This note proves (in the theorem below) a conjecture made by the author last year through the pages of the Departmental Problem Book. This arose in connection with some other investigations of arithmetic functions.

Let σ*(N) denote the sum of the unitary divisors of N, that is,

It is easily seen that σ*(N) is multiplicative. In fact σ*(l) = 1 if N > 1 has the prime decomposition . Let us define a positive integer to be unitary perfect whenever σ*(N) = 2N. The first four such numbers are 6, 60, 90 and 87, 360. In a recent abstract [ l ] published by one of us, the last of these numbers was overlooked. No other unitary perfect numbers are known to the authors.

P. Erdös and R. Rado [1] proved that to each pair of positive integers n and k, with k ≥ 3, there corresponds a least positive integer φ(n, k) such that if is a family of more than φ(n, k) sets, each set with n elements, then some k of the sets have pair-wise the same intersection.

If {avα: or = 1, 2, …, N }, with Nv → ∞ as v → ∞, is a double sequence of real numbers with the property that , then

1.1

is known in statistical literature as the Wald- Wolfowitz- Noether condition and it plays an important role in the proofs of certain types of central limit theorems (see e. g., [ l ], [2] ).

The line - graph of an ordinary graph G is that graph whose points can be put in one-to-one correspondence with the lines of G in such a way that two points of are adjacent if and only if the corresponding lines of G are adjacent. This concept originated with Whitney [ 5 ], has the property that its (point) chromatic number equals the line chromatic number of G, where the point (line) chromatic number of graph is the minimum number of colors required to color the points (lines) of the graph such that adjacent points (lines) are colored differently.

A graph is called Euler if it has no isolated vertices and every vertex has even or infinite degree. The graphs we consider may have multiple edges but no loops. Loosely speaking, we will be concerned with the conditional compactness of the set of all countable Euler subgraphs of a given graph.

As is well known, any module M over a ring possesses injective hulls, i. e., injective essential extensions [3], unique up to isomorphisms which map M identically, but the various proofs for this all require some transfinite arguments and hence provide little indication as to how these hulls may actually be constructed for a given module.

It is well known that Z(p∞) is isomorphic to each of its non-zero homomorphic images [3]. The aim of the present note is to generalize this fact about Z(p∞) to indecomposable infective modules over rings more general than the ring of integers which will include Dedekind domains as a special case.

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 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.

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.

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]

It is a well-known fact in the theory of Appell's hypergeometric function of two variables F4, defined by

1

where |x|1/2 + |y|1/2 < 1, that it can be expressed interms of products of ordinary hypergeometric functions when γ + γ' = α + β + 1.

It is well known that the determination of a (non-isotropic) curve in the euclidean 3-space with given curvature κ(S) and torsion τ(s), where s represents the arc-length, depends upon the integration of a Riccati equation; and that this can be performed only if a particular integral of the equation is known.

It is a commonplace that F is continuous on the cartesian square of the range of f if f is continuous and satisfies

1

say, for all real x, y (cf. e.g. [2]). A.D. Wallace has kindly called my attention to the fact, that this is trivial only if f is (constant or) strictly monotonic and asked for a simple proof of the strict monotonicity of f. The following could serve as such: if on an interval f is continuous, nonconstant and satisfies (1), then f is strictly monotonic there.

Let w = f(z) be holomorphic on the unit disk D = { z: | z | < 1}, with the additional restrictions that | f ( z ) | < l and , where denotes the (outer) angular limit of f (z) at z = 1. Let us now define and then focus our attention on the behaviour of g(z) in an arbitrary angular neighbourhood of z = 1. Whenever exists, this limit is commonly referred to as the angular derivative of f(z) at z = 1.