This work is devoted to the construction of approximate formulas for calculation of mathematical expectation of nonlinear functionals defined along the trajectories of random processes. Computation of mathematical expectation of functionals of random processes by the quadrature method is the task that depends heavily on a form in which the process is given. A lot of functional quadrature formulas are built in the cases where the characteristic functional of the process is known in explicit form. Some results are obtained in the cases where the process is the solution of the stochastic differential Itό equation. Recently, the author has proposed the approach to an approximate evaluation of mathematical expectation of a class of nonlinear random functionals based on the use of the Wiener chaos expansion. The article uses chaos expansion with respect to multiple Poisson – Ito integrals to construct functional quadrature formulas for calculating nonlinear functionals of the stochastic process defined on the probability space generated by the Poisson process. The formula is exact for the thirddegree symmetric functional polynomial, so the product formula of multiple Poisson – Ito integrals is used for construction.

 

This article is devoted to the problem of construction and research of the generalized interpolation Hermite –Birkhofftype formulas. For the scalar argument functions, the algebraic and trigonometric interpolation Hermite – Birkhoff-type polynomials, containing the value of the differential operator of special form at one of the nodes, are constructed. In the both cases, the differential operator of special form annuls the first basic functions of the corresponding Chebyshev system. Furthermore, the order of the differential operator does not depend on the number of nodes. For interpolation polynomials, the satisfaction theorems of interpolation conditions are proved. The classes of the polynomials, for which the interpolation formulas are exact, are determined. The trigonometric analogue of the Leibniz formula is constructed. This formula is used to prove the satisfaction theorem of interpolation conditions in the trigonometric case. The represenations and estimates of the interpolation error are obtained. In algebraic case, to obtain the representations and estimates of interpolation error, the consequence of Rolle’s theorem is used. In the trigonometric case, the integral representation of the interpolation error is utilized. The illustrative example of application of the trigonometric interpolation formula is constructed. The results can be used in the theoretical research as a basis for constructing both approximation methods of linear operators and approximate methods of solving some nonlinear operator equations that are available in nonlinear dynamics, mathematical physics.

 

This article is concerned with studying the classical solutions of boudary problems for the fourth-order nonstrictly hyperbolic equation with double characteristics. A classical solution is understood as a function that is defined everywhere in the domain closure and has all classical derivatives entering the equation and the problem conditions. The classical solution is built in analytical form for higher-order equations of interest for computational mathematics in testing numerical algorithms. Note that the correct formulation of mixed problems for hyperbolic equations not only depends on the number of characteristics, but also on their location. The operator appearing in the equation involves a composition of first-order differential operators. The equation is defined in the half-band of two independent variables. There are Cauchy’s conditions on the domain bottom and Dirichlet’s conditions and Neumann’s conditions on other boundary. Using the method of characteristics, the analytic solution of the considered problem is written. The uniqueness of the solutions is proved. In addition, it states: under what conditions a linear differential equation with constant fourth-order coefficients can be represented in the form of the nonstrictly hyperbolic equation considered in the article.

 

The mixed problem for the wave equation with one integral condition and one Dirichlet’s condition on the right boundary of the domain is considered in the one-dimensional case. It is proved that the fulfillment of the matching conditions is necessary and sufficient for existence and uniqueness of the classical solution of the given mixed problem under certain smoothness conditions for the given functions. The method of characteristics is used for analysis of the problem. This method is reduced to partitioning the original domain by characteristics line in sub-domains where the solution of the given problem is constructed with the help of initial, boundary and integral conditions. However, in some sub-domains the solution of the problem is reduced to Volterra’s second-type equation. For this equation, the theorems of correct solvability are fulfilled. Matching conditions are obtained by equating the values of the solution and its derivatives up to the second-order, including on characteristics. The obtained results allow building either the analytical solution of the given problem if Volterra’s equation solution can be constructed in explicit form, or the approximate solution with the help of numerical methods. However, in building the approximate solution, the additional conjugation conditions for solution and its derivatives should be introduced on characteristics.

