The conversion efficiency of a piezoelectric quartz crystal: Relation between input electrical power and frequency

The dependence of the conversion efficiency of a piezoelectric transducer on its frequency and the input electrical power has been examined and it is found that the efficiency is dependent not on the input power but on frequency and the nature of the liquid     

Overall values of conventional discrepancy indices for related and unrelated models of a non-centrosymmetric crystal with a centrosymmetric group

Theoretical expressions for the overall values of the conventional discrepancy indicesR(F) andR(I) are derived for a non-centrosymmetric crystal with a centrosymmetric group by taking the centrosymmetric group and a part of the other atoms in the unit cell as the trial structure. These results are used to obtain tables of values of these indices in terms of the parameter σ _1c ² and σ ₁ ² which define the fractional contribution to the local mean intensity from the centrosymmetric group and all the known atoms respectively. Contribution No. 561     

Malliavin's theorem for weak synthesis on nonabelian groups

Malliavin's celebrated theorem on the failure of spectral synthesis for the Fourier algebra A(G) on nondiscrete abelian groups was strengthened to give failure of weak synthesis by Parthasarathy and Varma. We extend this to nonabelian groups by proving that weak synthesis holds for A(G) if and only if G is discrete. We give the injection theorem and the inverse projection theorem for weak X-spectral synthesis, as well as a condition for the union of two weak X-spectral sets to be weak X-spectral for an A(G)-submodule X of VN(G). Relations between weak X-synthesis in A(G) and A(G×G) and the Varopoulos algebra V(G) are explored. The concept of operator synthesis was introduced by Arveson. We extend several recent investigations on operator synthesis by defining and studying, for a V^∞(G)-submodule M of B(L²(G)), sets of weak M-operator synthesis. Relations between X-Ditkin sets and M-operator Ditkin sets and between weak X-spectral synthesis and weak M-operator synthesis are explored.     

Generating Birth and Death Processes

Associated with an ordered sequence of an even number 2N of positive real numbers is a birth and death process (BDP) on {0, 1, 2,…, N} having these real numbers as its birth and death rates. We generate another birth and death process from this BDP on {0, 1, 2,…, 2N}. This can be further iterated. We illustrate with an example from tan(kz). In BDP, the decay parameter, viz., the largest non-zero eigenvalue is important in the study of convergence to stationarity. In this article, the smallest eigenvalue is found to be useful.     

On Semimonotone Matrices, $$R_0$$ -Matrices and Q-Matrices

Journal of Optimization Theory and Applications - In 1979, Pang proved that within the class of semimonotone matrices, $$R_0$$ -matrices are Q-matrices and conjectured that the converse is also...     

Diastereoselective [3+2] Annulation of Aromatic/Vinylic Amides with Bicyclic Alkenes through Cobalt‐Catalyzed C−H Activation and Intramolecular Nucleophilic Addition

A highly diastereoselective method for the synthesis of dihydroepoxybenzofluorenone derivatives from aromatic/vinylic amides and bicyclic alkenes is described. This new transformation proceeds through cobalt‐catalyzed C?H activation and intramolecular nucleophilic addition to the amide functional group. Transition‐metal‐catalyzed C?H activation reactions of secondary amides with alkenes usually lead to [4+2] or [4+1] annulation; to the best of our knowledge, this is the first time that a [3+2] cycloaddition is described in this context. The reaction proceeds under mild conditions and tolerates a wide range of functional groups. Mechanistic studies imply that the C?H bond cleavage may be the rate‐limiting step.     

Crystal and Molecular Structure of Desmodin

The crystal structure of the title compound has been determined from X‐ray diffraction. The compound crystallizes from benzene in the orthorhombic system, space group P2₁2₁2₁, with unit cell parameters: a = 8.485(2), b = 9.816(2), c = 22.597(4) Å, Z = 4, V = 1881.9(7) ų. The structure was determined by direct methods and refined to a final R‐factor of 0.04. Six membered rings B and E are planar. Ring A and ring C are in slightly distorted sofa conformation. Ring D is in envelope conformation. The structure is stabilised by weak intermolecular C‐H...O hydrogen bonds.