Recently the ℒ (q,λ) spaces have been investigated by many authors and the theory of these spaces has proved to be particularly important for research in partial differential equations (see for example [15], [16] and [18]).

The equations of elliptic type in these spaces were first studied by C. B. Morrey [8], [9], who applied his well-known imbedding theorems, and afterwards by S. Campanato [3], [4] with the aid of isomorphism theorems and the so-called fundamental inequalities due to him.

On the other hand, G. Stampacchia introduced the ℒ (q,λ) spaces of strong type [17], the structures of which are more general and complicated than those of ℒ (q,λ) Spaces in the usual sense, and greater part of them were characterized by him, L. C. Piccinini, Y. Furusho, the author and others (see [5], [11]-[14], [16] and [17]).

Given vectors and (or functions f(x) and g(x)) we define the Hölder Quotient Hpq by

1

or in case of functions by

2

Here ‖·‖p and ‖·‖q are the usual Lp and Lq norms. We assume throughout that

If p and q are both greater than one then they are positive but if we allow p and q to be less than one then one of them must be positive and the other one must be negative. This may cause a problem if for example, some value ai is zero and p is negative. In this case we use the convention that and

The hypercenter theorem [6] asserts that in a ring with no non-zero nil ideals an element commuting with a suitable power of each element of the ring must be central. In this paper we shall be concerned with a similar problem in the setting of rings with involution. Let R be a ring with involution *, let Z denote the center of R and let S = {x ∈ R|x = x *} be the set of symmetric elements in R. We define the symmetric hypercenter of R to be

What can one hope to say about H? That H need not equal Z is clear. For instance, in the ring R = F 2 of 2 X 2 matrices over a field, if * is the symplectic involution, all symmetric elements are central, hence H = R but Z ≠ R. Furthermore if R is a noncommutative ring in which every symmetric element is nilpotent then even in this case H = R and Z ≠ R follows.

In [18], the author initiated an investigation of compact, Banach-module-valued derivations of C *-algebras. In collaboration with C. A. Akemann [3] and S.-K. Tsui [16], he determined the structure of all compact and weakly compact, A-valued derivations of a C *-algebra A, and of all compact, B(H)-valued derivations of a C *-subalgebra of B(H), the algebra of bounded linear operators on a Hilbert space H. In this paper, we begin the study of weakly compact, B(H)-valued derivations of C *-subalgebras of B(H).

Let R be a C *-subalgebra of B(H), δ:R → B(H) a weakly compact derivation, i.e., a weakly compact linear map which has

Since δ has a unique weakly compact extension to a derivation of the closure of R in the weak operator topology (WOT) on B(H) (consult the proof of Theorem 3.1 of [16]), we may assume with no loss of generality that R is a von Neumann subalgebra of B(H). In this paper, we determine in Lemma 4.1 and Theorems 4.3 and 4.10 the structure of δ when R is type I, using I. E. Segal's multiplicity theory [14] for type I von Neumann algebras and results of E. Christensen [6], [7] on B(H)-valued derivations of von Neumann algebras.

Let X be a Riemann surface, and H ∞(X) the ring of bounded holomorphic functions in X. We offer here a question on divisibility in H∞ (X), and then give in Section 2 a condition in which the answer is yes (Corollary 2 to Lemma 1). In Section 3 we use part 2 to prove a theorem on the separation of points by H ∞(X). In Section 4 we study X/H∞ (X).

If f is meromorphic in X and z ∈ X, then by o(f, z) we mean the order of f at z. (We agree that o(f, z) = ∞ if f ≡ 0.) Let h be memomorphic in X; then h might be said to be of bounded type if h = f/g where f,g ∈ H∞(X), g ≠ 0.

E. Salow [8] introduced the concept of pre-Hjelmslev groups, a generalization of F. Bachmann's Hjelmslev groups [1] which leads to a more natural theory of homomorphisms and permits a simpler construction of algebraic models. Basically, both types of groups are the groups of motions of a metric plane, the so-called group plane. In such a plane there is a unique perpendicular through any point to any line and the product of three collinear points (three copunctal lines) is a point (a line). Our first section contains the precise definitions and some basic facts.

