One six-order partial differential equation in the presence of the Painleve property is considered in this work. Differential equations are the models of different physical processes such as tasks of nonlinear waves, processes of turbulence, drift waves in plasma, etc. Ablowitz’s hypothesis is widely used that all reductions of completely integrable partial differential equations lead to ordinary differential equations with the Painleve property. The Painleve property is the basis of classification and reduction to the canonical form of nonlinear partial differential equations, just like this property allows one to classify ordinary differential equations. The Painleve property classification of partial differential equations higher than the third order is still far from complete. This is due to the fact that the known methods of research give generally only necessary conditions for existence of the Painleve property. To prove the sufficiency, for example, it is possible to reduce the investigated equation by a suitable replacement to the equation, for which the presence of the Painleve property has already been found. Therefore, of particular interest are the methods allowing one to build the equations with the a priori Painleve property. Introduction contains the definition of the Painleve property for a partial differential equation known in the literature and describes the main method of research — resonance method. In the main part, the resonant structure is investigated and the fulfillment of necessary conditions for the presence of the Painleve property is checked. To achieve this goal, we solved the problems of constructing series representing the solution of the six-order partial differential equations containing six arbitrary functions. The convergence of the obtained series is proved by using majorant series. The terms of lesser weight are found, in the presence of which for the equation a necessary condition for existence of the Painleve property, as well as a suitable substitution reducing the obtained equation to the linear one will be satisfied. Rational solutions are built in terms of negative resonances with respect to the function φ.

The controllability of discrete systems with uncertainties is considered in this article. The concept of the ensemble of linear two-parameter discrete systems is introduced as a set of systems, whose coefficients belong to some given sets. The following problem is considered: for any initial state from the pre-assigned sets, one and the same control (universal control) is found for all ensemble systems reducing all solutions of these systems to a minimum zero neighborhood for finite time. In the case of interval uncertainty, finding the control is reduced to solving a nonlinear programming problem formulated in terms of the coefficients of the ensemble systems and the interval set of initial states. A constructive algorithm for building a desired control is proposed, an example is given.

The article relates to the classes of S-numbers in Mahler’s classification [1]. There are a number of the wellknown results on finding lower bounds in the theory of Diophantine equations. Some of them include a lower bound for approximation of rational numbers; a lower bound for approximation of algebraic numbers obtained by Liouville; a later improvement of the result of Liouville known as the Thue – Siegel – Roth theorem [2]. However, the bounds described above are considered ineffective in the sense that their proof does not give a way how to calculate them. In this case, these results and their proofs cannot be used to estimate the magnitude of the solutions of the corresponding Diophantine equations, but can be used to estimate the number of solutions of these equations. In the article, using the methods of metric theory of Diophantine approximations, we have considered individual and global lower bounds for polynomials [3]. A new global bound has been obtained for the unimprovability of Dirichlet’s theorem using the metric approach for finding a lower bound in a given interval for polynomials of a degree of no more than n and an additional condition for the modulus of the derivative of this polynomial.

The article is devoted to the algorithmic aspects of partial convexity, which is a generalization of the classical concept of convexity. Often, it is necessary to introduce the concept of partial convexity when studying many applied problems such as problems of VLSI design, image processing, database design, etc., since the requirement of traditional convexity is too restrictive for such problems. At the same time, the partial convexity preserves many useful properties of classical convexity, which makes it possible to find effective algorithms for solving these problems. A problem for recognizing directional convexity of the union of polyhedral sets in the n-dimensional linear space is investigated. We have established a necessary and sufficient attribute of the partial convexity for the union of polyhedral sets. Using this attribute of partial convexity, in the case of the finiteness of the set of directions of partial convexity, we have developed a polynomial recognition algorithm for the union of several polyhedral sets given by intersections of half-spaces, provided that the number of these sets is fixed. We note that earlier for the problem under consideration, a polynomial algorithm for solving it was known only for the case of the union of two polyhedral sets.

