We describe the order type of range sets compact ultrametrics and show that an ultrametrizable infinite topological space (X,τ) is iff are isomorphic for any two compatible with topology τ. It also shown separable every ultrametric, this topology, has at most countable set.

Generalized ultrametric spaces are a common generalization of preorders and ordinary ultrametric spaces, as was observed by Lawvere (1973). Guided by his enriched-categorical view on (ultra)metric spaces, we generalize the standard notions of Cauchy sequence and limit in an (ultra)metric space, and of adjoint pair between preorders. This leads to a solution method for recursive domain equations...

1. INTRODUCTION. When studying a metric space, it is valuable to have a mental picture that displays distance accurately. When the space is Z, Q, or R, we usually form such a picture by imagining points on the " number line ". When the space is X = Z 2 , Q 2 , R 2 , or C we use a planar picture in which nonempty discs (sets of the form {x ∈ X : d(x, b) ≤ γ } or {x ∈ X : d(x, b) < β}, with metri...

This paper explores the language classes that arise with respect to the head count of a finite ultrametric automaton. First we prove that in the one-way setting there is a language that can be recognized by a one-head ultrametric finite automaton and cannot be recognized by any k-head non-deterministic finite automaton. Then we prove that in the two-way setting the class of languages recognized...

A kinetics built upon q-calculus, the calculus of discrete dilatations, is shown to describe diffusion on a hierarchical lattice. The only observable on this ultrametric space is the “quasi-position” whose eigenvalues are the levels of the hierarchy, corresponding to the volume of phase space available to the system at any given time. Motion along the lattice of quasi-positions is irreversible....

We compared the properties of the error threshold transition in quasispecies evolution for three different topologies of the genome space. They are a) hypercube b) rugged landscape modelled by an ultrametric space, and c) holey landscape modelled by Bethe lattice. In all studied topologies the phase transition exists. We calculated the critical exponents in all the cases. For the critical expon...

It is shown that for every ε ∈ (0, 1), every compact metric space (X, d) has a compact subset S ⊆ X that embeds into an ultrametric space with distortion O(1/ε), and dimH(S) > (1− ε) dimH(X), where dimH(·) denotes Hausdorff dimension. The above O(1/ε) distortion estimate is shown to be sharp via a construction based on sequences of expander graphs.

Felsenstein's classical model for Gaussian distributions on a phylogenetic tree is shown to be toric variety in the space of concentration matrices. We present an exact semialgebraic characterization this model, and we demonstrate how structure leads methods maximum likelihood estimation. Our results also give new insights into geometry ultrametric

Ultrametric algorithms are similar to probabilistic algorithms but they describe the degree of indeterminism by p-adic numbers instead of real numbers. This paper introduces the notion of ultrametric query algorithms and shows an example of advantages of ultrametric query algorithms over deterministic, probabilistic and quantum query algorithms.

This paper presents results from lesion experiments on a modular attractor neural network model of semantic access. Real picture data forms the basis of perceptual input to the model. An ultrametric attractor space is used to represent semantic memory and is implemented using a biologically plausible variant of the Hopfield model. Lesioned performance is observed to be in agreement with neurops...