In [3] Rubel proved that if h(n) is an arithmetic function such that , L finite, then where μ(n) is the Mobius function. This result was extended to functions other than μ(n) in [4]; however, (as first pointed out to the author by Benjamin Volk), the order condition imposed there is unnecessary; in fact, utilizing the result of [3], the following slightly more general theorem has an almost trivial proof.

If R is a ring, a right (left) ideal of R is said to be large if it has non-zero intersection with each non-zero right (left) ideal of R [8]. If S is a set, let |S| be the cardinal number of S. We say a right (left) ideal I of a ring R is very large if |R/I| < < No. If a is an element of a ring R such that (a)r = {r ∊ R|ar = 0} is very large then we say a is very singular. The set of all very singular elements of a ring R is a two sided ideal of R. If R is a prime ring, then 0 is the only very singular element of R and a very large right (left) ideal of R is indeed large provided that R is not finite.

An automorphism Φ of a semigroup S is said to be an inner automorphism if there exists a unit u in S such that

for each a in S.

The aim of this paper is to extend the idea of a measure which takes on values in Euclidean n-space so as to allow it to assume infinite values while preserving its countable additivity over a given σ-ring. It is shown that in order to do this it is necessary to restrict the range of the measure to one infinite value.

In [1], p. 132, it is shown that there exist two disjoint subsets of a square each connecting a pair of opposite vertices. Our interest in obtaining covers by infinite collections of such sets was prompted by a question posed by R.B. Reed in [3], We are grateful also to Dr. M. Edelstein for his guidance and encouragement of our work on the subject.

It is an obvious property of convergence that implies that sn+k exists and equals s for k = -1 (left translativity ) and for k = 1 (right translativity). Not so for sumrnability.

The following proof of Caratheodory's Theorem, while not essentially new, seems to be natural and therefore of interest.

LEMMA 1. Let P denote a supporting hyperplane of the convex polytope K. Then P∩K is a convex polytope whose vertices are vertices of K.

An outstanding unsolved problem in the theory of rings is the existence or non-existence of a simple nil ring. Such a ring cannot be locally nilpotent as is well known [ 5 ]. Hence, if a simple nil ring were to exist, it would follow that there exists a finitely generated nil ring which is not nilpotent. This seemed an unlikely situation until the appearance of Golod's paper [ 1 ] where a finitely generated, non-nilpotent ring is constructed for any d ≥ 2 generators over any field.

It is well-known that Tychonoff's theorem on the product of compact spaces may be proved, for the special case of a countable number of metric spaces X1, X2…, Xn,…, in the following simple manner.

We write ω for the set of natural numbers (including zero) and for the set of all finite sequences of natural numbers. If n ∊ ω we write . If x is a function which takes its values in cu and whose domain of definition contains then we write for the element (x(0), …, x(n)) of .

Kaplansky's Lemma [3] states: the number of k-combinations of 1, 2,…, n with no two consecutive integers in any selection is Using this, Abramson [l; lemma 3] solves the problem: find the, number of k-combinations so that no two integers i and i+2 appear in any selection. (This is generalized by Abramson in [2].) An interesting solution, also using Kaplansky's lemma, is obtained as follows.

If n = 2m, we choose s from among the m even integer s, no two consecutive, and k- s from among the m odd integers, no two consecutive.

Let f(x) be a real-valued function defined on an interval Ia: [ 0, a ]. For each point x in Ia we form the series , where u0 and un+1 = f(un) for n ≥ 0. If the series converges, x will be called a point of convergence; if this series diverges, x will be called a point of divergence. La this note several properties of sets of convergence2 will be obtained.

The triangular matrix (A) = (X ), where n = 0, 1, 2,…; k = 0, 1, 2, …; and λn, k = 0 for k > n is regular (in the sense of defining a regular sequence to sequence transform) if

• for every fixed k ;

• independently of n;

• 

We prove the following

THEOREM 1. Let f1(z), f2(z) be entire functions of exponential type τ1, τ2; respectively. Suppose that for certain constants K1, K2,

on the real line.

The purpose of this paper is to prove and generalize the following theorem: Given any topological space X, of all the T2 spaces Z which are continuous images of X, there is a maximal one Y; that is, one over which all others factor, as in Figure 1.

In 1958, Egerváry and Turán [3] proposed and solved the problem of finding a stable interpolation process of minimal degree on a finite interval. Later [4] they investigated the same problem for an infinite interval with a suitable modification of the definition of stability. For the interval (-∞, ∞) their definition naturally differs from the one for the semi-infinite interval.

Let X and Y be topological spaces. If Y is a uniform space then one of the most useful function space topologies for the class of continuous functions on X to Y (denoted by C) is the topology of uniform convergence. The reason for this usefulness is the fact that in this topology C is closed in YX (see Theorem 9, page 227 in [2]) and consequently, if Y is complete then C is complete. In this paper I shall show that a similar result is true for the function space of connectivity functions in the topology of uniform convergence and for the function space of semi-connectivity functions in the graph topology when X×Y is completely normal. In a subsequent paper the problem of connected functions will be discussed.

In this note we consider a system of autonomous differential equations

1.1

where f: En → En is a continuously differentials Le mapping for n ≥ 2. We shall assume that f(0) = 0 and that the origin is locally asymptotically stable.

The following notations will be employed throughout this note.

The object of the present note is to obtain a new generating function for the Jacobi polynomials defined by [4, page 268]

Considering Cauchy's functional equation

f(z1+z2)=f(z1)+ f(z2),

where f(z) is an entire function of z, we have the following functional equation:

(1) |f(x+iy)|=|f(x)+f(iy)|,

where x and y are real.