The numerical results for functional integrals with respect to the conditional Wiener measure, generated by the Hamiltonian of a harmonic oscillator, the Hamiltonian of an anharmonic oscillator and the Hamiltonian of a one-dimensional rectangular well, are obtained in the work. Numerical results are obtained using the method based on the expansion in eigenfunctions of the Hamiltonian generating a functional integral. Evaluation of eigenvalues used in the expansion is based on counting the number of matches of signs of terms of the Sturm sequence. Therefore this method is stable to the accumulation of errors and is well implemented on a computer. The proposed method is more effective than the previously known methods for evaluation of functional integrals over the space of functions given on long intervals.

Algorithms designed for implementation on parallel computers with distributed memory consist of computational macro operations (calculation grains) and communication operations specifying the data arrays exchange between computing nodes. The major difficulty is how to find an efficient way to organize the data exchange. To solve this problem, it is first necessary to identify information dependences between macro operations and then to generate the communication operations caused by these dependences. To automate and simplify the process of code generation, it is necessary to formalize communication operations. The formalization is known for the case of homogeneous information dependences. Such formalization uses the vectors of global dependences as a representation of dependences between the calculation grains. Also, there is a way that makes it possible to obtain the data arrays exchange, but it requires the usage of tools to work with polyhedra and does not formalize communication operations. This article presents a formalization method and a method of inclusion of communication operations into the algorithm structure (receiving and sending data arrays) in case of a parallel algorithm with affine dependences. The usage of functions determining the relationship between macro operations allowed obtaining explicit representations of communication operations. This work is a generalization of the formalization of the operations of sending data in a parallel algorithm, where operations are not divided into macro operations, as well as a generalization of some aspects of obtaining the communication operation method.

The quantum dynamics of a two-level quantum-mechanical system subjected to the external monochromatic action beyond the rotating wave approximation was investigated. It was shown that under the condition of exact resonance on the trajectories of the Bloch vectors, special points are manifested under different initial conditions. These points are classified as cusps singularities. It is revealed that at such points, the instantaneous rotation axis, relative to which the Bloch vector rotates, reverses its direction. There is a movement stop. For a nonzero frequency detuning, the cusp singularities vanish. A numerical analysis of the singularities of the trajectories of the Bloch vector without rotating wave approximation was supplemented by a study based on the use of the Floquet methods. Within the framework of this approach, recurrence relations for the spectral components of the probability amplitudes were obtained and analyzed. An analytic expression was found for the two values of quasi-energies within a fourth order of magnitude in the interaction energy. It was shown that to obtain a singular behavior of the trajectories of the Bloch vector, it is sufficient to confine by four spectral harmonics in the Floquet expansion. The results obtained are important for achieving the accuracy when performing coherent transformations with two-level systems in the cases where the rotating wave approximation is inapplicable.

Thermographic methods based on IR images are actively used in medical practice for the early diagnosis of diseases, in clinical procedures and surgical operations. One of the weak points of the above methods is that the images are noisy due to the thermal influence of a tissue layer located between a human organ under examination and a thermal imager and they therefore carry only indirectly information about the temperature regime of internal organs. In order to improve the accuracy of remote thermographic methods, as applied to the human skin, the method of temperature estimation due to the presence of heat sources of different origin in the tissue has been developed. This method is based on using the linear systems theory approach and the calculation results of a point source. Sources in the shape of a sphere, a cylinder of different orientation, straight lines, and a circle are considered. Specific features of thermal fields from heat sources are indicated. Simple computational models are obtained for a temperature from a spherical source inside and outside it in an infinite medium. Temperatures from a heated line of finite and infinite length are compared. Using the example of a cylindrical heat source, the temperature inside and outside the source is analyzed in detail as a function of thermal physical parameters, occurrence depth and sizes. Particular attention is paid to comparing the results of the temperature distribution on the surface of the skin and its sizes. The given results can be used for temperature correction in thermographic studies.

