Abstract:
The properties of the similarity transformation in percolation theory in the complex plane of the percolation probability are studied. It is shown that the percolation problem on a two-dimensional square lattice reduces to the Mandelbrot transformation, leading to a fractal behavior of the percolation probability in the complex plane. The hierarchical chains of impedances, reducing to a nonlinear mapping of the impedance space onto itself, are studied. An infinite continuation of the procedure leads to a fixed point. It is shown that the number of steps required to reach a neighborhood of this point has a fractal distribution.

Abstract:
The electrostatics of 2D system with complicated inner boundary is studied. The object which we call "monster" is built by an iterative process of multiple conformal mapping of the circle exterior. The procedure leads to the figures built from circle with branching curved cuts (i), to the multiple tangential near-round circles (ii) and the Mandelbrot map (iii). The polarizability of a monster in the homogeneous external field is found. The light scattering cross-section was expressed through the polarizability of a monster.

Abstract:
One-dimensional tight-binding lattice, single site of which possesses harmonically vibrating level is studied. The states of non-interacting electrons incident with fixed energy from infinity are considered. It is shown that at definite conditions the site reflects electrons {\it absolutely and elastically} (high-frequency blockade states). The problem is treated both numerically and (in the case of narrow band) analytically. The results are compared with the free-electron 1D problem with vibrating $\delta$-functional potential. Together with the blockade states the local and reflectionless states are examined. Possible realization of the system as a lattice of quantum dots is discussed.

Abstract:
We demonstrate the possibility of existence of indirect moving Wannier-Mott excitons in graphene. Electron-hole binding is conditioned by the trigonal warping of conic energy spectrum. The binding energies are found for the lowest exciton states. These energies essentially depend on the value and direction of exciton momentum and vanish when the exciton momentum tends to the conic points. The ways to observe the exciton states are discussed. The opportunity of experimental observation of zero-gap excitons by means of external electron scattering is examined.

Abstract:
A one-dimensional system with two $\delta$-like barriers or wells bi-chromaticaly oscillating at frequencies $\omega$ and $2\omega$ is considered. The alternating signal leads to the direct current across the structure (even in a symmetric system). The properties of this quantum pump are studied in a wide range of the system parameters.

Abstract:
The 2D "Swiss-cheese" model of conducting media with round insulator inclusions is studied in the 2nd order of inclusion concentration and near the percolation threshold. The electric field distribution function is found to have power asymptotics for fields much exceeding the average field, independently on the vicinity to the threshold, due to finite probability of arbitrary proximity of inclusions. The strong field in the narrow necks between inclusions results in the induced persistent anisotropy of the system. The critical index for noise density is found, determined by the asymptotics of electric field distribution function.

Abstract:
The low-temperature equilibrial state of a system of small metal grains, embedded into insulator, is studied. We find, that the grains may be charged due to the fluctuations of the surface energy of electron gas in grains, rather than quantization of electron states. The higherst-occupied level in a grain fluctuates within the range of order of charging energy below the overall chemical potential. The system, called a gapless Hubbard insulator, has no overall energy gap, while the transfer of an electron on finite distances costs finite energy. The ionization energy is determined mostly by the intragrain Coulomb repulsion, rather than a weak intergrain interaction, responsible for the Coulomb gap. The hopping transport in the system is studied. The hopping energy is determined by the charging energy. At low temperature the transport has gapless character.

Abstract:
The stationary current induced by a strong running potential wave in one-dimensional system is studied. Such a wave can result from illumination of a straight quantum wire with special grating or spiral quantum wire by circular-polarized light. The wave drags electrons in the direction correlating with the direction of the system symmetry and polarization of light. In a pure system the wave induces minibands in the accompanied system of reference. We study the effect in the presence of impurity scattering. The current is an interplay between the wave drag and impurity braking. It was found that the drag current is quantized when the Fermi level gets into energy gaps.

Abstract:
The steady current induced by electromagnetic field in a 2D system with asymmetric scatterers is studied. The scatterers are assumed to be oriented cuts with one diffusive and another specular sides. Besides, the existence of isotropic impurity scatterers is assumed. This simple model simulates the lattice of half-disk which have been studied numerically recently. The model allows the exact solution in the framework of the kinetic equation. The static current response in the second order of electric field is obtained. The photogalvanic tensor contains both responses to linear and circular polarization of electromagnetic field. The model possesses non-analyticity with regards to the rate of impurity scattering.

Abstract:
Curvature of quantum wire results in intrasubband absorption of IR radiation that induces stationary photovoltage in presence of circular polarization. This effect is studied in ballistic (collisionless) and kinetic regimes. The consideration is concentrated on quantum wires with curved central part. It is shown, that if mean free path is shorter than length of the curved part the photovoltage does not depend on the wire shape, but on the total angle of rotation of wire tangent. It is not the case when mean free path is finite or large. This situation was studied for three specific shapes of wires: "hard angle", "open book" and "$\Omega$-like".