Вісник Київського національного університету імені Тараса Шевченка. Серія: фізико-математичні науки
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The development of software for simulation of random processes with a given accuracy and reliability
Get PDFToday, the theory of random processes and time series prediction is widely used in various fields of science, not only in natural fields. That is why one of the urgent problems is to build a mathematical model of a random process and study its properties. Numerical modeling tasks become especially important due to the powerful capabilities of computer technology, which allows you to create software modeling tools and predict the behavior of a random process. There are different methods of modeling random processes and fields. In some works related to the modeling of random processes, the issues of accuracy and reliability have not been studied. In [1, 2, 3] for various stochastic processes and fields this problem was investigated. In this paper the question of accuracy and reliability of the constructed model is considered. This means that we first build the model and then test it using some adequacy tests with known accuracy and reliability. We also find the estimators of the model parameters using methods of moments. All theoretical results are applied to develop software for model construction of stochastic processes.Key words: simulation, accuracy, reliability, estimation.Pages of the article in the issue: 83 - 87Language of the article: Ukrainia
Investigation of the periodic axisymmetric flow of a viscoelastic fluid through a cylindrical tube
Get PDFA generalized Womersley model of a nonstationary axisymmetric flow of a viscous incompressible fluid through a tube of circular cross-section to periodic pressure fluctuations at the inlet of the tube is obtained due for the case of a fluid with complicated rheology. The rheological parameters of the fluid are viscosity and four relaxation coefficients for strains and stresses of the first and second order. Such rheology is proper to the non-Newtonian viscoelastic fluids with mesostructure, namely technical and biological micro/ nanofluids. It was shown that with the increase of the relaxation coefficients of the first/second order the flow rate, the average and maximum velocities decrease/increase, accordingly. Simultaneous changes in these parameters can lead to complex changes in the velocity profile, especially for higher harmonics. The studied regularities can explain the deviations of the flow parameters of different micro/nanofluids from the values predicted by the classical Womersley solution for a homogeneous Newtonian fluid, which does not take into account viscous dissipation during the rearrangement of the fluid mesostructure.Key words: Womersley problem, viscoelastic fluids, rheological properties, mathematical modeling.Pages of the article in the issue: 49 - 52Language of the article: Ukrainia
From subgaussianity to stochastic approximation and modelling
Get PDFThe modern theory of subgaussian random variables and processes was created by independent efforts of several research schools from France, USA and Ukraine. Professor Yu.Kozachenko was a founder and leading figure of this research direction of the Ukrainian probability school. An outline of Professor Yu.Kozachenko\u27s contribution to the theory of sub-Gaussian random variables and processes is presented. The class of -subgaussian random variables is introduced and its key property is discussed. Then it is demonstrated how these results can be used in stochastic approximation and modeling. In particular, applications to approximation of trajectories of -subgaussian random processes with given accuracy and reliability are discussed. Two important clases of algorithms from the signal processing theory, the Shannon sampling method and wavelet decompositions, are used as examples. Some personal memories of the author about Yu. Kozachenko are included at the end of the paper.Key words: Yu.Kozachenko, subgaussian, stochastic approximation, stochastic modelling.Pages of the article in the issue: 84 - 88Language of the article: Ukrainia
On Hasse diagrams connected with the poset (1, 2, 7)
Get PDFRepresentations of posets introduced in 1972 by L. O. Nazarova and A. V. Roiter, arise when solving many problems in various fields of mathematics. One of the most important problem in the theory of representations of any objects is a description of the cases of representation finite type and representation tame type. The first of these problems for posets was solved by M. M. Kleiner, and the second L. O, Nazarova. M. M. Kleiner proved that a poset has finite type if and only if it does not contain subsets of the form (1, 1, 1, 1), (2, 2, 2), (1, 3, 3), (1, 2, 5) and (?, 4), which are called the critical sets. A generalization of this criterion to the tame case was obtained by L. O. Nazarova. The corresponding sets are called supercritical and they consist of the posets (1, 1, 1, 1, 1), (1, 1, 1, 2), (2, 2, 3), (1, 3, 4), (1, 2, 6) and (?, 5). V. M. Bondarenko proposed a generalization of the critical and supercritical posets, calling them 1-oversupercritical. This paper studies the combinatorial properties of one of such sets.Key words: poset, graph, critical and supercritical poset, 1-oversupercritical poset, anti-isomorphism, Hasse diagrams, minimax equivalence.Pages of the article in the issue: 16 - 19Language of the article: Englis