A limited number of samples and an impossible a priori control of a desired parameter value stipulate how it is important to solve the problem of selecting a training set for calibration by the multivariate spectral analysis in order to reduce a calibration error. Possible variants of a training subset selection from small data sets are shown for temperature calibration with fluorescence spectra of Yb³⁺:CaF₂ recorded in the range of 880–1120 nm with a resolution of about 0.2 nm for the temperature range from 66 to 150 °C and at a step of 2 °C. The methods applied are the uniform distribution, the Kennard and Stone algorithm and the cluster analysis in principal component space. The effect of the method choice on the calibration accuracy has been evaluated. The application of the principal component analysis gives the possibility to select spectra without a priori knowledge of a temperature, to which the fluorescence spectra correspond. The root-mean-square error of the predicted temperature value is shown to be 3.98 ° C for the uniform distribution of the training subset samples over the space of the first principal component and 1.07 ° C for the Kennard and Stone algorithm. The minimum root-mean-square prediction error of 0.98 °C is shown to be achieved with the training subset selection by the hierarchical cluster analysis in the space of the principal components of the spectra studied.

In the article, the previously constructed geometrical model, which describes the first-order phase transition dynamics in the configuration Finsler space of the Langmuir monolayer, is developed. The behavior both of a Cartan vector and a Berwald curvature of the Finsler space is studied. The Berwald curvature is found to change significantly during the first-order phase transition from liquid to crystal. The correspondence between the Berwald curvature behavior and the dynamics of monolayer thermodynamic parameters: surface pressure and compressibility, is established. The agreement between theoretical dependences and experimental data is shown. An approximate analytical expression is found for compressibility, as a function of the Berwald curvature at low compression rates. Comparison of numerical simulation results with the experimental isotherms reveals that the formation of phase nuclei with large relaxation times determines the phase transition dynamics during the monolayer formation with large compression rates.

Charge transport in the active volume of the cylindrical ionization fission chamber (FC) in the current mode has been studied. The model is based on the continuity equations for ions and electrons, as well as on Poisson’s equation for the electric field. The source for the continuity equations is calculated taking into account a correct distribution of the initial ionization density in the active volume of the chamber and the charge, mass, and energy dependent distributions of the fission fragments. The distributions of the ion and electron density and electric fields inside the active volume are found for two types of chambers – miniature chambers and “large” chambers with the consideration of the space charge. The correct algorithm for calculation of the beginning of the plateau of the current-voltage characteristic – the minimum voltage on the FC to provide the stationary operation of the chamber is given. It is shown that the often used condition of the electric field absence at the anode E (ra ) = 0 to determine this value is incorrect, since it leads to complex values of the electric field inside the chamber active volume. Neglecting the processes of ion diffusion and recombination, the sensitivity and output current of the chamber in the stationary mode are calculated. Calculations have been carried out for miniature and “large” chambers. It has been shown that the use of the approximation for the generation density of ion pairs by the fission fragment along its track to be constant, often used in practice for “large” chambers, leads to significant errors when estimating the densities of ions, electrons and electric fields inside the FC; at that, the sensitivity may differ by an order of magnitude.

The article is devoted to the investigation of surface erosion of ZrN/SiN_x multilayer films irradiated with He ions (30 keV) and annealed in vacuum at 600 °C. It was found that multilayer ZrN/SiN_x films remain resistant to blistering and flacking when irradiated with He ions (30 keV) up to a dose of 8∙10¹⁶ cm^–2. The influence of the thickness of the crystalline and amorphous layer on the surface erosion nature and degree of multilayer films as a result of post-radiation annealing at 600 °C are revealed. The possible mechanisms of blistering and flexing in multilayer systems ZrN/SiN_x are discussed in the work.

A 2D layer of spherical, crystalline Ge nanodots embedded in a SiO₂ layer was formed by low pressure chemical vapor deposition combined with furnace oxidation and rapid thermal annealing. The samples were characterized structurally by using transmission electron microscopy in plan-view and cross-section geometries. It was found that the formation of highdensity Ge dots took place due to oxidation induced by the Ge segregation. Electrical properties were controlled by measuring C–V and I–V characteristics after the formation of MOS capacitors in different oxidation conditions and the ambient medium. A strong evidence of the charge storage effect on the crystalline Ge-nanodot layer was demonstrated by the hysteresis behavior of the high-frequency C–V curves. It is shown that dry oxidation followed by its reduction increases the hysteresis value compared to wet oxidation conditions. This hysteresis behavior is discussed taking into account the decrease in the Ge concentration and a possible effect of low temperature GeO evaporation is followed by wet oxidation.