We show that all torsion free groups which are virtual surface bundle groups of type I in Johnson's trichotomy may be realised by aspherical closed smooth 4-manifolds. (This was already known for type II.)

We characterise the compact composition operators from any Mobius invariant Banach space to VMOA, the space of holomorphic functions on the unit disk U that have vanishing mean oscillation. We use this to obtain a characterisation of the compact composition operators from the Bloch space to VMOA. Finally, we study some properties of hyperbolic VMOA functions. We show that a function is hyperbolic VMOA if and only if it is the symbol of a compact composition operator from the Bloch space to VMOA.

We give a characterisation for the vanishing of the η-invariant of prime order automorphisms of hyperelliptic Riemann surfaces through the mapping torus construction. To this end, we introduce a notion of s-symmetry for finite order surface automorphisms.

In this paper a full classification of piecewise monotonic maps from the point of view of the topological sequence entropy is established.

A well-known result due to S.N. Bernstein is that sequence of Lagrange interpolation polynomials for |x| at equally spaced nodes in [−1, 1] diverges everywhere, except at zero and the end-points. In this paper we present a quantitative version concerning the divergence behaviour of the Lagrange interpolants for |x|3 at equidistant nodes. Furthermore, we present the exact rate of convergence for the interpolatory parabolas at the point zero.

By adding 1 to elements of the nilradical and Jacobson radical of a ring with identity, normal subgroups of the group of units are obtained. In this paper we record observations about complementation of these subgroups in the group of units of a ring, identifying large classes where complementation takes place and some examples where it fails.

We study the structure of function fields of plane quartic curves by using projections. Taking a point P ∈ ℙ2, we define the projection from a curve C to a line l with the centre P. This projection induces and extension field k (C)/k (ℙ1). By using this fact, we study the field extension k (C)/k (ℙ1) from a geometrical point of view. In this note, we take up quartic curves with singular points.

Weak solutions of the degenerate elliptic differential equation Lu := −div(A (x)∇u)+b·∇u+Vu = f, with |b|2ω−1, V, f in some appropriate function spaces, will be shown to be Hölder continuous.

Let E/F be a totally real quadratic extension of a totally real algebraic number field. The author has in an earlier paper considered automorphic forms defined with respect to a quaternion algebra BE over E and a theta lift from such quaternionic forms to Hilbert modular forms over F. In this paper we construct adelic forms in the same setting, and derive explicit formulas concerning the action of Hecke operators. These formulas give an algebraic foundation for further investigations, in explicit form, of the arithmetic properties of the adelic forms and of the associated zeta and L-functions.

This paper presents two properties of the Julia set of a random iteration system of rational functions, which are similar to the well-known results in the classical case.

In this paper, we focus our attention on the extrinsic shape of geodesics on geodesic spheres in a given Riemannian manifold M. We characterise real and complex space forms in the class of smooth Riemannian manifolds from this point of view.

We prove that a Banach space X has the Radon-Nikodym property if, and only if, every weak*-lower semicontinuous convex continuous function f of X* is Gâteaux differentiable at some point of its domain with derivative in the predual space x.

Let M be a left R-module and  be an M-natural class with some additional conditions. It is proved that every direct sum of M-injective left R-modules in  is  -continuous (-quasi-continuous) if and only if every direct sum of M- injective left R-modules in  is M-injective.

A new measure of weak noncompactness is introduced. A logarithmic convexity-type result on the behaviour of this measure applied to bounded linear operators under real interpolation is proved. In particular, it gives a new proof of the theorem showing that if at least one of the operators T: Ai → Bi, i = 0, 1 is weakly compact, then so is T : Aθ,p → Bθ,p for all 0 < θ < 1 and 1 < P < ∞.

We consider tensor product operators and discuss their best constants in preservation inequalities concerning the usual moduli of continuity. In a previous paper, we obtained lower and upper bounds on such constants, under fairly general assumptions on the operators. Here, we concentrate on the l∞-modulus of continuity and three celebrated families of operators. For the tensor product of k identical copies of the Bernstein operator Bn, we show that the best uniform constant coincides with the dimension k when k ≥ 3, while, in case k = 2, it lies in the interval [2, 5/2] but depends upon n. Similar results also hold when Bn is replaced by a univariate Szász or Baskakov operator. The three proofs follow the same pattern, a crucial ingredient being some special properties of the probability distributions involved in the mentioned operators, namely: the binomial, Poisson, and negative binomial distributions.

Properties of minimal idempotents in contractible and reflexive amenable Banach algebras are exploited to prove that such a kind of Banach algebra is finite demensional if each maximal ideal is contained in a maximal left or a maximal right ideal that is complemented as a Banach subspace. This result covers several known results on this subject.

It is shown that for each point P in the interior of the polynomial hull of a disc fibration X over the unit sphere ∂n there exists an H∞ analytic disc with boundary in X and passing through p.

This paper proves that every direct summand N of a direct sum of indecomposable injective submodules of a module is the sum of a direct sum of indecomposable injective submodules and a sum of indecomposable injective submodules of Z2(N).

We give a necessary and sufficient condition for the existence of an Edwards-Walsh resolution of a complex. Our theorem is an extension of Dydak-Walsh's theorem to all simplicial complexes of dimension ≥ n + 2. We also determine the structure of an Abelian group with the Edwards-Walsh condition, (which was introduced by Koyama and the author).

In a recent paper I have shown that Carlitz's q-Bernoulli number can be represented as an integral by the q-analogue μq of the ordinary p-adic invariant measure. In the p-adic case, J. Satoh could not determine the generating function of q-Bernoulli numbers. In this paper, we give the generating function of q-Bernoulli numbers in the p-adic case.