Several separation axioms, defined in terms of continuous functions, were examined by van Est and Freudenthal [3], in 1951. Since that time, a number of new topological properties which were called separation axioms were defined by Aull and Thron [1], and later by Robinson and Wu [2], This paper gives a general definition of separation axiom, defined in terms of continuous functions, and shows that the standard separation axioms, and all but one of these new topological properties, fit this definition.

Let C by a Cayley-Dickson algebra over an algebraically closed field F of characteristic 0. A multiplication table for a basis of this 8-dimensional alternative algebra can be found in [3], page 137, where we take α = β = γ = — 1.

The origin and importance of multiparameter Sturm-Liouville problems in mathematical physics has recently been discussed by Atkinson [1, sects. 3 and 4], [2, introduction]. In spite of the importance of such problems, and in spite of the work done in this field by such early investigators as Klein and Hilbert, Atkinson points out that in recent years this field has been relatively neglected in contrast to the single parameter case. As an example, he states that as opposed to the one parameter case, the detailed behaviour of the eigenvalues and eigenfunctions in the multiparameter Sturm-Liouville case is still far from clear.

This paper establishes the existence of an infinite set of zeros for the solution of a certain functional differential equation. The primary condition assuring this oscillatory behavior is expressed in terms of the magnitude of the delay.

A previous result of the author concerning the parametric representation of infinitely many solutions of the title equation is strongly improved. New classes each containing infinitely many solutions of the equation for specified values of d are stated explicitly. The method of solution hinges heavily on solving the generalized Pell’s equation x2—Dy2=c.

Carathéodory has shown that if x1, x2,…, xm, (m finite) are points of Rn and if some

then ∃μ∈Ω with at most n+1 nonzero components and for which (See [5]). The authors of [2] have extended this result to include the case where m= +∞. In theorems 1 and 2 below we establish somewhat similar results for the case in which Ω is further restricted by a finite system of linear inequalities (or equalities).

In this note all Banach space are assumed to be real and separable and their norms will be denoted by || ||. The canonical bilinear form between a Banach space B and its topological dual B′ will be denoted by 〈x, y〉, x ∊ B, y ∊ B′.

Rings (all of which are assumed to be associative) with no non-zero nilpotent elements will be called reduced rings; R is a reduced ring if and only if x2=0 implies x=0, for all x∈R. In 2. we prove that the following conditions on an annihilator ideal I of a reduced ring are equivalent: I is a maximal annihilator, I is prime, I is a minimal prime, I is completely prime. A characterization of reduced rings with the maximum condition on annihilators is given in 3.

Consider the 1st order linear system

1

when An(t) is real valued and continuous on I, ai, i+1(t) ≠ 0, aij(t) ≡ 0 for j ≥ i + 2, t ∊ I and i = 1, …, n — l. In what follows, components cj of a vector c will be identifiable by the superscript. The purpose of this note is to give a simple induction proof of the following theorem of Frobenius [1].

Throughout this paper all semigroups will have identity, and all S-systems (operands) will be right unitary S-systems. All homomorphisms will be S-homomorphisms unless specified.

The concept of standard representations of simple Lie algebras was introduced by I. Z. Bouwer [1], One of the difficulties was that of existence. The order zero standard representations are simply those having a dominant weight vector and these have been completely characterized, for example in [2].

Let M be a differentiable manifold of class C∞, with a given (1, 1) tensor field J of constant rank such that J2=λI (for some real constant λ). J defines a class of conjugate (G-structures on M. For λ>0, one particular representative structure is an almost product structure. Almost complex structure arises when λ<0. If the rank of J is maximum and λ=0, then we obtain an almost tangent structure. In the last two cases the dimension of the manifold is necessarily even. A Riemannian metric S on M is said to be related if one of the conjugate structures defined by S has a common subordinate structure with the G-structure defined by S. It is said to be J-metric if the orthogonal structure defined by S has a common subordinate structure.

Let f∈Lp (1≤p<∞) 2π-periodic,

(1)

and let us consider the rth order moduli of continuity of f

(2)

R. de Vore stated the following conjectures:

(a) Let ηv→0 decreasingly,

(3)

In this article, we are concerned with the following inequality

(1)

where 0<pi<1, 0<q<1, (i=l, 2,…,n), n is a fixed positive integer, n≥2 and f(p)≠0 for <p<l.

This inequality was first considered by A. Renyi, who gave the general differentiate solution of (1) for n≥3, [1]. With the help of this inequality one can characterize Renyi’s entropy [2].

We shall state later the Renyi’s result, which will be a special case of the Theorem 3.

In this paper we consider the equation

(1.1) (r(t)y′(t))′+p(t)f(y(t)) = 0

under the conditions

((H0): the real valued functions r, r′ and p are continuous on a non-trivial interval J of reals, and r(t)>0 for t∈J;

and

(H1):f:R→R is continuously differentiable and odd with f'(y)>0 for all real y. We also consider the equation

(1.2) y″(t)+m(t)y′(t)+n(t)f(y(t)) = 0

under the conditions (H1) and

(H2): the real valued functions m and n are continuous on a non-trivial interval J of reals.

We answer the following problem posed by Herrlich in the affirmative: “Can the Freudenthal compactification be regarded as a reflection in a sensible way?” This is accomplished by exploiting the one-to-one correspondence between proximities compatible with a given Tihonov space and compactifications of that space. We give topological characterizations of proximally continuous functions for the proximities associated with the Freudenthal and Fan-Gottesman compactifications.

Z. Opial [11] proved in 1960 the following theorem:

Theorem 1. If u is a continuously differentiable function on [0, b], and if u(0)= u(b)=0 and u(x) > 0 for x ∊ (0, b), then

1

where the constant b/4 is the best possible.

We name a functional equation with restricted argument one in which at least one of the variables is restricted to a certain discrete subset of the domain of the other variable(s). In particular, the subset may consist of a single element.

The purpose of this paper is to present a functional equation satisfied only by cosine functions.

Let F be a field of characteristic 0 and G a group such that each element of the group ring F[G] is either (right) invertible or a (left) zero divisor. Then G is locally finite.

This answers a question of Herstein [1, p. 36] [2, p. 450] in the characteristic 0 case. The proof can be informally summarized as follows: Let gl,…,gn be a finite subset of G, and let

1—x is not a zero divisor so it is invertible and its inverse is 1+x+x3+⋯. The fact that this series converges to an element of F[G] (a finite sum) forces the subgroup generated by g1,…,gn to be finite, proving the theorem. The formal proof is via epsilontics and takes place inside of F[G].

It is well known that in the Euclidean plane there are seven distinct frieze patterns, i.e. seven ways to generate an infinite design bounded by two parallel lines. In the hyperbolic plane, this can be generalized to two types of frieze patterns, those bounded by concentric horocycles and those bounded by concentric equidistant curves. There are nine such frieze patterns; as in the Euclidean case, their symmetry groups are and