Free subalgebras of Lie algebras close to nilpotent

We prove that for every automata algebra of exponential growth, the associated Lie algebra contains a free subalgebra. For n\geq 1, let L_{n+2} be a Lie algebra with generator set x_1,..., x_{n+2} and the following relations: for k\leq n, any commutator of length $k$ which consists of fewer than k different symbols from {x_1,...,x_{n+2}} is zero. As an application of this result about automata algebras, we prove that for every n\geq 1, L_{n+2} contains a free subalgebra. We also prove the similar result about groups defined by commutator relations

Application of High-Voltage Discharges for Disinfecting Water

A three-factor experiment is conducted on the disinfection of water by treatment with high-voltage discharges formed to achieve an electro-hydraulic effect, in order to detect optimal conditions and rules for the course of the processes under study. In the study, a high-voltage installation with an electro-hydraulic spark gap, an EnSURE luminometer (Hygiena) for measuring the level of hygiene of water and its solutions, test tubes for determining the total number of ATP in AquaSnap Total brand water (AQ100X) are used as materials and equipment. The influence of design parameters and exposure modes of an electro-hydraulic installation on the properties of water as a result of the generation of high-voltage discharges is investigated; experimental data are revealed for measuring the level of microbiological contamination of the water sample, which, according to the analysis of the data obtained, is reduced, which can serve as the basis for the possibility of the potential use of the effects of high-voltage discharges as a method of preparing water under irrigation in greenhouses; optimal ratios of factors for disinfecting a pond water sample from a source of artificial origin are revealed: operating voltage 19.9 kV, capacitance 0.1445 μF and the number of discharges 2861 pieces

The images of non-commutative polynomials evaluated on 2×22\times 2 matrices

Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field $K$ of any characteristic. It has been conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set of scalar matrices, or the set $sl_n(K)$ of matrices of trace 0, or all of $M_n(K)$. We prove the conjecture for $n=2$