In this paper, the author introduces the notion of Ω-Polish spaces (which includes the Polish spaces and a large class of Banach spaces) and extends Castaing's selection theorem (1966) for closed-valued measurable thin multifunctions from a measurable space into an Ω-Polish space. He also extends Robertson's theorem (1974) in the same way.

If m and n are natural numbers satisfying 1 ≦ m < n let denote the least integer k such that the statement: ‘Every (0,1) matrix with n columns, with constant row-sum m, and with at least k distinct rows, has rank n’ is true. Then = +1 for m ≧ 2, n ≧ m2 + 2. Further, for 1 ≦ m < n.

We give a partial answer to the following questions: Given an additive category and a class of sequences, under what conditions is the universal functor to the abelian category faithful and what other sequences are taken to exact sequences?

The “answers” to these questions appear as theorems 2.2 and 3.2 respectively and amount to a weakening of the condition that there be enough relative projectives.

We then characterize those additive categories with exactness which have the property that all relative exact sequences are determined by a small set of functors into the category of abelian groups.

The absence of limit cycles in the system (S): x′ = − y + dx2 + exy + fy2, y′ = x, (′ ≡ d/dt), for 0 ≦ f2 < 2, was shown by Yeh et al. (1963). The general case was established by Kukles and Šakhova using various special results and relying on an extraneous auxiliary system. In this paper we give a new proof of the theorem; in it, we draw mainly on the basic properties of the characteristic exponent of a cycle and use an intrinsic system, viz. the case d = − f(≠ 0) of (S) as an auxiliary system.

The representation theory of categories is used to embed each promonoidal monad in a monoidal biclosed monad. The existence of a promonoidal structure on the ordinary Eilenberg- Moore category generated by a promonoidal monad is examined. Several results by previous authors (notably A. Kock and F. E. J. Linton) are reproved and extended.

In this article a completion process for promonoidal categories is used to determine a sufficient condition for the existence of a promonoidal convolution structure on the category of functors between two promonoidal categories. This, in turn, leads to a partial closed structure on the 2-category of promonoidal categories, promonoidal functors, and promonoidal natural transformations.

Let G be a transitive permutation group on a finite set of n points, and let P be a Sylow p-subgroup of G for some prime p dividing |G|. We are concerned with finding a bound for the number f of points of the set fixed by P. Of all the orbits of P of length greater than one, suppose that the ones of minimal length have length q, and suppose that there are k orbits of P of length q. We show that f ≦ kp − ip(n), where ip(n) is the integer satisfying 1 ≦ ip(n) ≦ p and n + ip(n) ≡ 0(mod p). This is a generalisation of a bound found by Marcel Herzog and the author, and this new bound is better whenever P has an orbit of length greater than the minimal length q.

Hattori (1960) defined a right R-module A to be torsion-free if for all a ∈ A and x ∈ R, ax = 0 implies that there exist elements {x1, x2, …, xn} ⊆ R with xix = 0 for all 1 ≦ i ≦ n and {a1, a2, …, an} ⊆ A such that a = aixi. Left torsion-free is defined similarly. It is shown that for a ring R, these torsion-free modules are the torsion-free class of a hereditary torsion theory, corresponding to a perfect topology, if and only if the left flat epimorphic hull of R is a regular ring which is both left and right torsion-free. A class of right semi-hereditary rings for which the torsion-free modules of Hattori satisfy the above property are found and this class of rings is discussed.

The essential cover Mk of a class M is defined as the class of all essential extensions of rings belonging to M. M is called essentially closed if Mk = M. Every class M has a unique essential closure, i.e. a smallest essentially closed class containing M.

Let M be a hereditary class of (semi)prime rings. Then M is proved to be a (weakly) special class if and only if M is essentially closed. A main result is that Mk is the smallest (weakly) special class containing M. Further it is shown that the upper radical UM determined by M, is hereditary if and only if UM has the intersection property with respect to Mk.

Given a symmetric balanced incomplete block design or a finite plane, we recursively construct balanced incomplete block designs by taking unions of certain blocks and points of the original design to be the blocks of the new design.

The bichromaticity β(B) of a bipartite graph B has been defined as the maximum order of a complete bipartite graph onto which B is homomorphic. This number was previously determined for trees and even cycles. It is now shown that for a lattice-graph Pm × Pm the cartesian product of two paths, the bichromaticity is 2 + {mn/2}.

Let X be any Tychonoff space and βX the Stone—Čech compactification of X. Let F(βX) be the Graev free group of βX and let be the subspace topology on the Graev group F(X). Our results demonstrate that this topology is useful and behaves extremely well; the behavior of the free topology still remains enigmatic.

There are various applications, some of which clarify the free topology on F(X), while others improve various results recently published.

It is shown that if every nilpotent 2 × 2 matrix over a ring has nilpotent transpose, then the commutator ideal must be contained in the Jacobson radical, thus generalizing a result of R. S. Gupta, who considered the division ring case. Moreover, if the nilpotent elements form an ideal or if the ring satisfies a polynomial identity, then the above property of the transpose implies that in fact the commutator ideal must be nil.

Throughout this paper A will denote an f-ring i.e. a lattice-ordered ring in the sense of Birkhoff and Pierce (1956) in which for all x, y, z ∈ A, x ∧ y = 0 implies x ∧ zy = 0 = x ∧ yz.

In this note we prove a discrete analogue to the following Paley–Weiner theorem: Let f be the restriction to the line of a bounded analytic function in the upper half plane; then the spectrum of f is contained in ([0, ∈). The discrete analogue of complex analysis is the theory of discrete analytic functions invented by Lelong-Ferrand (1944) and developed by Duffin (1956) and others.

In this note we define the Artin radical of a Noetherian ring and describe some of its applications.