On the asymptotics of the solution to a singularly perturbed hyperbolic system of equations with several spatial variables in the critical case

A complete asymptotic expansion of the solution to an initial value problem for a singularly perturbed hyperbolic system of equations in several spatial variables is constructed and justified. A specific feature of the problem is that its solution has a spike zone in a neighborhood of which the asymptotics is described by a parabolic equation.     

The oscillations of a rod in an inhomogeneous elastic medium

The local length-dependence of the natural frequencies and forms of plane transverse oscillations of a thin inhomogeneous rod in an elastic medium with a variable stiffness and arbitrary elastic-fastening boundary conditions is investigated. It is established that the presence of an external elastic medium, described by the Winkler model, can lead to an anomalous effect – an increase in the natural frequencies of lower oscillation modes as the length of the rod increases continuously. The extremely fine properties of this change as a function of the length, the mode number and the method of fastening are revealed. The oscillations in the case of standard methods of fastening are investigated separately. Simple examples, which illustrate the anomalous dependence of the natural oscillation frequencies of the rod in an extremely inhomogeneous elastic medium with different boundary conditions are calculated.     

Heterometallic Co(III)(4)Fe(III)(2) Schiff Base Complex: Structure, Electron Paramagnetic Resonance, and Alkane Oxidation Catalytic Activity

The heterometallic complex [Co(4)Fe(2)OSae(8)]·4DMF·H(2)O (1) was synthesized by one-pot reaction of cobalt powder with iron chloride in a dimethylformamide solution of salicylidene-2-ethanolamine (H(2)Sae) and characterized by single crystal X-ray diffraction analysis, magnetic measurements, high frequency electron paramagnetic resonance (HF-EPR), and M?ssbauer spectroscopies. The exchange coupling in the Fe(III)-Fe(III) pair is of antiferromagnetic behavior with J/hc = -190 cm(-1). The HF-EPR spectra reveal an unusual pattern with a hardly detectable triplet signal of the Fe(III) dimer. The magnitude of D (ca. 13.9 cm(-1)) was found to be much larger than in related dimers. The catalytic investigations disclosed an outstanding activity of 1 toward oxidation of cycloalkanes with hydrogen peroxide, under mild conditions. The most efficient system showed a turnover number (TON) of 3.57 × 10(3) with the concomitant overall yield of 26% for cyclohexane, and 2.28 × 10(3)/46%, respectively, for cyclooctane. A remarkable turnover frequency (TOF) of 1.12 × 10(4) h(-1) (the highest initial rate W(0) = 3.5 × 10(-4) M s(-1)) was achieved in oxidation of cyclohexane. Kinetic experiments and selectivity parameters led to the conclusion that hydroxyl radicals are active (attacking C-H bonds) species. Kinetic and electrospray ionization mass spectrometry (ESI-MS) data allowed us to assume that the trinuclear heterometallic particle [Co(2)Fe(Sae)(4)](+), originated from 1 in solution, could be responsible for efficient generation of hydroxyl radicals from hydrogen peroxide.     

An unprecedented heterotrimetallic Fe/Cu/Co core for mild and highly efficient catalytic oxidation of cycloalkanes by hydrogen peroxide

An unprecedented hexanuclear heterotrimetallic Fe/Cu/Co complex bearing two Cu(mu-O)2Co(mu-O)2Fe cores is easily prepared by self-assembly and acts as a remarkable catalyst for the peroxidative oxidation of cycloalkanes under mild conditions.     

Relative stability of onium states of nitrogen,oxygen, sulfur,and selenium atoms in cations formed in the elimination processes

Using the HF, B3LYP, and MP2 methods in the 6−31+G(d) basis with zero-point energy corrections, quantum chemical calculations of energy characteristics were carried out for the model molecules CH₂=XH⁺ (X = NH, O, S, Se) formed from the molecules YCH₂XH (Y = NH₂, OH, SH, SeH, Cl) either through the elimination of Y-anions or as a result of the abstraction of neutral species YH from these molecules upon their interaction with a proton. Stabilities of the onium states of nitrogen atom were shown to be considerably higher than those of the onium states of chalcogen derivatives, with the lowest stability shown by the models with the Cl atom being in the onium state. The results have been formulated in the form of stability series of the onium compounds and allowed one to interpret certain features of the aminomethylation reaction (the Mannich reaction), hydroxy- and alkoxymethylation, thio- and selenomethylation, and chloromethylation (the Blanc reaction).     

Stereoselective synthesis of and atropisomerism in 4-pyridyl-3-(1-pyridinio)-3,4-trans-1,2,3,4-tetrahydropyridines and their transformation products

A series of 2-oxo-4-pyridyl-3-(1-pyridinio)-5-cyano-3,4-trans-1,2,3,4-tetrahydropyridine-6-6-olates were prepared by condensation of pyridinium ylides with α,β-unsaturated carbonyl compounds or more conveniently by a three-component condensation of pyridinium ylides, pyridine aldehydes, and ethyl cyanoacetate 3 and/or 6 . This analogues of the above tetrahydropyridines were prepared in a similar way starting from suitable substrates. Spectroscopic data revealed that the reaction leads to trans-isomers around the C³-C⁴ bond and is atroposelective. The conformation of and tautomerism in the tetrahydropyridines are discussed in the light of ¹H NMR data. The reaction of 5-cyano-3-(3-methyl-1-pyridinio)-2-oxo-4-(3-pyridyl)-1,2,3,4-tetrahydropyridine-6-thiolate 10c with phenacyl bromide was found to give 3-hydroxy-5-oxo-7-(3-pyridyl)-6-(3-methyl-1-pyridinio)-3-phenyl-8-cyano-6,7-trans-2,3,6,7-tetrahydrothiazol[3,2a]pyridine bromide, the crystal and molecular structure of which has been determined by X-ray crystallographic analysis.