Supercritical holes for the doubling map

For a map $S:X\to X$ and an open connected set ($=$ a hole) $H\subset X$ we define $\mathcal J_H(S)$ to be the set of points in $X$ whose $S$-orbit avoids $H$. We say that a hole $H_0$ is supercritical if (i) for any hole $H$ such that $\bar{H_0}\subset H$ the set $\mathcal J_H(S)$ is either empty or contains only fixed points of $S$; (ii) for any hole $H$ such that \barH\subset H_0 the Hausdorff dimension of $\mathcal J_H(S)$ is positive. The purpose of this note to completely characterize all supercritical holes for the doubling map $Tx=2x\bmod1$.Comment: This is a new version, where a full characterization of supercritical holes for the doubling map is obtaine

One class of linear Fredholm integral equations with functionals and parameters

The theory of linear Fredholm integral-functional equations of the second kind with linear functionals and with a parameter is considered. The necessary and sufficient conditions are obtained for the coefficients of the equation and those parameter values, in the nighbohood of which the equation has solutions. The leading terms of the asymptotics of the solutions are constructed. The constructive method is proposed for constructing a solution both in the regular case and in the irregular one. In the regular case, the solution is constructed as a Taylor series in powers of the parameter. In the irregular case, the solution is constructed as a Laurent series in powers of the parameter. Constructive theory and method is demonstrated on the model example

Main results of atmospheric fine structure parameter observation in the lower thermosphere

The capabilities of the radiometeor method of wind measurement increase with the increase of the transmitted power of radar stations fitted with goniometric systems which enables the observation of shower meteors along with sporadic background. In shower observations the meteor zone reflecting area narrows to the echo surface which is perpendicular to the flux radiant. Favorable conditions are created for singling out atmospheric disturbances in which the wave front is parallel to the echo surface which plays, in this case, the role of a frequency filter. For the first time this technique allowed wave disturbances with periods of approx. greater than 4 min. to be measured, with about a 99 percent probability of exceeding the level of the turbulence noise, during the Geminid and Perseid showers. Maximum values of such wave disturbance amplitudes were about 15 to 20 m/s, with lifetimes up to 2 hrs

Super Landau Models on Odd Cosets

We construct d=1 sigma models of the Wess-Zumino type on the SU(n|1)/U(n) fermionic cosets. Such models can be regarded as a particular supersymmetric extension (with a target space supersymmetry) of the classical Landau model, when a charged particle possesses only fermionic coordinates. We consider both classical and quantum models, and prove the unitarity of the quantum model by introducing the metric operator on the Hilbert space of the quantum states, such that all their norms become positive-definite. It is remarkable that the quantum n=2 model exhibits hidden SU(2|2) symmetry. We also discuss the planar limit of these models. The Hilbert space in the planar n=2 case is shown to carry SU(2|2) symmetry which is different from that of the SU(2|1)/U(1) model.Comment: 1 + 33 pages, some typos correcte