The problems concerning the physical process studies that lead to mathematical models based on a parabolic type equation are of great practical importance.When solving the parabolic equation, a transition from the one-dimensional equation to the multidimensional one gives a rise to some difficulties. The trouble is that amount of computations significantly increases in the transition from the one-dimensional problems to the multidimensional ones In this regard a task to build the economical difference schemes for a numerical solution of multidimensional problems is a challenge.A difference scheme that approximates the problem over time is economical, if it is unconditionally stable and the desirable number of arithmetic operations in the transition from layer to layer is proportional to the number of nodes in each time layer.The paper dwells on the construction of a locally one-dimensional (economical) difference scheme for the approximate solution of a parabolic type equation of a general form in a multidimensional domain, the main idea of which is to reduce a complex problem to a sequential solution of boundary value problems of a simpler structure. At the same time, there is a construction of the economical, unconditionally stable difference scheme for each of the intermediate problems. To have a numerical solution of the problem, there is a construction of the locally one-dimensional difference scheme of A.A. Samarsky. Using a method of energy inequalities allows us to obtain a priori estimates in the differential and difference interpretations, whence it follow that there are the uniqueness, stability, as well as the convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem. For a bi-dimensional problem, a numerical solution algorithm is constructed.

Numerical calculation uses to describe the operation of matrix permutation algorithms based on cyclic shifts of rows and columns. This choice of discrete transformation algorithms justified by the convenience of the cellular automaton (CA) formulation, which is used. Obtained Empirical formulas for the permutation period and for the last algorithm, which period formula is recurrent. For a base scheme period has the asymptotics: for a matrix with pairwise different elements. Despite the complexity of the scheme, the other two modifications only give a polynomial growth of period, no higher than 3. Fourth scheme has a non-trivial period dependence, but no higher than the exponential. In some cases algorithms make special permutations: rotate, reflect, and rearrange blocks for the matrix . These formulas are closely related to individual cells paths. And paths connected with the influence of the boundaries that gives branching the matrix order by subtraction class modulo 3,4 or 12. Visualizations of these paths make in the extended CA-field. Two mixing analyze as a parameter of CA dynamics on matrix permutations (compared to the initial). For all schemes and most branches, the behavior of these metrics shows in graphs and histograms (conditional density distribution) showing how often the permutation period occurs with the specified interval of metrics. The practical aim of this work is in the field of pseudorandom number generation and cryptography.

An adequacy of mathematical modelling of the technical object dynamics is always a challenge, and a user can make up his mind about the modelling package using simple test schemes. The article presents the schemes that are quite common from the point of view of an inexperienced user, but have specifics in mathematical modelling. The features of the schemes are that therein failures of equilibrium equations (such as the first Kirchhoff law) and continuity (such as the second Kirchhoff law) can occur. These features can lead to incorrect results when using the OpenModelica package, while the domestic PA9 and PRADIS packages used in modelling these schemes give the correct result. The article presents two simple schemes, the simulation result of which can be a priori estimated, and provides simulation results for three packages - OpenModelica, PA9, and PRADIS.

The paper deals with a problem of the obfuscated JavaScript code detection and classification based on Abstract Syntax Trees (AST) coloring. Colors of the AST vertexes and edges are assigned with regard to the types of the AST vertexes specified by the program lexical and syntax structure and the programming language standard. Research involved a few stages. First of the all, a non-obfuscated JavaScript programs dataset was collected by the public repositories evaluation. Secondly, obfuscated samples were created using eight open-source obfuscators. Classifier models were built using an algorithm of gradient boosting on the decision trees (GBDT). We built two types of the classifiers. The first one is the model that classifies the program according to the type of the obfuscator used, i.e. based on what obfuscator created the sample. The second one tries to detect samples obfuscated by the obfuscator whose samples are not observed during training. The quality of the obtained models is on par with the known published results. The feature engineering method proposed in the paper does not require a preliminary analysis of the obfuscators and obfuscating transformations. In the final part of the paper we analyze a quality of models estimated, discussing the certain statistical properties of the obfuscated and non-obfuscated samples obtained and corresponding colored ASTs. Analysis of generated samples of obfuscated programs has shown that the method proposed in the paper has some limitations. In particular, it is difficult to recognize minifiers or other obfuscating programs, which change the lexical structure to a greater extent and the syntax to a lesser extent. To improve the quality of detection of this kind of obscuring transformations, one can built combined classifiers using both the method based on the AST coloring and the additional information about lexemes and punctuation, for example, entropy of identifiers and strings, proportion of characters in upper and lower case, usage frequency of certain characters etc.

