Volume 27, Nº 3 (2025)

In this paper  we consider classical iterated function systems (IFS) consisting of a finite number of contracting mappings for a complete metric space. The main goal is to study the class of IFSs whose attractors are Cantor sets, i.e. perfect totally disconnected sets. Important representatives of this class are totally disconnected IFSs introduced by Barnsley. We have proposed other definitions of a totally disconnected IFS and proved their equivalence to the Barnsley definition. Sufficient conditions for IFS to be totally disconnected are obtained. It is shown that injectivity of mappings from an IFS implies the perfection of the attractor and its uncountability. Also it is proved that if the mappings from an IFS are injective and the sum of their contraction coefficients is less than one, then the attractor is a Cantor set. In general case, these conditions do not guarantee totally disconnectedness of an IFS. Meanwhile, it is shown that if an IFS consists of two injective mappings and the sum of their contraction coefficients is less than one, then the IFS is totally disconnected. Examples of IFS attractors are constructed, demonstrating that conditions of the proven theorems are only sufficient but not necessary.

The paper contains sufficient conditions for the action of generalized and linear generalized partial integral Romanovsky operator in the space of continuous functions defined on an n-dimensional parallelepiped. Continuity of these operators is established in case of their action in the space of continuous functions and, in a more general case of continuous kernels of operators with values in the space of measurable and Lebesgue integrable functions. Estimates are obtained for the norms of the operators mentioned. The dependence of the estimate for the norm of a linear Romanovsky type operator with generalized partial integrals on the space dimension and on the norm of continuous kernels of generalized partial integral Romanovsky operators with values in the space of measurable and Lebesgue integrable functions is shown. The properties established are applied to the study of linear generalized partial integral equations of Romanovsky type, in particular, to the study of generalized partial integral equation of n-connected Markov chains.

The paper proposes a new square-root gradient-based parameter identification method for discrete-time linear stochastic state-space systems with unknown input signals. A new algorithm is developed for calculating the values of the identification criterion and its gradient. The approach is based on a square-root modification of the Gillijns – De Moor method and uses numerically stable matrix orthogonal transformations. Unlike the existing solutions, this paper uses original methods for differentiating matrix orthogonal transformations. A new sensitivity model is constructed and theoretically justified, that allows calculating the values of the identification criterion gradient by using partial derivatives of state vector estimates based on identified parameters. The main results include new equations for the square-root sensitivity model and a square-root algorithm for calculating the values of the identification criterion and its gradient. Numerical experiments were performed in MATLAB for example of solving the numerical identification problem of a stochastic diffusion model with unknown boundary conditions. The effectiveness of the proposed algorithm is confirmed by comparison of gradient-based and gradient-free methods. The results of numerical experiments demonstrate efficiency of approach proposed which can be used to solve practical problems of identifying the parameters of mathematical models represented by discrete-time state-space linear stochastic systems, in the absence of any prior information about the input signals.

The dynamics of this process is modeled based on experiments on the capture and movement of a load by a system of self-organizing particles in a magnetic field. The transport system of particles is a structure in the form of one closed chain (two-dimensional case) or two parallel closed chains (three-dimensional case). As a result of the action of the external field, the particles are set in rotation and move translationally. Hydrodynamic interaction between all particles and the load, which does not interact with the external field, is taken into account. The mathematical model includes equations of viscous fluid hydrodynamics and particle dynamics in the approximation of small Reynolds numbers. Numerical modeling and visualization of the obtained results are performed using a specially developed software package. The performed numerical calculations confirmed the possibility of capture and transfer of the load in the case of location of the particle system and the load in the same plane. In the three-dimensional case, the capture of cargo and its movement by the transport structure in the form of parallel chains of particles does not lead to the capture of cargo. The obtained results are in qualitative agreement with the experiments. The proposed model can be used to calculate the dynamics of a system of particles self-organizing into closed chains in a liquid in the presence of foreign bodies.