THE ENERGY ASPECT OF THE RECIPROCAL INTERACTIONS OF PAIRS OF TWO DIFFERENT VIBRATION MODES OF A CLAMPED ANNULAR PLATE

The standardized mutual active and reactive sound power of a clamped plate, representing the energy aspect of the reciprocal interactions of two different in vacuo modes, has been computed. It was assumed that the vibrations are axisymmetric, elastic and time harmonic, the plate's transverse deflection is small as compared with the plate's size, and that the vibration velocity is small as compared with the acoustic wavenumber generated. The Kirchhoff-Love theory of a perfectly elastic plate was used. The integral formulae for the mutual sound power were transformed into their Hankel representations which made possible their subsequent computation. A closed path integral was used to express the integral in its Hankel representation to compute the mutual active sound power. The asymptotic stationary phase method was used to compute the two magnitudes, i.e., the mutual active and reactive sound power. The results obtained are the asymptotic formulae valid for the acoustically fast waves. The oscillating as well as the non-oscillating terms have been identified in the formulae to make possible their further separate analysis. The availability of the asymptotic formulae makes possible some fast numerical computations of the mutual sound power. Moreover, the formulae presented herein, together with those for the individual modes known from the literature, make a complete basis for further computations of the total sound power of the plate's damped and forced vibrations in fluid.     

The crystal structure of CuVo3(II), a high-pressure ilmenite phase

The crystal structure of a second high-pressure copper vanadate phase, CuVO₃(II), has been determined and refined by full-matrix least-squares procedures using automatic diffractometer data to a residual R = 0.042 (R_w = 0.051). The space group is rhombohedral, $\text{R}\text{3}$, with hexagonal unit cell a = 4.966(2) and c = 14.084(5) Å [a_R = 5.501(2) Å and α = 53.66(3)°]. The structure is the fully ordered ilmenite-type and, on the basis of published magnetic data and the interatomic distances, the valence distribution Cu⁺V⁵⁺O₃ is proposed. This represents a unique example of Cu⁺ in an octahedral environment.     

Enantioselective synthesis of axially chiral 1-(1-naphthyl)isoquinolines and 2-(1-naphthyl)pyridines through sulfoxide ligand coupling reactions

Racemic 1-(1′-isoquinolinyl)-2-naphthalenemethanol rac-12 was prepared through a ligand coupling reaction of racemic 1-(tert-butylsulfinyl)isoquinoline rac-7 with the 1-naphthyl Grignard reagent 10. Resolution of rac-12 was achieved through chromatographic separation of the Noe-lactol derivatives 14 and 15, providing (R)-(−)-12 of >99% ee and (S)-(+)-12 of 90% ee. The ligand coupling reaction of optically enriched sulfoxide (S)-(−)-7 (62% ee) with Grignard reagent 10 furnished rac-12, with the absence of stereoinduction resulting from competing rapid racemisation of the sulfoxide 7. Reaction of optically enriched (S)-(−)-7 with 2-methoxy-1-naphthylmagnesium bromide was also accompanied by racemisation of the sulfoxide 7, and furnished optically active (+)-1-(2′-methoxy-1′-naphthyl)isoquinoline (+)-3b in low enantiomeric purity (14% ee). The absolute configuration of (+)-3b was assigned as R using circular dichroism spectroscopy, correcting an earlier assignment based on the Bijvoet method, but in the absence of heavy atoms. Optically active 2-pyridyl sulfoxides were found not to undergo racemisation analogous to the 1-isoquinolinyl sulfoxide 7, with the ligand coupling reactions of (R)-(+)- and (S)-(−)-2-[(4′-methylphenyl)sulfinyl]-3-methylpyridines, (R)-(+)-17 and (S)-(−)-17, with 2-methoxy-1-naphthylmagnesium bromide providing (−)- and (+)-2-(2′-methoxy-1′-naphthyl)-3-methylpyridines, (−)-18 and (+)-18, in 53 and 60% ee, respectively. The free energy barriers to internal rotation in 3b and 18 have been determined, and the isoquinoline (R)-(−)-12 examined as a ligand in the enantioselectively catalysed addition of diethylzinc to benzaldehyde; (R)-(−)-12 was also converted to (R)-(−)-N,N-dimethyl-1-(1′-isoquinolinyl)-2-naphthalenemethanamine (R)-(−)-19, and this examined as a ligand in the enantioselective Pd-catalysed allylic substitution of 1,3-diphenylprop-2-enyl acetate with dimethyl malonate.     

Gårding's Theory of Hyperbolic Polynomials

This paper presents a simple, self‐contained account of Gårding's theory of hyperbolic polynomials, together with a recent convexity result of Bauschke‐Güler‐Lewis‐Sendov and an inequality of Gurvits. This account begins by establishing some new results. The first concerns the existence of a pointwise arrangement of the eigenvalues so that they become global real analytic functions. The second asserts that the associated “branches” are independent of the choice of hyperbolic direction. © 2013 Wiley Periodicals, Inc.     

Thermal Characterizations of Inorganic and Organometallic Polymers

This review emphasizes the breadth of metallic and metallic-like polymers evaluated as to thermal properties. Techniques usefully applied to particular systems are noted with the aim of suggesting their application to other systems.     

Low-lying energy levels of the FeH molecule

Multireference configuration interaction (MRCI) and complete active space second-order perturbation theory (CASPT2) calculations are performed on Fe₂ and Fe^? ₂. Although it is not possible to definitively identify the ground states of Fe₂ and Fe^? ₂, the calculations suggest that the ground state of Fe^? ₂ in ⁸Σ^? _u derived from 3d¹³4σ² _g4σ² _u and that the states observed in photodetachment are the ⁹Σ^? _g and ⁷Σ^? _g states with a 3d¹³4σ² _g4σ¹ _u occupation, but that the ground state of Fe₂ is ⁷Δ_u(3d¹⁴4σ² _g) and is not observed in the photo-detachment spectra.