The study of nonstationary rarefied gas flows is currently paid much attention. Such interest to these problems is caused by the creation of pulsed jets used for the deposition of thin films and special coatings on solid surfaces. However the problems of nonstationary rarefied gas flows have not been studied sufficiently fully because of their large computational complexity. In this paper the computational aspects of investigating the nonstationary flows of a reflected gas from a wall and flowing through a suddenly formed gap is considering. The objective of this study is to analyze the possible numerical kinetic approaches for solving such nonstationary problems and to identify the difficulties encountered in solving.When studying the gas flows in strong rarefaction regimes one should consider the Boltzmann kinetic equation, but its numerical implementation is rather laborious. 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 of the solutions of the model equations and the Boltzmann equation. For this purpose two auxiliary problems are considered: reflection of the gas flow from the wall and outflow of the free jet into the rarefied background gas. Numerical solution of these problems shows a weak dependence of the solution on the type of the collision operator in the rarefied region, but a strong dependence on the velocity grid step . The detailed velocity grid is necessary to avoid non-monotonous behavior of macroparameters caused by the “ray effect”. To reduce numerical costs on detailed grid a hybrid method based on the synthesis of model equation and the Boltzmann equation is proposed. Such approach can be promising since it reduces the domain in which the Boltzmann collision integral should be used.The results presented in this paper were obtained using two different software packages Unified Flow Solver (UFS) [13] and Nesvetay 3D [14-15]. Note that UFS uses the discrete ordinate method for velocity space on a uniform grid and a hierarchical adaptive mesh refinement in physical space. The possibility of calculating both the Boltzmann equation and model equations is realized. The Nesvetay 3D complex was created to solve the Shakhov model equation, (S-model) and makes it possible to calculate on non-structured non uniform grids in velocity and physical spaces.Translated from Russian. Original text: Mathematics and Mathematical Modeling. 2018. no. 4. Pp. 27-44.

Ferroelectrics, due a number of characteristics, behave as hereditary materials with fractal structure. To model mathematically the systems with so-called memory effects one can use the fractional time-derivatives. The pyro-electric properties of ferroelectrics arouse interest in developing the fractional-differential approach to simulating heat conductivity process.The present study deals with development and numerical implementation of fractal heat conductivity model for hereditary materials using the concepts of fractional-differential calculus applied to the simulation of intensive heating processes in ferroelectrics.The paper proposes a mathematical model governed through mixed initial-boundary value problem for partial differential equation containing a fractional time-derivative as well as nonlinear temperature dependence on the heat capacity. To solve the problem the computational algorithm was designed which is based on an analog of the Crank – Nicolson finite difference scheme combining with the Grunwald – Letnikov formula for fractional time-derivative approximation. The approximation of Neumann boundary condition is included into the finite difference problem statement using scheme of fictitious mesh points. The total system of linear algebraic equations is solved by sweep method.The designed application program allows one to perform the computer simulation of heat conductivity process in hereditary materials. The model verification was performed for numerical solving test problem with known analytical solution. The results of computational experiments are demonstrated for the example of estimating heat distribution in a typical ferroelectric crystal of TGS (triglycine sulfate) near the temperature of phase transition. The fractional derivative order was approximately evaluated to be ~0.7 at variation of this parameter. We applied the comparison of fractal model implementation results with experimental data related to the time when the ferroelectric crystal is heated to Curie temperature. These findings demonstrate that one needs to use the modified models at the analysis of the field effects arising in hereditary materials.

The paper deals with modeling a Piezoelectric Generator (PEG) that includes piezoactive elements, inertial mass, plate and rack. The PEG under consideration can be an element of the energy storage device in the capacity of the source of energy provided from vibrations of elements of structures and machines.The main objective of the paper is to gain the PEG efficiency by finding the optimal geometric parameters for finding the highest output potential.The elastic and piezoceramic media are modeled within the framework of the linear theory of electroelasticity. As a research tool, CAE package ACELAN is used in which three-dimensional and axisymmetric device models are built. The numerical experiments performed a modal and harmonic analysis that enabled us to identify the most effective operating frequencies.

