I. D. Borisov (obituary)

On December 3, 2020, a senior researcher at the Department of Applied Mathematics of V. N. Karazin Kharkiv National University Ivan Dmitrovich Borisov passed away. The bright memory of Ivan Dmitrievich Borisov, a real scientist and a wonderful man, forever remains in the hearts of his colleagues, students and friends.

Implicit linear difference equations over a non-Archi-medean ring

Over any field an implicit linear difference equation one can reduce to the usual explicit one, which has infinitely many solutions ~ one for each initial value. It is interesting to consider an implicit difference equation over any ring, because the case of implicit equation over a ring is a significantly different from the case of explicit one. The previous results on the difference equations over rings mostly concern to the ring of integers and to the low order equations. In the present article the high order implicit difference equations over some other classes of rings, particularly, ring of polynomials, are studied. To study the difference equation over the ring of integer the idea of considering p-adic integers ~ the completion of the ring of integers with respect to the non-Archimedean p-adic valuation was useful. To find a solution of such an equation over the ring of polynomials it is naturally to consider the same construction for this ring: the ring of formal power series is a completion of the ring of polynomials with respect to a non-Archimedean valuation. The ring of formal power series and the ring of p-adic integers both are the particular cases of the valuation rings with respect to the non-Archimedean valuations of some fields: field of Laurent series and field of p-adic rational numbers respectively. In this article the implicit linear difference equation over a valuation ring of an arbitrary field with the characteristic zero and non-Archimedean valuation are studied. The sufficient conditions for the uniqueness and existence of a solution are formulated. The explicit formula for the unique solution is given, it has a form of sum of the series, converging with respect to the non-Archimedean valuation. Difference equation corresponds to an infinite system of linear equations. It is proved that in a case the implicit difference equation has a unique solution, it can be found using Cramer rules. Also in the article some results facilitating the finding the polynomial solution of the equation are given.

To the 75th anniversary of Academician of the NAS of Ukraine A. A. Borisenko

May 24, 2021 turned 75 years since the birth of the famous mathematician, academician of the National Academy of Sciences of Ukraine Oleksandr Andriyovych Borisenko. The editorial board welcomes. According to the interviews of O.A. Borisenko: ''Half a century in geometry. To the 75th anniversary of corresponding member of NAS of Ukraine O.A. Borisenko. (2021). Bulletin of the National Academy of Sciences of Ukraine, (5), 95–102.''

Controllability of systems of linear partial differential equations

In a number of papers, the controllability theory was recently studied. But quite a few of them were devoted to control systems described by ordinary differential equations. In the case of systems described by partial differential equations, they were studied mostly for classical equations of mathematical physics. For example, in papers by G. Sklyar and L. Fardigola, controllability problems were studied for the wave equation on a half-axis. In the present paper, the complete controllability problem is studied for systems of linear partial differential equations with constant coefficients in the Schwartz space of rapidly decreasing functions. Necessary and sufficient conditions for complete controllability are obtained for these systems with distributed control of the special form: u(x,t)=e-αtu(x). To prove these conditions, other necessary and sufficient conditions obtained earlier by the author are applied (see ``Controllability of evolution partial differential equation''. Visnyk of V. N. Karasin Kharkiv National University. Ser. ``Mathematics, Applied Mathematics and Mechanics''. 2016. Vol. 83, p. 47-56). Thus, the system $$\frac{\partial w(x,t)}{\partial t} = P\left(\frac\partial{i\partial x} \right) w(x,t)+ e^{-\alpha t}u(x),\quad t\in[0,T], \ x\in\mathbb R^n, $$ is completely controllable in the Schwartz space if there exists α>0 such that $$\det\left( \int_0^T \exp\big(-t(P(s)+\alpha E)\big)\, dt\right)\neq 0,\quad s\in\mathbb R^N.$$ This condition is equivalent to the following one: there exists $\alpha>0$ such that $$\exp\big(-T(\lambda_j(s)+\alpha)\big)\neq 1 \quad \text{if}\ (\lambda_j(s)+\alpha)\neq0,\qquad s\in\mathbb R^n,\ j=\overline{1,m},$$ where $\lambda_j(s)$, $j=\overline{1,m}$, are eigenvalues of the matrix $P(s)$, $s\in\mathbb R^n$. The particular case of system (1) where $\operatorname{Re} \lambda_j(s)$, $s\in\mathbb R$, $j=\overline{1,m}$, are bounded above or below is studied. These systems are completely controllable. For instance, if the Petrovsky well-posedness condition holds for system (1), then it is completely controllable. Conditions for the existence of a system of the form (1) which is not completely controllable are also obtained. An example of a such kind system is given. However, if a control of the considered form does not exists, then a control of other form solving complete controllability problem may exist. An example illustrating this effect is also given in the paper.

