This is a special issue of the journal devoted to the jubilee of the outstanding Ukrainian and Russian mathematician, one of the founder of our journal Professor Rostislav Grigorchuk, who was 50 years of age on February 23, 2003. http://hdl.handle.net/123456789/74 2026-04-16T01:26:05Z 2026-04-16T01:26:05Z Vorobets, Yaroslav http://hdl.handle.net/123456789/82 2020-01-24T21:24:52Z 2003-01-01T00:00:00Z

On faithful actions of groups and semigroups by orientation-preserving plane isometries Vorobets, Yaroslav Feitful representations of two generated free groups and free semigroups by orientation-preserving plane isometries constructed.

2003-01-01T00:00:00Z Verbitsky, Oleg http://hdl.handle.net/123456789/81 2020-01-24T21:27:09Z 2003-01-01T00:00:00Z

Structural properties of extremal asymmetric colorings Verbitsky, Oleg Let ­ be a space with probability measure μ for which the notion of symmetry is defined. Given A µ ­, let ms(A) denote the supremum of μ(B) over symmetric B µ A. An r-coloring of ­ is a measurable map  : ­ ! {1, . . . , r} possi- bly undefined on a set of measure 0. Given an r-coloring Â, let ms(­; Â) = max1·i·r ms(Â−1(i)). With each space ­ we associate a Ramsey type number ms(­, r) = infÂms(­; Â). We call a col- oring  congruent if the monochromatic classes Â−1(1), . . . , Â−1(r) are pairwise congruent, i.e., can be mapped onto each other by a symmetry of ­. We define ms?(­, r) to be the infimum of ms(­; Â) over congruent Â. We prove that ms(S1, r) = ms?(S1, r) for the unitary circle S1 endowed with standard symmetries of a plane, estimate ms?([0, 1), r) for the unitary interval of reals considered with central symmetry, and explore some other regularity properties of extremal colorings for various spaces.

2003-01-01T00:00:00Z Szaszkowski, Zbigniew http://hdl.handle.net/123456789/80 2020-01-24T21:25:28Z 2003-01-01T00:00:00Z

Dynamics of finite groups acting on the boundary of homogenous rooted tree Szaszkowski, Zbigniew Criterion of embedding of finite groups into the automorphism groups of a homogenous rooted tree of a spherical index n is formulated. The sets of natural numbers which are the lengths of all orbits of finite groups acting on the boundary of tree are described.

2003-01-01T00:00:00Z Reznykov, I. I. http://hdl.handle.net/123456789/79 2020-01-24T21:25:11Z 2003-01-01T00:00:00Z

On 2-state Mealy automata of polynomial growth Reznykov, I. I. We consider the sequence of 2-state Mealy au- tomata over the finite alphabets, that have polynomial growth or- ders and define the infinitely presented automatic transformation semigroups.

2003-01-01T00:00:00Z