Journal of the Belarusian State University. Mathematics and Informatics
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Current trends in computer data analysis
The current trends in computer data analysis presented at the conference «Computer data analysis and modelling: stochastics and data science» and the congress «Information systems and technologies» are described.
Copyright (c) 2026 Journal of the Belarusian State University. Mathematics and Informatics https://creativecommons.org/licenses/by-nc/4.0
Limit theorem on convergence to the local time of a Brownian excursion
An integer random walk {Si, i ≥ 0} with zero drift and finite variance σ2 is investigated. For a random pro-cess that assigns, to a variable u > 0, the number of hits of the specified random walk into the state \[\left\lfloor {u\sigma \sqrt n } \right\rfloor \] up to time n and is considered under the condition that S1 > 0, …, Sn – 1 > 0, Sn ≤ 0, a functional limit theorem concerning convergence in distribution of the process to the local time of a Brownian excursion is proved.
Probability Theory and Mathematical Statistics
local time of conditional Brownian motions
Local information geometry for statistical analysis of high-order binary Markov chains
At the 14th International conference «Computer data analysis and modelling: stochastics and data analysis» the author presented some new results on asymptotic properties of statistics derived from purely random (uniformly distributed) binary sequence of increasing length. These results are obtained by methods of information geometry applied to manifolds of Markov probability distributions on the set of infinite binary sequences. In this paper the underlying information-geometric theory and the technics of proofs are describe in more detail. By few examples, it is shown how finding probabilistic and statistical properties comes down to geometric and combinatorial computations.
On well ve-covered and well ev‑covered graphs
In this paper, the classes of well vе-covered graphs and well ev-covered graphs are investigated. A graph is called well vе-covered (well ev-covered, respectively) if all its minimal independent vе-dominating (minimal independent ev-dominating, respectively) sets have the same cardinality. It is shown that the problems of recognising the graph classes under consideration are co-NP-complete even for some restrictions of these classes. Characterisations, in terms of forbidden induced subgraphs, for the maximal hereditary subclasses of well vе-covered graphs and well ev-covered graphs are found. The computational complexity of problems related to minimum independent vе-dominating and minimum independent ev-dominating sets is established.
Discrete Mathematics and Mathematical Cybernetics
Sequential statistical hypotheses testing for discrete random data with block structure
An important mathematical problem of computer data analysis – the problem of sequential statistical testing of simple hypotheses on probability distributions of discrete random data with a block structure – is considered. Explicit expressions of the sequential tests statistics, transparent for interpretation and convenient for computer realisation, are derived. An approach to calculate the performance characteristics – error probabilities of the decisions and mathematical expectations of the random number of observations required to guarantee the requested accuracy for decision rules – is developed. Asymptotic expansions for the mentioned performance characteristics under distortion of the hypothetical probability model are constructed.
mathematical expectation of the random number of observations
Reliability of two-level testing approach in the NIST SP 800-22 test suite
The two-level approach for testing random number generators involving the well-known NIST SP 800-22 test suite, i. e. counting the sequences passing a basic test and checking the p-values distribution with a chi-square test, was considered. Such an approach may increase the reliability of the test. However, it is sensitive to the approximation error introduced by the computing of p-values. In this paper it is shown that for AES-based sequences two-level testing approach is not reliable too. Systematic error in the computing of the p-values is dependent only on the accuracy of approximation of the exact distribution of statistic by its theoretical counterpart and the number of bits in the analysed sequences n. For a reliable second-level test, this error should be smaller, or at least, approximately equal to $\frac{\sigma}{N} = \frac{1}{k}\sqrt {\frac{{k\; - \;1}}{N}}$, where $\sigma = \sqrt {\frac{1}{k}\left( {1\; - \;\frac{1}{k}} \right)N}$ is the standard deviation from the mean number of particles in a bin for equiprobable scheme of allocation N particles in k bins. Such heuristic assumptions and carried out experiments suggest that for example in the second-level test of the Frequency test of NIST SP 800-22 test suite with n = 220 the number of tested sequences N should not exceed 26 000. To completely eliminate the systematic error appearing in the Frequency test when determining the number of bin from k bins (disjoint subintervals of the interval [0, 1]), to which the p-value belongs the two-sided estimates of the quantiles of the binomial law are proposed.
