Let f1, …, fN ∈ A(D). It is shown that the ideal I(f1,…, fN) generated by the functions f1 (j = 1,…, N) equals the ideal

if and only if the functions fj have no common zero on the boundary of the unit disk D.

Let F be an arbitrary field, and f(x) a polynomial in one variable over F of degree ≥ 1. Given a polynomial g(x) ≠ 0 over F and an integer m > 1 we give necessary and sufficient conditions for the existence of a polynomial z(x) ∈ F[x] such that z(x)m ≡ g(x) (mod f(x)). We show how our results can be specialised to ℝ, ℂ and to finite fields. Since our proofs are constructive it is possible to translate them into an effective algorithm when F is a computable field (for example, a finite field or an algebraic number field).

Let M be a compact complex manifold and ∇ an arbitrary complex (not necessarily Riemannian) connection. In this paper we study the relation between the geometry of (M, ∇) and the topology of M, that is, we are interested in the following problem: To what extent does the topology of M determine the relations between the group of holomorphically projective transformations, the group of projective transformations and the group of affine transformations on M? Under assumptions on the Ricci-type tensors of ∇ and Chern numbers of M we show that a holomorphically projective transformation and a projective transformation are in fact affine transformations on M. A family of interesting examples of connections of this kind are constructed. Also, the case when M is a Kähler manifold is studied.

The concept of superspace is fundamental for some recent physical theories, notably supersymmetry, and a mathematical feedback for it is provided by superanalysis and supergeometry. We survey the state of affairs in superanalysis, shifting our attention from supermanifold theory to “plain” superspaces. The two principal existing approaches to superspaces are sketched and links between them discussed. We examine a problem by Manin of representing even geometry (analysis) as a collective effect in infinite-dimensional purely odd geometry (analysis), by applying the technique of nonstandard (infinitesimal) analysis.

Sufficient conditions involving uniform multisplittings are established for the convergence of relaxed and AOR versions of asynchronous or chaotic parallel iterative methods for solving a large scale nonsingular system of linear equations Ax = b.

In my earlier paper [1] I made the claim that there are three groups in Hall and Senior's book “The Groups of Order 2n(n ≤ 6)” that are in error (groups 64/140, 64/141, 64/143). However, it has been pointed out to me by Franz Lemermeyer that I made the unfortunate oversight of using the definition [x, y] = xyx−1y−1 for the commutator whereas Hall and Senion use the definition [x, y] = x−1y−1xy (see [2]). With this correction there is no problem with the above three groups in Hall and Senior.