Sylow's theorem for Moufang loops

For finite Moufang loops, we prove an analog of the first Sylow theorem giving a criterion of the existence of a p-Sylow subloop. We also find the maximal order of p-subloops in the Moufang loops that do not possess p-Sylow subloops.Comment: 21 page

Infinite dimensional Lie algebra associated with conformal transformations\ud of the two-point velocity correlation tensor from isotropic turbulence

We deal with homogeneous isotropic turbulence and use the two-point velocity correlation tensor field (parametrized\ud by the time variable t) of the velocity fluctuations to equip an affine space K3 of the correlation vectors by a family\ud of metrics. It was shown in Grebenev and Oberlack (J Nonlinear Math Phys 18:109–120, 2011) that a special form of this\ud tensor field generates the so-called semi-reducible pseudo-Riemannian metrics ds2(t) in K3. This construction presents the\ud template for embedding the couple (K3, ds2(t)) into the Euclidean space R3 with the standard metric. This allows to introduce\ud into the consideration the function of length between the fluid particles, and the accompanying important problem to\ud address is to find out which transformations leave the statistic of length to be invariant that presents a basic interest of the\ud paper. Also we classify the geometry of the particles configuration at least locally for a positive Gaussian curvature of this\ud configuration and comment the case of a negative Gaussian curvature.This work was supported by FAPESP (grant No 11/50984-1), DFG Foundation (grant No OB 96/32-1) and partially by RFBR (grant No 11-01-12075-OFIM-2011)

The Extended Symmetry Lie Algebra and the Asymptotic Expansion of the Transversal Correlation Function for the Isotropic Turbulence

The extended symmetry of the functional of length determined in an affine space K3 of the correlation vectors for homogeneous isotropic turbulence is studied. The two-point velocity-correlation tensor field (parametrized by the time variable t) of the velocity fluctuations is used to equip this space by a family of the pseudo-Riemannian metrics dl2(t) (Grebenev and Oberlack (2011)). First, we observe the results obtained by Grebenev and Oberlack (2011) and Grebenev et al. (2012) about a geometry of the correlation space K3 and expose the Lie algebra associated with the equivalence transformation of the above-mentioned functional for the quadratic form dlD22(t) generated by dl2(t) which is similar to the Lie algebra constructed by Grebenev et al. (2012). Then, using the properties of this Lie algebra, we show that there exists a nontrivial central extension wherein the central charge is defined by the same bilinear skew-symmetric form c as for the Witt algebra which measures the number of internal degrees of freedom of the system. For the applications in turbulence, as the main result, we establish the asymptotic expansion of the transversal correlation function for large correlation distances in the frame of dlD22(t)

Lie algebra methods for the applications to the statistical theory of turbulence

Approximate Lie symmetries of the Navier-Stokes equations are used for the applications to scaling phenomenon arising in turbulence. In particular, we show that the Lie symmetries of the Euler equations are inherited by the Navier-Stokes equations in the form of approximate symmetries that allows to involve the Reynolds number dependence into scaling laws. Moreover, the optimal systems of all finite-dimensional Lie subalgebras of the approximate symmetry transformations of the Navier-Stokes are constructed. We show how the scaling groups obtained can be used to introduce the Reynolds number dependence into scaling laws explicitly for stationary parallel turbulent shear flows. This is demonstrated in the framework of a new approach to derive scaling laws based on symmetry analysis [11]-[13]