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The paper deals with determining a temperature field of an isotropic solid with inclusion represented as a spherical layer that absorbing penetrating radiation. A hierarchy of simplified analogues of the basic model of the heat transfer process in the system under study was developed, including a “refined model of concentrated capacity”, a “concentrated capacity” model, and a “truncated model of concentrated capacity”. Each of the mathematical models of the hierarchy is a mixed problem for a second-order partial differential equation of the parabolic type with a specific boundary condition that actually takes into account the spherical layer available in the system under study.The use of the Laplace integral transform and the well-known theorems of operational calculus in analytically closed form enabled us to find solutions to the corresponding problems of unsteady heat conduction. The “concentrated capacitance” model was in detail analysed with the object under study subjected to the radiation flux of constant density. This model is associated with a thermally thin absorbing inclusion in the form of a spherical layer. It is shown that it allows us to submit the problem solution of unsteady heat conduction in the analytical form, which is the most convenient in terms of both its practical implementation and a theoretical assessment of the influence, the spherical layer width has on the temperature field of the object under study.Sufficient conditions are determined under which the temperature field of the analysed system can be identified with a given accuracy through the simplified analogues of the basic mathematical model. For simplified analogues of the basic model, the paper presents theoretical estimates of the maximum possible error when determining the radiated temperature field.

When fighting against terrorism in modern armed conflicts, combat vehicles, including tanks, are widely used. To minimise own losses of vehicles and personnel for overthrowing enemy is a relevant task. To solve it, the paper considers certain events in the history of the Great Patriotic War, which are associated with battle of tanks that spring an ambush. A mathematical model of the battle is built. The state graph of the system is given. Using this graph, a probability of tank kills and a ratio of mathematical expectations of losses have been calculated. This mathematical model generalizes the models, previously published in this journal, based on the Markov chain apparatus. The paper gives an example of calculations for this model in the particular case in which experimental data are used as a basis. The ratios of mathematical expectations of losses of the warring parties are obtained. Further, we consider the mathematical models, in which it is assumed that probabilities for tank crews to provide operations of targets detection in firing are known. With technology development and its mathematical support it becomes increasingly more real. The formulas to obtain the probability of tank kills are given according to the graph of states using the known probabilities of transition from one state to another. In each of the three mathematical models under consideration there is a graph of the system state, which allows calculation of the tank kills probability. We have analysed the models to prove a significant dependence of the loss ratio of the warring parties on the number of firing positions used by the tank in ambush in case re-siting is unnoticeable for the enemy. The authors-considered models that use the examples of historical events confirm that the tactics of organising and conducting ambushes in tank battles can be successfully used nowadays, when the technology intensiveness of the opposing forces significantly grows. The obtained results can be applied to organise and conduct tank ambushes in modern armed conflicts and fight against terrorist army.

Block ciphers form one of the main classes of cryptographic algorithms. One of the challenges in development of block ciphers, like any other cryptographic algorithms, is the analysis of their cryptographic security. In the course of such analysis, statistical testing of block ciphers is often used. The paper reviews literature on statistical testing of block ciphers.The first section of the paper briefly and informally discusses approaches to the definition of the concept of a random sequence, including the Kolmogorov, von Mises, and Martin-Löf approaches and the unpredictability-related approach. However, all these approaches to the definition of randomness are not directly applicable in practice.The second section describes statistical tests of binary sequences. It provides brief descriptions of the tests included in the DieHard, NIST STS, RaBiGeTe statistical test suites.The third section provides the appropriate information to present further the operation modes of block ciphers.The fourth section deals with techniques for statistical testing of block ciphers. Usually such techniques lie in the fact that based on the block cipher under test, various generators of the pseudorandom sequences are built, with their output sequences being tested using any suite of statistical tests. The approaches to the construction of such generators are given.The paper describes the most known statistical test technique for block ciphers among the submitted for the AES competition. It is a technique the NIST uses for statistical testing of ciphers. In addition, there are other techniques mentioned in the literature.In conclusion the paper states that there is a need to develop new techniques for statistical testing of block ciphers.The paper support was provided from the Russian Foundation for Basic Research in the framework of the research project No. 16-07-00542 supported

For the processes described by dynamical systems, closed trajectories of dynamical systems are in line with periodic oscillations. Therefore, there is a considerable interest in describing the bifurcations of the generation of closed trajectories from equilibrium when the parameters change. In typical one-parameter and two-parameter families of smooth dynamical systems on a plane, closed trajectories can be generated only from equilibrium – weak focus. In mathematical modeling in the theory of automatic control, in mechanics and in other applications, piecewise smooth dynamical systems are often used. For them, there are other bifurcations of the generation of closed trajectories from equilibrium. The paper describes one of them, which is a typical family of dynamical systems specified by a piecewise smooth vector field on a two-dimensional manifold depending on two small parameters. It is assumed that for zero values of the parameters the vector field has a singular point O on the line of discontinuity of the field, and the point O is stable; in one half-neighborhood of the point O the field coincides with a smooth vector field for which the point O is a weak focus with positive (negative) first Lyapunov value, and in the other half-neighborhood it coincides with a smooth vector field directed at the points of the line of discontinuity inside the first of the semi-neighborhoods. The paper describes bifurcations in the neighborhood of the point O as the parameters change, in particular, indicating the regions of the parameters for which the vector field has a stable closed trajectory.

