Let X and Y be two independent normal variates each distributed with zero mean and a common variance. Then the quotient X/Y has the Cauchy distribution symmetrical about the origin. Of particular interest in recent years has been the converse problem and examples of non-normal distributions with a Cauchy distribution for the quotient have been illustrated by Mauldon [9], Laha [2; 3; 4] and Steck [10].

A tournament Tn with n nodes is a complete asymmetric digraph [2]. A set S of arcs of a tournament is called consistent if the tournament contains no oriented cycles composed entirely of arcs of S [1]. The object of this note is to provide a new lower bound for f(n), the greatest integer k such that every tournament with n nodes contains a set of k consistent arcs. Erdös and Moon [1] showed that where [x] denotes the largest integer not exceeding x, and the second inequality holds for any fixed ∈ > 0 and all sufficiently large n.

In this paper we introduce the concept of Boolean near-rings. Using any Boolean ring with identity, we construct a class of Boolean near-rings, called special, and determine left ideals, ideals, factor near-rings which are Boolean rings, isomorphism classes, and ideals which are near-ring direct summands for these special Boolean near-rings.

Blackett [6] discusses the near-ring of affine transformations on a vector space where the near-ring has a unique maximal ideal. Gonshor [10] defines abstract affine near-rings and completely determines the lattice of ideals for these near-rings. The near-ring of differentiable transformations is seen to be simple in [7], For near-rings with geometric interpretations, see [1] or [2].

Sufficient conditions will be derived for the linear elliptic partial differential equation

(1)

to be nonoscillatory in an unbounded domain R in n-dimensional Euclidean space En. The boundary ∂R of R is supposed to have a piecewise continuous unit normal vector at each point. There is no essential loss of generality in assuming that R contains the origin. Otherwise no special assumptions are needed regarding the shape of R: it is not necessary for R to be quasiconical (as in [2]), quasicylindrical, or quasibounded [1].

We consider the following cosine and sine functional equations:

(1)

(2)

where f is an entire function of a complex variable z and x, y are complex variables [1; 2; 3].

In this note, we characterize quasi-Frobenius rings by a weakened form of the usual condition, that every ideal is an annihilator ideal.

We then apply this result to pure rings in the sense of Cohn and to dominant rings, a concept arising in the study of ring epimorphisms. All rings considered have a unit element.

In [1], Coxeter gave a complete enumeration of the regular maps on a torus. The maps consist of two families of type {4, 4}b, c and {3, 6}b, c (and their duals). b and c are non-negative integers, which determine the maps uniquely. The maps are irreflexible if and only if bc(b - c) ≠ 0.

On surfaces of genus h > 1, irreflexible regular maps are rather exceptional. The simplest surface of negative characteristic which admits irreflexible regular maps is the orientable surface of genus 7. This was shown by the author [4, Theorem 3. 1 ]. The corresponding map was discovered by J. R. Edmonds [2, p. 388].

In this paper we consider a special class of linear operators defined on a cone K in a Banach space X. This class of operators is the natural generalization of a class of operators which has applications in the theory of interpolation spaces. In particular, using the criteria developed in Theorem 1, it is possible to characterize those sequence spaces X such that every linear operator A of weak types (p, p) and (q, q) is a continuous mapping of X into itself. For details of this we refer the reader to [3].

In 1957, Mordell [3] proved

Theorem. If p is an odd prime there exist non-negative integers x, y ≤ A p3/4 log p, where A is a positive absolute constant, such that

(1.1)

provided (abc, p) = 1.

Recently Smith [5] has obtained a sharp asymptotic formula for the sum where r(n) denotes the number of representations of n as the sum of two squares.

Call an m × n array an m × n; k array if its mn entries come from a set of k elements. An m × n; 1 array has mn like entries. We write

(1)

if every m × n; k array contains a p × q; 1 sub-array. The negation of (1) is written

and means that there is an m × n; k array containing no p × q; 1 sub-array. Relations (1) are called "polarized partition relations among cardinal numbers" by P. Erdös and R. Rado [2]. In this note we prove the following theorems.

The main object of this paper is to give an explicit object in the study of projective covers in the category of compact Hausdorff spaces and continuous maps studied in [2] and [5]. be a projective cover of the Stone-Čech compactification βX of a completely regular Hausdorff space X. Here, it will be shown that the maximal ideal space endowed with the Stone topology of the maximal ring of quotients of the ring C(X) of all real valued continuous functions on X is homeomorphic to K.

Let X be a random variable defined by a Bernoullian probability function

(1)

The probability function of the restricted random variable which is truncated away from zero is then

(2)

The divisor 1 - qN arises from the condition excluding zero.

A topological space X is called a direct limit of a family (Xα) of subspaces of X if and only if

(1)

(2)

If X is a direct limit of an increasing sequence (Xn) of closed subspaces then it is well known and easy to prove that X is a T1-space resp. a T4-space provided all Xn are T1-spaces resp. T4-spaces.

Let A be an algebraic function field with a constant field k which is an algebraic number field. For each prime p of k, we consider a local completion kp and set Ap = Ak ꕕ kp. Then we have the question:

Is it true that A/k is a rational function field (i.e., A is a purely transcendental extension of k) if Ap/kp is so for every p ? In this note we shall discuss the question in a slightly different and hence easier case.

Let Qt = [0, 1] be equipped with the topology consisting of Qt, the empty set and all subsets of Qt of the form [0, x), 0 < x ≦ 1. R. Nielsen and C. Sloyer [1, p. 514] proved that every T0-space can be embedded in for a suitable F. The purpose of this note is to generalize this result.

In his paper "Necessary and sufficient statistics for a family of probability distributions", Dynkin (1951) establishes the important concept of rank for such a family with this conclusion: "If the rank is infinite, then the family has no non-trivial sufficient statistic in any size of sample." His concept of rank is based on a theorem, Theorem 2 described below, which has been pointed out by Brown (1964) to be invalid under its hypotheses. This note shows that Dynkin's Theorem 2 remains valid under its original hypotheses provided that the set (in Dynkin's notation) Δ - S is countable.