For finitely generated groups Γ and ultrahomogeneous countable relational structures M we study the space Rep(Γ,M) of all representations of Γ by automorphisms on M equipped with the topology it inherits seen as a closed subset of Aut(M)Γ. When Γ is the free group Fn on n generators this space is just Aut(M)n, but is in general significantly more complicated. We prove that when Γ is finitely ge...

Based on the synergetic approach, the theory of a single avalanche formation is presented for the simplest sandpile model. The stationary values of sand velocity, slope and avalanche intensity are derived as functions of control parameter (externally driven sandpile tilt). The statistical ensemble of avalanche intensities is considered to investigate diffusion in ultrametric space of hierarchic...

We study hierarchical clustering schemes under an axiomatic view. We show that within this framework, one can prove a theorem analogous to one of Kleinberg (2002), in which one obtains an existence and uniqueness theorem instead of a non-existence result. We explore further properties of this unique scheme: stability and convergence are established. We represent dendrograms as ultrametric space...

The author reviews the theory and practice of determining what parts of a data set are ultrametric. He describes the potential relevance of ultrametric topology as a framework for unconscious thought processes. This view of ultrametric topology as a framework that complements metric-based, conscious, Aristotelian logical reasoning comes from the work of the Chilean psychoanalyst, Ignacio Matte ...

Let G be a group and X be a G-space with the action G×X → X, (g, x) 7→ gx. A subset F of X is called a kaleidoscopical configuration if there exists a coloring χ : X → C such that the restriction of χ on each subset gF , g ∈ G, is a bijection. We present a construction (called the splitting construction) of kaleidoscopical configurations in an arbitrary G-space, reduce the problem of characteri...

Let G be a group and X be a G-space with the action G X ! X, (g; x) 7! gx. A subset F of X is called a kaleidoscopical con guration if there exists a coloring : X ! C such that the restriction of on each subset gF , g 2 G, is a bijection. We present a construction (called the splitting construction) of kaleidoscopical con gurations in an arbitrary G-space, reduce the problem of characterization...

Morphological semigroups for functions on length spaces have recently studied [2], whose basic ingredients are the convolution in the (max,+)-algebra (or supremal convolution), the metric distance and a convex shape function. Links with Hamilton–Jacobi PDE on length spaces appears also naturally. These semigroups lead to powerful scale-space properties for multiscale filtering, regularization a...

This paper introduces hierarchical quasiclustering methods, a generalization of hierarchical clustering for asymmetric networks where the output structure preserves the asymmetry of the input data. We show that this output structure is equivalent to a finite quasi-ultrametric space and study admissibility with respect to two desirable properties. We prove that a modified version of single linka...

In this paper, we analyze the preservation of original semantic similarity among objects when dimensional reduction is applied on the original data source and a further clustering process is performed on dimensionally reduced data. An experiment is designed to test Baire, or longest common prefix ultrametric, and K-Means when prior random projection is applied. A data matrix extracted from a cu...