So far, there are no efficient algorithms to solve a problem of finding the low degree annihilators for arbitrary Boolean function. In the paper we present a new algorithm to find affine annihilators for an arbitrary Boolean function. We start with considering the identity fg ≡ 0 or the arbitrary Boolean function f and its possible affine annihilator g. We use a special representation of the Boolean function in sum of its sub-functions to reduce degrees of considering functions in previous identity. As a result, we establish equivalence between the identity fg ≡ 0 for Boolean functions of n variables and the system of Boolean equations of n-1 variable.An algorithm for finding the affine annihilators for the arbitrary Boolean function f must find all the affine functions g so that fg ≡ 0. Our algorithm is based on reducing the problem of finding the affine annihilators for the Boolean function f of n to the similar problem for its sub-functions of n-1 variable. The presented algorithm has the following advantages:An input function can be presented in different ways;Output can be also presented in different ways;The algorithm can be effectively parallelized.It should be noted that the result we have obtained is not final and highlights some development directions: first, to study the impact of its input and output on the efficiency of the algorithm of various representations and, second, to use our idea of constructing the algorithm for development of algorithms, which allow finding the 2nd, 3rd, etc. degree annihilators for a specified Boolean function.

The paper studies linear differential operators in derivatives with respect to one variable. Such operators include, in particular, operators defined on infinite prolongations of evolutionary systems of differential equations with one spatial variable. In this case, differential operators in total derivatives with respect to the spatial variable are considered. In parallel, linear differential operators with one independent variable are investigated. The known algorithms for reducing the matrix to a stepwise or diagonal form are generalized to the operator matrices of both types. These generalizations are useful at points, where the functions, into which the matrix components are divided when applying the algorithm, are nonzero.In addition, the integral operator is defined as a multi-valued operator that is the right inverse of the total derivative. Linear operators that involve both the total derivatives and the integral operator are called integro-differential. An invertible operator in the integro-differential sense is an operator for which there exists a two-sided inverse integro-differential operator. A description of scalar differential operators that are invertible in this sense is obtained. An algorithm for checking the invertibility in the integro-differential sense of a differential operator and for constructing the inverse integro-differential operator is formulated.The results of the work can be used to solve linear equations for matrix differential operators arising in the theory of evolutionary systems with one spatial variable. Such operator equations arise when describing systems that are integrable by the inverse scattering method, when calculating recursion operators, higher symmetries, conservation laws and symplectic operators, and also when solving some other problems. The proposed method for solving operator equations is based on reducing the matrices defining the operator equation to a stepwise or diagonal form and solving the resulting scalar operator equations.

To study an arithmetic nature of the values of hyper-geometric functions (and their derivatives including those with respect to parameter), it is common practice to use one either Siegel's method or the method based on the effective construction of the linear approximating form. The main distinction between these methods consists in the mode of construction of the first approximating form. Applying Siegel's method allows us to construct such a form by means of a pigeonhole principle that makes it possible to establish, in certain cases, the algebraic independence of the values of corresponding functions. The Siegel's method can be usually applied just while considering hyper-geometric functions with rational parameters. The effective method has a certain advantage here, for in some cases this method enables us to carry out corresponding investigation also for the functions with irrational parameters. Another advantage of the effective method is that it provides obtaining of high- accuracy quantitative results. By quantitative results one usually implies the estimates of the moduli of the linear forms in the values of corresponding functions. The effective method has made it possible to obtain, in some cases, estimates being precise regarding the height of such forms with calculation of the corresponding constants. A drawback of the effective method is the narrow domain of its applications. The effective construction of the linear approximating form, which initiates reasoning, is always a challenge. So far, an effective construction of the approximating form for the product of powers of hyper-geometric functions (with the rare exceptions) failed.In both aforementioned methods one proves previously linear independence (in Siegel's method, as a rule, algebraic independence,) of the functions under consideration. Such a proof is often considered as an independent result.In this paper, by means of the method especially elaborated for this purpose we establish linear independence of some hyper-geometric functions and their derivatives (including those with respect to parameter) over the field of rational fractions. Subsequently, it will be possible to apply this result to investigate arithmetic properties of the values of such functions. Herewith we mean the application of the effective method to achieve the sufficiently accurate quantitative result.

The study of non-stationary rarefied gas flows is, currently, attracting a great deal of attention. Such an interest arises from creating the pulsed jets used for deposition of thin films and special coatings on the solid surfaces. However, the problems of non-stationary rarefied gas flows are still understudied because of their large computational complexity. The paper considers the computational aspects of investigating non-stationary movement of gas reflected from a wall and flowing through a suddenly formed gap. The study objective is to analyse the possible numerical kinetic approaches to solve such problems and identify the difficulties in their solving. When modeling the gas flows in strong rarefaction one should consider the Boltzmann kinetic equation, but its numerical implementation is rather time-consuming. In order to use more simple approaches based, for example, on approximation kinetic equations (Ellipsoidal-Statistical model, Shakhov model), it is important to estimate the difference between the solutions of the model equations and of the Boltzmann equation. For this purpose, two auxiliary problems are considered, namely reflection of the gas flow from the wall and outflow of the free jet into the rarefied background gas.A numerical solution of these problems shows a weak dependence of the solution on the type of the collision operator in the rarefied region, but at the same time a strong dependence of a behavior of the macro-parameters on the velocity grid step. The detailed velocity grid is necessary to avoid a non-monotonous behavior of the macro-parameters caused by so-called ray effect. To reduce computational costs of the detailed velocity grid solution, a hybrid method based on the synthesis of model equations and the Boltzmann equation is proposed. Such an approach can be promising since it reduces the domain in which the Boltzmann collision integral should be used.The article presents the results obtained using two different software packages, namely a Unified Flow Solver (UFS) [13] and a Nesvetay 3D software complex [14-15]. Note that the UFS uses the discrete ordinate method for velocity space on a uniform grid and a hierarchical adaptive mesh refinement in physical space. The possibility to calculate both the Boltzmann equation and the model equations is realized. The Nesvetay 3D software complex was created to solve the Shakhov model equation (S-model) for calculations based on non-structured non-uniform grids, both in velocity space and in physical one.