Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics

Approximation of classes of Poisson integrals by Fejer means

O. G. Rovenska — 2023-05-16

The work is devoted to the investigation of problem of approximation of continuous periodic functions by trigonometric polynomials, which are generated by linear methods of summation of Fourier series.
The simplest example of a linear approximation of periodic functions is the approximation of functions by partial sums of their Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. Therefore, a many studies is devoted to the research of the approximative properties of approximation methods, which are generated by transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for the whole class of continuous functions. Particularly, Fejer means have been widely studied in the last time. One of the important problems in this field is the study of asymptotic behavior of the upper bounds over a fixed classes of functions of deviations of the trigonometric polynomials.
The aim of the work systematizes known results related to the approximation of classes of Poisson integrals of continuous functions by arithmetic means of Fourier sums, and presents new facts obtained for particular cases.
The asymptotic behavior of the upper bounds on classes of Poisson integrals of periodic functions of the real variable of deviations of linear means of Fourier series, which are defined by applying the Fejer summation method is studied. The mentioned classes consist of analytic functions of a real variable, which are narrowing of bounded harmonic in unit disc functions of complex variable. In the work, asymptotic equalities for the upper bounds of deviations of Fejer means on classes of Poisson integrals were obtained.

Construction of controllability function as time of motion

V. I. Korobov — 2023-06-08

This article is devoted to the controllability function method in admissible synthesis problems for linear canonical systems. The work considers methods of constructing such control so that the controllability function is time of motion of an arbitrary point to the origin. A canonical controlled system of linear equations $\dot{x}_i=x_{i+1}, i=\overline{1,n-1}, \dot{x}_n=u$ with control constraints $|u| \le d$ is considered. The controllability function $\Theta$ can be found as the only positive solution of the implicit equation $2a_0\Theta=(D(\Theta)FD(\Theta)x,x)$, where $D(\Theta)= diag(\Theta^ {-\frac{-2n-2i+1}{2}})_{i=1}^n$. Matrix $F=\{f_{ij}\}_{i,j=1}^n$ is positive definite and $a_0>0$ is chosen so that the control constraints are satisfied. The controllability function is motion time if $\dot{\Theta}= -1$. From this condition, an equation is obtained, the solution of which is considered in this work.
Unlike previous works on this topic, no additional restrictions are imposed on the appearance of matrix $F$. The task of this article is to find the parameters set of the matrix $F$ and the column vector $a$, which satisfy the obtained equation and for which the controllability function is the time of movement from the point $x$ to the origin. In this way, we get a family of controls depending on this parameters such that the trajectory of system steers the origin in finite time. In general case, difficulties may arise when finding the solution of Cauchy problem of the corresponding system. Canonical system can be reduced to Euler's equation, for which a characteristic equation can be found, and therefore a trajectory in an explicit form.
Two-dimensional, three-dimensional and four-dimensional canonical systems are considered. In each case, the matrix equation is solved and sets of parameters for which the controllability functions value will be the time of movement of an arbitrary point to the origin are found. Conditions on parameters are obtained from positive definiteness of the matrix $F$. Some parameters and an arbitrary initial point are chosen and the solution of Cauchy problem in analytical form is found.

A review on rheological models and mathematical problem formulations for blood flows

N. M. Kizilova — 2023-07-06

A review on constitutive equations proposed for mathematical modeling of laminar and turbulent flows of blood as a concentrated suspension of soft particles is given. The rheological models of blood as a uniform Newtonian fluid, non-Newtonian

shear-thinning, viscoplastic, viscoelastic, tixotropic and micromorphic fluids are discussed. According to the experimental data presented, the adequate rheological model must describe shear-thinning tixotropic behavior with concentration-dependent viscoelastic properties which are proper to healthy human blood. Those properties can be studied on the corresponding mathematical problem formulations for the blood flows through the tudes or ducts. The corresponding systems of equations and boundary conditions for each of the proposed rheological models are discussed. Exact solutions for steady laminar flows between the parallel plates and through the circular tubes have been obtained and analyzed for the Ostwald, Hershel-Bulkley, and Bingham shear-thinning fluids. The influence of the model parameters on the velocity profiles has been studied for each model. It is shown, certain sets of fluid parameters lead to flattening of the velocity profile while others produce its sharpening around the axis of the channel.

It is shown, the second-order terms in the viscoelastic models give the partial derivative differential equations with high orders in time and mixed space-time derivatives. The corresponding problem formulations for the generalized rhelogical laws are derived. Their analytical solutions in the form of a normal mode are obtained. It is shown, the dispersion equations produce an additional set for the speed of sound (so called second sound) in the fluid. It is concluded, the most general rheological model must include shear-thinning, concentration and second sound phenomena

Liver regeneration after partial hepatectomy: the upper optimality estimate

V. V. Karieva — 2023-06-07

This publication investigates one of the fundamental problems of mathematical biology, specifically the development of mathematical models for the dynamics of complex biosystems that have a satisfactory explanatory and predictable power. A necessary condition for the development of such models is to find a solution for the problem of identifying the objective principles and rules of regulation of the "cellular system", which determines among all the possibilities exactly the "real path" of its dynamics observed in the experiment.

One of the promising approaches to solving this problem is based on the hypothesis that the regulation of processes for support/restoration of the dynamic homeostasis of tissues and organs of the body occurs according to certain principles, and criteria of optimality, which have developed due to the natural selection of the body during its previous evolution.

It is quite difficult to solve this problem at the current time due to the many uncertainties in the paths of the previous evolution of the organism, the dynamics of changes in external conditions, as well as the high computational complexity of solving such a problem.

Instead of this, we have proposed a simplified formulation of the problem of searching for regulation control strategies, which gives us an upper estimate of optimality for the processes of maintaining/restoring dynamic homeostasis of the liver. The upper estimate of the optimality of regulation and testing of hypotheses for the model of liver regeneration was considered in the case of partial hepatectomy and was solved by Python software methods.

The result shows that in the case of partial hepatectomy, the liver regeneration strategies obtained in numerous experiments for the problem of the upper optimality estimate qualitatively coincide with the processes of liver regeneration that can be observed during biological experiments.

In plenty of experiments following hypotheses were also tested: how significant is the contribution of the process of controlled apoptosis, and how other processes (polyploidy, division, and formation of binuclear hepatocytes) affect the strategy of liver regeneration.