Abstract:
On the basis of a three-dimensional non stationary model of a convective cloud with detailed description of dynamic, thermodynamic and microphysical processes, numerical experiments were conducted to study the formation of parameters of convective clouds under unstable stratification of the atmosphere. Numerical experiments have been carried out to study the formation of convective processes in the atmosphere. The thermo hydrodynamic parameters in the zone of a thunderstorm cloud are determined, and regions with a vortex motion of air are identified. The main flows feeding the convective cloud in the mature stage are determined. Due to the means of visualization, the areas of formation and growth of precipitation particles are identified. In a three-dimensional form, the interaction of dynamic and thermodynamic processes is analyzed. The interaction of fields is manifested in the form of deformation of fields of thermodynamic parameters under the influence of dynamic processes. Trajectories of air streams around a cloud and the trajectories of drops in a cloud are determined. The results of numerical experiments confirm that dynamic processes significantly influence the formation of fields of thermodynamic parameters in the cloud, which also determine the course of microphysical processes and the nature of the growth of precipitation particles.

Abstract:
The Cauchy problem for the Fokker-Plank-Kolmogorov equation with a nonlocal nonlinear drift term is reduced to a similar problem for the correspondent linear equation. The relation between symmetry operators of the linear and nonlinear Fokker-Plank-Kolmogorov equations is considered. Illustrative examples of the one-dimensional symmetry operators are presented.

Abstract:
The Cauchy problem for the Fokker-Plank-Kolmogorov equation with a nonlocal nonlinear drift term is reduced to a similar problem for the correspondent linear equation. The relation between symmetry operators of the linear and nonlinear Fokker-Plank-Kolmogorov equations is considered. Illustrative examples of the one-dimensional symmetry operators are presented.

Abstract:
The state of the physics of convective clouds and cloud seeding is discussed briefly. It is noted that at the present time there is a transition from the stage of investigation of “elementary” processes in the clouds to the stage of studying the formation of macro- and microstructural characteristics of clouds as a whole, taking into account their system properties. The main directions of the development of cloud physics at the upcoming stage of its development are discussed. The paper points out that one of these areas is the determination of the structure-forming factors for the clouds and the study of their influence on their formation and evolution. It is noted that one of such factors is the interaction of clouds with their surrounding atmosphere, and the main method of studying its role in the processes of cloud formation is mathematical modeling. A three-dimensional nonstationary model of convective clouds is presented with a detailed account of the processes of thermohydrodynamics and microphysics, which is used for research. The results of modeling the influence of the wind field structure in the atmosphere on the formation and evolution of clouds are presented. It is shown that the dynamic characteristics of the atmosphere have a significant effect on the formation of macro- and microstructural characteristics of convective clouds: the more complex the structure of the wind field in the atmosphere (i.e., the more intense the interaction of the atmosphere and the cloud), the less powerful the clouds are formed.

This paper uses a new statistical method for studying self-organization
in order to explain the origins of global catastrophes, namely the exceeding
of the critical
level of the planet’s organization. This method differs from
traditional methods, because it uses specific notions such as the “entrostat”
and “critical level of the organization of the open system”.

The reasons for
introducing the concept of the entrostat in statistical physics are examined.
The introduction of the entrostat has allowed researchers to show the
possibility of self-organization in open systems within the understanding of
entropy as a measure of disorder. The application of the laws written down for
the entrostat has also allowed us to formulate the “synergetic open system”
concept. A nonlinear model of the activity of a medium-sized company in the
market is presented. In the course of the development of this model, the
concept of the entrostat was used. This model includes the equation of a firm’s
market activity and condition of its stability. It is shown that this stability
depends on the income of the average buyers of the firm’s goods and furthermore
that the equation estimating the firm’s market activity includes the scenario
of a subharmonic cascade, which ends in chaos for the majority of market
participants, i.e., in an economic
crisis. The feature of this paper is that the decision containing the scenario
of the subharmonic cascade is found analytically (instead of numerically, as is
customary in the current scientific literature).

Abstract:
Following V. Drinfeld and G. Olshansky, we construct Manin triples $(\fg, \fa, \fa^*)$ such that $\fg$ is different from Drinfeld's doubles of $\fa$ for several series of Lie superalgebras $\fa$ which have no even invariant bilinear form (periplectic, Poisson and contact) and for a remarkable exception. Straightforward superization of suitable Etingof--Kazhdan's results guarantee then the uniqueness of $q$-quantization of our Lie bialgebras. Our examples give solutions to the quantum Yang-Baxter equation in the cases when the classical YB equation has no solutions. To find explicit solutions is a separate (open) problem. It is also an open problem to list (\`a la Belavin-Drinfeld) all solutions of the {\it classical} YB equation for the Poisson superalgebras $\fpo(0|2n)$ and the exceptional Lie superalgebra $\fk(1|6)$ which has a Killing-like supersymmetric bilinear form but no Cartan matrix.

Abstract:
A concept of semiclassically concentrated solutions is formulated for the multidimensional nonlinear Schr\"odinger equation (NLSE) with an external field. These solutions are considered as multidimensional solitary waves. The center of mass of such a solution is shown to move along with the bicharacteristics of the basic symbol of the corresponding linear Schr\"odinger equation. The leading term of the asymptotic WKB-solution is constructed for the multidimensional NLSE. Special cases are considered for the standard one-dimensional NLSE and for NLSE in cylindrical coordinates.

Abstract:
The complex WKB-Maslov method is used to consider an approach to the semiclassical integrability of the multidimensional Gross-Pitaevskii equation with an external field and nonlocal nonlinearity previously developed by the authors. Although the WKB-Maslov method is approximate in essence, it leads to exact solution of the Gross-Pitaevskii equation with an external and a nonlocal quadratic potential. For this equation, an exact solution of the Cauchy problem is constructed in the class of trajectory concentrated functions. A nonlinear evolution operator is found in explicit form and symmetry operators (mapping a solution of the equation into another solution) are obtained for the equation under consideration. General constructions are illustrated by examples.

Abstract:
The complex WKB-Maslov method is used to consider an approach to the semiclassical integrability of the multidimensional Gross-Pitaevskii equation with an external field and nonlocal nonlinearity previously developed by the authors. Although the WKB-Maslov method is approximate in essence, it leads to exact solution of the Gross-Pitaevskii equation with an external and a nonlocal quadratic potential. For this equation, an exact solution of the Cauchy problem is constructed in the class of trajectory concentrated functions. A nonlinear evolution operator is found in explicit form and symmetry operators (mapping a solution of the equation into another solution) are obtained for the equation under consideration. General constructions are illustrated by examples.