We give a lower bound and an upper bound for the dimension of the homology of 2-step nilpotent Lie algebras with adjoint coefficients. We conjecture, that the upper bound and the actual dimension are asymptotically equivalent.

The set-valued optimisation problem with constraints is considered in the sense of super efficiency in locally convex linear topological spaces. Under the assumption of nearly cone-subconvexlikeness, by applying the separation theorem for convex sets, Kuhn-Tucker and Lagrange necessary conditions for the set-valued optimisation problem to attain its super efficient solutions are obtained. Also, Kuhn-Tucker and Lagrange sufficient conditions are derived. Finally two kinds of unconstrained programs equivalent to set-valued optimisation problems are established.

The Poincaré inequality is generalised to metric-measure spaces which support a strong version of the doubling condition. This generalises the Poincaré inequality for manifolds whose Ricci curvature is bounded from below and metric-measure spaces which satisfy the measure contraction property.

Suppose X is a set with |X| = p ≥ q ≥ ℵ0 and let B = BL(p, q) denote the Baer-Levi semigroup defined on X. In 1984, Howie and Marques-Smith showed that, if p = q, then BB−1 = I(X), the symmetric inverse semigroup on X, and they described the subsemigroup of I(X) generated by B−1B. In 1994, Lima extended that work to ‘independence algebras’, and thus also to vector spaces. In this paper, we answer the natural question: what happens when p > q? We also show that, in this case, the analogues BB−1 for sets and GG−1 for vector spaces are never isomorphic, despite their apparent similarities.

This paper is devoted to the study of a new kind of proper efficiency in terms of proximal normal cones for vector minimisation. This new notion called proximal proper efficiency is used to obtain a scalar characterisation when a set related to the criterion set is a nonconvex set. Proximal proper efficiency is related with the well known notions of Benson and Borwein proper efficiency which are defined in the literature in terms of tangent cones. The study is further extended to characterise Benson and Borwein proper efficiency in terms of normal cones assuming convexity of a related set.

In this paper we investigate identities related to centralisers in rings and algebras. We prove, for example, the following result. Let A be a semisimple H* -algebra and let T: A → A be an additive mapping satisfying the relation T(xm+n+1) = xmT(x)xn for all x ∈ A and some integers m ≥ 1, n ≥ 1. In this case T is a left and a right centraliser.

Metrisable Choquet simplexes with the set of extreme points being an Fσ-set are characterised by means of the behaviour of the space of affine Baire-one functions.

We show that any residue class λ modulo p can be represented in the form n1! +…+ nℓ! ≡ λ (mod p) with ℓ = O((log p)3 log log p).

The purpose of the paper is to show that weak order convergence of a net of Dedekind complete Riesz space-valued σ-measures is uniform over uniformly bounded, uniformly equicontinuous classes of functions. The paper ends with generalizing Ulam's theorem for tightness of positive, finite Borel measures to Riesz space-valued σ-measures.

We investigate Carlson type inequalities for finite sums, that is, inequalities of the form to hold for some constant C independent of the finite, non-zero set a1,…,am of non-negative numbers. We find constants C which are strictly smaller than the sharp constants in the corresponding infinite series case. Moreover, corresponding results for integrals over bounded intervals are given and a case with any finite number of factors on the right-hand side is proved.

In this paper we study a general variational inequality model with set-valued quasimonotone operators, a model which includes several variational inequalities and equilibrium problems. We establish unifying conditions for existence of solutions in a topological vector space setting. Applications to parametric equilibrium models and to a contact problem are given.

In this note we show that k-convex functions on ℝn are twice differentiable almost everywhere for every positive integer k > n/2. This generalises Alexsandrov's classical theorem for convex functions.

We study the behaviour of solutions in a neighbourhood of the origin for a certain type of non-autonomous system of partial differential equations whose linear approximation is non autonomous.

We describe a type of Lie colour algebra, which we call generic, whose universal enveloping algebra is a domain with finite global dimension. Moreover, it is an iterated Ore extension. We provide an application and show Gröbner basis methods can be used to study universal enveloping algebras of factors of generic Lie colour algebras.

Lower and upper bounds for CM-dimension, called CM*-dimension and CM*-dimension, will be defined for any finitely generated module M over a local Noetherian ring R. Both CM* and CM*-dimension reflect the Cohen–Macaulay property of rings. Our results will show that these dimensions have the expected basic properties parallel to those of the homological dimensions. In particular, they satisfy an analog of the Auslander-Buchsbaum formula.