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.

In this survey article we give an exposition of some work on the power structure of p-groups; especially work of the author's. The titles of the three sections are: Section 1. p-groups which have regular power structure; Section 2. Some weaker power structure properties; Section 3. p-central series of p-groups.

This paper contains a Hukuhara – type theorem for nonlinear Volterra integral equations in locally convex spaces.

Let {zj} be an interpolation sequence in the open unit disc and {wj} a bounded sequence. In this note, it is shown that there is a function F in H∞ + C satisfying ‖F‖ ≤ 1 and as j → ∞ if and only if there exists a compact matrix [tij] such that on where .

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.

Gilbert Baumslag, B.H. Neumann, Hanna Neumann, and Peter M. Neumann successfully exploited their concept of discrimination to obtain generating groups of product varieties via the wreath product construction. We have discovered this same underlying concept in a somewhat different context. Specifically, let V be a non-trivial variety of algebras. For each cardinal α let Fα(V) be a V-free algebra of rank α. Then for a fixed cardinal r one has the equivalence of the following two statements:

(1) Fr(V) discriminates V. (1*) The Fs(V) satisfy the same universal sentences for all s≥r. Moreover, we have introduced the concept of strong discrimination in such a way that for a fixed finite cardinal r the following two statements are equivalent:

(2) Fr(V) strongly discriminates V. (2*) The Fs(V) satisfy the same universal formulas for all s ≥ r whenever elements of Fr(V) are substituted for the unquantified variables. On the surface (2) and (2*) appear to be stronger conditions than (1) and (1*). However, we have shown that for particular varieties (of groups) (2) and (2*) are no stronger than (1) and (1*).

If au Asplund space is weakly countably determined, then it admits an equivalent Fréchet differentiable norm and is weakly compactly generated. If on an Asplund space there exists an equivalent Gâteaux differentiable norm, then its dual has a projectional resolution of identity.

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.

We solve the question raised by Barbosa and do Carmo as to whether there exists a complete, noncompact stably immersed surface in R3 with nonzero constant mean curvature. We show that such a surface is necessarily minimal, that is, its mean curvature is zero.

Let us consider the abstract equation

(0.1) L (ɛ) u + F (ɛ,u) = 0,

where F (ɛ,u) = 0 (|u|2) for ɛ near zero. In this paper we define a multiplicity depending only on L (ɛ) allowing us to obtain the following result: “Odd multiplicity entails bifurcation and, if the multiplicity is even, it is possible to find F (ɛ,u) such that the only solution to (0.1) near the origin are the trivial ones”.

Müntz's theorems give necessary and sufficient conditions for a sequence of powers in one variable to be complete in the spaces of all real-valued continuous functions or square integrable functions with the usual norms.

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.

Completeness properties of (i) the category Alg(T) of T-algebras over a functor T: X → X and (ii) the subcategory XT in the case where T = (T, μ, η) is a monad, are investigated. It is known that if X is compact, then each XT is compact; we present a functor T: Set → Set such that Alg(T) is non-compact, although it is hypercomplete. If T either preserves epis or has a rank, we prove that Alg(T) and XT are topologically algebraic over X provided X satisfies mild additional hypotheses. Nevertheless, a natural monad over the category of Δ-comp1ete posets is exhibited such that its category of algebras is solid, but not topologically algebraic, over Set.

In this note, we establish some results concerning the total torsion and the total absolute torsion of certain non-closed stereographically projected analytic curves. The method of proof involves only elementary techniques of integration, a periodicity argument and Liouville's Theorem.

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.

We present a unified treatment of the band circulant matrices which occur in the periodic spline and histospline projection theory with equispaced knots. Explicit bounds for the norm of these matrices are given.

A continuous mapping f of a compact real interval I to itself which is chaotic in the sense of Li and Yorke, cannot have a scrambled set residual in an subinterval of I. This shows that chaos in this case cannot be large from a topological point of view.

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.

In this note translation-invariant Dirichlet forms on a commutative hypergroup are studied. The main theorem gives a characterisation of an invariant Dirichlet form in terms of the negative definite function associated with it. As an illustration constructions of potentials arising from invariant Dirichlet forms are given. The examples of one- and two-dimensional Jacobi hypergroups yield specifications of invariant Dirichlet forms, particularly in the case of Gelfand pairs of compact type.