List of Departments
Topmost Scientific Results

- Department of Astrophysics and Elementary Particles
- Department of High-Density Energy Physics
- Department of Theory of Quantum Processes in Nanosystems
- Department of Mathematical Methods in Theoretical Physics
- Department of Theory of Nonlinear Processes in Condensed Matter
- Department of Synergetics
- Department of Theory of Nuclei and Quantum Field Theory
- Department of Computer Maintenance

Department of Synergetics

- An approach to the geometric description of elementary particles is proposed. The nature of supersymmetry is explained and the mechanism of spontaneous symmetry breaking of a system with different degrees of freedom is proposed. This approach more fully reflects the main idea of geometrization of interaction, which is the basis of field theory.
*Acad. of the NAS of Ukraine B.I. Lev* - Based on the model of the respiratory mechanism of cyanide destruction by microorganisms, the influence of the cell wall on the destruction process in conditions with and without an initial short-term pulsed electric field treatment is presented. An initial short-term treatment with a pulsed electric field has been shown to increase the permeability of the cyanide through the cell wall and the rate of activation of the respiratory chains.
*Acad. of the NAS of Ukraine B.I. Lev, V.S. Yakovliev* - With the help of a mathematical model, the process of oxidative phosphorylation and ATP synthesis of the cell is investigated. The scenario of the occurrence of auto-periodic and chaotic regimes of the system depending on the amount of ATP dissipation was found. The invariant measure of the strange attractor is calculated and the histograms of its projection onto the phase space planes of the system are constructed. Recommendations have been made on how to biochemically get rid of the chaotic regime and restore the stability of self-organization of the cell biosystem.
*V.I. Grytsay* - A general mathematical model of the chain of metabolic processes of glycolysis-gluconeogenesis was built. With the help of spectral analysis of the kinetics of one of the variables of the system, the scenario of the occurrence of autoperiodic and chaotic regimes of the system is investigated. The invariant measure of the odd attractor is calculated and the histograms of its projection onto the phase space planes of the system are constructed. Recommendations were made - how to biochemically get rid of the chaotic regime and restore the stability of the attractor of the cell's biosystem.
*V.I. Grytsay* - An analysis of Betti curves was carried out depending on the redshift. It is shown that the Wasserstein distance between persistence diagrams for one-dimensional homologies can be used to estimate the difference in σ8 between two universe simulations. The dependence of the shape of Betti curves on redshift was studied.
*V.B. Tymchyshyn* - The theory of photodetection for the resonator mode using a Josephson photomultiplier is constructed. It has been demonstrated that such a device has a high degree of discrimination between the numbers of photons, which opens up ways of using it in the latest quantum technologies.
*A.A. Semenov, E.V. Stolyarov* - The kinetic theory of large-scale electric fluctuations in a collisional weakly ionized dusty plasma is formulated with due regard to the grain charging dynamics. The dependencies of the charging frequencies and effective collision frequencies for electrons and ions on dusty plasma parameters are studied in detail. To describe resonant properties of collective fluctuations, the analysis of the ion-acoustic wave spectrum in dusty plasmas is presented for a wide range of ion collisions for both non-isothermal and isothermal plasmas. The electron density correlation spectra are calculated for various values of the grain density, grain size, and ion collisionality.
*Acad. of the NAS of Ukraine A.G. Zagorodny* - A new mathematical method of constructing equivalent martingale measures to the original measure for the evolution of risk assets that has memory and takes into account the phenomenon of clustering has been developed. A theorem on the integral representation for them in terms of a set of point martingale measures is established. Found the formulas for the fair price of the superhedge for put and call options. The theorem on the invariance of a set of equivalent martingale measures with respect to a certain class of transformations is proved, which makes it possible to build models of financial markets.
*N.S. Gonchar* - The principles of sustainable economic development of economic systems at the macroeconomic level are formulated. The theorem on the necessary and sufficient conditions for the functioning of economic systems with strictly positive added value for the production model "expenditure - output" has been established. A formula was found for the probability of functioning of economic systems with strictly positive added value for the production model "costs - output".
*N.S. Gonchar* - Normal fluctuations of the internal degrees of freedom (Higgs and Goldstone modes) of three-band superconductors were obtained, taking into account both the internal proximity effect and the intergradient interaction. The spectrum of in-phase Higgs oscillations was obtained, and it was shown that, unlike oscillations in single-band superconductors, they can be stable.
*K.V. Grigorishin* - A new formula for the concentration of the smell was obtained, in which the increase in the selectivity of the olfactory receptor neuron due to fluctuations will be the highest.
