Let D ⊂⊂ Cn be a bounded domain with smooth boundary ∂D, and let F be a bounded holomorphic function on D. A generalization of the classical theorem of Fatou says that the set E of points on ∂D at which F fails to have nontangential limits satisfies the condition σ (E) = 0, where a denotes surface area measure. We show in the present paper that this result remains true when σ is replaced by 1-dimensional Lebesgue measure on certain smooth curves γ in ∂D. The condition that γ must satisfy is that its tangents avoid certain directions.

The Green's function G(x, ξ, λ) associated with the differential equation is of importance in the theory of the expansion of an arbitrary function in terms of the solutions of the differential equation. It is proved that this function is unique if q(x) ≧ — Ax2— B, where A and B are positive constants or zero. A similar theorem is proved for the Green's function G(x, y, ξ, η, λ) associated with the partial differential equation

In this paper, we study two generalizations of the Stirling numbers of the first and second kinds, inspired from their combinatorial interpretation in terms of 0-1 tableaux. They are the 𝔄-Stirling numbers and the partial Stirling numbers. In particular, we give a q and a p, q-analogue of convolution formulae for Stirling numbers of the second kind, due to Chen and Verde-Star, and we extend these formulae to Stirling numbers of the first kind. Included in this study are the a, d-progressive Stirlingnumbers, corresponding to 0-1 tableaux with column lengths from an arithmetic progression ﹛a + id﹜i≥0.

An identity of L. Carlitz for a bibasic hypergeometric series is used to find some summation formulas for series in which the bases are either q and q2 or q and q3,0 < q < 1 In general, these series are neither balanced nor very-well-poised in the usual sense

This paper continues the study appearing in (9) and (10) of the combinatorial properties of a matrix A of m rows and n columns, all of whose entries are 0's and l's. Let the sum of row i of A be denoted by ri and let the sum of column j of A be denoted by Sj. We call R = (r1, … , rm) the row sum vector and S = (s1 . . , sn) the column sum vector of A. The vectors R and S determine a class

1.1

consisting of all (0, 1)-matrices of m rows and n columns, with row sum vector R and column sum vector S. The majorization concept yields simple necessary and sufficient conditions on R and S in order that the class 21 be non-empty (4; 9). Generalizations of this result and a critical survey of a wide variety of related problems are available in (6).

In the present paper, we shall prove the following result.

Theorem A. The groups PSL(2, p) can be presented by the following system of generators and relations:

1.1

This theorem considerably improves earlier results of Bussey, Frasch, and Todd (cf. 2, pp. 93-96). The presentation of Frasch reads as follows:

1.2

Let p be a prime number. Let k be a field of characteristic different from p and containing the p-th roots of unity. Let be a finite group. Let L/k be a finite normal extension with Galois group and let c be a 2-cocycle on with coefficients in , where acts trivially on By Emb(L/k, c) we denote the question of the existence of a finite normal extension M of k, such that M contains L, such that [M: L] = p, and such that, denoting by the Galois group of M/k, the extension is given by the class of c.

Studies of optimization problems have led in recent years to definitions of several types of generalized directional derivatives. Those derivatives of primary interest in this paper were introduced and investigated by F. M. Clarke ([5], [6], [7], [8]), J. B. Hiriart-Urruty ([12]), Lebourg ([16], [17]), R. T. Rockafellar ([23], [24], [26], [27]), Penot ([21], [22]) among others.

In an attempt to explore in more detail relationships between various types of generalized directional derivatives we discovered some unexpected results which were not observed by the above mentioned authors. We are able to give simple conditions which characterize directionally Lipschitzian functions defined on a Baire metrizable locally convex topological vector space.

The purpose of this paper is to show how Fred hoi m criteria for Toeplitz operators, whose symbols lie in an algebra,A, may often be generalized to cover a larger symbol algebra generated by A and SO, the slowly oscillating functions. Mere A and SO are algebras of continuous functions on the real line, so that we are concerned principally with the effect of a single discontinuity in the symbol function.

We shall treat the cases when A is the almost periodic functions, the semi-almost periodic functions and the multiplicatively periodic functions. Sufficient criteria for Fredholmness are obtained in Section 5. The more difficult task of establishing necessary and sufficient criteria is only achieved here for the slowly oscillating almost periodic functions and this is done in Section 6.

On replacing the parameter n in Bessel's differential equation

1.1

by the operator y(∂/∂y), the partial differential equation Lu = 0 is constructed, where

1.2

This operator annuls u(x, y) = v(x)yn if, and only if, v(x) satisfies (1.1) and hence is a cylindrical function of order n. Thus every generating function of a set of cylindrical functions is a solution of Lu = 0.

