Some minimax inequalities are first proved both in the compact case and in the non-compact case using the concept of escaping sequences introduced by Border. Applications are given to deduce a generalisation of the Gale-Nikaido-Debreu Lemma due to Mehta and Tarafdar and to obtain a new generalisation of the Gale-Nikaido-Debreu Lemma from which the corresponding generalisation due to Grandmont is derived.

Let G be a nondiscrete locally compact Hausdorff abelian group. It is shown that if G contains an open torsion subgroup, then every proper dense subgroup of G is contained in a maximal subgroup; while if G has no open torsion subgroup, then it has a dense subgroup D such that G/D is algebraically isomorphic to R, the additive group of reals. With each G, containing an open torsion subgroup, we associate the least positive integer n such that the nth multiple of every discontinuous character of G is continuous. The following are proved equivalent for a nondiscrete locally compact abelian group G:

(1) The intersection of any two dense subgroups of G is dense in G.

(2) The intersection of all dense subgroups of G is dense in G.

(3) G contains an open torsion subgroup, and for each prime p dividing the positive integer associated with G, pG is either open or a proper dense subgroup of G.

Finally, we construct a locally compact abelian group G with infinitely many dense subgroups satisfying the three equivalent conditions stated above.

The Cahn-Hilliard system, a natural extension of the single Cahn-Hilliard equation in the case of multicomponent alloys, will be shown to generate a dissipative semigroup on the phase space ℋ = [H2(Ω)]m. Following Hale's ideas and based on the existence and form of the Lyapunov functional, our main result will be the existence of a global attractor on a subset of ℋ. New difficulties specific to the system case make our problem interesting.

This paper deals with the equivalence between u−density and m−density for the subrings of C(X). It was proved by Kurzweil that such equivalence holds for those subrings that are closed under bounded inversion. Here an example is given in C(N) of a u−dense subring that is not m−dense. It is deduced that the two types of density coincide only in the trivial case where these topologies are the same, that is, if and only if X is a pseudocompact space.

In this paper we show that the method of calculating the Gaussian period polynomial which originated with Gauss can be replaced by a more general method based on formulas for Lagrange resolvants. The period polynomial of cyclic sextic fields of arbitrary conductor is determined by way of example.

In this paper, some necessary and sufficient conditions for oscillation of a first order delay differential equation with oscillating coefficients of the form

are established. Several applications of our results improve and generalise some of the known results in the literature.

Recently, Bhargava, Adiga and Somashekara made use of Ramanujan's 1Ψ1 summation formula to prove a convolution identity for certain coefficients generated by the quotient of two infinite products. As special cases of this identity, they deduced several results (connected especially with the generalised Frobenius partition functions) including, for example, the convolution identities given earlier by Kung-Wei Yang. In this sequel to the aforementioned works, we provide a complete answer to an interesting question raised by Bhargava, Adiga and Somashekara in connection with one class of their convolution identities.

We define the notion of Darboux integrability for linear second order partial differential operators,

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We then build on certain geometric results of E. Vessiot related to the theory of Monge characteristics to show that the Darboux integrable operators L can be used to obtain a solution of the A2 Toda field theory. This solution is parametrised by four arbitrary functions. The approach presented in this paper thereby represents an alternative means of linearising the A2 Toda equations and may be contrasted with the known linearisation via the Lax pair.

In this paper, we characterise all primitive ideals of the quantised enveloping algebra Uq[sl(2, ℂ)] of the complex Lie algebra sl(2, ℂ) and show how they are similar to those of U[sl(2, ℂ)], the enveloping algebra of sl(2, ℂ).

We show that a Banach space has modulus of convexity of power type p if and only if best approximants to points from straight lines are uniformly strongly unique of order p. Assuming that the space is smooth, we derive a characterisation of the best simultaneous approximant to two elements, and use the characterisation to prove that p–type modulus of convexity implies order p strong unicity of the simultaneous approximant.

Let D be a skew field with uncountable centre K. The main result in the present paper is as follows: If D satisfies a non-trivial generalised power central rational identity, then D is finite dimensional over K. As a corollary we obtain the following result. Let a be an element of D such that (a−1x−1ax)q(x) ∈ K for all x ∈ D ﹨ {0} where q(x) is a positive integer depending on x. Then a ∈ K.

Let M be a finite quasigroup of order n. For any integer k ≥ 2, let H(k, M) be the smallest positive integer h such that there exist h subsets Ai (i = 1, 2, …, h) such that Ai … Ah = M and |Ai| = k for every i = 1, 2, …, h. Define H(k, n) = max H(k, M). It is proved in this paper that

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Every orthogonal array of strength s and of prime-power (or perhaps infinite) order q, has a well-defined collection of ranks r. Having rank r means that it can be constructed as a cone cut by qs hyperplanes in projective space of dimension r over a field of order q.

Constructive, simple proofs for the existence, regularity, continuous dependence and dynamical properties of a repelling invariant curve for a discrete dynamical system of the plane with an attracting fixed point with real eigenvalues are given. These proofs can be used to generate a numerical algorithm to find these curves and to compute explicitly the dependence of the curve with respect to the system.

Two Lockett sections of Fitting classes of supersoluble groups are determined. One of these sections has only one member, a minimal supersoluble nonnilpotent Fitting class.

We extend the results of N.K. Ribarska and A.V. Arhangel'skiĭ to the class of strongly countably complete spaces. And we show another characterisation of Eberlein and Radon-Nikodým compact spaces.

If f(x) is defined on [−1, 1], let H¯1 n(f, x) denote the Lagrange interpolation polynomial of degree n (or less) for f which agrees with f at the n+1 equally spaced points xk, n = −1 + (2k)/n (0 ≤ k ≤ n). A famous example due to S. Bernstein shows that even for the simple function h(x) = │x│, the sequence H¯1 n (h, x) diverges as n → ∞ for each x in 0 < │x│ < 1. A generalisation of Lagrange interpolation is the Hermite-Fejér interpolation polynomial H¯mn (f, x), which is the unique polynomial of degree no greater than m(n + 1) – 1 which satisfies (f, Xk, n) = δo, pf(xk, n) (0 ≤ p ≤ m − 1, 0 ≤ k ≤ n). In general terms, if m is an even number, the polynomials H¯mn(f, x) seem to possess better convergence properties than the H¯1 n (f, x). Nevertheless, D.L. Berman was able to show that for g(x) ≡ x, the sequence H¯2n(g, x) diverges as n → ∞ for each x in 0 < │x│. In this paper we extend Berman's result by showing that for any even m, H¯mn(g, x) diverges as n → ∞ for each x in 0 < │x│ < 1. Further, we are able to obtain an estimate for the error │H¯mn(g, x) – g(x)│.

It is proved that under the following condition, the sum f of the double trigonometric series with coefficients cjk is integrable and the rectangular partial sums smn(f, x, y) converge to f in L1 norm:

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It is shown that if a partially-ordered topological space X admits a finite-point T2-ordered compactification, then it admits a countable T2-ordered compactification if and only if it admits n−point T2-ordered compactifications for all n beyond some integer m.

In this paper, we investigate a general class of variational inequalities. Existence and multiplicity results are obtained by using minimax principles for lower semicontinuous functions due to A. Szulkin.