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.