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An issue related to using cable products for building structures is to ensure their stability and durability when operating within wide limits. Therefore, the object of research was a change in the properties of the polymer sheath of the cable during the formation of a swollen coating layer under the influence of high temperature. It is proved that in the process of thermal action on the flame retardant coating, the process of thermal insulation of the cable involves the formation of particulate products on the surface of the sample. Under the action of the burner flame, a temperature was reached on the surface of the sample, which led to a swelling of the coating of more than 16mm. The measured temperature on the inverse surface of the sample was no more than 160°C, which indicates the formation of a barrier for temperature. In this regard, a calculation and experimental method for determining thermal conductivity when using a flame retardant as a coating has been developed, which makes it possible to estimate the coefficients of temperature conductivity and thermal conductivity under high-temperature action. According to the experimental data and established dependences, the coefficients of temperature conductivity and thermal conductivity of wood were calculated, which are 214.4·10–6m2/s and 0.62W/(m∙K), respectively, due to the formation of a heat-insulating swollen layer. The maximum possible temperature penetration through the thickness of the coating was assessed. A temperature was created on the surface of the sample, which significantly exceeds the ignition temperature of the polymer sheath of the cable, and, on a non-heated surface, does not exceed 160°C. Thus, there is reason to argue about the possibility of directed adjustment of the fire protection processes of an electrical cable by using coatings capable of forming a protective layer on the surface of the material, which inhibits the rate of heat transfer.

The test methods for flame propagation of long elements of the electrical wiring system, in particular, cables, cable conduits and ducts, are analyzed, and differences in them are found in the test conditions and criteria for evaluating the resistance to flame propagation. Using a substrate of a wooden board covered with a layer of tissue paper with an areal density of (21±9)g/m2, adopted for testing other elements of the electrical wiring system, a cable was identified that is not resistant to flame propagation. It is proposed to use this substrate for testing the flame propagation of cables instead of a substrate made of a double layer of filter paper with a surface density of (80±15)g/m2. In one of three experiments, a cable that was not resistant to flame propagation was found based on the criterion of the presence of ignition of the substrate located under it. To reduce the risk of making an incorrect decision on compliance, it is proposed that the assessment of long elements of the wiring system be carried out according to the rules established for cable ducts, trays and ladders in EN 50085-1 and IEC 61537. For the AVVG cable with an outer diameter of 10mm to 60mm, when it touches the blue flame cone of 1kW, the correlation coefficient of the dependence of the length of the charred part on the diameter was 0.969. For a distance of 100mm between the sample and the burner along its axis, a correlation coefficient of 0.985 was obtained. It is proposed to test cables under the second condition recommended in IEC 60695-11-2. For two conduits, flame propagation was revealed when exposed to a 1kW flame for 120 s and 240 s. However, for these pipelines, flame propagation did not occur under standard conditions of exposure to such a flame for 20 s and 25 s. To identify long elements of the wiring system that are not resistant to flame propagation, it is proposed to test them at a duration of exposure to a flame of 1kW, established for cables in IEC 60332-1-2

We show that if $G$ is a non-elementary word hyperbolic group, mapping class group of a hyperbolic surface or the outer automorphism group of a nonabelian free group then $G$ has $2^{\aleph_0}$ many continuous ergodic invariant random subgroups. If $G$ is a nonabelian free group then $G$ has $2^{\aleph_0}$ many continuous $G$-ergodic characteristic random subgroups. We also provide a complete classification of characteristic random subgroups of free abelian groups of countably infinite rank and elementary $p$-groups of countably infinite rank.

Let R be the ring of algebraic integers in a number field K and let Λ be a maximal order in a finite dimensional semisimple K-algebra B. Building on our previous work [3], we compute the smallest number of algebra generators of Λ considered as an R-algebra. This reproves and vastly extends the results of P.A.B. Pleasants, who considered the case when B is a number field. In order to achieve our goal, we obtain several results about counting generators of algebras which have finitely many elements. These results should be of independent interest.

Let G be one of the lamplighter groups $${({\Bbb Z}/p{\Bbb Z})^n} \wr {\Bbb Z}$$ and Sub(G) the space of all subgroups of G. We determine the perfect kernel and Cantor-Bendixson rank of Sub(G). The space of all conjugation-invariant Borel probability measures on Sub(G) is a simplex. We show that this simplex has a canonical Poulsen subsimplex whose complement has only a countable number of extreme points. If F is a finite group and Γ an infinite group which does not have property (T), then the conjugation-invariant probability measures on Sub( $$F \wr \Gamma $$ ) supported on $$_{ \oplus \Gamma }F$$ also form a Poulsen simplex.

The techniques of modules and actions of groups on rooted trees are applied to study the subgroup structure and the lattice subgroup of lamplighter type groups of the form ℒn,p = (ℤ/pℤ)n ≀ ℤ for n ≥ 1 and p prime. We completely characterize scale invariant structures on ℒ1,2. We determine all points on the boundary of binary tree (on which ℒ1,p naturally acts in a self-similar manner) with trivial stabilizer. We prove the congruence subgroup property (CSP) and as a consequence show that the profinite completion [Formula: see text] of ℒ1,p is a self-similar group generated by finite automaton. We also describe the structure of portraits of elements of ℒ1,p and [Formula: see text] and show that ℒ1,p is not a sofic tree shift group in the terminology of [T. Ceccherini-Silberstein, M. Coornaert, F. Fiorenza and Z. Sunic, Cellular automata between sofic tree shifts, Theor. Comput. Sci.506 (2013) 79–101; A. Penland and Z. Sunic, Sofic tree shifts and self-similar groups, preprint].