*A.К. Vidybida, O.V. Shchur* - The influence of inhibitory autopsy on the activity of neurons is analytically described. Conditions are formulated for a set of non-adaptive models of impulse neurons with delayed inhibitory feedback, instead of considering a specific neuron model. The neuron is stimulated by a stochastic point process of restoration of excitatory impulses. The probability density function (PDF) p(t) of the output inter-spike intervals (ISIs) of such a neuron is determined exactly without any approximations. It is expressed in terms of the ISIs PDF for the recovery input and the ISIs PDF for the same neuron without any feedback. The obtained results are applied to a subset of neural models with a threshold of 2, when the time intervals between input pulses are distributed according to the Erlang-2 distribution. In this case, we have clearly found the model-independent initial part of the ISI PDF p(t), defined in some initial interval [0;T2] of ISI7 values.
*A.К. Vidybida, O.V. Shchur* - Computer simulations and an approach based on the backtracking of sample plasma particles were used to simulate the interaction of the Ganymede satellite with the plasma flow in the Jupiter magnetosphere and ion precipitation on the surface of Ganymede. Maps of H+, O+, S+++ ion deposition on the surface corresponding to different particle energies were obtained. The existence of a shielded equatorial region on Ganymede is shown. At low particle energies, Ganymede's magnetosphere can partially shield the equatorial surface from impacting ions and thus deflect them toward the poles. In general, the polar regions are more accessible to the influence of particles, since there are open lines of force of the magnetic field at the poles of the satellite.
*A.P. Fomina* - The model of the influence of plasma effects on the formation of vortices near the poles of the planets is considered. The plasma flow is controlled by the magnetic field and affects mainly the poles of the planets along the converging lines of force of the magnetic field. At the same time, the electron flow is compressed. There is a relative transverse polarization of ion and electron flows, with a crossed configuration of electric transverse and longitudinal/vertical magnetic fields. It was studied how relative drift and imbalances affect the formation of vortices. The mechanism of the formation of vortex structures was studied using the example of Jupiterian auroral processes, in particular, within the framework of the interaction between Io and Jupiter. The dynamics of vortices in the configuration of electric and magnetic fields is described. It is shown how the movement of vortices depends on the occurrence of electron density disturbances.
*A.P. Fomina*

- Some thermodynamic relations are obtained for self-gravitating systems. Based on the relations obtained equation for geometry of matter distribution is derived.
*academician B.I. Lev, academician A.G. Zagorodny* - A new formula is obtained for estimating selectivity of olfactory receptor neuron through selectivity of its receptor proteins with thermal fluctuations taken into account
*A.Vidybida* - An analog of Bell inequality for nonclassical correlations of radiation fields has been obtained. Such correlations cannot be simulated as a statistical mixture of classical fields. I has been shown that such correlations can be observed even in scenarios when local realism is not violated
*A. Semenov* - The bacteria metabolic process of open nonlinear dissipative system far from equilibrium point is modelled using classical methods of synergetics. The distribution of point density of trajectory intersection of phase space cells with maximum invariant measure and convergence in time of its average value was obtained. It is shown that the value of an invariant measure can be a characteristic of the transitional process of adaptation of cell metabolic process to change outside environment.
*V.Н. Grytsay* - A mathematical model of international trade is introduced and the problem is formulated. An overview of mathematical algorithms for solving this problem is provided. All positive solutions of the system of equations are described, the right-hand side of which is a vector from the interior of the polyhedral cone.
*N. S. Gonchar*

- Based on the self-consistent Hartree-Fock approximation, the nonstationary equation is obtained for the one-particle wave function describing the Bose-Einstein condensate in a rarefied gas of spin-zero bosons. A rarefied gas of bosons is exposed to the static external field, which ensures its finite ground state. The derived equation allows one to correctly determine the ground state energy in the stationary case.
*Acad. of the NAS of Ukraine A.G. Zagorodny* - It is shown that in the general case the conditions of cyclotron resonances should be formulated with regard to the influence of intense electromagnetic fields, if present, on the particle motion. Moreover, the greater is the wave field strength, the more significantly such effects are pronounced. These effects lead to the appearance of qualitatively new features in the particle dynamics. Such features include regularization of the particle dynamics, particularly a stepwise structure appears in the time dependencies of momenta and energy.
*Acad. of the NAS of Ukraine A.G. Zagorodny* - The theory of the behavior of colloidal particles in the deformedelastic field of the liquid crystal is constructed. The general expression of the intrinsic energy of a particle in a deformed liquid crystal is obtained. The force acting on a single colloidal particle by the elastic medium in the deformation of the director's field is determined.
