The purpose of this paper is to generalize the results of Sherbert on Lipschitz algebras and to study the relationship between homomorphisms of these algebras and continuous maps of the underlying metric spaces. In Sections 1, 2, and 3 we associate with each metric space a class of Lipschitz-type algebras and extend Sherbert's results in [7] to this class; in particular Sherbert's theorem 5.1 is extended to non-compact metric spaces (3.3, 3.4, 3.5).

Let p denote a prime and n a positive integer ≥ 2. Let Nn(p) denote the number of polynomials xn + x + a, a = 1, 2,…, p-l, which are irreducible (mod p). Chowla [5] has made the following two conjectures:

Conjecture 1. There is a prime p0(n), depending only on n, such that for all primes p ≥ p0(n)

Let X and Y be two independent normal variates each distributed with zero mean and a common variance. Then the quotient X/Y has the Cauchy distribution symmetrical about the origin. Of particular interest in recent years has been the converse problem and examples of non-normal distributions with a Cauchy distribution for the quotient have been illustrated by Mauldon [9], Laha [2; 3; 4] and Steck [10].

In 1957 Hájek [1] proved that the extreme bilateral derivates of an arbitrary finite real valued function of a real variable, are Borel measurable of class ≦ 2. It was later shown by Staniszewska [3] that Hájek's result is the best possible (even among the class of functions satisfying a Lipschitz condition). Staniszewska exhibited a Eipschitz function whose extreme bilateral derivates are not in Borel class 1. Staniszewska's proof makes use of a result of Zahorski's [4] concerning kernel functions.

The pair (S, M) is a matroid if S is a finite set and M a collection of subsets of S such that (1) every subset of a set of M is in M, and (2) all maximal sets in M have a common cardinality. The span of a set A ⊂ S is Γ(A) where y ∈ Γ (A) if and only if y ∈ A or there is A' ⊂ A, A' ∈ M and {y} ∪ A' ∉ M. A maximal set in M is called a base.

G. Legrand [1] studied a generalization of the almost complex structures [2] by considering a linear operator J acting on the complexified space of a differentiable manifold Vm satisfying a relation of the form J2 = λ2 (identity) where λ is a nonzero complex constant. Such structures are called π-structures. A π-structure is defined on Vm by the knowledge of two fields, of proper supplementary subspaces T1 and T2 of the complexified tangent space at x ∈ Vm, such that dim(T1) = n1; dim(T2) = n2; n1 + n2 = m. In the remaining case, λ= 0, H. A. Eliopoulos [3] introduced almost tangent structures and discussed euclidean structures compatible with almost tangent structures [4]. In a similar way the purpose of this paper is to study singular riemannian structures compatible with π structures, briefly Rπ-structures.

Nous nous proposons ici d'étendre certains résultats de la théorie des anneaux commutatifs à deux classes d'anneaux non- commutatifs. Il s'agit, d'une part, des anneaux inversifs à droite, définis par Thierrin [4], et des anneaux réversifs. Les premiers possèdent la propriété xyR = yxR, les seconds la propriété xRy = yRx, ce pour tout x, y ∈ R. Nous considérons aussi la réunion des parties d'un anneau qui possèdent respectivement l'une ou l'autre propriété.

By a group theoretic class we mean a class of groups which contains the trivial group, denoted E, and any group isomorphic to a group in the class. Let I be a group theoretic class. Following P. Hall [4, p. 533], we define EI, CI, SI, QI, and NoI to be the (group theoretic) classes consisting of extensions of I groups by I groups, cartesian products of I groups, subgroups of I groups, homorphic images of I groups and products of two normal I subgroups of a group, respectively. If T is one of the above operations on classes of groups and TI = I, we say X is T closed.

The implicit function theorem has applications at all levels of mathematics from elementary calculus (implicit differentiation) to finding periodic solutions of systems of differential equations ([1, Chapter 14] and [4], for example).

In 1961 W. S. Loud [3] studied the case of two equations in three unknowns. He considered only cases where up to third order derivatives were involved and only those cases where the derivative of the solutions at the critical point existed. Coddington and Levinson [1] consider a specific singular case involving n equations in n + m unknowns. In general the number of distinct critical cases involving up to third derivatives for such a general system is not known.

