Dedicated to Edwin Hewitt

If G is any Hausdorff topological group and βG is the Stone-Čech compactification then where |G| denotes the cardinalty of G It is known that if G is a discrete group then and if G is the additive group of real numbers with the Euclidean topology, then |βG| = 2|G|. In this paper the cardinality and weight of βG, for a locally compact group G, is calculated in terms of the character and Lindelöf degree of G The results make it possible to give a reasonably complete description of locally compact groups G for which |βG| = 2|G| or even |βG| = |G|.

For p ∈ (0, ∞) define Qp0(∂Δ) as the space of all Lebesgue measurable complex-valued functions f; on the unit circle ∂Δ for which ∫∂Δf;(z)|dz|/(2π) = 0 and

as the open subarc I of ∂Δ varies. Note that each Qp,0(∂Δ) lies between the Dirichlet space and Sarason's vanishing mean oscillation space. This paper determines the extreme points of the closed unit ball of Qp,0(∂Δ) equipped with an appropriate norm.

In this note we give a bound for the weighted star discrepancy of Faure-Niederreiter (0, m, s)-nets in prime-power base b ≥ s – 1 for the special sequence of weights given by γi = 1/i2, i ≥ 1.

Let E be a Banach space ordered by a solid and normal cone K, and normed by the Minkowski functional of an order interval [–p, p], p ∈ K∘. We derive global one-sided estimates for quasimonotone increasing functions f : [0, T) × E → E with respect to the norm, and the distance to the line generated by p, under conditions of f; in direction p.

We prove large time Gaussian bounds for the derivatives of the semigroups kernel associated with complex, second-order, subelliptic operators on Lie groups of polynomial growth.

For all finite p-groups, we describe the Jacobson radicals of incidence rings of group automata.

We consider the logistic equation − Δu = a (x) u − b (x) up on all of RN with possibly unbounded coefficients near infinity. We show that under suitable growth conditions of the coefficients, the behaviour of the positive solutions of the logistic equation can be largely determined. We also show that certain linear eigenvalue problems on all of RN have principal eigenfunctions that become unbounded near infinity at an exponential rate. Using these results, we finally show that the logistic equation has a unique positive solution under suitable growth restrictions for its coefficients.

In this paper, we discuss the relationship among the concepts of an S-strictly quasiconcave vector-valued function introduced by Benson and Sun, a C-strongly quasiconcave vector-valued function and a C-strictly quasiconcave vector-valued function in a topological vector space with a lattice ordering. We generalize a main result obtained by Benson and Sun about the closedness of an efficient solution set in multiple objective programming. We prove that an efficient solution set is closed and connected when the objective function is a continuous S-strictly quasiconcave vector-valued function, the objective space is a topological vector lattice and the ordering cone has a nonempty interior.

In this article we consider index 1 invariant subspaces M of the operator of multiplication by ζ(z) = z, Mζ, on the Bergman space of the unit disc . It turns out that there is a positive sesquianalytic kernel lλ defined on  ×  which determines M uniquely. Here we study the boundary behaviour and some of the basic properties of the kernel lλ. Among other things, we show that if the lower zero set of M, (M), is nonempty, the kernel lλ for fixed λ ∈  has a meromorphic continuation across \(M), where  is the unit circle. Furthermore we consider some special type of kernels lλ and by studying their structure we obtain information for the invariant subspaces related to them. Lastly, and after introducing the general for the invariant subspaces related to them. Lastly, and after introducing the general vector valued setting, we discuss some analogous results for the case of , where m is a positive integer.

Let be the function-indexed Kaplan-Meier integral process constructed from the random censorship model. We study a uniform version of the law of large numbers of Glivenko-Cantelli type for {Un} under the bracketing entropy condition. The main result is that the almost sure convergence and convergence in the mean of the process Un holds uniformly in ℱ. In proving the result we shall employ the bracketing method which is used in the proof of the uniform law of large numbers for the complete data of the independent and identically distributed model.

Let be the Kaplan-Meier integral process constructed from the random censorship model. We prove a uniform central limit theorem for {Un} under the bracketing entropy condition and mild conditions due to the censoring effects. We also prove a sequential version of the uniform central limit theorem that will give a functional law of the iterated logarithm of Strassen type.

In this paper we consider a nonlinear parabolic equation with gradient dependent nonlinearities of the form

0 < p, q and a, b ∈ ℝ, with homogeneous boundary condition in a bounded domain Ω ⊆ ℝ,N. In the case 0 < p, q ≤ 1 we prove the existence of solution for suitable initial data. A comparison theorem for the solutions with respect to supersoultions and subsolutions is proved. Using these result, uniqueness and boundedness of solutions is studied.

In this paper we show that there do not exist any real hypersurfaces in a complex two dimensional Grassmannians G2(ℂn+2) with parallel shape operator.

Turbnikov/s property (U, λ, α, β) is investigated. In particular, it is shown that property (U, λ, α, α, − 1) with α > 1 is equivalent to α-uniform smoothness. It s also shown that property (U, 1, α, 1) with α > 1 is equivalent to the space being a Hilbert space. The dual property (U*, γ α α − 1) is also introduced and it is shown that a Banach space X has (U*, γ α, α − 1) if and only if X is α-uniformly convex.

An Abelian torsion group is determined by its automorphism group if and only if its locally cyclic component is determined by its automorphism group. We describe the locally cyclic groups that are determined by their automorphism groups.

In [1] there is an error, as pointed out to us by J.M. Isidro. In that paper we consider the set Rm of all m-rotations, that is, Rm = {czm: |c| = 1}; the set Aut() of the automorphisms of the unit disc  and the set of the m-automorphisms of , Autm () = {ψ ∘ R ∘ ϕ : ψ, ϕ ∈ Aut(), R ∈ Rm}. We asserted that

where ϕa(Z) = (a − z)/(1 − āz). Equality (1) is not true. For instance, it suffices to consider f (z) = ϕ¼ ∘ ∘ z2 ∘ ϕ½. We have that f ∈ Aut2 (), f (0) = (0) and, after calculations, f (z) = (7z2 − 6z)/(6z − 7). Then it is evident that will never hold f (z) = ϕ0 ∘ R ∘ ϕ0 with R ∈ R2, because ϕ0 is equal to identity.