A space has the shrinking property if, for every open cover {Va | a ∈ A}, there is an open cover {Wa | a ∈ A} with for each a ∈ A.lt is strangely difficult to find an example of a normal space without the shrinking property. It is proved here that any ∑-product of metric spaces has the shrinking property.

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 Poincaré map of mixed exterior algebra generalizes the Hodge star operator and it plays a central rôle in the proofs of many classical identities of linear algebra. The principal purpose of this paper is to derive a new formula for it. This formula is useful in circumstances when the definition is too implicit. Several applications are discussed.

A left FPF ring is a ring R such that every finitely generated faithful left R -module generates the category of left R-modules. It is shown that such rings split into R = A⊕B, where A is a two sided ideal, and A contains the left singular ideal of R as an essential submodule. If R is FPF on both sides B is two sided too, and R is the product of A and B. An example shows this is the best possible and that right FPF does not imply left FPF.

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.

Some inclusion theorems are obtained relating the absolute summability of divergent integrals of the form under three summability methods: Abelian A(x), Abelian A(lnx) and Stieltjes S(x).

We discuss the construction of the spaces Lp(μij), 1 ≤ p ≤ ∞, where {μij} is an n × n positive matrix measure, correct a mistake in the literature concerning those spaces and develop an interpolation theory for them.

Let {Xn} be a sequence of random variables, not necessarily independent or identically distributed, put and Mn =max0≤k≤n|Sk|. Effective bounds on in terms of assumed bounds on , are used to identify conditions under which an extended-valued stopping time τ exists. That is these inequalities are used to guarantee the existence of the stopping time τ such that E(ST/aτ) = supt ∈ T∞ E(|Sτ|/at), where T∞ denotes the class of randomized extended-valued stopping times based on S1, S2, … and {an} is a sequence of constants. Specific applications to stochastic processes of the time series type are considered.

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.

If H is an arbitrary harmonic function defined on an open set Ω⊂ℂ, then the curvature of the level curves of H can be strictly maximal or strictly minimal at a point of Ω. However, if Ω is a doubly connected domain bounded by analytic convex Jordan curves, and if H is harmonic measure of Ω with respect to the outer boundary of Ω, then the minimal curvature of the level curves of H is attained on the boundary of Ω.

Let A = (aij) be an idempotent latin square of order n, n ≥ 3, in which aii = i, 1 ≤ i ≤ nc. A set S ⊆ N = {1, 2, …, n} is a cover of A if (N × N)\{(i, i):i ∉ S} = {(i, j): i ∊ S, j ∊ N} ∪ {(j, i): i ∊ S, j ∊ N} ∪ {(i, j): aij ∊ S}. A cover S is minimum for A if |S| < |T| for every cover T of A and we write c(A) = |S|. We denote by c(n) the maximum value of c(A) over all idempotent latin squares A of order n and in this paper show that (7n/10)-3.8 ≤ c (n) < n - n1/3 + 1 for all n ≥ 15. The problem of determining c(n) was first raised by J. Schönheim.

Let R be a prime ring and d≠0 a derivation of R. We examine the relationship between the structure of R and that of d(R). We prove that if R is an algebra over a commutative ring A such that d(R) is a finitely generated submodule then R is an order in a simple algebra finite dimensional over its center.

We construct an explicit continuous function F such that for each point x, every extended real number is a derived number of F at x and F has an infinite left and an infinite right derived number at x.

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.

Let H(x) be a homogeneous polynomial in n indeterminates over an algebraically closed field K. A necesssary and sufficient condition is given for H(x) to admit a factorization of the form

a, b∈ Kn, and “∘” is the usual inner product. This condition involves the linear derivatives of H(x).

Solutions of Xk + Yk = Zk in invertible pairwise commuting rational 2 × 2 matrices are determined for k = 3, 4, 6, 9, from the analogous results of A. Aigner for algebraic number fields.

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.

The question of what sets of integers may be the score sets of bipartite tournaments was posed recently by K. B. Reid. The main theorem of this paper establishes a sufficient condition for pairs of sets to be bipartite score sets. This simple condition yields an immediate affirmative answer for a large class of pairs of sets.

A graph is k -cycled if all its cycles are integral multiples of an integer k ≥ 2. We determine the structure of refinements of Kn and Kn, m which are k-cycled.

A popular problem today in orthogonal polynomials is that of classifying all second order differential equations which have orthogonal polynomial solutions. We show that the Krall polynomials satisfy a second order equation of the form

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