TOTAL ENERGY OF HARMONIC OSCILLATOR WITH IMPULSE ACTION

The problem of finding the total energy of a harmonic oscillator with pulsed action at fixed moments of time is considered. Both for the case of the homogeneous equation of harmonic oscillations and for the case of the equation of harmonic oscillations in the presence of external perturbation, formulas for the total energy of the oscillatory system are obtained. The case of periodic impulse effects is analyzed. The conditions under which in this oscillatory system there are periodic modes are specified. It is shown that under the fulfillment of these conditions on the values of impulse action and external perturbation, the total energy of the vibrational system is also a periodic function of the time variable.

THE LAW OF COEXISTENCE OF HOMOCLINIC TRAJECTORIES FOR MAPS OF THE SPACE OF CONTINUOUS FUNCTIONS

The paper deals with problem of ordering coexistence of homoclinic trajectories for maps of segment into itself, which are generated by a certain continuous function of the segment.

VERIFICATION OF THE CONTROL ALGORITHM FOR RESERVOIRS WITH LIQUID BASED ON THE COMPENSATION OF THE FORCE RESPONSE IN DIFFERENT RANGES OF MANIFESTATION OF NONLINEARITIES

Problem about motion of a reservoir with liquid with a free surface is considered based on the compensation of a force response of the liquid on reservoir walls. Such an approach is selected since usual methods of control of mechanical system motion are mostly intended for linear systems of relatively small dimension. However, models of dynamics of the combined motion of reservoirs with liquid are described with relatively high-dimensional nonlinear systems ordinary differential equations. For obtaining the mathematical model of combined motion of a reservoir with liquid with a free surface we use the Hamilton–Ostrogradskiy variational principle, for which it is possible to determine analytically all internal forces of interaction of system component parts. Namely using this algorithm, we determine the main vector of forces of the liquid pressure on reservoir walls (force response of liquid). The algorithm of the motion control of the reservoir with liquid is based on the inclusion of the compensation of the liquid force response to controlling actions, this reduces the motion of the system reservoir–liquid, where the effect of forces from oscillating liquid on the reservoir motion is eliminated. This algorithm was tested for problems of impulse and vibration disturbance of the translational motion of the system in the horizontal plain. We consider the disturbance of the system motion by a force rectangular impulse applied to the reservoir wall, the duration of the impulse is lesser than a quarter of the period of a liquid free oscillations according to the first normal mode. Amplitudes of the impulse were selected with the purpose of analysis of the behavior of the controlled system in different ranges of manifestation of nonlinearities. We state the problem to verify the accuracy of this algorithm for three ranges of manifestation of nonlinear properties in the system, namely, for the linear range (amplitudes of waves on a free surface h do not exceed 0,1 of the radius of a free surface (  <0,1R); for the weakly nonlinear range (  <0,2R) and for the strongly nonlinear range with maximum amplitudes of waves about  =0,32R. Numerical modeling enables the determination of errors of developed algorithm, which does not exceed 0,5 %, although they insignificantly increase with the increase of amplitudes of oscillations on a free surface of liquid. At the same time perturbations on a free surface of liquid for the controlled motion are always greater than for the uncontrolled motion.

NILPOTENT MODULES OVER POLYNOMIAL RINGS

Let K be an algebra ically closed field of characteristic zero, K[X ] the polynomial ring in n variables. The vector space Tn = K[X] is a K[X ] -module with the action i = xi 'x  v v for vTn . Every finite dimensional submodule V of Tn is nilpotent, i.e. every f  K[X ] acts nilpotently (by multiplication) on V . We prove that every nilpotent K[X ] -module V of finite dimension over K with one-dimensional socle can be isomorphically embedded in the module Tn . The groups of automorphisms of the module Tn and its finite dimensional monomial submodules are found. Similar results are obtained for (non-nilpotent) finite dimensional K[X ] -modules with one dimensional socle.

