Global convergence property of modified Levenberg-Marquardt methods for nonsmooth equations

In this paper, we discuss the globalization of some kind of modified Levenberg-Marquardt methods for nonsmooth equations and their applications to nonlinear complementarity problems. In these modified Levenberg-Marquardt methods, only an approximate solution of a linear system at each iteration is required. Under some mild assumptions, the global convergence is shown. Finally, numerical results show that the present methods are promising.     

苯胺电化学聚合机理的研究

主要应用电化学方法研究了苯胺及其电化学聚合产物─—聚苯胺的循环伏安曲线，同时讨论了苯胺的存在对４－氨基二苯胺循环伏安曲线的影响。结果表明：随着苯胺聚合的进行，聚苯胺同时发生降解；苯胺电化学聚合按“白催化机理”进行。     

On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers

Let A_n=Circ(F₁,F₂,…,F_n) and B_n=Circ(L₁,L₂,…,L_n) be circulant matrices, where F_n is the Fibonacci number and L_n is the Lucas number. We prove that A_n is invertible for n > 2, and B_n is invertible for any positive integer n. Afterwards, the values of the determinants of matrices A_n and B_n can be expressed by utilizing only the Fibonacci and Lucas numbers. In addition, the inverses of matrices A_n and B_n are derived.     

A flexible enantioselective approach to 3,4-dihydroxyprolinol derivatives by SmI2-mediated reductive coupling of chiral nitrone with ketones/aldehydes

A flexible enantioselective approach to polyhydroxylated prolinol derivatives was described, which is based on the samarium diiodide-mediated reductive coupling of the chiral nitrone (3S,4R)-8, derived from d-isoascorbic acid with aldehydes/ketones. Thereby, polyhydroxyprolinol derivatives 9a–e and 9h–j were obtained from aromatic ketones and aliphatic aldehydes in good to excellent yields of 65–91%. These reductive hydroxyalkylations are highly diastereoselective in establishing the C-4 stereogenic center. By this way, the asymmetric syntheses of (?)-8a-epi-swainsonine (4) and (?)-8,8a-di-epi-swainsonine (5) have been achieved.