An s-arc in a graph is a vertex sequence (α0,α1,…,αs) such that {αi−1,αi} ∈ EΓ for 1 [les ] i [les ] s and αi−1 ≠ αi+1 for 1 [les ] i [les ] s − 1. This paper gives a characterization of a class of s-transitive graphs; that is, graphs for which the automorphism group is transitive on s-arcs but not on (s + 1)-arcs. It is proved that if Γ is a finite connected s-transitive graph (where s [ges ] 2) of order a p-power with p prime, then s = 2 or 3; further, either s = 3 and Γ is a normal cover of the complete bipartite graph K2m,2m, or s = 2 and Γ is a normal cover of one of the following 2-transitive graphs: Kpm+1 (the complete graph of order pm+1), K2m,2m − 2mK2 (the complete bipartite graph of order 2m+1 minus a 1-factor), a primitive affine graph, or a biprimitive affine graph. (Finite primitive and biprimitive affine 2-arc transitive graphs were classified by Ivanov and Praeger in 1993.)

The distribution on the torus ℝ/ℤ of a set of fractions of the form

(formula here)

is investigated, where q is a large integer, m is the inverse of m modulo q, R(x) is a rational function defined modulo q, and [Uscr ], [Mscr ], [Nscr ] are subsets of {1,…,q}. Under some natural assumptions, it is shown that the set [Rscr ] is uniformly distributed on ℝ/ℤ.

We prove that the mod [pfr ]-reduction of a Shimura subvariety of the moduli space of abelian varieties has a Zariski dense ordinary locus for almost all height 1 primes [pfr ] of the reflex field. We show further that the ordinary locus is empty at primes which are not of height 1.

In this paper, what is already known about defect 2 blocks of symmetric groups is used to deduce information about the corresponding blocks of Schur algebras. This information includes Ext-quivers and decomposition numbers, as well as Loewy structures of the Weyl modules, principal indecomposable modules and tilting modules.

It is shown that for each compactly generated totally disconnected locally compact group G, there is a finite number of prime numbers, p1, p2, …, pn, such that the scale function s : G → N satisfies s(x) = p1a1(x)p2a2(x)…, pnan(x), where x ∈ G.

Let φ be an analytic function from [ ] to the symmetrized bidisc

(formula here)

We show that if φ(0) = (0,0) and φ(λ) = (s, p) in the interior of Γ, then

(formula here)

Moreover, the inequality is sharp: we give an explicit formula for a suitable φ in the event that the inequality holds with equality. We show further that the inverse hyperbolic tangent of the left-hand side of the inequality is equal to both the Caratheodory distance and the Kobayashi distance from (0,0) to (s, p) in int Γ

We establish the peak point conjecture for uniform algebras generated by smooth functions on two-manifolds: if A is a uniform algebra generated by smooth functions on a compact smooth two-manifold M, such that the maximal ideal space of A is M, and every point of M is a peak point for A, then A = C(M). We also give an alternative proof in the case when the algebra A is the uniform closure P(M) of the polynomials on a polynomially convex smooth two-manifold M lying in a strictly pseudoconvex hypersurface in Cn.

The dual of a reflexive separable hereditarily indecomposable complex Banach space of Gowers and Maurey admits no chaotic continuous linear operator in the sense of Devaney defined on it.

We prove uniqueness for the nonharmonic trigonometric series [sum ]∞k=0akeiλkx under the weaker condition (*) where [sum ]∞k=0 [mid ]ak[mid ]/exp[mid ]λk[mid ]γ < ∞, for some 0 < γ < 1. In other words, if {λk}∞0 satisfies the above condition (*), and if [sum ]∞k=0akeiλkx = 0, then ak = 0 for all k = 0, 1,…. Finally, we state an improvement of Zygmund's uniqueness result as a corollary.

Let M be a simply connected closed manifold of dimension greater than 4 which does not admit a metric with positive scalar curvature. We give necessary conditions for M to admit a scalar-flat metric. These conditions involve the first Pontrjagin class and the cohomology ring of M. As a consequence, any simply connected scalar-flat manifold of dimension greater than 4 with vanishing first Pontrjagin class admits a metric with positive scalar curvature. We also describe some relations between scalar-flat metrics, almost complex structures and the free loop space.

We present a simple example of an integrable function for which the integral is not differentiable almost everywhere in the strong sense. A first example of such a function was given by S. Saks in 1935. Our construction is considerably more simple, due to the use of a remarkable theorem of P. X. Gallagher.

In a previous paper, the author proved a recent Wiener–Hopf formula, established by Alili and Doney, by means of a path transform. We derive here two other results using this path transform.

We study closed subsets in the plane which intersect each line in at least m points and at most n points, for which we try to minimize the difference n – m. It is known that m cannot be equal to n. The results in this paper show that for every even number n there exist closed sets in the plane for which m = n – 2.

Brian Hartley began his algebraic career as one of Philip Hall's research students in Cambridge. He obtained his Ph.D. in 1964, spent two post-doctoral years in the USA and, on his return to the United Kingdom, accepted a lectureship in the newly established Mathematics Department at Warwick University; there he was promoted to a readership in 1973. He was appointed to a chair of pure mathematics at the University of Manchester in 1977 and was Head of the Mathematics Department there from 1982–4. He was elected to the London Mathematical Society in 1968 and served on Council from 1987–9. He won an EPSRC Senior Research Fellowship, but died on 8 October 1994, a few days after taking it up. He travelled widely and took a lively interest in other cultures and languages. His intellectual energy, enthusiasm for algebra, direct manner and dry sense of humour endeared him to the many mathematical friends he made around the world. He was devoted to mathematics and gave generously of his time and energy in support of younger colleagues.