In Sobolev spaces and its subspaces considering boundary conditions theorems of existence of the generalized solutions of the conjugation problems of the Poissoʹn equation were proved. In these proofs the mollifiers operators with variable step are used. 

In this paper, we construct by means of JWKB-method a solution of the differential system emerging in the Riemann – Hilbert problem, with components satisfying the defined relations. 

By means of the method of characteristics and the Duhamel’s method we have derived a closed-form expression for a unique classical solution of a mixed problem for the inhomogeneous equation of vibration of a semibounded string with the first characteristic directional derivative in the boundary condition where all coefficients are time-dependent. The characteristic nature of this first directional derivative means that it is directed in terms of the critical characteristic of the vibration equation. We have found the necessary and sufficient conditions for the right-hand side of the equation and the initial and boundary data for single-valued everywhere solvability of this mixed problem in the set of classical solutions. 

Asymptotic properties of the maximum likelihood estimators of parameters for a binomial conditionally autoregressive model of spatio-temporal data are studied. The asymptotic normality is proved and the asymptotic covariance matrix is found for the estimators. The results of computer experiments are presented.

The problem of statistical determination of the nearest stationary time series is considered. The decision rules in a space of autoregressive coefficients are proposed and their efficiency is analytically investigated. The case of two classes is studied. 

This work considers solving a particular type of blocking flowshop scheduling problem. In the environment under consideration there are only two machines (denoted as M₁ and M₂). For each job its execution time on one of these machines equals to zero. For each job with a non-zero execution time on machine M₁ (M₂) there is a due date D₁ (D₂ respectively). The authors prove that the problem is NP-hard and propose a pseudo-polynomial time algorithm that solves it. 

In this paper we consider the mixed problem of biwave equation. Using method of characteristics the analytical solution of the mixed problem for the equation is under construction. We prove the uniqueness of the solution and conditions of the coordination of initial and boundary condition are deduced. 

The laser ablation efficiency of cornea at 213 nm radiation was investigated and optimized. It is shown that self-absorption of radiation at 213 nm in a thin cloud of ablated material takes place and strongly influences the ablation efficiency. Ablation products are being blown out in the lateral direction along the cornea surface, as a result of which the pulse fluence at the pit boundary F_bnd is more than the threshold fluence F_th. As a consequence, a pit radius r_bnd is less than a laser beam radius r_th corresponding to the threshold fluence F_th that should be taken into account in planning ablation surgery at 213 nm radiation. 

The matrix consisting of 33 miniature LEDs of Rebel Z-series emitting in the blue-green region of the spectrum was created. The maximum emission power density of the matrix was ~ 18 W/cm2 at continuous injection current. A computer model describing the temperature distribution in the matrix of LEDs depending on the supplied electrical power was developed. The simulation results are consistent with the results of measurement of the matrix LEDs temperature by a thermal imager, as well as with the results of determination of the LED active region temperature by the optical method.

In this paper different techniques of surface enhanced Raman scattering (SERS) by inorganic art pigments have been tested. Every pigment has each own dependence of the enhancement factor on nanoparticles sizes. The relative position of analyte and metal nanoparticles has no significant impact on the value of the enhancement factor. However, applying nanoparticles on top of the pigment is the most simple and less time-consuming method for sample preparation. In addition, the use of gold and silver nanoparticles provide comparable values of the intensity in the SERS spectra, but the use of gold nanostructures allows reducing the fluorescent background. So, the interpretation of spectra becomes easier.

The paper considers a new method for generation of Airy dynamic beams. Using SLM, the possibility of production of incomplete Airy beams due to light refraction by a sinusoidal phase grating is confirmed experimentally. The properties of such beams are studied. 

A method for determination of the steady-state Raman gain coefficients in crystals has been proposed. The method is based on comparing the numerical simulation data and the experimental Raman generation results. The proposed approach has been tested for barium nitrate in the field of spectrally limited nanosecond laser pulses. Its application for potassiumgadolinium tungstate has revealed the values of the Raman gain coefficients to be 14 ± 3 and 11 ± 3 cm/GW at a wavelength  of 532 nm for the E║N_m and  E║N_g  sample orientations, respectively.

We investigated the methods of transformation of coherent monochromatic light beams in biaxial crystals under the conditions of conical refraction. Also, we developed the method for determining optimal parameters of optical elements to create a converter that forms radially or azimuthally polarized laser radiation, as well as conical light beams with dislocation of a first-order wave front depending on the material of the used crystal and laser radiation wavelength.

The studies performed in this work have shown that the radiation exposure to LED sources with an emission band maximum of about 465 nm corresponding to the absorption spectrum of bilirubin is able to cause damaging effects on animal cells in culture, being in a logarithmic growth stage, that have been pre-incubated with bilirubin. The photobiological effect is caused by photo dynamic processes with participation of singlet oxygen generated by a triplet-excited sensitizer. The type of cell survival depending on an energy dose indicates that the sensitizer serves as bilirubin but not its photoproducts. 

The transformation peculiarities of oxygen-containing radiation-induced defects in Czochralski-grown silicon irradiated with fast electrons or neutrons upon subsequent high temperature annealing at T ≥ 450 ºC have been investigated by means of IR absorption. It is found that in the temperature range 450–700 ºC the vacancy-oxygen-related complexes VO_m (m ≥ 5) are formed. These complexes are responsible for the appearance of a number of vibrational absorption bands in the wave number range 980–1115 cm^–1.