Suppose throughout that

and that {μn}(n≥ 0) is a sequence of real numbers. The (generalized) Hausdorff moment problem is to determine necessary and sufficient conditions for there to be a function x in some specified class satisfying

.

We extend the Picard's theorem to ordinary differential equation of generalized order α, 0 ≤ α ≤ l, and prove a global existence and uniqueness theorem by using the Banach contraction principle.

Let w(z) be regular in the unit disc U:|z|<l, with w(0) = 0 and let h(r, s, t) be a complex function defined in a domain D of C3. The author determines conditions on h such that if

z∈U, then |w(z)|< 1 for z ∈ U and n= 0, 1, 2, …. Here Dnw(z) = (z/(l-z)n+1*w(z), where * stands for the Hadamard product (convolution). Some applications of the results to certain differential equations are given.

Let E be a locally convex tvs, F a normed space and the space of continuous linear mappings from E into F In this paper, we investigate the continuous convergence structure (c-structure) on. denotes the resulting convergence vector space (cvs).

The c-structure is by definition the coarsest cvs structure on making evaluation a continuous mapping.

Let 0<a1<…<an be integers and (a, b) denotes the greatest common divisor of a, b. R. L. Graham [1] has conjectured that

for some i and j. In a recent paper Weinstein [2] has improved Winterle's result [3] and has proven the following interesting theorem:

If A is the sequence a1< … <an where ak = P, a prime for some k and , then

.

D. E. Blair [1] has introduced the notion of K-manifolds as an analogue of the even dimensional Kähler manifolds and of the odd dimensional quasi-Sasakian manifolds. These manifolds have been studied with respect to a positive definite metric. In this paper, we study a more general case of if-manifolds carrying an arbitrary non-degenerate metric, in particular, a metric of Lorentz signature. This theory is then applied within the frame-work of general relativity. Using the Ruse-Synge classification [8, 9] of non-null electromagnetic fields with source, we develop a geometric proof for the existence of either two space like or one space like and one time like Killing vector fields on the space-time manifold.

J. H. E. Cohn (1) has shown that

are the only square Fibonacci numbers in the set of Fibonacci numbers defined by

If n is a positive integer, we shall call the numbers defined by

(1)

pseudo-Lucas numbers.

Let (X, d) be a metric space and Y and Z subsets of X. We say that Z is a bisector in Y and write Y⊳Z iff Y⊃Z and there are two distinct points y1, y2 ∈ Y such that Z = ={z:d(z, y1) = d(z, y2) and z∈Y}. By a reduced bisector chain in (X, d) of length n we understand a chain X = such that dim Xn≤0 and dimXn-1>0). By r(X, d) we denote the maximum length of reduced bisector chains in (X, d). For a metrizable topological space X we introduce the topological invariant r(X) as the minimum of r(X, d) taken over the set of all metrizations d of X. We prove that the function r(X) coincides with the dimension of X on the class of compact metric spaces.

It is well known that the real number field can be characterized as an ordered field satisfied the “least upper bound” property.

Using the idea of n -ordered set, introduced in [3], and generalizing the notion of l.u.b. in a suitable way, it is possible to give a similar categorical definition of the complex field.

With these extended meanings, the main theorem of this paper (Theorem 7 in the text) is stated almost identically to the one for the real field. Any directly two-ordered field, in which the "supremum property" holds, is isomorphic to the complex field.

Throughout R will denote an associative ring with identity. Let Zℓ(R) be the left singular ideal of R. It is well known that Zℓ(R) = 0 if and only if the left maximal ring of quotients of R, Q(R), is Von Neumann regular. When Zℓ(R) = 0, q(R) is also a left self injective ring and is, in fact, the injective hull of R. A natural generalization of the notion of injective is the concept of left continuous as studied by Utumi [4]. One of the major obstacles to studying the relationships between Q(R) and R is a description of J(Q(R)), the Jacobson radical of Q(R). When a ring is left continuous, then its left singular ideal is its Jacobson radical. This facilitates the study of the cases when either Q(R) is continuous or R is continuous.

