Articles

Physical education is the foundation of a healthy lifestyle. The effectiveness and correctness of physical exercises can't be ensured without the supervision of an expert. The article proposes a system architecture based on hybrid intelligence, in particular, methods for human pose estimation and fuzzy logical inference, which makes it possible to determine the indicators necessary to evaluate exercise performance and form personal recommendations with adjustments to achieve correct performance.

The paper examines a multispecies model of population dynamics of biological communities proposed by U. Dieckmann and R. Law in a space of arbitrary dimension. A system of nonlinear integral equations, arising in the problem of finding the equilibrium state after the three-parameter power-2 closure of the third spatial moments, is investigated. After preliminary transformations, this system is represented as a single operator equation in a Banach space of functional matrices of a special form, which allows describing the equilibrium state as the fixed point of a certain operator. Conditions are formulated that are sufficient to ensure that small changes in the closure parameters lead to small changes in the fixed points set of the operator, in terms of the distance induced by the norm of the Banach space.

A detailed description of the implementation of an algorithm for constructing a variable-structure stabilizing controller for a switched linear discrete system whose operating modes can have different dynamic orders is presented. A neural network is proposed as an observer of the closed-loop system's active modes. Simulation results of the constructed stabilization system are presented.

This work considers the problem of numerical solution of the multicomponent Smoluchowski coagulation equation using a low-rank representation in the tensor train (TT) format. An adaptive time-stepping approach based on a second-order Runge–Kutta method with local error control is proposed and tested. Numerical experiments are carried out for ballistic and generalized Brownian kernels in the two-dimensional case. The use of adaptive time stepping significantly reduces computational time compared to constant step size maintaining low relative error. The advantages of adaptive time stepping are most pronounced in the numerical solution of the problem of finding a steady-state solution of the two-component coagulation equation with a particle source. The results demonstrate the efficiency of adaptive methods for the multicomponent coagulation problems in combination with tensor decompositions.

Universal Boolean functions, determing any linear function by part of their values, were considered earlier. In this paper we studied the problem of constructing functions, determing by part of their values any arbitrary pair of linear functions, that are not negation of each other and differ by the value in zero. We obtained a sequence of such functions with the domain O(n^2) depending on n ⩾ 7 variables.

Each finite binary chain of zeros and ones will be assigned a number, which will be called a measure of the randomness of this chain. The larger this measure, the more "random" the chain will be. A method for constructing chains with the largest measure is proposed.

The paper finds optimal control strategies for energy storages with low efficiency of energy exchange with the electricity network and estimates investment returns for several types of hydrogen energy storages. A method for computation of the optimal storage volume is obtained. The paper shows a high investment attractiveness of hydrogen energy storages in the medium term.

This paper demonstrates the process of deriving macroscopic equations of gas dynamics from a stochastic microscopic model describing the behavior of gases from hard spheres. As the Knudsen number decreases, the resulting equations acquire a form similar to quasi-gasdynamic (QGD) equations with regularizing components that differ from the classical Navier–Stokes system.

A classic example of a simple discrete problem of the motion of a gas consisting of absolutely elastic spheres is considered. This example illustrates the key stages of constructing mathematical models for large systems using stochastic differential equations (SDE) and their computer implementation.

Ludwig Boltzmann derived his fundamental equation based on the balance of the distribution function describing the gas as a continuous medium in the phase space of coordinates and velocities, that is, using the Eulerian approach. We consider a gas as a set of individual particles, the dynamics of which is described by the SDE system, which means the Lagrange formalism.

The proposed model is self-sufficient and does not require setting additional equations of state or configurable parameters.

The article describes the process of forming a feature space for a system that allows monitoring the performance of physical exercises and developing personal recommendations for correcting their performance based on the rules used by an expert trainer. Human Pose Estimation (HPE) methods are used to digitally represent exercise performance. The feature space should represent HPE data (coordinates of human body parts) as a set of feature values for making recommendations (for example, exercise intensity, user endurance). The feature space plays a central role in the development of such systems: the richer it is, the better the recommendations can be implemented.