We describe how a finite-state automorphism of a regular rooted tree changes the Bernoulli measure on the boundary of the tree. It turns out that a finite-state automorphism of polynomial growth, as defined by S. Sidki, preserves a measure class of a Bernoulli measure, and we write down the explicit formula for its Radon-Nikodym derivative. On the other hand, the image of the Bernoulli measure under the action of a strongly connected finite-state automorphism is singular to the measure itself.

In this note, we construct torsion-free countable, amenable, weakly mixing groups, which answer a question of V. Bergelson. Some results related to verbal subgroups and crystallographic groups are also presented.

It is shown that the long-time dynamics (the global attractor) of the 2D Navier-Stokes system is embedded in the long-time dynamics of an ordinary differential equation, called a determining form, in a space of trajectories which is isomorphic to C1b (R;RN) for sufficiently large N depending on the physical parameters of the Navier-Stokes equations. A unified approach is presented, based on interpolant operators constructed from various determining parameters for the Navier-Stokes equations, namely, determining nodal values, Fourier modes, finite volume elements, finite elements, and so on. There are two immediate and interesting consequences of this unified approach. The first is that the constructed determining form has a Lyapunov function, and thus its solutions converge to the set of steady states of the determining form as the time goes to infinity. The second is that these steady states of the determining form can be uniquely identified with the trajectories in the global attractor of the Navier-Stokes system. It should be added that this unified approach is general enough that it applies, in an almost straightforward manner, to a whole class of dissipative dynamical systems. © 2014 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

We construct a faithful self-similar action of the discrete Heisenberg group with the following properties: This action is self-replicating, finite-state, level-transitive, and noncontracting. Moreover, there exist orbital Schreier graphs of action on the boundary of the tree with different degrees of growth. Побудовано точну самоподiбну дію дискретної групи Гейзенберга з наступними властивостями. Дія є рекурентною, скінченностановою, сферично транзитивною, нестискуючою та існують графи Шрайєра дії групи на межі дерева з різними степенями зростання.

In this paper we study algebraic and asymptotic properties of generating sets of algebras over orders in number fields. Let $A$ be an associative algebra over an order $R$ in an algebraic number field. We assume that $A$ is a free $R$-module of finite rank. We develop a technique to compute the smallest number of generators of $A$. For example, we prove that the ring $M_3(\mathbb{Z})^{k}$ admits two generators if and only if $k\leq 768$. For a given positive integer $m$, we define the density of the set of all ordered $m$-tuples of elements of $A$ which generate it as an $R$-algebra. We express this density as a certain infinite product over the maximal ideals of $R$, and we interpret the resulting formula probabilistically. For example, we show that the probability that 2 random $3\times 3$ matrices generate the ring $M_3(\mathbb{Z})$ is equal to $(\zeta(2)^2 \zeta(3))^{-1}$, where $\zeta$ is the Riemann zeta-function., Comment: 58 pages; a minor typo was corrected in formula (14)

Let $G$ be a finitely generated torsion-free nilpotent group and $\phi:H\rightarrow G$ be a surjective homomorphism from a subgroup $H, Comment: 11 pages

Let $A$ be an expanding integer $n\times n$ matrix and $D$ be a finite subset of $Z^n$. The self-affine set $T=T(A,D)$ is the unique compact set satisfying the equality $A(T)=\cup_{d\in D} (T+d)$. We present an effective algorithm to compute the Lebesgue measure of the self-affine set $T$, the measure of intersection $T\cap (T+u)$ for $u\in Z^n$, and the measure of intersection of self-affine sets $T(A,D_1)\cap T(A,D_2)$ for different sets $D_1,D_2\subset Z^n$., 5 pages, 1 figure

Classification of groups generated by 3-state automata over a 2-letter alphabet started in the first paper (see http://www.arxiv.org/abs/math/0612178) is continued., Comment: 26 pages, 31 figures

The determining modes for the two-dimensional incompressible Navier-Stokes equations (NSE) are shown to satisfy an ordinary differential equation (ODE) of the form dv/dt = F(v), in the Banach space, X, of all bounded continuous functions of the variable s∈R with values in certain finite-dimensional linear space. This new evolution ODE, named determining form, induces an infinite-dimensional dynamical system in the space X which is noteworthy for two reasons. One is that F is globally Lipschitz from X into itself. The other is that the long-term dynamics of the determining form contains that of the NSE; the traveling wave solutions of the determining form, i.e., those of the form v(t, s) = v0(t + s), correspond exactly to initial data v0 that are projections of solutions of the global attractor of the NSE onto the determining modes. The determining form is also shown to be dissipative; an estimate for the radius of an absorbing ball is derived in terms of the number of determining modes and the Grashof num...

This is the second in a series of papers presenting results on the classification of groups generated by 3-state automata over a 2-letter alphabet. Among the examples treated here, one can find automata generating the free product of 3 cyclic groups of order 2, a virtually free abelian group of rank 3, a solvable group of derived length 3, some virtually torsion-free weakly branch groups, and other interesting self-similar groups.

Let $G$ be a group and $\phi:H\to G$ be a contracting homomorphism from a subgroup $H, Comment: 25 pages, 2 figures