Let Ω ⊂ ℝN be a bounded domain. We study existence and uniqueness of principal eigenvalues for the Dirichlet periodic parabolic problem with weight Lu = λmu in Ω × ℝ when the independent coefficient of the differential operator L is not necessarily positive.

The present paper proposes new criteria for compactness of a composition operator Cφf = f ∘ φ on the Bloch space and the little Bloch space. For the case when φ is univalent, a criterion given by K. Madigan and A. Matheson is generalised.

In this paper we describe a construction of a large class of hyperconvex metric spaces. In particular, this construction contains well-known examples of hyperconvex spaces such as ℝ2 with the “river” metric or with the radial one.

Further, we investigate linear hyperconvex spaces with extremal points of their unit balls. We prove that only in the case of a plane (and obviously a line) is there a strict connection between the number of extremal points of the unit ball and the hyperconvexity of the space.

Some additional properties concerning the notion of hyperconvexity are also investigated.

Weighted Hadamard product characterisations of Bloch functions in the unit ball of ℂn are studied. In particular, we prove that f belongs to the Bloch space ℬ(Bn) if and only if the non-weighted Hadamard products of f and g belong to BMOA(Un) for all f in , a subspace of the Hardy space H1(Bn).

We give an example of an Engel structure on the 4-dimensional Euclidean space which admits a compact characteristic leaf. We also show that every Engel structure on an open 4-manifold can be modified so that the resulting structure has a compact characteristic leaf.

Let  be a certain set of nonnegative symmetric matrices, such as the set of symmetric doubly stochastic matrices or the set, of symmetric permutation matrices. It is proven that a linear transformation mapping  onto  must be of the form X ↦ PtX P for some permutation matrix P except for several low dimensional cases.

We continue the investigation started by A. Dubickas of the numbers which are differences of two conjugates of an algebraic integer over the field Q of rational numbers. Mainly, we show that the cubic algebraic integers over Q with zero trace satisfy this property and we give a characterisation for those for which this property holds in their normal closure. We also prove that if a normal extension K/Q is of prime degree, then every integer of K with zero trace is a difference of two conjugates of an algebraic integer in K if and only if there exists an integer of K with trace 1.

We show that the subspace An(X) of the free Abelian topological group A(X) on a Tychonoff space X is locally compact for each n ∈ ω if and only if A2(X) is locally compact if an only if F2(X) is locally compact if and only if X is the topological sum of a compact space and a discrete space. It is also proved that the subspace Fn(X) of the free topological group F(X) is locally compact for each n ∈ ω if and only if F4(X) is locally compact if and only if Fn(X) has pointwise countable type for each n ∈ ω if and only if F4(X) has pointwise countable type if and only if X is either compact or discrete, thus refining a result by Pestov and Yamada. We further show that An(X) has pointwise countable type for each n ∈ ω if and only if A2(X) has pointwise countable type if and only if F2(X) has pointwise countable type if and only if there exists a compact set C of countable character in X such that the complement X \ C is discrete. Finally, we show that F2(X) is locally compact if and only if F3(X) is locally compact, and that F2(X) has pointwise countable type if and only if F3(X) has pointwise countable type.

In this paper we show that if E is a separable Banach space, F is a reflexive Banach space, and n, k ∈ ℕ, then every continuous polynomial of degree n from E into F has at least one element of best approximation in the Banach subspace of all continuous k–homogeneous polynomials from E into F.

Recently, B.-Y. Chen obtained an inequality for slant surfaces in complex space forms. Further, B.-Y. Chen and one of the present authors proved the non-minimality of proper slant surfaces in non-flat complex space forms. In the present paper, we investigate 3-dimensional proper contact slant submanifolds in Sasakian space forms. A sharp inequality is obtained between the scalar curvature (intrinsic invariant) and the main extrinsic invariant, namely the squared mean curvature.

It is also shown that a 3-dimensional contact slant submanifold M of a Sasakian space form (c), with c ≠ 1, cannot be minimal.

We describe all group gradings of the matrix algebra M2(k), where k is an arbitrary field. We prove that any such grading reduces to a grading of type C2, a grading of type C2 × C2, or to a good grading. We give new simple proofs for the description of C2-gradings and C2 × C2-gradings on M2(K).

In this paper we establish existence theorems for vector quasi-equilibrium problems in Hausdorff topological vector spaces both under compactness and noncompactness assumptions.

We explore the role of weighted functional Stolarsky means in providing bounds in generalised Hadamard-type inequalities for g-convex functions. Refinements are given for Levinson's inequality and the generalised Hadamard inequality. Applications are made to multivariate weighted functional Stolarsky means.

The monoid n of uniform block permutations is the factorisable inverse monoid which arises from the natural action of the symmetric group on the join semilattice of equivalences on an n-set; it has been described in the literature as the factorisable part of the dual symmetric inverse monoid. The present paper gives and proves correct a monoid presentation forn. The methods involved make use of a general criterion for a monoid generated by a group and an idempotent to be inverse, the structure of factorisable inverse monoids, and presentations of the symmetric group and the join semilattice of equivalences on an n-set.

We prove that the following are equivalent for a locally compact group G:

(i) G is amenable;

(ii) M(G) is Connes-amenable;

(iii) M(G) has a normal, virtual diagonal.

We prove that Carnot groups with real model filiform Lie algebras are not rigid. Consequently non-trivial smooth contact and quasiconformal mappings exist in abundance.

Let S be a semigroup contained in a locally compact Abelian group G. Let Ĝ denote the Bohr compactification of G. We say that a sequence contained in S is Hartman uniform distributed on G if

for any character χ in Ĝ. Suppose that (Tg)g∈s is a semigroup of measurable measure preserving transformations of a probability space (X, β, μ) and B is an element of the σ-algebra β of positive μ measure. For a map T: X → X and a set A ⊆ X let T−1A denote {x ∈ X: Tx ∈ A}. In an earlier paper, the author showed that if k is Hartman uniform distributed then

In this paper we show that ≥ cannot be replaced by =. A more detailed discussion of this situation ensues.