In this paper, the poset $BX$ of formal balls is studied in fuzzy partial metric space $(X,p,*)$. We introduce the notion of layered complete fuzzy partial metric space and get that the poset $BX$ of formal balls is a dcpo if and only if $(X,p,*)$ is layered complete fuzzy partial metric space.

We introduce a concept of tree-graded metric space and we use it to show quasi-isometry invariance of certain classes of relatively hyperbolic groups, to obtain a characterization of relatively hyperbolic groups in terms of their asymptotic cones, to find geometric properties of Cayley graphs of relatively hyperbolic groups, and to construct the first example of finitely generated group with a ...

Let X be a geodesic metric space. Gromov proved that there exists ε0 > 0 such that if every sufficiently large triangle ∆ satisfies the Rips condition with constant ε0 · pr(∆), where pr(∆) is the perimeter ∆, then X is hyperbolic. We give an elementary proof of this fact, also giving an estimate for ε0. We also show that if all the triangles ∆ ⊆ X satisfy the Rips condition with constant ε0 · p...

In this work, we study the asymptotic geometry of the mapping class group and Teichmuller space. We introduce tools for analyzing the geometry of “projection” maps from these spaces to curve complexes of subsurfaces; from this we obtain information concerning the topology of their asymptotic cones. We deduce several applications of this analysis. One of which is that the asymptotic cone of the ...

If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X)...

The strict minimum message length (SMML) principle links data compression with inductive inference. The corresponding estimators have many useful properties but they can be hard to calculate. We investigate SMML estimators for linear regression models and we show that they have close connections to hyperbolic geometry. When equipped with the Fisher information metric, the linear regression mode...

We define and study a quantale-valued Wijsman structure on the hyperspace of all non-empty closed sets of a quantale-valued metric space. We show its admissibility and that the metrical coreflection coincides with the quantale-valued Hausdorff metric and that, for a metric space, the topological coreflection coincides with the classical Wijsman topology. We further define an index of compactnes...

Abstract. This article is about chromatic numbers of hyperbolic surfaces. For a metric space, the d-chromatic number is the minimum number of colors needed to color the points of the space so that any two points at distance d are of a different color. We prove upper bounds on the d-chromatic number of any hyperbolic surface which only depend on d. In another direction, we investigate chromatic ...

Based on previous results that study the completion of fuzzy metric spaces, we show that every intuitionistic fuzzy quasi-metric space, using the notion of fuzzy metric space in the sense of Kramosil and Michalek to obtain a generalization to the quasi-metric setting, has a bicompletion which is unique up to isometry.