The transforms discussed here are the quasi-Hausdorff (Theorem 1), the [J, ƒ(x)] (Theorem 2) and the Borel integral means (Theorem 3). We are concerned here with whether or not the limit-points of these transforms of bounded sequences form connected sets. Such a set is one which cannot be decomposed into the union of two disjoint nonempty open sets.

In [3, p. 77] Ryser notes the importance of the minimum of the permanent function on the class of (0, 1)-matrices having exactly k ones in each row and column. In [4] a lower bound was found for the minimum of the permanent on the class Λn of n × n (0, 1)-matrices with exactly three l's in each row and column. The purpose of our work is to improve this result, in particular we show that minA ∊ Λn (per A) ≥ 3(n-1).

Let H be a complex Hilbert space and let K = H ⊕ H. Then K can be identified with the set of all column matrices

equipped with componentwise addition and scalar multiplication and the scalar product

Consider a one-dimensional random walk model where a particle starting at the origin at any instant either takes a jump through a unit distance to the right with probability p1, or stays at the same position with probability p0, or else takes a jump through either of 1, 2, … μ, units of distance to the left with probabilities p-1, p-2, …, p-μ respectively.

By a tournament Tn on n vertices, we shall mean a directed graph on n vertices for which every pair of distinct vertices form the endpoints of exactly one directed edge (e.g., see [5]). If x and y are vertices of Tn we say that x dominates y if the edge between x and y is directed from x to y. In 1962, K. Schütte [2] raised the following question: Given k > 0, is there a tournament Tn(k) such that for any set S of k vertices of Tn(k) there is a vertex y which dominates all k elements of S. (Such a tournament will be said to have property Pk.)

After the publication of the important paper by Rota [9] on Möbius functions a large number of papers have appeared in which the ideas are applied or generalized in various directions, the papers by Crapo [3], Smith [10] and Tainiter [11] are some of them. The theory of Möbius functions is now recognized as a valuable tool in combinatorial and arithmetical research.

It is shown that if a chainable continuum has exactly two arc components, then one of them is an arc and the other is a half-ray.

Let A be a ring (associative) with unity, and let denote the category of unital left A-modules. If is a strongly complete Serre class in then (see [3], and also [6]) there is an exact functor S: , where is the quotient category , and is an abelian category.

In connection with a statistical problem concerning the Galtontest Cśaki and Vincze [1] gave for an equivalent Bernoullian symmetric random walk the joint distribution of g and k, denoting respectively the number of positive steps and the number of times the particle crosses the origin, given that it returns there on the last step.

Sanderson (Canad. Math. Bull., 8 (1965), 505–513), considering a nonempty collection Σ of left ideals of a ring R, with unity, defined the concepts of “Σ-injective module” and “Σ-essential extension” for unital left modules. Letting Σ be an idempotent topologizing set (called a σ-set below) Σanderson proved the existence of a “Σ-injective hull” for any unital left module and constructed an Utumi Σ-quotient ring of R as the bicommutant of the Σ-injective hull of RR. In this paper, we extend the concepts of “Σinjective module”, “Σ-essentialextension”, and “Σ-injective hull” to modules over arbitrary rings. An overring Σ of a ring R is a Johnson (Utumi) left Σ-quotient ring of R if RR is Σ-essential (Σ-dense) in RS. The maximal Johnson and Utumi Σ-quotient rings of R are constructed similar to the original method of Johnson, and conditions are given to insure their equality. The maximal Utumi Σquotient ring U of R is shown to be the bicommutant of the Σ-injective hull of RR when R has unity. We also obtain a σ-set UΣ of left ideals of U, generated by Σ, and prove that Uis its own maximal Utumi UΣ-quotient ring. A Σ-singular left ideal ZΣ(R) of R is defined and U is shown to be UΣ-injective when Z Σ(R) = 0. The maximal Utumi Σ-quotient rings of matrix rings and direct products of rings are discussed, and the quotient rings of this paper are compared with these of Gabriel (Bull. Soc. Math. France, 90 (1962), 323–448) and Mewborn (Duke Math. J. 35 (1968), 575–580). Our results reduce to those of Johnson and Utumi when 1 ∊ R and Σ is taken to be the set of all left ideals of R.

A topological space X is called maximally resolvable if it admits a largest possible family of pairwise disjoint, “maximally dense” subsets. More precisely, if Δ(X) denotes the least among the cardinal numbers of the nonvoid open subsets of X, then X is maximally resolvable if it has isolated points or there exists a family {Rα}α < Δ(X) of subsets of X, called a maximal resolution for X, such that ∪{Rα | α < Δ(X)} = X, Rγ ∩ Rδ==ϕ if γ ≠ δ, and, for each a and each nonvoid open subset V of X, the cardinality of Rα ∩ V is not less than Δ(X).

Let X be a topological space. We say that cat X ≤ n if there exists a map ϕ: X → T1(X, …, X) such that jϕ≃Δ: X → Xn+1, where T1(X, …, X) is the “fat wedge”, j is the inclusion and Δ is the diagonal map. This is an example of a right structure system. This right structure system leads to an associated weak structure system, namely weak category in this particular case.

A three-term recurrence

1

where An (n ≥ 0), Bn (n ≥ 0) and Cn (n ≥ 0) are real numbers for which AnCn+1 ≠0 (n ≥ 0), generates a sequence {Pn} of real polynomials in which Pn has degree exactly n.

Let Σ be a left amenable semigroup, and let {Tσ: σ ∊ Σ} be a representation of Σ as a semigroup of positive linear contraction operators on L1(X, , p). This paper is devoted to the study of existence of a finite equivalent invariant measure for such semigroups of operators.

In contrast to the very complicated structure of the lattice of equational classes of commutative semigroups (see [5]), the lattice of equational classes of commutative monoids (semigroups with unit) is isomorphic with N × N* with a unit adjoined, where N is the lattice of natural numbers with the usual order and N* is the lattice of natural numbers ordered by division. (See [4].) However, the lattice of equational classes of commutative semigroups-with-zero is not so simple to describe.

In [3, p. 149], J. Lambek gives a proof of a theorem, essentially due to Grothendieck and Dieudonne, that if R is a commutative ring with 1 then R is isomorphic to the ring of global sections of a sheaf over the prime ideal space of R where a stalk of the sheaf is of the form R/0P, for each prime ideal P, and . In this note we will show, this type of representation of a noncommutative ring is possible if the ring contains no nonzero nilpotent elements.

In this brief exposition we collect several results on rings of formal power series with coefficients from a field or a ring with some special properties. The results that are catalogued below are mostly algebraic in nature.

This paper deals with the convergence of iterations of the equation

1.1

with u0 ∊ Rn, both as a root and a point of repulsion. Here u and ϕ are real n-vectors and ϕ is continuously differentiate with respect to u. By taking u0 as an initial approximation to u0 in a neighbourhood of u0 we define the sequence of iterates {uk+1} by

1.1

For the sake of convenience we shall, sometimes, call the equation (1.1) an iteration.

In 1927 van der Waerden [6] proved that given positive integers k and l, there exists an integer W such that if 1, 2, …, W are partitioned into k or fewer classes, then at least one class contains an l-term arithmetic progression (l-progression). Let W(k, l), be the smallest such integer W. It would be of interest to find a reasonable upper estimate for W(k, l), say one that could be written down.

In [4], A. Sade defines the singular direct product for quasigroups. In this paper we use the singular direct product to construct quasigroups satisfying the identity x(xy)=yx. In particular, we show that the singular direct product preserves the identity x(xy)=yx.