Binap-silver-catalyzed enantioselective multicomponent 1,3-dipolar cycloaddition of azomethines ylides derived from ethyl glyoxylate

The enantioselective binap–silver catalyzed multicomponent 1,3-dipolar cycloaddition using ethyl glyoxylate, phenylalanine ethyl ester, and maleimides is described. The employment of basic silver carbonate allows the reaction to take place in the absence of an extra base giving high yields and ee. In addition, low-level calculations regarding the importance of the benzyl substituent at the α-position of the amino ester justify the expected absolute configuration of the final cycloadducts and the observed high enantiodiscrimination.

On the Analytic Solutions of the Functional Equations w 1 f(a 1 z) + w 2 f(a 2 z) + ... + w n f(a n z) = 0

In this paper, it is showed that, given an integer number n ≥ 2, each zero of an exponential polynomial of the form w1az1+w2az2+⋯+wnazn, with non-null complex numbers w 1,w 2,…,w n and a 1,a 2,…,a n , produces analytic solutions of the functional equation w 1 f(a 1 z) + w 2 f(a 2 z) + ... + w n f(a n z) = 0 on certain domains of C, which represents an extension of some existing results in the literature on this functional equation for the case of positive coefficients a j and w j.