*Acad. of the NAS of Ukraine B.I. Lev* - A fundamental link between nonclassicality of quantum states in quantum optics and violations of Einstein-Podolsky-Rosen-Bell principles of physical reality is established. This leads to an analog of Bell inequalities. Their violation is a sufficient condition of nonclassicality. It is shown that this technique is effective in several practical scenarios, where other methods are not applicable. For example, it can be used to demonstrate nonclassicality of photocounting statistics for squeezed vacuum states. Measurements can be conducted with realistic detectors characterized by imperfect resolution between numbers of photons.
*A.A. Semenov* - For a neuron with delayed inhibitory feedback, stimulated by a renewal stochastic process, convergence of the output interspike intervals statistics to a stationary mode, as well as the uniqueness of the final stationary mode are studied.
*O.V. Shchur, A.K. Vidybida* - A new, mathematically transparent proof of the possibility of using adsorption-desorption noise to increase the selectivity of the electronic nose has been found.
*A.K. Vidybida* - The class of evolution of risky assets, for which a spot martingale measure is the only possible one, is described on a discrete probability space. In terms of the evolution of risk assets the necessary and sufficient conditions are given, under which the spot martingale measure is the only martingale measure for a given evolution of risk asset. A self-financing trading strategy is built for each contingent claim, according to which this contingent claim can be completely hedged. The formula for the fair price of a contingent claim is established. The latter is crucial in building non-arbitrage models of complete financial markets.
*N.S. Gonchar*

- It is presented a new approach based on the application of a non-equilibrium statistical operator that allows the inhomogeneous distributions of the particles and the temperature to be taken into account. The method uses the saddle-point procedure to find dominant contributions to the partition function of the system and enables all its thermodynamic parameters to be determined. Probable peculiarities in the behavior of the systems with interaction - such as gravitational systems, systems with Coulombic repulsion, and so forth - under various thermodynamic conditions are predicted. A new approach is proposed to describe non-equilibrium systems in the energy space, which is an extension of the Gibbs approach to macroscopic systems under non-equilibrium conditions.
*Acad. of the NAS of Ukraine A.G. Zagorodny, Cor.-member of the NAS of Ukraine B.I. Lev* - A new method (arithmetic entropy) for analyzing big sequential data sets, like trajectories of dynamical systems, similar to the permutation entropy one, is proposed. The characteristic features of this method are as follows: it preserves information about equal values, if any, in the embedding vectors; it is exempt from combinatorics; and it delivers the same entropy value as does the permutation method, provided the embedding vectors do have no equal components. In the latter case, this method is computationally more efficient and may be used instead of the permutation one. If the embedding vectors have equal components, this method could be more precise in discriminating similar data sets.
*A.K. Vidybida* - For a class of neuronal models, the probability density function of interspike intervals for a neuron with delayed feedback inhibition stimulated with stochastic renewal point process has been obtained. It has been applied to a case of a neuron with threshold 2 stimulated with Erlang-2 stream of random impulses. For the leaky integrate-and-fire neuron stimulated with the stream of input impulses that forms the stochastic point Poisson process, the moment-generating function of the probability distribution is calculated explicitly. The latter one, according to Curtiss theorem, determines completely the distribution itself. In particular, explicit expressions for the moments of all orders are derived from the moment-generating function.
*A.K. Vidybida, O.V. Shchur* - Based on BCS model with the external pair potential that was formulated in my earlier work, the analogous model with electron-phonon coupling and Coulomb coupling is proposed. The generalized Eliashberg equations in the regime of renormalization of the order parameter are obtained. High temperature asymptotics and effect of Coulomb pseudopotential on them are investigated: as in the BCS model the order parameter asymptotically tends to zero as temperature rises, but the accounting of the Coulomb pseudopotential leads to existence of critical temperature. The effective Ginzburg-Landau theory is formulated for such model.
*K.V. Grigorishin* - A phase-parametric diagram of self-oscillating modes depending on the dissipation of cholesterol of the blood is constructed. The spectral analysis reveales a bifurcation scenario for doubling the period of self-oscillations, until eventually, due to discontinuity in the formation of the mixing funnel, chaotic modes of strange attractors appear.
*V.I. Grytsay* - We propose to use the statistical approach to the description of the steady states of the quasi stationary systems with the elements of the quantum field theory methods as a basis to explain the appearance of the cosmological scalar field. Particularly, we apply two fundamental principles, i .e., the H-theorem and least-energy principle to show principal possibility of the scalar field origination.Thus, we propose the possible physical justification of the spontaneous cosmological scalar field generation.
