Let F: X → Y be a multifunction from X to Y. Then, given measure-theoretic or topological structures on X and Y, it is possible in various ways to define the measurability of F. The selection problem is to determine which structures on X and Y and which definitions of measurability of F ensure that F will have a measurable selector. This problem has been studied recently in papers by Castaing (2) and Kuratowski and Ryll-Nardzewski (6). In the latter paper, the problem is studied for its own interest. The former uses solutions of the problem to obtain general Filippov-type theorems. (See, for example, the corollaries following Theorems 2 and 3 of Castaing's paper.) For other material on Filippov's results see, among others, (3; 4; 5; 7; 9).

In (2), Fitzpatrick and Ford defined a class of spaces which they called order totally paracompact. This class of spaces is significant since the order totally paracompact metric spaces constitute the largest known class of spaces for which the small and large inductive dimensions coincide. In this paper we shall consider a class of spaces containing the class of order totally paracompact spaces; we shall call these spaces order paracompact. We shall establish the relationship of this class to the more familiar classes of spaces and obtain an invariance theorem for order paracompact spaces analogous to theorems obtained for various other classes.

In what follows, no separation axioms are assumed unless specifically mentioned. Thus, for example, a Lindelöf space is not taken to be Hausdorff without making the further assumption. Therefore, a Lindelöf space is not necessarily paracompact.

In this paper, we wish to characterize those abelian groups whose integral homology groups vanish in some positive dimension. We obtain a complete characterization provided the dimension in which the homology vanishes is odd; in fact, we prove that the only abelian groups which possess a vanishing homology group in an odd dimension are, up to isomorphism, subgroups of Qn, where Q denotes the additive group of rational numbers. The case of vanishing in an even dimension is much more complicated. We exhibit a class of groups whose homology vanishes in even dimensions and is otherwise very nice, namely the subgroups of Q/Z, and then show that unless we impose further restrictions, there exist abelian groups which possess the homology of subgroups of Q/Z without being isomorphic to a subgroup of Q/Z.

In this note we study the question of automorphisms of the integral group ring Z(G) of a finite group G. We prove that if G is nilpotent of class two, any automorphism of Z(G) is composed of an automorphism of G and an inner automorphism by a suitable unit of Q(G), the group algebra of G with rational coefficients. In § 3, we prove that if two finitely generated abelian groups have isomorphic integral group rings, then the groups are isomorphic. This is an extension of the classical result of Higman (2) for the case of finite abelian groups. In the last section we give a new proof of the fact that an isomorphism of integral group rings of finite groups preserves the lattice of normal subgroups. Other proofs are given in (1;4).

1. Let U denote an n-dimensional vector space over a field F and let Gnr denote the set of non-zero decomposable r-vectors of the Grassmann product space ΛrU. Let T be a linear transformation of ΛrU into itself which maps Gnr into itself. If F is algebraically closed, or if T is non-singular, then the structure of T is known. In this paper we show that if T is singular, then the image of ΛrU has a very special form with dimension equal to the larger of the integers r + 1 and n – r + 1. We give an example to show that this can occur.

The theory of generalized Frattini subgroups of a finite group is continued in this paper. Several equivalent conditions are given for a proper normal subgroup H of a finite group G to be a generalized Frattini subgroup of G. One such condition on H is that K is nilpotent for each normal subgroup K of G such that K/H is nilpotent. From this result, it follows that the weakly hyper-central normal subgroups of a finite non-nilpotent group G are generalized Frattini subgroups of G.

Let H be a generalized Frattini subgroup of G and let K be a subnormal subgroup of G which properly contains H. Then H is a generalized Frattini subgroup of K.

Let ϕ(G) be the Frattini subgroup of G. Suppose that G/ϕ(G) is nonnilpotent, but every proper subgroup of G/ϕ(G) is nilpotent. Then ϕ(G) is the unique maximal generalized Frattini subgroup of G.

The main theorem of this paper names seven ring classifications which coincide with the class of rings named in the title. Three of them are (1) rings over which every module is rationally complete, (2) left and right perfect rings over which every module is corationally complete, and (3) right perfect rings over which every right module is rationally complete. Corational completeness (introduced in this paper) and rational completeness are generalizations of projectivity and injectivity, respectively. One recalls that a proper subclass of the rings investigated in this paper, the artinian rings with zero radical, is known to have one-sided characterizations in terms of the injectivity (projectivity) of modules. An example proves, however, that the class of rings over which every right module is rationally complete properly contains the rings of the title.