If x and y are independent p component column vectors, and the conditional distribution of x, given x+y = z, is known, what can be said about the distributions of x and y? This problem has been solved by Seshadri (1966) in the particular case when the conditional distribution of x, given x+y = z, is multivariate normal. In fact Seshadri′s paper implicitly contains a characterization of the multivariate linear exponential distribution

(1)

where A(x) is a function of x not involving the p component column vector w of constant terms.

All homology and cohomology groups are reduced. is the class of finite abelian groups whose orders are prime to p. is the class of abelian groups whose orders are products of primes less than a. If G is a finitely generated abelian group, Gp is the quotient of G by the subgroup of G made up of all elements whose orders are prime to p.

A tournament Tn with n nodes is a complete asymmetric digraph [2]. A set S of arcs of a tournament is called consistent if the tournament contains no oriented cycles composed entirely of arcs of S [1]. The object of this note is to provide a new lower bound for f(n), the greatest integer k such that every tournament with n nodes contains a set of k consistent arcs. Erdös and Moon [1] showed that where [x] denotes the largest integer not exceeding x, and the second inequality holds for any fixed ∈ > 0 and all sufficiently large n.

The Bernstein polynomials

(1.1)

and the Bernstein power series

(1.2)

have been the subject of much research (e. g. [1; 2; 3; 6; 7; 8]). It is the purpose of this paper to demonstrate the relationship between these linear operators and certain classes of moment sequences defined below.

The application of Borel summability to Fourier series has been discussed by Takahashi and Wang [8] and Sahney [5]. Sahney [6] and Sinvhal [7] obtained sufficient conditions for the Borel summability of the derived Fourier series and its conjugate series, respectively. Kathal [3] obtained different conditions in the case of the conjugate series. In this paper we give a new criterion for Borel summability of Fourier series.

The original motivation for this work was the problem of determining whether the signum function of a real valued continuous function defined on the real line is Riemann integrable. This problem is considered in § 2 where an example of an infinitely differentiable function is presented which possesses a non-Riemann integrable signum function. Moreover, it is shown that, for any ∈ > 0, it is possible to construct such an example for which the set of points of analyticity has Lebesgue measure which is less than ∈. This appears to be a more interesting property than the one originally sought.

For a finite abelian group G let λ(G) be the least positive integer such that λ(G)G = 0. Let be the least integer such that λ(G) | (λ(G) divides ) and if 2 | λ(G) then 4 | . For a finitely generated abelian group G let GT be the subgroup of G made up of all elements of G of finite order, and let GF = G/GT. For a simply-connected C-W complex X, let be the smallest class of abelian groups containing the groups .

The K transformation (also called the Meijer transformation) has been extended by Zemanian [1; 2] to a class of generalized functions, For , he defined the K transform of f by

(1)

In [2, Section 6.6] the following inversion theorem for the K transform of f is proven:

(2)

in the sense of weak convergence in D'(I). Here, σ is any fixed real number greater than σf, μ is zero or a complex number with positive real part and D'f(I) is the space of Schwartz distributions on I = (0, ∞).

In this paper we introduce the concept of Boolean near-rings. Using any Boolean ring with identity, we construct a class of Boolean near-rings, called special, and determine left ideals, ideals, factor near-rings which are Boolean rings, isomorphism classes, and ideals which are near-ring direct summands for these special Boolean near-rings.

Blackett [6] discusses the near-ring of affine transformations on a vector space where the near-ring has a unique maximal ideal. Gonshor [10] defines abstract affine near-rings and completely determines the lattice of ideals for these near-rings. The near-ring of differentiable transformations is seen to be simple in [7], For near-rings with geometric interpretations, see [1] or [2].

A finite sequence (pk) = (p3, p4,…) of non-negative integers shall be called realizable provided there exists a 3-valent 3-polytope P which has pi. i-gonal faces for every i. P is called a realization of (pk).

For realizability of a sequence (pk), from Euler's formula follows

(*) as a necessary condition.

Sufficient conditions will be derived for the linear elliptic partial differential equation

(1)

to be nonoscillatory in an unbounded domain R in n-dimensional Euclidean space En. The boundary ∂R of R is supposed to have a piecewise continuous unit normal vector at each point. There is no essential loss of generality in assuming that R contains the origin. Otherwise no special assumptions are needed regarding the shape of R: it is not necessary for R to be quasiconical (as in [2]), quasicylindrical, or quasibounded [1].