consider fields of characteristic two and decide when k is algebraically closed in a function field of a quadratic k-form. For quadratic forms in three variables this has recently been done by Ohm.

Given two subsets X and Y of ℝ each with at least two points, we describe the surjective linear isometries between the spaces of functions of bounded variation BV(X) and BV(Y): namely, if T : BV(X) → BV(Y) is such an isometry, then there exist α ∈ ℂ, |α| = 1, and a monotonic bijective map h : Y → X such that (Tf)(y) = αf(h(y)) for every f ∈ BV(X) and every y ∈ Y.

If is a variety of groups which are nilpotent of class c then is generated by its free group of rank c. It is proved that under certain general conditions cannot be generated by its free group of rank c - 2, and that under certain other conditions is generated by its free group of rank c - 1. It follows from these results that if is the variety of all groups which are nilpotent of class c, then the least value of k such that the free group of of rank k generates is c - 1. This extends known results of L.G. Kovács, M.F. Newman, P.P. Pentony (1969) and F. Levin (1970).

Let X be a real or complex Banach space and let U be a Banach space with an unconditional basis. We show that the projective tensor product of U and X, UX, has the complete continuity property (respectively, the analytic complete continuity property) whenever U and X have the complete continuity property (respectively, the analytic complete continuity property). More general versions of these results are also obtained. Moreover, the techniques applied here to the projective tensor product, can also be used to establish some Banach space properties of the Fremlin projective tensor product.

A maximal-clique partition of a graph G is a way of covering G with maximal complete subgraphs, such that every edge belongs to exactly one of the subgraphs. If G has a maximal-clique partition, the maximal-clique partition number of G is the smallest cardinality of such partitions. In this paper the existence of maximal-clique partitions is discussed – for example, we explicitly describe all graphs with maximal degree at most four which have maximal-clique partitions - and discuss the maximal-clique partition number and its relationship to other clique covering and partition numbers. The number of different maximal-clique partitions of a given graph is also discussed. Several open problems are presented.

Let p be a prime number ≥ 5, and n a positive integer > 1. This note is concerned with the diophantine equation x4 − y4 = nzp. We prove that, under certain conditions on n, this equation has no non-trivial solution in Z if p ≥ C(n), where C(n) is an effective constant.

It is shown that the inequality

holds for an infinity of integers p, q and that here the factor ½ may not be replaced by a smaller number. Corresponding best possible inequalities are given for the numbers e±2/t, where t is a positive integer.

Finite groups G possessing a proper subgroup U such that for each element g of G there exists an automorphism of G mapping g into U are considered. The question of how the structure of U determines the structure of G is examined. For example, if G is soluble and U is nilpotent then G is nilpotent.

A group G ≤ GLK(V) is called K-primitive if there exists no non-trivial decomposition of V into a sum of K-spaces which is stabilised by G. We show that if V is a finite vector space and G a K-primitive subgroup of GLK(V) whose order is coprime to |V|, we can bound the order of G by |V|log2(|V|) apart from one exception. Later we use this result to obtain some lower bounds on the number of p–singular elements in terms of the group order and the minimal representation degree.

Let $S\subseteq \mathbb{P}^{d}$ be an anticanonically embedded surface of degree $d\geq 3$ . In this note, we classify stable Ulrich bundles on $S$ of rank two. We also study their moduli spaces.

A group G is said to be a B(n,k) group if for any n-element subset A of G, ∣A2∣≤k. In this paper, characterizations of B(5,16) groups and B(5,17) groups are given.

It is shown that, if U is a subvariety of the join of a nilpotent variety and a metabelian variety and if V is a variety with a finite basis for its laws, then UV also has a finite basis for its laws. The special cases U nilpotent and U metabelian have been established by Higman (1959) and Ivanjuta (1969) respectively. The proof here, which is independent of Ivanjuta's, depends on a rather general sufficient condition for a product variety to have a finite basis for its laws.

Let R be a commutative ring possessing (non-zero) zero-divisors. There is a natural graph associated to the set of zero-divisors of R. In this article we present a characterisation of two types of R. Those for which the associated zero-divisor graph has diameter different from 3 and those R for which the associated zero-divisor graph has girth other than 3. Thus, in a sense, for a generic non-domain R the associated zero-divisor graph has diameter 3 as well as girth 3.

Let P be an extraspecial p-group which is neither dihedral of order 8, nor of odd order p3 and exponent p. Let G be a finite group having P as a Sylow p-subgroup. Then the mod-p cohomology ring of G coincides with that of the normaliser NG (P).

In this note we examine the relationship of a distributive lattice to its lattice of ideals. Our main result is that a distributive lattice and its lattice of ideals share exactly the same collection of finite sublattices. In addition we give a related result characterizing those finite distributive lattices L which can be embedded in a lattice L′ whenever they can be embedded in its lattice of ideals T(L′).

For every $q\in (0,1)$, we obtain the Herglotz representation theorem and discuss the Bieberbach problem for the class of $q$-convex functions of order $\unicode[STIX]{x1D6FC}$ with $0\leq \unicode[STIX]{x1D6FC}<1$. In addition, we consider the Fekete–Szegö problem and the Hankel determinant problem for the class of $q$-starlike functions, leading to two conjectures for the class of $q$-starlike functions of order $\unicode[STIX]{x1D6FC}$ with $0\leq \unicode[STIX]{x1D6FC}<1$.

The purpose of this paper is to show that a well-known method for constructing “queer” sets in models of ZF set theory is also applicable to certain algebraic structures. An infinite set is called “quasi-minimal” if every subset of it is either finite or cofinite. In Section 1 I set out the two systems of set theory to be used in this paper, and illustrate the technique in its most fundamental form by constructing a model of set theory containing a quasi-minimal set. In Section 2 I show that by choosing the parameters appropriately, one can use this technique to obtain models of set theory containing groups whose carriers are quasi-minimal. In the third section various independence results are deduced from the existence of such models: in particular, it is shown that it is possible in ZF set theory to have an infinite group that satisfies both the ascending and descending chain conditions. The quasi-minimal groups constructed in Section 2 were all elementary abelian; in Section 4 it is shown that this was not just chance, but that in fact all quasi-minimal groups must be of this type. Finally in Section 5 permutations and permutation groups on quasi-minimal sets are examined.