The article solves a problem of developing a scheme to provide a secret key exchange over an open communication channel. The basic idea of creating such a scheme is well known. It is based on a concept of the one-way function. This refers to the functions whose values are calculated much easier than the inverse function values. When developing the one-way functions a recognition algorithm of words equality in groups with conditions of small cancellation C (3) - T (6) is used. In this case, the group is represented by a set of its generating and determining relations. All the work to accomplish development of algorithms and evaluate their complexity is carried out using the group diagrams of equality. The existence of such diagrams is proved in the well-known van Campen lemma. The paper result is that the proposed scheme for the exchange of secret keys has the following properties. Direct algorithms have a linear complexity, and a complexity of the inverse algorithms is exponential. It should be noted that the algorithms complexity was estimated by the areas of the corresponding group diagrams, which are determined by the number of areas they include. The constructed secret key represents some element of a pre-selected group with conditions C (3) – T (6). It can be represented in an infinite number of ways by words in the alphabet from the generators of the group. Thus, the remaining obstacle to the practical application of the key exchange scheme developed is the ambiguity of the secret key record. Finding a common representative as the lexicographically shortest word in the class of equal words turns out to be too difficult. Thus, this question remains open. Although the task of exchanging secret keys itself can be formally considered as solved.

A finite-element model developed for quasi-static processes describes irreversible processes of deformation and polarization occurring in polycrystalline ferroelectric media due to the effect of intense electric fields and mechanical loads. The paper presents external parameters such as strain and polarization as a sum of residual and reversible parts. Using the incremental theory, the virtual work law, and the constitutive relations for reversible and irreversible components, a system of linear algebraic equations was built for the increments of nodal values of the main variables, namely the displacement vector and the electric potential, during the transition from one equilibrium state to another.The constructed constitutive relations connect the reversible parts of the strain and polarization with the stresses and the electric field in the form of linear tensor equations. It is shown that the physical characteristics depend on the residual parameters so that the coefficients of elastic compliance and dielectric constant linearly depend on the principal values of residual strain, and the piezoelectric modules depend linearly on the module of residual polarization. The constitutive relations for the increments of the residual parameters are determined as element values for each finite element from the equations in differentials. Ultimately, the task is reduced to a system of linear algebraic equations, the matrix and right sides of which depend on the residual parameters and are determined at each equilibrium state. As a result, the non-linearity of the problem is replaced by solving a sequence of linear problems until the external loads reach their final values.The model is implanted into a finite-element complex, which allows us to determine the fields of residual strain and polarization, the physical characteristics of a partially polarized body, and local anisotropy for the case of complete and partial polarization.

This article is a sequel to the earlier articles, which describe the invertible ordinary differential operators and their generalizations. The generalizations are invertible mappings of filtered modules generated by one differentiation, and are called invertible D-operators. In particular, invertible ordinary linear differential operators, invertible linear difference operators with periodic coefficients, maps defined by unimodular matrices, and C-transformations of control systems are invertible D-operators. C-Transformations are those invertible transformations for which the variables of one system are expressed in terms of the variables of the other system and their derivatives.In the article we consider the invertible D-operators whose inverses are D-operators of the same type. In previous papers, a classification of invertible D-operators was obtained. Namely, a table of integers was associated to each invertible D-operator. These tables were described in a clear elementary-geometric language. Thus, to each invertible D-operator one assigns an elementary-geometric model, which is called a d-scheme of squares. The class of invertible D-operators having the same d-scheme was also described. In this paper, the invertible D-operators whose d-schemes consist of a single square are called unicellular. It is proved that any unicellular operator in some bases is given by an upper triangular matrix that differs from the identity matrix only by the first row. The main result is representation of the arbitrary invertible D-operator as a composition of unicellular operators. The minimum number of unicellular operators in such a composition is equal to the number of squares of the d-scheme of the original D-operator. As in previous papers, the used method is based on the description of d-schemes in the language of spectral sequences of algebraic complexes.The results obtained can be useful in the transformation and classification of control systems, in particular to describe flat systems.