Control of multiscale systems with constraints. 1. Basic principles of the concept of evolution of systems with varying constraints

Physical fundamentals of the self-organizing theory for the system with varying constraints are considered. A variation principle, specifically the principle of dynamic harmonization as a generalization of the Gauss-Hertz principle for the systems with varying internal structure is formulated. In compliance with this principle the system evolves through dynamics of the processes leading to harmonization of the internal multiscale structure of the system and its connections with external actions as a result of minimizing the dynamic harmonization function. Main principles of the shell model of self-organization under the action of the dominating entropic disturbance are formulated.Comment: First par

Distribution of averages in a correlated Gaussian medium as a tool for the estimation of the cluster distribution on size

Calculation of the distribution of the average value of a Gaussian random field in a finite domain is carried out for different cases. The results of the calculation demonstrate a strong dependence of the width of the distribution on the spatial correlations of the field. Comparison with the simulation results for the distribution of the size of the cluster indicates that the distribution of an average field could serve as a useful tool for the estimation of the asymptotic behavior of the distribution of the size of the clusters for "deep" clusters where value of the field on each site is much greater than the rms disorder.Comment: 15 pages, 6 figures, RevTe

On the classification of discrete Hirota-type equations in 3D

In the series of recent publications we have proposed a novel approach to the classification of integrable differential/difference equations in 3D based on the requirement that hydrodynamic reductions of the corresponding dispersionless limits are `inherited' by the dispersive equations. In this paper we extend this to the fully discrete case. Our only constraint is that the initial ansatz possesses a non-degenerate dispersionless limit (this is the case for all known Hirota-type equations). Based on the method of deformations of hydrodynamic reductions, we classify discrete 3D integrable Hirota-type equations within various particularly interesting subclasses. Our method can be viewed as an alternative to the conventional multi-dimensional consistency approach.Comment: 29 page