Volume 11, Issue 1

We examine properties of the right ideal structure of right distributive domains. Right distributive domains R are exactly those rings whose localizations at maximal right ideals M are right chain domains R M - On the one hand, the paper focuses on the question in which way properties of R are carried over to R M and vice versa. We examine the problem under which conditions two-sided ideals of R are again two-sided in the extension R M (Lemma 2.2). Further, we observe the relationship between completely prime resp. semiprime ideals of R and the extended ideals in R M . On the other hand, we prove in particular that for any maximal right ideal M = R\S M the right-S M -saturation of a completely semiprime ideal I M of R is completely prime (Theorem 2.9). A central role is played by waists of right distributive rings which are right ideals comparable to each other ideal, in particular there exists a largest waist W which is completely prime. We present a representation theorem in terms of ideals in R W . We apply these results to the Jacobson radical J(R) of a right distributive domain R. Illustrative examples are given.

The main object of the paper is to study the distance between Banach spaces introduced by Kadets. For Banach spaces X and Y, the Kadets distance is defined to be the infimum of the HausdorlT distance d{B x , B y ) between the respective closed unit balls over all isometric linear embeddings of X and Y into a common Banach space Z. This is compared with the Gromov-Hausdorff distance which is defined to be the infimum of d(B x , B y ) over all isometric embeddings into a common metric space Z. We prove continuity type results for the Kadets distance including a result that shows that this notion of distance has applications to the theory of complex interpolation.

It is proved that a virtually free pro-p group G having a free pro-p subgroup of index p satisfies a pro-p version of the Dyer-Scott structure theorem (Comm. Alg. 3(3) (1975), 195-201). The pro-2 case had been settled by W. Herfort and P. Zalesskii in (manuscr. math. 93, 457-464 (1997)). A proof for (topologically) finitely generated G has been given by C. Scheiderer. A consequence of our result is that for any automorphism of order p" of a free pro-p group its fixed point group is a free factor. The main theorem generalizes Serre’s well known result, stating that any virtually free torsion free pro-p group is free pro-p (Topology 3, 413-420, (1965)).

On an arbitrary compact Riemann surface, necessary and suflicient conditions are found for the existence of semistable vector bundles with slope between zero and one and a prescribed number of linearly independent holomorphic sections. Existence is achieved by minimizing the Yang-Mills-Higgs functional.

This paper concerns 4-dimensional (topological locally compact connected) elation Laguerre planes that admit large automorphism groups. In particular, it is shown that such a plane is classical if its automorphism group is at least 11-dimensional. Furthermore, the elation Laguerre planes admitting simple Lie groups of automorphisms are investigated and various characterizations of the classical complex Laguerre plane and the semi-classical Laguerre planes are obtained.

The logarithmic Sobolev inequality is developed as a geometric asymptotic estimate With respect to Lebesgue measure. Two short geometric arguments are given to derive (1) the logarithmic Sobolev inequality from the isoperimetric inequality and (2) Nash’s inequality With an asymptotically sharp constant from the logarithmic Sobolev inequality. In addition, the Fisher information form of the logarithmic Sobolev inequality is obtained directly from the isoperimetric inequality. A new formulation of the logarithmic Sobolev inequality is given for hyperbolic space which can be interpreted as an “uncertainty principle” in this setting.