ORDINALITY OF ISOMETRY GROUPS OF HAMMING SPACES OF PERIODIC SEQUENCES

We study Hamming spaces (known also as measure algebras). For all Steinitz numbers s , we find cardinalities of the groups of isometries of Hamming spaces of s -periodic sequences and the group of automorphisms of such space and we prove that are both cardinalities equal to Pic 1.

RANDOM GRAVITY WAVES IN TWO-LAYER GIDRODYNAMIC SYSTEM

The article is devoted to the study of the propagation of random gravitational waves in a three-dimensional hydrodynamic system half-space– half-space. An overview of studies on the analysis of the propagation of random waves in different systems is given. Mathematical statement of the problem contains second-order differential equations with respect to velocity potentials, kinematic and dynamic conditions on the contact surface. To study the problem, the field of deviations and the potentials of the wave velocities are presented in the form of expansions in Fourier-Stiltjes integrals. Stochastic amplitudes of the corresponding fields are expressed through the amplitude of the deviation field in the form of recurrent relations. Using the expansion in series in a small parameter for the stochastic field amplitude variations, the dynamic equation in integral form has been received. It should be noted that the use of a small parameter makes it possible to control the contribution of the nonlinearity of the corresponding terms. Subintegral functions of two- and three-wave interaction are obtained in symmetrized form. Based on the obtained equation, a linear dispersion relationship is derived. In the two-dimensional case, it degenerates into the dispersion relationship obtained by A. Naifehfor deterministic wave motions in a two-layer system. Using the equations for the amplitude of the deviation field and the ensemble averaging procedure, the equation for the spectrum of the first harmonics is obtained. The reliability of the obtained results is confirmed by a comparison with previous studies of the problem of propagation of random surface gravitational waves performed in the works of Masuda and others. The obtained results can be used in the study of the propagation of random internal waves in the oceans.

LEBESGUE STRUCTURE OF DISTRIBUTION OF GENERALIZED BERNOULLI CONVOLUTIONS OF SPECIAL TYPE

The paper deals with the problem of deepening the Jessen-Wintner theorem for generalized Bernoulli convolutions of a special kind. The main attention is paid to the case when the terms of a random series acquire three values: 0, 1, 2. In the case when the probability that the term of a random series becomes 2 is 0, the corresponding generalized Bernoulli convolutions coincide with classic Bernoulli convolutions, which were actively studied domestic scientists (Pratsovyty M., Turbin G., Torbin G., Honcharenko Ya., Baranovsky O., Savchenko I. and others) as well as foreign researchers (Erdos P., Peres Y., Schlag W, Solomyak B., Albeverio S. and others). The problem of deepening the Jessen-Wintner theorem concerning the necessary and sufficient conditions for the distribution of a probably convergent random series with discrete additions to each of the three pure types, is extremely difficult to formulate and is not completely solved even for classical Bernoulli convolutions. The results of the study are a deepening in relation to the analysis of the Lebesgue structure of random series formed by s-expansions of real numbers. In the case when the corresponding Bernoulli convolution is generated by the sequence 3-n, we have a random variable with independent triple digits, which was studied by scientists in different directions: Lebesgue structure (Chaterji S., Marsaglia G.), topological-metric structure of the distribution spectrum (Pratsovityi M., Turbin G.), fractal analysis of the distribution carrier (Pratsovyty M., Torbin G.), asymptotic properties of the characteristic function at infinity (Honcharenko Ya., Pratsovyty M., Torbin G.). The paper presents certain sufficient conditions for the absolute continuity and singularity of the distribution, with certain restrictions on the stochastic distribution matrix and the asymptotics of the values of the random terms of the series. In the case when the Lebesgue measure of the set of realizations of the generalized Bernoulli convolution is different from zero, it is possible together with Levy's theorem to formulate criteria for belonging of the Bernoulli convolution distribution to each of the three pure Lebesgue types, namely: purely discrete, purely continuous or purely singular.

