Volume 17, Issue 2

The second main theorem in Nevanlinna theory is proved when targets are rational functions. We use Ahlfors’ theory of covering surfaces for a proof.

Let T be a right chain ring with non-zero maximal ideal J . In this paper we study subrings R of T containing J and determine conditions for R to be a right distributive (right Bezout) ring. As a consequence we obtain a structure theorem for semiprime right distributive (right Bezout) rings with non-zero left waists.

The article proves that there are smooth one fixed point actions of the alternating group of degree 5 on the 8-dimensional sphere. It follows that a sphere has a smooth one fixed point action of some finite group if and only if the dimension of the sphere is greater than or equal to 6.

Here we describe the locally Cohen-Macaulay pure one-dimensional non-degenerate subschemes Y ⊂ P 3 without trisecant lines or with only finitely many trisecant lines. If Y has no trisecant line, then either it is a rational normal curve or it is the complete intersection of two quadric surfaces.

We consider a sequence of vaguely convergent measures μ n = σ n dx → σ dx = μ on ℝ d and a sequence of symmetric Dirichlet forms { ℰ n }, ℰ n ( f, g ) = ∑ d i,j=1 ∫ ℝ d a n i,j ∂ i f ∂ j g dx , where every ℰ n is defined on L 2 ( σ n dx ). We apply a functional analytic theory of Mosco convergence on changing L 2 -spaces recently developed by K. Kuwae and T. Shioya and obtain some new results about convergence of { ℰ n } and weak convergence of finite dimensional distributions of associated stochastic processes.

A study is made of real Lie algebras admitting compatible complex and product structures, including numerous 4-dimensional examples. Any Lie algebra ɡ with such a structure is even-dimensional and its complexification has a hypercomplex structure. In addition, ɡ splits into the direct sum of two Lie subalgebras of the same dimension, and each of these is shown to have a left-symmetric algebra (LSA) structure. Interpretations of these results are obtained that are relevant to the theory of both hypercomplex and hypersymplectic manifolds and their associated connections.

For maps f, g : X → Y  between closed orientable n -manifolds, we investigate conditions for which N(f, g)  = R(f, g) where N(f, g) and R(f, g) denote the Nielsen and Reidemeister coincidence numbers respectively. In particular, we give necessary and suffcient conditions for the equality to hold when Y is a solvmanifold or when both X and Y are infrasolvmanifolds.

We characterize holomorphic convexity of a complex space X with non trivial holomorphic functions in terms of 1-convexity of its hypersurfaces and H 1 ( X , O X ).

In this paper we study the relation between the Lefschetz number and the Nielsen number of an Anosov diffeomorphism on a flat manifold. As a first result we obtain that for each n ≥ 4 and each k satisfying 2 ≤ k ≤ n − 2, there exists a flat n -dimensional manifold M having first Betti number b 1 (M) = k and admitting an Anosov diffeomorphism f on M with N ( f ) ≠ | L ( f )|. On the other hand, in almost all cases one can also construct on the same manifold M an Anosov diffeomorphism g with N ( g ) = | L ( g )|. Analogous results are obtained in the case of primitive flat manifolds M , i.e. with b 1 ( M ) = 0. Since flat manifolds with b 1 ( M ) = 1 or b 1 ( M ) = n − 1 admit no Anosov diffeomorphisms, and the class of flat manifolds with b 1 ( M ) = n consists entirely of tori, a complete picture is obtained.