When studying the solutions of elliptic boundary value problems in a bounded, smoothly bounded domain D⊂Rn we often encounter the formula

where u(x)∈C2(D)∩C′() is a solution of the second order self-adjoint elliptic equation

and denotes differentiation along the inward normal to ∂D at x∈∂D.

In this paper, the author continues his investigation, initiated in (4) and (5), into the nature of certain “arithmetical” functions associated with the factorisation of normalised non-zero polynomials in the ring GF[q, X1, …, Xk], where k ≧ 1, GF(q) is the finite field of order q and X1, …, Xk are indeterminates. By normalised polynomials we mean that exactly one polynomial has been selected from equivalence classes with respect to multiplication by non-zero elements of GF(q). With this normalisation GF[q, X1, …, Xk] becomes a unique factorisation domain. The constant polynomial will be denoted by 1. By the degree of a polynomial A in GF[q, X1, …, Xk], we shall mean the ordered set (m1, …, mk), where mi is the degree of A in Xi, 1 ≦ i ≦.k. We shall assume that A(≠ 1), a typical polynomial in GF[q, X1, … Xk], has prime factorisation

where P1, …, Pr are distinct irreducible polynomials (i.e. primes).

Dieudonné (4) has constructed an example of a Banach space X and a complete Boolean algebra of projections on X such that has uniform multiplicity two, but for no choice of x1, x2 in X and non-zero E in is EX the direct sum of the cyclic subspaces clm {Ex1:E∈} and clm {Ex2:E∈}. Tzafriri observed that it could be deduced from Corollary 4 (9, p. 221) that the commutant ′ of is equal to A(), the algebra of operators generated by in the uniform operator topology. A study of (3) suggested the direct proof of the second property given in this note. From this there follows a simple proof that has the first property.

In a paper (1) by Harding there is a tacit invitation to seek the connection between the following two problems:

(i) Find the number, ηk(N), of regions into which a k-dimensional space is partitioned by a set of N (k- l)-dimensional hyperplanes.

(ii) Find the number, vk(N), of distinct partitions of a given set of N points in a k-dimensional space E that can be induced by (k- 1)-dimensional hyperplanes.