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|.

A number of Banach space properties have been shown to imply the weak fixed point property. If the dual of a Banach space were to possess some of these properties then the original space can been shown to satisfy related conditions.

In this paper we describe the irreducible decomposition of the Magnus representation of the Torelli group.

Let χ be a space of homogeneous type of infinite measure. Let T be a singular integral operator which is bounded on Lp (χ) for some p, 1 < p < ∞. We give a sufficient condition on the kernel of T so that when a function b ∈ BMO(χ), the commutator [b, T](f) = T (bf) – bT (f) is bounded on Lp spaces for all p, 1 < p > ∞. Our condition is weaker than the usual Hörmander condition. Applications include Lp-boundedness of the commutators of BMO functions and holomorphic functional calculi of Schrödinger operators, and divergence form operators on irregular domains.

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.

We study the concepts of continuous shadowing and continuous inverse shadowing with respect to various classes of admissible pseudo orbits, and characterize hyperbolicity and structural stability using the notion of continuous inverse shadowing.

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

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.

The authors provide precise lower bounds for the completely bounded norm of the operator Ta,b: ß(H) → ß(H) defined by Ta,b (x) = axb + bxa and the injective norm of the corresponding tensor. Further, they compute the norm of the operator x ↦ a*xb + b*xa acting on the space of all conjugate-linear operators on H.

The notion of approximations relative to a functor is introduced and several characterisations of relative (dual) Maschke functors are given by using them. As an application, the injective objects in the category of comodules over a coring are described.

Let T be an eventually expansive transformation on the unit interval satisfying the Markov condition. The T is an ergodic transformation on (X, ß, μ) where X = [0, 1), ß is the Borel σ-algebra on the unit interval and μ is the T invariant absolutely continuous measure. Let G be a finite subgroup of the circle group or the whole circle group and φ: X → G be a measurable function with finite discontinuity points. We investigate ergodicity of skew product transformations Tφ on X × G by showing the solvability of the coboundary equation φ(x) g (Tx) = λg (x), |λ| = 1. Its relation with the uniform distribution mod M is also shown.

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.

We introduce a new geometric coefficient related to the Jordan-von Neumann constant. This leads to improved versions of known results and yields new ones on super-normal structure for Banach spaces.

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

A fixed point theorem and two homotopy invariant results are presented for generalized contractive maps defined on complete gauge spaces.

We obtain a basic B,-Y. Chen's inequality for a C-totally real submanifold in a (κ,μ)-contact space form involving intrinsic invariants, namely the scalar curvature and the sectional curvatures of the submanifold on left hand side and the main extrinsic invariant, namely the squared mean curvature on the right hand side. Inequalities between the squared mean curvature and Ricci curvature and between the squared mean curvature and κ-Ricci curvature are also obtained. These results are applied to get corresponding results for C-totally real submanifolds in a Sasakian space form.

In this paper, we introduce a new class of set-valued mappings in a non-convex setting called D-KKM mappings and prove a general D-KKM theorem. This extends and improves the KKM theorem for several families of set-valued mappings, such as (X, Y), C(X, Y), C (X, Y), C (X, Y) and C (X, Y). In the sequel, we apply our theorem to get some existence results for maximal elements, generalised variational inequalities, and price equilibria.

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.