Perturbation theory and derivatives of matrix eigensystems

We sketch some recent results in the perturbation theory of the matrix eigenvalue problems Ax = λx and Ax = λBx for multiple eigenvalues. Two quite different approaches — generalizing the Bauer-Fike Theorem and differentiating the eigensystems — have been used.     

Synthesis and reactivity of sulfur and silyl substituted α-alkylidene-β-lactams

Synthetically valuable α-alkylidene-β-lactams are produced from the addition of chlorosulfonyl isocyanate to allenyl sulfides.     

Regioselective openings of 2,3-epoxy acids with organocuprates

Lithium organocuprates react with 3-alkyl-2,3-epoxy acids (glycidic acids) to afford the products of ring-opening in good yield. The regiochemistry of ring-opening is dependent on the stereochemistry of the epoxy acid.     

Comonomer and stereosequence distributions in high polymers

Comonomer and stereosequence distributions in linear high polymers are considered from the point of view of the general theory of stationary random sequences. Although the discussion is centered about binary polymerizations which may be regarded as stationary Markov processes of low order, attention is paid to distinguish results valid for any stationary random process including stationary non-Markov processes from those special to a particular Markov process. Alternative derivations of copolymer composition equations are given along with a corrected version of a previously suggested scheme of relating conditional probabilities of different orders. The nature and relative merits of the several measures proposed in the literature to characterize comonomer and stereosequence distributions are also discussed in some detail.     

Scattering of light by disordered spherulites

The scattering of light by a two-dimensional spherulite of radius R is calculated when there is disorder of optic axis orientation with respect to the radius. Special cases are considered when (1) the disorder occurs in the radial direction only, (2) the disorder occurs in the angular direction only, (3) there is combined radial and angular disorder, and (4) the optic axis makes a constant angle with the radius but there is disorder in the twist angle about the axis. In all of these calculations, a correlation function for disorder is defined and the scattering pattern depends on the ratio of the associated correlation distance to the size of the spherulite. With decreasing correlation distance, the azimuthal dependence of the scattering becomes less and there is a change in the variation of scattered intensity with scattering angles in a manner dependent upon the type of disorder.     

关于连贯与错排计数问题的评注

近两年来,连贯与错排的计数问题引起了国内某些学者的兴趣,并对之进行了一系列研究。张忠辅、蔡茂诚、林诒勋在[5]中给出了错排问题的一个计数公式;钟集[6],陈义、刘明华[7]分别利用递归方法研究了这类问题的计数。但均未能得到简洁而实用的计数公式。     

Field equations for gravity and matter

The possibility is explored that Einstein's equation straightforwardly generalized to include Fermi-type coordinates and torsion can serve as the field equation for some other fields besides the gravitational field. We find that (i) the electromagnetic field and (ii) the instanton solution of the Yang-Mills gauge field equation satisfy the generalized field equations.