*Acad. of the NAS of Ukraine A.G. Zagorodny, Cor. Member of the NAS of Ukraine B.I. Lev* - A two-component fluid model for describing anomalous heat and particle transport in a fusion plasma is proposed. This model is introduced on the basis of the kinetic theory with the accuracy up to the second order in terms of the ratio of the field energy to the thermal particle energy and is equivalent to the kinetic description within the accuracy of the approximation used. The advantage of the model is the ability to take into account external and internal transport barriers without the use of additional approximations. The kinetic description on the basis of which the fluid model is proposed includes a complete transport matrix for both particle flows and heat flux. The proposed model gives the correct scaling of the plasma confinement time from the heating power.
*Acad. of the NAS of Ukraine A.G. Zagorodny* - The transport of passive scalars in a two-dimensional random velocity field is studied using numerical simulation and analytical method. Solutions of statistical equations recover the simulation results in the transition from diffusion to convection without involving fitting parameters.
*V.I. Zasenko, Acad. of NAS of Ukraine A.G. Zagorodny, O.V. Cherniak* - A class of evolutions of risky assets having memory on the basis of discrete Brownian motion of two types is constructed: 1) the price of a risky asset for the period under review cannot fall to zero though strictly positive; 2) the price of a risky asset for the period under review may fall to zero. It is proved that the considered class of evolution describes an incomplete non-arbitrage market. For such evolutions, a family of martingale measures equivalent to the initial probability measure is completly described. On the basis of discrete geometric Brownian motion a parametric random process that describes the evolution of assets, whose price cannot fall to zero as close as possible for the considered evolution time, is constructed. The theorem on the fair price of a contract with a standard type of call and put options is proved. A characteristic feature of this formula is that the price of the super-hedge is less than the price of the underlying asset. The interval of non-arbitrage prices is calculated.
*N.S. Gonchar*

- It is proposed the new approach to the description of liquid crystal under the assumption that colloidal particle can be treated as a source of possible deformation of the ground state of the director field distribution.The ground state imposes restrictions on possible deformations and determines the peculiarity of the interaction between colloidal particles. On the basis of this approach, both the Coulomb-like interaction between dipole particles in a cholesteric liquid crystal and a crucial change in the character of the interaction in a smectic liquid crystal had been predicted.
*Corr. Member of the NAS of Ukraine B.I. Lev* - The exact solution of the classical and quantum cosmological model with exponential dependence on fundamental scalar field potential in homogeneous and in-homogeneous cases has been obtained. A new solution of the Universe evolution model was found at the inhomogeneous change of scalar field.
*Corr. Member of the NAS of Ukraine B.I. Lev, A.P. Rebesh* - The simple model for explanation experimental result about rotation of elongate particle at external electric field has been proposed. The model based on collective behavior of assembly of the particles with induced dipole moment.
*Corr. Member of the NAS of Ukraine B.I. Lev* - The theory of collective diffusion effects in a system of colloidal particles in a liquid crystal is proposed. The specifics of diffusion which can be observed experimentally are described. The dependence of the diffusion coefficient on the temperature and particle density is found. It is shown that collective diffusion in a system of colloidal particles in a liquid crystal arises from the elastic distortion of the director field generating the interparticle interaction.
*Acad. of the NAS of Ukraine A.G.Zagorodny, Corr. Member of the NAS of Ukraine B.I. Lev* - On the basis of non-equilibrium statistical operator with a view to describe a behavior of self-gravitating systems it has been proposed the new approach, which takes into account a possible spatially nonuniform distribution of particles, chemical potential, and temperature. Statistical induced dynamic and behavioral peculiarities of self-gravitating systems under different conditions have been theoretically predicted.
*Corr. Member of the NAS of Ukraine B.I. Lev* - Explicit expressions are found for the stationary probability distribution of a magnetic field, generated by moving charges of the plasma environment, and stationary probability distribution of force, acting on a charged particle in this environment with regard for its interaction with electric and magnetic fields in plasma.
*Acad. of the NAS of Ukraine A.G. Zagorodny, Corr. Member of the NAS of Ukraine B.I. Lev, V.B. Tymchyshyn* - The kinetic theory of electric fluctuations in a collisional weakly ionized dusty plasma is formulated with due regard to the grain charging dynamics. The correlation functions of electron and ion density are obtained by considering their collisions with neutrals described within the Bhatnagar-Gross-Krook model. The electron density correlation spectra in isothermal and nonisothermal plasma are calculated for various values of grain density, grain size, and ion collisionality.
