Upper bounds on the number of determining modes, nodes, and volume elements for a 3D magenetohydrodynamic-$\alpha$ model

In this paper we give upper bounds on the number of determining Fourier modes, determining nodes, and determining volume elements for a 3D MHD-$\alpha$ model. Here the bounds are estimated explicitly in terms of flow parameters, such as viscosity, magnetic diffusivity, smoothing length, forcing and domain size.     

Construction of a stabilizing control and solution to a problem about the center and the focus for differential systems with a polynomial part on the right side

The stationary differential systems with polynomial right sides are considered. Necessary and sufficient conditions are formulated when a given domain is a domain of asymptotic stability and the origin of coordinates is either the focus or the center. The problem of construction of a stabilizing control in a form of polynomial is studied.