Uniform partition extensions,a generating functions perspective

In this paper, a bivariate generating function CF(x, y) =f(x)-yf(xy)1-yis investigated, where f(x)= n 0fnxnis a generating function satisfying the functional equation f(x) = 1 + r j=1 m i=j-1aij xif(x)j.In particular, we study lattice paths in which their end points are on the line y = 1. Rooted lattice paths are defined. It is proved that the function CF(x, y) is a generating function defined on some rooted lattice paths with end point on y = 1. So, by a simple and unified method, from the view of lattice paths, we obtain two combinatorial interpretations of this bivariate function and derive two uniform partitions on these rooted lattice paths.     

Extraction-chromatographic behavior of Zr(IV) and Hf(IV) on TRU and LN resins in mixtures of HNO3 and HF

The behavior of group-4 homologs Zr and Hf on extraction-chromatographic sorbents LN resin and TRU resin in mixtures of HF and HNO₃ is considered. Distribution coefficients of the elements in the mixtures of 5·10⁻⁴ M–1 M HF and 0.01 M–5 M HNO₃ are determined. Strong retention of both elements was found on LN resin in the range of concentrations c(HF) ≤ 0.01 M for all concentrations of HNO₃. Retention tends to gradually disappear while increasing c(HF) to 0.5 M. On TRU resin retention is observed only in solutions with c(HNO₃) ≥ 2 M and c(HF) ≤ 0.01 M. The possibility of separating Zr(IV) and Hf(IV) on LN resin is illustrated in two different acid mixtures, whereas their separation on TRU resin under the conditions studied in this work is difficult. The results obtained can be used to isolate analytes from multicomponent mixtures during analytical tasks, as well as to separate them from each other.

     

Stability of O3 + N2, O3 + O2 and O3 + CO2 double hydrates: DFT study

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Site-Selective Supramolecular Complexation Activates Catalytic Ethane Oxidation by a Nitrido-Bridged Iron Porphyrinoid Dimer

Development of supramolecular methods to further activate a highly reactive intermediate is a fascinating strategy to create novel potent catalysts for activation of inert chemicals. Herein, a supramolecular approach to enhance the oxidizing ability of a high-valent oxo species of a nitrido-bridged iron porphyrinoid dimer that is a known potent molecular catalyst for light alkane oxidation is reported. For this purpose, a nitrido-bridged dinuclear iron complex of porphyrin-phthalocyanine heterodimer 3 ⁵⁺, which is connected through a fourfold rotaxane, was prepared. Heterodimer 3 ⁵⁺ catalyzed ethane oxidation in the presence of H₂O₂ at a relatively low temperature. The site-selective complexation of 3 ⁵⁺ with an additional anionic porphyrin (TPPS⁴⁻) through π–π stacking and electrostatic interactions afforded a stable 1:1 complex. It was demonstrated that the supramolecular post-synthetic modification of 3 ⁵⁺ enhances its catalytic activity efficiently. Moreover, supramolecular conjugates achieved higher catalytic ethane oxidation activity than nitrido-bridged iron phthalocyanine dimer, which is the most potent iron-oxo-based molecular catalyst for light-alkane oxidation reported so far. Electrochemical measurements proved that the electronic perturbation from TPPS⁴⁻ to 3 ⁵⁺ enhanced the catalytic activity.     

Room-Temperature,Metal-Free,and One-Pot Preparation of 2H-Indazoles through a Mills Reaction and Cyclization Sequence

The Mills reaction and cyclization of readily available 2-aminobenzyl alcohols and nitrosobenzenes using thionyl bromide provided 2H-indazoles in up to 88 % yields. In the metal-free process, acetic acid played a crucial role for the both Mills reaction and cyclization. A brominated 2H-indazole could also be obtained through the one-pot sequence.