Recently, E.B. Saff and A. Sharma proved a result on Walsh's type equiconvergence for certain sequences of rational interpolants having uniformly distributed poles and nodes of interpolation. Here we examine the sensitivity of this result to a slight perturbation of the poles and nodes. Some problems related to our work are also formulated.

We differentiate certain refined Ahlfors-Weill local sections of the Bers projections. This yields reproducing formulas for holomorphic functions – which are then shown to be naturally related to Bers' important and well-known reproducing formulas.

Here we initiate the study of the following problem. Let Ω be a compact domain in a Riemannian manifold such that ∂Ω is of minimum area for the contained volume. Can ∂Ω be approximated by smooth hypersurfaces of positive mean curvature? It reduces to the question of whether or not a stable (or minimizing) hypercone in a Euclidian space can be approximated by smooth hypersurfaces of positive mean curvature. The positive solution to the problem may be useful for studying the curvature and topology of Ω.

We show in this paper that such approximation is possible provided that the given minimal cone satisfies some additional hypothesis.

In this paper we show that the hypercone over S2 × S4 is strictly area-minimizing in ℝ8. We also show the existence of smooth embedded stable hypersurfaces in ℝ8 which are not area-minimizing.

Let M be a closed, connected smooth and 3-connected mod 2 (that is Hi(M;ℤ2) = 0, 0 < i ≤ 3) manifold of dimension n = 7 + 8k. Using a combination of cohomology operations on certain cohomology classes of M and on the Thom class of the stable normal bundle of M we show that under certain conditions M immerses in R2n−8. This extends previously known results for such a general manifold when the number of 1's in the dyadic expansion of n is less than 8.

The class of quasi-retracts of AR's (AQR's) was first considered by Stallings, who observed that every AQR has the fixed point property. More recently, it has been shown that the class of AQR's is closed with respect to the operations of taking cones, suspensions, or products with AR's, and that no AQR separates a Euclidean space. In this paper we show that every AQR is acyclic, and we obtain a simple, direct characterization of the AQR's in a certain class of acyclic continua.

A modified product rule method for solving Cauchy-type singular integral equations is proposed. It is based on interpolating the unknown and some of its higher derivatives at any prespecified points. At these nodes the value of the unknown can be calculated directly by solving the discretized linear system. No need of further interpolatory formulae arises, as is the case with other quadrature methods.

Every finite loopless undirected graph G is isomorphic to an induced subgraph of a suitable finite direct power of the undirected graph G0 with two adjacent vertices 0,1 and one loop at vertex 1. The least natural number m such that G can be represented in this way is called its G0-dimension. We give some upper and lower bounds of this dimension depending on certain other graph invariants and determine its exact values for some special classes of graphs. Some methods to determine a concrete G0-representation, that is an embedding of G into , are presented. Moreover we show that the problem of determining the G0-dimension of a graph is NP-complete.

Let E and F be locally convex spaces and let K be a compact Hausdorff space. C(K,E) is the space of all E-valued continuous functions defined on K, endowed with the uniform topology.

Starting from the well-known fact that every linear continuous operator T from C(K,E) to F can be represented by an integral with respect to an operator-valued measure, we study, in this paper, some relationships between these operators and the properties of their representing measures. We give special treatment to the unconditionally converging operators.

As a consequence we characterise the spaces E for which an operator T defined on C(K,E) is unconditionally converging if and only if (Tfn) tends to zero for every bounded and converging pointwise to zero sequence (fn) in C(K,E).

We prove the following theorem: if G is a locally compact Hausdorff group such that each of its proper closed subgroups has only a finite number of closed subgroups, then G is topologically isomorphic to the circle group.

We show that a diagonal theorem of P. Antosik can be used to give a proof of the Closed Graph Theorem for normed spaces which does not depend upon the Baire Category Theorem.

Parabolic equations with forcing terms are studied and sufficient conditions are given that all solutions of boundary value problems are oscillatory in a cylindrical domain.

In this paper we study both the implicit and the explicit complementarity problem using some special and interesting connections between the complementarity problem and fixed point theory in Hilbert space.

We give a description of factorisation in the ring of generalised power sums (the sequence of Taylor coefficients of rational functions regular at infinity) with a view to giving detailed bounds on the order of generalised power sum factors and roots of such sums.