Showing papers in "Pacific Journal of Mathematics in 1974"

Self-Adjoint Extensions of Symmetric Subspaces

Abstract: A theory of self-adjoint extensions of closed symmetric linear manifolds beyond the original space is presented. It is based on the Cayley transform of linear manifolds. Resolvent and spectral families of such extensions are characterized. These extensions are also determined by means of analytic contractions between the "deficiency spaces" of the original symmetric linear manifold.

Nonsolvable finite groups all of whose local subgroups are solvable. II.

Abstract: TABLE OF CONTENTS

On the irrationality of certain series.

Abstract: We conjectured that the series (1.2) are irrational under the single assumption that {a,) is monotonic and we observed that some such condition is needed in view of the possible choices a, = cp(n) + 1 or a, = O(N) + 1. These particular choices do not satisfy the hypothesis lim inf ~,+,/a, > 0 but we do not know whether that hypothesis which is weaker than that of the monotonicity of a, would suffice. In this note we obtain various improvements and generalizations of Theorem 1.1, in particular by relaxing the growth conditions on the a, and using more precise results in the distribution of primes. In 0 2 we obtain some general conditions for the rationality of series of the form C b,/(al, a. a, a,) which are modifications of [2, Lemma 2.291. In 5 3 we use a result of A. Selberg [3] on the regularity of primes in intervals to obtain improvements and generalizations of Theorem 1.1.

Dual generalizations of the Artinian and Noetherian conditions

Abstract: An Artίnίan module can be characterized in terms of certain properties of its factor modules. A module Mis Artinian if and only if the following two conditions hold for M: (I) Every nonzero factor module of M contains a minimal submodule. (A) The socle of every factor module ofM is finitely generated. The dual to the factor module is the submodule. We state the dual of (I): (Π) Every nonzero submodule ofM contains a maximal submodule. We call a module with property (Π) a Max module and one with property (I) a Min module. Every Noetherian module is a Max module but not conversely. This paper investigates these generalizations of the Artinian and Noetherian conditions and the relationships among them. Throughout this paper M denotes a right module over an arbitrary

Some remarks on harmonic measure in space.

Abstract: This investigation was motivated in part by the work of Hunt and Wheeden, [5], [6]. In these papers they consider Lipschitz domains, that is, domains whose boundaries are locally representable by graphs of Lipschitz functions. One of their main results is that a positive harmonic function defined on a Lipschitz domain has a nontangential limit at all points of the boundary except possibly those that belong to a set of harmonic measure zero. In the classical case where the domain is taken to be the half-space of R, the nontangential limit is known to exist at H' almost every point of the boundary, c.f., [2], [3]. We will show that for domains Ω satisfying a geometric measure theoretic condition, H~ (restricted to the boundary of Ω) and harmonic measure have the same null sets. Therefore, for these domains, the results of Hunt and Wheeden will represent a generalization of the classical case. By use of the conformal mapping theorem it is not difficult to prove, for a domain in R whose boundary is a simple closed rectifiable curve, that harmonic measure and H measure have the same null sets. In § 4 it will be shown that the analog of this does not hold in R. We give an example of a topological 2-sphere whose boundary has finite H measure and has a tangent plane at each point, but for which H measure is not absolutely continuous with respect to harmonic measure.

A new class of infinite sphere packings

Abstract: The oscillatory or Apollonian packing in two dimensions is well known and is described for example in [13]. Recently we investigated the three dimensional osculatory packing of a sphere [4] However, the results of that paper indicate that, for N > 3, ΛΓ-dimensional osculatory packings are irregular and not invariant under inversion as is the case for N = 2 and 3. In this paper we introduce a class of packings which we call discrete packings, and produce some examples. This class is analogous to the class of lattice packings which appear in the theory of packings of equal spheres. We shall use the systems of polyspherical coordinates developed in [4]. Section 2 contains a description of these as well as the proofs of some additional results needed here. The idea of the separation Δ{X, Y) between two spheres X and Y will again play an important role. In § 3 we consider inversively generated configurations obviously generalizing the construction used in [4]. That is, we begin with a 'cluster' of (N + 2) disjoint spheres and by successive inversions replace the spheres one at a time with new spheres in such a way that the separations between the spheres in the new cluster are the same as for the initial cluster. In terms of polyspherical coordinates the necessary inversions are represented by matrices which preserve a certain indefinite quadratic form. Repetition of the process leads to a configuration of spheres in EN which may or may not be a packing, depending on the initial cluster. In § 4 we give sufficient conditions under which an inversively generated configuration is a packing. The conditions force the separations between the spheres in the configuration to lie in a discrete subset of the rational numbers, hence the name 'discrete packing'. In addition to the two and three dimensional osculatory packings, we give examples of discrete packings for dimensions 2, 3, 4, 5, and 9. We do not know yet whether such packings exist in all dimensions. The examples we have found are given in § 6. The packings described in § 4 are not in general osculatory; that is, the largest possible sphere is not generated at each step. However,