№ 2 (65) (2024)
Mathematics
On one Control Problem of a Variable Structure With Fractional Caputo Derivatives
Аннотация
We consider an optimal control problem with a variable structure, described in different time intervals by various ordinary nonlinear fractional differential equations. Using an analogue of the incremental method, a necessary condition for first-order optimality is proved. In the case of convex control domains, a linearized maximum condition is proved, and in the case of open control domains, an analogue of the Euler equation is obtained.
5-16
Spectral Properties of the "Reaction-Diffusion" System Operators and Bifurcations Signs
Аннотация
The article discusses differential equations that arise when modeling reaction-diffusion systems. Questions about the stability of equilibrium points in critical cases, as well as about bifurcations in the vicinity of such points, are studied. The main attention is paid to the linearized problem operators spectral properties study. The spectrum discreteness was established, the root properties and invariant subspaces were studied, and formulas for eigenfunctions were proposed. As an application, questions about the multiple equilibrium bifurcation signs and Andronov –Hopf bifurcation in the vicinity of equilibrium points are discussed. Examples are given to illustrate the proposed approaches effectiveness in studying stability and bifurcations problems.
17-25
About Automorphisms of Graphs With Intersection Arrays {"44" ,"40" ,"12" ;"1,5" ,"33" } and {"48" ,"35,9" ;"1,7" ,"40" }
Аннотация
Distance-regular graph Γ of diameter 3 with strongly regular graphs Γ2 and Γ3 has intersection array {r(c2+1)+a3, r c2, a3 + 1; 1, c2, r(c2 + 1)} (M.S. Nirova). For distance-regular graph Γ of diameter 3 and degree 44 there are exactly 7 feasible intersection arrays. For each of them graph Γ3 is strongly regular. For intersection array {44, 30, 5; 1, 3, 40} we have a3 = 4, c2 = 3, r = 10, Γ2 has parameters (540,440,358,360) and Γ3 has parameters (540,55,10,5). Graph does not exist (Koolen-Park). For intersection array {44, 35, 3; 1, 5, 42} graph Γ3 has parameters (375,22,5,1) and does not exist (its neighbourhood of vertex is the union of isolated 6-cliques). In this paper it is found futomorphisms of graphs with intersection arrays {44,40,12; 1,5,33 and 48,35,9; 1,7,40}.
26-33
Mechanics
Control Systems With Thyristor Converters
Аннотация
The article discuses systems whose control circuit contains thyristor converters or a special type of indirect control system. An example of constructing self-oscillations in a system describing the behavior of electrical devices using such elements in a control loop is given. The criterion of orbital stability and the method of synthesis of stabilizing controls are proposed. The conditions for the existence of orbital asymptotic stability and Lyapunov stability for periodic solutions with a given period are presented. A control loop has been synthesized that ensures the existence of such solutions. Examples of self-oscillations are given for structurally linear systems. Solutions for self-oscillations of a given period are constructed.
34-41
The Gyrostat Motion Around the Inertia Center in the Semi-Euclidean Space
Аннотация
The inertial motion of a gyrostat in a semi-Euclidean space with given index and defect is studied. A gyrostat with a constant gyrostatic moment moves so that its carrier rotates around a fixed center of inertia. Criteria for the existence of regular motions are obtained as a condition for the presence of axial structural-kinetic symmetry of the gyrostat. The properties of nutational, precessional, vibrational-rotational motions are studied and their description in configuration and phase spaces is given. The quadrature dependences of the gyrostat motion parameters in elliptic functions of time are determined. Parametric equations of hodographs of angular velocity and kinetic momentum vectors are found. The study was carried out for the case of the gyrostat angular momentum eigenvector.
42-53
Computer science
A Net Virus Spreading in a Local Computer Network Modeling With Using Percolation Theory Methods
Аннотация
A net virus spreading in a local computer network is investigated in this paper. Two percolation models were proposed to describe two types of networks: wired and wireless. The percolation threshold corresponds to the fraction of infected computers in the network at which the network loses its operability. Algorithms for lattice filling by occupied nodes, for distributing occupied nodes into clusters, for searching a percolation cluster and for the percolation threshold determining were developed and implemented for the models. A numerical experiment was conducted to estimate the percolation threshold and its dependence on various virus characteristics.
54-60
Comparative Evaluation of Clustering Methods in Working With Big Data
Аннотация
The paper considers the problems of using cluster analysis methods in the tasks of processing, analyzing and storing structured and unstructured large-volume data and evaluates the feasibility of their use in various aspects of working with Big Data. The aim of the work is to identify the most preferred of the common data clustering algorithms. To do this, the task was set to conduct a comparative evaluation of the following popular algorithms: hierarchical clustering, k-means, DBSCAN, OPTICS and CURE. The algorithmic complexity of the methods is considered, the stability of algorithms to noise and emissions is analyzed, as well as the potential possibilities of visualizing their results and the scope of economic application are indicated. Conclusions are drawn about the advantages and disadvantages of each presented algorithm when used in the field of Big Data and about the most preferred methods of cluster analysis in various aspects of working with big data.
61-67