*Acad. of the NAS of Ukraine A.G. Zagorodny* - The theory of tokamak transport due to drift waves is revised. The development of the theory has been analized, starting with simple theories from the 1950s till today advanced theories using improved fluid models taking full account of kinetic effects in the frequency regime of drift waves, also including the effects of zonal flows, and fully nonlinear kinetic theory itself. An exact fluid closure makes it possible to use fluid theory completely without expansion, thus maintaining the fluid resonances due to magnetic drifts, that gives us a new normalization of drift wave equations and enables to recover nonlinear (Dimits) upshifts, spinup of poloidal rotation in internal transport barriers and the L-H transition.
*Acad. of the NAS of Ukraine A.G. Zagorodny* - On the basis of the developed analytical method, as well as by numerical simulation, the dependences of the transport coefficient of charged particles across a magnetic field on a frequency of a random electric field were found. It is shown that the results of the calculations are consistent with the data of direct numerical simulation over a wide frequency range. The results obtained can be useful for optimizing the performance of plasma devices in which processes of anomalous diffusion are significant.
*V.I. Zasenko, O.M. Cherniak* - A new mechanism for the selectivity enhancement of secondary neurons of the olfactory system is proposed, which operates at low odor concentrations when previously known mechanisms are ineffective.
*A.K. Vidybida* - For a class of neuronal models the spiking statistics of inhibitory neuron with delayed feedback stimulated with stochastic renewal point process is described. The explicit formulas without applying diffusion approximation have been found.
*A.K. Vidybida, O.V. Shchur* - A mathematical model of the open dissipative system with localized metabolic process involving anaerobic bacteria is presented. The general map of its metabolic paths is constructed. The synergetic method to study the self-organization of a dynamical chaos in metabolic processes in a cell and the whole organism is developed.
*V. Grytsay* - A model is worked out of the biochemical process of cellulose hydrolysis associated with the growth of cells and biosynthesis of enzymes. The model proposed describes an active medium that is an open system with the input and dissipation of both substance and energy in terms of a dynamical system of 6 nonlinear differential equations. The results obtained contribute to the general theory of nonequilibrium systems and refine the universality of the self-organization laws in nature.
*A.S. Zhokhin* - Strategies optimizing continuous-variable teleportation of a quantum-light mode through the turbulent atmosphere have been proposed.
*A.A. Semenov* - The principle of equilibrium of the open economy with its environment is formulated: the national currency exchange rate relative to the USA dollar is an indicator of the equilibrium the economy to its environment. Using this principle the canonical money circulation equation in the economy is established. On the basis of this equation the principle of the sustainable economy development is formulated. Outside of such a state the economy can be in the state of recession or in the state being close to it. On this basis the classification of the equilibrium states of the economy are given.
*M.S. Gonchar, O.P. Dovzhyk*

- In terms of the nonequilibrium statistical operator, the fundamental scalar field is shown to provide a natural representation of the repulsive interaction that produces scattering in the system and thus motivates law of entropy increasing.
*Corr. Member of the NAS of Ukraine B.I. Lev* - It is proposed a model of the first-order phase transition in the liquid crystals on the graphene surface with impurities which can change the orientation conditions. Experiments which confirm the theoretical predictions are performed out.
*Corr. Member of the NAS of Ukraine B.I. Lev* - Noncongruent first-order liquid-gas phase transition is studied within the quantum van der Waals model with two chemical potentials which regulate two conserved charges.
*M.I. Gorenstein, R.V. Poberezhnyuk* - Taking an excitatory neuron with delayed feedback as an example it is proven that statistics of activity of a neuron with such a feedback cannot be presented as Poisson process.
*O.V. Shchur, A.K. Vidybida* - Calculation of electromagnetic field energy in a medium with temporal and spatial dispersion outside thetransparency domain is discussed. It is shown that charged particle contribution to the energy of electromagnetic perturbations in the general case can be described in terms of a bilinear combination of the dielectric polarizability of the medium. The explicit form of such contribution is found. The relations obtained are used to generalize the Planck law and Kirchhoff law to the case of an absorptive medium with spatial dispersion.
*Acad. of the NAS of Ukraine A.G. Zagorodny* - On the basis of microscopic approach consistent kinetic theory is formulated for low temperature dilute gases of hydrogen-like atoms in an external electromagnetic field. Within the proposed approach the set of kinetic equations for Wigner distribution functions is obtained for free fermions of both sorts and their bound states — hydrogen-like atoms with regard to the influence of external and self-consistent (mean) fields.
