Stokes eigenmodes in square domain and the stream function–vorticity correlation

The Stokes eigenmodes in the square are numerically determined and their symmetry properties are identified. The spectra evolution laws are in excellent qualitative agreement with the theoretical asymptotic predictions proposed by Constantin and Foias (in “Navier–Stokes equations”, University of Chicago Press, 1988), . The slopes are reported here and are found to be specific to the eigenmodes symmetry family. The dynamic equilibria are analyzed and show a linear relationship between the vorticity and the stream function in the core of the eigenmodes. These features of the Stokes eigenmodes confined in the square are shared by the fully periodic Stokes eigenmodes.     

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.     

Palladium-catalyzed synthesis of 2,3-dihydro-2-ylidene-1,4-benzodioxins

Palladium-catalyzed condensation of benzene-1,2-diol with various propargylic carbonates afforded regio- and stereoselectively 2,3-dihydro-2-ylidene-1,4-benzodioxins. The reaction is suggested to proceed by the formation of a (sigma-allenyl)palladium complex, followed by the intermolecular attack of the phenoxide ion on this complex to generate a new (sigma-allyl)palladium complex in equilibrium with the corresponding (eta(3)-allyl)palladium complex. Intramolecular attack of the phenoxide ion afforded the corresponding benzodioxan compound. This last attack occurs predominantly at the more electrophilic end of the (eta(3)-allyl)palladium intermediate. The Z- or E-stereochemistry of the products was established by (1)H NMR and proton NOE measurements and also by X-ray analysis on an example. The Z-stereochemistry generally observed is in agreement with the formation of this (eta(3)-allyl)palladium intermediate. However, in the case of tertiary propargylic carbonates, the E-stereochemistry generally observed could be explained by an intramolecular attack of the phenoxide ion on the intermediate (sigma-allyl)palladium complex, in slow equilibrium with the (eta(3)-allyl)palladium complex.     

A simple and efficient enantioselective synthesis of 2-alkylidene-3-alkyl-1,4-benzodioxanes by palladium-catalyzed annulation of benzene-1,2-diol and propargylic carbonates

[reaction: see text] Benzene-1,2-diol reacts with various propargylic carbonates in the presence of a palladium catalyst and a chiral atropoisomeric diphosphine to give 2-alkylidene-3-alkyl-1,4-benzodioxanes in good yields and 56-97% enantiomeric excess.     

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.