http://hdl.handle.net/123456789/25 2026-04-15T22:51:06Z http://hdl.handle.net/123456789/82 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 http://hdl.handle.net/123456789/81 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 http://hdl.handle.net/123456789/80 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 http://hdl.handle.net/123456789/79 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