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Estimate of the module of analogue Weyl’s trigonometrical sum in ring of Gaussian numbers
Computer Research and Modeling, 2010, v. 2, no. 4, pp. 343347The ring of Gaussian numbers is considered. The estimation of the module of some analogue of Weyl's trigonometrical sum with summation on Gaussian numbers is proved by methods of the analytical number theory. Multiplicative norm of Gaussian numbers is less than some integer.

Computer studies of polynomial solutions for gyrostat dynamics
Computer Research and Modeling, 2018, v. 10, no. 1, pp. 725Views (last year): 15.We study polynomial solutions of gyrostat motion equations under potential and gyroscopic forces applied and of gyrostat motion equations in magnetic field taking into account Barnett–London effect. Mathematically, either of the above mentioned problems is described by a system of nonlinear ordinary differential equations whose right hand sides contain fifteen constant parameters. These parameters characterize the gyrostat mass distribution, as well as potential and nonpotential forces acting on gyrostat. We consider polynomial solutions of Steklov–Kovalevski–Gorjachev and Doshkevich classes. The structure of invariant relations for polynomial solutions shows that, as a rule, on top of the fifteen parameters mentioned one should add no less than twenty five problem parameters. In the process of solving such a multiparametric problem in this paper we (in addition to analytic approach) apply numeric methods based on CAS. We break our studies of polynomial solutions existence into two steps. During the first step, we estimate maximal degrees of polynomials considered and obtain a nonlinear algebraic system for parameters of differential equations and polynomial solutions. In the second step (using the above CAS software) we study the solvability conditions of the system obtained and investigate the conditions of the constructed solutions to be real.
We construct two new polynomial solutions for Kirchhoff–Poisson. The first one is described by the following property: the projection squares of angular velocity on the nonbaracentric axes are the fifth degree polynomials of the angular velocity vector component of the baracentric axis that is represented via hypereliptic function of time. The second solution is characterized by the following: the first component of velocity conditions is a second degree polynomial, the second component is a polynomial of the third degree, and the square of the third component is the sixth degree polynomial of the auxiliary variable that is an inversion of the elliptic Legendre integral.
The third new partial solution we construct for gyrostat motion equations in the magnetic field with Barnett–London effect. Its structure is the following: the first and the second components of the angular velocity vector are the second degree polynomials, and the square of the third component is a fourth degree polynomial of the auxiliary variable which is found via inversion of the elliptic Legendre integral of the third kind.
All the solutions constructed in this paper are new and do not have analogues in the fixed point dynamics of a rigid body.

Direct multiplicative methods for sparse matrices. Newton methods
Computer Research and Modeling, 2017, v. 9, no. 5, pp. 679703Views (last year): 7. Citations: 1 (RSCI).We consider a numerically stable direct multiplicative algorithm of solving linear equations systems, which takes into account the sparseness of matrices presented in a packed form. The advantage of the algorithm is the ability to minimize the filling of the main rows of multipliers without losing the accuracy of the results. Moreover, changes in the position of the next processed row of the matrix are not made, what allows using static data storage formats. Linear system solving by a direct multiplicative algorithm is, like the solving with $LU$decomposition, just another scheme of the Gaussian elimination method implementation.
In this paper, this algorithm is the basis for solving the following problems:
Problem 1. Setting the descent direction in Newtonian methods of unconditional optimization by integrating one of the known techniques of constructing an essentially positive definite matrix. This approach allows us to weaken or remove additional specific difficulties caused by the need to solve large equation systems with sparse matrices presented in a packed form.
Problem 2. Construction of a new mathematical formulation of the problem of quadratic programming and a new form of specifying necessary and sufficient optimality conditions. They are quite simple and can be used to construct mathematical programming methods, for example, to find the minimum of a quadratic function on a polyhedral set of constraints, based on solving linear equations systems, which dimension is not higher than the number of variables of the objective function.
Problem 3. Construction of a continuous analogue of the problem of minimizing a real quadratic polynomial in Boolean variables and a new form of defining necessary and sufficient conditions of optimality for the development of methods for solving them in polynomial time. As a result, the original problem is reduced to the problem of finding the minimum distance between the origin and the angular point of a convex polyhedron, which is a perturbation of the $n$dimensional cube and is described by a system of double linear inequalities with an upper triangular matrix of coefficients with units on the main diagonal. Only two faces are subject to investigation, one of which or both contains the vertices closest to the origin. To calculate them, it is sufficient to solve $4n – 4$ linear equations systems and choose among them all the nearest equidistant vertices in polynomial time. The problem of minimizing a quadratic polynomial is $NP$hard, since an $NP$hard problem about a vertex covering for an arbitrary graph comes down to it. It follows therefrom that $P = NP$, which is based on the development beyond the limits of integer optimization methods.

