In this paper the results of [5] are extended to classes of lattices. We assume familiarity with [5], but we recall for convenience the principal definitions and notations. If is a category and if is a direct [resp., inverse] limit system in , then is the direct [resp., inverse] limit of (determined only up to isomorphism in ). If is an inverse limit system of sets or universal algebras, let denote the canonical construction of inverse limit described for example in [1, Chapter 3].

We consider effective homogeneous spaces M = G/H where G is a compact connected Lie group, H is a closed subgroup and G acts effectively on M (i.e., H contains no non-trivial subgroup normal in G). It is known that dim G ≦ m2/2 + m/2 where m = dim M and that if dim G = m2/2 + m/2, then M is diffeomorphic to the standard sphere Sm or the standard real projective space RPm [1]. In addition it has been shown that for fixed m there are gaps in the possible dimensions for G below the maximal bound [4; 5].

In [11] and [12] we investigated the process of localization of right Noetherian rings R at prime ideals. We shall now extend these investigations to semiprime ideals N of R.

In Section 2 we show that localizing at the injective right R-module E(R/N) is the same as localizing with respect to the multiplicative set

Let ϵ stand for the set of nonnegative integers (numbers), V for the class of all subcollections of ϵ (sets), Λ for the set of isols, and ΛR for the set of regressive isols. A function, f, is a mapping from a subset of ϵ into ϵ and δf and ρf denote the domain and range of f respectively. The relation of inclusion is denoted by ⊂ and that of proper inclusion by ⊊. The sets α and β are recursively equivalent (written α ≃ β), if δf = α and ρf = β for some function f with a one-to-one partial recursive extension.

The classical comparison and interlacing theorems of Sturm were originally proved for the equations

under the assumption that all coefficients are real-valued, continuous, and p > 0, P > 0. Atkinson [1, Chapter 8] has carried out the standard theory for eigenvalue problems involving (1), under the more general hypothesis

A collection of subspaces of a Hilbert space is called a nest if it is totally ordered by inclusion. The set of all bounded linear operators leaving invariant each member of a given nest forms a weakly-closed algebra, called a nest algebra. Nest algebras were introduced by J. R. Ringrose in [9]. The present paper is concerned with generating nest algebras as weakly-closed algebras, and in particular with the following question which was first raised by H. Radjavi and P. Rosenthal in [8], viz: Is every nest algebra on a separable Hilbert space generated, as a weakly-closed algebra, by two operators? That the answer to this question is affirmative is proved by first reducing the problem using the main result of [8] and then by using a characterization of nests due to J. A. Erdos [2].

The commutative Gelfand-Naimark Theorem says that any commutative C*-algebra A is isomorphic to the ring C0(M, C) of all continuous complex-valued functions tending to zero outside of compact sets of a locally compact Hausdorff space M. A very important part of this theorem is an intrinsic and also a complete characterization of M as exactly the primitive ideal space of A in the hull-kernel (or weak-star) topology. In the non-commutative case, A ≌ Γ0(M, E)—the ring of sections tending to zero outside of compact subsets of a locally compact Hausdorff space M with values in the stalks or fibers E.

Let G be a discrete group and let H = L2(G), with norm | |. Let B(H) be the ring of bounded operators on H with the norm

The right regular representation of G on H induces an injection ρ : C[G] → B(H), and W(G) is the closure of the image of ρ in the weak operator topology on B(H) (C = complex numbers). Using ρ, we identify C[G] with its image in W(G).

In [6], by means of convex functions Φ :R→R, Hardy, Littlewood and Pólya proved a theorem characterizing the strong spectral order relation for any two measurable functions which are defined on a finite interval and which they implicitly assumed to be essentially bounded (cf. [6, the approximation lemma on p. 150 and Theorem 9 on p. 151 of their paper]; see also L. Mirsky [10, pp. 328-329] and H. D. Brunk [1,Theorem A, p. 820]).

In this paper, we are concerned with the following problem: Let S be a finite set and Sm* ⊂ 2S a collection of subsets of S each of whose members has m elements (m a positive integer). Let f be a real-valued function on S and, for p ∊ Sm*, define f(P) as Σs∊pf (s). We seek the minimum (or maximum) of the function f on the set Sm*.

The Traveling Salesman Problem is to find the cheapest polygonal path through a given set of vertices, given the cost of getting from any vertex to any other. It is easily seen that the Traveling Salesman Problem is a special case of this system, where S is the set of all edges joining pairs of points in the vertex set, Sm* is the set of polygons, each polygon has m elements (m = no. of points in the vertex set = no. of edges per polygon), and f is the cost function.

