Free evolution for quantum particle in generic ultrametric space is considered. We prove that if mean zero wave packet is localized in some space domain then its evolution remains localized in the same domain. In the present note we consider ultrametric quantum mechanics with real time and ultrametric space. Quantum mechanics with ultrametric (p–adic and adelic) space was considered in many wor...

We consider ultrametric space with a measure, such that the measure of any ball is positive. A family of orthonormal bases of ultrametric wavelets in the space of quadratically integrable functions is introduced. Pseudodifferential operators (PDO) in the ultrametric space are investigated. We prove that these operators are diagonal in the introduced bases of ultrametric wavelets and compute the...

A family of orthonormal bases of ultrametric wavelets in the space of quadratically integrable with respect to arbitrary measure functions on general (up to some topological restrictions) ultrametric space is introduced. Pseudodifferential operators (PDO) on the ultrametric space are investigated. We prove that these operators are diagonal in the introduced bases of ultrametric wavelets and com...

Both preorders and ordinary ultrametric spaces are instances of generalized ul-trametric spaces. Every generalized ultrametric space can be isometrically embedded in a (complete) function space by means of an ultrametric version of the categorical Yoneda Lemma. This simple fact gives naturally rise to: 1. a topology for generalized ultrametric spaces which for arbitrary preorders corresponds to...

Γ-ultrametric spaces are spaces which satisfy all the axioms of an ultrametric space except that the distance function takes values in a complete lattice Γ instead of R≥0. Γ-ultrametric spaces have been extensively studied as a way to weaken the notion of an ultrametric space while still providing enough structure to be useful (see for example [17], [18], [8]). The many uses of Γ-ultrametric sp...

We show that every locally compact Polish group is isomorphic to the isometry group of a proper separable metric space. This answers a question of Gao and Kechris. We also analyze the natural action of the isometry group of a separable ultrametric space on the space. This leads us to a structure theorem representing an arbitrary separable ultrametric space as a bundle with an ultrametric base a...

A new way of construction of ultrametric spaces of quite general form by completion of directed trees with respect to the metric, defined in the special way, is proposed. We introduce a measure on the constructed ultrametric space, such that the measure of any ball is positive. A family of orthonormal bases of ultrametric wavelets in the space of quadratically integrable with respect to the def...

We investigate the relationships between tolerance relations, equivalence relations, and ultrametrics. The set of spheres associated to an ultrametric space has a tree structure that reflects a hierarchy on the set of equivalences associated to that space. We show that every ultrametric defined on a finite space is a linear combination of binary ultrametric and we introduce the notion of ultram...

Gaussian random field on general ultrametric space is introduced as a solution of pseudodifferential stochastic equation. Covariation of the introduced random field is computed with the help of wavelet analysis on ultrametric spaces. Notion of ultrametric Markovianity, which describes independence of contributions to random field from different ultrametric balls is introduced. We show that the ...

Ultrametric approach to the genetic code and the genome is considered and developed. p-Adic degeneracy of the genetic code is pointed out. Ultrametric tree of the codon space is presented. It is shown that codons and amino acids can be treated as p-adic ultrametric networks. Ultrametric modification of the Hamming distance is defined and noted how it can be useful. Ultrametric approach with p-a...