For a linear-quadratic tracking problem with a target trajectory described by a linear dynamics, convergence of reinforcement learning using policy iteration method is investigated. A proof of convergence is given, and convergence rate is analyzed. Namely, it is proven that the policy iteration method converges and has a quadratic convergence rate under relaxed constraints on the initial control.

In this paper, we propose a compact textual representation of scanned 3D objects. This representation can be obtained from the original point cloud by extracting geometric primitives, forming a topological graph depicting the relative positions of the primitives in the object, and then converting the graph into a string of characters. Our experiments show that many objects can be described quite accurately as a set of geometric primitives. Furthermore, the proposed representation significantly reduces the memory footprint compared to the original point cloud. The object recognition problem can be solved by searching the database for the object whose textual representation is closest in terms of the Levenshtein distance to the textual representation of the object being recognized. Computational experiments show that the proposed representation enables achieving a fairly high recognition accuracy.

It is shown that any scaled mixture of multivariate normal distributions can be a stationary distribution in the multivariate stochastic difference equation (SDE) scheme—a first-order multivariate autoregression with random coefficients. A correspondence is established between the resulting mixture and the behavior of the coefficients generating the stationary distribution.

In this paper, we study the spectral problem for the Sturm-Liouville equation. One of the boundary conditions is a homogeneous Dirichlet condition, and the other is a condition of the third kind, and contains the square of the spectral parameter. A biorthogonal system to the system of eigenfunctions of the task is constructed. The problems of the basis property and the Riesz basis property of system of eigenfunctions are investigated.

In this paper, we consider a numerical solution of the Cauchy problem for a second-order differential equation calculated using the interpolation (implicit) multistep Stormer method. A guaranteed error estimate is found using ellipsoids that contain exact and numerical solutions. The use of ellipsoids avoids the wrapping effect that occurs when using interval analysis methods. It is proposed to modernize the method of obtaining a more accurate estimate of small terms in the recalculation of ellipsoids, which was previously proposed for solving linear differential equations, to the case of nonlinear equations. Formulas for recalculating ellipsoids have been obtained to obtain a guaranteed estimate of the error of the Stormer interpolation multistep method. The results of estimating the error in the numerical solution of nonlinear second-order differential equations on large intervals are presented, demonstrating the effectiveness of the proposed method.

We consider universal functions for subclasses of linear Boolean functions. For classes of linear functions, keeping a constant, using universal functions of small number of variables, we obtained lower and upper bounds on the size of range of definition. For the classes of linear self-dual and linear self-dual, keeping constants, we constructed universal functions with minimal power of range o definition for an arbitrary number of variables.

The paper studies the asymptotic behaviour of the mean total reserve of an insurance company in the case of a random number of clients. We use results on asymptotic deficiency and relative efficiency of procedures based on samples of fixed and random sizes. Zero-truncated Delaporte and generalized Poisson distributions are considered as models for the random number of clients. Asymptotic expansions of inverse moments are obtained for these distributions and the coefficients in the asymptotic formulas are computed. It is shown that the asymptotic deficiency of the scheme with a random number of clients can be expressed in terms of distribution parameters and in particular cases equals 1 and (1−θ)⁻². An empirical illustration is given based on data from the U.S. National Flood Insurance Program.

This work is devoted to studying methods for modeling and forecasting the value of underlying assets based on market option prices. The use of options allows one to recover probability distributions of future asset prices and construct their volatility surfaces, which is necessary for modeling asset dynamics and valuing derivative financial instruments. Non-parametric approaches based on the extraction of risk-neutral probability density functions from option prices are considered, as well as parametric implied volatility surface models, such as SVI and SABR, which provide input for constructing the dynamic Dupire local volatility model. Dynamic models of asset prices with calibration based on market option quotes are also discussed, including the Heston model and discrete ARMA-GARCH models.

In the article, the asymptotics of the coefficients of the generating functions are found with a certain accuracy, which can be used to calculate the powers of layers of certain types of partially ordered sets, as well as to calculate the values of the sums of boundary functionals when estimating the number of antichains in such sets. In addition, applications of the obtained results are considered using examples.

