Inšthis paper, the system ofštwo vortices inšanšannular region isšshown tošbešintegrable inšthe sense ofšLiouville. Ašfew methods for analysis ofšthe dynamics ofšintegrable systems are discussed and these methods are then applied tošthe study ofšpossible motions ofštwo vortices ofšequal inšmagnitude intensities. Using the previously established fact ofšthe existence ofšrelative choreographies, the absolute motions ofšthe vortices are classified inšrespect tošthe corresponding regions inšthe phase portrait ofšthe reduced system.

With the help ofšmathematical modelling, wešstudy the dynamics ofšmany point vortices system onšthe plane. For this system, wešconsider the following cases:
?švortex rings with outer radius $r = 1$šand variable inner radius $r_0$,
?švortex ellipses with semiaxesš$a$, $b$.
The emphasis isšonšthe analysis ofšthe asymptotic $(t ? ?)$ behavior ofšthe system and onšthe verification ofšthe stability criteria for vorticity continuous distributions.

This study isšthe continuation ofšthe computer experiment [1] with particles ofšgas inšašone-dimensional tube, described earlier. Inšthis paper wešgive investigation results for the statistical properties ofšašrelativistic gas inšašone-dimensional tube. Itšisšshown that this system reaches the state ofšthermodynamical equilibrium whose distribution function isšdetermined byšthe relativistic energy ofšparticles. The system ofšparticles inšašone-dimensional tube isšdescribed byšanalogy with the billiards inšašpolygon.

With the help ofšmathematical modeling, wešstudy the behavior ofšašgas ($\sim10^6$ particles) inšašone-dimensional tube. For this dynamical system, wešconsider the following cases:
?šcollisionless gas (with and without gravity) inšaštube with both ends closed, the particles ofšthe gas bounce elastically between the ends,
?šcollisionless gas inšaštube with its left end vibrating harmonically inšašprescribed manner,
?šcollisionless gas inšaštube with ašmoving piston, the piston?s mass isšcomparable tošthe mass ofšašparticle.
The emphasis isšonšthe analysis ofšthe asymptotic ($t??$)) behavior ofšthe system and specifically onšthe transition tošthe state ofšstatistical oršthermal equilibrium. This analysis allows preliminary conclusions onšthe nature ofšrelaxation processes.
Atšthe end ofšthe paper the numerical and theoretical results obtained are discussed. Itšshould bešnoted that not all the results fit well the generally accepted theories and conjectures from the standard texts and modern works onšthe subject.

The generalized model of formation of a new phase is considered. The basic stages of process of growth are gathered in a model at phase transition of the first sort. The numerical solution of the kinetic equation of Fokker–Planck is received. Dependence of the solution on parametres of system is investigated. Areas of applicability of assumptions made by Zeldovich, Lifshits and Slezov are revealed. Also it is shown, that depending on parametres of system it is possible to reserve both equilibrium distribution, and automodelling distribution of Lifshits–Slezov. At some values of parametres the equation has the oscillatory solution.