On some nonlinear problems in mathematical physics with exponents close to critical ones

The paper deals with several problems having the following common feature. The differential equation under consideration contains a power function of the solution or of its gradient, with the exponent depending on a small parameter. This dependence is such that a certain property of the solutions corresponding to the positive values of the parameter may disappear as the latter tends to zero. This phenomenon is shown to be lacking in the case where not only the said exponent, but also the coefficients of the equation depend on the small parameter in a suitable way. Several theorems are proved on sharp conditions, ensuring such uniformity of various properties exhibited by the solutions. Bibliography: 44 titles. Dedicated to Olga Arsenievna Oleinik This paper has been supported in part by the Russian Foundation for Basic Research (Project 93-013-1744) and by the International Science Foundation (Grant MIE000). Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 19, pp. 000-000, 0000.     

Motion of a helium atom through a comparable-diameter channel in the Frenkel-Kontorova model

The motion of a helium atom through a channel with a diameter comparable to the diameter of this atom is investigated. Such channels are observed in materials of the quartz type. The inclusion of the local law of momentum conservation in the analysis of the interaction between the He atom and the atoms forming the channel wall makes it possible to reduce the problem to one-dimensional motion similar to the motion of a dislocation in the Frenkel-Kontorova model. In the model proposed here, the activation energy of the complex formed by the He atom and the displacements of atoms of the channel wall is calculated. The obtained energy is expressed in terms of the shear modulus of the material forming the channel and the helium atom polarizability, which depends on the state of the helium atom.     

4a,6,7,8a-Tetrahydro-1,3-dithiolo[4',5':5,6]-1,4-dithiino [2,3-b]-1,4-dithiine-2-thione as a precursor for synthesis of new organic π-donors

[2+4[-Cycloaddition of 2,3-dihydro-1,4-dithiine and oligomeric 1,3-dithiol-2,4,5-trithione results in the formation of 4a,6,7,8a-tetrahydro-1,3-dithiolo[4',5':5,61-[1,4[dithiino[2,3-b[1,4-dithiine-2-thione, which is a precursor for synthesis of new organic rz-donors.Translated fromIzvestiya Akademi Nauk Seriya Khimicheskaya No. 3, pp. 775–776, March, 1996.     

Synthesis and some properties of cyclic acetals of malonaldehyde

The Lewis acid-catalyzed addition of cyclic orthoformates to vinyl ethyl ether, which leads to the formation of malonaldehyde acetals, was studied. It is shown that of the linear-cyclic malonaldehyde acetals, 2-(2,2-diethoxyethyl)- and 5,5-dimethyl-2-(2,2-diethoxyethyl)-1,3-dioxanes are stable. The transacetalization of 1,1,3,3-tetraethoxypropane with 1,2- and 1,3-diols, which leads to the formation of cyclic malonaldehyde acetals, was studied. The physicochemical constants of the acetals were determined, and their ¹H and ¹³C NMR spectra are described.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 4, pp. 459–463, April, 1983.     

Protection of the carboxy group in the form of the 2-cyanoethyl ester in synthesis of peptides

With the aim of studying the suitability of the 2-cyano group for the protection of carboxylic functions in peptide synthesis, we have obtained the 2-cyanoethyl esters of a number of amino acids and have studied their behavior under the conditions of peptide synthesis. The synthesis of the pentapeptide leucine-enkaphalin has been performed with the use of 2-cyanoethyl protection for C-terminal carboxy groups throughout. The physicochemical characteristics of the compounds synthesized are given.All-Union Scientific-Research Institute of Molecular Biology Kol'tsovo, Novosibirsk oblast Vector Scientific Industrial Association of the USSR Minmedbioprom, Novosibirsk. Translated from Khimiya Prirodnykh Soedinenii, No. 3, pp. 412–416, May–June, 1988.     

Supermolecular structures and flow birefringence in polymer solutions

Experimental studies of supermolecular structures and localized flow birefringence in solutions of high-molecular weight polymer are described. Advantage is taken of poly(ethylene oxide) and polyisobutylene. Supermolecular structures are examined with the aid of optical microscopy using freeze-dried samples of the polymer solutions. Birefringence is investigated that arises in planar elongational flow in a cross-slot cell. Flow velocities at which the onset of the localized birefringence occurs are determined. Then these velocities are correlated with viscoelastic characteristics of the solutions. The presence of a liquid-crystalline fibrillar network in the polymer solutions exhibiting flow birefringence is ascertained. The fibrils are birefringent objects. The fibrils are birefringent objects. The localized birefringence phenomenon is explained in term of the orientation of the fibrils in elongational flow. It has been shown that the onset of localized birefringence occurs at a critical Weissenberg number, the value of which is close to unity.