A Banach space X is called a twisted sum of the Banach spaces Y and Z if it has a subspace isomorphic to Y such that the corresponding quotient is isomorphic to Z. A twisted Hilbert space is a twisted sum of Hilbert spaces. We prove the following tongue-twister: there exists a twisted sum of two subspaces of a twisted Hilbert space that is not isomorphic to a subspace of a twisted Hilbert space. In other words, being a subspace of a twisted Hilbert space is not a three-space property.

We generalise a result of Hosaka and Kato by proving that if the set of periodic points of a continuous map of a tree is closed then each chain recurrent point is a periodic one. We also show that the topological entropy of a tree map is zero if and only if the w-limit set of each chain recurrent point (which is not periodic) contains no periodic points.

In this article we define symmetric geodesies on conformal compactifications of Euclidean Jordan algebras and classify symmetric geodesics for the Euclidean Jordan algebra of all n × n symmetric real matrices. Furthermore, we show that the closed geodesics for the Euclidean Jordan algebra of all 2 × 2 symmetric real matrices are realised as the torus knots in the Shilov boundary of a Lie ball.

We investigate how varying the parameters of t-(ν, κ, λ) designs affects the sizes of smallest defining sets. In particular, we consider the effect of varying each of the parameters t, ν and λ. We establish a number of new bounds for the sizes of smallest defining sets and find the size of smallest defining sets for an infinite family of designs. We also show how one of our results can be applied to the problem of finding critical sets of Latin squares.

Let L be a possibly infinite-dimensional Lie algebra of maximal class. We show that if L admits the structure of a Lie p-algebra then the dimension of L can be at most p + 1. Furthermore, this bound is best possible.

We obtain a new characterisation of finite representability of operators and present new results about uniformly convexifying, Rademacher cotype and Rademacher type operators on some classical Banach spaces, including JB* -triples and spaces of analytic functions.

Let be a homogeneous tree, o be a fixed reference point in , and be the closed ball of radius N in centred at o. In this paper we characterise the image under the Helgason–Fourier transformation ℋ of , the space of functions supported in , and of , the space of rapidly decreasing functions on . In both cases our results are counterparts of known results for the Helgason–Fourier transformation on noncompact symmetric spaces.

If U, V are the generators of a rational or irrational rotation C*-algebra then an automorphism φ of the algebra is determined by φ(U) = λUaVc and φ(V) = μUbVd where λ, μ are complex numbers of modulus 1 and a, b, c, d are integers with ad − bc = 1. If ad − bc = −1, then these formulae determine an antiautomorphsm of the algebra. The classification of such automorphisms and antiautomorphisms up to conjugacy by arbitrary automorphisms is studied and an almost complete classification is obtained.

In this paper, we extend our theories on the dynamics of transcendental semigroups. Several properties of Fatou and Julia sets of transcendental semigroups will be explored.

We study triangles and cyclic quadrilaterals which have rational area and whose sides form geometric or arithmetic progressions. A complete characterisation is given for the infinite family of triangles with sides in arithmetic progression. We show that there are no triangles with sides in geometric progression. We also show that apart from the square there are no cyclic quadrilaterals whose sides form either a geometric or an arithmetic progression. The solution of both quadrilateral cases involves searching for rational points on certain elliptic curves.

Stowe's Theorem on the stability of the fixed points of a C2 action of a finitely generated group Γ is generalised to C1 actions of such groups on Banach manifolds. The result is then used to prove that if φ is a Cr action on a smooth, closed, manifold M satisfying H1(Γ, Dr−1(M)) = 0, then φ is locally rigid. Here, r ≥ 2 and Dk(M) is the space of Ck tangent vector fields on M. This generalises a local rigidity result of Weil for representations of a finitely generated group Γ in a Lie group.

A new coincidence point theorem is proved for a pair of multivalued mappings operating between G-convex spaces. From this theorem, a generalisation of the classical Fan-Glicksberg fixed point theorem is established.

In this paper, we establish some estimates for the distance between adjacent zeros of the solutions of the first order delay differential equation

and the first order advanced differential equation

where P ∈ C([t0, ∞), [0, ∞)) and τ ∈ R+. Our results improve recent results in the literature.

An extension of the classical summation formula of Plana is obtained. The extension is obtained by using the zeros of a Bessel function of the first kind.

Hoping for a hand-shake between methods from diophantine approximation theory and transcendance theory, we show how zeros estimates from transcendance theory imply Roth's type lemmas (including the product theorem). We also formulate some strong conjectures on lower bounds for linear forms in logarithms of rational numbers with rational coefficients, inspired by the subspace theorem and which would imply, for example, the abc conjecture.

Given two subsets X and Y of ℝ each with at least two points, we describe the surjective linear isometries between the spaces of functions of bounded variation BV(X) and BV(Y): namely, if T : BV(X) → BV(Y) is such an isometry, then there exist α ∈ ℂ, |α| = 1, and a monotonic bijective map h : Y → X such that (Tf)(y) = αf(h(y)) for every f ∈ BV(X) and every y ∈ Y.

We show that for every integer k ≥ 3 and every prime p ≤ 5 there is a group with soluble length k and order .