4,932 research outputs found
The torsion subgroup of the additive group of a Lie nilpotent associative ring of class 3
Let be the free unital associative ring freely
generated by an infinite countable set . Define a
left-normed commutator by , . For , let be the two-sided ideal in generated by all commutators . Let be the two-sided ideal of
the ring generated by all elements and .
It has been recently proved in arXiv:1204.2674 that the additive group of
is a direct sum where
is a free abelian group isomorphic to the additive group of and is an elementary abelian
-group. A basis of the free abelian summand was described explicitly in
arXiv:1204.2674. The aim of the present article is to find a basis of the
elementary abelian -group .Comment: 23 pages; extended introduction, additional reference
The subalgebra of graded central polynomials of an associative algebra
Let be a field and let be the free unital
associative -algebra on the free generating set .
A subalgebra (a vector subspace) in is called a
-subalgebra (a -subspace) if for all endomorphisms
of . For an algebra , its central polynomials
form a -subalgebra in . Over a field of
characteristic there are algebras whose algebras of all central
polynomials are not finitely generated as -subspaces in . However, no example of an algebra such that is not
finitely generated as a -subalgebra is known yet.
In the present paper we construct the first example of a -graded unital
associative algebra over a field of characteristic whose algebra of all -graded central polynomials is not finitely generated as a
-subalgebra in the free -graded unital associative -algebra . Here and are sets of even and odd free generators of ,
respectively. We hope that our example will help to construct an algebra
whose algebra of (ordinary) central polynomials is not finitely
generated as a -subalgebra in .Comment: 8 page
The additive group of a Lie nilpotent associative ring
Let Z be the free unitary associative ring freely generated by an infinite
countable set X = {x_1, x_2,...}. Define a left-normed commutator [x_1, x_2,
..., x_n] by [a,b] = ab - ba, [a,b,c] = [[a,b],c]. For n \ge 2, let T^(n) be
the ideal in Z generated by all commutators [a_1,a_2,..., a_n] (a_i \in
Z). It can be easily seen that the additive group of the quotient ring Z
/T^(2) is a free abelian group. Recently Bhupatiraju, Etingof, Jordan, Kuszmaul
and Li have noted that the additive group of Z /T^(3) is free abelian as
well. In the present note we show that this is not the case for Z /T^(4).
More precisely, let T^(3,2) be the ideal in Z generated by T^(4) together
with all elements [a_1, a_2, a_3][a_4, a_5] (a_i \in Z). We prove that
T^(3,2)/T^(4) is a non-trivial elementary abelian 3-group and the additive
group of Z /T^(3,2) is free abelian.Comment: 13 pages. Proposition 1.5, Remarks 1.6 and 2.3 and some references
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A comment on "Ab initio calculations of pressure-dependence of high-order elastic constants using finite deformations approach" by I. Mosyagin, A.V. Lugovskoy, O.M. Krasilnikov, Yu.Kh. Vekilov, S.I. Simak and I.A. Abrikosov
Recently, I. Mosyagin, A.V. Lugovskoy, O.M. Krasilnikov, Yu.Kh. Vekilov, S.I.
Simak and I.A. Abrikosov in the paper: "Ab initio calculations of
pressure-dependence of high-order elastic constants using finite deformations
approach"[Computer Physics Communications 220 (2017) 2030] presented a
description of a technique for ab initio calculations of the pressure
dependence of second- and third-order elastic constants. Unfortunately, the
work contains serious and fundamental flaws in the field of finite-deformation
solid mechanics.Comment: 3 pages, 0 figure
Limit T-subspaces and the central polynomials in n variables of the Grassmann algebra
Let F be the free unitary associative algebra over a field F on the set X
= {x_1, x_2, ...}. A vector subspace V of F is called a T-subspace (or a
T-space) if V is closed under all endomorphisms of F. A T-subspace V in F
is limit if every larger T-subspace W \gneqq V is finitely generated (as a
T-subspace) but V itself is not. Recently Brand\~ao Jr., Koshlukov, Krasilnikov
and Silva have proved that over an infinite field F of characteristic p>2 the
T-subspace C(G) of the central polynomials of the infinite dimensional
Grassmann algebra G is a limit T-subspace. They conjectured that this limit
T-subspace in F is unique, that is, there are no limit T-subspaces in F
other than C(G). In the present article we prove that this is not the case. We
construct infinitely many limit T-subspaces R_k (k \ge 1) in the algebra F
over an infinite field F of characteristic p>2. For each k \ge 1, the limit
T-subspace R_k arises from the central polynomials in 2k variables of the
Grassmann algebra G.Comment: 22 page
On identities in the products of group varieties
Let be the variety of groups satisfying the law . It is
proved that for every sufficiently large prime , say , the
product cannot be defined by a finite set of identities.
This solves the problem formulated by C.K. Gupta and A.N. Krasilnikov in 2003.
We also find the axiomatic and the basis ranks of the variety . For this goal, we improve the estimate for the basis rank of the product
of group varieties obtained by G. Baumslag, B.H. Neumann, H. Neumann and P.M.
Neumann long ago.Comment: 9 page
Influence of polymer-pore interactions on translocation
We investigate the influence of polymer-pore interactions on the
translocation dynamics using Langevin dynamics simulations. An attractive
interaction can greatly improve translocation probability. At the same time, it
also increases translocation time slowly for weak attraction while exponential
dependence is observed for strong attraction. For fixed driving force and chain
length the histogram of translocation time has a transition from Gaussian
distribution to long-tailed distribution with increasing attraction. Under a
weak driving force and a strong attractive force, both the translocation time
and the residence time in the pore show a non-monotonic behavior as a function
of the chain length. Our simulations results are in good agreement with recent
experimental data.Comment: 4 pages, 5 figures, Submitted to Phys. Rev. Let
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