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.

In this paper various conditions are given under which the strict singularity (respectively, strict cosingularity) of a linear relation implies the strict singularity (respectively, strict cosingularity) of its conjugate.

We give a simple proof of the identity The proof uses only a few well-known properties of the cubic theta functions a(q), b(q) and c(q). We show this identity implies the interesting definite integral .

Let κ ≥ 2 be an integer. We show that there exist infinitely many positive integers N such that the number of κ-full integers in the interval (Nκ, (N + 1)κ) is at least (log N)1/3+ο(1). We also show that the ABC-conjecture implies that for any fixed δ > 0 and sufficiently large N, the interval (N, N + N1−(2+δ)/κ) contains at most one κ-full number.

Using Moore's ergodicity theorem, S.G. Dani and S. Raghavan proved that the linear action of SL(n, ℤ) on ℝn is topologically (n − l)-transitive; that is, topologically transitive on the Cartesian product of n − 1 copies of ℝn. In this paper, we give a more direct proof, using the prime number theorem. Further, using the congruence subgroup theorem, we generalise the result to arbitrary finite index subgroups of SL(n, ℤ).

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.

In this paper we establish two minimax theorems of Sion-type in G-convex spaces. As applications we obtain generalisations of some theorems concerning compatibility of some systems of inequalities.

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.

We prove a Seidel-Walsh-type theorem about the universality of a sequence of derivation-composition operators generated by automorphisms of the unit disk in the setting of the higher order Hardy spaces. Moreover, some related positive or negative assertions involving interpolating sequences and sequences between two tangent circles are established for the class of bounded functions in the unit disk. Our statements improve earlier ones due to Herzog and to the first and third authors.

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.

The goal of this paper is to study the multiplicity of positive solutions of a class of quasilinear elliptic equations. Based on the mountain pass theorems and sub-and supersolutions argument for p-Laplacian operators, under suitable conditions on nonlinearity f(x, s), we show the follwing problem: , where Ω is a bounded open subset of RN, N ≥ 2, with smooth boundary, λ is a positive parameter and ∆p is the p-Laplacian operator with p > 1, possesses at least two positive solutions for large λ.

Let SL(2, Γn) be the n-dimensional Clifford matrix group and G ⊂ SL(2, Γn) be a non-elementary subgroup. We show that G is the extension of a subgroup of SL(2, ℂ) if and only if G is conjugate in SL(2, Γn) to a group G′ which satisfies the following properties:

(1) there exist loxodromic elements g0, h ∈ G′ such that fix(g0) = {0, ∞}, fix(g0) ∩ fix(h) = ∅ and fix(h) ∩ ℂ ≠ ∅;

(2) tr(g) ∈ ℂ for each loxodromic element g ∈ G′.

Further G is the extension of a subgroup of SL(2, ℝ) if and only if G is conjugate in SL(2, Γn) to a group G′ which satisfies the following properties:

(1) there exists a loxodromic element g0 ∈ G′ such that fix(g0) ∩ {0, ∞} ≠ ∅;

(2) tr(g) ∈ ℝ for each loxodromic element g ∈ G′.

The discreteness of subgroups of SL(2, Γn) is also discussed.

Let Ω1 and Ω2 be two domains in the complex plane with a nonempty intersection. Suppose that μj are locally extremal Beltrami coefficients in Ωj (j = 1, 2) respectively. In 1980, Sheretov posed the problem: Will the coefficient μ defined by the condition μ(z) = μj(z) for z ∈ Ωj, j = 1, 2, be locally extremal in Ω1 ∪ Ω2? We give a counterexample to show that μ may not be locally extremal and not even be extremal.

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.

Julia sets of meromorphic functions are uniformly perfect under some suitable conditions. So are Julia sets of the skew product associated with finitely generated semigroup of rational functions.

S.G. Dani and S. Raghavan showed the linear action of Sp(2n,ℤ) on the space of symplectic p-frames for p ≤ n is topologically transitive. We give an alternative proof, from the prime number theorem and the congruence subgroup theorem, and show the action of every finite index subgroup of Sp(2n, ℤ) is topologically transitive.

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.

Let f: [0, 1] × ℝ2 → ℝ be a function satisfying the Carathéodory conditions and t (1 − t) e (t) ∈ L1(0, 1). Let ai ∈ ℝ and ξi ∈ (0, 1) for i = 1, …, m − 2 where 0 < ξ1 < ξ2 < … < ξm−2 < 1. In this paper we study the existence of C[0, 1] solutions for the m-point boundary value problem The proof of our main result is based on the Leray-Schauder continuation theorem.

In this paper we prove the following result: Let R be a 2-torsion free semiprime ring. Suppose there exists an additive mapping T: R → R such that T(xyx) = T(x)yx − xT(y)x + xyT(x) holds for all pairs x, y ∈ R. Then T is of the form 2T(x) = qx + xq, where q is a fixed element in the symmetric Martindale ring of quotients of R.

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.