In [1] Higman and Neumann asked the questions whether the Frattini subgroup of a generalized free product can be larger than the amalgamated subgroup and whether such groups necessarily have maximal subgroups. In [4] Whittemore gave answers to the special cases of generalized free products of finitely many free groups with cyclic amalgamation and of generalized free products of finitely many finitely generated abelian groups. In this paper we shall study the Frattini subgroups of generalized free products of any groups with cyclic amalgamation.

The purpose of this paper is to continue the work of [7]. Throughout the paper all notations shall have the same meanings as those in [7]; that is, the ring R is commutative with identity, B(R) is the set of idempotents in R, Spec B(R) is the set of prime ideals in B(R), and Ue for e in B(R) denotes the set {x/x in Spec B(R) with 1—e in x}. Then from [6] we know that {Ue/e in B(R)} forms a basic open set for a topology imposed in Spec B(R) and this topological space is totally disconnected, compact and Hausdorff.

We are concerned with the nth (n≥3) order linear differential equation

1

where the coefficients are continuous on (∞, ∞). Our main result is to give conditions under which the two-point boundary value function rij(t) (see Definition 2) are strictly increasing continuously differentiable functions of t. Levin [1] states without proof a similar theorem concerning just the monotone nature of the rij(t) but assumes that the coefficients in (1) satisfy the standard differentiability conditions when one works with the formal adjoint of (1). Bogar [2] looks at the same problem for an nth-order quasi-differential equation where he makes no assumption concerning the differentiability of the coefficients in the quasi differential equation that he considers. Bogar gives conditions under which the rij(t) are strictly increasing and continuous. The different approach of the author to this problem also enables the author to establish the continuous differentiability of the rij(t) and to express the derivatives in terms of the principal solutions Uj(x, t), j=0, 1,…,n-1 (see Definition 4).

This paper, by generalizing von Neumann's proof of the Radon-Nikodym and Lebesgue decomposition theorems [3], obtains analogous results for positive linear functional on a C* algebra. The concept of "absolute continuity" used and the Radon-Nikodym portion of the resulting theorem are due to Dye [2].

The Kantorovitch polynomials of a summable function s, defined on [0, 1], are

where

and

They are the analogue for summable functions of the Bernstein polynomials Bnf(x), and they possess similar properties [1].

In the theory of deformations of compact complex manifolds, the hypothesis of constancy of the dimension of diverse structural cohomology groups pertaining to a fibre plays an important role (see, for instance, [3, Propositions 2.5 and 2.7, Theorems 2.2 and 2.3, and Definition 6.1]). This paper is the first of two devoted to the investigation of conditions under which constancy of the dimension of given cohomology groups is assured, and more generally, to the study of the variation of that dimension.

In [2] Griffiths introduces extendible forms in a holomorphic deformation. We consider in this paper a differentiate family, and besides extendible, also co-extendible and transportable forms (see §5), and deduce from their existence conclusions about the variation of the dimension of the corresponding structural cohomology groups. It is left to a subsequent paper to give more explicit conditions by means of cohomology operations, and to deal with some applications.

It is shown that a topological space has a unique compatible quasi-uniformity if its topology is finite. Examples are given to show the converse is false for T1 and for normal second countable spaces. Two sufficient conditions are given for a topological space to have a compatible quasi-uniformity strictly finer than the associated Császár-Pervin quasi-uniformity. These conditions are used to show that a Hausdorff, semi-regular or first countable T1 space has a unique compatible quasi-uniformity if and only if its topology is finite. Császár and Pervin described, in quite different ways, quasi-uniformities which induce a given topology. It is shown that, for a given topological space, Császár and Pervin described the same quasi-uniformity.

Throughout this paper, (X, T, π) is a topological transformation group [1], L={x∊X:xt=x for some t∊{e}} and 0=X—L is nonempty; standard topological concepts are used as defined in [2].

The problem to be considered here has been studied in [3] and [6]. In [3], X is assumed to be a compact metric space, and each t e T satisfies a convergence condition on certain subsets of X. Under these conditions, Kaul proved that if T is equicontinuous on 0, then the group properties of discontinuity, proper discontinuity, and Sperner's condition (see Definition 1) are equivalent.

The author has defined [2] the approximately continuous Denjoy integral (AD-integral) which includes exactly the general Denjoy integral and the AP-integral defined by Burkill [2].

