In this paper we examine the nature of rings R with unit having an automorphism ϕ ≠ 1 such that x — ϕ(x) is 0 or invertible for every x ∊ R. We show that the only examples of such rings are R = D, R = D2, and R = D ⊕ D, where D is a division ring. Furthermore, for the case D2, we describe the division rings that are possible.

The principle of duality for Boolean algebra states that if an identity ƒ = g is valid in every Boolean algebra and if we transform ƒ = g into a new identity by interchanging (i) the two lattice operations and (ii) the two lattice bound elements 0 and 1, then the resulting identity ƒ = g is also valid in every Boolean algebra. Also, the equational theory of Boolean algebras is finitely based. Believing in the cosmic order of mathematics, it is only natural to ask whether the equational theory of Boolean algebras can be generated by a finite irredundant set of identities which is already closed for the duality mapping. Here we provide one such equational basis.

The local class group of a Krull domain A is the quotient group G(A) = CI(A)/Pic(A). A Krull domain A is locally factorial if and only if G(A) = 0. In this paper, we characterize the Krull domains for which G(A) is a torsion group. We evaluate the local class group of several examples and finally, we explain why every abelian group is the local class group of a Krull domain.

Let T be an ergodic measure preserving transformation with the following property: there exists a positive integer n and a finite partition α such that the number of atom of is one more than that of , and the probability of at least one of the atoms is irrational. Then there exists a unique (up to conjugacy) transformation S such that there is a partition β with S restricted to isomorphic to T restricted to and the number of atoms in is one more than the number of atoms in for all m ≥ n. Moreover this transformation has discrete spectrum with at most two generators. If there are two generators, one of them must be a root of unity.

The main results of this paper are the following: (1) An extension theorem for a uniform semigroup-valued measure on a ring to the generated σ-ring. This result unifies the classieal Hahn-Carathéodory theorem, the extension theorem of Sion and a more recent result of Weber.

(2) A theorem stating that every monocompact additive uniform semigroup-valued set function on a semiring is σ-additive. This result generalizes several earlier theorems of Alexandroff, Dinculeanu-Kluvanek, Glicksberg, Huneycutt, Mallory, Marczewski, Millington and Topsøe.

An oscillation criterion for an even order equation: x(n) + q(t)ƒ(x(t)), x[g(t)]) = 0 is provided. This criterion is an extension of a result established by Yeh for the second order equation ẍ + q(t)ƒ(x(t)), x[g(t)]) = 0.

In the first part we construct a subset H of positive measure in the unit interval and a zero-sequence {an} so that H contains no homothetic copy of {an}. In Theorem 2 we prove that if ε > 0 and a zero-sequence {an} are given then there exists a set A of measure less than ε so that covers the interval. An application of this result is Theorem 3: for any sequence {an} and ε > 0 there is a set H of measure 1 - ε such that for no N and c is {an + c}n ≥ N contained by H.

It is well-known that a topological space is a Gleason part of some complex function algebra if and only if it is completely regular and σ-compact. In the present paper, a Gleason part of a real function algebra is characterized as a completely regular σ-compact topological space which admits an involutoric homeomorphism.

It is reasonable to expect that, under suitable conditions, Padé approximants should provide nearly optimal rational approximations to analytic functions in the unit disc. This is shown to be the case for ez in the sense that main diagonal Padé approximants are shown to converge as expeditiously as best uniform approximants. Some more general but less precise related results are discussed.

The question as to the existence of nontrivial compact or weakly compact multipliers between spaces of functions on groups has been investigated for several years. Until now, however, no general method which is applicable to a large class of function spaces seems to be known

In this paper we prove that the existence of nontrivial compact multipliers between Banach function spaces on which a group acts is related to the existence of nonzero almost periodic functions.

Let E be a Banach lattice,

where x1, …, xn ∊ E. We study properties of constants

A characterization of AM-spaces is obtained which generalizes the result of Abramocič, Positselskiĭ, Yanovskii. Asymptotic estimates of φp for some classical finite dimensional lattices are given.

Let X be a separable metric space and let be a family of countably many self-maps of X. Then there is a countable subalgebra of the Boolean algebra of regular open subsets of X which is a base for X such that for each the function defined by Φf(B) = (f-1(B))-0 is a homomorphism.

This paper deals with the differential equation: ẍ + μF(ẋ) + x = ƒ( X, ẋ, t/Tμ) for μ ≫ 1 where F is a piecewise linear function and f is a periodic function of period μT, where T is to be chosen. It is established that periodic forced vibrations exist in an annular domain R(μ) constructed for the free vibration (ƒ ≡ 0), provided ƒ is not of higher order than Subsequently with ƒ = A cos (2πt/μT*), an asymptotic treatment of the forced vibration problem is carried out, by finding the proper initial conditions and the proper period μT* of f. Finally it is concluded that μT* is close to the period of the free vibration.

G. B. Gurevič enumerated all the orbits of GL8(ℂ) in Λ3(ℂ8). There are precisely 23 orbits (including the trivial orbit). For each of these orbits, we determine its closure (for the ordinary topology).

Let d(G) denote the minimal number of generators of the finite p-group G, r(G) the maximum over all subgroups H of G of d(H) and ra(G) the maximum over all abelian subgroups H of G of d(H). If G is of class two it is clear that

By considering properties of the stability number of graphs we construct examples which show that any value of r(G) within these bounds can occur.

This article introduces the concept of a condensed domain, that is, an integral domain R for which IJ = {ij: i ∊ I, j ∊ J} for all ideals I and J of R. This concept is used to characterize Bézout domains (resp., principal ideal domains; resp., valuation domains) in suitably larger classes of integral domains. The main technical results state that a condensed domain has trivial Picard group and, if quasilocal, has depth at most 1. Special attention is paid to the Noetherian case and related examples.

In this paper, we construct, for any open cover of a metric space, a partition of unity consisting of a family of uniformly continuous functions.

A Banach space is fully 2-rotund if (xn) converges whenever ‖xn + xm‖ converges as m, n → ∞ and locally uniformly rotund if xn → x whenever ‖xn‖ and ‖(xn + x)/2‖ → ‖x‖.

We show that I2 with the equivalent norm

is fully 2-rotund but not locally uniformly rotund, thus answering in the negative a question first raised by Fan and Glicksberg in 1958.

Simple structure theorems or representation theorems make their appearance at a very elementary level in subjects such as algebra, but one infrequently encounters such theorems in elementary topology. In this note we offer one such theorem, for finite topological spaces, which involves only the most elementary concepts: discrete space, indiscrete space, product space and homeomorphism. Our theorem describes the structure of those finite spaces which are homogeneous.

A short proof is given of a result of Burde giving the parity of the number of quadratic residues (mod p) in the interval (0, p/4), where p ≡ 1(mod 4) is prime.