In 1996, Cromwell and Nutt conjectured that α(L) − c (L) = 2 for a link L if and only if L is alternating. In this paper we calculate that α(L) = c (L) for some non-alternating pretzel links L, define a new invariant ρ(L) of adequate links L and show that for each non-negative integer n, there is a prime adequate knot K such that α(K) − c (K) = −2n. We conjecture that α(L) − c (L) = 2ρ(L) for any adequate link L.

For transcendental meromorphic functions of finite order, we prove that there exist iterated orbits which tend to the Borel directions. This gives a relation between the value distribution theory and the iteration theory of meromorphic functions.

We discuss Poisson transforms which carry sections of certain vector bundles to mixed automorphic forms, and identify vector bundles whose sections are liftings of holomorphic forms on families of Abelian varieties via Poisson transforms.

We consider a space X of polynomial type and a self-adjoint operator on L2(X) which is assumed to have a heat kernel satisfying second-order Gaussian bounds. We prove that any power of the operator has a heat kernel satisfying Gaussian bounds with a precise constant in the Gaussian. This constant was previously identified by Barbatis and Davies in the case of powers of the Laplace operator on RN. In this case we prove slightly sharper bounds and show that the above-mentioned constant is optimal.

We consider continuous time and discrete time Halanay-type inequalities for nonautonomous scalar systems with discrete and distributed delays. The results obtained generalise the existing results of Halanay and improve certain results of Baker and Tang. Furthermore, it is shown that the discrete time inequalities which are analogues of continuous time inequalities preserve the stability conditions corresponding to the continuous time Halanay-type inequalities.

H. Mohebi and M. Radjabalipour raised a conjecture on the invariant subspace problem in 1994. In this paper, we prove the conjecture under an additional condition, and obtain an invariant subspace theorem on subdecomposable operators.

We give a necessary and sufficient condition for a set of elements to be a generating set of a quotient group F/N, where F is the free group of rank n and N is a normal subgroup of F. Birman's Inverse Function Theorem is a corollary of our criterion. As an application of this criterion, we give necessary and sufficient conditions for a set of elements of the Burnside group B (n,p) of exponent p and rank n to be a generating set.

We establish an existence theorem for weak saddle points of a vector valued function by making use of a vector variational inequality and convex functions.

In this paper we modify the classical Gröbner basis technique and prove the existence of a characteristic polynomial in two variables associated with a finitely generated module over a Weyl algebra. We determine invariants of such a polynomial and show that some of the invariants are not carried by the Bernstein dimension polynomial of the module.

For a given nonlinear partial differential equation defined on a bounded domain with irregular boundary, the available analytical tools are very limited in relation to the study of positive solutions. In this paper wer first use weak convergence methods to show that for an elliptic equation of a certain type, classical positive solutions on nearby smooth domains approach a generalised positive solution on the given domain. The idea is then applied to sublinear elliptic problems to obtain existence and uniqueness results.

We consider 3-dimensional conformally flat hypersurfaces of E4 with constant Gauss-Kronecker curvature. We prove that those with three different principal curvatures must necessarily have zero Gauss-Kronecker curvature.

Let G be a finitely generated soluble group. The main result of this note is to prove that G is nilpotent-by-finite if, and only if, for every pair X, Y of infinite subsets of G, there exist an x in X, y in Y and two positive integers m = m (x,y), n = n (x, y) satisfying [x, nym] = 1. We prove also that if G is infinite and if m is a positive integer, then G is nilpotent-by-(finite of exponent dividing m) if, and only if, for every pair X, Y of infinite subsets of G, there exist an x in X, y in Y and a positive integer n = n (x,y) satisfying [x,nym] = 1.

A ring R with unity is called a (quasi-) Baer ring if the left annihilator of every (left ideal) nonempty subset of R is generated (as a left ideal) by an idempotent. Armendariz has shown that if R is a reduced PI-ring whose centre is Baer, then R is Baer. We generalise his result by considering the broader question: when does the (quasi-) Baer condition extend to a ring from a subring? Also it is well known that a regular ring is Baer if and only if its lattice of principal right ideals is complete. Analogously, we prove that a biregular ring is quasi-Baer if and only if its lattice of principal ideals is complete.

Given a variety ν and ν-algebras A and B, an algebraic formationF: A ⇉ B is a ν-homomorphism FL R × A → B, for some ν-algebra R, and the resulting functions F (r,-): A → B for r ∈ R are termed formable. Firstly, as motivation for the study of algebraic formations, categorical formations and their relationship with natural transformations are explained. Then, formations and formable functions are described for some common varieties of algebras, including semilattices, lattices, groups, and implication algebras. Some of their general properties are investigated for congruence modular varieties, including the description of a uniform congruence which provides information on the structure of B.

We show that induction of covariant representations for C*-dynamical systems is natural in the sense that it gives a natural transformation between certain crossed-product functors. This involves setting up suitable categories of C*-algebras and dynamical systems, and extending the usual constructions of crossed products to define the appropriate functors. From this point of view, Green's Imprimitivity Theorem identifies the functors for which induction is a natural equivalence. Various special cases of these results have previously been obtained on an ad hoc basis.

Let Hn be the Heisenberg group of dimension 2n + 1. Let ℒ1,…,ℒn be the partial sub-Laplacians on Hn and T the central element of the Lie algebra of Hn. We prove that the operator m (ℒ1,…,ℒn,−iT) is bounded on Lp (Hn), 1 < p < +∞, if the function m satisfies a Marcinkiewicz-type condition in Rn+1.

In this note, we find all solutions of the diophatine equation x2 + 3m = yn, where (x, y, m, n) are non-negative integers with x ≠ 0 and n ≥ 3.

A variety of fixed point results are presented for weakly sequentially upper semicontinuous maps. In addition an existence result is established for differential equations in Banach spaces relative to the weak topology.

We introduce a property formally weaker than weak uniform rotundity, which we call equatorial weak uniform rotundity. We show that an equatorially weakly uniformly rotund norm need not be weakly locally uniformly rotund. Nevertheless, we show that an equatorially weakly uniformly rotund Banach space is an Asplund space.

We show how some results of the theory of iterated function systems can be derived from the Tarski–Kantorovitch fixed–point principle for maps on partialy ordered sets. In particular, this principle yields, without using the Hausdorff metric, the Hutchinson–Barnsley theorem with the only restriction that a metric space considered has the Heine–Borel property. As a by–product, we also obtain some new characterisations of continuity of maps on countably compact and sequential spaces.