The aim of this paper is to give a constructive definition of the AD-integral.

In this paper we characterize the 2 × 2 invertible matrices over a Boolean ring, and, using this characterization, show that every invertible matrix has order dividing 6. This suggests that GL2 of a Boolean ring is built up out of copies of the symmetric group S3. Indeed, if B is a finite Boolean ring, then GL2(B) turns out to be a direct sum of copies of S3. If B is infinite, then GL2(B) is more difficult to calculate; we present here descriptions of GL2(B) for the "extreme" cases of countable Boolean rings—namely, the Boolean ring which is generated by its atoms and the atomless Boolean ring. The former provides a negative answer to the question of whether the functor GL2(⋅) preserves inverse limits; the latter is a corollary of a theorem which states that, under certain circumstances, GL2(⋅) preserves direct limits. It turns out, in addition, that every invertible matrix is a product of elementary ones, as is the case for matrices over a Euclidean domain.

We find several equivalent conditions for an A*-algebra with dense socle to be completely continuous. Such an A*-algebra is modular annihilator [10]. We also study modular annihilator A*-algebras with the weak (βk)-property and obtain a necessary and sufficient condition for such algebras to be dual.

Lehmer [3] and Reisel [7] have devised tests for determining the primality of integers of the form A2n—1. Tables of primes of these forms may be found in [7] and Williams and Zarnke [10]. Little work, however, seems to have been done on integers of the form N=2A3n—1. Lucas [6] gave conditions that were only sufficient for the primality of N. Recently Lehmer [4] has given a method for determining the primality of an integer N if the factorization of N+1 is known.

The R1 axiom was first introduced by Davis in [1]. It is strictly weaker than the T2 axiom. Murdeshwar and Naimpally, in [4], have weakened the T2 hypothesis to R1 in some well-known theorems. We show that in many topological spaces the R1 axiom and regularity are equivalent. Also, the definition of local compactness given in [4] can be weakened to the usual definition and still get the same results.

The notion of a bitopological space was first introduced by Kelley in [3]. Fletcher, Hoyle, and Patty discuss pairwise compactness for bitopological spaces in [2]. One of our main results is that a bitopological space (X, P, Q) is pairwise compact if and only if each ultrafilter v on X, containing a proper P closed set and a proper Q closed set, has a common P and Q limit.

Let X be any completely regular Hausdorff topological space, and let βX denote its Stone-Čech compactification. This note is devoted to proving the following result:

5. THEOREM. Let X be realcompact and noncompact. Then βX—X is not connected im kleinen at any point.

In [2] and [3] the homology of reduced product spaces of certain type of polyhedra was studied. Let Xn=X×X×…×X be the Cartesian product of n copies of a topological space X.

A simple but useful result in the measure theory for product spaces can be stated as follows:

Theorem A. A necessary and sufficient condition that a measurable subset E of X×Y has measure zero is that almost every X-section (or almost every Y-section) has measure zero (see [1, §36]).

We will show, in this short note, that a similar result also holds for the exponential of measure spaces. Before proceeding any further, we describe briefly here the exponential construction of a measure space.

Let V[0, 2π] denote the class of all normalized functions F of bounded variation in [0, 2π] such that F(x) = 2-1{F(x+0)+F(x-0)} and F(x+2π)-F(x) = F(2π) — F(0) for all x and let {Cn} be the sequence of Fourier-Stieltjes coefficients of F. Wiener [9] (cf. Bari [1, p. 212], Zygmund [10, p. 108]) proved the following theorem.

Using a well-known theorem of Burnside on permutation groups of prime degree we offer new and simplified proofs of Theorems A, B, B' below for the case q=p a prime.

Bochner has introduced the idea of almost automorphy in various contexts (see for example [1] and [2]). We shall use the following definition:

A measurable real valued function f of a real variable will be called almost automorphic if from every given infinite sequence of real numbers we can extract a subsequence {αn} such that

• (i) exits for every real t but no kind of uniformity of convergence is stipulated;

• (ii) exits for every t;

• (iii) for every t.

For the Hermite polynomials Hn(z) defined by

1

it is easy to see from the Rodrigues formula that

2

where, as usual, Dz=d/dz.

In recent papers ([1], [2]) Carlitz has proved the following formulae:

3

4

where, on the right-hand side of (4), the range of each summation is from i= 1 to i=k(k=1,2,…).