A nonlinear two-phase mathematical model of production at a mining plant is proposed, where ore flow is represented as two phases connected to each other through concentration of useful substance contained in the ore. The process of material balancing is studied, which leads to optimization problem with nonlinear constraints. The interior point method is used to solve the problem. Known measured values of useful substance concentrations are taken into account, as well as the results of indirect measurements, ranges of flow values, unknown concentration values, limits on stored resources. Test experiments are performed, and calculation results are demonstrated.

Initial boundary-value problem for the evolutional equation with involution of the space variable and a non-local boundary condition is studied. The uniqueness of the problem’s solution is obtained and classes of initial functions which provide solution’s existence and stability are described.

A certain characterization of the logistic distribution is proposed by the property of identical distribution of two linear forms.

This work addresses the problem of single-variable function approximation within the class of nonlinear approximations with two sets of parameters. The approximating function is linear with respect to the first parameter (which is constrained to be positive), while each term of the approximating function represents a given function with nonlinear dependence on the second parameter. The computational algorithm is based on residual minimization in Hilbert space. The first key aspect of the developed approach involves effectively reducing the problem to a standard linear best approximation problem by discretizing the second parameter over an extended set of points within the admissible segment. The second key feature lies in determining the set of linear approximation parameters at each iteration of the classical non-negative least squares (NNLS) method. Numerical results are presented to demonstrate the capabilities of this computational algorithm for nonlinear function approximation.

This work examines a modification of a spatially distributed Lanchester model, which describes an antagonistic game between two groups in a two-dimensional domain. The model originally emerged in 1916 to describe combat scenarios between two military forces, characterizing events of the First World War. The model accounts for directed strikes, defined by a velocity vector, and the initial displacement of troop concentrations using a step function. The system includes diffusion terms, nonlinear reactions, Gaussian white noise, and a topographic obstacle (a lake of arbitrary shape) that restricts troop movement. The numerical solution is implemented using the finite element method based on domain triangulation to handle irregular domains. The simulation results demonstrate an increase in spatial heterogeneity under the influence of noise and the obstacle, as well as a significant impact of directed movement and initial displacement on conflict dynamics. Stability analysis confirms the system’s stability, including an analysis specific to the finite element method. Sensitivity analysis of the parameters and computational error estimation have been added. Visualization illustrates the evolution of troop densities over time. The advantages of the proposed approach over existing methods include better handling of irregular domains and model scalability.

The paper considers the asymptotic behaviour of the distribution functions of the statistics based on the samples with random sizes in the testing problems conserning univariate parameters. An asymptotic comparison of the tests is carried out in terms of the necessary additional number of such factors (asymptotic deficiency). Two examples illustrating the obtained results are presented. These examples concern truncated Poisson and binomial distributions are considered.

The paper considers a method of stabilized hard thresholding in the problem of inverting linear homogeneous operators using wavelet decomposition. In a data model with additive Gaussian noise, an analysis of the unbiased estimate of the mean square risk of this method is carried out. Under the assumption of a long-term dependence between noise coefficients, conditions are given under which the unbiased risk estimate is strongly consistent and asymptotically normal.

The problem of Hausdorff approximation by finite sets of the solution and the value of a multicriteria mixed strategy bimatrix game using a representation based on linear scalarization is considered. For the case of 2×2 matrices, explicit formulas are obtained for constructing nodes of a δ-net on the product of simplices of scalarization parameters. It is proved that the set of the equilibrium values obtained for the net converges in the Hausdorff metric to the solution of the initial game at δ →0. Possibile appearance of degenerate bimatrix games in scalarization is taken account. Examples are given for two-criteria 2×2×2 games.

In this paper, the definitions of generalized Student’s distributions are extended to a wider set of parameters of these distributions and multiplication theorems are given that allow the generalized Student’s and Lomax’s distributions to be represented as scale mixtures of the same distributions but with larger parameters. A similar result is obtained for beta distributions. Analogs of multiplication theorems are obtained for the classical Student’s and Lomax’s distributions as corollaries; in particular, it is shown that the Student’s distribution can be represented as a scale mixture of the Student’s distribution with a large number of degrees of freedom. A representation of strictly stable distributions concentrated on the positive semiaxis is also obtained as scale mixtures of a special distribution that is not stable. This alternative representation complements the multiplication theorem for such strictly stable laws.