*Acad. Of the NAS of Ukraine A.G. Zagorodny* - The dispersion relation for ion-acoustic waves in collisional weakly ionized dusty (complex) plasma is formulated with due regard to the grain charge fluctuations. The dependencies of the charging frequencies and effective collision frequencies on dusty plasma parameters are studied in detail. The analysis of the ionacoustic wave spectrum is presented for the wide range of the ion collisionality for both nonisothermal and isothermal plasmas.
*Acad. Of the NAS of Ukraine A.G.Zagorodny* - On the basis of the developed analytical method, the dependence of the magnetized plasma particle transverse diffusion coefficient on the correlation time of random electric fields is obtained. The results of calculations are in agreement with the data of numerical simulation in a wide range of correlation times.
*V.I. Zasenko, O.M. Cherniak* - It is proved that a risky asset, which evolves according to the law of discrete geometric Brownian motion, in contrast to the continuous case generates an incomplete financial market. New convenient research formula for a fair price of super-hedge for a broad class of payment functions is found.
*N.S. Gonchar* - The internal factors that form the exchange rate "hryvnia/dollar" are established, and proposals are made regarding the macroeconomic policy of the state to stabilize inflation.
*N.S. Gonchar, V.H. Kozyrski, A.S. Zhokhin, O.P. Dovzhik* - A mathematical model of the kinetics of a polyenzyme biosystem with nonstationary kinetics of a strange attractor was constructed.
*A.S. Zhokhin* - It has been demonstrated that the electron-vibron interaction and the local pairing of electrons on the 3-fold degenerate molecular orbital causes superconductivity of alkali-doped fullerides which would have to be antiferromagnetic Mott insulators.
*K.V. Grigorishin* - The analytical description of the domain structure dynamics with complex morphology and phase diagrams of ferroelectric nanoparticles has been developed in the framework of Landau-Ginzburg-Devonshire phenomenological theory.
*A.N. Morozovska*

- We report the first direct observation of Coulomb-like elastic interactions between colloidal particles in a nematic liquid crystal predicted theoretically. The interaction of two particles located at the boundary of twist and parallel aligned regions is observed. We demonstrate that such particles produce deformation elastic charges responsible for Coulomb-like interactions.
*Corr. Member NASU B.I. Lev* - The mechanism of the colloidal particle motion is proposed for liquid crystals with deformed director field. It is shown that the translation motion of colloidal particles depends on the specific deformation of the elastic field.
*Corr. Member NASU B.I. Lev* - A new approach which employs a non-equilibrium statistical operator that takes into account inhomogeneous distribution of particles and temperature is proposed. The method involves a saddle-point procedure to find the dominant contributions to the partition function. The equation of state for self-gravitating systems has been determined.
*Corr. Member. NASU B.I. Lev* - A formalism to describe an equilibrium based on consideration of a macroscopic but finite volume has been proposed. The presence of a Bose-Einstein condensate is taken into account through an off-diagonal long-range order in the transition to the thermodynamic limit. On this basis, the degenerate Bose-gas with delta-shaped interaction potentials has been described in term of self-consistent Hartree-Fock approximation. The explicit expressions for the energy spectra of single particle and collective excitation have been obtained.
*Acad. NASU A.G. Zagorodny* - The exact solution is found for the model of nonlinear oscillator with the coordinate-dependent mass, or velocity-dependent elastic modules. The Poincare cross-sections are studied and the conditions of the stochastic behavior of the oscillator motion are found experimenatally.
*Acad. NASU A.G. Zagorodny, Corr. Member NASU B.I. Lev, V.B. Tymchyshyn* - For wide class of neuronal models it is proven mathematically rigorous that activity of fast CL-type inhibitory neuron with delayed feedback stimulated with a renewal stochastic stream is non-Markovian one.
*A.K. Vidybida* - A physical mechanism is proposed for information reduction in a neuronal network through merging of different trajectories into a single one with subsequent entrainment onto a periodic regime. It is established that neuronal firing is a necessary prerequisite for merging. For a wide class of neuronal models for spiking inhibitory neuron with fast CL-type inhibitory delayed feedback a possibility to present the output statistics through the statistics for of that same neuron without feedback and the delay line characteristics. Exact mathematical expressions for such a representation are derived.
*A.K. Vidybida, O.V. Shchur* - The metabolic process in a cell is modeled using the Fourier transformation. The histograms of the invariant measures of chaotic attractors are constructed. Necessary number of harmonics, which represent uniquely the most complicated mode of a strange attractor, has been found. The dependence of the attractor type on the distributions and the harmonics amplitudes in the Fourier spectrum has been studied. The harmonics forming the laminar and turbulent parts of the attractor trajectory are identified. The projection histograms of the invariant measures of the main types of the system strange attractors have been constructed, and their dependence on the phase portrait has been determined.
