The work establishes the estimates of linear independence, transcendence, and algebraic independence measures of the values at algebraic points of a set of E-functions, satisfying linear differential equations with coefficients from C(z), rather close to their natural bounds.

In this note we give several new characterizations of arbitrary pseudocompact spaces, that is spaces characterized by the property that all continuous real-valued functions on the space are bounded.

Commutative idempotent quasigroups with a sharply transitive automorphism group G are described in terms of so-called Room maps of G. Orthogonal Room maps and skew Room maps are used to construct Room squares and skew Room squares. Very general direct and recursive constructions for skew Room maps lead to the existence of skew Room maps of groups of order prime to 30. Also some nonexistence results are proved.

R. Paré and W. Schelter (1978) have extended the Cayley-Hamilton theorem by showing that for each n<1 there is an integer k such that all n x n matrices over any (possibly noncommutative) ring satisfy a monic polynomial of degree k. We give a lower bound for this degree, namely π(n), which is defined as the shortest possible length of a sequence with entries from {1, 2, …, n}.

In this paper a general concept of regularity for rings is defined. It is shown that every regularity determines in a natural way a subradical and a radical for rings. A wide class of regularities is constructed: the class of polynomial regularities. All well-known regularities, such as the Perlis-Jacobson regularity, the von Neumann regularity and many others, belong to this class. Special attention is paid to regularities which are elementary in the sense that the so-called unic and nullic polynomial regularities can be thought of as intersections of the elementary ones.

Let λj (1 ≦ j ≦ 4) be any nonzero real numbers which are not all of the same sign and not all in rational ratio and let pj be polynomials of degree one or two with integer coefficients and positive leading coefficients. The author proves that if exactly two pj are of degree two then for any real n there are infinitely many solutions in primes pj of the inequality . where 0 <β < (√(21)–1)∖5760.

The Moore-Penrose inverse of a general bordered matrix is found under various conditions. The Moore-Penrose inverses obtained by Hall and Hartwig (1976) are shown to be special cases of these more general results.

The notion of duality of functors is used to study and characterize spaces satisfying the Radon-Nikodym property. A theorem of equivalences concerning the Radon-Nikodym property is proved by categorical means; the classical Dunford-Pettis theorem is then deduced using an adjointness argument. The functorial properties of integral operators, compact operators, and weakly compact operators are discussed. It is shown that as an instance of Kan extension the weakly compact operators can be expressed as a certain direct limit of ordinary hom functors. Characterizations of spaces satisfying the Radon-Nikodym property are then given in terms of the agreement of dual functors of the functors mentioned above.

Let P be a finite, connected partially ordered set containing no crowns and let Q be a subset of P. Then the following conditions are equivalent: (1) Q is a retract of P; (2) Q is the set of fixed points of an order-preserving mapping of P to P; (3) Q is obtained from P by dismantling by irreducibles.

In this paper we generalize a recent result of Freedman (1973) concerning the cardinality of the type set of a rank two torsion-free abelian group. We show that if A is such a group and A supports a non-trivial associative ring then the type set of A contains at most three elements.