We present an efficient algorithm for checking language equivalence of states in top-down deterministic finite tree automata (DFTAs). Unlike string automata, tree automata operate over hierarchical structures, posing unique challenges for algorithmic analysis. Our approach reduces the equivalence checking problem to that of checking the solvability of a system of language-theoretic equations which specify the behavior of a DFTA. By constructing such a system of equations and systematically manipulating with it through substitution and conflict detection rules, we develop a decision procedure that determines whether two states accept the same tree language. We formally prove the correctness and termination of the algorithm and establish its worst-case time complexity as O(n²) under the RAM (Random Access Machine) model of computation augmented with pointers.

The aim of equivalence testing is to verify that two parameters are sufficiently close or, alternatively, that the parameter of interest lies within two pre-specified limits. The two one-sided tests procedure is arguably the most widely known approach for assessing equivalence in the pharmaceutical field. Using a model that accounts for missing data, it is shown analytically that the type I error rate may exceed the nominal significance level. A refined estimate of this error is also obtained. For the standard 2x2 crossover design, a method is proposed that enables control of the type I error in the presence of missing data.

This study presents a modified vector autoregression (VAR) method for forecasting the quality metrics of overlay channels. The modification involves the introduction of weighted coefficients for time series quantiles, with two distinct approaches proposed for calculating these weights: exponential weighting (EVAR) and linear weighting (LVAR). Experimental results demonstrate that the proposed method improves prediction accuracy by 2.6% to 25.2% compared to classical AR and VAR methods, albeit at the expense of higher computational complexity.

It was proved earlier that product xy is a universal function for the class of linear functions depending on two arguments if k=6l±1. Then the criterion of existence of a universal polinomial was obtained for any number of arguments and any k. In this paper we show that product xy is universal for any class of linear functions over Galua field GF(p^m), where p is any prime and m is natural, m⩾2.

A detailed description of the implementation of an algorithm for constructing a variable-structure stabilizing controller for a switched interval linear system with slow switches inaccessible to observation is proposed. It is proposed to use a neural network as an observer of the active modes of a closed system. The simulation results of the constructed stabilization system are presented.

The unification problem for parameterized finite state machines consists in finding for two given automata such values of their parameters that these automata will compute the same transduction relations. This presents a quadratic time algorithm for computing the most general unifiers of parameterized deterministic finite state machines. This algorithm is based on the Martelli–Montanari unification algorithm for terms and the equivalence checking algorithm for deterministic finite state machines

The efficiency of network services depends on how applications are distributed in the network. At the same time, these services are to be provided with a certain quality of services and experience for users, i.e.:

1) access to information “at any time anywhere”;

2) seemless integration of informational systems;

3) timely monitoring and analysis of different data sources;

4) migration from a centralized scheme “data download with cleaning followed by analysis and distribution” to the scheme “distributed placement and preprocessing of data and, if necessary, subsequent loading, analysis and distribution”.

One of the key preconditions to provide an acceptable level of services is forecasting the execution time of network services on a variety of virtualization platforms and physical equipment.

The work analyzes approaches to predicting temporal characteristics of network services. The methods are based on operational log data of services and physical equipment, take into account the current and predictable state of hardware and software resources. Among the solutions considered are machine learning models, including a random forest, multilayer perceptrons, and convolutional neural networks.

5G NR communication systems use antenna arrays for directional transmission and reception, which in turn contributes to increased communication performance and efficiency. For duplex systems with frequency division of channels, feedback about the channel status is important. The code page for the configuration of the base station’s antenna array and the antenna ports of the client device is selected through the exchange of reports. The essence of the work is to optimize the exchange procedure, apply and study data compression methods.

Using the technique of Boolean algebras, all 55 implicatively implicit extensions of systems of one-place functions of three-valued logic are defined. It is established that the specified extensions are defined by systems of one-place functions containing from one to three functions.