In 1908, Lindelöf showed that if w = f(x) is a bounded holomorphic function in a sector S: |arg z| < θ1, and if f(z) has an asymptotic value w0 as z tends to ∞ along a half-ray in S; then f(z) tends uniformly to w0 as z tends to ∞o within any sector |arg z| ≦ θ, 0 ≦ θ < θ1. Montel (8) later replaced the condition that f(z) be bounded by the condition that f(z) be meromorphic and omitted three values. The following is an immediate consequence of the Lindelöf-Montel theorem.

THEOREM 1. Let w = f(x) be a function meromorphic in the sector | arg z| < θ1, and let f(z) tend to a value w0 as z → ∞ along the positive real axis.

The Lefschetz theorem on hyperplane sections, as proved by Andreotti and Frankel (1), depends upon the following result.

THEOREM. If M is a non-singular affine algebraic variety of real dimension 2k of complex n-space, then

This theorem, which is interesting in itself, has been strengthened by Milnor (7), who showed that M has the homotopy type of a k-dimensional CW-complex.

In this paper we generalize the above theorem in two directions. First, we replace complex n-space by some other complete simply connected Riemannian manifold which either has non-positive curvature or is a compact symmetric space. Secondly, we allow M and to be quasi-Kâhlerian (see below) instead of Kählerian.

We first introduce some notation. Let M and be C∞ Riemannian manifolds with M isometrically immersed in . Denote by 〈, 〉 the metric tensor of either M or .

In this paper, we seek to determine the greatest lower bound of the essential spectrum of self-adjoint singular differential operators of the form

1

where 0 ≦ x < ∞. In the event that this bound is + ∞, our results will yield criteria for the discreteness of the spectrum of (1).

Such bounds have been established by Friedrichs (3) for Sturm-Liouville operators of the form

and our techniques will be closely related to those of (3). However, instead of studying the solutions of

2

directly, we shall exploit the intimate connection between the infimum of the essential spectrum of (1) and the oscillation properties of (2).

Given a homomorphically closed class of (not necessarily associative) rings , the lower radical property determined by is the least radical property for which all rings in are radical. Recently (7) a process of constructing the lower radical property from a class of associative rings has been given which terminates after a countable number of steps. In this process, an ascending chain of classes

is obtained and the property of being a ring in the class is the lower radical property determined by . In Theorem 1 we give another characterization of the rings in the class , λ ∈ {1, 2, …, omega;0}, and a procedure for constructing the lower radical determined by in an arbitrary associative ring is given.

Consider the following result.

PROPOSITION. Let D be a finite-dimensional central division algebra over a field F, and let Dn be the algebra (over F) of all n × n matrices with entries in D. Let A and B be in Dn, and suppose that BX = XB for every X in Dn such that XA = AX. Then B is a polynomial in A with coefficients in F.

The case D = F is a well-known classical result. Recently, the particular case where D is the algebra of real quaternions was established by Cullen and Carlson (2). In this note, the general proposition is proved by reduction to the classical case by way of tensor products.

In this paper we shall work in the category of countable CW-complexes with base point and base point preserving maps. All homotopies shall also respect base points. For simplicity, we shall frequently use the same symbol for a map and its homotopy class. Given spaces X, Y, we denote the set of homotopy classes of maps from X to Y by [X, Y]. We have an isomorphism τ: [∑X, Y] → [X, Ω Y] taking each map to its adjoint, where ∑ is the suspension functor and Ω is the loop functor. We shall denote τ(1 ∑x) by e′ and τ-1(1Ωx) by e.

Let V denote a unitary vector space with inner product (x, y). A self-adjoint linear map T: V → V is positive (positive definite) if (Tx, x) ≧ 0 ((Tx, x) ≧ 0) for all x ≠ 0 in V. We write S ≧ T(S > T) if S and T are self-adjoint and S – T ≧ 0 (S – T > 0). If U is a unitary vector space, a function f: Hom(V, V) → Hom(U, U) is monotone idf S ≧ T implies that f(S) ≧ f(T). If both U and V are taken to be the n-dimensional unitary space Cn of n-tuples of complex numbers with standard inner product, then f is a monotone matrix junction, a notion introduced for a more restrictive class of functions by Löwner (3) which has important applications in pure and applied mathematics. For orientation we refer the reader to (1), where several interesting examples of monotone and related functions are displayed in detail.

