Two point sets H and K are said to be strongly separated if there exist two mutually exclusive domains DH and DK containing H and K respectively such that either and are mutually exclusive or · is R. L. Moore has shown [2, Theorem 153, Chapter I] that if S is a normal Moore space and H and K are two mutually separated point sets then H and K are strongly separated. In this paper it is shown that if 5 is a Moore space, (1) H and K are two mutually separated point sets and (2) the closure of the set of all boundary points of H which do not belong to is compact, then H and K are strongly separated.

Let R be a ring with identity, and let C be a small, nonempty category. We denote the category of right R-modules by AbR and the category of contravariant functors C → AbR by AbRC*. The limit functor

is left exact, and its kth right derived functor is denoted by colimk. The R-cohomological dimension of C is defined by

If there is a unitary ring homomorphism R→S, then it is not difficult to show that cdsC ≦ cdRC.

A graph X is a set V(X) (the vertices of X) with a system E(X) of 2-element subsets of V(X) (the edges of X). Let X, Y be graphs and f : V(X) → V(Y) a mapping; then/ is called a homomorphism of X into F if [f(x),f(y)] ∈ E(Y) whenever [x,y] ∈ E(X). Endomorphisms, isomorphisms and automorphisms are defined in the usual manner.

Much work has been done on the subject of representing groups as groups of automorphisms of graphs (i.e., given a group G, to find a graph X such that the group of automorphisms of X is isomorphic to G). Recently, this was related to category theory, the main question being as to whether every monoid (i.e., semigroup with 1) can be represented as the monoid of endomorphisms of some graph in a given category of graphs.

Let F be a linear, but not necessarily closed, subspace of L2[X, dm], where (X,,m) is a σ-finite measure space with the Borel subsets of the locally compact space X. If u and v are measureable functions, then v is called a normalized contraction of u if and Assume that F is stable under normalized contractions, that is, if u ∈ F and v is a normalized contraction of u, then v ∈ F.

The lattices of all closed subspaces of separable, infinitedimensional Hilbert space (real, complex, and quaternionic) share the following purely lattice-theoretic properties. Each is complete, orthocomplemented, atomistic, irreducible, separable, M-symmetric, and orthomodular [2]. We will call any lattice possessing these seven properties a Hilbert lattice. The general situation which motivates the investigations of this paper concerns infinite-dimensional Hilbert lattices (the dimension of a Hilbert lattice being the cardinality of any maximal family of orthogonal atoms). There are several lattice theoretic properties, possessed by the three canonical lattices, whose only known proofs involve the analytic properties of the underlying Hilbert space, that is, there is no known purely lattice-theoretic proof of these properties.

By a Noetherian (Artinian) ring = (R; + , —, 0, ·) we mean an associative ring satisfying the ascending (descending) chain condition on left ideals. An arbitrary ring is said to be equationally compact if every system of ring polynomial equations with constants in is simultaneously solvable in provided every finite subset is. (The reader is referred to [2; 8; 13; 14] for terminology and relevant results on equational compactness, and to [4] for unreferenced ring-theoretical results.) In this report a characterization of equationally compact Artinian rings is given - roughly speaking, these are the finite direct sums of finite rings and Prüfer groups; as consequences it is shown that an equationally compact ring satisfying both chain conditions is always finite, as is any Artinian ring which is a compact topological ring.

Let X and Y be sets, let λ be a bounded positive measure on X, and let μ be a bounded positive measure on Y. Furthermore let M be a subalgebra of L∞(λ), let p ∈ (0, ∞), and let A be a linear transformation of M into Lp(μ) such that

for all f in M.

In § 2 of this paper we will prove the following theorem.

Herein, a ring is not required to have an identity. All rings are associative but not necessarily commutative. However, we specialize to the commutative case for some of our results. The paper is concerned primarily with rings having the property that all unbounded ideals or all unbounded homomorphic images are isomorphic to the ring. We say that a ring R is bounded if nR = 0 for some positive integer n; alternately, R, with or without 1, is said to have finite characteristic. Unbounded rings having the property that all proper subrings are bounded were characterized in [8].