Relaxation model of viscous heatconducting gas
Computer Research and Modeling, 2022, v. 14, no. 1, pp. 2343A hyperbolic model of a viscous heatconducting gas is presented, in which the Maxwell – Cattaneo approach is used to hyperbolize the equations, which provides finite wave propagation velocities. In the modified model, instead of the original Stokes and Fourier laws, their relaxation analogues were used and it is shown that when the relaxation times $\tau_\sigma^{}$ и $\tau_w^{}$ tend to The hyperbolized equations are reduced to zero to the classical Navier – Stokes system of nonhyperbolic type with infinite velocities of viscous and heat waves. It is noted that the hyperbolized system of equations of motion of a viscous heatconducting gas considered in this paper is invariant not only with respect to the Galilean transformations, but also with respect to rotation, since the Yaumann derivative is used when differentiating the components of the viscous stress tensor in time. To integrate the equations of the model, the hybrid Godunov method (HGM) and the multidimensional nodal method of characteristics were used. The HGM is intended for the integration of hyperbolic systems in which there are equations written both in divergent form and not resulting in such (the original Godunov method is used only for systems of equations presented in divergent form). A linearized solver’s Riemann is used to calculate flow variables on the faces of adjacent cells. For divergent equations, a finitevolume approximation is applied, and for nondivergent equations, a finitedifference approximation is applied. To calculate a number of problems, we also used a nonconservative multidimensional nodal method of characteristics, which is based on splitting the original system of equations into a number of onedimensional subsystems, for solving which a onedimensional nodal method of characteristics was used. Using the described numerical methods, a number of onedimensional problems on the decay of an arbitrary rupture are solved, and a twodimensional flow of a viscous gas is calculated when a shock jump interacts with a rectangular step that is impermeable to gas.

Bank slope evolution in trapezoidal channel riverbed
Computer Research and Modeling, 2022, v. 14, no. 3, pp. 581592A mathematical model is formulated for the coastal slope erosion of sandy channel, which occurs under the action of a passing flood wave. The moving boundaries of the computational domain — the bottom surface and the free surface of the hydrodynamic flow — are determined from the solution of auxiliary differential equations. A change in the hydrodynamic flow section area for a given law of change in the flow rate requires a change in time of the turbulent viscosity averaged over the section. The bottom surface movement is determined from the Exner equation solution together with the equation of the bottom material avalanche movement. The Exner equation is closed by the original analytical model of traction loads movement. The model takes into account transit, gravitational and pressure mechanisms of bottom material movement and does not contain phenomenological parameters.
Based on the finite element method, a discrete analogue of the formulated problem is obtained and an algorithm for its solution is proposed. An algorithm feature is control of the free surface movement influence of the flow and the flow rate on the process of determining the flow turbulent viscosity. Numerical calculations have been carried out, demonstrating qualitative and quantitative influence of these features on the determining process of the flow turbulent viscosity and the channel bank slope erosion.
Data comparison on bank deformations obtained as a result of numerical calculations with known flume experimental data showed their agreement.

NINE: computer code for numerical solution of the boundary problems for nonlinear differential equations on the basis of CANM
Computer Research and Modeling, 2012, v. 4, no. 2, pp. 315324Views (last year): 1. Citations: 1 (RSCI).The computer code NINE (Newtonian Iteration for Nonlinear Equation) for numerical solution of the boundary problems for nonlinear differential equations on the basis of continuous analogue of the Newton method (CANM) is presented. Numerov’s finitedifference appproximation is applied to provide the fourth accuracy order with respect to the discretization stepsize. Algorithms of calculating the Newtonian iterative parameter are discussed. A convergence of iteration process in dependence on choice of the iteration parameter has been studied. Results of numerical investigation of the particlelike solutions of the scalar field equation are given.

Forecasting methods and models of disease spread
Computer Research and Modeling, 2013, v. 5, no. 5, pp. 863882Views (last year): 71. Citations: 19 (RSCI).The number of papers addressing the forecasting of the infectious disease morbidity is rapidly growing due to accumulation of available statistical data. This article surveys the major approaches for the shortterm and the longterm morbidity forecasting. Their limitations and the practical application possibilities are pointed out. The paper presents the conventional time series analysis methods — regression and autoregressive models; machine learningbased approaches — Bayesian networks and artificial neural networks; casebased reasoning; filtrationbased techniques. The most known mathematical models of infectious diseases are mentioned: classical equationbased models (deterministic and stochastic), modern simulation models (network and agentbased).

