In this paper we obtain several lower bounds for the degree of commutativity of a p-group of maximal class of order pm. All the bounds known up to now involve the prime p and are almost useless for small m. We introduce a new invariant b which is related with the commutator structure of the group G and get a bound depending only on b and m, not on p. As a consequence, we bound the derived length of G and the nilpotency class of a certain maximal subgroup in terms of b. On the other hand, we also generalise some results of Blackburn. Examples are given in order to check the sharpness of the bounds.

A continuous form of Hölder's inequality is established and used to extend the inequality of Chuan on the arithmetic-geometric mean inequality.

Let R be a prime ring of characteristic not 2, C be the extended centroid of R, and f: R → R be an additive map. Suppose that [f(x), x2] = 0 for all x ∈ R. Then there exist λ ∈ C and an additive map ζ: R → C such that f(x) = λx + ζ(x) for all x ∈ R. In particular, if f(x)2 = x2 for all x ∈ R, then ζ = 0 and either λ = 1 or λ= -1.

Lots of formulas for series of zeta function have been developed in many ways. We show how we can apply the theory of the double gamma function, which has recently been revived according to the study of determinants of Laplacians, to evaluate some series involving the Riemann zeta function.

We show that continuous triangular maps of the square I1, F: (x, y) → (f(x), g(x, y)), exhibit phenomena impossible in the one-dimensional case. In particular: (1) A triangular map F with zero topological entropy can have a minimal set containing an interval {a} × I, and can have recurrent points that are not uniformly recurrent; this solves two problems by S.F. Kolyada.

(2) In the class of mappings satisfying Per(F) = Fix(F), there are non-chaotic maps with positive sequence topological entropy and chaotic maps with zero sequence topological entropy.

One of the well-known generalisations of the Hölder inequality was given by Beckenbach. An inverse to this inequality for the discrete case has appeared in the literature. Here we give a simple proof of the inverse to the Beckenbach inequality that is applicable to both the integral and discrete cases.

In this paper we determine the mod-2 cohomology rings and the 2-primary part of the integral cohomology rings of finite groups with semi-dihedral Sylow 2-subgroups. The method used here is algebraic and can be considered as elementary.

An associative ring R is said to have stable range one if for any a, b ∈ R satisfying aR + bR = R, there exists y ∈ R such that a + by is right (equivalently, left) invertible. Call a ring R strongly π-regular if for every element a ∈ R there exist a number n (depending on a) and an element x ∈ R such that an = an+1x. It is an open question whether all strongly π-regular rings have stable range one. The purpose of this note is to prove the following Theorem: If R is a strongly π-regular ring with the property that all powers of every nilpotent von Neumann regular element are von Neumann regular in R, then R has stable range one.

It is shown that the dynamical game theoretic mating behaviour of males and females can be modelled by a planar system of autonomous ordinary differential equations. This system occurs in modelling “the battle of the sexes” in evolutionary biology. The existence of a heteroclinic cycle and a continuous family of periodic orbits of the system is established; then the dynamical characteristics of a time-periodic perturbation of the system are investigated. By using the well-known Melnikov's method, a sufficient condition is obtained for the perturbed system to have a transverse heteroclinic cycle and hence to possess chaotic behaviour in the sense of Smale. Finally, subharmonic Melnikov theory is used to obtain a criterion for the existence of subharmonic periodic orbits of the perturbed system.

An new extension of Hölder's inequality is derived. This is shown to follow from a generalisation of Steffensen's inequality.

Let f: X → X be a map of a continuum X. Let P(f) denote the set of all periodic points of f and R(f) denote the set of all recurrent points of f. In [2], Coven and Hedlund proved that if f: I → I is a map of the unit interval I = [0, 1], then CI(P(f)) = CI(R(f)). In [7], Ye generalised this result to maps of a tree. It is natural to ask whether the result generalises to maps of a dendrite. (A dendrite is a locally connected continuum which contains no simple closed curve.) The aim of this paper is to show that the answer is negative.

