Consider the graph Mn, where n = 2r ≥ 6, consisting of a polygon of length n and all n/2 chords joining opposite pairs of vertices. This graph has 2r vertices which we denote by 1, 2, 3,..., 2r, and the 3r (undirected) edges

We call Mn the n-ladder, defined thus far only for n even. The three smallest n-ladders with n even are shown in Figure 1.

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).

Let n and k be integers with n ≥ k ≥ 3. Denote by f(n, k) the largest positive integer for which there exists a set S of f (n, k) integers satisfying (i) and (ii) no k members of S have pairwise the same greatest common divisor. The problem of determining f(n, k) appears to be difficult. Erdős [2[ proved that there is an absolute constant c > 1 such that for every ∈ > 0 and every fixed k

1

provided n > no (k, ∈). In [l[ it i s proved that for every ∈ > 0 and every fixed k

The main object of this paper is to show that the existence of a particular kind of isomorphism between the integral group rings of two finite groups implies that the groups themselves are isomorphic. The proof employs certain types of linear forms which are first discussed in general. These linear forms are in some way related to the bilinear forms used by Weidmann [3] in showing that groups with isomorphic character rings have the same character table, and a shorter and, in a sense, more natural proof of this result is included here as another application of these linear forms.

According to the still unproved conjecture of Borsuk [1] a bounded subset A of the Euclidean n-space En is a union of n + 1 sets of diameters less than the diameter D of A. Since A can be imbedded in a set of constant width D, [2], it may be assumed that A is already of constant width. If in addition A is smooth, i. e., if through every point of its boundary ∂A there passes one and only one support plane of A, then the truth of Borsuk′s conjecture can be proved very easily [3]. The question arises whether Borsuk′s conjecture holds also for arbitrary smooth convex bodies, not merely for those of constant width. Since it is not known whether a smooth convex body K can be imbedded in a smooth set of constant width D, the answer is not immediate. In this note we show that the answer is affirmative.

Le radical R(A) d′un anneau A peut être caract érisé de plusieurs manières (Jacobson [1]). Rappelons seulement ici deux des caractérisations les plus importantes:

(1) R(A) est l′intersection de tous les idéaux à droite (à gauche) modulaires maximaux de A.

(2) R(A) est un idéal quasi - régulier contenant tout idéal à droite (á gauche) quasi - régulier.

L′objet de cet article est de donner quelques caractérisations du radical d′un anneau.

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.

If one considers the theories of Hurewicz, - Serre - or other fibrations in the categories of topological or pointed topological spaces, one can see that many of the fundamental theorems can be formulated and proven in the general case of categories for which certain functors and natural transformations are given. And, since fibrations may be defined either by a cylinder or a path space construction, we shall give in an analogous way two different definitions of fibrations in the general case.

In connection with various problems concerning polynomials

on the unit interval, the Tchebycheff polynomial

is known to play a very important role [11, problem 34].

We use the terminology of [1]. Let D be a strongly connected digraph on n points and containing m lines, and let A = A(D) be the corresponding adjacency matrix, so that A is an n x n 0-1 matrix containing m unit elements. We recall that A and D are said to be p-cyclic if p is the greatest common divisor of the lengths of all directed cycles of D. Clearly, the larger the value of p, the smaller the value of m must be; in this note we make the latter and related statements precise.

To each tournament Tn with n nodes n there corresponds the automorphism group G(Tn) consisting n of all dominance preserving permutations of the set of nodes. In a recent paper [3], Myron Goldberg and J. W. Moon consider the maximum order g(n) which the group of a tournament with n nodes may have. Among other results they prove that

1

2

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.

In this note we consider a relatively ancient stability problem: the behaviour of solutions of the second order differential equation ẍ + f(s) x = 0, where f(s) tends to plus infinity as s tends to plus infinity. An extensive survey of the literature concerning this problem and a resume of results may be found in [ l ]. More recently McShane et a l. [2] have shown that the additional assumption f(s) ≥ 0 is not sufficient to guarantee that all solutions tend to zero as s tends to infinity. Our aim is to demonstrate a new criterion for which all solutions do have the above property. This criterion overlaps many of the cases heretofore considered.

In [3], the author proved that a ring R with identity is right noetherian and right injective if and only if R is a direct sum of a finite number of uniform right ideals, which are completely primary in the sense of that paper. In this paper, we shall determine the structure of such rings in the case where the sum of the isomorphic uniform components are twosided ideals. The ring is found to be a direct sum of total matrix rings over local rings.

Let R be a ring with an identity 1, and R′ a ring anti - isomorphic to R. Let V be an R-module as well as an R′-module. We assume that 1 a = a for all elements a in V and that V satisfies the minimum condition for R-submodules. Elements of R will be denoted by α, β, …, and those of V by a, b, … Elements of R′ will be a α′, β′, …, where α′ fcorresponds to α by the anti - isomorphism. A mapping f of V x V to R is called a bilinear mapping of V to R if it satisfies the following.

If f(x) is a polynomial with integral coefficients then the integer r is said to be a residue of f(x) modulo an integer m if the congruence

is soluble for x; otherwise r is termed a non-residue. When m is a prime p, Mord ell [4] has shown that the least nonnegative residue ℓ of f(x) (mod p) satisfies

Examining certain problems in physics M. Hosszu [l] obtained the functional equation

(1)

where x, y, f are real.

In another paper M. Hosszu [2] proved that the equation (1) is equivalent to the functional equation of Cauchy; i. e., to the equation

(1)

under the assumption that x is real and f is real and continuous.

Throughout this paper (X, d) will be a metric space with metric d, and h a homeomorphism of X onto itself. For any real number r > 0, and p ∊ X, U(p, r) will denote the open r - sphere about p. Any point p ∊ X is called regular [3] if for any given ∊ > 0 there exists a δ > 0 such that d(p, y) <δ implies d(hn(p), hn(y)) < ∊ for all integers n, where hn denotes the iterates of h for n > 0, of h-1 for n < 0, and h0 is the identity. Any point of X which is not a regular point i s called an irregular point. Let I(h) denote the set of all the irregular points of X and R(h) = X-I(h). Lim inf and Lim sup are defined as in [4].

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.

Moser and Zayachkowski [1] have discussed certain planar lattice paths from (0, 0) to (m, n). In this note we consider analogous paths in three dimensional space. For basic definitions see reference [2]. Throughout this note each of m, n and k is a positive integer and if S is a finite set, |S| will denote the number of elements of S.

Each path under consideration here is such that each of its steps is of one of the following types: