On a free group of transformations defined by an automaton

We prove that three automorphisms of the rooted binary tree defined by a certain 3-state automaton generate a free non-Abelian group of rank 3. Both authors are supported by the NSF grants DMS-0308985 and DMS-0456185. Yaroslav Vorobets is supported by a Clay Research Scholarship.     

Isospectrality and galois projective geometries

We construct a series of pairs of domains in the plane and pairs of surfaces with boundary that are isospectral but not isometric. The construction is based on the existence of finite transformation groups that are spectrally equivalent but not isomorphic. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 660–664, May, 1998. This research was partially supported by the Russian Foundation for Basic Research and by INTAS under joint grant No. 96-01-00043.     

On the commutator group of the group of interval exchange transformations

We study the group of interval exchange transformations and obtain several characterizations of its commutator group. In particular, it turns out that the commutator group is generated by elements of order 2.     

Billiards in rational polygons: Periodic trajectories, symmetries, and d-stability

Periodic trajectories of billiards in rational polygons satisfying the Veech alternative, in particular, in right triangles with an acute angle of the form π/n with integern are considered. The properties under investigation include: symmetry of periodic trajectories, asymptotics of the number of trajectories whose length does not exceed a certain value, stability of periodic billiard trajectories under small deformations of the polygon. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 66–75, July, 1997. Translated by V. N. Dubrovsky     

Photoelectrochemical properties of the composites based on TiO2 nanotubes,CdSe and graphene oxide

Research on Chemical Intermediates - The photoelectrochemical properties of electrodes based on a nanocomposite consisting of TiO2 nanotubes and CdSe films modified with graphene oxide have been...     

Spectroscopic,electrocatalytic, and photoelectrochemical characteristics of mixed-ligand bis(β-dicarbonylato) phthalocyanine complexes of zirconium(IV) and hafnium(IV)

The electronic absorption (EAS) and fluorescence spectra and electrocatalytic and photoelectrochemical characteristics of mixed-ligand complexes of zirconium(IV) and hafnium(IV) based on phthalocyanines in solutions and in thin films were investigated. It was established that a significant bathochromic shift of the Q band and redistribution of the intensities in the absorption bands are observed in the electronic absorption spectra. It was shown that films of the synthesized compounds are promising for use as photosensitizers and electrochemical sensors for oxygen in solution. __________ Translated from Teoreticheskaya i éksperimental’naya Khimiya, Vol. 44, No. 3, pp. 133–137, May–June, 2008.     

Potentiometric and Spectrophotometric Determination of the Ionic Composition of Concentrated Sodium Polysulfides

The concentrations of ionic species in polysulfide electrolytes of photoelectrochemical converters, calculated from the results of potentiometric and spectrophotometric measurements in concentrated polysulfide solutions, are presented. The possible processes that occur at the semiconducting CdSe photoelectrode and involve electrically active species are discussed.     

On a substitution subshift related to the Grigorchuk group

We study the dynamics of a substitution subshift given by the substitution a → aca, b → d, c → b, d → c, which is related to the Grigorchuk group. This dynamical system is shown to be, up to a countable set, conjugate to the binary odometer.