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.     

Über Sesquiselenide der Lanthanoide: Einkristalle von Ce2Se3 im C‐, Gd2Se3 im U‐ und Lu2Se3 im Z‐Typ

On Sesquiselenides of the Lanthanoids: Single Crystals of C‐type Ce₂Se₃, U‐type Gd₂Se₃, and Z‐type Lu₂Se₃ Single crystals of lanthanoid sesquiselenides (M₂Se₃; here: M = Ce, Gd, Lu) are accessible through conversion of the elements (lanthanoid and selenium) in molar ratios of 2:3 within seven days at 850 °C from evacuated silica ampoules if equimolar amounts of NaCl serve as a flux. In the case of Ce₂Se₃ (a = 897.74(6) pm) und Gd₂Se₃ (a = 872.56(5) pm) the cubic C‐type (I4¯3d, Z = 5.333) forms as dark red beads, whereas the orthorhombic Z‐type (Fddd, Z = 16) emerges for Lu₂Se₃ (a = 1125.1(1), b = 798.06(8), c = 2387.7(2) pm) as orange‐yellow bricks. Upon oxidation of monochloride hydrides (MClH_x or A_yMClH_x; M = Ce, Gd, Lu; x = 1; A = Li, Na; y = 0.5) with selenium in arc‐welded tantalum ampoules the same main products appear with C‐Ce₂Se₃ and Z‐Lu₂Se₃, even with a surplus of NaCl or LiCl as fluxing agent. In the case of Gd₂Se₃, however, black‐red needles of the orthorhombic U‐type (Pnma, Z = 4; a = 1118.2(1), b = 403.48(4); c = 1097.1(1) pm) are yielded instead of C‐Gd₂Se₃. C‐Ce₂Se₃ crystallizes in a cation‐deficient Th₃P₄‐type structure (Ce₂S₃ type) according to Ce_2.667□_0.333Se₄ (Z = 4) or with Z = 5.333 for the empirical formula Ce₂Se₃. Here, Ce³⁺ is coordinated by eight Se^2— anions trigon‐dodecahedrally. In U‐Gd₂Se₃ (U₂S₃ type) two crystallographically independent Gd³⁺ cations with coordination numbers of 7 (Gd1) and 7+1 (Gd2), respectively, are present, exhibiting mono‐ or bicapped trigonal prisms as coordination polyhedra. The crystal structure of Z‐Lu₂Se₃ (Sc₂S₃ type) shows two different Lu³⁺ cations as well, which now both reside in octahedral coordination of six Se^2— anions each.     

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.