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.     

Isolation and Crystal Structure of Novel by-products Observed During the Formation of bis(thiosemicarbazones)

During the condensation of 4-methyl-4-phenyl-3-thiosemicarbazide (3) with 2,3-butanedione an unexpected by-product, 8a-methyl-6,8a-dihydro-8-methyl-2-[N-methyl-N-phenylamino]-5H-1,3,4-thiadiazolo[3,2-d][1,2,4]triazine-5-thione, (5) was obtained in addition to the desired bis(thiosemicarbazone) (4). An X-ray crystal structure was obtained and is presented together with a proposed mechanism for its formation. A brief investigation of the coordination chemistry of bicycle (5) was carried out. The formation of cyclic compounds during the synthesis of analogues with extended backbones is also described.     

Gold complexes with potentially tri- and tetradentate phosphinothiolate ligands

Reactions of [Au(PPh3)Cl], (Bu4N)[AuCl4] and the organometallic gold complex [Au(damp-C1,N)Cl2] (damp- = 2-(N,N-dimethylaminomethyl)phenyl) with the potentially tri- and tetradentate proligands PhP(C6H3-SH-2-R-3)2 (H2L1a, R = SiMe3; H2L1b, R = H) and P(C6H4-SH-2)3 (H3L2) result in the formation of mono- or dinuclear gold complexes depending on the precursor used. Monomeric complexes of the type [AuL1Cl] are formed upon the reaction with [Au(damp-C1,N)Cl2], but small amounts of dinuclear [AuL1]2 complexes with gold in two different oxidation states, +1 and +3, have been isolated as side-products. The dinuclear compounds are obtained in better yields from [AuCl4]-. A dinuclear complex having two Au(III) centers can be isolated from the reaction of [Au(PPh3)Cl] with H3L2, whereas from the reaction with H2L1b the mononuclear [Au(Ph3P)HL1b] is obtained, which contains a three-coordinate gold atom. Comparatively short gold-gold distances have been found in the dinuclear complexes (2.978(2) and 3.434(1) A). They are indicative of weak gold-gold interactions, which is unusual for gold(III).     

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.