No 3 (2016)

Venture capital now is a significant part of investments in innovative projects. In this article it is proposed dynamic model of a two-sector economy with venture investment. There are five economic agents: the population, the banks, resellers and producers, divided into two sectors the traditional and innovative. The article gives microdescription of firms in both sectors. The main sector companies at any given time are founded by moving into it one of the venture sector firms, where several firms are being created in each moment. Innovative firms parameters are set by the normal distribution. Venture investor takes loans of the banking system, and is fully in control of gains and losses of the venture sector. People invest money in the firms which join the traditional sector to gain their share in profits. The article describes the process of exiting the innovation sector by firms and their sale. Also it is described the process of liquidation of unprofitable firms in both sectors. The results of numerical experiments with a closed mathematical model are presented. The proposed model shows the exponential growth in which stands a distinctive kink in the time when the investments of the population reaching the level of bank loans.

The new algorithms and programs, implemented in Maple, for solving waveguide-type multichannel scattering and eigenvalue problems for systems of the second-order ODEs with N х N matrix piecewise constant coefficients on the axis are proposed. New algorithm and program for solving the boundary-value problems by method of matching the fundamental solutions (MMFS) of the system of ODEs at the points of discontinuity of potentials are elaborated. In each of the subintervals of an axis the general solution of the system of ODEs are sought in the form of linear combination of 2N fundamental solutions with unknown coefficients. Each fundamental solution explicitly dependent on spectral parameter and eigenvalues and eigenvectors of algebraic eigenvalue problems with N х N matrix of constant potentials. From the condition of continuity for the solutions and their derivatives at the discontinuity points of the potentials, the system of algebraic equations is followed. In the case of bound or metastable state problem the obtained system of algebraic equations contains nonlinear dependence of unknown spectral parameter. For solving such nonlinear problem symbolic-numerical algorithm is formulated. The benchmark calculations of bound, metastable and scattering states of BVPs for systems of the second-order ODEs obtained using program of the MMFS are compared with those obtained using program of the finite element method.

The analysis of the flow of turbulent natural convection boundary layer near a heated vertical plate was done. On the basis of comparison of criteria of similarity determined relative influence of viscous and convective forces in this region. The approximate equations describing the flow characteristics in the near wall region takes into account the relative influence of viscous and convective forces. Using the analogy between equations of forced turbulent boundary layer and obtained the equations for the near wall region was found corresponding profiles of vertical velocity and excess temperature. On the basis of the profiles of vertical velocity and excess temperature were built of the velocity field and temperature in the near wall region. In the fields of velocities obtained an expression describing the friction in turbulent wall region of a turbulent boundary layer free convection flow. Based on the analogy with a forced turbulent boundary layer and the flow in the near wall region of natural convection turbulent boundary layer near a vertical plate has been proposed to use the Blasius formula for finding the values of the turbulent shear stress on the wall. A review of the results was done.

The problem of a short ion beam passing through the plasma layer is considered in this paper. In such a process the plasma electrons capture by the beam takes place. The charge compensation is necessary during the charged beams transport with the purpose to prevent their dispersion under the influence of the own Coulomb field. It is important to investigate the methods of the beam charge compensation methods for the reason of their numerous applications. Specifically the active investigations of the intensive ion beams interactions with the thermonuclear targets in the controlled fusion problem are performed last years. In this paper the one-dimensional electrostatic approximation is used and the conditions of its applicability are presented. The electron movement in the ion beam field with the model density distribution is considered. It is shown by the numerical simulation using the particle-in-cell method that during the short ion beam passing through the plasma layer the capture of the part of plasma electrons by the beam field takes place. But unlike the hydrodynamical description used by other authors this process has the essentially kinetic nature moreover the collective electric field is compared with the beam field. The beams of accelerated electrons are formed under the influence of the total field leading to the nonlinear regime of the beam instability and strong heating of the plasma electrons. It is shown that the oscillating field caused by the plasma oscillations on the plasma boundaries affects essentially on the electron capture. The numerical simulation of the beam passing through the plasma layer on the time intervals compared with ion plasma period is carried out. The particle-in-cell method is applied in this case for the ion movement calculation. It was supposed that the electrons have the Boltzmann density distribution in the self-consisted field. The boundary problem for the Poisson equation which becomes nonlinear in such a statement was solved numerically by the shooting method. It was demonstrated the formation of the stationary structures of the ion phase space holes type for the electron temperature much larger the ion one.