A characterization of absolute retracts

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 33, No. 2, pp. 215–217, March–April, 1992.     

Traveling waves in a nonlinear medium with cubic nonlinearity

Unlike linear nondispersive media, which allow propagation of wave packets of arbitrary forms, nonlinear media admit only certain profiles of traveling waves. Here we examine media with Duffing oscillators, i.e., with bound electrons for which an equilibrium disturbance causes forces proportional to the first and third powers of deviation. We show that the linearly polarized traveling plane waves such media can transmit have profiles modulated as Jacobi elliptic functions. When discussing propagation across an interface between different media, only incidence from the side of the linear medium is considered. Even in this case, to launch a traveling wave in the nonlinear medium, a severe restriction must be imposed on the incident wave’s amplitude.     

An estimate of the curvature of level curves in a class of univalent functions

In this note a class of normalized functions which are univalent in the unit disk is considered. An exact solution of the problem of an upper estimate of curvature of level curves for functions of this class is obtained. In particular, an exact solution of this problem for univalent functions with real coefficients at the points of the interval (0, 1) is obtained.Translated from Matematicheskie Zametki, Vol. 19, No. 3, pp. 381–388, March, 1976.The author is thankful to the referee for taking interest in this note.     

Statistics of the fragments forming with the destruction of solids by explosion

The article sets forth one of the possible approaches to the construction of a size distribution function of the fragments. The Rozin-Rammler law for the distribution is obtained from general theoretical probability consideration. The theoretically obtained distribution function was verified in a large number of experiments. The experimental data are in good agreement with the theoretical deductions.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 87–100, March–April, 1971.