Complete convergence for partial weighted sums of negatively orthant dependent random variables
Negatively orthant dependence is regarded as a generalisation of independence for random variables, introduced by K. Joag-Dev and F. Proschan. Numerous researchers have investigated inequalities and laws of large numbers for such sequences of random variables. In particular, the concept of complete convergence, defined by P. L. Hsu and H. Robbins, has attracted significant attention. Complete convergence for partial weighted sums of negatively orthant dependent random variables dominated by a random variable X is established. Sufficient conditions for this type of convergence are provided under mild assumptions on the weights and the moments of random variable X.
Mild solution of a mixed problem for a third-order semilinear hyperbolic equation with the wave operator
For a semilinear hyperbolic equation of the third-order given in the first quadrant we study a mild solution of a mixed problem in which the initial conditions are specified on the spatial half-line and the mixed conditions are specified on the time half-line. The operator in the equation is a composition of the wave operator and the transport operator. Mild solution is defined as a solution to coupled integral equations that are satisfied by a classical solution. It is shown that under some smoothness conditions on the initial and boundary data the problem under consideration admits the existence and uniqueness of the mild solution. It is established that the twice continuously differentiable mild solution is the limit of classical solutions to the problem under study.
Differential Equations and Optimal Control
non-linear hyperbolic equation of the third-order
Generalised Newton ‒ Kantorovich method under the modified regular smoothness condition
The article deals with the generalised Newton – Kantorovich method for solving non-linear operator equations of the form f(x) + g(x) = 0 in Banach spaces, where f is the operator satisfying the regular smoothness condition; g is the non-differentiable operator satisfying Lipschitz condition. The main convergence theorem is proved under the modified regular smoothness condition in which increments of the operator f derivative are majorised by the increments of a scalar function.
Multi-model approach with vector search in operational management critical IT service
In the work are investigated current problems related to the task of operational management of critical IT services. To improve the quality of management, a new method is proposed, extending the dynamic local approximation approach. It implements linking vector representations of load patterns with trainable neural network models, which allows promptly constructing a predictor for a new pattern based on vector search for semantically close models in the library. Original principles of designing a control system are outlined, which for adaptation to new, uncertainty-related load patterns uses approximation of data obtained in real-time mode. The results of experiments confirmed the effectiveness of the proposed approach and showed that when processing irregular and complex loads, hybrid proactive algorithms in a number of indicators are more effective than reactive ones.
Estimation of the parameters of a second order Markov modulated Poisson process using change-point detection procedure
The problem of estimating the parameters of a second order Markov modulated Poisson process is considered. A new two-staged algorithm for their estimation is developed. In the first stage initial estimates of the intensity parameters are constructed using the method of moments and maximum likelihood method. In the second stage these estimates are used in the cumulative sums change-point detection algorithm to determine time intervals, during which the process intensity is constant; after that estimates of all parameters are constructed. The developed algorithm is compared with a well-known algorithm for finding maximum likelihood estimates (the EM algorithm). The proposed method is non-iterative, ensuring relatively low computational complexity.
Detection of image objects for autonomous localisation of unmanned aerial vehicles and map formation of the surrounding space
The problem of autonomous localisation of an unmanned aerial vehicle and construction of a map of its environment is considered. The low stability of keypoint detectors to changing weather and time conditions of image formation in the surrounding space is demonstrated compared to the stability of line detectors. To improve the stability of unmanned aerial vehicles localisation it is proposed to use image regions (segments) with uniform brightness as key identification objects. An algorithm for segmentation based on local minima of the brightness gradient of halftone images obtained from the original colour images of the unmanned aerial vehicle’s surrounding space has been developed. To equalise the growth rates of local minima a wave segmentation algorithm with automatic stopping is proposed. This algorithm is based on identifying new initial and additional growth points of regions using a monotonically varying threshold and taking into account the rate of change of the brightness gradient along the region growth trajectory in the stopping criterion. It is shown that the proposed algorithms provide more stable localisation of areas, when changing weather and time conditions for image formation in comparison with known algorithms for extracting lines and key points.
