One form of Dirichlet's theorem on simultaneous diophantine approximation asserts that if α1, α2, …, αn are any real numbers and m ≥ 2 is an integer, then there exist integers q, p1, p2, …, pn such that 1 ≤ q < m and |qαi.-pi| ≤ m–1/n holds for 1 < i < n. The paper considers the problem of the extent to which this theorem can be improved by replacing m–1/n by a smaller number. A general solution to this problem is given. It is also shown that a recent result of Kurt Mahler [Bull. Austral. Math. Soc. 14 (1976), 463–465] amounts to a solution of the case n = 1 of the above problem. A related conjecture of Mahler is proved.

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.

A group A is said to be SQ-universal if every countable group is embeddable in some quotient of A. It is well-known that the number of non-isomorphic factor groups of a countable sp-universal group B is the power of the continuum. Since B has such an abundance of non-isomorphic quotients it is natural to ask what types of quotients B must have and what types B need not have. This note is concerned with these questions.

Let (α β) denote the class of functions

analytic in the unit disc Δ ≡{z: |z| < 1} and satisfying

for some α, β (0 ≤ α < 1, 0 < β ≤ 1) and for all z ∈ Δ. In the present paper, sharp coefficient estimates for functions in (α, β) have been obtained. The results thus obtained not only generalize the corresponding results of Thomas H. MacGregor (Michigan Math. J. 10 (1963), 277–281), A.V. Boyd (Proc. Amer. Math. Soc. 17 (1966), 1016–1018) and others, but also give rise to analogous results for various other subclasses of starlike functions.

The article contains the basic theory resulting from the presentation of a dense functor N: A → C by means of an expansion K(k, C) ० NJk ≅ C the term dense functor being used instead of the equivalent term left-adequate functor. Results by various authors on the density type of a functor are formulated in the V-context for V symmetric monoidal closed, and elementary proofs are given. In particular a characterisation theorem containing the well-known results of Beck and Ulmer is established.

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.

This paper presents a necessary condition for a ribbon link to be an homology boundary link and gives a consequent simple counterexample to the conjecture of Smythe that the vanishing of the first Alexander polynomial characterizes homology boundary links among all 2-component links.

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.

Let G be a finite group, p a prime divisor of |G|, and T a p–subgroup of G. Define σ(T) to be the number of Sylow p–subgroups of G containing T. Call T a central p–Sylow intersection if for some Σ ⊆ Sylp (G), T = ∩(S | S є Σ), and if, in addition, T contains the center of a Sylow p–subgroup of G. This work is inspired and motivated by work of G. Stroth [J. Algebra 37 (1975), 111–120]. Generalizing an argument of his we describe finite groups in which every central p–Sylow intersection T with p–rank(T) > 2 satisfies σ(T) ≤ p.

Related methods yield the description of finite groups in which every central p–Sylow intersection T with p–rank(T) ≥ 2 satisfies σ(T) ≤ 2p.

This paper presents three arguments, one involving orientability, and the others Milnor duality and, respectively, the injectivity of cup product into H2 for an abelian group and free finite group actions on homotopy 3-spheres to show that there are high dimensional knot groups which are not the groups of knotted 2-spheres in S4, thus answering a question of Fox (“Some problems in knot theory”, Topology of 3-manifolds and related topics”, 168–176 (Proceedings of the University of Georgia Institute, 1961. Prentice-Hall, Englewood Cliffs, New Jersey, 1962).

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.

The characterizations of the closed ranges of the multiplication operators and the composition operators on L2 (λ) are reported in this paper.

The classical Wiener-Pitt phenomenon for measures may be formulated as an existence theorem for Fourier multipliers with irregular, spectral properties and the result has been refined in various ways over the years. The most recent development is due to Zafran, who exhibits abnormal spectral behaviour in multipliers whose transforms vanish at infinity and which operate on Lp where 1 < p < ∞, p ≠ 2. Zafran's methods use Littlewood-Paley theory, interpolation, and the construction of a certain class of measures. We show here how the constructive element of his proof may be considerably simplified and sharpened.

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 note, Conjecture Cl of Toby Lewis (1976), concerning a reciprocal pair of characteristic variables, is shown to be false.

Converses are proved for the Osgood (the Principle of Uniform Boundedness), Dini, and other well known. theorems. The notion of a continuous step function on a topological space is defined and a class of spaces identified for which each lower semicontinuous function is the pointwise limit of a monotonically increasing sequence of step functions.

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.