This paper shows that an arbitrary scale mixture of normal laws can be a stationary distribution of a stochastic difference equation (first-order autoregressive scheme) with random coefficients.

An example is given of what a (random) diffusion coefficient should look like for a particular mixture to be a stationary distribution.

Automated assessment of a group of autonomous agents’ spatial structure reduces development costs for group control systems and eliminates the human factor in analyzing group behavior. To construct a metric for the spatial regularity of an agent group, it is proposed to use a repeatability characteristic of the spatial structure based on autocorrelation methods. The paper examines various methods for calculating autocorrelation using a matrix of pairwise distances between agents, and their application to different types of spatial structures.

The paper considers an approach to solving the problem of noise removal in a large array of sparse data under weak dependence. The approach is based on the method of controlling the false discovery rate (FDR). For this approach, the rate of convergence of the mean–square risk estimator to the normal law is obtained.

The particle method is a numerical method for modeling large systems based on their Lagrangian description.

The discontinuous particle method is of the “particle–particle” type and consists of two main stages: predictor and corrector. At the predictor stage, a particle shift occurs. At the corrector stage, a partner for interaction is selected among the neighbors of the particle, most influencing the local dynamics of the system. The “discontinuity” of the method lies in the method of density correction only one of the interacting particles, due to which the restoration of the distribution density occurs in a minimal region defined by only two selected particles, which leads to smearing of the front by only one particle.

The novelty of the method presented in this article is that The density of the particles is put in the foreground, not their shape. The criterion for restructuring is the preservation of the projection of the mass onto the plane passing through the centers masses of interacting particles. The neighbor for density correction is selected using the “impact parameter”. The density is constructed using two selected interacting particles, which makes it possible to reduce a two-dimensional problem to a one-dimensional one.

The effectiveness of the method is presented using the Crowley test as an example. It is shown that the Runge–Kutta method at the predictor stage significantly increases the accuracy of the numerical solution.

Our Lagrangian approach to constructing the particle method contrasts with another frequently used particle-particle method, the smoothed particle method (SPH).

The inverse Sturm–Liouville problem consists in determining the coefficient (potential) in the stationary Schrцdinger equation on a segment based on a set of eigenvalues. The paper considers a numerical solution to the inverse problem based on a finite set of the first eigenvalues of two Sturm–Liouville problems. The remaining eigenvalues are set according to the classical asymptotics.

The method of solving the inverse spectral problem is based on a one-to-one correspondence between the inverse spectral problem and the nonstationary inverse problem for a telegraphic equation with a variable coefficient (potential). The reduction to a non-stationary problem is carried out analytically by inverting the Laplace transform according to the Mellin formula. An explicit formula for the reaction function in the inverse scattering problem is obtained.

The inverse scattering problem for the telegraphic equation is to determine an unknown coefficient from the reaction function. This problem is solved numerically by the inversion of a difference scheme. The paper presents the results of solving a series of inverse Sturm–Liouville problems. In conclusion, it is noted that the number of given frequencies corresponds to the number of harmonics in the expansion of a desired potential.

The author considers an inverse coefficient problem for a model of sorption dynamics. The inverse problem is reduced to a nonlinear operator equation for an unknown coefficient. The differentiability of the nonlinear operator is proved. The Newton–Kantorovich method and the modified Newton–Kantorovich method are constructed for the numerical solution of the inverse problem. The results of numerical calculations are presented.

When describing the group behavior of high-frequency traders, a boundary value problem arises based on the concept of mean field games. The system consists of two coupled partial differential equations: the Hamilton–Jacobi–Bellman equation, which describes the evolution of the average payoff function in backward time, and the Kolmogorov–Fokker–Planck equation, which describes the evolution of the probability density distribution of traders in forward time. The system is ill-conditioned due to the turnpike effect. Under certain assumptions, it is possible to reduce the system to a set of Riccati equations; however, the question of the well-posedness of the reduced problem remains open. This work investigates this question, specifically the conditions for the existence and uniqueness of the solution to the boundary value problem depending on the model parameters.

