Suppose U and V are to m × n matrices over the complex field. We obtain a representation for the Moore–Penrose inverse of the sum U + V. A well-known result of Cline is then derived as a special case of a corollary of this representation.

If T = (tij) is an n x n matrix with non-negative integer entries, we define a transformation T: Cn → Cn by z' = Tz where

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The now well-known Coale–Lopez theorem, and variants of it, state that, under certain conditions, the product of the first r members of an infinite inhomogeneous sequence of nonnegative matrices approaches, as r → ∞, the class of matrices which, apart from scalar multiples, have only one distinct column. The aim of the present paper is to lay down conditions under which such products approach the class of matrices which, apart from scalar multiples, have no more than d distinct columns. A stronger result is then obtained by considering stochastic matrices instead of just nonnegative ones.

In this note, a one sided aspect of regular variation is considered, and some different results which can be obtained by bounding the ratio f(xλ)/f(x) above or below are given.

The problem considered is that of approximating irrationals α by rationals p/q where p and q avoid certain congruence classes mod 2k for certain integers k. Results are obtained which give close bounds on a number c such that |α - p/q| < c/q2 has infinitely many solutions where p and q can be expressed as the sum of three squares.

Extension theorems are proved for smooth functions on a coherent real analytic space for which local defining functions exist which are finitely determined in the sense of J. Mather, (1968), and for smooth functions invariant under the action of a compact lie group G. thus providing the main step in the proof that smooth infinitesimal stability implies smooth stability in the appropriate categories. In addition the remaining step for the category of CxG-manifolds of finite orbit type is filled in.

An (r, 1)–design is a pair (V, F) where V is a v–set and F is a family of non-null subsets of V (b in number) which satisfy the following. (1) Every pair of distinct members of V is contained in precisely one member of F. (2) Every member of V occurs in precisely r members of F. A pseudo parallel complement PPC(n,α) is an (n + 1, 1)–design with v = n2 – αn and b ≦ n2 + n − α in which there are at least n – α a blocks of size n. A pseudo intersecting complement PIC(n, α) is an (n + 1, 1)–design with v = n2 − αn + α − 1 and b ≦ n2 + n – α in which there are at least n − α + 1 blocks of size n − 1. It has previously been shown that for α ≧ 4, every PIC(n, α) can be embedded in a PPC(n, α − 1) and that for n > (α4 − 2α3 + 2α2 + α − 2)/2, every PPC(n, α) can be embedded in a finite projective plane of order n. In this paper we investigate the case of α = 3 and show that any PIC(n, 3) is embeddable in a PPC(n,2) provided n ≥ 14.

We denote the complete graph on n vertices by Kn. A proper k–colouring of Kn is a way of assigning colours from a set of k colours to the edges of Kn in such a way that no monochromatic triangles are formed. It is known that there are precisely two proper 3-colourings of K16 each of which has exactly one proper 3-colouring of K15 embedded in it. We show that these two are the only proper 3-colourings of K15.

In this paper we introduce and study the notion of dual continuous (d–continuous) modules. A decomposition theorem for a d–continuous module is proved; this generalizes all known decomposition theorems for quasi-projective modules. Besides we study the structure of d–continuous modules over some special types of rings.