Y Trinuclear Cu-II structural isomers coordination, magnetism, electrochemistry and catalytic activity towards the oxidation of alkanes

The reaction of the Schiff base (3,5-di-tert-butyl-2-hydroxybenzylidene)-2-hydroxybenzohydrazide (H3L) with copper(II) nitrate, acetate or metaborate has led to the isomeric complexes [Cu-3(L)(2)(MeOH)(4)] (1), [Cu-3(L)(2)(MeOH)(2)]2MeOH (2) and [Cu-3(L)(2)(MeOH)(4)] (3), respectively, in which the ligand L exhibits dianionic (HL2-, in 1) or trianionic (L3-, in 2 and 3) pentadentate 1O,O,N:2N,O chelation modes. Complexes 1-3 were characterized by elemental analysis, IR spectroscopy, single-crystal X-ray crystallography, electrochemical methods and variable-temperature magnetic susceptibility measurements, which indicated that the intratrimer antiferromagnetic coupling is strong in the three complexes and that there exists very weak ferromagnetic intermolecular interactions in 1 but weak antiferromagnetic intermolecular interactions in both 2 and 3. Electrochemical experiments showed that in complexes 1-3 the Cu-II ions can be reduced, in distinct steps, to Cu-I and Cu-0. All the complexes act as efficient catalyst precursors under mild conditions for the peroxidative oxidation of cyclohexane to cyclohexyl hydroperoxide, cyclohexanol and cyclohexanone, leading to overall yields (based on the alkane) of up to 31% (TON = 1.55x10(3)) after 6 h in the presence of pyrazinecarboxylic acid.

Spectral and weak polynomial completeness for the product of nonsingular matrices

Let F be a field with at least four elements. In this paper, we identify all the pairs (A, B) of n x n nonsingular matrices over F, satisfying the following property: for every monic polynomial f (x) = x(n) + a(n-1)x(n-1) +... + a(1)x + a(0) over F, with a root in F and a(0) = (-1)(n) det(AB), there are nonsingular matrices X, Y is an element of F-nxn such that XAX(-1)Y BY-1 has characteristic polynomial f (x).

From weak Allee effect to no Allee effect in Richards’ growth models

Population dynamics have been attracting interest since many years. Among the considered models, the Richards’ equations remain one of the most popular to describe biological growth processes. On the other hand, Allee effect is currently a major focus of ecological research, which occurs when positive density dependence dominates at low densities. In this chapter, we propose the dynamical study of classes of functions based on Richards’ models describing the existence or not of Allee effect. We investigate bifurcation structures in generalized Richards’ functions and we look for the conditions in the (β, r) parameter plane for the existence of a weak Allee effect region. We show that the existence of this region is related with the existence of a dovetail structure. When the Allee limit varies, the weak Allee effect region disappears when the dovetail structure also disappears. Consequently, we deduce the transition from the weak Allee effect to no Allee effect to this family of functions. To support our analysis, we present fold and flip bifurcation curves and numerical simulations of several bifurcation diagrams.