The Widder-Post real inversion operator [4[ is defined by

1.1

k = 1, 2,…. Utilizing this inversion operator one can obtain the following representation theorem (see e.g. [4[ Chapter VII, Theorem 15a).

The basic theory of the Hausdorff means for double sequences was developed some thirty - three years ago by C.R. Adams [1], and independently by F. Hallenbach [3], Yet today, many of the properties of these means remain largely uninvestigated. The calculations here, although clearly more complex, for the most part break down into obvious modifications of the calculations in the one dimensional case.

It is the object of this note to extend to general harmonic structures a theorem due to Privaloff [2] concerning the definition of harmonic functions. The notation is that of [8, 9, 10], where many of the definitions not given here will be found.

The main purpose of this note is to give a characterization of almost continuous functions similar to that of approximately continuous functions ([3], §235, p. 312). The characterization of approximately continuous real functions has been used to show that each approximately continuous function is almost continuous ([4], §6). Thus combining these results we see that the classes of almost continuous, approximately continuous and continuous functions form a descending chain.

K. Morita has shown that, given two rings R and S, there is an isomorphism between the category of left R-modules and the category of left S-modules if and only if there exists an R-S bimodule U such that

(1) U is a progenerator in the category of left R-modules, and

(2) S ≅ (EndR U)opp as rings.

The purpose of this paper is to extend results of Ball [1] concerning maximal subgroups of the group S(X) of all permutations of the infinite set X. The basic idea is to consider S(X) as a group of operators on objects more complicated than X. The objects we consider here are subspaces of the Stone-Čech compactification of the discrete space X and the Boolean algebra of “big setoids” of X.

In [1[ Trypanis has proved the following extension of Fermat's theorem: If a is any integer, paprime, , then

1

This result is to be understood in the sense that a(p-1)/pn-1=p1/pnα where α is an algebraic integer.

The densest lattice packings of spheres in Euclidean spaces En of n dimensions are known for n ≤ 8 (for full n — references see [6]). However, it i s not known for any n ≥ 3 whether there can be any non-lattice sphere packing with density exceeding that of the corresponding densest lattice packing.

It is an old problem to find how a collection of congruent plane figures should be arranged without overlapping to cover the largest possible fraction of the plane or some region of the plane. If similar figures of arbitrary different sizes are permitted, Vitali's theorem ([7] p. 109) guarantees that packings which cover almost all points are possible. It is natural to study the diameters of figures used in such a packing and we will investigate this for the case of a closed disk packed with smaller open disks.

Holsztyński [1] called a map f : X → Y f r om a space X into a space Y ‘universal for all maps of X into Y f if for all maps g : X → Y there exists a point x ∊ X such that f(x) = g(x) , i. e., if f has a coincidence with all maps from X into Y. As the word ‘universal’ is already widely used with different meanings, we prefer the more precise term ‘coincidence producing’ for these maps. Such maps must clearly be surjective.

An orientation of a graph G is an assignment of a unique direction to each line of G. The result is called an oriented graph. Two orientations of a graph are regarded as equivalent if the resulting oriented graphs are isomorphic as directed graphs. For example, the graph C3 consisting of a cycle of length 3 (a triangle) shown in Figure 1(a), has exactly two orientations D1 and D2; see Figure 1(b) and (c).

Let M be a metric space with metric ρ which has the following properties.

1. M is finitely compact.

2. There exists a geodesic in M joining each two points of M.

3. For all a, b∈M, ρ(a, b) is equal to the length of a geodesic joining a and b.

A topological space is an R1 - space if and only if x, y ε X, and such that U∩V = ϕ and x ε U, y ε V, where Ā denotes closure of the set A. Let be an R1 - space.

Let ℓ, m, n be three mutually distinct lines in the projective plane. The (ℓ, m, n)-Pappus condition can be described as follows.

Let A, B, C, A', B', C' be any six mutually distinct points such that A, B, C lie on ℓ; A', B', C' lie on m; and none of these points lies on ℓ∩m, m∩n, or n∩ℓ. If the points AB'∩BA' and BC'∩CB' both lie on n, then the point AC'∩CA' also lies on n.

In this note we prove what we believe to be a new result concerning matrices, namely, that if an n × n matrix with rational entries has a finite order then this order is bounded. We also give an estimate for this bound and an application.

Let X be a random variable whose frequency function is

1.1

Form (1.1) is Stacy′s [3] generalization of the gamma distribution. The familiar gamma, chi, chi-squared, exponential and Weibull variâtes are special cases, as are certain functions of normal variate - viz., its positive even powers, its modulus, and all positive powers of its modulus.