The motivation for this paper lies in the following remarkable property of certain probability distributions. The distribution law of the r. v. (random variable) X is exactly the same as that of 1/ X, and in the case of a r. v. with p. d. f. (probability density function) f(x; a, b) where a, b are parameters, the p. d. f. of 1/X is f(x; b, a). In the latter case the p. d. f. of the reciprocal is obtained from the p. d. f. of X by merely switching the parameters. The existence of random variables with this property is perhaps familiar to statisticians, as is evidenced by the use of the classical 'F' distribution. The Cauchy law is yet another example which illustrates this property. It seems, therefore, reasonable to characterize this class of random variables by means of this rather interesting property.

A collineation is a one-one mapping of a projective plane onto itself, taking points into points, lines into lines and preserving incidence, ([1], p. 247). A perspective collineation (sometimes called a central collineation) is a collineation which leaves invariant all points on a line h called the axis, and all lines through a point H called the centre. The perspective collineation is an elation if H is incident with h; otherwise it is a homology.

A well-known theorem of Copping [2] states that a conservative matrix with a bounded left inverse cannot evaluate a bounded divergent sequence. (Definitions are given in the next paragraph.) A proof was given by Parameswaran [3, Theorem 6. 1], using only the simplest Banach-space ideas. This proof, however, is valid only for co-regular methods; it was stated in [3, Theorem 6. 2] that a co-null matrix cannot have a bounded left inverse, but the proof there given is incorrect, as it uses for co-null methods a theorem established only for co-regular. It would be desirable to have a short independent proof of this known result, which excludes co-null matrices from consideration in Copping' s theorem. This is furnished by the slightly more general result given below.

For any group G and integer n, let Pn:G→G be the function defined by Pn(g) = gn for all g ϵ G. If G is abelian then Pn is a homomorphism for all n. Conversely, it is well known (and easy to show) that if P2 or P-1 is a homomorphism then G is abelian. As the groups Gn described below show, for every n other than 2 and -1 there exist non-abelian groups for which Pn is a homomorphism.

A ring R is called a prime ring [1] if and only if a·R·b = 0 implies that a = 0 or b = 0 for all a, b ϵ R. Hence if R is a prime ring and a is a non-zero element of R, a·R ≠ 0 and R·a ≠ 0. In the present note we prove that a prime ring with a maximal annihilator right ideal has no non-zero nil right or left ideal.

In this note we wish to present an alternative proof for the following well-known theorem [1, Theorem 16]: every convex polytope X in Euclidean n-dimensional space Rn is the intersection of a finite family of closed half-spaces. It will be supposed that the converse of this theorem has been verified by conventional arguments, namely: every bounded intersection of a finite family of closed half-spaces in Rn is a convex polytope [cf. 1, Theorem 15].

A graph Gn consists of n nodes some pairs of which are joined by a single edge. A complete k-graph has k nodes and edges. Erdos [1] proved that if a graph Gn has edges, then it contains at least complete 3-graphs if it contains any at all. The main object of this note is to extend this result to complete k-graphs.

The following question was asked by R. A. Hirschfeld in [1]: "E and F are Banach spaces, F reflexive, D is a subset of E and T: D→F a nonlinear contraction, i.e.,

Can T be extended to a contraction (for E = F = Hilbert space the answer is yes)."