A characterization of finite homogeneous ultrametric spaces and generated by unrooted labeled trees has been made in terms representing trees. Finite having perfect strictly n-ary have characterized special graphs connected with the space. detailed survey some classes spaces, which were considered last ten years, given their hereditary properties studied. More precisely, we are interested follo...

It is well known that one can use an adaptation of the inverse-limit construction to solve recursive equations in the category of complete ultrametric spaces. We show that this construction generalizes to a large class of categories with metric-space structure on each set of morphisms: the exact nature of the objects is less important. In particular, the construction immediately applies to cate...

Ultrametric trees are trees whose leaves lie at the same distance from the root. They are used to model the genealogy of a population of particles co-existing at the same point in time. We show how the boundary of an ultrametric tree, like any compact ultrametric space, can be represented in a simple way via the so-called comb metric. We display a variety of examples of random combs and explain...

We study hierachical segmentation in the framework of edgeweighted graphs. We define ultrametric watersheds as topological watersheds null on the minima. We prove that there exists a bijection between the set of ultrametric watersheds and the set of hierarchical edgesegmentations.

In this paper, the concept of ultrametric structure is intertwined with SLAM procedure. A set pre-existing transformations has been used to create a new simultaneous localization and mapping (SLAM) algorithm. We have developed two parallel algorithms that implement time-consuming Boolean space dissimilarity matrix. The resulting matrix an important input vector quantization (VQ) step in process...

The natural join and the inner union combine in different ways tables of a relational database. Tropashko [18] observed that these two operations are the meet and join in a class of lattices—called the relational lattices— and proposed lattice theory as an alternative algebraic approach to databases. Aiming at query optimization, Litak et al. [12] initiated the study of the equational theory of...

It is well known that one can use an adaptation of the inverse-limit construction to solve recursive equations in the category of complete ultrametric spaces. We show that this construction generalizes to a large class of categories with metric-space structure on each set of morphisms: the exact nature of the objects is less important. In particular, the construction immediately applies to cate...

General non-degenerate p-adic operators of ultrametric diffusion are introduced. Bases of eigenvectors for the introduced operators are constructed and the corresponding eigenvalues are computed. Properties of the corresponding dynamics (i.e. of the ultrametric diffusion) are investigated.

We show that Möbius maps between ultrametric spaces are local similarities. The proof is based on the notions of chordal ultrametric and normalization, which we introduce in this paper.