The dual of Hp of the unit disk for 0 < p < 1 has been characterized by Duren, Romberg and Shields (see [3]). The present paper is concerned with the analogous result for Hp(R+n+1) in the sense of Stein and Weiss (see [11]). In this connection it may be recalled that the dual of H1 has been characterized by Fefferman (see [4]).

If for each permutation the number of partial orderings fixed by that permutation is known, it is possible to count the number of non-isomorphic partial orderings on a finite set using a lemma of Burnside. In this paper it is shown that knowledge of the numbers of partial orderings fixed by permutations will enable the number of non-isomorphic pre-orderings to be counted also.

A space X is locally planar if each point of X has a neighborhood which is embeddable in the plane. If X is a closed, locally planar subset of E3, we will say that X is locally tame if each point of X has a neighborhood in X which lies on a tame disk in E3; if every cell-like subset of X has such a neighborhood, we say that X is strongly locally tame.

L'enveloppe de translations d'un demi-groupe présente un intérêt particulier pour les deux raisons suivantes : d'une part, elle apparaît d'une façon naturelle dans la construction des extensions idéales des demi-groupes, et d'autre part, elle est souvent utilisée pour des caractérisations abstraites des demi-groupes. Afin de l'appliquer aux problèmes particuliers, il est important de fournir des constructions qui, pour un demi-groupe donné sous une forme ou une autre, soient aussi explicites que possible. Un exemple d'une telle construction est celle de l'enveloppe de translations d'un demi-groupe de matrices de Rees (voir [4]) et ses nombreuses applications [5 ; 6 ; 7 ; 8].

An invertible measure preserving transformation T on the unit interval I generates a unitary operator U on the space L2(I) of Lebesque square integrable functions given by (Uf)(x) = f(Tx) for all f in L2(I) and x in I. By definition

for all f , g in L2(I), the bar denoting complex conjugation.

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.

We let , where the xi are positive integers and i ∈ N, the set of all integers. We define , where where . We let = [0; x1, x2, … ] where

We let

and define

The range of L(ξ) is known as the Lagrange spectrum and the range of M(ξ) as the Markov spectrum. It is known that both are closed and that the Markov spectrum includes the Lagrange spectrum.

For k ≧ 2 denote by Vk the class of functions f regular in and having the representation

(1.1)

where μ is a real-valued function of bounded variation on [0, 2π] with

(1.2)

Vk is the class of functions with boundary rotation at most kπ.

In [2], we have briefly described, as examples of the general theory developed there, a scale of symmetric Cesaro-Perron integrals, namely SCnP-integral for n = 1, 2, 3, … . The purpose of this paper is to consider the integrals in a greater detail.

As a preliminary, we prove some lemmas, which are also interesting for their own sake, concerning the de la Vallée Poussin derivatives in Section 1, and we also state two deep theorems concerning the n-convex functions in Section 2.

We begin with some definitions and a formulation of the problem treated in subsequent sections. Also included in this section is a brief indication of some of the known results dealing with the n-omino enumeration problem. Some of what follows together with more details may be found in [3] or [4].

A function denned in a domain D is n-valent in D if f(z) — w0 has at most n zeros in D for each complex number w0. Let denote the class of nonconstant, holomorphic functions f in the unit disc that are n-valent in each component of the set . MacLane's class is the class of nonconstant, holomorphic functions in the unit disc that have asymptotic values at a dense subset of |z| = 1.

This paper continues the study, begun in [7], of the spectral theory of non-self-ad joint second order ordinary differential operators on a half-line. The case of a ‘Very small” potential was studied in [4; 5; 6]. The case considered in [7], and in the present paper, is that where the potential is not so small.

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.

Let

(1.1)

where pk ∈ C(α, β) and - ∞ ≦ α < β ≦ ∞ . A solution of (1.1) is a nontrivial function y ∈ Cn(α, β), a neighborhood of β is an interval of the form (γ, β), α ≦ γ < β, and a neighborhood of α is an interval of the form (α, γ), α < γ ≦ β.

Let α be a real vector bundle over a finite CW complex X and let T(α;X) be its associated Thorn complex. We propose to study the S-type (stable homotopy type) of Thorn complexes in the framework of the Atiyah-Adams J-Theory. Therefore we focus our attention on the group JR(X) which is defined to be the group of orthogonal sphere bundles over X modulo stable fiber homotopy equivalence.

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.

Erdös and Hajnal [1] have introduced an edge partition relation for graphs

(1)

which means that whenever the edges of G are separated into two sets, E1 and E2, there exists a subgraph G’ of G such that G’ is isomorphic to Hi and the edges of G’ are all in Ei. for i = 1 or 2. A class of graphs has the G-R (Galvin-Ramsey) property [2] if for each H in there exists a G in which satisfies G→(H,H).

In the present paper a general form of integral limit laws for additive functions is obtained. Our limit law contains Kubilius’ results [5] on his class H. In the proof we make use of characteristic functions (Fourier transforms), which reduces our problem to finding asymptotic formulas for sums of multiplicative functions. This requires an extension of previous results in order to enable us to take into consideration the parameter of the characteristic function in question. We call this extension a parametric mean value theorem for multiplicative functions and its proof is analytic on the line of [4].

The problem of Hermite-Birkhoff interpolation is to determine: what n + 1 interpolatory conditions imposed on a polynomial P(x) of degree n and its derivatives determine the polynomial uniquely. It is customary now to indicate the imposed conditions by means of a matrix E = (eij) which is called the incidence matrix for the problem.