Real-time gait recognition system based on lightweight dual-modal pose estimation
A lightweight approach for joint 2D and 3D gait analysis is proposed. A dual-modal gait recognition system is adopted, achieving high-precision real-time analysis through a 2D and 3D collaborative estimation architecture. The system employs MediaPipe for real-time 2D keypoint detection and gait metric extraction. An improved lightweight architecture MTSA-former with 2.5 mln parameters is used for 3D pose estimation. This model utilises cascaded processing incorporating temporal and spatial modules, along with an adjacency matrix based on skeletal topological structure to model physical constraints between joints. A local-global hybrid spatial modelling strategy fusing graph convolutional networks is implemented to achieve efficient 3D pose estimation, attaining high performance while maintaining lightweight characteristics. This approach ensures accurate and efficient 3D pose reconstruction in real time.
Asymptotic non-negative correlation of some statistics used in testing the quality of random number generators
The problem of testing the hypothesis, stating that the tested sequence consists of independent random variables with a given polynomial distribution, is considered. To solve this problem four types of statistics that are generalisations of statistics of tests of the NIST STS, TestU01 and other packages are used. It is proved that many of these statistics, including 10 statistics of the NIST STS, are asymptotically non-negatively correlated, and some of them, in particular some of the 10 statistics of the NIST STS, are asymptotically uncorrelated.
asymptotically non-negatively correlated statistics
Managerial decision analysis using small language models
This paper proposes a method for the automatic assessment of managerial decision quality based on free-form textual responses to Russian-language business cases using a fine-tuned DistilBERT model. The novelty of the study lies in a context-dependent classification algorithm with concatenated input (case description + response) and a post-processing technique that explicitly links predictions to specific managerial competencies or red flags. For the first time, high-accuracy evaluation of managerial decisions in Russian (accuracy = 0.986 8, F1-score = 0.987 0) with interpretable competency-based feedback has been achieved, unlike prior work focused on general text or sentiment analysis. The approach enables the development of automated systems for managerial assessment, personnel selection and personalised leadership development in human resources and corporate training.
Smoke and camera tampering detection system based on single-board computer for real-time fire monitoring
An edge-based architecture for a video surveillance system with integrated smoke and camera tampering detection is presented. The system consists of parallel-executing modules responsible for configuration management and video data processing, including receive, analysis, and transmission. Smoke detection is performed using computer vision algorithms based on the sequential extraction of discriminative visual features: motion direction, colour characteristics, and impact on background visibility. Camera tampering is detected by evaluating scene sharpness through an edge-detection filter. High-resolution video processing is performed on a single-board computer, with computational tasks distributed across the cores of a multi-core central processor to optimise performance. The proposed approach significantly reduces the load on the central node of the surveillance infrastructure, which is a critical advantage for scalable deployments involving numerous cameras. Experimental results demonstrate high detection accuracy for smoke and tampering, as well as stable real-time operation of the implemented architecture.
On Shemetkov’s problem of describing critical formations in the class of τ-closed σ-local formations
The critical τ-closed σ-local formations of finite groups are studied, where σ is some partition of the set of all primes <span lang="RU" style="font-size: 12.0pt; font-family: 'Euclid Math Two'; mso-ascii-font-family: 'Times New Roman'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; mso-hansi-font-family: 'Times New Roman'; mso-bidi-font-family: 'Times New Roman'; mso-ansi-language: RU; mso-fareast-language: RU; mso-bidi-language: AR-SA; mso-char-type: symbol; mso-symbol-font-family: 'Euclid Math Two'; mso-no-proof: yes;">P, τ is an arbitrary subgroup functor. A description of minimal τ-closed σ-local non-H-formations is obtained for an arbitrary σ-local formation of classical type, i. e. a σ-local formation that has a σ-local definition all of whose non-Abelian values are σ-local formations. Thus, the problem of describing critical formations in the class of τ-closed σ-local formations proposed by L. A. Shemetkov (1980) is solved.
