In this paper, we offer an efficient parallel algorithm for solving the Graph Isomorphism Problem. Our goal is to construct a suitable vertex substitution or to prove the absence of such. The problem is solved for undirected graphs without loops and multiple edges, it is assumed that the graphs can be disconnected. e question of the existence or absence of an algorithm for solving this problem with polynomial complexity is currently open. Therefore, as for any time-consuming task, the question arises of accelerating its solution by parallelizing the algorithm. We used the RPM ParLib library developed by the author as the main tool to program the algorithm. This library allows us to develop effective applications for parallel computing on a local network in the .NET Framework. Such applications have the ability to generate parallel branches of computation directly during program execution and dynamically redistribute work between computing modules. Any language with support for the .NET Framework can be used as a programming language in conjunction with this library. For our experiments, we developed some C# applications using this library. The main purpose of these experiments was to study the acceleration achieved by recursive-parallel computing. Specially generated random regular graphs with varying degrees of vertices were used as initial data. A detailed description of the algorithm and its testing, as well as the results obtained, are also given in the paper.

In this paper, we consider the notion of a direct type algorithm introduced by V. A. Bondarenko in 1983. A direct type algorithm is a linear decision tree with some special properties. the concept of a direct type algorithm is determined using the graph of solutions of a combinatorial optimization problem. e vertices of this graph are all feasible solutions of a problem. Two solutions are called adjacent if there are input data for which these and only these solutions are optimal. A key feature of direct type algorithms is that their complexity is bounded from below by the clique number of the solutions graph. In 2015-2018, there were five papers published, the main results of which are estimates of the clique numbers of polyhedron graphs associated with various combinatorial optimization problems. the main motivation in these works is the thesis that the class of direct type algorithms is wide and includes many classical combinatorial algorithms, including the branch and bound algorithm for the traveling salesman problem, proposed by J. D. C. Little, K. G. Murty, D. W. Sweeney, C. Karel in 1963. We show that this algorithm is not a direct type algorithm. Earlier, in 2014, the author of this paper showed that the Hungarian algorithm for the assignment problem is not a direct type algorithm. us, the class of direct type algorithms is not so wide as previously assumed.

The experiment aimed at finding a distribution of path lengths between nodes in the global network and an estimation of parameters of that distribution is described.In particular, the method of measurement of path length with traceroute utility of the GNU/Linux system and limitations on the selection of nodes imposed by traceroute are described. e measurement results are provided and high values of skewness and kurtosis for all resulting distributions are noted. Simulation model of this experiment was developed to test the experiment validity in the determination of distribution parameters in the global network. This model is also described. It is shown that high values of skewness and kurtosis of the measured distributions are not the result of the measurement technique, therefore the global network could not be described by the Barabasi–Albert ´ model. Several most viable hypotheses explaining diffierences in skewness and kurtosis of experimentally obtained pathlength distribution estimations and values derived from the Barabasi–Albert model are listed. Results of diffierent hypotheses ´ simulations are provided. It is shown that the most fitting hypothesis is that definitive influence on skewness and kurtosis of path-length distribution estimations is caused by the quasi pre-fractal structure of the global network.

Traceability schemes which are applied to the broadcast encryption can prevent unauthorized parties from accessing the distributed data. In a traceability scheme a distributor broadcasts the encrypted data and gives each authorized user unique key and identifying word from selected error-correcting code for decrypting. The following attack is possible in these schemes: groups of c malicious users are joining into coalitions and gaining illegal access to the data by combining their keys and identifying codewords to obtain pirate key and codeword. To prevent this attacks, classes of error-correcting codes with special c-FP and c-TA properties are used. In particular, c -FP codes are codes that make direct compromise of scrupulous users impossible and c -TA codes are codes that make it possible to identify one of the aackers. We are considering the problem of evaluating the lower and the upper boundaries on c, within which the L-construction algebraic geometric codes have the corresponding properties. In the case of codes on an arbitrary curve the lower bound for the c-TA property was obtained earlier; in this paper, the lower bound for the c-FP property was constructed. In the case of curves with one infinite point, the upper bounds for the value of c are obtained for both c-FP and c-TA properties. During our work, we have proved an auxiliary lemma and the proof contains an explicit way to build a coalition and a pirate identifying vector. Methods and principles presented in the lemma can be important for analyzing broadcast encryption schemes robustness. Also, the c-FP and c-TA boundaries monotonicity by subcodes are proved.

