Let $A$ be a locally finite, N-graded, noetherian algebra with a balanced dualizing complex. If $A$ is a Hopf algebra, then $A$ has finite injective dimension.

A very important misprint occurs in the statement of the main theorem of the paper entitled ‘Weak behaviour of Fourier-Neumann series’ Glasgow Math. J. 45 (2003), 97–104. The statement in lines 3–7 of p. 99 should be replaced by the following statement.

Born in London on 14 July 1947, Robert Winston Keith Odoni was the eldest of the five children of Walter Anthony and Lois Marie Theresa Odoni. The family name is Italian: grandfather Alfred Odoni had come to England from Switzerland in the 1920s and set up a manufacturing business in Birmingham, later transferring to London. Robert's father continued the business (then making cycle stands) until the 1970s. His mother was from Boulder in Colorado: they had met in America whilst he was on wartime naval service.

We prove that for a negatively pinched ($-b^2\le\cK\le -1$) topologically tame 3-manifold $\skew5\tilde{M}/\Gamma$, all geometrically infinite ends are simply degenerate. And if the limit set of $\Gamma$ is the entire boundary sphere at infinity, then the action of $\Gamma$ on the boundary sphere is ergodic with respect to harmonic measure, and the Poincaré series diverges when the critical exponent is 2.

We derive variational formulae for natural first order energy functionals and obtain criteria for the stability of isometric immersions. This generalizes known results for the classical energy, the $p$-energy and the exponential energy.

Page 188 in Ramanujan's lost notebook is devoted to a certain class of infinite series connected with Euler's pentagonal number theorem. These series are represented in terms of Ramanujan's famous Eisenstein series $P, Q$, and $R$. The purpose of this paper is to prove all the formulas on page 188 and to show that one of them leads to an interesting, new recurrence formula for $\sigma (n)$, the sum of the positive divisors of $n$.

We define a class of equations that are not amenable but are type K and are therefore solvable over torsion-free groups. Moreover, we show that these new equations are solvable over all groups.

In this paper, a dual version of the Mountain Pass Theorem and the Generalized Mountain Pass Theorem are extended to functions that are locally Lipschitz only. An application involving elliptic hemivariational inequalities is next examined.

We show that surfaces such that the natural projections of the unit normal bundles are harmonic morphisms are composed of minimal points and totally umbilical points. As its application, we find a harmonic map from the torus to the complex quadric in $CP^3$ such that the projection map of the associated sphere bundle constructed by Gudmundsson is not a harmonic morphism. This contrasts sharply with the situation for holomorphic maps. We also establish sufficient conditions for reducing the codimension of an isometric immersion with conformal second fundamental form.

For $p\ge 2$ we introduce the notion of an almost $p$-structure on vector-bundles which generalizes the notion of an almost-complex structure and investigate the existence of almost $p$-structures on spheres and complex projective spaces.

For a subset $\Lambda$ of the dual group of a compact metrizable abelian group, we introduce the type I-, II-, and III-$\Lambda$-Riemann–Lebesgue property of a Banach space. As an application we use these properties to characterize Rajchman sets.

We prove two comparison theorems between the time derivative of a real function $u(x, t)$ such that $u(\cdot,t)$ belongs to L$^1 (\Omega)$ for all $t$, and the time derivative of the vector function $\skew2\hat{u}(t) = u(\cdot, t)$.

We show that the order of an automorphism of an arithmetic Riemann surface of genus $g$ is not greater than $2g\,{-}\,2$, provided $g$ is large enough. This bound is an arithmetic analog of the classical Wiman bound. We prove that it is sharp and attained for any genus but in contrast to the general case the automorphisms of maximal order act without fixed points. This allows us to consider the automorphisms which act on arithmetic Riemann surfaces and have a given number of fixed points. For these automorphisms we describe the asymptotic behavior of their orders.

For a class of second order nonlinear differential equations, sufficient conditions are presented to ensure that some, respectively all solutions are asymptotic to lines.

Using the classification by Dotti and Fino [3] we show the existence of an HKT metric on a neighbourhood of the centre of any 8-dimensional nilpotent Lie group $G$ with invariant hypercomplex structure. This metric exists globally if the hypercomplex structure is abelian, and in these cases we construct an HKT structure on a neighbourhood of the zero section of the cotangent bundle $T^{*}G$ extending the HKT metric on $G$.