The purpose of this note is to continue the author's study of the automorphisms of certain factors of type II1 Namely, those factors arising from the left regular representation of a free nonabelian group. Our main result shows that the outer conjugacy classes of automorphisms of such a factor are not countably separated. This had previously been shown only when the number of free generators was assumed to be infinite.

Let R be a commutative ring with identity and let X be an indeterminate. In [5] and [16] it was shown that if f, g∈R[X], then for some integer n≥l, c(f)n+1c(g) = c(g)nc(fg), where c(h) denotes the additive subgroup of JR generated by the coefficients of h ∈ R[X]. Actually the statements in [5J and [16] are not so general as this; however, the proofs are. Specifically, in [16] Mertens considers only the case that R is a polynomial ring over the integers, but this gives the result for any ring by specializing the coefficients of f and g. In [5] Dedekind considers only the case that JR is a ring of algebraic integers, but his proof is completely general. Further, the above formula then holds if one lets c(h) denote the S-submodule of R generated by the coefficients of h ∈ R[X], S a subring of R, and it is in this form that it usually appears, especially the case S = R. Dedekind′s very elegant proof is reproduced in [15, p. 9, Lemma 6.1].

A hypergraph is a subtree of a tree (SOFT) hypergraph if there exists a tree T such that X=V(T) and for each there is a subtree Ti of T such that Ei = V(Ti). It is shown that H is a SOFT hypergraph if and only if has the Helly property and , the intersection graph of is chordal. Results of Berge and Gavril have previously shown these to be necessary conditions.

There are now a number of Vitali covering properties which have been defined to handle problems arising in differentiation theory. Although some of these have received a unified treatment, as for example in the setting of Orlicz spaces in [1, p. 168], the underlying simplicity can be lost and the intimate connection with the original weak Vitali covering property of de Possel obscured. In this note we present an exposition of a family of covering properties and show how the original methods of de Possel in [4] can be pushed to provide an exact solution of the problem of determining necessary and sufficient covering properties for a basis which is known to differentiate a given class of integrals.

The primary aim of this paper is to extend Barnes [1], Marshall-Olkin [6], and Nehari [8] inequalities as applications of some results introduced in [10] by the author.

Since several results from various sources are adopted here, a unified notation is required in order to simplify our subsequent arguments. To this end, let Lp = Lp(S, ∑, μ), p>0 (unless otherwise stated), be the space of all pth power non-negative integrable functions over a given finite measure space (S, ∑, μ) (where S may be regarded as a bounded subset of real numbers).

In this paper we answer an open question of C. J. Earle ([2] §3.3 remarks (a) and (b)) in several cases. We first give some definitions and state some results which are given in greater detail in [2].

We let X be a smooth surface of genus g ≥ 2 and let M (X) be the space of smooth complex structures with the C∞ topology. If μ∈M(X) let Xμ denote the Riemann surface determined by μ.

The Chinese Remainder Theorem states that if I and J are comaximal ideals of a ring R, then A/(I∩J)A is isomorphic to A/IA×A/JA for any left R-module A. In this paper we study the converse; when does A/(I∩J)A and A/IA×A/JA isomorphic imply that I and J are comaximal?

Let us say that an order embedding of an uncountable regular cardinal in a linearly ordered set is continuous if it preserves the suprema (for all smaller limit ordinals). This makes the embedding a homeomorphism for the two order topologies; and if the image has no supremum, it is a closed subspace. Since uncountable regular cardinals fail to be paracompact, a linearly ordered set can be paracompact only if it admits no such embedding or anti-embedding. Conversely, Gillman and Henriksen have shown that this suffices (Trans. A.M.S. 77 (1954) pp. 352 ff).

In this note, we patch up the proof of a Theorem due to Handel on the characterization of homotopy epimorphisms ([6], 2.2) and generalize a Theorem due to Ibisch on attaching disk-bundles to Dold fibrations ([7], Satz 1).

We work throughout in the category TopB of spaces over B for some fixed topological space B.

In this note we prove two results about derivations. The first result holds in any ring, whereas the second one is valid in prime rings. From the nature of the results one would expect them to be known, but we have not been able to locate them in the literature.