The dual of the category of De Morgan algebras is described in terms of compact totally ordered-disconnected ordered topological spaces which possess an involutorial homeomorphism that is also a dual order-isomorphism. This description is used to study the coproduct of an arbitrary collection of De Morgan algebras and also to represent the coproduct of two De Morgan algebras in terms of the continuous order-preserving functions from the Priestley space of one algebra to the other algebra, endowed with the discrete topology. In addition, it is proved that the coproduct of a family of Kleene algebras in the category of De Morgan algebras is the same as the coproduct in the subcategory of Kleene algebras if and only if at most one of the algebras is not boolean.

We obtain a general transcendence theorem for the solutions of a certain type of functional equation. A particular and striking consequence of the general result is that, for any irrational number w, the function

takes transcendental values at all algebraic points α with 0 < |α| < 1.

The aim of this paper is to extend a recent result of S.A. Husain and V.M. Sehgal [Bull. Austral. Math. Soc. 13 (1975), 261–267]. The condition that the function should be continuous in Husain and Sehgal, op. cit., is replaced by a semicontinuity condition. Moreover, a different proof is given, the last part of which requires no continuity or semicontinuity condition whatsoever.

Let G be a finite group. A nontrivial proper subgroup M of G is called a CC-subgpoup if M contains the centralizer in G of each of its nonidentity elements. In this paper groups containing a CC-subgroup of order divisible by 3 are completely determined.

Pseudo-autonomous linear differential equations are defined. A linear differential equation with bounded coefficient matrix is pseudo-autonomous if and only if it is almost reducible. A linear differential equation with recurrent coefficient matrix is pseudo-autonomous if and only if it has pure point spectrum.

The formula of [E.] Meissel [Math. Ann. 2 (1870), 636–642] is generalized to arbitrary arithmetic progressions. Meissel's formula is applicable not only to computation of π(x) for large x (recently x = 1013), but also is a sieve technique (see MR36#2548), useful for studying the subtle effect of primes less then or equal to x1/2 on the behavior of primes less than or equal to x. The same is true of the generalized Meissel, with the added advantage that the behavior of primes less than or equal to x can be studied in arbitrary progressions.

Let R be a ring with an identity and for each x, y in R, (xy)k = xkyk for three consecutive positive integers k. It is shown in this note that R is a commutative ring.

It is known that, for an arbitrary convergence space X, the vector space generated by X is dense in LcCc (X) where both C(X) and its dual space carry the continuous convergence structure. In this note, a natural analogue formulated for the operator space L(Cc(X), Cc(X)) is considered, namely: is the vector space generated by the composition operators associated to the continuous mappings in C(X, X) dense in Lc (Cc (X), Cc (X)) ? This question is answered in the negative by a counterexample.

In this paper, we obtain an explicit form of the currently best known inequality for linear forms in the logarithms of algebraic numbers. The results complete our previous investigations (Bull. Austral. Math. Soc. 15 (1976), 33–57) which were conditional on a certain independence condition on the algebraic numbers. The extra work needed to obtain unconditional results centres on the properties of multiplicative relations in number fields. In particular, we show that a set of multiplicatively dependent algebraic numbers always satisfies a relation with relatively small exponents.

Until recently most authors have devoted their research to the theory of perturbed systems under continuous perturbations. In this paper, Liapunov's second method is employed to investigate sufficient conditions for integral and integral asymptotic stability of ordinary differential systems with respect to impulsive perturbations.

Various existing models of the optimal control of production rates of manufacturing firms are discussed. A new model is derived by considering the combined effects of: the inventory level of the firm, the shipment sent from the firm, the shipment rate, the Orders received by the firm, the demand rate, the rate of change of the demand rate, the production rate, the advertising expenditure, and the level of unfilled Orders. Further, a new version of shortage cost is introduced. The well-known Pontryagin maximum principle and transversality condition are used to obtain the optimal production rate and the optimal advertising expenditure. A numerical example is given for Illustration.

Let R be the field of real numbers. For α in R, let ‖α‖ be the distance of α from the nearest integer. The following conjecture of Kurt Mahler [Bull. Austral. Math. Soc. 14 (1976), 463–465] is proved.

Let m, n be two positive integers n ≥ 2m. Let S be a finite or infinite set of positive integers with the following properties:

(Q1) S contains the integers m, m+1, …, n−m;

(Q2) every element of S satisfies

Then

In this article closed variety structures generated by a monoidal class of identities are examined. This leads to closed enrichments of various results of Y. Diers [Ann. Soc. Sci. Bruxelles Sér. I 90 (1976), 159–172] on varieties of a category.