Mathematical Logic, Algebra and Number Theory
A finite-difference interpretation of a nonlocalproblem with a parameterised first-order differential equation
In this work the nonlocal initial value problem and corresponding finite-difference interpretation for the first-order linear ordinary differential equation with a parameter at the derivative is studied. The nonlocal initial value condition is given by terms of multipoint linear combination. The difference scheme with exponential fit is proposed on a uniform mesh. The article identifies the requirements on the location of nonlocal data carriers in the multipoint condition, on the values of corresponding coefficients and on the parameter variation interval, under which a uniform on parameter stability of classical and difference solutions, as well as a uniform on parameter convergence of difference solution to classical solution are proved. The identification and proof of such conditions, which provide a uniform on parameter approximation of the nonlocal initial value problem classical solution by the solution of exponentially fitted difference scheme, define the novelty of the current work.
Copyright (c) 2025 Journal of the Belarusian State University. Mathematics and Informatics https://creativecommons.org/licenses/by-nc/4.0
The article analyses the methods for studying the isochronism of autonomous systems on a plane for different types of singular points: centre, weak focus and strong focus. The difficulty of determining a weak isochronous focus is explained. Examples of systems in the Poincaré – Dulac normal form, which show that the requirement to be isochronous imposes weak restrictions on a system with a weak focus, are given. A theorem on the form of a normalising change of a strongly isochronous strong focus is proved. Isochrones of a strongly isochronous strong focus of a system with quadratic nonlinearities are found. Phase portraits of the obtained isochronous foci and their isochrones are constructed.
RLD -YOLO: new method for object detection in unmanned aerial vehicle images using YOLOv11 neural network
Unmanned aerial vehicle images are widely used now in many applications. However, these images face many challenges such as dense distribution of small objects, variable object scales, and inconspicuous edge features, leading to missed and false object detection. To address these issues, this paper proposes a lightweight enhanced solution based on the improved YOLOv11n model RLD-YOLO. The algorithm combines RepConv structural reparameterisation technology, retaining multi-branch feature expression capabilities during training and automatically converting to an efficient single-branch structure during inference. It designs the LKAConv large kernel attention module to enhance the feature capture ability of small targets through 7 × 7 depthwise separable convolution and spatial attention mechanism. It introduces the DASI dynamic adaptive fusion module to optimise multi-scale feature interaction through learnable weight allocation. Experimental results show that the improved RLD-YOLO object detection algorithm, which integrates LKAConv, RepConv, and DASI, increases mAP50 and mAP50-95 by 2.02 and 1.17 % respectively on the VisDrone2019-DET dataset. The post-processing speed is optimised by 9.09 %. Although the pre-processing time increases due to feature enhancement operations, the critical inference stage still maintains a real-time performance of 1.7 ms/frame. The RLD-YOLO model, fused with LKAConv, RepConv, and DASI, is very suitable for the task of small object target detection in unmanned aerial vehicle images.
Theoretical Foundations of Computer Science
Asymptotic analysis of the statistical estimators of Shannon entropy of binary s-tuples
The asymptotic probability distribution of the statistical estimate of Shannon entropy of s-tuples H(s) and the asymptotic joint probability distribution of the statistical estimates of Shannon entropy of s- and (s+1)-tuples H(s), H(s+1) for a uniformly distributed random binary sequence with increasing length are found. It is proved that as the value of s increase, the correlation coefficient of the statistical estimates of Shannon entropy of s- and (s+1)-tuples H(s), H(s+1) tends to zero. The theoretical results are illustrated by computer experiments.
