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- Документ(In)homogeneous invariant compact convex sets of probability measures(2020) Natalia Mazurenko, Mykhailo ZarichnyiIt is proved that for any iterated function system of contractions on a complete metric space there exists an invariant compact convex sets of probability measures of compact support on this space. A similar result is proved for the inhomogeneous compact convex sets of probability measures of compact support.
- ДокументDynamics and exact solutions of the generalized Harry Dym equation(2020) Ruslan MatviichukThe Harry Dym equation is the third-order evolutionary partial differential equation. It describes a system in which dispersion and nonlinearity are coupled together. It is a completely integrable nonlinear evolution equation that may be solved by means of the inverse scattering transform. It has an infinite number of conservation laws and does not have the Painleve property. The Harry Dym equation has strong links to the Korteweg – de Vries equation and it also has many properties of soliton solutions. A connection was established between this equation and the hierarchies of the Kadomtsev – Petviashvili equation. The Harry Dym equation has applications in acoustics: with its help, finite-gap densities of the acoustic operator are constructed. The paper considers a generalization of the Harry Dym equation, for the study of which the methods of the theory of finite-dimensional dynamics are applied. The theory of finite-dimensional dynamics is a natural development of the theory of dynamical systems. Dynamics make it possible to find families that depends on a finite number of parameters among all solutions of evolutionary differential equations. In our case, this approach allows us to obtain some classes of exact solutions of the generalized equation, and also indicates a method for numerically constructing solutions.
- ДокументOn semiconvexity of open sets with smooth boundary in the plane(2020) Tetiana OsipchukThe present work considers properties of classes of generally convex sets in the plane known as 1-semiconvex and weakly 1-semiconvex. More specifically, it is proved that open, weakly 1-semiconvex but not 1-semiconvex set with smooth boundary in the plane consists of not less than four connected components.