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 $n=2$

The complexes with property of uniform ellipticity

This paper is devoted to construction of finitely presented infinite nil semigroup with identity $x^9=0$. This construction answers to the problem of Lev Shevrin and Mark Sapir. The paper is quite long so the proof is separated into geometric, combinatorial and finalization parts. In the first part we construct uniformly elliptic space. Space is called {\it uniformly elliptic} if any two points $A$ and $B$ at the distance of $D$ can be connected by the system of geodesics which form a disc with width $\lambda\cdot D$ for some global constant $\lambda>0$. In the second part we study combinatorial properties of the constructed complex. Vertices and edges of this complex coded by finite number of letters so we can consider semigroup of paths. Defining relations correspond to pairs of equivalent short paths on the complex. Shortest path in sense of natural metric correspond nonzero words in the semigroup. Words which are not presented as paths on complex and words correspond to non shortest paths can be reduced to zero. In the third part we make a finalization. In particular, we show that word containing ninth degree word can be reduced to zero by defining relations. The present paper contains first part of the proof. This work was carried out with the help of the Russian Science Foundation Grant N 17-11-01377. The first author is the winner of the contest Young Mathematics of Russia .Comment: 32 pages, 12 figures, in Russia

О равномерно рекуррентных словах, порождаемых перекладыванием отрезков, в том числе с изменением ориентации

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Stochastic Processes Occurring during the Transition of Technical State of the Structure

The possibility of applying the theory of stochastic processes for evaluating the dynamic pattern of different states is studied for critical-duty structures. Heterogeneous Markovian processes of technical state transition for metallurgical overhead crane structure, Markovian theorem and Kolmogorov-Chapman equation are analyzed. Markovian chain is reviewed at t →∞, i.e. under marginal steady-state (stabilized) condition. Real values of limit probabilities are obtained for the structure of the metallurgical overhead crane under review. The proposed approach redefines and elaborates the existing methods and procedures for evaluating the technical state of structures and reduces the level of ambiguity associated with such kind of problems

Nonlinear Dynamics of Heavy Structures

At the moment, not enough attention is paid to different aspects of nonlinear dynamics for heavy structures. In this article we attempt to create a mathematical model for finding a frame (field) with predictable dynamic pattern of load-carrying capability for a heavy structure based оn the parameters of its reliable (failure-free, low-risk) operation. It is difficult to find a solution for this problem now but the following algorithm can be applied. Small dimension projection is first obtained for orthonormal vectors determining the structural load-carrying capability. Then we use available methods to find a field where any relationship (functional, logical) can be obtained between the rules (wild cards) and the load-carrying capability displayed by a heavy structure. This article carries on the cycle of activities on structural risk analysis involving heavy structures. Numerical and calculated data are based on previous studies. The analysis is performed on a metallurgical overhead crane. The obtained findings are used for adopting various engineering solutions at different stages of heavy structure operation