On the permutability of Bäcklund transformations: 1 infinitesimal Bäcklund transformations of the sine-Gordon equation

Letu′=B _a u be the Bäcklund transformation of the sine-Gordon equation, we prove that $$B_{a + \varepsilon } B_a^{ - 1} u = u + \varepsilon \sum\limits_{n = 0}^\infty {2D^{ - 1} } \frac{{\delta G_{n + 1} }}{{\delta u}}a^{2n} ,$$ where {G _n} is an infinite set of conserved densities of the sine-Gordon equation and η ⁿ ≡D ^?1δG _n /δu are just the symmetries obtained by Olver [17]. Basing upon this expansion, we prove the equivalence between the permutability of the infinitesimal Bäcklund transformations and the involution of the conserved densities of the sine-Gordon equation.     

Derivatives of 2H-pyrazolo[4,5-b]-1,4-thiazine

Derivatives of 2H-pyrazolo[4,5-b]-1,4-thiazine, which tend to undergo dimerization at the 2-methylene group were synthesized by reaction of 5-amino-3-methyl-4-thio-1-phenylpyrazole with -bromoacetophenones.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 2, pp. 265–268, February, 1989.     

Asymmetric intramolecular Friedel–Crafts reaction catalyzed by a spiropyrrolidine organocatalyst: Enantioselective construction of indolizine and azepine frameworks

The asymmetric intramolecular Friedel–Crafts type Michael reaction of α,β-unsaturated aldehyde with pyrrole, catalyzed by a spiropyrrolidine (SP)-type organocatalyst, has been accomplished, which allows the construction of a series of azepine and indolizine frameworks with high to excellent enantioselectivities (up to 98% ee). Moreover, the substrate scope could be extended to generate a quaternary center in indolizine frameworks (up to 96% ee).     

Growth of Solutions of Some Linear Difference Equations with Meromorphic Coefficients

In this paper, we investigate the properties of solutions of some linear difference equations with meromorphic coefficients, and obtain some estimates on growth and value distribution of these meromorphic solutions.     

Copper-catalyzed aerobic oxidative dehydrogenation for conversion of 2-(alkylthio)-1,4-dihydropyrimidines to 2-(alkylthio)pyrimidines

A simple and efficient aerobic oxidative dehydrogenation reaction method for the conversion of 2-(alkylthio)-1,4-dihydropyrimidines to 2-(alkylthio)pyrimidines using copper catalyst with no additives, such as an oxidant, acid, or base, has been developed. The reaction was successful with a wide range of 2-(alkylthio)-1,4-dihydropyrimidine substrates.