Speed of convergence of complementary probabilities on finite group

Let function P be a probability on a finite group G, i.e. $P(g)\geq0\ $ $(g\in G),\ \sum\limits_{g}P(g)=1$ (we write $\sum\limits_{g}$ instead of $\sum\limits_{g\in G})$. Convolution of two functions $P, \; Q$ on group $G$ is \linebreak $ (P*Q)(h)=\sum\limits_{g}P(g)Q(g^{-1}h)\ \ (h\in G)$. Let $E(g)=\frac{1}{|G|}\sum\limits_{g}g$ be the uniform (trivial) probability on the group $G$, $P^{(n)}=P*...*P$ ($n$ times) an $n$-fold convolution of $P$. Under well known mild condition probability $P^{(n)}$ converges to $E(g)$ at $n\rightarrow\infty$. A lot of papers are devoted to estimation the rate of this convergence for different norms. Any probability (and, in general, any function with values in the field $R$ of real numbers) on a group can be associated with an element of the group algebra of this group over the field $R$. It can be done as follows. Let $RG$ be a group algebra of a finite group $G$ over the field $R$. A probability $P(g)$ on the group $G$ corresponds to the element $ p = \sum\limits_{g} P(g)g $ of the algebra RG. We denote a function on the group $G$ with a capital letter and the corresponding element of $RG$ with the same (but small) letter, and call the latter a probability on $RG$. For instance, the uniform probability $E(g)$ corresponds to the element $e=\frac{1}{|G|}\sum\limits_{g}g\in RG. $ The convolution of two functions $P, Q$ on $G$ corresponds to product $pq$ of corresponding elements $p,q$ in the group algebra $RG$. For a natural number $n$, the $n$-fold convolution of the probability $P$ on $G$ corresponds to the element $p^n \in RG$. In the article we study the case when a linear combination of two probabilities in algebra $RG$ equals to the probability $e\in RG$. Such a linear combination must be convex. More exactly, we correspond to a probability $p \in RG$ another probability $p_1 \in RG$ in the following way. Two probabilities $p, p_1 \in RG$ are called complementary if their convex linear combination is $e$, i.e. $ \alpha p + (1- \alpha) p_1 = e$ for some number $\alpha$, $0 <\alpha <1$. We find conditions for existence of such $\alpha$ and compare $\parallel p ^ n-e \parallel$ and $\parallel {p_1} ^ n-e \parallel$ for an arbitrary norm ǁ·ǁ.

BVI-noise generation by wing-shaped helicopter blade

Aerodynamic noise includes a number of noise components, among which rotational noise and vortex noise (BVI-noise) make the largest contribution to the overall noise generated. Rotation noise depends on the magnitude of the velocity of the incoming blade and prevails over other noise components at significant Mach Mach numbers. Unlike rotation noise, vortex noise is evident at low helicopter flight speeds, moderate Mach numbers. In the formation of this type of noise,an important role is played by the longitudinal geometry. Therefore, recently the shape of the helicopter blade is chosen close to existing natural forms, which are as balanced as possible. One of these may be a wing-shaped blade. In this work, the problem of generating BVI noise by the wing shaped blade of a helicopter is posed and solved. The mathematical model of the problem is constructed on the previously proposed by the author and successfully tested system of aeroacoustic equations for the general case. Estimated features in this system are pulsations of sound pressure and sound potential. The calculated data of these quantities, as well as their derivatives, were used to study near and far sound fields. In particular, the dependence of the density ripple distribution is revealed from the blade geometry, the angle of attack and the blade angle to the oncoming flow. Increasing flow velocity contributes to the emergence of transverse ripples on the surface blades that dominate the longitudinal ripples by level. An interesting feature noticed in the calculations is that there are calculations for moderate Mach numbers M=0.2,0.3 situations, at certain angles of blade placement to the stream and angles of attack where rotation noise dominates eddy noise. For values Mach numbers M>0.4 rotation noise plays a major role in blade noise generation. The noise level generated is in the range 50dB≤L≤60dB, which is lower by 5-6dB for the Blue Edge blade, as well as the rounded blade. In addition, activation of the high-frequency region in the frequency spectrum of noise was observed f≈840Hz. The results of the calculations show that the blade of the wing-shaped is low-noise in the mode of maneuvers at small flight speeds.

Different strategies in the liver regeneration processes. Numerical experiments on the mathematical model

It is considered the generalized mathematical model which describes the processes of maintaining / restoring dynamic homeostasis (regeneration) of the liver and obviously depends on the control parameters. The model is a system of discrete controlled equations of the Lotka – Volterra type with transitions. These equations describe the controlled competitive dynamics of liver cell populations’ (hepatic lobules) various types in their various states and controlled competitive transitions between types and states. To develop this model there were accepted such assumptions: homogeneous approximation; independence of biological processes; small toxic factors. In the mathematical model the process of the liver regeneration occurs due to hyperplasia processes, replication, polyplodia and division of binuclear hepatocytes into mononuclear and controlled apoptosis. All these processes are necessary for adequate modeling of the liver regeneration. For example, single and constant toxic functions show that the above processes are not able to cope with the toxic factors that are accumulated in the body. The process of restoring the body’s functional state requires the non-trivial strategy of the liver regeneration. Numerical calculations revealed that the mathematical model corresponds to biological processes for different strategies of the liver regeneration. Based on the calculations in the case of partial hapatectomy it is concluded that the mixed strategy of regeneration should be used for the regeneration process. Henceforward it is planned to extend the mathematical model in the case of the liver regeneration, which occurs under the influence of strong toxins, that is, using the stem cells and fibrosis. It is also supposed to justify the principles and criteria for optimal regulation of the processes of maintaining / restoring liver’s dynamic homeostasis.

