Subexponential estimations in Shirshov's height theorem

In 1993 E. I. Zelmanov asked the following question in Dniester Notebook: "Suppose that F_{2, m} is a 2-generated associative ring with the identity x^m=0. Is it true, that the nilpotency degree of F_{2, m} has exponential growth?" We show that the nilpotency degree of l-generated associative algebra with the identity x^d=0 is smaller than Psi(d,d,l), where Psi(n,d,l)=2^{18} l (nd)^{3 log_3 (nd)+13}d^2. We give the definitive answer to E. I. Zelmanov by this result. It is the consequence of one fact, which is based on combinatorics of words. Let l, n and d>n be positive integers. Then all the words over alphabet of cardinality l which length is greater than Psi(n,d,l) are either n-divided or contain d-th power of subword, where a word W is n-divided, if it can be represented in the following form W=W_0 W_1...W_n such that W_1 >' W_2>'...>'W_n. The symbol >' means lexicographical order here. A. I. Shirshov proved that the set of non n-divided words over alphabet of cardinality l has bounded height h over the set Y consisting of all the words of degree <n. Original Shirshov's estimation was just recursive, in 1982 double exponent was obtained by A.G.Kolotov and in 1993 A.Ya.Belov obtained exponential estimation. We show, that h<Phi(n,l), where Phi(n,l) = 2^{87} n^{12 log_3 n + 48} l. Our proof uses Latyshev idea of Dilworth theorem application.Comment: In Russian, 22 pages. The current version of this paper differs from the previous versions by the better estimation. English version of the article is located at http://arxiv.org/abs/1207.298

Effect of perturbed flow on the transition from the supersonic laminar boundary layer to the turbulent

Results of experimental studies on the effect of various factors on the transition of a supersonic boundary layer are discussed. It is shown that in supersonic wind tunnels, a significant effect on the transition of the boundary layer on a model is exerted by the scale of acoustic perturbations, which is proportional to the boundary layer displacement thickness of the working section. Experimental data obtained over a wide range of variation of flow parameters in aerodynamically similar test installations with different dimensions of the working section are generalized by means of a correlation parameter based on the displacement thickness

Antiferromagnetic state in bilayer graphene

Motivated by the recent experiment of Velasco Jr. {\em et al.} [J. Velasco Jr. {\em et al.}, Nat. Nanotechnology 7, {\bf 156} (2012)], we develop a mean-field theory of the interaction-induced antiferromagnetic (AF) state in bilayer graphene at charge neutrality point at arbitrary perpendicular magnetic field B. We demonstrate that the AF state can persist at all $B$. At higher $B$, the state continuously crosses over to the AF phase of the $\nu=0$ quantum Hall ferromagnet, recently argued to be realized in the insulating $\nu=0$ state. The mean-field quasiparticle gap is finite at B=0 and grows with increasing B, becoming quasi-linear in the quantum Hall regime, in accord with the reported behavior of the transport gap. By adjusting the two free parameters of the model, we obtain a simultaneous quantitative agreement between the experimental and theoretical values of the key parameters of the gap dependence -- its zero-field value and slope at higher fields. Our findings suggest that the insulating state observed in bilayer graphene in Ref. 1 is antiferromagnetic (canted, once the Zeeman effect is taken into account) at all magnetic fields.Comment: 5 pages, 3 figs; v3: published versio