2026.2

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.