In the combinatorial category, a two-complex X is said to be Kervaire if any set of equations modelled on X, over any group, has a solution in a larger group. The Kervaire-Laudenbach conjecture speculates that if H2(X) = 0 then X is Kervaire. We show that the validity of this conjecture would imply that all aspherical two-complexes are Kervaire. In particular, any two-complex homotopically equivalent to a two-manifold (≠ S2, RP2) would be Kervaire. We show that this is indeed the case for certain such two-complexes. We generalise this to staggered two-complexes, and, more generally, one-relator extensions of Kervaire complexes. We obtain similar results for diagrammatically reducible two-complexes. Our proofs make use of covering spaces.

We prove that the number of factors of length k in any paperfolding sequence is equal to 4k once k ≥ 7.

Let X be a Hausdorff locally convex space. We show that convergence of a net of continuous linear functionals on X with respect to a given polar topology on its continuous dual X′ can be explained in terms of the convergence of the corresponding net of its graphs in X × R, and the corresponding net of level sets at a fixed height in X, with respect to a natural generalisation of Attouch-Wets convergence in normable spaces.

In this paper we extend the concept of invexity to set-valued maps and study vector optimisation problems with invex set-valued data. Necessary and sufficient optimality conditions are established in terms of contingent derivatives. Wolfe type dual problems are constructed via two recently developed approaches which guarantee the zero-gap duality property.

For a set E in a metric space X the index of non-separability is β(E) = inf{r > 0: E is covered by a countable-family of balls of radius less than r}.

Now, for a set-valued mapping Φ from a topological space A into subsets of a metric space X we say that Φ is β upper semi-continuous at t ∈ A if given ε > 0 there exists a neighbourhood U of t such that β(Φ(U)) < ε. In this paper we show that if the subdifferential mapping of a continuous convex function Φ is β upper semi-continuous on a dense subset of its domain then Φ is Fréchet differentiable on a dense Gδ subset of its domain. We also show that a Banach space is Asplund if and only if every weak* compact subset has weak* slices whose index of non-separability is arbitrarily small.

Let A be a right alternative algebra, and [A, A] be the linear span of all commutators in A. If [A, A] is contained in the left nucleus of A, then left nilpotence implies nilpotence. If [A, A] is contained in the right nucleus, then over a commutative-associative ring with 1/2, right nilpotence implies nilpotence. If [A, A] is contained in the alternative nucleus, then the following structure results hold: (1) If A is prime with characteristic ≠ 2, then A is either alternative or strongly (–1, 1). (2) If A is a finite-dimensional nil algebra, over a field of characteristic ≠ 2, then A is nilpotent. (3) Let the algebra A be finite-dimensional over a field of characteristic ≠ 2, 3. If A/K is separable, where K is the nil radical of A, then A has a Wedderburn decomposition

Functions of bounded kth variation and absolutely kth continuous functions are considered on sets and various properties are studied.

In this paper, we consider Ishikawa's iteration scheme to compute fixed points of real Lipschitz functions. Two general convergence theorems are obtained. Our results generalise the result of Hillam.

Let α > 0 and let f[α](z) be the αth fractional derivative of an analytic function f on the unit disc D. In this paper we show that f ∈ BMOA if and only if |f[α](z)|2 (l - |z|2)2α−1dA(z) is a Carleson measure and f ∈ VMOA if and only if |f[α](z)|2 (1 − |z|2)2α−1dA(z) is a vanishing Carleson measure, where A denotes the normalised Lebesgue measure on D. Hence a significant extension of familiar characterisations for analytic functions of bounded and vanishing mean oscillation is obtained.

The differentiability, of a specified strength, of a convex function at a point, is shown to be characterised by the convergence of subdifferentials in the appropriate topology on the dual space. This is used to prove that if each gauge is densely differentiable then so is each convex function. The generic version of this is equivalent to a conjecture which, for Gateaux differentiability and Banach spaces, is the long standing open question of whether X × ℝ is Weak Asplund whenever X is. Some progress is made towards a resolution.

In this note, we shall prove a new variational inequality in non-compact sets and as an application, we prove a generalisation of the Schauder-Tychonoff fixed point theorem.

We obtain new sufficient conditions for the oscillation of all solutions of the neutral differential equation with variable coefficients

where P, Q, R ∈ C([t0, ∞), R+), r ∈ (0, ∞) and τ, σ ∈ [0, ∞). Our results improve several known results in papers by: Chuanxi and Ladas; Lalli and Zhang; Wei; Ruan.

We consider the sequence Λ = {0 < λ2 < λ2 < …}, for which λn → +∞. We denote by PD(Λ) the class of Dirichlet's series having the form F(s) = defined in the half plan Re s > 0 converging absolutely and Re F ≥ 0. If N0 = {0, 1,2, …} then the class PD(N0) coincides with the Caratheodory's class P. In this paper some classical results holding for the class P are generalised in any class PD(Λ). In special cases for the sequence Λ extreme problems are examined in the class PD(Λ).

A number of Szegö-type prediction error formulas are given for two-parameter stationary random fields. These give rise to an array of elementary inequalities and illustrate a general duality relation.