A divisor d of a positive integer n is a unitary divisor if d and n/d are relatively prime. An integer is said to be unitary perfect if it equals the sum of its proper unitary divisors. Subbarao and Warren [2] gave the first four unitary perfect numbers: 6, 60, 90 and 87360. In 1969,1 reported [3] that

is also unitary perfect. The purpose of this paper is to show that this last number, which for brevity we denote by W, is indeed the next unitary perfect number after 87360.

Let B be the set of positive integers prime to d which are representable by some primitive, positive, integral binary quadratic form of discriminant d. It is the purpose of this note to show that the following asymptotic estimate for the number of integers in B less than or equal to x can be proved using only elementary arguments:

(1)

where c1 is the positive constant given in (17) below. (Using the deeper methods of complex analysis James [2] has proved this result with the error term ((log x)-1/2) replacing ((log log x)-1). Heupel [1] using a transcendental method as in James [2] improved this to ((log x)-1).)

Let S be a locally compact Borel subsemigroup of a locally compact semigroup G. It is shown that the algebra of all "absolutely continuous' measures on S is isometrically order isomorphic to the algebra of all measures in M(G) which are "concentrated" and "absolutely continuous" on S.

A family of sets is said to have property if there exists a set B such that B ∩ F ≠ Ø and B ⊅ F for every F ∊ . Such a B will be called suitable with respect to F. It is known (see [3]) that for each positive integer n there exists a family of sets satisfying the following conditions:

1. (a) |F| = n for each F ∊

2. (b) |F ∩ G| ≤ for F, G ∊ F ≠ G

3. (c) does not have property

A bounded basis {xn} of a Banach space X is called perfectly homogeneous if every bounded block basic sequence {yn} of {xn} is equivalent to {xn}. By a result of M. Zippin [4], a basis in a Banach space is perfectly homogeneous if and only if it is equivalent to the unit vector basis of c0 or lp, 1 ≤ p < + ∞. A basis {xn} of a Banach space X is called symmetric, if every permutation {xσ(n)} of {xn} is a basis of X, equivalent to the basis {xn}. It is clear that every perfectly homogeneous basis is symmetric.

Let A be an associative, but not necessarily commutative, ring with identity, and J = J(A) its Jacobson radical. A (unital) module is regular iff every submodule is pure (see (1)). The regular socle R(M) of a module M is the sum of all its submodules which are regular. These concepts have been introduced and studied in (2).

A theorem of M. Katětov asserts that a bounded uniformly continuous function f on a subspace Q of a uniform space P has a bounded uniformly continuous extension to all of P. In this note we give new proofs of two special cases of this theorem: (i) Q is totally bounded, and (ii) P is a locally compact group and Q is a subgroup, both P and Q having the left uniformity.

Let Y be a linear space over the complex plane C, and let F be a mapping on the complex linear space Y ⊕ C into subsets of C with the following properties: for y ∊ Y, λ and μ ∊ C, F(y + μ) is a nonempty and bounded subset of C, F(λy + μ) = λF(y) + μ and F(μ) = {μ}. We shall write f(y + μ) = sup{|λ + μ|: λ ∊ F(y)}, the radius of F(y + μ), y ∊ Y and μ ∊ C. The convex hull (resp. the closure) of a subset M of C is denoted by conv M (resp. ).

The purpose of this note is to give a short proof of a generalisation of a theorem of Ryser, Theorem 10.2.3 of [1], concerning matrices that occur in the theory of symmetric block designs.

The two main results of matrix theory required in the proof given below are:

1. (1) If B, C are square matrices such that BC = zI where z is a non-zero complex number, then CB = zI

2. (2) A matrix S which is both symmetric (i.e. S' = S) and skew-symmetric (i.e. S' = —S)is zero.

In a recent article [1] vacuum field solutions of the Brans-Dicke [2] field equations were found, the space-time metric in each solution being of the Friedmann type. Most of these solutions existed only for specific values of the parameter ω and, in particular, the two largest sets of solutions corresponded to the values and . Peters [3, 4] has shown that when all solutions of the Brans-Dicke vacuum equations are conformai to space-times with vanishing Ricci tensor. The purpose of this note is to investigate the possible geometric consequences of the value .

Problems involving Egyptian fractions (rationals whose numerator is 1 and whose denominator is a positive integer) have been extensively studied. (See [1] for a more complete bibliography). Some of the most interesting questions, many still unsolved, concern the solvability of

where k is fixed.

In [2] Rav proved necessary and sufficient conditions for the solvabilty of the above equation, as a consequence of some other theorems which are rather complicated in their proofs. In this note we give a short, elementary proof of this theorem, and at the same time generalize it slightly.