Moduli of families of locally quasisymmetric surfaces

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 3, pp. 9–15, May–June, 1989.     

A dynamical model of optimal investment in R&;D

We provide steps towards a welfare analysis of a two-country endogenous growth model where a relatively small follower absorbs part of the knowledge generated in the leading country. To solve a suitably defined dynamic optimization problem, an appropriate version of the Pontryagin maximum principle is developed. The properties of optimal controls and the corresponding optimal trajectories are characterized by the qualitative analysis of the solutions of the Hamiltonian system arising through the implementation of the Pontryagin maximum principle. We find that for a quite small follower, optimization produces the same asymptotic rate of innovation as the market. However, relative knowledge stocks and levels of productivity differ in general. Thus, policy intervention has no effect on growth rates but may affect these relative levels. The results are different for not so small follower economies.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 9 , Suzdal Conference-3, 2003.     

Quasiconformal extension from curvilinear triangles

We consider the problem of extension of a quasi-Möbius (quasisymmetric) mapping from a planar curvilinear triangle to a quasiconformal automorphism of the extended plane. Existence of an extension of the sort from the polygons whose sides are quasiarcs was proven by A. K. Varisov but without any upper estimates for the quasiconformality coefficient of the extended mapping. The application of another method makes it possible to obtain, for triangular domains, an upper estimate of dilation of the quasiconformal extension depending only on the distortion function of the initial quasi-Möbius mapping, on the upper estimate of the turning of the sides and the Möbius-invariant characteristic of nondegeneracy of a curvilinear triangle which is introduced in this article and is analogous to the relative radius of the inscribed circle.     

Transformation of a Si(111) stepped surface during the formation of a Si(3 × 1)Ca superstructure

The morphological changes of a Si(111) stepped surface during the formation of a Si(3 × 1)Ca surface structure have been studied using ultra high vacuum reflection electron microscopy (UHV REM). Monoatomic step clustering during surface reconstruction and an influence of adatom electromigration on step clustering have been found. The mechanism and conditions of step cluster formation during superstructural transitions are discussed.     

Gluing of quasisymmetric imbeddings in the problem of quasiconformal extension

We prove a general theorem on gluing of quasisymmetric imbeddings. Using this theorem, we solve the problem of quasiconformal extension from a one-parameter family of quasiconformal triangles of a special type that do not have a general estimate of quasiconformal convexity.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 6, pp. 737–744, June, 2004.