The present paper seeks to put into a clearer focus the roles of certain axioms in axiomatic studies of finite dimensional Cartesian spaces over various classes of ordered fields. Following Tarski [3], we define a Cartesian space as follows.

In a recent note [3] in this department C. Clark has shown that Rellich's theorem on the compactness of the imbedding is valid if G is the "spiny urchin" domain obtained by removing from the plane the union of the sets Sk (k = 1, 2,…) defined in polar coordinates by

A near-ring is a triple (R, +,.) such that (R, +) is a group, (R,.) is a semigroup and. is left distributive over +; i.e. w(x + z) = wx + wz for each w, x, z in R. A near-field is a nearring such that the nonzero elements form a group under multiplication. Zassenhaus [3] showed that if R is a finite near-field, then (R, + ) is abelian. B.H. Neumann [1] extended this result to all near-fields. Recently another proof of this important result was given by Zemmer [4]. The purpose of this note is to give another generalization of the Zassenhaus theorem. In fact, we shall prove the following.

Consider the functional equation

(1)

assumed valid for all real x, y and h. Notice that (1) can be written

(2)

a difference analogue of the wave equation, if we interpret etc., (i. e. symmetric h differences), and that (1) has an interesting geometric interpretation. The continuous solutions of (1) were found by Sakovič [5].

The purpose of this note is to present a short proof for the following theorem.

Let A and B be two complex m x n matrices. If B*A = 0 and AB* = 0 then rank(A + B) = rank(A) + rank(B).

Let A† and B† be the generalized inverses of A and B, respectively, in the sense of Penrose [ 1]. Now,

Problem 7 of Section VI of H. T. Croft's "Research Problems" (August, 1967 edition) is by J. H. Conway:

A is a finite set of integers {ai}. A + A denotes {ai + aj}, A - A denotes {ai - aj}. Prove that A - A always has more members than A + A, unless A is symmetrical about 0.

First note that 'about 0' is redundant and erroneous.

For p a prime ≡ 1 (mod n), where n is an odd positive integer, let k(p, n) denote the least integer k such that the numbers xn and (-x)n, where x = 1,2,…, k, yield all the non-zero n-th power residues (mod p) (possibly with repetitions). Clearly

Let ψ be the outer measure generated by a gauge g and a sequential covering class C of closed sets of a metric space X, and let D be the resulting strong upper density function. That is, for each A ⊆ X and each x ∈ X:

the infimum being taken over all countable subcollections {Iα} of C such that and, for all Iα in the subcollection, d(Iα) < ε where d(Iα) is the diameter of d(Iα).

The André-Poincaré "probléme du scrutin" [9] can be stated as follows: In an election between two candidates A polls m votes, B polls n, m > n. If the votes are counted one by one what is the probability that A leads B throughout the counting? Many derivations and interpretations of the solution have been given and a convenient summary of methods till 1956 can be found in Feller [1]. So numerous are the generalizations of ballot problems and their applications since this date that we do not even attempt an enumeration here.

The following question has been asked by J. Dénes [2]: If n - 1 elements of the diagonal of an n × n array are prescribed, is it possible to complete the array to form an n × n latin square?" It is known that if n diagonal elements are given such a completion is not always possible.

That the answer to Dénes' question is yes follows directly from a theorem of M. Hall Jr. [1].

In [2, page 151], J. Lambek proposes the following exercise: With any maximal right ideal M of a ring R with 1 associate the ideal . Show that OM, is the right annihilator of the injective hull of the right R-module R/M. The purpose of this note is to show that the above statement is true for a much larger class of right ideals than that of maximal regular right ideals of a ring.

We prove that the Frattini subgroups are trivial for finite groups whose orders are not divisible by squares of a prime.

If G is a group, we define its Frattini subgroup ϕ(G) as the intersection of all the maximal subgroups of G. An element x ∈ G is a nongenerator of G if whenever G = < X, x >, where X ⊆ G, then G = < X >.