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.     

Effect of the Scanning Step on the Surface Quality of Microdevices in Femtosecond-Laser Micromachining and its Optimization

Direct writing technique is usually used in femtosecond-laser two-photon micromachining. The size of the scanning step has important influence on the surface quality and machining efficiency of microdevices. According to the mechanism of two-photon polymerization combining the distribution function of the light intensity and the free-radical-concentration theory, we establish the mathematical model of coverage of solidification unit and analyze the effect of coverage on the machining quality and efficiency of microdevices. Using the principle of exposure equivalence, we also obtain the analytic expression of the relationship between the surface-quality characteristic parameters of microdevices and the scanning step and carry out the numerical simulation and experiment. The results show that the scanning step has little influence on the surface quality of the line when it is much smaller than the size of the solidification unit. However, when the scanning step is gradually increasing, the smoothness of the line surface is reduced rapidly, and the surface quality becomes worse drastically. For stereo-device processing with different slopes, we propose a kind three-dimensional continuous-variable scanning method and provide the calculation expression of the scanning step. From the experimental example of fabricating a spherical structure, we show that, compared with the fixed-step scanning method, the continuous-variable scanning-step method should be used to improve the surface quality of microdevices.