Let A be an algebraic function field with a constant field k which is an algebraic number field. For each prime p of k, we consider a local completion kp and set Ap = Ak ꕕ kp. Then we have the question:

Is it true that A/k is a rational function field (i.e., A is a purely transcendental extension of k) if Ap/kp is so for every p ? In this note we shall discuss the question in a slightly different and hence easier case.

The purpose of this note is to give a new characterization of Prüfer domains using the concept of ring epimorphism, and to indicate some connections with well-known properties of Prüfer domains. All rings are commutative and have a unit element.

Let A, B, denote two non-void finite complexes (= subsets) of the torsion free abelian group G,

Let d(A),… denote the maximum number of linearly independent elements of A,… and let n = n(A, B) denote the number of elements of A + B whose representation in the form a + b is unique. In the preceding paper, Tarwater and Entringer [1] proved that n ≥ d(A).

The purpose of this paper is two-fold. Sections 2 and 3 are motivated by an observation that certain theorems concerning "diminishing orbital diameters" (introduced in [1]) are true under weaker assumptions. Specifically, we investigate the relationship between that concept and alternate conditions such as "asymptotic regularity", and in the process we sharpen some metric space results established in [1;5]. Mention is made in these sections of examples which show that certain additional weakenings of our hypotheses cannot be made, but we include in detail only the one which seemed to us most intricate.

Let Qt = [0, 1] be equipped with the topology consisting of Qt, the empty set and all subsets of Qt of the form [0, x), 0 < x ≦ 1. R. Nielsen and C. Sloyer [1, p. 514] proved that every T0-space can be embedded in for a suitable F. The purpose of this note is to generalize this result.

Let be the lower central series of a group G, where K2(G) = [G, G] and inductively Kn+1(G) = [Kn (G), G]. A theorem of Blackburn ([1], Hilfssatz) states that

Theorem 1. The exponent of Kn+1 (G)/Kn+2 (G) divides the exponent of Kn (G)/Kn+1 (G).

In the theory of surface area one meets situations where a function z = f(x, y) which is defined and continuous on a closed rectangle E, is partially differentiable on E except on a subset of E of Lebesgue measure zero.

Let (X, d) be a metric space and f a mapping of X into itself. D.F. Bailey [l] considered a class of mappings f satisfying the condition: ∀x, y ∈ X, x ≠ y,

(1.1)

where I+ denotes the set of positive integers.

Let d be a positive integer and let p be a prime > d. Set q = pm, where m ≥ 1, and let I (q, d) denote the number of distinct primary irreducible polynomials of degree d over GF(q). It is a simple deduction from the well-known expression for I(q, d) that

(1)

where d* is the largest positive integer < d which divides d if d > 1, and d* is 0 if d = 1.

It is shown that if v is an odd prime power, other than a prime of the form 22n + 1, then there exists a Room square of order v + 1.

A room square of order 2n, where n is a positive integer, is an arrangement of 2n objects in a square array of 2 side 2n - 1, such that each of the (2n - 1)2 cells of the array is either-empty or contains exactly two distinct objects; each of the 2n objects appears exactly once in each row and column; and each (unordered) pair of objects occurs in exactly one cell.

The following theorem appears in [1].

Let R be a closed simply connected region of the inversive plane bounded by a Jordan curve J, and let J be divided into three closed arcs A1, A2, A3. Then there exists a circle contained in R and having points in common with all three arcs.

A family of sets is said to possess property if there exists a set such that and for every We consider the following question raised by P. Erdös |1|: let n and N be positive integers, n ≥ 2 and N ≥ 2n - 1 and let S be a set of N elements; what is the least integer (provided such an integer exists), for which there exists a family of subsets of S satisfying

1. (a) |F| = n for each

1. (b)

1. (c) does not have property

1. (d) if and then has property

In his paper "Necessary and sufficient statistics for a family of probability distributions", Dynkin (1951) establishes the important concept of rank for such a family with this conclusion: "If the rank is infinite, then the family has no non-trivial sufficient statistic in any size of sample." His concept of rank is based on a theorem, Theorem 2 described below, which has been pointed out by Brown (1964) to be invalid under its hypotheses. This note shows that Dynkin's Theorem 2 remains valid under its original hypotheses provided that the set (in Dynkin's notation) Δ - S is countable.

In [1] Cutler proved the following theorem.

Theorem. If G and K are abelian groups such that nG ≅ nK for some positive integer n, then there are abelian groups U and V such that U ⊕ G ≅ V ⊕ K and nU = 0 = nV.

Cutler's proof is long and fairly involved. Walker [3] obtains the theorem rather elegantly as a corollary of his results on n-extensions. We give here a proof that is extremely simple both in conception and execution. Our proof relies on the notion of p-basic subgroups introduced by Fuchs in [2]. Therefore we shall first recall certain pertinent facts from [2].

Let A be an H-space and K a space. It is well known that [K, A] is a loop. Suppose A has a comultiplication as well, that is, cat A < 2. Then we shall prove that [K, A] is a Moufang loop. This generalises a result of C. W. Norman who proved this for the case where A is the circle, the 3-sphere or the 7-sphere. It also improves the known result that [K, A] is a diassociative loop if A has a comultiplication as well, since Moufang loops are diassociative.