Analytical solution and adaptation to the parameter estimation of the SIR model
Get PDFThe article deals with analytical solution and adaptation to the parameter estimation of the SIR model of the epidemic. By a special replacement of the exponential function by inverse proportionality, the approximate general solution of the SIR model is found. It is spoken in detail about the process of integration of ordinary differential equations of the SIR model. The equality of the sum of the obtained analytical solutions and population size is checked. The obtained solutions are simple and understandable. To parametrically estimate the SIR model, its general solution is adapted to paired linear regressions. The article is of interest for students, graduate students and scientists involved in mathematical epidemiology.Key words: SIR model, parameter estimation, analytical solution.Pages of the article in the issue: 40 - 43Language of the article: Ukrainia
Stokes-polarimetry of ultrathin Au and Sn island films
Get PDFThe optical properties of ultrathin Au and Sn islet films, obtained by the methods of magnetron sputtering and thermal evaporation, respectively, are considered in this paper. By measuring the Stokes vector of the beam reflected from the samples, polarized and depolarized radiation components were separated. The conditions of the polarization degree dependence on the surface structure for a series of islet films with different morphologies are analyzed. To determine the morphological structure of the metal layer, methods of atomic force microscopy and resistivity measurement were also employed. The parameters of discontinuous film, obtained by optical and non-optical methods, are compared. It is established that with an increase in the angle of radiation incidence onto the samples, the polarization degree of the reflected beam decreases. Such behavior can be explained by the Mie theory of light scattering by particles. The magnitude of depolarizing action of the samples also depends on the morphology of their surface, correlating with the number of inequalities on it. The applied method of Stokes polarimetry, thus, allows one to obtain additional information on the structure of the surface, which is its advantage.Key words: polarimetry, Stokes vector, island film, depolarization, gold, tin.Pages of the article in the issue: 122-129Language of the article: UkrainianThe optical properties of ultrathin Au and Sn islet films, obtained by the methods of magnetron sputtering and thermal evaporation, respectively, are considered in this paper. By measuring the Stokes vector of the beam reflected from the samples, polarized and depolarized radiation components were separated. The conditions of the polarization degree dependence on the surface structure for a series of islet films with different morphologies are analyzed. To determine the morphological structure of the metal layer, methods of atomic force microscopy and resistivity measurement were also employed. The parameters of discontinuous film, obtained by optical and non-optical methods, are compared. It is established that with an increase in the angle of radiation incidence onto the samples, the polarization degree of the reflected beam decreases. Such behavior can be explained by the Mie theory of light scattering by particles. The magnitude of depolarizing action of the samples also depends on the morphology of their surface, correlating with the number of inequalities on it. The applied method of Stokes polarimetry, thus, allows one to obtain additional information on the structure of the surface, which is its advantage.Key words: polarimetry, Stokes vector, island film, depolarization, gold, tin.Pages of the article in the issue: 122 - 129Language of the article: Ukrainia
Maghetohydrodynamic flows of micro/nano fluids through thin capillaries