Several papers of George Glauberman have appeared which analyze the structure of a finite p-group which contains two isomorphic maximal subgroups. The usual setting for an application of these results is a finite group, a p-subgroup, and an isomorphism of this p-group induced by conjugation. In this paper we prove a stronger version of Glauberman's Theorem 8.1 [1].

A dissectible polyhedron is a natural extension of a concept whose history dates back to at least 1758 and Euler [7]—the concept of a dissection of a polygon. An interesting historical survey of dissections of a polygon is given by Brown [4]. Some approaches to the classical problem have been given by Moon and Moser [9] and by Guy [8] ; the latter provides an approach which is the basis of the work in this paper. A summary of enumeration results on dissections of polygons and polyhedra by automorphism groups has been given by the authors [2].Recent extensions of the problem have been investigated in a series of papers by Brown and Tutte [3; 5; 14; 15] and b y Takeo [10; 11; 12; 13].

In this paper we consider representations of groups over the field of the complex numbers.

The nth-Kronecker power σ⊗n of an irreducible representation σ of a group can be decomposed into the constituents of definite symmetry with respect to the symmetric group Sn. In the special case of the general linear group GL(N) in N dimensions the decomposition of the defining representation at once provides irreducible representations of GL(N) [9; 10; 11].

In [7; 8; 11] it is shown that an appropriate map of a planar surface into a 3-manifold can be replaced by an embedding. In [1 ; 4 ; 6 ; 7 ; 9 ; 10] conditions are given so that a "non-trivial" map of a planar surface (2-sphere) can be replaced by a non-trivial embedding of a planar surface (2-sphere). In this paper we give conditions on an annular map which guarantee the existence of a non-trivial embedding of an annulus. It is reported that F. Waldhausen has proved a similar but stronger "annulus theorem".

The purpose of this article is to show how the main result of Easton [1] can be obtained as a special case of a general theory, which was developed in [6], of Boolean-valued models of ZF when the Boolean algebra is a proper class in the ground model. Indeed [1] was the motivating example for [6], Thus the present article together with [6] contain a presentation of Easton's forcing argument in the context of Boolean-valued models. This presentation is not, however, an automatic translation of Easton's argument from the language of forcing to that of Boolean-valued models. In fact, we hope to illuminate the "black magic" referred to in Rosser [8, p. 169].

Let S1 . . . ,Sr be r line segments, each of non-zero length, in n-dimensional euclidean space Rn. If a polytope Z is defined as the vector (Minkowski) sum

(1) Z = S1 + . . . + Sr,

then the segments Si will be called the components of Z. Since we do not wish to exclude the possibility that some of the components may be parallel, the polytope Z may be written in the form (1) in many different ways. For this reason it is convenient to define a zonotope to be the polytope Z together with some specified set of components {S1 , . . . , Sr}. Figures 1, 2 and 3 show some zonotopes of 1, 2 and 3 dimensions with 4, 5 and 6 components.

In this note a number of compactifications are discussed within the class of artinian rings. In [1] the following was proved:

Theorem. For an artinian ring R the following are equivalent:

(1) R is equationally compact.

(2) R+ ≃ B ⊕ P, where B is a finite group, P is a finite direct sum of Prüfer groups, and R · P = P · R = {0}.

(3) R is a retract of a compact topological ring.

In a fundamental paper on torsion-free abelian groups, R. Baer [1] proved that the group P of all sequences of integers with respect to componentwise addition is not free. This means precisely that P is not a direct sum of infinite cyclic groups. However, E. Specker proved in [9] that P has the property that any countable subgroup is free. Since an infinite abelian group G is called -free if each subgroup of rank less than is free, these results are equivalent to: P is -free but not free.

The decidability of the elementary theory of abelian groups, and their complete classification by elementary properties (i.e. those formalizable in the lower predicate calculus (LPC) of formal logic), were established by W. Szmielew [13]. More general results were proved by Eklof and Fischer [2], and G. Sabbagh [12]. The rather formidable "high-power" techniques used in obtaining these remarkable results, and the length of the proofs (W. Szmielew's proof takes about 70 pages) triggered off several attempts at simplification. M. I. Kargapolov's proof [3] unfortunately turned out to be erroneous (cf. J. Mennicke's review in the Journal of Symbolic Logic, vol. 32, p. 535).

This paper is divided into two sections. In the first, we characterize the subgroups N of a reduced abelian primary group for which all pure N-high subgroups are bounded. This condition on pure N-high subgroups occurs in several instances, for instance, all pure N-high subgroups of a primary group G are bounded if G is the smallest pure subgroup of G containing N; all N-high subgroups are bounded if N ≠ 0 and all N-high subgroups are closed in the p-adic topology.