Geometry of word equations in simple algebraic groups over special fields

This paper contains a survey of recent developments in investigation of word equations in simple matrix groups and polynomial equations in simple (associative and Lie) matrix algebras along with some new results on the image of word maps on algebraic groups defined over special fields: complex, real, p-adic (or close to such), or finite.Comment: 44 page

From Thompson to Baer-Suzuki: a sharp characterization of the solvable radical

We prove that an element $g$ of prime order $>3$ belongs to the solvable radical $R(G)$ of a finite (or, more generally, a linear) group if and only if for every $x\in G$ the subgroup generated by $g, xgx^{-1}$ is solvable. This theorem implies that a finite (or a linear) group $G$ is solvable if and only if in each conjugacy class of $G$ every two elements generate a solvable subgroup.Comment: 28 page

Equations in simple Lie algebras

Given an element $P(X_1,...,X_d)$ of the finitely generated free Lie algebra, for any Lie algebra $g$ we can consider the induced polynomial map $P: g^d\to g$. Assuming that $K$ is an arbitrary field of characteristic $\ne 2$, we prove that if $P$ is not an identity in $sl(2,K)$, then this map is dominant for any Chevalley algebra $g$. This result can be viewed as a weak infinitesimal counterpart of Borel's theorem on the dominancy of the word map on connected semisimple algebraic groups. We prove that for the Engel monomials $[[[X,Y],Y],...,Y]$ and, more generally, for their linear combinations, this map is, moreover, surjective onto the set of noncentral elements of $g$ provided that the ground field $K$ is big enough, and show that for monomials of large degree the image of this map contains no nonzero central elements. We also discuss consequences of these results for polynomial maps of associative matrix algebras.Comment: 22 page

Nonlinear magnetoelectric effect in a ferromagnetic-piezoelectric layered structure induced by rotating magnetic field

The magnetoelectric (ME) effect induced by a rotating magnetic field, h, in the presence of a dc magnetic field, H 0, is investigated in a disk-shaped ferromagnetic FeBSiC - piezoelectric lead zirconate titanate bilayer structure. It is found that, due to the nonlinear field-dependence of magnetostriction λ(H) in the ferromagnetic layer, voltage harmonics are generated. These harmonics have a specific dependence of their amplitude and phase on H 0 and h, which is different from the case of excitation with a linearly polarized field. A theory is developed that describes characteristics of the ME effect for the cases of weak h ≪ H 0 and strong h ≫ H 0 excitation fields. The effect can be employed in designing highly sensitive sensors of permanent and alternating magnetic fields. </p

Organising Waste Management in Marsabit County

This Bachelor’s Thesis was commissioned by HAMK Sheet Metal Center and is a part of HAMK’s contribution to the project of Sustainable Housing in Marsabit County, Kenya. The aim of the thesis was to create a technical plan for the implementation of a sustainable and low-tech waste management system that can be executed in hot and arid climatic conditions of Marsabit. The thesis consists of detailed descriptions of a number of small- and big-scale solutions along with a set of policies and strategies that should be implemented in order to ensure effective waste management. As a result of the thesis, recommendations are provided on how to utilize the described techniques in tandem to achieve better efficiency of the waste management system