Copper-organic frameworks assembled from in situ generated 5-(4-pyridyl) tetrazole building blocks: synthesis, structural features, topological analysis and catalytic oxidation of alcohols

Two new metal- organic compounds {[Cu-3(mu(3)-4-(p)tz)(4)(mu(2)-N-3)(2)(DMF)(2)](DMF)(2)}(n) (1) and {[Cu(4ptz) (2)(H2O)(2)]}(n) (2) {4-ptz = 5-(4-pyridyl)tetrazolate} with 3D and 2D coordination networks, respectively, have been synthesized while studying the effect of reaction conditions on the coordination modes of 4-pytz by employing the [2 + 3] cycloaddition as a tool for generating in situ the 5-substituted tetrazole ligands from 4-pyridinecarbonitrile and NaN3 in the presence of a copper(II) salt. The obtained compounds have been structurally characterized and the topological analysis of 1 discloses a topologically unique trinodal 3,5,6-connected 3D network which, upon further simplification, results in a uninodal 8-connected underlying net with the bcu (body centred cubic) topology driven by the [Cu-3(mu(2)-N-3)(2)] cluster nodes and mu(3)-4-ptz linkers. In contrast, the 2D metal-organic network in 2 has been classified as a uninodal 4-connected underlying net with the sql [Shubnikov tetragonal plane net] topology assembled from the Cu nodes and mu(2)-4-ptz linkers. The catalytic investigations disclosed that 1 and 2 act as active catalyst precursors towards the microwave-assisted homogeneous oxidation of secondary alcohols (1-phenylethanol, cyclohexanol, 2-hexanol, 3-hexanol, 2-octanol and 3-octanol) with tert-butylhydroperoxide, leading to the yields of the corresponding ketones up to 86% (TOF = 430 h(-1)) and 58% (TOF = 290 h(-1)) in the oxidation of 1-phenylethanol and cyclohexanol, respectively, after 1 h under low power ( 10 W) microwave irradiation, and in the absence of any added solvent or additive.

Dynamical analysis in growth models: Blumberg’s equation

We present a new dynamical approach to the Blumberg's equation, a family of unimodal maps. These maps are proportional to Beta(p, q) probability densities functions. Using the symmetry of the Beta(p, q) distribution and symbolic dynamics techniques, a new concept of mirror symmetry is defined for this family of maps. The kneading theory is used to analyze the effect of such symmetry in the presented models. The main result proves that two mirror symmetric unimodal maps have the same topological entropy. Different population dynamics regimes are identified, when the intrinsic growth rate is modified: extinctions, stabilities, bifurcations, chaos and Allee effect. To illustrate our results, we present a numerical analysis, where are demonstrated: monotonicity of the topological entropy with the variation of the intrinsic growth rate, existence of isentropic sets in the parameters space and mirror symmetry.