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 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.

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 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.

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 give direct proofs of meromorphic continuality on the whole complex plane of certain zeta functions ZP, Q (s) and Z (P/Q, s) associated with a pair of polynomials P, Q. We calculate ZP, Q (−q) (q a non-negative integer) and give explicit formulas for the residues of Z (P/Q, s) at poles.

We prove that each Polish space is homeomorphic to the ω–limit set of a bounded solution of an autonomous equation x′ = f (x) in l2 × c0, in which f is Lipschitz continuous on bounded sets.

In this paper, we count two-block configurations in a general setting. In particular, no restriction is put on the pair repetition factor (that is, the parameter Λ) of the block designs being considered. Besides giving formulae for the counts of the two-block BTD (V, B, R, 3, Λ) configuration classes, a basis for the formulae is given and shown to have size four. Details of the special cases where Λ equals two and three are also presented, along with 2-block BIBD (v, b, r, 3, λ) configuration results.

We discuss the problem of finding an analogue of the concept of topological space in supergeometry, motivated by the search for a procedure to compactify supermanifolds along odd coordinates. In particular, we examine the topologies arising naturally on the sets of points of locally ringed superspaces, and show that in the presence of a nontrivial odd sector such topologies are never compact. The main outcome of our discussion is the following new observation: not only the usual framework of supergeometry (the theory of locally ringed spaces), but the more general approach of the functor of points, need to be further enlarged.

By using various kinds of moduli of smoothness, it is established that the multivariate variants of the well-known singular integrals of Picard, Poisson-Cauchy, Gauss-Weierstrass and their Jackson-type generalisations satisfy the “global smoothness preservation” property. The results are extensions of those proved by the authors for the univariate case.

In this paper potential symmetries are sought of the inhomogeneous nonlinear diffusion equations ut = x1−M[xN−1f (u) ux]x. The functional forms of f (u) that admit such symmetries are completely classified. A complete list is presented of the symmetries, which depend on the values of the parameters M and N. We give examples of similarity solutions using potential symmetries. In some cases, the potential symmetries enable us to convert non–invertible mappings of nonlinear partial differential equations to linear ones.