Packings by unequal spheres in three dimensional space have interested many authors. This is to some extent due to the practical applications of such investigations to engineering and physical problems (see, for example, [16; 17; 31]). There are a few general results known concerning complete packings by spheres in N-dimensional Euclidean space, due mainly to Larman [20; 21]. For osculatory packings, although there is a great deal of specific knowledge about the two-dimensional situation, the results for higher dimensions, such as [4], rely on general methods which do not give particularly precise information.

Let be the class of functions f : R2 → R satisfying (1) f is continuous, (2) for all t, x, y ∈ R, (4) f is locally Lipschitzian. The classical Sturm theorems, the Leighton-Wintner oscillation theorem, and perturbation theorems have been established by Bihari in [1; 2; 3] for the equation , .

This paper investigates the question of strong oscillation of

(1.1)

for p ∈ C[0, ∞ ), f ∈ . The equation is viewed as “almost linear” because of the homogeneity and sign conditions on f.

In [8] the notion of a reparameterizing isomorphism of transformation groups, henceforth called an R-isomorphism, is introduced generalizing Ura's type-2 isomorphism (see [13]). We have shown [8] that an R-isomorphism is weaker than a transformation group isomorphism. For example, although R-isomorphisms preserve pointwise almost periodicity and minimality they do not preserve the existence of slices [7] or almost periodicity. This suggests that R-isomorphisms might be a useful classification tool in topological dynamics.

R. M. Thrall [10] introduced QF — 1, QF — 2 and QF — 3 rings as generalizations of quasi-Frobenius rings. (For definitions, see section 1. It should be noted that all rings considered are assumed to be left and right artinian.) He proved that QF — 2 rings are QF — 3 and asked whether all QF — 1 rings are QF — 2, or, at least, QF — 3. In [9] we have shown that QF — 1 rings are very similar to QF — 3 rings. On the other hand, K. Morita [6] gave two examples of QF — 1 rings, one of them not QF — 2 and therefore not QF — 3, the other one QF — 3, but not QF — 2.

Let G be a group, ZG its integral group ring and Δ = Δ(G) the augmentation ideal of ZG. Denote by Gi the ith term of the lower central series of G. Following Passi [3], we set . It is well-known that (see, for example [1]). In [3] Passi shows that if G is an abelian group then , the second symmetric power of G.

In this paper we consider the question of whether there is a natural class of semilattices associated with a fixed equational class of lattices. We give four classes of semilattices which may be obtained from a given equational class of lattices and show that for the distributive case they coalesce into one class. An incomplete set of relations is given for the general case.

If G is a torsion–free group and F is a field, is the group ring F[G] a ring without zero divisors? This is true if G is an ordered group or various generalizations thereof - beyond this the question remains untouched. This paper proves a related result.

A category A is binding (or universal) if any full category of algebras is isomorphic to a full subcategory of A. There are many binding categories: the category of all commutative rings with unit and all unit-preserving homomorphisms [1], the category of bounded lattices [2], the category of semigroups [3], the category A(1, 1) of all algebras with two unary fundamental operations and the category of directed graphs [4], the category of all commutative groupoids [11] and many others.

Let {λn} be an increasing sequence of positive numbers. The question of the closure in Lp(0, ∞ ) (1 ≦ p ≦ ∞ ) of the linear span of the sequence has been considered by several authors, notably by Boas (1) and Fuchs [3;4].

In the development of any mathematical theory it is often advisable to test the known theoretical results in particular cases, for in this way we are able to judge the state of the subject, its strengths and its weaknesses. It is the purpose of this paper to put forward three remarkable graphs which can be used to indicate the progress being made in research in graph theory. We shall undertake three tasks - the construction of the graphs themselves, the investigation of their properties (insofar as the theory allows), and the indication of possible lines for further enquiry.

Let denote the class of functions f(z) = z + that are analytic and univalent in and will denote the collection of f ∈ that map U onto a domain that is respectively starlike with respect to the origin and convex.

In [4, p. 85] Hayman used Steiner symmetrization to solve a problem, a special case of which is the following.