We consider a billiard system inside a ring formed by two ellipses with common foci F_1 and F_2 affected by Coulomb forces, condensed in F_1 and F_2 with charges γ_1 and γ_2 , respectively. The reflection from the boundary of the domain is supposed to be essentially elastic. This billiard is Liouville integrable in terms of piecewise smoothness. An explicit formula of the additional first integral was found. We study the topology of the Liouville foliation of this system in several cases: γ_1<0,γ_2=0 (Kepler case); γ_1>0,γ_2=0 ; γ_1=γ_2 . For each of these cases, Fomenko and Fomenko–Zieschang invariants are calculated.
Luminosity points of sets are studied; these are the points for which the Kolmogorov (solar) condition for a best approximant holds. We obtain sufficient conditions for a given point of a set to be a luminosity point of this set. The results are formulated in terms of points of monotone connectedness of the set. We consider the characteristics of luminosity points which involve individual characteristics of both the space and the set (points of monotone connectedness of a set and rotund points of the unit sphere of the space). Prox-monotone path-connected sets are introduced; for such a set M , if a point y is a nearest point from this set for some point x∉ M , then y is a point of monotone connectedness of M . We also show that each boundedly compact prox-monotone path-connected set (prox-Menger connected set) is a sun. Applications to the space c_0 are also given.
In the paper, theorems on the decomposition of the Wiener measure with respect to the orbits of the action of the diffeomorphism group are proved.
The spreading of a droplet of incompressible fluid over a substrate under the action of capillary forces and gravity is studied within a phase-field framework. An asymptotic analysis shows that the solution of the phase-field system (the regularized problem) converges to the solution of the corresponding limit problem when a regularization parameter tends to zero. The theoretical results are illustrated by numerical simulations for hydrophobic and hydrophilic substrates.
