Vol 64, No 11 (2019)

An analog of an optical waveguide in the form of a chain of metallic nanoparticles has been investigated using modified cylindrical dielectric resonators in the microwave band. Similar to the plasma oscillations in spherical nanoparticles, the two lowest resonances of the dielectric resonator correspond to the dipole and quadrupole oscillatory modes. It is shown that a waveguide consisting of seven resonators exhibits high frequency selective properties and relatively low loss if the resonances of the quadrupole oscillatory modes are used to form its passband. The characteristics of the investigated waveguide remain almost unchanged at its bending by 90°, and the cross section of localization of the main part of the energy propagating in the waveguide is smaller than the electromagnetic wavelength by a factor of five, which approximately corresponds to the optical waveguides based on plasma oscillations in nanoparticles.

The connection matrix of the Ising model on a \(d\)-dimensional hypercube is investigated. In addition to the interactions between the nearest neighbors, the interactions between the next-nearest neighbors are taken into account. For such a matrix, the exact relations for the eigenvalues and eigenvectors, which are reasonably simply expressed through the corresponding characteristics of the one-dimensional Ising model, are obtained. Both periodic and free boundary conditions are considered.

The problem of creating a stable satellite orbit about the asteroid Apophis as a homogeneous triaxial ellipsoid is investigated. The motion of the spacecraft around the asteroid has been simulated by taking into account the main perturbations (from the attraction of celestial bodies, the asteroid’s nonsphericity, and the solar radiation pressure) and the asteroid’s spin. It is shown that there is a possibility of creating a stable satellite orbit about the asteroid. This orbit can be used for a detailed study of the asteroid’s characteristics and the improvement of Apophis’ orbit through ground-based radio measurements of the motion of a satellite equipped with a radio beacon.

A waveguide occupies a domain G with several cylindrical ends. The waveguide is described by a nonstationary equation of the form \(i{{\partial }_{t}}f = \mathcal{A}f\), where \(\mathcal{A}\) is a selfadjoint second order elliptic operator with variable coefficients (in particular, for \(\mathcal{A} = - \Delta \), where Δ stands for the Laplace operator, the equation coincides with the Schrödinger equation). For the corresponding stationary problem with spectral parameter, we define continuous spectrum eigenfunctions and a scattering matrix. The limiting absorption principle provides expansion in the continuous spectrum eigenfunctions. We also calculate wave operators and prove their completeness. Then we define a scattering operator and describe its connections with the scattering matrix.

In this paper, we propose a method to initiate combustion of kerosene in a supersonic air flow in part of a combustion chamber with a constant cross section and experimentally confirm the possibility of its implementation. An intensive combustion mode is implemented using a bunch of pulses with variable energy and is maintained when the exposure is turned off after the combustion is arranged in the diverging part of the combustion chamber.