We prove that a non linearly compact valuation domain R admits a proper immediate extension S. This is the main point of Kaplansky’s classical result that a valuation domain is linearly compact if and only if it is maximal. In fact, Kaplansky’s original proof, as well as later versions of it, do not show that R and S have the same residue field.

We survey and extend recent work on integrally closed overrings of two-dimensional Noetherian domains, where such overrings are viewed as intersections of valuation overrings. Of particular interest are the cases where the domain can be represented uniquely by an irredundant intersection of valuation rings, and when the valuation rings can be chosen from a Noetherian subspace of the Zariski-Rie...

Let V (resp. D) be a valuation domain (resp. SFT Prüfer domain), I a proper ideal, and V̂ (resp. D̂) be the I-adic completion of V (resp. D). We show that (1) V̂ is a valuation domain, (2) Krull dimension of V̂ = dimV I+1 if I is not idempotent, V̂ ∼= V I if I is idempotent, (3) dim D̂ = dimD I + 1, (4) D̂ is an SFT Prüfer ring, and (5) D̂ is a catenarian ring. Throughout this paper, all rings are assu...

We endow the collection of ω-bifinite domains with the structure of a probability space, and we will show that in this space the collection of all universal domains has measure 1. For this, we present a probabilistic way to extend a finite partial order by one element. Applying this procedure iteratively, we obtain an infinite partial order. We show that, with probability 1, the cpo-completion ...

a cartan manifold is a smooth manifold m whose slit cotangent bundle 0t *m is endowed with a regularhamiltonian k which is positively homogeneous of degree 2 in momenta. the hamiltonian k defines a (pseudo)-riemannian metric ij g in the vertical bundle over 0 t *m and using it, a sasaki type metric on 0 t *m is constructed. a natural almost complex structure is also defined by k on 0 t *m in su...

The research is financed by by the National Natural Science Foundation of China (11071001) and the Research Fund for the Doctoral Program of Higher Education of China(20093401110001) Abstract In this paper, we study weighted pseudo-almost periodic solutions to general neutral integral equations. By applying the properties of weighted pseudo-almost periodic functions and the fixed point theorem,...

In this paper, we shall deal with μ-pseudo almost automorphic solutions to a semi-linear fractional differential equation by a new concept of μ-pseudo almost automorphic functions presented recently. First we establish some new properties of μ-pseudo almost automorphic functions, and then apply the obtained results to prove some existence theorems combined with the Leray-Schauder alternative th...