coming from an ionized phosphorus center weakly coupled to a cluster of two neutral phosphorus atoms.Note added in proof. Dr. J. J. Pearson made a calculation of J without neglecting the anisotropy of the envelope function. The results will be published in a separate article.The Frenkel-Kontorova dislocation model is modified by replacing the sinusoidal substrate force by one which is piecewise linear. Exact solutions are found for the static configuration and for the Peierls stress, a P . Good agreement is found between these values of ap and those obtained previously for a two-dimensional Rosenstock-Newell model. The atoms of the linear chain are then considered in random motion corresponding to thermal equilibrium and under an applied stress a<.ap. The time required for motion of the dislocation from one position of stable equilibrium to an adjacent one is computed by means of a rate-theory formulation adapted to the present type of problem in which the positions of all the atoms in the chain are required to vary in passing over the potential barrier. The theoretical transition times for an infinite chain are compared with analog computer results for a six-atom chain and reasonably good agreement is found.

Abstract. For a Cohen-Macaulay ring R, we exhibit the equivalence of the bounded derived categories of certain resolving subcategories, which, amongst other results, yields an equivalence of the bounded derived category of finite length and finite projective dimension modules with the bounded derived category of projective modules with finite length homologies. This yields isomorphisms of K-theory and Witt groups (amongst other invariants) and improves on terms of associated spectral sequences and Gersten complexes.

A modified Frenkel—Kontorova dislocation model, with a piecewise linear substrate potential, is extended to two dimensions to describe the entire slip plane of the dislocation with one kink. Accurate solutions are found, with the aid of several analytic techniques and a digital computer, for the static configuration under applied stress. The kink Peierls stress σk P, the kink width w, and the kink energy F, are determined for a range of values of γ, the perfect crystal shear strength. σk P is found to be very sensitive to γ, and to have maxima of the order of 10−6 to 10−5, in units of the shear modulus. w and E are less sensitive to γ; w ranges from ∼4 to ∼30 lattice parameters, while E varies from several hundred to several thousand degrees Kelvin for typical crystal parameters.

For an associative ring $R$, the projective level of a complex $F$ is the smallest number of mapping cones needed to build $F$ from projective $R$-modules. We establish lower bounds for the projective level of $F$ in terms of the vanishing of homology of $F$. We then use these bounds to derive a new version of The New Intersection Theorem for level when $R$ is a commutative Noetherian local ring.Comment: To appear in the Journal of Algebra. In this new version, the paper has been rewritten to study projective levels, and to account for the existence of balanced big Cohen-Macaulay algebra

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