It is a well known result (see [4, p. 108]) that if R is a ring and Q(R) its maximal right quotient ring, then Q(R) is (von Neumann) regular if and only if every large right ideal of R is dense. This condition is equivalent to saying that the singular ideal of R is zero. In this note we show that the condition loses its magic in the theory of semigroups.

The classical uniqueness theorem of Nagumo [1] for ordinary differential equations is as follows.

Theorem. If f(t, y) is continuous on 0≤t≤1, -∞<y<∞ and if

then there is at most one solution to the initial value problem y'=f(t, y), y(0)=0.

The following theorem is due to J. L. Walsh (see [2, Theorem 17, 2a]):

Theorem. If all the zeros of f1(z)=zn+a1zn-1+ … + an lie in or on the circle C1 with centre c1 and radius r1 and if all the zeros of f2(z)=zn+b1zn-1+ … + bn lie in or on the circle C2 with centre c2 and radius r2, then each zero of the polynomial

lies in at least one of the circles Γk with centre γk and radius ρk, where

and where the ωk (k= 1, 2,…, n) are the nth roots of λ.

The purpose of this note is to give a basis with respect to which the companion matrix of an equation (over a field of any characteristic) is in Jordan canonical form.

Let k be a field. Define a k-linear map Di:k[X]→k[X] by where the integer is the binomial coefficient n!;/i!;(n —i)!;. We adopt the usual convention that if i>9 or j<0. Then D=(D0, D1D2,…) is a higher derivation (see [1, p. 192]).

Let A be a k-dimensional Euclidean region having unit volume. An m-colors pattern is a partition PA of A into m sets Ai, i—1,…, m with positive volume. PA is called a random pattern if in addition the partition of A is a realization of a random process with the following stationarity and isotropy properties:

• (i) for all points s ∊ A, P(s ∊ Ai) =pi9, i = 1,..., m

• (ii) for all pair of points s, s'∊ A with distance |s—s|=d between them, P(s' ∊ Ai | s ∊ Aj)Pij(d) i,j= 1,..., m.

P. Erdös asked the following problem: Does there exist an infinite sequence of integers a1<…satisfying for every x≥1

1

so that every integer is of the form 2k+ai [1]. The analogous questions can easily be answered affirmatively if the powers of 2 are replaced by the rth power.

In this note we give a simple affirmative answer to the problem of Erdôs. Let c2 be a sufficiently small absolute constant. Our sequence A consists of all the integers of the form

2

We give an alternative construction of a Hausdorff compactification due to Fell [2]. We say that a space is compact if it has the Heine-Borel property, locally compact if each point has a fundamental system of compact neighbourhoods. The interesting spaces from the point of view of this paper, are the non-Hausdorjf ones since for locally compact Hausdorff spaces Fell's compactification is the usual one-point compactification. The motivation for the compactification comes from the theory of continuous fields of C*-algebras: the primitive spectrum of a C*- algebra A is a locally compact T0 space X and Fell [3] realizes A as an algebra of fields of operators over the compactification of X. This note is based on a discussion of the author with Professor Fell.