A poset can be regarded as a category in which there is at most one morphism between objects, and such that at most one of Hom(c, c′) and Hom(c′, c) is nonempty for c 6= c′. If we keep in place the latter axiom but allow for more than one morphism between objects, we have a sort of generalized poset in which there are multiplicities attached to the covering relations, and possibly nontrivial au...

The goal of this paper is twofold. First, it consists of an introduction to the basic features of hyperbolic geometry, and the geometry of an important class of functions of the hyperbolic plane, isometries. Second, it identifies a group structure in the set of isometries, specifically those that preserve orientation, and deals with the topological properties of their discrete subgroups. In the...

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In this paper we introduce a new functor H ab(K,G) from the category of connected reductive groups over a field K of characteristic 0 to abelian groups. We call H ab(K,G) the first abelian Galois cohomology group of G. The abelian group H ab(K,G) is related to the pointed setH (K,G) by the abelianization map ab:H(K,G)→ H ab(K,G). When K is a local field or a number field, the map ab 1 is surjec...

We construct a compact nonpositively curved squared 2-complex whose universal cover contains a flat plane that is not the limit of periodic flat planes. AMS Classification 20F67; 20F06

In this paper, we prove the existence of Hermitian-Einstein metrics for holomorphic vector bundles on a class of complete Kähler manifolds which include Hermitian symmetric spaces of noncompact type without Euclidean factor, strictly pseudoconvex domains with Bergman metrics and the universal cover of Gromov hyperbolic manifolds etc. We also solve the Dirichlet problem at infinity for the Hermi...

Let H denote the set {f1, f2, ..., fn}, 2 the collection of all subsets of H and F ⊆ 2 be a family. The maximum of |F| is studied if any k subsets have a non-empty intersection and the intersection of any l distinct subsets (1 ≤ k < l) is empty. This problem is reduced to a covering problem. If we have the conditions that any two subsets have a non-empty intersection and the intersection of any...