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.

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.

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.

In this paper, the strongly Cohen–Macaulay ideals of second analytic deviation one are characterized in terms of the depth properties of the powers of the ideal in the ‘standard range’. This provides an explanation of the behaviour of certain ideals that have appeared in the literature.

Let [Gfr ] be a finite-dimensional semisimple Lie algebra over the complex numbers. Let A be the finite-dimensional algebra of a (regular or singular) block of the BGG-category [Oscr ] . By results of Soergel, A has a combinatorial description in terms of a subalgebra C0 of the coinvariant algebra C. König and Mazorchuk have constructed an embedding from C0-mod into the category [Fscr ](Δ) of A-modules having a Verma flag. This is the main tool for the classification of [Fscr ] (Δ) into finite, tame and wild representation types presented here. As a consequence a classification of A-mod into finite, tame and wild representation types is obtained, thus re-proving a recent result of Futorny, Nakano and Pollack.

This paper improves a theorem of Bergweiler on the zeros of functions f ○ g — R with f meromorphic, g entire and R rational. The proof yields a simple inequality for the number of zeros and T(r, g).

It is shown that, for λ > ω, λ-orthogonality classes in locally λ-presentable categories coincide with full subcategories that are closed under limits and under λ-directed colimits. This comes as a surprise, since the statement is known to be false for λ = ω.

It is known that, if f is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set, J(f), are equal. In this paper it is shown that, for a hyperbolic transcendental meromorphic function f, the packing and upper box dimensions of J(f) are equal, but can be strictly greater than the Hausdorff dimension of J(f).

Let f be analytic in the unit disc, and let it belong to the Hardy space Hp, equipped with the usual norm ∥f∥p. It is known from the work of Hardy and Littlewood that for q > p, the constants

[formula here]

with the usual extension to the case where q = ∞, have C(p, q) < ∞ .The authors prove that

[formula here]

and

[formula here]

If we consider the 14-sided hyperbolic polygon of Felix Klein that defines his famous surface of genus 3, then we observe that we have a uniform, unifacial dessin whose automorphism group is transitive on the edges, but not on the directed edges of the dessin. We show that Klein's surface is the unique platonic surface with this property.

Let X be a reflexive Banach space, and let C ⊂ X be a closed, convex and bounded set with empty interior. Then, for every δ > 0, there is a nonempty finite set F ⊂ X with an arbitrarily small diameter, such that C contains at most δ · |F| points of any translation of F. As a corollary, a separable Banach space X is reflexive if and only if every closed convex subset of X with empty interior is Haar null.

The weak compactness of the composition operator CΦ(f) = f ○ Φ acting on the uniform algebra of analytic uniformly continuous functions on the unit ball of a Banach space with the approximation property is characterized in terms of Φ. The relationship between weak compactness and compactness of these composition operators and general homomorphisms is also discussed.

This paper presents explicit constructions of universal ℝ-trees as certain spaces of functions, and also proves that a 2ℵ0-universal ℝ-tree can be isometrically embedded at infinity into a complete simply connected manifold of negative curvature, or into a non-abelian free group. In contrast to asymptotic cone constructions, asymptotic spaces are built without using the axiom of choice.

It is proved that the cone length or strong category of a product of two co-H-spaces is less than or equal to two. This yields the following positive solution to a problem of Ganea. Let α ∈ π2p(S3) be an element of order p, p a prime [ges ] 3, and let X(p) = S3∪αe2p+1. Then X(p) × X(p) is the mapping cone of some map φ : Y → Z where Z is a suspension.

The paper considers the asymptotic behaviour of the heat content of a riemannian manifold M, when Dirichlet conditions are imposed on a small compact subset of M.

Explicit relationships are given connecting ‘almost’ isospectral Sturm–Liouville problems with eigen-value dependent, and independent, boundary conditions, respectively. Application is made to various direct and inverse spectral questions.