 

The object of research is the Markov queuing network with infinite-server queues. The disciplines of the customer’s service in queuing systems (QS) are FIFO (first come – first served), service rates of customers are distributed exponentially with their own rates for each QS in each line of QS. The purpose of the research is to obtain sufficient conditions for representability of non-stationary state probabilities of such a network operating within the heavy-traffic regime in the multiplicative form. In the introduction, the field of applications of Markov networks with infinite-server queues has been described; the relevance of this work has also been indicated; a brief overview of the previous results on this subject has been given. In the main part, the network has been shown; the system of Kolmogorov’s difference-differential equations for the state probabilities of the network conditions has been derived. The main result of this article is as follows, i.e. the multiplicative form of the non-stationary state probabilities of the above-mentioned Markov network operating within the heavy-traffic regime is formulated and proved as a theorem. The obtained results can be used for modeling the behavior of information and computer systems and networks, transportation systems, insurance companies, banking networks and other facilities, the stochastic models which are the queuing networks.

 

The present article is devoted to considering measurable set-valued random functions that are adapted to a fixed filtration of σ-algebras and the values of which are closed subsets of some complete separable metric space. For such functions, a criterion of measurability and adaptation is proved, which is analogous to Castain’s well-known criterion of measurability of set-valued functions. A theorem on existence of measurable and adapted selectors of set-valued random functions, which approximate some measurable adapted random function, is obtained. This theorem is improved in the case of set-valued functions with compact values. The generalization of Filippov’s theorem about the inverse function to the set-valued measurable random functions is proved. The obtained results can be useful both for proving the existence and for considering the properties of the solutions of stochastic differential inclusions.

The actual information security problem of developing statistical tests of the hypothesis about a discrete uniform distribution (‘pure randomness’) of output sequences of cryptographic generators is considered. For the entropy functionals of Shannon, Renyi and Tsallis, the point statistical estimators based on the principle of ‘plug-in’ frequency statistics are constructed. The asymptotic probability distribution of the constructed point estimators is found when the ‘pure randomness’ hypothesis in asymptotics is valid, meaning that the number of observed data is comparable with the number of estimated parameters. With the use of the probability distributions of point estimators, the interval statistical estimators of considered information entropy functionals are constructed. On the basis of interval estimators, the decision rules for statistical testing of the hypothesis about the ‘pure randomness’ of the observed discrete sequence are developed. The results of computer experiments, in which the developed statistical tests are applied to the output sequence of cryptographic generators, are given. In these experiments, the output binary sequence was transformed to the sequence of alphabet with a larger dimension by combining the s neighboring elements in the s-grams.

 

The task of deformation of elastic anisotropy in a specific nonlinear elastic-plastic model is considered. According to a given criterion, the excessive growth of anisotropy causes the unexpectably early appearance of macrocracks due to plastic deformation. The elastic properties of material are described by the generalized Murnaghan law of elasticity. Initially, the material is assumed to be isotropic, and the values of anisotropy parameters are zero. The defining equation for the potential energy density of elastic deformation (stress potential) is written in the general form of anisotropy – triclinic. Possible restrictions for transversely isotropic, orthotropic and monoclinic materials were under search. For triclinic material, all seventy seven parameters can be nonzero. For monoclinic material, forty five parameters can be nonzero, and for other types of anisotropy – twenty nine. For transversely isotropic material, the restrictions in the form of homogeneous linear equations are found. Also, the restrictions on cubic-isotropic materials are found, which can be used only in the theory of elasticity, as this anisotropy is nondeformation. The second defining equation in finite form for the Cauchy stress tensor is written. An active elastic-plastic process takes place through an alternate alternation of plastic and elastic material states. The growth of anisotropy occurs in the plastic state (in flow). We introduce three differential equations in flow: for voltage potential, stress tensor and anisotropy parameters. The nonnegative parameter of the anisotropy growth is determined. The system of equations yields the measure speed of elastic distortions and the growth parameter to implement the minimization procedure. The suitability of the last equation to describe the derived constraints is checked. It is found that all of them are performed, but for the part of restrictions for transversely isotropic material. Therefore, for uniaxial loadings this equation should be complemented by twelve homogeneous linear equations.

