Antoniouk O. Analytic methods in statistic mechanics, nonlinear diffusion and complexity theory of time series.

The thesis is dedicated to the investigation of mathematical objects that arise due to the rigorous description of statistic mechanics models, nonlinear diffusion and discrete time series which are the result of recording EEG and ECG EKG signals. In this manuscript it is proposed and developed the method of nonlinear estimates on variations, which is applied to the investigation of smoothness and raise of smoothness properties for the semigroup, associated with the Ising stochastic model of infinite number of interacting particles. The geometric invariant approach to the regularity for the solutions of nonlinear diffusion equations on noncompact Riemannian manifolds is developed. The Least Action Principle for the weighted porous media equation is proved. It is obtained the asymptotic expansion for the solution of the heat equation in non-cylindric domain with cuspidal singularity on the boundary. The Kolmogorov-Sinai entropy is characterized in terms of ordinal partitions.