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.     

Ascorbate-induced oxidation of formate by peroxodisulfate: product yields,kinetics and mechanism

The slow reaction between peroxodisulfate and formate is significantly accelerated by ascorbate at room temperature. The products of this induced oxidation, CO₂ and oxalate (C₂O^2– ₄), were analyzed by several methods and the kinetics of this reaction were measured. The overall mechanism involves free radical species. Ascorbate reacts with peroxodisulfate to initiate production of the sulfate radical ion (SO^– ₄), which reacts with formate to produce carbon dioxide radical ion (CO^– ₂) and sulfate. The carbon dioxide radical reacts with peroxodisulfate to form CO₂ or self-combines to form oxalate. Competition occurring between these two processes determines the overall fate of the carbon dioxide radical species. As pH decreases, protonation of the carbon dioxide radical ion tends to favor production of CO₂.     

Synthesis of the 4-methyl-1,2-oxazetidine-4-carboxylic acid moiety of the originally proposed halipeptin A and B structures

O-Alkylation of N-hydroxycarbamate 6 with iodo ester 5 affords 15, which was elaborated to mesylate 4. Intramolecular N-alkylation affords methyl N-Boc-4-methyl-1,2-oxazetidine-4-carboxylate (3). The geminal coupling constant of the methylene protons is 8.5 Hz, which is much smaller than the 12.0 Hz reported for halipeptins A and B. This confirms that the halipeptins do not contain an oxazetidinecarboxylic acid as originally proposed in structure 1, but a thiazoline as in the revised structure 2. The unusual O-alkylation of 5 probably proceeds by an electron transfer mechanism.     

Transition-state effects in acid-catalyzed aryl epoxide hydrolyses

The hydronium ion-catalyzed hydrolyses of 5-methoxyindene 1,2-oxide and of 6-methoxy-1,2,3,4-tetrohydronaphthalene-1,2-epoxide were each found to yield 75-80% of cis diol and only 20-25% of trans diol as hydrolysis products. The relative stabilities of the cis and trans diols in each system were determined by treating either cis or trans diols with perchloric acid in water solutions and following the approach to an equilibrium cis/trans mixture as a function of time. These studies establish that the trans diol in each system is more stable than the corresponding cis diol. Thus, acid-catalyzed hydrolysis of each epoxide, which proceeds via a carbocation intermediate, yields the less stable cis diol as the major product. Transition-state effects, presumably of a hydrogen-bonding nature, selectively stabilize the transition state for attack of water on the intermediate 2-hydroxy-1-indanyl carbocation leading to the less stable cis diol in this system. Transition-state effects must also be responsible for formation of the less stable cis diol as the major product in the acid-catalyzed hydrolysis of 5-methoxy-1,2,3,4-tetrahydronaphthalene 1,2-epoxide. However, in this system steric effects at the transition state may be more important than hydrogen bonding in determining the cis/trans diol product ratio. The synthesis of 5-methoxyindene 1,2-oxide and a study of its rate of reaction as a function of pH in water and dioxane-water solutions are reported. Both an acid-catalyzed reaction leading to only diol products and a pH-independent reaction yielding 71% of 5-methoxy-2-indanone and 29% of diols are observed; the half-life of its pH-independent reaction in water is only 2.4 s.     

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.