Estimates for the Rate of Convergence in Ordinary Differential Equations under the Action of Random Processes with Fast Time

We study the procedure of averaging in the Cauchy problem for an ordinary differential equation perturbed by a certain Markov ergodic process. We establish several estimates for the rate of convergence of solutions of the original problem to solutions of the averaged one. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 435–457, April, 2005.     

A Stochastic Analog of Bogolyubov's Second Theorem

We establish an estimate for the rate at which a solution of an ordinary differential equation subject to the action of an ergodic random process converges to a stationary solution of a deterministic averaged system on time intervals of order for some 0 < ρ < 1. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 879–894, July, 2005.     

Macrocyclic amidoacylhydrazones

Substituted dibenzo[d,j][1,2,6,9,13,14]hexaazahexadeca-2,4,10,12-tetraene-7,8-15,16-tetraones have been obtained from hydrazones of ortho-acyltosylanilines. The structure of the former has been demonstrated by IR, PMR, and mass spectra and by alternative synthesis.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 7, pp. 991–995, July, 1985.     

Crystal and molecular structures of <Emphasis Type="Italic">trans</Emphasis>-2-(4′-dimethylaminobenzylideneacetyl)-5,5-dimethylcyclohexane-1,3-dione

The crystal and molecular structures of a cyclic β-triketone, namely, trans-2-(4′-dimethylaminobenzylideneacetyl)-5,5-dimethylcyclohexane-1,3-dione (I), are determined using X-ray diffraction analysis. It is established that the compound in the crystalline state exists in a diketo-enol form stabilized by intramolecular hydrogen bonds. The specific features of the structure and the physicochemical and fluorescence properties of the compound are discussed.     

On the role of umklapp processes in the scattering of delocalized positronium by acoustic phonons in ionic crystals

The effect of umklapp processes in the scattering of delocalized positronium atoms by acoustic phonons in ionic crystals is analyzed. It is shown that at temperatures above which this effect becomes large, umklapp scattering renormalizes the constant of the deformation potential of positronium. This renormalization has recently been observed in experiments measuring the momentum distribution of delocalized positronium in a MgF₂ crystal. Pis’ma Zh. éksp. Teor. Fiz. 69, No. 3, 215–219 (10 February 1999)     

A transient two-photon–one-photon double resonance on interband transitions in crystals: II. Calculation of absorbed of light energy

The dependences of the energy that is absorbed in a crystal exposed to the action of a femtosecond light pulse on the radiation intensity at a fixed pulse duration and on the pulse duration at a fixed pulse energy have been determined under the conditions of a two-photon–one-photon double resonance on interband transitions in the crystal.