Incorporating fast charging stations accessibility into an urban traffic assignment model based on the Frank – Wolfe algorithm
The growing use of electric vehicles necessitates the adaptation of transport models to account for the specifics of their operation. The aim of this work is to modify the classic traffic assignment model based on the Frank – Wolfe algorithm, widely used in transportation planning, to accurately reflect electric vehicle charging processes in urban environments. The novelty of the research lies in the development of an approach that, unlike existing approaches, comprehensively addresses three key features: the requirement for an electric vehicle needing to charge to route through exactly one charging station; the possibility of routing along paths with cycles to access charging stations and subsequently reach the destination, considering urban road network constraints; the behavioural aspect of a user’s decision to forgo charging if the total time costs (including travel to the station and queuing) exceed a certain threshold. The paper also proposes a time-cost function that accounts for the specifics of the charging process, including queues and the number of charging spots. The results demonstrate that the classic traffic assignment model based on the Frank – Wolfe algorithm can indeed be modified to incorporate these features. It is noted that the proposed model may exhibit multiple equilibrium flow distributions, dependent on initial conditions. The practical significance of the work lies in its potential for more accurate modelling of traffic flows and the load on urban charging infrastructure.
Distribution of finances taking into account statisticalinformation on the reliability of the items produced
This paper examines the problem of distribution of finances among project contractors taking into account statistical information on the reliability of the items produced. Mathematical programming formulations necessary for constructing an optimal strategy of distribution are presented. To account for the influence of the spent amount of money on the number of component failures, it is proposed to build a linear or generalised linear regression, the coefficients of which are used in mathematical programming formulations.
About the Ricci flow on three-dimensional non-unimodular Lie groups with semisymmetric equiaffine connection
The Ricci flow on three-dimensional non-unimodular Lie groups with semisymmetric equiaffine connection is studied. The Ricci flow equation in the coordinate system proposed by J. Milnor is reduced to systems of algebraic and differential equations. A solution to the Ricci flow equation in the class of left-invariant Milnor metrics is found. The results of works by K. Onda, D. Knopf and K. McLeod concerning the Ricci flow on three-dimensional Lie groups in the case of Levi-Civita connectivity are generalised.
three-dimensional non-unimodular Lie groups
A generalisation of the Steiner – Lehmus theorem and critical values transcendence of its parameters
The internal n-line of a triangle is a segment from the vertex to the opposite side dividing this side into segments proportionally to the nth powers of the adjacent sides. An analogue of the Steiner – Lehmus theorem for the internal n-lines of a triangle is considered. All values n ∈ R for which the mentioned analogue of the Steiner – Lehmus theorem holds are found. Also all values n ∈ R for which there exists a non-equilateral triangle with three equal internal n-lines are determined. The transcendence of positive critical values of n of the generalised Steiner – Lehmus theorem is proved.
Solution of the hypersingular integro-differential equation on a curve located in the angular domain
We solve a new linear integro-differential equation on a closed curve on the complex plane. Some restrictions are placed on the curve location and on coefficients of the equation. The equation contains hypersingular and regular integrals. Initially, it is reduced to the Riemann – Carleman boundary value problem for analytic functions, which has a partial form and an unconventional formulation. Next, two linear differential equations with constant coefficients in domains of the complex plane with additional conditions on the solution are solved. All solvability conditions of the original equation are stated explicitly. When they are performed, the solution to the original equation is given explicitly.