 

Using the methods of computational chemistry, we calculated matrices A_KL describing hyperfine interactions (HFI) between the electron spin of the color ‘nitrogen-vacancy’ center (NV center) in a diamond and a ¹³C nuclear spin located somewhere in the Н-terminated carbon cluster C₅₁₀[NV]H₂₅₂ hosting the NV center. The rates W₀ of the ¹³C spin flip-flops induced by anisotropic HFI are calculated systematically for all possible locations of ¹³C in the cluster. It is shown that in the cluster, there are specific positions of nuclear ¹³C spin, in which it almost does not undergo such flip-flops due to small off-diagonal elements in corresponding matrices A_KL. Spatial locations of the ¹³C stability positions in the cluster are discovered and characteristic splitting values in the spectra of optically detected magnetic resonance (ODMR) for the stable NV–¹³C systems are calculated, which can be utilized to identify them during their experimental search for use in emerging quantum technologies. It is shown that the positions of the ¹³C nuclear spin located on the NV center symmetry axis are completely stable (W₀ = 0). The characteristics of eight ‘axial’ NV–¹³C systems are elucidated. The presence of additional ‘non-axial’ near-stable NV–¹³C spin systems also exhibiting very low flip-flop rates (W₀ → 0) due to a high local symmetry of the spin density distribution resulting in vanishing the off-diagonal HFI matrix A_KL elements for such systems is revealed for the first time. Spatially, these ‘non-axial’ stable NV–¹³C systems are located near the plane passing through the vacancy of the NV center and being perpendicular to the NV axis. Analysis of the available publications showed that apparently, some of the predicted stable NV–¹³C systems have already been observed experimentally.

 

The task of a spatial distribution of ion pairs in the active volume of the ionization fission chamber has been studied. The formula of a spatial distribution of ion pairs in cylindrical fission chambers, in which a fissile material layer is coated on the internal side of the external electrode, is derived. It is calculated in two ways: counting the number of ion pairs created in the infinitesimal volume inside the active volume of the chamber by all the trajectories, which emanate from a radiator. In the first case, the infinitesimal volume is a sphere. In the second case, it is arbitrary in shape. The formula has evaluated correctly the density of ion pairs at any point in the active volume of the fission chamber. The dependence of the initial density of ion pairs on a radial distance to a radiator created a typical fission fragment – ion Sr in the center of the chamber and the spatial distribution of the initial density of ion pairs along the chamber have been studied.

 

The evaluation results of hyperspectral data correlation in spatial and spectral domains are presented by the example of the hypercube AVIRIS Moffett Field, and the key features of hyperspectral data are formulated. The basic approaches to lossless compression and the algorithms, which can be applied in Earth remote sensing, are considerеd. They are the prediction (linear prediction, fast lossless, spectral oriented least squares, correlation-based conditional average prediction, M-CALIC), the lookup tables (lookup table, locally averaged interband scaling lookup tables), the 3D wavelets (3D-SPECK). A compression algorithm of hyperspectral data is proposed with regard to the advantages and disadvantages of specific implementations of the analyzed algorithms in remote sensing. The main algorithm stages are the preprocessing (for each spectral channel, it is executed independently), the reduction of a correlation level in the spectral area and the entropy coder. The test results of the developed algorithm are given in comparison to the alternative codecs on the AVIRIS test set (Cuprite, Jasper Ridge, Low Altitude, Moffet Field) that prove the efficiency of the proposed algorithm: parallel processing, low computing cost (low latency instructions are used, no division and multiplication), small random access memory requirements (the memory is used only for storage of the hypercube). In the context of the above advantages, the hardware implementation of the algorithm is allowed for on board the aircraft.