We verify a conjecture of Vershik by showing that Hall's universal countable locally finite group can be embedded as dense subgroup in the isometry Urysohn space and automorphism random graph. In fact, we show same for all groups known infinite ultraextensive spaces. These include, addition, rational space, ultrametric spaces, Kn-free graph n≥3. Furthermore, actions on metric spaces or some rel...

In this paper we provide an answer to a question found in [3], namely when given a quasi-metric ρ, if one examines all quasi-metrics which are pointwise equivalent to ρ, does there exist one which is most like an ultrametric (or, equivalently, exhibits an optimal amount of Hölder regularity)? The answer, in general, is negative, which we demonstrate by constructing a suitable Rolewicz–Orlicz sp...

Phrase structure trees have a hierarchical structure. In many subjects, most notably in taxonomy such tree structures have been studied using ultrametrics. Here syntactical hierarchical phrase trees are subject to a similar analysis, which is much simpler as the branching structure is more readily discernible and switched. The occurrence of hierarchical structure elsewhere in linguistics is men...

This paper has two main objectives. The first is to introduce a notion of entropy that well suited for the analysis nonlinear input-output systems have Chen-Fliess series representation. latter defined in terms its generating over noncommutative alphabet. idea assign an as element graded vector space. second objective describe originating from interconnected arise context control theory. It sho...

2014 A probability distribution has been proposed recently by one of us as an order parameter for spin glasses. We show that this probability depends on the particular realization of the couplings even in the thermodynamic limit, and we study its distribution. We also show that the space of states has an ultrametric topology. J. Physique 45 (1984) 843-854 MAI 1984, Classification Physics Abstra...

We give a new characterization of the Baire class 1 functions (defined on an ultrametric space) by proving that they are exactly the pointwise limits of sequences of full functions, which are particularly simple Lipschitz functions. Moreover we highlight the link between the two classical stratifications of the Borel functions by showing that the Baire class functions of some level are exactly ...

We review the theory and practice of determining what parts of a data set are ultrametric. It is assumed that the data set, to begin with, is endowed with a metric, and we include discussion of how this can be brought about if a dissimilarity, only, holds. The basis for part of the metric-endowed data set being ultrametric is to consider triplets of the observables (vectors). We develop a novel...

The classical morphological segmentation paradigm is based on the watershed transform, constructed by flooding the gradient image seen as a topographic surface. For flooding a topographic surface, a topographic distance is defined from which a minimum distance algorithm is derived for the watershed. In a continuous formulation, this is modeled via the eikonal PDE, which can be solved using curv...

We investigate several unconventional models of finite automata and algorithms. We show that two-way alternating automata can be smaller than fast bounded-error probabilistic automata. We introduce ultrametric finite automata which use p-adic numbers to describe the branching process of the computation. We examine the size complexity of all the abovementioned automata for the counting problem. ...

We prove that for every ε ∈ (0, 1) there exists Cε ∈ (0,∞) with the following property. If (X, d) is a compact metric space and μ is a Borel probability measure on X then there exists a compact subset S ⊆ X that embeds into an ultrametric space with distortion O(1/ε), and a probability measure ν supported on S satisfying ν (Bd(x, r)) 6 (μ(Bd(x,Cεr)) 1−ε for all x ∈ X and r ∈ (0,∞). The dependen...