Let \(Q_n\) be a partition of the interval \([0,1]\) defines as
\(
\begin{array}{l}
Q_1 =\{0,q^2,q,1\}.
  \\
Q_{n+1}' = qQ_n \cap q^2Q_n, \ \
Q_{n+1}'' = q^2+qQ_n \cap qQ_n, \ \
Q_{n+1}'''= q^2+qQ_n \cap q+q^2Q_n,
  \\
Q_{n+1} = Q_{n+1}'\cup Q_{n+1}'' \cup Q_{n+1}''',
  \end{array}
\)
where \(q^2+q=1\).
The sequence  \(d= 1,2,1,0,1,2,1,0,1,0,1,2,1,0,1,2,1,\dots\) defines as follows.
\(
\begin{array}{l}
  d_1=1, \ d_2=2,\ d_4 =0;
 d[2F_{2n}+1 : 2F_{2n+1}+1] = d[1:2F_{2n-1}+1];\\
 \quad  n = 0,1,2,\dots;\\
d[2F_{2n+1}+2 : 2F_{2n+1}+2F_{2n-2}] = d[2F_{2n-1}+2:2F_{2n}];\\
d[2F_{2n+1}+2F_{2n-2}+1 : 2F_{2n+1}+2F_{2n-1}+1] = d[1:2F_{2n-3}+1];\\
d[2F_{2n+1}+2F_{2n-1}+2 : 2F_{2n+2}] = d[2F_{2n-1}+2:2F_{2n}];\\
 \quad n = 1,2,3,\dots;\\
  \end{array}
\)
where \(F_n\) are Fibonacci numbers (\(F_{-1} = 0, F_0=F_1=1\)).
The main result of this paper.
\({\bf Theorem.}
\\
Q_n' = 1 - Q_n''' =\left \{ \sum_{i=1}^k q^{n+d_i}, \ k=0,1,\dots, m_n\right\},
\\
Q_n'' = 1 - Q_n'' = \left\{q^2 + \sum_{i=m_n}^k  q^{n+d_i}, \ k=m_n-1,m_n,\dots, m_{n+1} \right\},
\\\)
where \(m_{2n} = 2F_{2n-2}, \ m_{2n+1} = 2F_{2n-1}+1\).

The goal of the paper is to develop an algorithm for matching the shapes of images of objects based on the geometric method of de Rham currents and preliminary affine transformation of the source image shape. In the formation of the matching algorithm, the problems of ensuring invariance to geometric image transformations and ensuring the absence of a bijective correspondence requirement between images segments were solved. The algorithm of shapes matching based on the current method is resistant to changes of the topology of object shapes and reparametrization. When analyzing the data structures of an object, not only the geometric form is important, but also the signals associated with this form by functional dependence. To take these signals into account, it is proposed to expand de Rham currents with an additional component corresponding to the signal structure. To improve the accuracy of shapes matching of the source and terminal images we determine the functional on the basis of the formation of a squared distance between the shapes of the source and terminal images modeled by de Rham currents. The original image is subjected to preliminary affine transformation to minimize the squared distance between the deformed and terminal images.

In this work, we study a Markov model of cyber threats that act on a computer system. Within the framework of the model the computer system is considered as a system with failures and recoveries by analogy with models of reliability theory. To estimate functionally-temporal properties of the system we introduce a parameter called the lifetime of the system and defined as the number of transitions of the corresponding Markov chain until the first hit to the final state. Since this random variable plays an important role at evaluating a security level of the computer system, we investigate in detail its random distribution for the case of mutually exclusive cyber threats; in particular, we derive explicit analytical formulae for numerical characteristics of its distribution: expected value and dispersion. Then we generalize substantially the Markov model dropping the assumption that cyber threats acting on the system are mutually exclusive. This modification leads to an extended Markov chain that has (at least qualitatively) the same structure as the original chain. This fact allowed to generalize the above analytical results for the expected value and dispersion of the lifetime to the case of non-mutually exclusive cyber threats. At the end of the work the Markov model for non-mutually exclusive cyber threats is used to state a problem of finding an optimal configuration of security remedies in a given cyber threat space. It is essential that the formulated optimization problems belong to the class of non-linear discrete (Boolean) programming problems. Finally, we consider an example that illustrate the solution of the problem on selecting the optimal set of security remedies for a computer system.

