Singular numbers of the monodromy operator and sufficient conditions of the asymptotic stability of periodic system of differential equations with fixed delay

To obtain sufficient conditions for the asymptotic stability of linear periodic systems with fixed delay commensurable with the period of coefficients, singular numbers of the monodromy operator are used. To find these numbers, a self-adjoint boundary value problem for ordinary differential equations is applied. We study the motion of eigenvalues of this boundary value problem under a variation of a parameter. Obtaining sufficient conditions for the asymptotic stability is reduced to finding the bifurcation value of the parameter for the boundary value problem.     

Regioselective catalytic addition of terminal acetylenes to methyl esters of 1-alkylcyclopropene-3-carboxylic acids

Conclusions Terminal acetylenes and HCN in the presence of CuCl add regioselectively as CH-acids to the unsaturated three-membered ring of methyl esters of 1-alkylcyclopropene-3-carboxylic acids to form the methyl esters of the corresponding 3-alken-5-ynoic acids or 4-cyano-3-alkyl-3-butenoic acids in yields up to 60%.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 10, pp. 2401–2405, October, 1985.These results were partially reported at the Seventh All-Union Conference on the Chemistry of Acetylene [1].     

Formation of unsaturated bicyclic γ-butyrolactone by oxidative dimerization of 2-methyl-2-ethynylcyclopropanecarboxylic acid

Conclusions The oxidative dimerization of a mixture of the trans and cis isomers of 2-methyl-2-ethynylcyclopropanecarboxylic acid in the presence of CuCl and NH₄Cl gives, along with bis(trans-1-methyl-2-carboxyl-1-cyclopropyl)diacetylene (53% yield), also the corresponding dilactone, and specifically bi(1-methyl-4-oxo-3-oxabicyclo[3.1.0]-2-hexalidene)methyl (14% yield), which is probably formed via the intramolecular cyclization of the intermediately formed bis(cis-1-methyl-2-carboxy-1-cyclopropyl) diacetylene.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 5, pp. 1164–1166, May, 1981.     

Approximating characteristic equations for autonomous systems of differential equations with aftereffect

In this paper we construct approximating polynomial characteristic equations for a linear autonomous system with aftereffect. The procedures for constructing approximating characteristic equations use analytic representations of resolvents of infinitesimal operators and the theory of characteristic determinants and perturbation determinants in a separable Hilbert space.