We call a comb a map f : I → [0,∞), where I is a compact interval, such that {f ≥ ε} is finite for any ε. A comb induces a (pseudo)-distance d̄f on {f = 0} defined by d̄f (s, t) = max(s∧t,s∨t) f . We describe the completion Ī of {f = 0} for this metric, which is a compact ultrametric space called comb metric space. Conversely, we prove that any compact, ultrametric space (U, d) without isolated p...

We describe a Coq formalization of constructive ω-cpos, ultrametric spaces and ultrametric-enriched categories, up to and including the inverse-limit construction of solutions to mixed-variance recursive equations in both categories enriched over ω-cppos and categories enriched over ultrametric spaces. We show how these mathematical structures may be used in formalizing semantics for three repr...

A class of statistical models is proposed which aims to recover latent settings structures in social networks. Settings may be regarded as clusters of vertices. The measurement model builds on two assumptions. The observed network is assumed to be generated by hierarchically nested latent transitive structures, expressed by ultrametrics. It is assumed that expected tie strength decreases with u...

The preservation of weak pullbacks is studied for a functor M 1 on the category UMS of ultrametric spaces and nonexpansive mappings. The functor M 1 associates with an ultrametric space its collection of Borel probability measures with compact support. By application of the Max-ow Min-cut Theorem of graph theory a mediating morphism for a weak pullback diagram can be constructed in SET. It is s...

We study hierarchical clusterings of metric spaces that change over time. This is a natural geometric primitive for the analysis of dynamic data sets. Specifically, we introduce and study the problem of finding a temporally coherent sequence of hierarchical clusterings from a sequence of unlabeled point sets. We encode the clustering objective by embedding each point set into an ultrametric spa...

This paper addresses the informational asymmetry for con structing an ultrametric evolutionary tree from upper and lower bounds on pairwise distances between n given species We show that the tallest ultrametric tree exists and can be constructed in O n time while the existence of the shortest ultrametric tree depends on whether the lower bounds are ultrametric The tallest tree construction algo...

The category of 1-bounded compact ultrametric spaces and non-distance increasing functions (KUM’s) have been extensively used in the semantics of concurrent programming languages. In this paper a universal space U for KUM’s is introduced, such that each KUM can be isometrically embedded in it. U consists of a suitable subset of the space of functions from [0, 1) to IN, endowed with a “prefix-ba...

and when this happens we say that d(x, y) is an ultrametric. One can check that an ultrametric space is totally disconnected, which is to say that it does not contain a connected subset with more than two elements. Let us say that a subset E of a metric space (M, d(x, y)) is chain connected if for every pair of points u, v ∈ E and every ǫ > 0 there is a finite chain w1, . . . , wl of points in ...

It is proved that for any $0<\beta<\alpha$, bounded Ahlfors $\alpha$-regular space contains a $\beta$-regular compact subset embeds biLipschitzly in an ultrametric with distortion at most $O(\alpha/(\alpha-\beta))$. The bound on the asymptotically tight when $\beta\to \alpha$. main tool used proof regular form of skeleton theorem.