u = \[h, g, g, g], [h, gO] . Note that, with the obvious order on the given free generating set, u is a basic commutator. A routine calculation shows that if the image of u under an arbitrary endomorphism of G is expressed in terms of basic commutators, in this expression u itself will occur with square exponent (and, of course, only commutators of weight 6 occur with nonzero exponent). Consequ...

The basic higher order commutator theorem is formulated for the real interpolation methods associated with the quasi-power parameters, that is, the function spaces on which Hardy inequalities are valid. This theorem unifies and extends various results given by Cwikel, Jawerth, Milman, Rochberg, and others, and incorporates some results of Kalton to the context of commutator estimates for the re...

An autosampler/injector commutator for flow injection analysis (FIA) was constructed with electronic components of used equipments. The apparatus is controlled by commercially available multifunctional interface (PCL711B) connected to a personal computer, and the software was written in Visual Basic language. The system was applied to water analysis and it presented satisfactory results. The lo...

In this paper we determine the structure of (c,1,...,1) polynilpotent multiplier of certain class of groups. The method is based on the characterizing an explicit structure for the Baer invariant of a free nilpotent group with respect to the variety of polynilpotent groups of class row (c,1,...,1).

Comtrans algebras are unital modules over a commutative ring R, equipped with two basic trilinear operations: a commutator [x, y, z] satisfying the left alternative identity [x, x, y] = 0, (1.1) and a translator 〈x, y, z〉 satisfying the Jacobi identity 〈x, y, z〉+ 〈y, z, x〉+ 〈z, x, y〉 = 0, (1.2) such that together the commutator and translator satisfy the comtrans identity [x, y, x] = 〈x, y, x〉....

Basic ideas about noncommuting coordinates are summarized, and then coordinate noncommutativity, as it arises in the Landau problem, is investigated. I review a quantum solution to the Landau problem, and evaluate the coordinate commutator in a truncated state space of Landau levels. Restriction to the lowest Landau level reproduces the well known commutator of planar coordinates. Inclusion of ...

The basic commutators derived from the generators of a free group were introduced by Philip Hall and studied extensively by Marshall Hall, Jr. There exists a natural linear ordering for these commutators. It is the purpose of this paper to show that this ordering is, in a certain sense, invariant under "multiplication," i.e., under the process of forming the commutator with the same basic commu...

Introduction In the theory of groups, the important concepts of Abelian group, solvable group, nilpotent group, the center of a group and centraliz-ers, are all defined from the binary operation [x, y] = x −1 y −1 xy. Each of these notions, except centralizers of elements, may also be defined in terms of the commutator of normal subgroups. The commutator [M, N] (where M and N are normal subgrou...