For fixed k ≧ 2, let Vk denote the class of normalized analytic functions

such that z ∈ E = {z; |z| <1} are regular and have f′(0) = l,f′(z) ≠ 0, and

1

Let Sk be the subclass of Vk whose members f(z) are univalent in E. It was pointed out by Paatero (4) that Vk coincides with Sk whenever 2 ≦ k ≦ 4. Later Rényi (5) showed that in this case, f(z) ∈ Sk is also convex in one direction in E. In (6) I showed that the Bieberbach conjecture

holds for functions convex in one direction.

If A is a set with only a finite number of elements, we write |A| for the number of elements in A. Let p be a large prime and let m be a positive integer fixed independently of p. We write [pm] for the finite field with pm elements and [pm]′ for [pm] – {0}. We consider in this paper only subsets H of [pm] for which |H| = h satisfies

1.1

If f(x) ∈ [pm, x] we let N(f; H) denote the number of distinct values of y in H for which at least one of the roots of f(x) = y is in [pm]. We write d(d ≥ 1) for the degree of f and suppose throughout that d is fixed and that p ≧ p0(d), for some prime p0, depending only on d, which is greater than d.

In 1946 Montgomery and Samelson (11) introduced a generalization of the notion of a differentiable group action with one type of orbit besides fixed points. Such an object is essentially a locally trivial fibering except on a certain singular set over which fibres are pinched to points. In recent years there has been a fair amount of research on these MS-fiberings and similar singular fiberings. This paper is another effort in this direction. For a fairly complete bibliography of the literature, the reader should consult the references, and in particular, (5).

Let f: Mn → Sp, with Mn a closed connected n-manifold and Sp the unit p-sphere with standard differentiable structure, be the projection map of a smooth MS-fibering with finite non-empty singular set.

Throughout this paper, S will be a ring (not necessarily commutative) with an identity element ls ≠ 0s. We shall use R to denote a second ring, and ϕ: S→ R will be a fixed ring homomorphism for which ϕ1S = 1R.

In (7), Higman generalized the Casimir operator of classical theory and used his generalization to characterize relatively projective and injective modules. As a special case, he obtained a theorem which contains results of Eckmann (3) and of Higman himself (5), and which also includes Gaschütz's generalization (4) of Maschke's theorem. (For a discussion of some of the developments of Maschke's idea of averaging over a finite group, we refer the reader to (2, Chapter IX).) In the present paper, we define the Casimir operator of a family of S-homomorphisms of one R-module into another, and we again use this operator to characterize relatively projective and injective modules.

By a graph we mean a pair (X, R) where X is a non-void set and R ⊂ X × X. A mapping f: X → Y is called a compatible map (or morphism) from (X, R) into (Y, S) if 2f(R) ⊂ S, where 2f: X2 → Y2 is defined by 2f((x1, x2)) = (f(x1),f(x2)). The set of all compatible maps from (X, R) into itself forms a monoid (semigroup with a unit element) under composition, which is denoted by M(X, R). A graph (X1, R1) is said to be a full subgraph of (X, R) if X1 ⊂ X and R1 = R ∩ (X1 × X1). A graph (X, R) is said to be without loops if (x, x) ∉ R for all x ∈ X.

Two elements A, B of a lattice are said to form a modular pair when (X ∨ A) Λ B = X ∨ (A Λ B) holds for all X ≦ B, and are said to form a dual-modular pair when (X Λ A) ∨ B = X Λ (A ∨ B) holds for all X ≧ B.

We are concerned here with a particular lattice, the lattice of closed subspaces of a normed linear space, and with a question posed by Mackey in 1945 (6, p. 206, problem 2), namely:

“Are there any incomplete normed linear spaces in whose lattices of closed subspaces modularity and d-modularity are equivalent?”.

The principal result of this paper is the following.

Let L/K be a field extension of characteristic p ≠ 0. If L/K is purely inseparable and of bounded exponent, then the property that L has a subbasis over K (11, p. 436) is of significance in the theory of higher derivations (11) and in the theory of Hopf algebras (9; 10). In this case, where L/K is of bounded exponent, it has been shown independently in (1; 9; 5) that L/K having a sub-basis is equivalent to the property that L/K is modular (9, p. 401). Our aim in this paper is to extend and apply these properties for L/K purely inseparable and of unbounded exponent.

In Theorem 1, we give several conditions on a p-basis of L which are equivalent to the property that L/K has a sub-basis. In Theorem 2, we give a sequence of implications starting with L/K has a sub-basis.

I t was shown in (2) that if

(1) L is a locally compact connected topological lattice and if

(2) L is topologically contained in R2, the Euclidean plane, then each compact subset of L has an upper bound and a lower bound in L. I t was also asked whether this result could be proved without assuming condition (2). In this note, we show that this result continues to hold if condition (2) is weakened to: L is finite-dimensional.

In (11), it was shown that the centre of a compact topological lattice is totally disconnected. We shall prove t h a t this result is also true even in a locally compact, locally convex topological lattice with 0 and 1. This yields that any locally compact topological Boolean algebra is totally disconnected.