*V.I. Grytsay* - Free-space optical communication links are promising channels for establishing secure quantum communication. We study the transmission of nonclassical light through a turbulent atmospheric link under diverse weather conditions, including rain or haze. Based on previous results, we study theoretically the transmission of quadrature squeezing and Gaussian entanglement under these weather conditions.
*D.Yu. Vasylyev* - We proposed a method to calculate the type of a lattice formed by grains in dusty plasma and to estimate its potential energy. Basically, this task is complicated by the interparticle potential that appertains to "catastrophic potentials". This kind of potentials needs special approaches to avoid divergences during potential energy calculations. We developed all the necessary modifications to appropriate methods. It has been shown that the obtained potential energy expression could be used to determine lattice parameters and these parameters comply to known experimental data.
*Acad. NASU A.G. Zagorodny, Corr. Member. NASU B.I. Lev, V.B. Tymchyshyn* - The diffusion of charged particles in a static random electric field across the magnetic field is considered. A new closure of the statistical equation describing particle diffusion in a frozen field is proposed, taking into account the effects of particle trapping and finite Larmor radius without the assumption of its smallness. A direct numerical simulation was performed and it was shown that the statistical characteristics found as solutions of the analytical model are in agreement with the simulation results in a wide range of Larmor radii.
*Acad. NASU A.G. Zagorodny, V.I. Zasenko, O.M. Cherniak* - It has been presented a general concept of bank work description, based on investment strategy both in risk and risk-free assets, dynamics of deposit attraction and incomes from operational activity, dynamics of liability payments,. Using these factors, a stochastic model of bank capital evolution is constructed. The problem of assessment of bankruptcy probability, depending on the choice of strategy investment, the deposit attraction on bank accounts, the restrictions for the bank liability payments is solved. A quantitative characteristic of management quality is introduced. Under the condition that the quantitative characteristic is satisfactory the conditions for the factors introduced above are found under that a bank can function infinitely long time with the sufficiently small bankruptcy probability.
*M.C. Gonchar, W.H. Kozyrski, A.S. Zhokhin, L.S. Terentieva*

- It has been shown that interband mixing of gradients of two order parameters (drag effect) in an isotropic bulk two-band superconductor plays important role - such a quantity of the intergradients coupling exists that the two-band superconductor is characterized with a single coherence length and a single Ginzburg-Landau (GL) parameter. Other quantities or neglecting of the drag effect lead to existence of two coherence lengths and dynamical instability due to violation of the phase relations between the order parameters. Thus so-called type-1.5 superconductors are impossible. An approximate method for solving of set of GL equations for a multi-band superconductor has been developed: using the result about the drag effect it has been shown that the free-energy functional for a multi-band superconductor can be reduced to the GL functional for an effective single-band superconductor.
*K.V. Grigorishin* - A general concept of the long-range elastic interactions in continuous medium is proposed. The interaction is determined as a consequence of symmetry breaking of the elastic field distribution produced by the topological defect as isolated inclusions. It is proposed to treat topological defects as the source of elastic field that can be described in terms of this field. The source is considered as a nonlinear object which determines the effective charge of the field at large distances in the linear theory. The models of the nonlinear source are proposed.
*A.G. Zagorodny, B.I. Lev* - We have introduced the elliptic-beam model for the transmittance probability distribution (PDT) of the atmospheric quantum channel. This model adequately accounts the loss and fading of the signal due to the atmospheric distortions of the beam centroid position and beam deformation. It appears that our model applies to weak, weak-to-moderate, and strong turbulence and hence to the most important regimes in atmospheric communication scenarios (PRL,117, 090501б (2016)). As an example we have studied the influence of atmospheric turbulence on quantum nonlocality for the scenario of counterpropagation of photonic pair as well as the influence so-called double-click events, which are caused by dark counts, stray light, and multiphoton entangled pairs.
*D.Yu. Vasylyev* - Method of preparation of multimode sub-Poisson light was developed using control photodetection of multimode twin-beam squeezed vacuum states and imposing a condition on the main photodetector. The method was shown useful upon imperfect detection efficiency. The numerical modeling was carried out followed by the experimental verification of the method for bright nonclassical light in collaboration with Max-Planck Institute for the Science of Light in Erlangen.