Let G be a convex function of m variables, let ω be a domain in ℝn, and let LG(ω) denote the vector-valued Orlicz space determined by G. We give an imbedding theorem for the space of weakly differentiable functions u provided with the norm ∥(u, Du)∥G, where m = n + 1 and Du denotes the gradient of u. This theorem is a variant of an imbedding theorem by N.S. Trudinger for the completion of in the norm ∥Du∥G, where m=n.

We present a duality theorem. We give a necessary and sufficient condition for any set of algebraic relations to entail the set of all algebraic relations in Davey and Werner's sense. The main result of the paper states that for a finite algebra a finite set of algebraic relations yields a duality if and only if the set of all algebraic relations can be obtained from it by using four types of relational constructs. Finally, we prove that a finite algebra admits a natural duality if and only if the algebra has a near unanimity term operation, provided that the algebra possesses certain 2k-ary term operations for some k. This is a generalisation of a theorem of Davey, Heindorf and McKenzie.

It is shown that a projective CS right module M over a ring R is a direct sum of uniform modules of composition lengths at most 2 if (i) every finitely generated direct summand of M is continuous and (ii) every non-zero M-singular right R-module contains a non-zero M-injective submodule. In particular, a right continuous ring R is semisimple if R is right weakly SI, that is, if every non-zero singular right R-module contains a non-zero injective submodule.

Some structure theorems for totally disconnected groups are described. These theorems produce a certain positive integer valued function, called the scale function, on each totally disconnected group. The scale function has properties connected to the structure of the group and can be used to prove some conjectures of Hofmann and Mukherjea.

The aim of this paper is to give a new precise formula for h(n, A), where A is a finite non-abelian simple group, h(n, A) is the maximum number such that Ah(n, A) can ke generated by n elements, and n ≥ 2. P. Hall gave a formula for h(n, A) in terms of the Möbius function of the subgroup lattice of A; the new formula involves a concept called cospread associated with that of spread as explained in Brenner and Wiegold (1975).

Let QT = ω x (0, T), where ω is a bounded domain in ℝn (n ≥ 3) having the cone property and T is a positive real number; let Y be a nonempty, closed connected and locally connected subset of ℝh; let f be a real-valued function defined in QT × ℝh × ℝnh × Y; let ℒ be a linear, second order, parabolic operator. In this paper we establish the existence of strong solutions (n + 2 ≤ p < + ∞) to the implicit parabolic differential equation

with the homogeneus Cauchy-Dirichlet conditions where u = (u1, u2, …, uh), Dxu = (Dxu1, Dxu2, …, Dxuh), Lu = (ℒu1, ℒu2, … ℒuh).

We establish a link between the two characterizations, independently obtained in 1991 by Puchs and Robson, of complete matrix rings in terms of the existence of nilpotent elements.

Our main result shows that a conformal mapping of hyperbolic n-space into another Riemannian manifold with scalar curvature bounded above by −n(n − 1) is necessarily distance decreasing. This is a generalisation of Ahlfors' version of the Schwarz-Pick lemma to Riemannian Geometry.

The Splitting Theorem says that any given Hamel basis for a (Hausdorff) barrelled space E determines topologically complementary subspaces Ec and ED, and that Ec is flat, that is, contains no proper dense subspace. By use of this device it was shown that if E is non-flat it must contain a dense subspace of codimension at least ℵ0; here we maximally increase the estimate to ℵ1 under the assumption that the dominating cardinal ∂ equals ℵ1 [strictly weaker than the Continuum Hypothesis (CH)]. A related assumption strictly weaker than the Generalised CH allows us to prove that ED is fit, that is, contains a dense subspace whose codimension in ED is dim (ED), the largest imaginable. Thus the two components are extreme opposites, and E itself is fit if and only if dim (ED) ≥ dim (Ec), in which case there is a choice of basis for which ED = E. Morover, E is non-flat (if and) only if the codimension of E′ is at least in E*. These results ensure latitude in the search for certain subspaces of E* transverse to E′, as in the barrelled countable enlargement (BCE) problem, and show that every non-flat GM-space has a BCE.