Linear Ω-Algebras

Abstract

The light emitted when pion irradiated glucose is dissolved in luminol solution has been found to be proportional to the pion beam depth dose distribution in water as determined by a TE ionisation chamber. The lyoluminescence of glucose overlaps very closely with the response profile of the ionisation chamber to the 170 MeV/c Ir--mesons giving a Bragg peak to plateau ratio of 3 : 1. In comparison, the thermoluminescence response of LiF (TLD-700) to pions has been found to deviate significantly from this ratio. The close tissue equivalence of glucose, non-toxicity and its excellent lyoluminescent retention properties are important advantages over currently used dosemeters in clinical pion therapy, especially when direct in v i m measurements are considered.Pion doses ranging between 0.5 Gy (50 rad) and 30 Gy (3000 rad) were measured with an accuracy *s0/o and reproducibility 3-5%.

Operads and varieties of algebras defined by polylinear identities

“…It is worth noticing that this theory includes as special cases the theory of Ω-algebras (without any additional structure), presented in many monographs (for example, in [17]), and the theory of linear multioperator algebras [6,7]. These cases correspond to Examples 2.1 and 2.3.…”

“…Some examples of varieties over commutative operads are provided by categories of modules over commutative rings and the category of convexors [2,3]. A variety of algebras over a commutative operad C is convenient for constructing (by analogy with the theory of linear multioperator algebras [6,7]) a theory of C-linear multioperator algebras; in particular, of algebras defined by C-polylinear identities. We set forth a sketch of such a theory in § 3.…”

Classical and quantum Nambu mechanics

“…) c. Related studies in mathematics Algebras which involve multi-linear products have also been considered at various times in the mathematical literature, partly as efforts to understand or generalize Jordan algebras [28,29,33] (cf. especially the "associator"), but more generally following Higgins' study in the mid-1950s [8,20,26,32]. This eventually culminated in the investigations of certain cohomology questions, by Schlesinger and Stasheff [43], by Hanlon and Wachs [24], and by Azcárraga, Izquierdo, Perelomov, and Pérez Bueno [4,5,6], that led to results most relevant to Nambu's work.…”

“…the discussion of the modified fundamental identity, (23) et seq.). The expansions in (8) and (10) also apply to the super-maximal case as well, where they provide vanishing theorems for the sums on the RHSs of those relations.…”

Identities for the Associator in Alternative Algebras

“…There are many different ways to approach this problem. Some of these are discussed in Kurosh (1969), Baranovich and Burgin (1975), Filippov (1985), Gnedbaye (1995aGnedbaye ( ,b, 1997, Hanlon and Wachs (1995), Bremner (1997Bremner ( , 1998, and Bremner and Hentzel (2000). All the identities presented in this paper can be regarded as candidates for a ternary analogue of the Jacobi identity.…”