The nonlinear Boltzmann equation for the model of rough spheres is considered. It describes the evolution of a rarefied gas of rotating rigid molecules. Explicit approximate solutions of this equation are constructed in a form of the linear combinations of two non-stationary inhomogeneous distributions. Such the distributions correspond to the vortical flows which can rotate about their axes and move translationally in any direction. The asymptotic behavior of the uniform-integral discrepancy between the sides of the equation is studied. In some cases this discrepancy can be done arbitrary small, so the correspondent solutions give the approximate description of the process of interaction between two vortical flows. 2000 Mathematics Subject Classification 76P05. Top of the page.
The problem of admissible control synthesis for the triangular system with the geometric restrictions on control and its derivatives is considered. The controllability function method is the basis of the investigation. The corresponding controls are constructed. The time of motion is calculated and the trajectories are given. The results are illustrated by the example. 2000 Mathematics Subject Classification 93B50. Top of the page.
In this work the L.Ronkin's theorem about the function $\log |f(z)|$ is extended to the class of the logarithmic subharmonic(plurisubharmonic) functions. 2000 Mathematics Subject Classification: 42A75, 31A05. Top of the page.
In the paper it is considered plurisubharmonic functions in $D \times \bf C$ where $D$ is a domain in the space $\bf C^n, n \ge 1.$ The measure of these functions are concentrated on the set $D \times \{w: \Im w = 0\}.$ For such functions it is established the connection between the behavior of the function at infinity along the chosen variable and the function of the distribution of its measure in the terms of polynomial asymptotic representations. 2000 Mathematics Subject Classification: 32U05, 31C10. Top of the page.
Ruled and SO(2)-invariant minimal surfaces are studied in the threedimensional Heisenberg group with leftinvariant metric $ds^2=dx^2+dy^2+(dz-xdy)^2.$ We prove nonexistence of totally geodesic surfaces. 2000 Mathematics Subject Classification: 53C40,53C42. Top of the page.
The functional models for the commutative system of the linear limited operators $A_1,$ $ A_2$ have been built in the case when this system of operators does not consist of dissipative operators, that is $ A_1x_1 + A_2x_2 \; $ is not a dissipative operator for every $ x_1, x_2 \in R, $ and $\dim G = 2,$ where G equals next $G = {\rm span} (A_k - A_k^*)h,$ $k = 1,2;$ $h\in H.$ 2000 Mathematics Subject Classification: 47A45. Top of the page.
The 2D incompressible viscous flow in a simple connected finite domain is considered. The method based on the R-functions method and the successive approximations one for numerical solution are proposed. The numerical results of computations are present at the paper. 2000 Mathematics Subject Classification: 42A70. Top of the page.
In the paper the algebraic structures generated by the representation of an affine control system as a series of nonlinear power moments are studied. Different gradings of the algebra of nonlinear power moments correspond to different methods of summing the series. The method of reduction of infinite-dimensional gradings to finite-dimensional ones for the construction of the canonical form of nonlinear control systems is proposed. 2000 Mathematics Subject Classification: 93B10, 93B25. Top of the page.
The paper analyzes the problems of saving the continuity of solutions of a difference equation Ô(hx)-Ô(x)=F(h, x), where the functions are given on a multiplicative group with values in a separable Frechet space. Weakly complete range of the solution and its boundedness on some relative dense set is required. Then from the continuity of the right member follows the continuity of the solution. The local variant is also surveyed. From these theorems of saving the continuity as a corollaries are gained the theorems of almost periodicity, Levitan almost periodicity, almost automorphy of the solutions of the difference equation and the theorems of an infinite integral and curvilinear integral, which is not depending on the path of integrations. The theorem of continuity comprises: the Bohl-Bohr theorem, Kadec theorem, Doss theorem, Boles theorem, Lubarsky theorem, theorems of the authors about the curvilinear integral. Simultaneously with this it transfers them on a wider class of the Frechet spaces and reduces the requirement for total boundedness, requiring boundedness on some relative dense set. 2000 Mathematics Subject Classification: 43A60. Top of the page.
Some results of investigation of wave motion of a viscous incompressible liquid in a system, that consists of a long thick-walled viscoelastic tube connected in series with two elements with given admittances, are presented. The system represents the model of the arterial bed of an inner organ that is included in the general circulatory system. In the framework of linearized motion equations of the liquid and tube’s wall the decision of the problem as expansion in terms of separate harmonics as well as expressions for input admittance of the system are obtained taking into account wave reflection at the end of the tube. The numerical decision of the dispersion equation is received and the influence of mechanical and geometrical parameters of the model on wave propagation and attenuation is investigated. The analysis of the input admittance of the system over a wide range of variation of the parameters that correspond to the arterial system of a human is carried out. It is shown, that the input admittance reaches its extreme at frequencies which are determined by tube’s length. The amplitude of the admittance strongly depends on the admittance of the terminal element and weakly depends on the density and viscosity of the liquid, the material of the wall, internal and external diameters. The results may be assumed as a basis of novel methods of pulse diagnostics of the inner organs’ state. 2000 Mathematics Subject Classification: 76D33, 76Z05, 74F10, 92C35. Top of the page.
The existence of the asymptotics of the basic functions of a generalized Taylor series is proved, and the first term of the asymptotic expansions of these fuctions is obtained. 2000 Mathematics Subject Classification: 42A70. Top of the page.