It is shown in § 2 that the partial differential equation Lu = 0 is invariant under a three-parameter Lie group. This group is then applied to the systematic determination of generating functions for Bessel functions, following the methods employed in two previous papers (4; 5).

A generalization of the definition of combinatorial species is given by considering functors whose domains are categories of finite sets, with various classes of relations as moronisms. Two cases in particular correspond to species for which one has notions of restriction and quotient of structures. Coalgebras and/or Hopf algebras can be associated to such species, the duals of which provide an algebraic framework for studying invariants of structures.

The observations of this paper evolved from the concept of 'asymptotic nonexpansiveness' introduced by two of the writers in a previous paper [10]. Let X be a Banach space and K ⊆ X. A mapping T : K → K is called asymptotically nonexpansive if for each x, y ∊ K

where {ki} is a fixed sequence of real numbers such that ki→1 as i → ∞ . It is proved in [10] that if K is a bounded closed and convex subset of a uniformly convex space X then every asymptotically nonexpansive mapping T : K → K has a fixed point. This theorem generalizes the fixed point theorem of Browder-Göhde-Kirk [2 ; 12 ; 16] for nonexpansive mappings (mappings T for which ||T(x) — T(y)|| ≦ ||x — y||, x, y ∊ K) in a uniformly convex space. (A generalization along similar lines also has been obtained by Edelstein [4].)

large class of simple stably projectionless C*-algebras are shown to arise as crossed products of simple purely infinite C*-algebras by trace scaling one-parameter automorphism groups. The Elliott invariant is computed for this class.

A construction of the free lattice generated by a partially ordered set P and preserving every least upper bound (lub) and greatest lower bound (glb) of pairs of elements existing in P has been given by Dilworth (2, pp. 124-129) and, when P is finite, by Gluhov (5).

The results presented here construct the free lattice FL generated by the partially ordered set P and preserving

(1) the ordering of P

(2) those lub's of a family of finite subsets of P which possess lub's in P

This paper is devoted to extending formulas for the geometric approximate subdifferential and the Clarke subdifferential of extended-real-valued functions on Banach spaces. The results are strong enough to include completely the finite dimensional setting.

The structure theory of hereditary noetherian prime (hnp) rings—in particular of Dedekind prime rings—has been recently developed by many authors including Eisenbud, Griffith, Michler and Robson; this theory extends some of the well-known results concerning commutative Dedekind domains. In this paper we study quasi-injective modules and quasi-projective modules over those (hnp) rings which are not right primitive and establish some results which extend the corresponding well-known results concerning commutative Dedekind domains. Let R be an (hnp) ring, which is not right primitive.

Generalized weighing (GW) matrices are orthogonal matrices whose nonzero entries are roots of unity. Several families are constructed with the aid of finite geometries which include as special cases interesting examples of conference matrices and weighing matrices.

We provide a framework for studying randomly coloured point sets in a locally compact second-countable space on which a metrizable unimodular group acts continuously and properly. We first construct and describe an appropriate dynamical system for uniformly discrete uncoloured point sets. For point sets of finite local complexity, we characterize ergodicity geometrically in terms of pattern frequencies. The general framework allows us to incorporate a random colouring of the point sets. We derive an ergodic theorem for randomly coloured point sets with finite-range dependencies. Special attention is paid to the exclusion of exceptional instances for uniquely ergodic systems. The setup allows for a straightforward application to randomly coloured graphs

At the Georgia conference in 1961 Fox presented a paper, “Knots and periodic transformations”, in which he asked which knots may be fixed by a periodic transformation of the 3-sphere. He distinguished eight cases according to the type of fixed point set of T and its relationship to the knot. Except for case a), all these cases have since received some attention and conditions have been given for knots to fall into each of these classes. In fact, the problem of deciding which knots fall into category d) (that is “periodic knots“; I will refer to them as cyclically periodic knots) has been the subject of at least six papers [2], [3], [13], [15], [17], [22], but measured by their effectiveness at determining the periods of the knots to nine crossings, the theorems contained in these papers are not entirely satisfactory.

Complete monotonicity of functions, Definition 3.1, is often proved by showing that their inverse Laplace transforms are nonnegative. There are relatively few simple functions whose inverse Laplace transforms can be expressed in terms of standard higher transcendental functions. Inverting a Laplace transform involves integrating a complex-valued function over a vertical line, and establishing the positivity of the resulting integral can be tricky. Sometimes asymptotic methods are helpful, see for example Fields and Ismail [6].