Let Δn be the set of all n × n, non-singular matrices of the form PD, where P is a permutation matrix and D is a diagonal matrix with complex entries. In (1, conjecture 12), Marcus and Mine asked: Is Δn a maximal group on which the permanent function is multiplicative? (that is, per AB = per A per B). The field over which the entries range was not mentioned in the conjecture; however, we assume that the complex number field was intended. Corollary 1 answers this in the affirmative. In fact, Δn is the only maximal group (or semigroup) on which the permanent is multiplicative. Let ρi be the set of all non-zero entries in the ith row and let λj be the set of all non-zero entries in the jth column.

Unless further restricted, P, Q, and R denote arbitrary partially ordered sets whose order relations are all written “≦” .

An isotone mapping ϕ: P → Q is said to be residuated if there is an isotone mapping ψ: Q → P such that

(RM 1) xϕψ ≧ x for all x i n P;

(RM 2) yψϕ ≦ for all y in Q.

Let Q* denote the partially ordered set with order relation dual to that of Q.

(A) The following conditions are equivalent:

(i) ϕ: P → Q* is a Galois connection;

(ii) ϕ: P → Q is a residuated mapping;

(iii) Max{z ∈ P: zy ≦ y} exists for all y in Q and is equal to yψ.

Since ψ is uniquely determined by ϕ, it will be denoted by ϕ+.

Hamilton (1) proved that a hereditarily unicoherent, hereditarily decomposable metric continuum has the fixed-point property for homeomorphisms. In this paper we shall generalize this result by showing that if X is a hereditarily unicoherent, hereditarily decomposable Hausdorff continuum and 5 is an abelian semigroup of continuous monotone functions from X into X, then S leaves a point of X fixed.

Let X be a Hausdorff continuum. X is unicoherent if, whenever X = A ∪ B, where A and B are subcontinua of X, A ∩ B is a continuum. If each subcontinuum of X is unicoherent, X is hereditarily unicoherent. X is decomposable if X is the union of two of its proper subcontinua. If each subcontinuum of X which contains more than one point is decomposable, X is hereditarily decomposable.

I. In the first part of this paper we introduce a new class of operators, mentioned in the title. It is easy to say that these are a generalization of self-adjoint operators for Hilbert space. This is deceptive since it implies that the definition of self-adjointness is forced into the unnatural setting of a Banach space. We feel that the definition of adjoint abelian preserves the obvious distinction between a space and its dual. Certain attractive properties of self-adjoint operators have already been singled out and carried over to Banach space. Specifically, we mention the notion of hermitian (see 17; 11), and spectral type operators (see 4). There is some comparison of these concepts later.

In a recent paper (3), Gránbaum has found a general and unifying setting for a number of properties of some special lines associated with a planar convex body. Besides various interesting results, two conjectures are stated and two kinds of convexity and polygonal connectedness are introduced.

In the present paper, we shall prove one of Gránbaum's conjectures (§ 3, Theorem 1); we consider the other in § 4 and establish some related results in §§ 5 and 6. Six-partite problems are studied in this general setting (§ 7) and a question raised by Ceder (2) is answered. We give a generalization of the notion of a continuous family of curves in § 8, and discuss some new kinds of connectedness in § 9.

A Blaschke product on the unit disc,

where |c|= 1 and kis a non-negative integer, is said to be interpolatingif the condition

C

is satisfied for a constant δ independent of m.A Blaschke product always belongs to the set I of inner functions; it has norm 1 and radial limits of modulus 1 almost everywhere. The most general inner function can be uniquely factored into a product BS,where Bis a Blaschke product and

for some positive singular measure μ(θ) on the unit circle. The discussion will be carried out in terms of the hyperbolic geometry on the open unit disc D,its metric

and its neighbourhoods N(x, ∈) = ﹛z′ ∈ D: Ψ(z, z′) < ∈ ﹜