Jasem, Milan
Published in
Mathematica Slovaca

In the paper weak isometries in directed groups are investigated. It is proved that for every weak isometry f in a directed group G the relation f(UL(x, y) ∩ LU(x, y)) = UL(f(x), f(y)) ∩ LU(f(x), f(y)) is valid for each x, y ∈ G. The notions of an orthogonality of two elements and of a subgroup symmetry in directed groups are introduced and it is s...

Ivanov, A. O. Nikolaeva, N. K. Tuzhilin, A. A.
Published in
Mathematical Notes

Gundyrev, I. A.
Published in
Siberian Advances in Mathematics

We study locally compact spaces with an intrinsic metric such that the group of metric similarities is transitive and the group of isometries is not transitive. We suggest an algebraic characterization of such spaces.

Vershik, A. M.
Published in
Functional Analysis and Its Applications

The notion of a homogeneous standard filtration of σ-algebras was introduced by the author in 1970. The main theorem asserted that a homogeneous filtration is standard, i.e., generated by a sequence of independent random variables (is Bernoulli), if and only if a standardness criterion is satisfied. The author has recently generalized the notion of...

Vershik, A. M. Malyutin, A. V.
Published in
Functional Analysis and Its Applications

We describe the full exit boundary of random walks on homogeneous trees, in particular, on free groups. This model exhibits a phase transition; namely, the family of Markov measures under study loses ergodicity as a parameter of the random walk changes. The problem under consideration is a special case of the problem of describing the invariant (ce...

Gundyrev, I. A.
Published in
Siberian Advances in Mathematics

We study locally compact spaces with an intrinsic metric such that the group of metric similarities is transitive and the group of isometries is not transitive. We confirm Berestovskiĭ’s conjecture on the structure of such spaces.

Vershik, A. M.
Published in
Functional Analysis and Its Applications

We suggest a new method for describing invariant measures on Markov compacta and on path spaces of graphs and, thereby, for describing characters of certain groups and traces of AF-algebras. The method relies on properties of filtrations associated with a graph and, in particular, on the notion of a standard filtration. The main tool is an intrinsi...

Korobkov, M. V.
Published in
Siberian Advances in Mathematics

We say that a domain U ⊂ ℝn is uniquely determined by the relative metric (which is the extension by continuity of the intrinsic metric of the domain on its boundary) of its Hausdorff boundary if any domain V ⊂ ℝn such that its Hausdorff boundary is isometric in the relative metric to the Hausdorff boundary of U, is isometric to U in the Euclidean ...

Gundyrev, I. A.
Published in
Russian Mathematics

In this article, we generalize partially the theorem of V. N. Berestovskii on characterization of similarity homogeneous (nonhomogeneous) Riemannian manifolds, i.e., Riemannian manifolds admitting transitive group of metric similarities other than motions to the case of locally compact similarity homogeneous (nonhomogeneous) spaces with intrinsic m...

Gorbatsevich, V. V.
Published in
Siberian Mathematical Journal

We study the algebraic conditions for all intrinsic metrics to be Finsler on a homogeneous space. These conditions were firstly found by Berestovskiĭ in terms of Lie algebras and their subalgebras (the corresponding subalgebras will be called strong). We obtain a description of the structure of strong subalgebras in semisimple solvable Lie algebras...