Two Theorems on Identities in Multioperator Algebras

Abstract

Illinois 601 15. USC M S . received 14th M a y 1970Ahtract. Starting from bare-ion pseudopotentials, net crystal potentials are evalutiied in direct space by (a) the usual linear screening technique and (b) the Thomas-Fermi method. For the metals considered (Na and AI), the potentials are quite similar, crossing outside the cores and never differing by more than 0.02 a.u. in the case of Na nor by more than 0.2 a.u. in the case of Al. The similarity of the results based on (a) and (b) suggest the use of the pseudopotential concept in conjunction with the rather general method (b) under conditions when (a) is questionable, that is whenever the zeroth order uniform electron gas solution is a poor physical approximation to the actual system under study (specific examples being molecules, insulating solids and interfaces).Methods (a) and (bj both yield shallower wells than those most often used as input information in band structure calculations and obtained by superposing atomic data.

Operads and varieties of algebras defined by polylinear identities

“…It is known that, for linear multioperator algebra over a field of characteristic zero, every variety can be defined by polylinear identities [18,19]. As for other commutative operads and the corresponding linear multioperator algebras, the question of how wide the class of varieties determined by polylinear identities is still remains open.…”