Get PDFSteady magnetohydrodynamic (MHD) flows of suspensions of conducting micro/nanoparticles through a thin tube of a circular cross-section in a transverse constant magnetic field driven by a constant pressure drops at the ends of the tube is studied. The governing MHD system of equations for a viscous incompressible micro/nanofluid in the non-induction approximation is solved with the second order velocity slip boundary condition at the wall of the tube. The material parameters of the fluid are considered as nonlinear functions of the particle concentration according to the mixture models of suspensions and electric conductivity theory. The velocity field, pressure, electric current and magnetic field have been computed as series expansions. The influence of two non-dimensional slip coefficients of the flow rate and wall shear stress is studied. Optimal concentrations of the micro/nanoparticles in the suspensions have been computed from the minimum entropy production condition for different slip conditions, material parameters, magnetic fields and flow regimes (Reynolds and Hartmann numbers).Key words: Microfluids, Nanofluids, Magnetohydrodynamics, Steady flows.Pages of the article in the issue: 32 - 36Language of the article: EnglishSteady magnetohydrodynamic (MHD) flows of suspensions of conducting micro/nanoparticles through a thin tube of a circular cross-section in a transverse constant magnetic field driven by a constant pressure drops at the ends of the tube is studied. The governing MHD system of equations for a viscous incompressible micro/nanofluid in the non-induction approximation is solved with the second order velocity slip boundary condition at the wall of the tube. The material parameters of the fluid are considered as nonlinear functions of the particle concentration according to the mixture models of suspensions and electric conductivity theory. The velocity field, pressure, electric current and magnetic field have been computed as series expansions. The influence of two non-dimensional slip coefficients of the flow rate and wall shear stress is studied. Optimal concentrations of the micro/nanoparticles in the suspensions have been computed from the minimum entropy production condition for different slip conditions, material parameters, magnetic fields and flow regimes (Reynolds and Hartmann numbers).Key words: Microfluids, Nanofluids, Magnetohydrodynamics, Steady flows.Pages of the article in the issue: 32 - 36Language of the article: Englis
Automorphism group of the variant of the lattice of partitions of a finite set
Get PDFIn this paper we consider variants of the lattice of partitions of a finite set and study automorphism groups of this variants. We obtain irreducible generating sets for of the lattice of partitions of a finite set.We prove that the automorphism group of the variant of the lattice of partitions of a finite set is a natural generalization of the wreath product. The first multiplier of this generalized wreath product is the direct product of the wreaths products, such that depends on the type of the variant generating partition and the second is defined by the certain set of symmetric groups.Key words: semigroup, variant, lattice, automorphism group.Pages of the article in the issue: 115 - 119Language of the article: Englis
Wave characteristics of the cardiorespiratory system under increasing of a heart rate
Get PDFThe heart rate influence on chaotic dynamics of the model of interaction between cardiovascular and respiratory subsystems has been studied. The considered model contains both direct and inverse mechanical connection between the heart and the waves generated by the movement of the chest. Applied numerical methods of the theory of dynamical systems are used. It is shown how an increase in heart rate can affect the wave characteristics of the heart.Key words: cardiovascular system, respiratory system, limited excitation, chaos.Pages of the article in the issue: 71 - 74Language of the article: Ukrainia
Characteristic Cauchy’s problem with prehistory in the case of nonlinear differential equations in partial derivatives
Get PDFWe have built a constructive method of investigation and approximate solution for nonlinear Gursa’s problem with prehistory. We have established sufficient condition of subsistence, existence of unity and constant signs solution of the investigated problem. At mathematical description to different nature process (gas sorption, the spread of moisture in the porous substances, pipes heating by a stream of hot water, drying by the airflow, etc. [1]) we often come to boundary value problems for nonlinear differential equations in partial derivatives, when not all output data are known, that is some of them need to be found from auxiliary nonlinear problems, which are mathematical models of processes that proceeded the research. These problems should be named as problems with prehistory. One approach to investigation and approximate solution to such a problem has been proposed in the current paper.Key words: two-sided method, sufficient conditions, "free" curves, characteristic Cauchy’s problem with prehistory.Pages of the article in the issue: 28 - 33Language of the article: Ukrainia