Stochastic sensitivity analysis of dynamic transformations in the “two prey – predator” model
Computer Research and Modeling, 2022, v. 14, no. 6, pp. 13431356This work is devoted to the study of the problem of modeling and analyzing complex oscillatory modes, both regular and chaotic, in systems of interacting populations in the presence of random perturbations. As an initial conceptual deterministic model, a Volterra system of three differential equations is considered, which describes the dynamics of prey populations of two competing species and a predator. This model takes into account the following key biological factors: the natural increase in prey, their intraspecific and interspecific competition, the extinction of predators in the absence of prey, the rate of predation by predators, the growth of the predator population due to predation, and the intensity of intraspecific competition in the predator population. The growth rate of the second prey population is used as a bifurcation parameter. At a certain interval of variation of this parameter, the system demonstrates a wide variety of dynamic modes: equilibrium, oscillatory, and chaotic. An important feature of this model is multistability. In this paper, we focus on the study of the parametric zone of tristability, when a stable equilibrium and two limit cycles coexist in the system. Such birhythmicity in the presence of random perturbations generates new dynamic modes that have no analogues in the deterministic case. The aim of the paper is a detailed study of stochastic phenomena caused by random fluctuations in the growth rate of the second population of prey. As a mathematical model of such fluctuations, we consider white Gaussian noise. Using methods of direct numerical modeling of solutions of the corresponding system of stochastic differential equations, the following phenomena have been identified and described: unidirectional stochastic transitions from one cycle to another, trigger mode caused by transitions between cycles, noiseinduced transitions from cycles to the equilibrium, corresponding to the extinction of the predator and the second prey population. The paper presents the results of the analysis of these phenomena using the Lyapunov exponents, and identifies the parametric conditions for transitions from order to chaos and from chaos to order. For the analytical study of such noiseinduced multistage transitions, the technique of stochastic sensitivity functions and the method of confidence regions were applied. The paper shows how this mathematical apparatus allows predicting the intensity of noise, leading to qualitative transformations of the modes of stochastic population dynamics.

Struggle against economic corruption in resource allocation
Computer Research and Modeling, 2019, v. 11, no. 1, pp. 173185Views (last year): 33. Citations: 1 (RSCI).A dynamic game theoretic model of struggle against corruption in resource allocation is considered. It is supposed that the system of resource allocation includes one principal, one or several supervisors, and several agents. The relations between them are hierarchical: the principal influences to the supervisors, and they in turn exert influence on the agents. It is assumed that the supervisor can be corrupted. The agents propose bribes to the supervisor who in exchange allocates additional resources to them. It is also supposed that the principal is not corrupted and does not have her own purposes. The model is investigated from the point of view of the supervisor and the agents. From the point of view of agents a noncooperative game arises with a set of Nash equilibria as a solution. The set is found analytically on the base of Pontryagin maximum principle for the specific class of model functions. From the point of view of the supervisor a hierarchical Germeyer game of the type Г_{2t} is built, and the respective algorithm of its solution is proposed. The punishment strategy is found analytically, and the reward strategy is built numerically on the base of a discrete analogue of the initial continuous time model. It is supposed that all agents can change their strategies in the same time instants only a finite number of times. Thus, the supervisor can maximize his objective function of many variables instead of maximization of the objective functional. A method of qualitatively representative scenarios is used for the solution. The idea of this method consists in that it is possible to choose a very small number of scenarios among all potential ones that represent all qualitatively different trajectories of the system dynamics. These scenarios differ in principle while all other scenarios yield no essentially new results. Then a complete enumeration of the qualitatively representative scenarios becomes possible. After that, the supervisor reports to the agents the rewardpunishment control mechanism.

From homogeneous to inhomogeneous electronic analogue of DNA
Computer Research and Modeling, 2020, v. 12, no. 6, pp. 13971407In this work, the problem of constructing an electronic analogue of heterogeneous DNA is solved with the help of the methods of mathematical modeling. Electronic analogs of that type, along with other physical models of living systems, are widely used as a tool for studying the dynamic and functional properties of these systems. The solution to the problem is based on an algorithm previously developed for homogeneous (synthetic) DNA and modified in such a way that it can be used for the case of inhomogeneous (native) DNA. The algorithm includes the following steps: selection of a model that simulates the internal mobility of DNA; construction of a transformation that allows you to move from the DNA model to its electronic analogue; search for conditions that provide an analogy of DNA equations and electronic analogue equations; calculation of the parameters of the equivalent electrical circuit. To describe inhomogeneous DNA, the model was chosen that is a system of discrete nonlinear differential equations simulating the angular deviations of nitrogenous bases, and Hamiltonian corresponding to these equations. The values of the coefficients in the model equations are completely determined by the dynamic parameters of the DNA molecule, including the moments of inertia of nitrous bases, the rigidity of the sugarphosphate chain, and the constants characterizing the interactions between complementary bases in pairs. The inhomogeneous Josephson line was used as a basis for constructing an electronic model, the equivalent circuit of which contains four types of cells: A, T, G, and Ccells. Each cell, in turn, consists of three elements: capacitance, inductance, and Josephson junction. It is important that the A, T, G and Ccells of the Josephson line are arranged in a specific order, which is similar to the order of the nitrogenous bases (A, T, G and C) in the DNA sequence. The transition from DNA to an electronic analog was carried out with the help of the Atransformation which made it possible to calculate the values of the capacitance, inductance, and Josephson junction in the Acells. The parameter values for the T, G, and Ccells of the equivalent electrical circuit were obtained from the conditions imposed on the coefficients of the model equations and providing an analogy between DNA and the electronic model.
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