The purpose of this note is to report the main results obtained by the authors in [8] and [9], respectively. The notion of a Smyth completable quasi-uniform space provides an efficient tool to give a topological foundation for many kinds of spaces which arise naturally in Theoretical Computer Science; in particular in Domain Theory (e.g. [5], [13] and [14]) and Complexity Theory (e.g. [9] and [...

In this article we introduce and investigate the concept of a partial quasi-metric and some of its applications. We show that many important constructions studied in Matthews’s theory of partial metrics can still be used successfully in this more general setting. In particular we consider the bicompletion of the quasi-metric space that is associated with a partial quasi-metric space and study i...

We generalize the construction of the formal ball model for metric spaces due to A. Edalat and R. Heckmann to obtain computational models for separated Q-categories. We fully describe (a) Yoneda complete and (b) continuous Yoneda complete Q-categories via their formal ball models. Our results yield solutions to two open problems in the theory of quasi-metric spaces: we show that (a) a quasi-met...

In the present paper, we prove some fixed point theorems of Hegedus contraction in some types of distance spaces, dislocated metric space, left dislocated metric space, right dislocated metric space and dislocated quasi-metric metric space which are generalized metrics spaces where self-distances are not necessarily zero.

We introduce the quasi-hyperbolicity constant of a metric space, rough isometry invariant that measures how space deviates from being Gromov hyperbolic. This number, for unbounded spaces, lies in closed interval $[1,2]$. The an hyperbolic is equal to one. For CAT$(0)$-space, it bounded above by $\sqrt{2}$. Banach at least two dimensional below $\sqrt{2}$, and non-trivial $L_p$-space exactly $\m...

We study a concept of 1-local retract in quasi-metric spaces. In this article, we generalize further known results about 1-local retract subsets from metric setting to quasi-metric point of view. In particular we show that any commuting family of nonexpansive self-mappings in a nonempty T0-quasi-metric space (X, d) for which Aq(X) is compact and normal has a common fixed point and the common fi...

In this paper it is proved that the volumes of the moduli spaces of polarized Calabi-Yau manifolds with respect to Weil-Petersson metrics are rational numbers. Mumford introduce the notion of a good metric on vector bundle over a quasi-projective variety in [10]. He proved that the Chern forms of good metrics define classes of cohomology with integer coefficients on the compactified quasi-proje...