It is shown that there exists a finitely generated infinite simple group of infinite commutator width, and that the commutator width of a finitely generated infinite boundedly simple group can be arbitrarily large. Besides, such groups can be constructed with decidable word and conjugacy problems.

In this paper, we prove the sharp function inequality for the multilinear commutator related to the singular integral operator with variable Calderón-Zygmund kernel. By using the sharp inequality, we obtain the Lnorm inequality for the multilinear commutator. 2000 Mathematics Subject Classification: 42B20, 42B25.

In this paper we consider the vector-valued extension of the Fefferman-Stein-Stronberg sharp maximal inequality under growth condition. As an application we obtain the vectorvalued extension of the boundedness of the commutator. Furthermore we prove the boundedness of the commutator.

We show that strongly protomodular categories (as the category Gp of groups for instance) provide an appropriate framework in which the commutator of two equivalence relations do coincide with the commutator of their associated normal subobjects, whereas it is not the case in any semi-abelian category.

Henriques and Kamnitzer defined a commutator in the category of crystals for complex semisimple Lie algebras based on Lusztig’s involution on a crystal. More recently, Kamnitzer and Tingley proved that the action of this commutator on the highest weight elements (which determines it) is given by Kashiwara’s involution on the Verma crystal. Both of these definitions depend on some maps of crysta...

In the paper we consider some commutator-type and power-type matrix equations in the group UT(∞,K) of infinite dimensional unitriangular matrices over a fieldK. We introduce a notion of a power outer commutator ω1k k (x1, ..., xk) and a power Engel commutator e1k k (x, y) as outer (respectively Engel) commutators modified by allowing powers of letters instead of letters alone. For a given infin...

We show that a suitably chosen position-momentum commutator can elegantly describe many features of gravity, including the IR/UV correspondence and dimensional reduction (‘holography’). Using the most simplistic example based on dimensional analysis of black holes, we construct a commutator which qualitatively exhibits these novel properties of gravity. Dimensional reduction occurs because the ...

This note presents purely mechanical proofs of the Levi commutator problem in group theory. The problem was solved first by using the theorem prover EQP, developed by William McCune at the Argonne National Laboratory. The fastest proof was found by using Peers-mcd, the Clause-Diffusion parallelization of EQP, developed by the author at the University of Iowa. 1 The Levi commutator problem The L...

in this paper, we generalize some transfer theorems.~in particular, we derive one of the main results of gagola(contemp math 524:49--60, 2010) from our results.

The class of commutator-free Magnus integrators is known to provide a favourable alternative to standard Magnus integrators, in particular for large-scale applications arising in the time integration of non-autonomous linear evolution equations. A high-order commutator-free Magnus integrator is given by a composition of several exponentials that comprise certain linear combinations of the value...