Direct Nucleophilic Substitution of Free Allylic Alcohols in Water Catalyzed by FeCl3⋅6 H2O: Which is the Real Catalyst?

The allylic substitution reaction, and particularly the direct allylic amination reaction, of free allylic alcohols in water catalyzed by FeCl3⋅6 H2O is described. This novel environmentally-friendly methodology allows the use of a wide variety of nitrogenated nucleophiles such as sulfonamides, carbamates, benzamides, anilines, benzotriazoles, and azides, generally giving good yields of the corresponding substitution products. The synthetic applicability of the process is also demonstrated because the reaction can be performed on gram-scale. Additionally, carbon nucleophiles such as silylated nucleophiles, aromatic compounds, and malonates also proved to be suitable for this transformation. Finally, the nature of the catalytic species present in aqueous media is unveiled, pointing towards the formation of hexaaquo iron(III) complexes.

A note on the real projection of the zeros of partial sums of Riemann zeta function

This paper proves that the real projection of each simple zero of any partial sum of the Riemann zeta function ζn(s):=∑nk=11ks,n>2 , is an accumulation point of the set {Res : ζ n (s) = 0}.

On the Result of Invariance of the Closure Set of the Real Projections of the Zeros of an Important Class of Exponential Polynomials

In this paper we provide the proof of a practical point-wise characterization of the set RP defined by the closure set of the real projections of the zeros of an exponential polynomial P(z) = Σn j=1 cjewjz with real frequencies wj linearly independent over the rationals. As a consequence, we give a complete description of the set RP and prove its invariance with respect to the moduli of the c′ js, which allows us to determine exactly the gaps of RP and the extremes of the critical interval of P(z) by solving inequations with positive real numbers. Finally, we analyse the converse of this result of invariance.