Conformal Mappings of Riemannian Spaces onto Ricci Symmetric Spaces

Mathematical Notes -     

Synthesis of (±)-biotin from a substituted 2,3-dehydrothiophane

(±)-Biotin has been synthesized from a mixture of cis- and allo-trisubstituted dehydrothiophanes by a scheme providing for the easy elimination of the accompanying allo-substituted thiophane in the form of an intermediate and not as the final isomeric allo-biotin. The imidazoline ring of biotin is formed from a dialkoxy-carbonyldiaminothiophane under the conditions of partial hydrolysis of one of the protective groups with the elimination of phosgene from the scheme of synthesis.All-Union Scientific-Research Vitamin Institute, Moscow. Translated from Khimiya Prirodnykh Soedinenii, No. 2, pp. 220–223, March–April, 1981.     

Study of the inversion of 4-benzamido-3-hydroxythiophane

Inversion during the acid hydrolysis of trans-4-benzamido-3-hydroxythiophane is accomplished through the intermediate formation of cis-2-phenyltetrahydrothieno[3,4-d]oxazoline. The latter has cis ring fusion, and the oxazoline ring is planar. Opening of the oxazoline ring is not accompanied by inversion and leads to cis-3,4-substituted thiophanes. Migration of the acyl group from the nitrogen to the oxygen atom was detected in substituted thiophanes.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No.10, pp. 1339–1344, October, 1971.     

Synthesis of oxazolidine analogs of biotin and epibiotin

dl-cis- and dl-trans-6-(4-Carboxybutyl)-2-oxo-cis-hexahydrothieno[3,4-d]oxazoles were synthesized by the action of phosgene on r-4-amino-c-3-hydroxy-c(or, respectively, )-2-(4-carboxybutyl)thiophan hydrohalides. It is shown that the acid hydrolysis of r-4-benzamido-t-3-hydroxy-c(or t)-2-(4-methoxycarbonylbutyl)thiophans is accompanied by inversion to give r-4-amino-c-3-hydroxy-c(or t)-2-(4-alkoxycarbonylbutyl)thiophans.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 4, pp. 459–465, April, 1976.     

Optimal accuracy approximation of functions and their derivatives

We consider the approximation of the function (x) and its derivative '(x) on [a, b] given that (x)C _2,N, i.e., belongs to the class of functions f(x) that satisfy the conditions f(x)L, f(x_i)=y_i, i=1,,N, where L and y_i are given real numbers and x_i are the nodes of an arbitrary grid, a=x₁<x₂<<X_N=b. A solution algorithm on the class of functions C_2,L,N is proposed which has optimal accuracy with a constant not exceeding 2. A bound on the approximation error of the function and its derivative is derived.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 57–61, 1985     

Nonlinear nonlocal problems for a parabolic equation in a two-dimensional domain

We establish the convergence of the Rothe method for a parabolic equation with nonlocal boundary conditions and obtain an a priori estimate for the constructed difference scheme in the grid norm on a ball. We prove that the suggested iterative process for the solution of the posed problem converges in the small. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 244–254, February, 1997.     

Rising vortex ring in a viscous fluid

Most theoretical results for thermals, whose motion is determined by the complex interaction between dynamics and buoyancy, have been obtained numerically [1–4]. The analytic solutions for a convection element have been limited to consideration of the self-similar regime [5]. At the same time, the preself-similar stage of development of a vortex ring of dynamic origin has been described analytically [6]. This approach is now extended to a rising vortex ring. In this case a modification of the traditional formulation of the problem makes it possible to obtain an analytic solution of the problem of a weak thermal in the form of unsteady temperature, vorticity and stream function fields that tend in the limit to the self-similar regime. The rate of ascent of the convective vortex ring is found. A solution is obtained for the two-dimensional analog of the problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–48, May–June, 1989.