A study of a quasilinear model of the particles of a suspension that are aggregated and settled in an inhomogeneous field

The mathematical model of the sedimentation process of suspension particles is usually a quasilinear hyperbolic system of partial differential equations, supplemented by initial and boundary conditions. In this work, we study a complex model that takes into account the aggregation of particles and the inhomogeneity of the field of external mass forces. The case of homogeneous initial conditions is considered, when all the parameters of the arising motion depend on only one spatial Cartesian coordinate x and on time t. In contrast to the known formulations for quasilinear systems of equations (for example, as in gas dynamics), the solutions of which contain discontinuities, in the studied formulation the basic system of equations occurs only on one side of the discontinuity line in the plane of variables (t; x). On the opposite side of the discontinuity surface, the equations have a different form in general. We will restrict ourselves to considering the case when there is no motion in a compact zone occupied by settled particles, i.e. all velocities are equal to zero and the volumetric contents of all phases do not change over time. The problem of erythrocyte sedimentation in the field of centrifugal forces in a centrifuge, with its uniform rotation with angular velocity ω = const is considered. We have studied the conditions for the existence of various types of solutions. One of the main problems is the evolution (stability) problem of the emerging discontinuities. The solution of this problem is related to the analysis of the relationships for the characteristic velocities and the velocity of the discontinuity surface. The answer depends on the number of characteristics that come to the jump, and the number of additional conditions set on the interface. The discontinuity at the lower boundary of the area occupied by pure plasma is always stable. But for the surface separating the zones of settled and of moving particles, the condition of evolution may be violated. In this case, it is necessary to adjust the original mathematical model.

Regularization of the electrostatics problem for three spheres and an electrostatic charge

A numerical-analytical algorithm for investigation of the potential of a sphere with a circular hole, surrounded by external and internal closed ribbon spheres, is constructed. The number of ribbons on the spheres is arbitrary. The ribbons on the spheres are separated by non-conductive, infinitely thin partitions. The partitions are located in planes parallel to the shear plane of the sphere with a hole. Each ribbon has its own independent potential. An electrostatic charge is placed between the outer sphere and the sphere with a hole in the axis of the structure. The full potential must satisfy, in particular, Maxwell’s equations, taking into account the absence of magnetic fields, satisfy the boundary conditions, have the required singularity at the point where the charge is placed. To solve this problem, we first used the method of partial domains and the method of separating variables in a spherical coordinate system. In this case, for the Fourier series, we use power functions and Legendre polynomials of integer orders. From the boundary conditions, using an auxiliary system of 3 equations with 4 unknowns, a pairwise system of functional equations of the first kind with respect to the coefficients of the Fourier series is obtained. The system is not effective for solving by direct methods. The method of inversion of the Volterra integral operator and semi-inversion of the matrix operators of the Dirichlet problem for the Laplace equation are applied. The method is based on the ideas of the analytical method of the Riemann - Hilbert problem. In this case, integral representations for the Legendre polynomials are used. A system of linear algebraic equations of the second kind with a compact matrix operator in the Hilbert space l`2 is obtained. The system is effectively solvable numerically for arbitrary parameters of the problem and analytically for the limiting parameters of the problem. Particular variants of the problem are considered.

One numerical approach to optimal control the linear heat conduction processes

It is proposed the generalized mathematical formulation of the problem about the optimal control for the heat conduction processes representing by the partial differential equation. The proposed formulation not includes the necessary clarifications about the conditions which must be satisfied by the current and required temperature fields. But, during the generalized solving of the formulated problem, it is established that the current and required temperature fields must be agreed with the mathematical model of the heat conduction so that to have possibilities to provide uniquely these temperature fields by means the control vector. To solve the problem about the optimal control for the heat conduction processes it is developed the numerical approaches based on reducing to the especially built ordinary differential equations and minimization problem. This reducing is based on discretisation the heat conduction by using the grid method and on defining the unknown control vector as the numerical solution of the especially built Cauchy problem. To satisfy the all limitations it is proposed to build the permissible velocity of the unknown control vector considering with the requirements of necessary switching in some moments of the time. The particular example of using the proposed generalized approaches is considered to illustrate their application technique. It is shown that the proposed generalized mathematical formulation is fully corresponded with the considered particular example. In this considered particular example, the resolving Cauchy problem can be built and the switching time can be found in the depending on the grid node choosing. It is shown that the transient time can be decrease almost twice due to optimizing the control in the particular example at least. All these results will allow giving the clear representation of the proposed approaches and the technique of their using to solve the engineering problems about the optimal control of the heat conduction processes in different industrial systems.