A classification of the ways in which an element of a free group can be expressed as a product of commutators or as a product of squares is given. This is then applied to some particular classes of elements. Finally, a question about expressing a commutator as a product of squares is addressed.

We study the general form of the * –commutator treated as a deformation of the Poisson bracket on the Grassman algebra. We show that, up to a similarity transformation , there are other deformations of the Poisson bracket in addition to the Moyal commutator (one at even and one at odd n, n is the number of the generators of the Grassman algebra) which are not reduced to the Moyal commutator by ...

Based on the concept of double central extension from categorical Galois theory, we study a notion of commutator which is defined relative to a Birkhoff subcategory B of a semi-abelian category A. This commutator characterises Janelidze and Kelly’s B-central extensions; when the subcategoryB is determined by the abelian objects inA, it coincides with Huq’s commutator; and when the category A is...

Let G be the first Grigorchuk group. We show that the commutator width of G is 2: every element g ∈ [G, G] is a product of two commutators, and also of six conjugates of a. Furthermore, we show that every finitely generated subgroup H ≤ G has finite commutator width, which however can be arbitrarily large, and that G contains a subgroup of infinite commutator width. The proofs were assisted by ...

Commutator errors arise in large-eddy simulation of incompressible turbulent flow from the application of non-uniform filters to the continuity – and Navier– Stokes equations. For non-uniform, high order filters with bounded moments the magnitude of the commutator errors is shown to be of the same order as that of the turbulent stress fluxes. Consequently, one cannot reduce the size of the comm...

Electroweak vacuum transition processes (sphalerons) in the early Universe provide a possible explanation of the baryon asymmetry. Combining this physics with the anomalous commutators of Adler and Boulware and renormalization group invariance, we argue that electroweak baryon number violation also induces a “topological condensate” in the vacuum. QCD sphaleron processes act to distribute the b...

A new notion of commutator was defined in any variety of Ω-groups A, relative to any fixed subvariety B of A [1]. This notion gives back the usual commutator of normal subgroups when A is the variety of groups and B the subvariety of abelian groups (see Section 2), but also the Peiffer commutator of normal precrossed submodules, when A is the variety of precrossed modules and B the subvariety o...

We investigate commutator operations on compatible uniformities. We define a commutator operation for uniformities in the congruence-modular case which extends the commutator on congruences, and explore its properties. Introduction The purpose of this paper is to generalize the commutator of congruences to a commutator of compatible uniformities. Commutator theory (on congruences) works best fo...