Continuous abstract wavelet transform on homogeneous spaces

The support of a wavelet transform associated with a square integrable irreducible representation of a homogeneous space is shown to have infinite measure. Assumptions are illustrated and supported by examples. The pointwise homogeneous approximation property for a wavelet transform has been investigated. An analogue of Heisenberg type inequality is also obtained for a wavelet transform on a wavelet group.

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.

Review of the works of V. N. Shchennikova on the study of the convergence of nonlinear almost periodic systems by the comparison method

The article provides an overview of the studies of V. N. Shchennikov on the problems of almost periodic convergence of nonlinear differential equations' systems. The problem of convergence established by linear or homogeneous approximation is considered. The conditions for convergence of complex systems are given, that are obtained by constructing Lyapunov vector functions and using the comparison method. It should be noted that in the course of the proof constructive estimates are made for the values of small parameters and interconnection functions. The dimensions of the region in which the limiting almost periodic mode is located are also specified. As an application, the problem of convergence in an electric circuit modeled by a second-order nonlinear differential equation with a small parameter is considered. In conclusion, possible applications and unsolved problems for new directions of research, on which V. N. Shchennikov worked in recent years, are discussed.

Mathematical model of liver regeneration processes: homogeneous approximation

This paper deals with the rules and the mechanisms regulation of liver regeneration. The generalized mathematical model was developed. This model has a explicit dependence on the control parameters. To solve this problem there were accepted such assumptions: homogeneous approximation; small toxic factors.

CALCULATION OF SWIRLED NONISOTHERMIC LAYER FLOW OF TWO-PHASE NON-NEWTONIAN MEDIUM ON A CONICAL SURFACE

We consider the non-isothermic layer flow of two-phase non-Newtonian medium on the inner surface of the conical tube. The flow regime is laminar , axisymmetric and steady. The rheological state of the medium is described by the generalized law Ostwald de Ville. We also took into account the dependence of the temperature of medium consistency. The conservation equations of mass, momentum and energy mechanics of heterogeneous medium is used in quasi-homogeneous approximation. The recorded in biconical coordinate system equations are solved by method of equal costs surfaces. The provisions of equal costs surfaces are determined from the condition of the flow of the medium constancy between them. Conservation equations, written on the flow lines, are simplified and take the form of ordinary differential equations on the longitudinal coordinate. So that to calculate the partial derivatives on the transverse coordinate, which are present in the right part of the differential equations, the grid solutions are presented in the form of series expansion. The system of constructed ordinary differential equations is solved numerically.