Let K be a field of characteristic p > 0 and let A be a separably generated algebraic function field of one variable with K as its exact constant field. Throughout this paper we shall use the following notations to classify differentials of A/K:

D(A) : the K-module of all differentials,

G(A) : the K-module of all differentials of the first kind,

R(A) : the K-module of all residue free differentials in the sense of Chevalley [2, p. 48],

E*(A) : the K-module of all pseudo-exact differentials in the sense of Lamprecht [7, p. 363], (compare the definition with our Lemma 8).

It is well-known that the space L1(Rn) of integrable functions on Euclidean space fails to be preserved by singular integral operators. As a result the rather large Lp theory of partial differential equations also fails for p = 1. Since L1 is such a natural space, many substitute spaces have been considered. One of the most interesting of these is the space we will denote by H1(Rn) of integrable functions whose Riesz transforms are integrable.

Cooper [1] called a complex-valued function f on the real line Rpositive definite for F, where F is a set of complex-valued functions on R, if the integral

exists as a Lebesgue integral and is nonnegative for every ϕ in F. Let us denote by P(F) the set of functions positive definite for F, by the set of functions in LP(R) with compact support, and by the set of functions which are locally in LP(R), i.e., for every compact subset K of R. Cooper showed that for any p ≧ 2 and that each function in is essentially bounded and hence equal almost everywhere to an ordinary continuous positive definite function in the sense of Bochner.

In recent years a number of inequalities have appeared which involve rearrangements of vectors in Rn and of measurable functions on a finite measure space. These inequalities are not only interesting in themselves, but also are important in investigations involving rearrangement invariant Banach function spaces and interpolation theorems for these spaces [2; 8; 9].

An O*-group is a group wherein every partial order can be extended to some full order.

THEOREM. Suppose the group G has a normal chain G = G1 ⊇ G2 ⊇ … such that

and each G/Gi is locally nilpotent and torsion-free. Then G can be embedded in thecomplete direct product G’ of divisible O*-groups.

If one wishes to construct a homomorphism of a group, an obvious approach is to begin with a homomorphism of a subgroup and attempt to extend it in some way. Any such extended homomorphism is determined by its action on elements of the group which, together with the subgroup, generate the whole group. The values assigned to such generators must then satisfy various functional equations. By a governing list for the extension problem, we shall mean a list of functional equations whose solutions describe all possible homomorphisms extending a given subgroup homomorphism.

Let 1 be a subdivision of [0, 1], and let denote the class of functions whose restriction to each sub-interval is a polynomial of degree at most k. Gaier [1] has shown that for uniform subdivisions △n (that is, subdivisions for which

if and only if f is a polynomial of degree at most k. Here, and subsequently, denotes the usual norm in Lp[0, 1], 1 ≦ p ≦ ∞, and we should emphasize that functions differing only on a set of Lebesgue measure zero are identified.

H. Scheid [4] has found necessary and sufficient conditions on a partially ordered set S(≦) which is a direct sum of a countable number of trees for a certain subalgebra G(+, *) of the incidence algebra F(+, *) to be an integral domain. In this paper we prove that under similar conditions on S, G(+, *) is actually a unique factorization domain or, failing this, that there is a subalgebra H(+, *) of F(+, *) which is a unique factorization domain and contains G. Similar results are then obtained as corollaries in the regular convolution rings of Narkiewicz.

Throughout this paper will denote a C*-algebra and will denote its second dual, which is identified with the enveloping von Neumann algebra of . A projection E in is said to be open if it supports a left ideal in , that is, if E = - for some left ideal in . Here the bar - means the stong closure. When has a unit, this definition coincides with the definition of Akemann [1, Definition II. 1]. In the sequel, we shall solely be concerned with two-sided ideals, and consequently central projections [4, I, § 3, Corollary 3 of Theorem 2]. Our aim is to show that is CCR if and only if the open central projections are strongly dense in the set of central projections on .

In this paper we continue our investigation of the class of right LCM domains which was introduced in [2]. A right LCM domain is an (not necessarily commutative) integral domain with unity in which the intersection of any two principal right ideals is again principal. In this note we study the right quotient rings of such a ring. In Section 1 we describe some of the characteristic properties of right quotient monoids with respect to which quotient rings are formed. Three particular types of quotient rings are described in Section 2. In Section 3 we relate the right ideals of a ring to those of its quotient ring.

The visibility function assigns to each point x of a fixed measurable set E in a Euclidean space En the Lebesgue outer measure of S(x), the set {y : rx + (1 — r)y ∊ E for every r in [0, 1]}.

The purpose of this paper is to determine sufficient conditions for the continuity of the function on the interor of a starshaped set.