The homomorphic image of a pre-Hjelmslev group can be more complicated than the pre-image. For instance, there may always be a unique line through two distinct points of the pre-image but not of the image. We study regular homomorphisms of pre-Hjelmslev groups, i.e., homomorphisms with the following property: If two lines intersect at exactly one point, their images will also have precisely one point in common.

In the literature about the definition of data types there exist many approaches using some concept of fixed point. Wand [13] and Lehmann, Smyth [9] e.g. constructed data types as least fixed points of functors F:K → K. Arbib and Manes [3] showed that some data types turn out to be the greatest fixed points of such endofunctors. In this paper we regard least and greatest fixed points that have a given property.

Let X be a compactum in En of dimension at most n – 2. In [9, Theorem 4.1] it was shown that there is an arbitrarily small homeomorphism h of En fixed outside any given neighborhood of X, so that h(X) has vertical order n – 1 provided n ≠ 3. If X is a 0-dimensional set or a tame 1-dimensional set in E3 then the result is still true. However, the examples of tangled continua of Bothe [2] and McMillan and Row [7] are not amenable to the techniques used in dimensions other than three. This prompted Wright [9] to make the following conjecture.

The “sets” in this paper are actually multi-sets. That is, we allow an element to occur several times in a set and distinguish between the number of elements in a set and the number of distinct elements in the set. On the few occasions when we need to avoid repetition we will use the term “ordinary set.“

Definition. Let G be a group and let S a set of elements of G. An r-sum in S is an ordered subset of S of cardinality r; the result of that r-sum is the product of its elements in the designated order.

Definition. If S is a set, r(x, S) denotes the number of times x appears in S and [x, S] is a set consisting of r(x, S) copies of x. An n-set or n-subset is a set consisting of n elements. Hence [x, S] is an r(x, S)-subset of S.

Let Φ be a regular closed C 2 curve on a sphere S in Euclidean three-space. Let H(S)[H(Φ) ] denote the convex hull of S[Φ]. For any point p ∈ H(S), let O(p) be the set of points of Φ whose osculating plane at each of these points passes through p.

1. THEOREM ([8]). If Φ has no multiple points and p ∈ H(Φ), then |0(p) | ≧ 3[4] when p is [is not] a vertex of Φ.

2. THEOREM ( [9]). a) If the only self intersection point of Φ is a doublepoint and p ∈ H(Φ) is not a vertex of Φ, then |O(p)| ≧ 2.

b) Let Φ possess exactly n vertices. Then

• (1) |O(p)| ≦ n for p ∈ H(S) and

• (2) if the osculating plane at each vertex of Φ meets Φ at exactly one point, |O(p)| = n if and only if p ∈ H(Φ) is not vertex.

Let be a complete Heyting algebra (CHA). An -valued set is a pair (X, δ) where X is a set and δ is a function from X × X to such that

for all x, y z in X. -valued sets form a category as follows: a morphism from (X, δ) to (Y, δ) is a function f from X × Y to such that

• (i) f(x) ∧ δ(x, x′) ≦ f(x′, y), f(x, y) ∧ δ(y, y′) ≧ f(x, y′),

• (ii) f(x, y) ∧ f(x, y′) ≦ δ(y, y′), and

• (iii)

A system of openly disjoint discs in the planeis said to form a 6-neighboured circle packing if every is tangent to at least 6 other elements of (It is evident that such a system consists of infinitely many discs.) The simplest example is the regular circle packing all of whose circles are of the same size and have exactly 6 neighbours. L. Fejes Tóth conjectured that the regular circle packing has the interesting extremal property that, if we slightly “perturb” it, then there will necessarily occur either arbitrarily small or arbitrarily large circles. More precisely, he asked whether or not the following “zero or one law” (cf. [3], [6]) is valid: If is a 6-neighboured circle packing, then

where r(C) denotes the radius of circle C, inf and sup are taken over all C ∊