It is well known in functional analysis that construction of \(k\)-order derivative in Sobolev space \(W_p^k\) can be performed by spreading the \(k\)-multiple differentiation operator from the space \(C^k.\) At the same time there is a definition of \((k,p)\)-differentiability of a function at an individual point based on the corresponding order of infinitesimal difference between the function and the approximating algebraic polynomial \(k\)-th degree in the neighborhood of this point on the norm of the space \(L_p\). The purpose of this article is to study the consistency of the operator and local derivative constructions and their direct calculation. The function \(f\in L_p[I], \;p>0,\) (for \(p=\infty\), we consider measurable functions bounded on the segment \(I\) ) is called \((k; p)\)-differentiable at a point \(x \in I\;\) if there exists an algebraic polynomial of \(\;\pi\) of degree no more than \(k\) for which holds \( \Vert f-\pi \Vert_{L_p[J_h]} = o(h^{k+\frac{1}{p}}), \) where \(\;J_h=[x_0-h; x_0+h]\cap I.\) At an internal point for \(k = 1\) and \(p = \infty\) this is equivalent to the usual definition of the function differentiability. The discussed concept was investigated and applied in the works of S. N. Bernshtein [1], A. P. Calderon and A. Sigmund [2]. The author's article [3] shows that uniform \((k, p)\)-differentiability of a function on the segment \(I\) for some \(\; p\ge 1\) is equivalent to belonging the function to the space \(C^k[I]\) (existence of an equivalent function in \(C^k[I]\)). In present article, integral-difference expressions are constructed for calculating generalized local derivatives of natural order in the space \(L_1\) (hence, in the spaces \(L_p,\; 1\le p\le \infty\)), and on their basis - sequences of piecewise constant functions subordinate to uniform partitions of the segment \(I\). It is shown that for the function \( f \) from the space  \( W_p^k \) the sequence piecewise constant functions defined by integral-difference \(k\)-th order expressions converges to  \( f^{(k)} \) on the norm of the space \( L_p[I].\) The constructions are algorithmic in nature and can be applied in numerical computer research of various differential models.

The growth of popularity of online platforms which allow users to communicate with each other, share opinions about various events, and leave comments boosted the development of natural language processing algorithms. Tens of millions of messages per day are published by users of a particular social network need to be analyzed in real time for moderation in order to prevent the spread of various illegal or offensive information, threats and other types of toxic comments. Of course, such a large amount of information can be processed quite quickly only automatically. that is why there is a need to and a way to teach computers to “understand” a text written by humans. It is a non-trivial task even if the word “understand” here means only “to classify”. the rapid evolution of machine learning technologies has led to ubiquitous implementation of new algorithms. A lot of tasks, which for many years were considered almost impossible to solve, are now quite successfully solved using deep learning technologies. this article considers algorithms built using deep learning technologies and neural networks which can successfully solve the problem of detection and classification of toxic comments. In addition, the article presents the results of the developed algorithms, as well as the results of the ensemble of all considered algorithms on a large training set collected and tagged by Google and Jigsaw.

Optimal portfolio selection is a common and important application of an optimization problem. Practical applications of an existing optimal portfolio selection methods is often difficult due to high data dimensionality (as a consequence of the large number of securities available for investment). In this paper, a method of dimension reduction based on hierarchical clustering is proposed. Clustering is widely used in computer science, a lot of algorithms and computational methods have been developed for it. As a measure of securities proximity for hierarchical clustering Pearson pair correlation coefficient is used. Further, the proposed method’s influence on the quality of the optimal solution is investigated on several examples of optimal portfolio selection according to the Markowitz Model. The influence of hierarchical clustering parameters (intercluster distance metrics and clustering threshold) on the quality of the obtained optimal solution is also investigated. The dependence between the target return of the portfolio and the possibility of reducing the dimension using the proposed method is investigated too. For each considered example in the paper graphs and tables with the main results of the proposed method - application which are the decrease of the dimension and the drop of the yield (the decrease of the quality of the optimal solution) - for a portfolio constructed using the proposed method compared to a portfolio constructed without the proposed method are given. For the experiments the Python programming language and its libraries: scipy for clustering and cvxpy for solving the optimization problem (building an optimal portfolio) are used.