The main goal of the paper is the construction of a triangular mapping F of the square with zero topological entropy, possessing a minimal set M such that F|M is a strongly chaotic homeomorphism, as well as other properties that are impossible for continuous maps on an interval.

To do this we define a parametric class of triangular maps on Q × I, where Q is an infinite minimal set on the interval, which are extendable to continuous triangular maps F: I2 → I2. This class can be used to create other examples.

We find the spectrum of the Berezin operator T on L∞(Bn), then we show that if f ∈ L∞(Bn) satisfies Sf = rf for some r in the unit circle, where S is any convex combination of the iterations of T, then f is M-harmonic.

Finally we decompose the subspace of L∞(Bn) where lim Tkf exists into the direct sum of two subspaces of L∞(Bn).

In this paper, we study the existence of solutions of generalised variational-like inequality problems by using a generalised form of the Fan-KKM-Theorem. We also introduce a gap function for generalised variational-like inequalities.

We construct Poincaré series and Eisenstein series for automorphic pseudodifferential operators, and show that the space of automorphic pseudodifferential operators associated to cusp forms is generated by Poincaré series. We also obtain explicit formulas for such Poincaré series and Eisenstein series.

Boundedness and compactness of little Hankel operators from H∞ to the Bloch space and the little Bloch space are characterised.

Let E(x) be a monic polynomial over the finite field q of q elements. A formula for the number of n × n matrices θ over q, satisfying E(θ) = 0 is obtained by counting the representations of the algebra q[x]/(E(x)) of degree n. This simplifies a formula of Hodges.

A characterisation is given of some of the parabolics found in Co3, and SP4 (9) using the amalgam method.

For any pair of integers p, q such that (p, q) = 1 and p ≥ 2q, the graph has vertices {0, 1, …, p − 1} and edges {ij: q ≤ |i − j| ≤ p − q}. These graphs play the same role in the study of circular chromatic number as that played by the complete graphs in the study of chromatic number. The graphs share many properties of the complete graphs. However, there are also striking differences between the graphs and the complete graphs. We shall prove in this paper that for many pairs of integers p, q, one may delete most of the edges of so that the resulting graph still has circular chromatic number p/q. To be precise, we shall prove that for any number r < 2, there exists a rational number p/q (where (p, q) = 1) which is less than r but arbitrarily close to r, such that contains a subgraph H with and . This is in sharp contrast to the fact that the complete graphs are edge critical, that is, the deletion of any edge will decrease its chromatic number and its circular chromatic number.

It is shown that meromorphic solutions of certain first-order nonlinear differential equations do not have wandering domains.

The structure of finite quotients of “large” subgroups of the multiplicative group of a finite dimensional division algebra is interesting and is related to the Margulis-Platonov conjecture. We develop machinery to handle such quotients and we conjecture that finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. The proofs rely on Wedderburn's Factorisation Theorem.

In this paper we prove some coincidence point theorems for nonself single-valued and multivalued maps satisfying a nonexpansive condition. These extend fixed point theorems for multivalued maps of a number of authors.

A relationship is pointed out between the results in a recent paper of Shin's and those in a previously published paper by M.J.C. Gover.

We prove generalised results for localised approximation by generalised translation networks. We also show the relationship between the minimum number of neurons in the generalised translation networks with one hidden layer and the desired accuracy where the target functions are in a subset V1, p ([−1, 1]s) of the Sobolev space W1, p([−1, 1]s).

It is known that the order of the isometry group on a compact Riemannian manifold with negative Ricci curvature is finite. We show by local nilpotent structures that a bound on the orders of the isometry groups exists depending only on the Ricci curvature, the conjugate radius and the diameter.

In this work we demonstrate that the lower radical class construction on a homomorphically closed class of associative rings generates a radical class for any class of associative rings. We also give a new description of the upper radical class using the construction on an appropriate generating class.

Let K be a planar, compact, convex set with circumradius R, diameter d, width w and inradius r, and containing no points of the integer lattice. We generalise inequalities concerning the ‘dual’ quantities (2R − d) and (w − 2r) to rectangular lattices. We then use these results to obtain corresponding inequalities for a planar convex set with two interior lattice points. Finally, we conjecture corresponding results for sets containing one interior lattice point.

Let G be a finite connected simple graph. The isomorphism classes of graph bundles and graph coverings over G have been enumerated by Kwak and Lee. Recently, Archdeacon and others characterised bipartite coverings of G and enumerated the isomorphism classes of regular 2p-fold bipartite coverings of G, when G is nonbipartite. In this paper, we characterise bipartite graph bundles over G and derive some enumeration formulas of the isomorphism classes of them when the fibre is a connected bipartite graph. As an application, we compute the exact numbers of the isomorphism classes of bipartite graph bundles over G when the fibre is the path Pn or the cycle Cn.

Let K be a planar convex body containing no points of the integer lattice. We give a new inequality relating the inradius and circumradius of K.