Mathematical models automata

The last twenty years of the last millennium are characterized by complex automatization of industrial plants. Complex automatization of industrial plants means a switch to factories, automatons, robots and self adaptive optimization systems. The mentioned processes can be intensified by introducing mathematical methods into all physical and chemical processes. By being acquainted with the mathematical model of a process it is possible to control it, maintain it at an optimal level, provide maximal yield of the product, and obtain the product at a minimal cost. Statistical methods in mathematical modeling of a process should not be opposed to traditional theoretical methods of complete theoretical studies of a phenomenon. The higher the theoretical level of knowledge the more efficient is the application of statistical methods like design of experiment (DOE). 

Another powerful tool is the cellular automata method invented by John (or Janos) von Neumann and Stanislaw Marcin Ulam (under the name of cellular spaces ). The cellular automata are mathematical models in which space and time both have a granular structure (similar to Monte Carlo simulations on lattices, in MD only time has such a structure). A cellular automaton consists of a periodic lattice of cells (nodes in space). In order to describe the system locally, we assume that every cell has its state representing a vector of N components. Each component is a Boolean variable i.e., a variable having a logical value (e.g., 0 for false and 1 for true ). 

Cellular automata can be constructed as simplified models of reaction-diffusion systems. Often the main features of an apparently very complex dynamics can be captured in a simple rule. The Greenbeig-Hastings rules [13] are an example of simple cellular automaton rules that model excitable media. We term such cellular automaton models classical cellular automata since they are constructed in the spirit of the original cellular automaton models of Von Neumann. There is a laige literature [14] on this topic that deals with the mathematical properties of different abstract cellular automaton rules [15], as well as studies that attempt to model rather detailed features of specific reaction-diffusion equations [16,17]. 

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