Transference inequalities for multiplicative diophantine exponents

We prove inequalities for multiplicative analogues of Diophantine exponents; these inequalities are similar to the ones known in the classical case. Particularly, we show that a matrix is badly approximable if and only if its transpose is badly approximable and establish some inequalities connecting multiplicative exponents with ordinary ones.     

Chiral Atropisomeric 2-Iodo-4,4′,6,6′-tetramethyl-2′-(diphenylphosphoryl)-1,1′-biphenyl-enantiodifferentiation by Rh2[MTPA]4-Adduct Formation and Conformational Analysis

Enantiodifferentiation of the chiral 2-iodo-4,4′,6,6′-tetramethyl-2′-(diphenylphosphoryl)-1,1′-biphenyl (2) can be accomplished easily by adding one mole equivalent of the enantiopure dirhodium complex Rh ^(II) ₂ [(R)-(+)-MTPA]₄ (Rh*). The internal competition of the two bindings sites in 2, the Ph ₂ P═O group, and the iodine atom was identified by variable-temperature ³¹ P NMR. Energy optimization (PM3) and ROESY spectroscopy of 2 in the absence and presence of Rh* reveal that 2 prefers a conformation in the adducts, which is the least stable one in the free molecule, i.e., adduct formation is accompanied by a rotation of the Ph ₂ P═O group about the C-2′─P bond.     

A thermodynamic analysis of stable and metastable equilibria in Fe-Mn-C alloys

Thermodynamic calculations and an analysis of phase transitions and reactions in the Fe-Mn-C system were performed taking into account all the possible stable and metastable states. Special attention was given to the concentration-temperature range of the existence of cementite-type carbides responsible for metal dusting. It was found that, at the temperature most characteristic of this transformation (873 K), cementite with the complex composition (Fe_{1 − y} Mn _y )₃C was a stable phase. An increase in temperature to 973 K slightly narrowed and deformed the region of (Fe_{1 − y} Mn _y )₃C stability. According to the results obtained, it was likely possible to create iron based metallic materials at manganese concentrations increased to 8–12 wt % stable toward destruction in active hydrogen- and carbon-containing gas media. This conclusion was substantiated experimentally.     

The physicochemical properties of <Emphasis Type="Italic">cis-trans</Emphasis> isomers of substituted epoxycyclohexanes and their molecular complexes with maleic anhydride

AM1, IR, and NMR studies of 1-methoxycarbonyl- and 1-acryloyloxymethyl-3,4-epoxycyclohexanes showed that the ratios between their trans and cis isomers were of 0.60: 0.40 and 0.45: 0.55, respectively. The equatorial forms were most favorable energetically (0.86 and 0.67 mole fractions). The trans-e and cis-e H-complexes of the epoxycyclohexanes with maleic anhydride were obtained; their heats of formation ranged from 0.8 to 3.3 kcal/mol according to AM1 calculations. The participation of the epoxide, carbonyl, and methyl or vinyl group in complex formation gave structures I, III, IIa or IIb, and IV with C-H⋯O (I–III) and C-H⋯π (IV) interactions. The vinyl group was a proton donor in complex II and a proton acceptor in complex IV. The presence of H-bonds in the structures caused the polarization of C=C bonds in 1-acryloyloxymethyl-3,4-epoxycyclohexane and maleic anhydride, which increased their reactivities. Structures IIIb and IV should exhibit the highest reactivity in copolymerization, and complex Ib should exhibit that in oxirane ring opening.     

On the theory of ferromagnetism of hexagonal diborides

The electronic structure and the magnetic properties of the MnB₂ and CrB₂ compounds with hexagonal AlB₂-type lattices were studied. The problem was treated in terms of the generalized Hubbard model with an infinite electron-electron repulsion energy in the same atom. Equations for spin magnetic susceptibility were derived and used to determine the conditions of ferromagnetic instability and construct the phase diagram of the existence of ferromagnetic ordering.     

Control of spectrum in bilinear systems

For a bilinear stationary control system, we obtain necessary and sufficient conditions for the solvability of the spectrum control problem for the case in which the coefficients have a special form.