Statistical inference often assumes that the distribution being sampled is normal. As observed, following the normal distribution assumption blindly may affect the accuracy of inference and estimation procedures. For this reason, many tests for normality have been proposed in the literature. This paper deals with the problem of testing normality in the case when data consists of a number of small independent samples such that in each small sample observations are independent and identically distributed while from sample to sample they have different parameters but the same type of distribution (call this multi-sample data). In this case it is necessary to use test statistics which do not depent on the parameters. A natural way to exclude the nuisance location parameter is to replace the observations within each small group by diferences. We obtain some estimates of stability of such a decomposition and study and compare the power of eight selected normality tests in the case of multi-sample data. The following tests are considered: the Pearson chi-square test, the Kolmogorov–Smirnov, the Cramer–von Mises, the Anderson-Darling, the Shapiro–Wilk, the Shapiro–Francia, the Jarque–Bera, and the adjusted Jarque–Bera tests. Power comparisons of these eight tests were obtained via the Monte Carlo simulation of sample data generated from several alternative distributions.

This article presents a new algorithm for calculating singularity (boundary) type of ribbon surface of generalized pseudo-Anosov homeomorphism using the surface’s combinatorial description provided with the so-called configuration. As an additional output the fundamental group relators of the ribbon surface are calculated for its co-presentation associated with a given ribbon partition. In comparison to a known algorithm, the one which is presented in this article does not involve any auxiliary sets nor recurrent functions.

The creation of cryptographic systems based on lattice theory is a promising direction in the field of post-quantum cryptography. The aim of this work is to obtain new properties of lattices through related objects — dense packings of equal spheres. The article proposes a method for constructing lattice packings of equal spheres corresponding to the packing density of the “Lambda” series in dimensions 1–24, using a series of coefficients to the height of a fundamental parallelepiped of dimension (n−1): 1/2, 1/3, 1/2, 0, 1/2, 1/3, 1/2, √−1, 1/2, 1/3, 1/2, 0, 1/2, 1/3, 1/2, √−1, 1/2, 1/3, 1/2, 0, 1/2, 1/3, 1/2. The construction of lattice packings of equal spheres using this method was carried out up to dimension 11 inclusive.

This paper solves the construction problem of embedding the complete rooted binary and ternary trees with k, k =1, 2, . . . , levels in rectangular lattices (RL) with minimum length and near minimum height. It is assumed that different vertices of the tree go to different (main) vertices of the RL, with the leaves of the tree going to the vertices of the RL located on its horizontal sides. It is also assumed that the edges of the tree go to simple (transit) chains of the RL, which connect the images of their end vertices and do not pass through other main vertices, with no more than 1 (correspondingly 2) transit chains passing through the same edge (the same vertex) of the RL.

Time optimal control problem with state constraint investigated in this article. The behavior of an object is described by a system of second-order linear differential equations. The coefficient matrix for state variables has various positive eigenvalues. The state constraint is linear. A admissible control is a piecewise continuous function that takes values from a given compact set. Sets of controllability to the origin are constructed for time intervals of various lengths. A study of the dependence of the solution of the problem on the parameter determining the state constraint was carried out.

In the work on real data, the modeling and assessment of the infection rate of a population of ixodid ticks with tick-borne encephalitis virus and Borrelia burgdorferi sensu lato is carried out using the maximum likelihood and moment methods, and their comparative analysis is given. A review of methods for solving direct and inverse problems of binary object propagation based on individual and group observations is made.

The paper describes a modified scenario method for risk estimation of financial instruments. It is based on the scenario method proposed by Jamshidian F. and Zhu Y. allowing to significantly reduce computation time of risk indicators on large portfolios compared to the Monte Carlo method. It is proposed to change the way of choosing scenarios and points of approximating distribution. It allows to improve the quality of approximation. More than that, the new method makes possible to remove restrictions on the type of distribution of portfolio price factors allowing to expand the scope of its application. The comparison of VaR (value at risk) estimation using original and modified method was performed on a financial portfolio consisting of an interest rate swap for cases when the values of the price factors have normal distribution, gamma distribution and Student’s t-distribution.