CHARACTER OF MANIFESTATION OF NONLINEARITIES FOR VIBRATION DISTURBANCE OF MOTION OF ELLIPSOIDAL RESERVOIR WITH LIQUID WITH A FREE SURFACE

The problem with vibration disturbance of the reservoir of ellipsoidal shape, partially filled with a liquid, is under consideration. For the construction of the model, we use the before developed method, based on the use of non-Cartesian parametrization of the domain, occupied by a liquid. And the method of the auxiliary domain for satisfying boundary conditions on tank walls above the unperturbed free surface of a liquid, where the liquid can pass in its perturbed motion. The liquid is considered as ideal incompressible. The mathematical model of the system is constructed based on the variational formulation of the problem in the form of the Hamilton–Ostrogradskiy principle. The motion of a liquid free surface is given in the form of decomposition with respect to normal modes of oscillations. Amplitude parameters of oscillations of a liquid free surface together with parameters of the translational motion of the reservoir form a complete independent system of parameters, for which the resolving system of ordinary differential equations is constructed. The constructed model includes nonlinear properties of the system and corresponds to the model of the combined motion of the liquid with the reservoir. According to its structure, the model has considerable similarities with the case of the cylindrical reservoir. The practical implementation of the method is done for vibration disturbance of the system motion in the horizontal plane for the case of extended and compressed ellipsoidal reservoirs. The analysis of the character of manifestation of the dynamical behavior of the system in different ranges of frequencies of motion disturbance shows that mainly this system behaves as a system with the soft type of nonlinearities. The system output to the steady mode of oscillations is not observed. Modulation of oscillations of a liquid free surface is considerably manifested for most modes. Increased attention is paid to the study of regularities of variation of a period of the oscillation modulation. It was ascertained that due to compression of the spectrum of liquid oscillations with the increase of the wavenumber, the simultaneous considerable effect of several frequencies is manifested in the system reservoir–liquid, which leads to complex modulation envelope lines.

NUMERICAL SOLUTION OF THE BOUNDARY PROBLEM FOR PARABOLIC EQUATION WITH NON-SELF-ADJOINT OPERATOR AND RELATED BOUNDARY CONDITIONS

A boundary value problem for a second-order parabolic equation with a non-self-adjoint operator is considered. Such problems are mathematicalmodels for a number of problems, describing convective-diffusion processes of matter transfer, breakdown mechanisms of laser activity in plasma, etc. While studying the physics of breakdown, one should take into account the avalanche-like increase in the number of free electrons due to multiphoton ionization processes under the influence of optical pulses. This requires the inclusion of related boundary conditions in the problem formulation. An important circumstance that must be taken into account when developing a method for solving the problem is fulfillment of a certain conservation law for its solution. To solve the boundary value problem an approach based on the finite difference method is proposed. The approximation of the equation and boundary conditions is constructed so that the difference scheme is completely conservative. It approximates the original problem with the second order in the spatial variable and in time, and it has the second order of convergence. To effectively solve a system of linear algebraic equations at each time layer, the sweep method for complex systems in combination with the non-monotonic sweep method for systems with a tridiagonal matrix is used. Software based on computer mathematics MATLAB is developed to perform numerical calculations. It is obtained an approximate solution of an applied problem for different instants of time, as well as values of an absorption coefficient, the change in sign of which determines the transition of the plasma in a laser-active state.

A GENERALIZED SOLUTION OF THE DIRICHLET PROBLEM FOR A MODEL ANISOTROPIC WEIGHTED EQUATION

The paper deals with the Dirichlet problem for a model nonlinear degenerate anisotropic elliptic second-order equation. Anisotropy and degeneration (with respect to the independent variables) is characterized by the presence of different exponents q1 , q2 and weighted functions |x|^q1 та |x|^q2 in the left side of the equation. The main result of the paper is theorem on the existence of the generalized solution of the Dirichlet problem under consideration.