Riemann – Carleman boundary value problem
Chebyshev spectral method for solving complete generalised Prandtl equation
This article is devoted to the problem of constructing computational schemes for solving Prandtl integro-differential equations that arise in many problems in mechanics. An approximate numerical method for solving singular integro-differential equations of the generalised Prandtl equation type has been developed. The proposed approximate computational schemes are based on representing the solution of the equation as an expansion over an orthogonal basis of Chebyshev polynomials. The use of known spectral relations has made it possible to obtain an analytical expression for the singular component of the equation. As a consequence, the developed method demonstrates excellent accuracy and exponential rate of convergence of the approximate solution in relation to the degree of interpolation polynomials. The computational qualities of this method are demonstrated using a test example. In particular, it is shown that a discrete model based on the representation of the solution as a decomposition by Chebyshev polynomials leads to a well-conditioned system of linear algebraic equations for the decomposition coefficients, and the convergence rate of the approximate solution error can reach a linear speed in relation to the degree of the interpolation polynomial.
orthogonal basis of Chebyshev polynomials
Conformal Killing equation on a 2-symmetric six-dimensional indecomposable Lorentzian manifold with trivial Weyl tensor
In this paper, we study the conformal analogue of the Killing equation on 2-symmetric six-dimensional indecomposable Lorentzian manifolds, and also study the properties of the conformal factor of this equation. For the case of conformally flat metrics, new non-trivial examples of conformal Killing vector fields with a variable conformal factor are constructed.
Heterogeneous blocked all-pairs shortest paths algorithm for clustered weighted graphs
New heterogeneous blocked algorithm of finding all-pairs shortest paths in a large directed weighted simple graph consisting of weakly connected dense clusters (subgraphs) of different sizes is proposed. The algorithm considers and actively exploits the input and output bridge-vertices and edges of each cluster to speed up computation and localise memory accesses. It divides all blocks of the cost adjacent matrix into four types (square diagonal, rectangular vertical cross, rectangular horizontal cross and rectangular peripheral) and uses a separate computation procedure for each type, considering the design features of the block itself and the way it is computed through other blocks. A theoretical justification of the advantages of the proposed algorithms, which reduce the execution time when searching for the shortest paths, is given. The validity of the formulated statements is also confirmed by the results of computational experiments. We have developed single-threaded implementations and multi-threaded OpenMP implementations of the proposed heterogeneous algorithm and two known homogeneous blocked algorithms of finding shortest paths. Computational experiments on multicore processors were performed on directed weighted random sparse graphs decomposed into weakly connected dense clusters of different sizes. The results are described for four clustered graphs, two of which have 4800 vertices (20 and 41 clusters, respectively) and two of which have 9600 vertices (40 and 80 clusters, respectively). On the MacBook M1 Max computer in the case of single-threaded implementations proposed heterogeneous blocked algorithm for clustered graph with bridge-vertices outperformed the known homogeneous blocked algorithm for the same graphs by a factor of 1.62–1.94; in the case of multi-threaded OpenMP implementations the speedup was 1.87–1.97. On a server with Intel Xeon E5-2620v4 processors heterogeneous algorithm outperformed the known homogeneous algorithm by a factor of 1.58 –1.66 for single-threaded implementations and by a factor of 1.29–1.64 for multi-threaded implementations. A comparison of proposed algorithm with the classical blocked Floyd – Warshall algorithm in which all blocks are of the same size showed a speedup of 4.17– 8.18 times in the case of single-threaded implementations and a speedup of 3.91– 6.36 times in the case of multi-threaded OpenMP implementations.
Methodology for information security threat analysis using digital twins
This paper presents a methodology for analysing information security threats in cyber-physical systems based on digital twins. The proposed approach involves formalising the system and threat space through a multi-layered structure, including technical, process, functional, organisational and sectoral layers. Next, dynamic threat modelling is conducted in a secure virtual environment of the digital twin, enabling the reproduction of attack scenarios and generation of synthetic data to train threat indicator detection algorithms. To identify anomalies, frequency analysis, machine learning and clustering methods are applied, ensuring adaptive and accurate detection of both known and previously unknown attacks. The methodology is verified using a smart grid example, demonstrating the effectiveness of training and testing algorithms on synthetic data that reflect normal and emergency operating modes. The results show the potential for creating self-adjusting information security systems with a high degree of adaptability and threat detection accuracy. The presented methodology provides iterative feedback between stages, enhancing the quality of threat modelling and detection.