We prove that an ultrametric space can be bi-Lipschitz embedded in R if its metric dimension in Assouad’s sense is smaller than n. We also characterize ultrametric spaces up to bi-Lipschitz homeomorphism as dense subspaces of ultrametric inverse limits of certain inverse sequences of discrete spaces.

We introduce the simple parametrization for the space of codons (triples of nucleotides) by 8 × 8 table. This table (which we call the dyadic plane) possesses the natural 2–adic ultrametric. We show that after this parametrization the genetic code will be a locally constant map of the simple form. The local constancy of this map will describe degeneracy of the genetic code. The map of the genet...

We establish universality and ultra-homogeneity of ( U , u GH ) the collection all compact ultrametric spaces endowed with so-called Gromov-Hausdorff . This result also gives rise to a novel construction R-Urysohn universal space for each countable subset R ⊂ ≥ 0 containing 0.

We introduce the simple parametrization for the space of codons (triples of nucleotides) by 8 × 8 diadic plane. This diadic plane possesses the natural 2–adic ultrametric. We show that after this parametrization the genetic code will be the locally constant map of the simple form defined on the diadic plane. The local constancy of this map will describe degeneracy of the genetic code. The map o...

and Applied Analysis 3 Example 1.2. The vector spaceC1 0, 1 consisting of all continuously differentiable real-valued functions on 0, 1 equipped with the modular ρ defined by ρ x max t∈ 0,1 |x t | max t∈ 0,1 ∣ ∣x′ t ∣ ∣, ( x ∈ C1 0, 1 ) 1.7 constitutes a complete modular space. The subset P { x ∈ C1 0, 1 : x t ≥ 0, ∀t ∈ 0, 1 } 1.8 is a unital cone in C1 0, 1 with unit 1. The cone P is not norma...

We consider certain spaces of linear signals equipped with a standard prefix relation and a suitably defined generalized distance function. We introduce a new class of abstract structures, which we call generalized ultrametric semilattices, and prove a representation theorem stating that every generalized ultrametric semilattice with a totally ordered distance set is isomorphic to a space of th...

We study percolation in the hierarchical lattice of order N where the probability of connection between two points separated by distance k is of the form ck/N , δ > −1. Since the distance is an ultrametric, there are significant differences with percolation in the Euclidean lattice. We consider three regimes: δ < 1, where percolation occurs, δ > 1, where it does not occur, and δ = 1 which is th...