The torsion subgroup of the additive group of a Lie nilpotent associative ring of class 3

Let $\mathbb Z \langle X \rangle$ be the free unital associative ring freely generated by an infinite countable set $X = \{ x_1,x_2, \dots \}$. Define a left-normed commutator $[x_1,x_2, \dots, x_n]$ by $[a,b] = ab - ba$, $[a,b,c] = [[a,b],c]$. For $n \ge 2$, let $T^{(n)}$ be the two-sided ideal in $\mathbb Z \langle X \rangle$ generated by all commutators $[a_1,a_2, \dots, a_n]$ $( a_i \in \mathbb Z \langle X \rangle )$. Let $T^{(3,2)}$ be the two-sided ideal of the ring $\mathbb Z \langle X \rangle$ generated by all elements $[a_1, a_2, a_3, a_4]$ and $[a_1, a_2] [a_3, a_4, a_5]$ $(a_i \in \mathbb Z \langle X \rangle)$. It has been recently proved in arXiv:1204.2674 that the additive group of $\mathbb Z \langle X \rangle / T^{(4)}$ is a direct sum $A \oplus B$ where $A$ is a free abelian group isomorphic to the additive group of $\mathbb Z \langle X \rangle / T^{(3,2)}$ and $B = T^{(3,2)} /T^{(4)}$ is an elementary abelian $3$-group. A basis of the free abelian summand $A$ was described explicitly in arXiv:1204.2674. The aim of the present article is to find a basis of the elementary abelian $3$-group $B$.Comment: 23 pages; extended introduction, additional reference

The subalgebra of graded central polynomials of an associative algebra

Let $F$ be a field and let $F \langle X \rangle$ be the free unital associative $F$-algebra on the free generating set $X = \{ x_1, x_2, \dots \}$. A subalgebra (a vector subspace) $V$ in $F \langle X \rangle$ is called a $T$-subalgebra (a $T$-subspace) if $\phi (V) \subseteq V$ for all endomorphisms $\phi$ of $F \langle X \rangle$. For an algebra $G$, its central polynomials form a $T$-subalgebra $C(G)$ in $F \langle X \rangle$. Over a field of characteristic $p > 2$ there are algebras $G$ whose algebras of all central polynomials $C (G)$ are not finitely generated as $T$-subspaces in $F \langle X \rangle$. However, no example of an algebra $G$ such that $C(G)$ is not finitely generated as a $T$-subalgebra is known yet. In the present paper we construct the first example of a $2$-graded unital associative algebra $B$ over a field of characteristic $p>2$ whose algebra $C_2 (B)$ of all $2$-graded central polynomials is not finitely generated as a $T_2$-subalgebra in the free $2$-graded unital associative $F$-algebra $F \langle Y,Z \rangle$. Here $Y = \{ y_1, y_2, \dots \}$ and $Z = \{ z_1, z_2, \dots \}$ are sets of even and odd free generators of $F \langle Y,Z \rangle$, respectively. We hope that our example will help to construct an algebra $G$ whose algebra $C(G)$ of (ordinary) central polynomials is not finitely generated as a $T$-subalgebra in $F \langle X \rangle$.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 adde

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 ${\cal B}_n$ be the variety of groups satisfying the law $x^n=1$. It is proved that for every sufficiently large prime $p$, say $p>10^{10}$, the product ${\cal B}_p{\cal B}_p$ 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 ${\cal B}_p{\cal B}_p$. 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