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.)

This paper proves that a difference field (E, σ) admits quantifier elimination if and only if E is an algebraically closed field, and σ is an integer power of the Frobenius automorphism.

Let [ ]P2,p (where 2 < p) be the class of all finite nilpotent groups of class 2 and of exponent p with an additional predicate for a subgroup of the centre that contains the commutator subgroup. The Fraïssé limit D of this class exists. Non-forking is described for Th(D).

Let F be a real quadratic field, and let R be an order in F. Suppose given, a polarized abelian surface (A; λ), defined over a number field k, with a symmetric action of R defined over k. This paper considers varying A within the k-isogeny class of A to reduce the degree of λ and the conductor of R. It is proved, in particular, that there is a k-isogenous principally polarized abelian surface with an action of the full ring of integers of F, when F has class number 1 and the degree of λ and the conductor of R are odd and coprime.

We give a survey on the use of Eva Kallin's lemma. This lemma gives a condition on two polynomially convex sets in [Copf ]n under which their union is polynomially convex. This result has proved to be a useful tool in different areas of complex function theory of several variables, for instance in the study of polynomial convexity of the union of totally real surfaces, and in approximation problems in function algebras.

In this paper we give two different proofs that the flat cover conjecture is true: that is, every module has a flat cover. The two proofs are of completely different nature, and, we hope, will have different applications. The first of the two proofs (due to the third author) is essentially an application of the work of P. Eklof and J. Trlifaj (work which is more set-theoretic). The second proof (due to the first two authors) is more direct, and has a model-theoretic flavour.

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 fundamental group of the double of the figure-eight knot exterior admits a faithful discrete representation into SO(4,1; R), for which the image group is separable on its geometrically finite subgroups.

Let V be a commutative valuation domain of arbitrary Krull-dimension (rank), with quotient field F, and let K be a finite Galois extension of F with group G, and S the integral closure of V in K. If, in the crossed product algebra K [midast ] G, the 2-cocycle takes values in the group of units of S, then one can form, in a natural way, a ‘crossed product order’ S [midast ] G ⊆ K [midast ] G. In the light of recent results by H. Marubayashi and Z. Yi on the homological dimension of crossed products, this paper discusses necessary and/or sufficient valuation-theoretic conditions, on the extension K/F, for the V-order S [midast ] G to be semihereditary, maximal or Azumaya over V.

This paper proves a conjecture of G. A. Jones, D. Singerman and K. Wicks, that a suborbital graph for the modular group is a forest if and only if it contains no triangles.

Any doubly stochastic finite matrix with integer entries is the sum of permutation matrices. A lower bound on how many of the permutations are equal is obtained.

A minimal surface of general type with pg(S) = 0 satisfies 1 [les ] K2 [les ] 9, and it is known that the image of the bicanonical map φ is a surface for K2S [ges ] 2, whilst for K2S [ges ] 5, the bicanonical map is always a morphism. In this paper it is shown that φ is birational if K2S = 9, and that the degree of φ is at most 2 if K2S = 7 or K2S = 8.

By presenting two examples of surfaces S with K2S = 7 and 8 and bicanonical map of degree 2, it is also shown that this result is sharp. The example with K2S = 8 is, to our knowledge, a new example of a surface of general type with pg = 0.

The degree of φ is also calculated for two other known surfaces of general type with pg = 0 and K2S = 8. In both cases, the bicanonical map turns out to be birational.

The most natural candidates to the definition of equisingularity of germs of complex space curves are discussed, showing that they coincide in the case of Lagrangean curves (projectivizations of conic Lagrangean varieties of a symplectic manifold of dimension four).

A construction is given of an infinite family of finite self-complementary, vertex-transitive graphs which are not Cayley graphs. To the authors' knowledge, these are the first known examples of such graphs. The nature of the construction was suggested by a general study of the structure of self-complementary, vertex-transitive graphs. It involves the product action of a wreath product of permutation groups.

The Stöhr–Voloch approach is used to obtain a new bound for the number of solutions in (Fq)2 of an equation f(X, Y) = 0, where f(X, Y) is an absolutely irreducible polynomial with coefficients in a finite field Fq.

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.

An alternative proof is given of a result, originally due to Guido Mislin, giving necessary and sufficient conditions for the inclusion of a subgroup to induce an isomorphism in mod p cohomology.

The asymptotic probability theory of conjugacy classes of the finite general groups leads to a probability measure on the set of all partitions of natural numbers. A simple method of understanding these measures in terms of Markov chains is given in this paper, leading to an elementary probabilistic proof of the Rogers–Ramanujan identities. This is compared with work on the uniform measure. The main case of Bailey's lemma is interpreted as finding eigenvectors of the transition matrix of a Markov chain. It is shown that the viewpoint of Markov chains extends to quivers.

We derive an identity connecting a theta function and a sum of Lambert series. As a consequence of this identity, we deduce a number of results of Jacobi, Dirichlet, Lorenz, Ramanujan and Rademacher.

The paper studies an isoperimetric problem for the Gaussian measure and coordinatewise symmetric sets. The notion of boundary measure corresponding to the uniform enlargement is considered, and it is proved that symmetric strips or their complements have minimal boundary measure.