*V.C. Usenko* - The size of air bubbles in nematic liquid crystals can be continuously decreased through the absorption of air molecules into the host liquid crystal. A bubble and its accompanying hyperbolic hedgehog point defect undergo a continuous asymmetric motion, while the bubble decreases in size. In this study, a mechanism is proposed to theoretically explain both the motion of the air bubble and the point defect observed experimentally. Anisotropic evaporation of air molecules may occur because of the symmetry breaking of the director configuration near the point defect. The motion of the center of the air bubble to the hyperbolic hedgehog point defect is induced by the anisotropic force due to evaporation of air molecules and Stokes drag force.
*B.I. Lev* - For leaky integrate-and-fire neuron with threshold 2 stimulated with Poisson stream, intensity of the output stochastic process is calculated without using the diffusion approximation as function of LIF's physical parameters and intensity of the input stochastic process.
*О.К. Vidybida* - Within a mathematical model, the process of interaction of the metabolic processes such as glycolysis and gluconeogenesis is studied. As a result of the running of two opposite processes in a cell, the conditions for their interaction and the self-organization in a single dissipative system are created. The reasons for the appearance of autocatalysis in the given system and autocatalytic oscillations are studied. With the help of a phase-parametric diagram, the scenario of their appearance is analyzed. The bifurcations of the doubling of a period and the transition to chaotic oscillations according to the Feigenbaum scenario and the intermittence are determined. The obtained strange attractors are created as a result of the formation of a mixing funnel. Their complete spectra of Lyapunov indices, KS-entropies, ―horizons of predictability,‖ and the Lyapunov dimensions of strange attractors are calculated. The conclusions about the reasons for variations of the cyclicity in the given metabolic process, its stability, and the physiological state of a cell are made.
*V.I. Grytsay* - We uncovered physical underpinnings of formation of colloidal elastic hexadecapoles and described the ensuing bonding inaccessible to elastic dipoles, quadrupoles and other nematic colloids studied previously.
*S.B. Tovkach* - Peculiarities of migration and capture of a quantum particle in the chain with traps are described. Nontrivial dependencies of the quantum yield of capture W(χ) on the trap intensity χ (in particular, the effect of 'imprisonment' of the particle in the chain with χ growing) and initial conditions of migration are revealed in the cases of infinite, semi-infinite and finite chains.
*L.N. Christophorov, A.G. Zagorodny* - The analytical approach for description of magnetized particle transport in random frozen isotropic electrostatic field is developed using the concept of subensembles. It is shown that more detailed description in terms of subensembles gives better quantitative agreement with results of direct numerical simulation.
*V.I. Zasenko, O.М. Cherniak, A.G. Zagorodny* - The electrostatic potential distribution around a solitary charged absorbing macroparticle embedded into weakly ionized strongly collisional plasma is studied in details on the bases of numerical solutions of plasma dynamic nonlinear equations within the drift-diffusion approximation. Calculation was done under assumption that a macroparticle absorbs electrons and ions from plasma environment; the dependence of accumulated charge on dust particle size as well as plasma parameters is found. The comparison of calculated potential and plasma particles distributions with analytical expressions is performed. It is shown that consistency of analytic approximations (the superposition of Debye and Coulomb potentials) with the results of calculations can be significantly improved by use of a screening length scaling which depends on a dusty particle size and is associated with a significant deviation of a charge density near the particle surface from the unperturbed value. The universal dependence of the scaling parameter for screening length and effective charges on macroparticle size is obtained.
*A.G. Zagorodny, A.I. Momot* - Mathematical model of rate of exchange and targeting inflation is constructed. Stochastic difference equation for the rate of exchange hryvnia/dollar is established. An equation of money circulation that relate money demand, nominal Gross National Product, rate of National Bank, price level is constructed. Money circulation velocity during 2012-2014 ages is found and it is proved that it grows. This fact certifies inflation growing in this period and falling of living standard.
*N.S. Gonchar, A.S. Zhokhin, W.H. Kozyrski, L.S. Terentieva* - A notion of local regular supermartingale relative to a convex set of equivalent measures is introduced. The theorem giving the necessary and sufficient conditions of validity of optional decomposition for local regular supermartingales is proved that is generalization of the well known Doob decomposition theorem for supermartingale. Complete description of local regular supermartingales relative to a convex set of equivalent measures is given. An important notion of complete convex set of equivalent measures is introduced and theorem stating that for all non negative supermartingale relative to this set of measures optional decomposition is valid. This result is important for Mathematical Finance. Fair price for contingent claim relative to a convex set of equivalent measures is introduced that is generalization of the notion in case as the set of equivalent measures is a set of equivalent martingale measures for a certain evolution of risk and non risk assets. A formula for fair price of standard contract with European option for arbitrary non arbitrage evolution as risk and non risk assets in incomplete market is found.
*N.S. Gonchar*