No 2 (2013)

The purpose of the present paper is to investigate the potentiality of the operator of differential difference equations and to construct the functional, if the given operator is a potential on a given set relatively to the some bilinear form. Method of construction of variational factor is suggested.

We study Riesz potentials in n-dimensional Euclidean space. They are constructed on rearrangement-invariant spaces as convolutions with kernels with general form, their description of the class of kernels is based by means of some non-negative, decreasing function Φ. Generalized Riesz potentials include classical Riesz potentials spaces. Here we consider as a “base” space RIS Lorentz type space Λ p, 1 < p < ∞. During consideration of the question of finding conditions for embeddings of Riesz type potentials in RIS we used criteria stated by M.L. Goldman, where the operator of Hardy type and inequalities for operators of this type are playing the key role. For the case of Riesz potentials, 1 < p < ∞, the condition of optimal embedding in RIS is established. The case of Riesz type potentials based on space L p, 1 < p < ∞, considered by the authors M.L. Goldman and O.M. Guselnikova, corresponds with the result of this work.

The structure of solutions in nonlinear dissipative dynamical systems described by differential equations, including systems with chaotic behavior is considered. It is shown that the structure of the solutions in these systems is provided by either set of limit cycles, or tori, and is determined by the spectrum of Floquet Exponents. Important role in forming of structures play limit cycles having a complex but not complex conjugate Floquet Exponents. Examples of using the concept of the structure of solutions of nonlinear differential equations in the study of the formation of solitary traveling waves and the phenomenon of turbulence are presented.

A numerical investigation of a thermoelastic waves arising in a nickel sample under the action of the uranium heavy ion high energy 700 MeV is made on the basis of a thermal spike model which is modified by adding to it move equations. It is shown that the arising maximal strength of the thermoelastic waves exceeds hundred times ultimate strength of a samples material on the axis of a track. The time dependence of the electron gas and ion lattice temperatures, of the thermoelastic waves strength are determined at a different distance from center of track. The time dynamic change is also determined of the maximal region within which thermoelastic strength exceeds ultimate strength of a samples material. By moving away from the center of a track along the radius, the amplitude of the thermoelastic waves will decrease inversely proportional to the radius.

Model with one spatial cylindrical variable for optimization of a technical device is presented. Device is in the form of a multilayer cylindrical sample with a pulse source operating at cryogenic temperatures. The results of numerical experiments are shown.

A method is proposed and software is developed for automatic recognition of gestures used in ASL fingerspelling. Static gestures are captured using the new generation 3D sensor Asus Xtion Pro Live. Gesture recognition is achieved by extracting and further comparing the normalized geometric skeletons of the hand. Hand skeletons are compared using Dynamic Time Warping algorithm, which has polynomial complexity.

Nonlinear sigma models with the potential of interactions (chiral cosmological models) are applied widely for description of various epochs of Universe evolution. For example, by considering an inflationary epoch, the essential progress in understanding of the essence of the phenomenon was achieved due to method of exact solutions construction for inflation. Obtained exact solutions are considered as background ones when cosmological perturbations are studied. However, the application of exact solutions construction method to modern Universe with dark energy and dark matter domination over radiation and baryonic matter, faces with considerable difficulties. Therefore one of the possibilities to overcome this difficulties is to investigate evolutionary behavior of kinetic interaction of the scalar fields, describing dark energy, considering the behavior of chiral metric components as a function of a scalar factor. In this way it becomes possible to describe both dark energy and dark matter based on special properties of sigma model inner space metric.

The numerical method for solution of the gas-dynamical equations in strict divergent form has been modified. The two-dimensional numerical code for perfect non-stationary gas-dynamical flows simulation on the polar grid is constructed. This code is based on the explicit quasimonotonic high resolution TVD-scheme.

The temperature dependences of DC and low-frequency AC conductivities of single crystals LiCu 2O 2 + δ in range from 4.5 K to 360 K were studied. The observed properties of conductivity indicated strong localization and influences of the spin, lattice and charge degrees of freedom on charge transport.

The problem of terahertz (THz) surface plasmon-polaritons (SPP) refractometry, i.e. determination of their complex refractive index κ = κ′ + i ⋅ κ′′ employing interferometric measurements, is considered in the paper. It is stated that one can determine both parts of κ provided the interference pattern formed by a reference bulk wave and the wave produced by the SPP is recorded. The idea was tested for SPP generated by monochromatic radiation (wavelength 140 μm) of Novosibirsk THz free-electron laser on gold samples covered with different thickness ZnS layers. Besides, intensity distribution of the SPP field in air over the track has been registered instantly with an uncooled vanadium oxide microbolometer focal plane array consisting of 320×240 sensitive elements. The results obtained are in good agreement with the theory provided the covering layer thickness is equal or exceeds 2 μm.