Theoretical Mathematics & Applications

Order, Chaos and Symmetry in the Deterministic Chains of Elements

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  • Abstract

    A new concept of order and chaos has been suggested, whereby any system’s level of order can be determined provided existence of the order establishment procedure. Compositions of open chains (words) are being analyzed. Determined hereby are compositions of words enjoying ideal symmetry, as well as the procedure of step-by-step transformation of open sequence of elements into the ideal-symmetry composition. The level of order in a word is determined by the number of steps in the defined procedure that transforms this word into the symmetric-state composition. To describe a word or a word composition the A matrix is being used, components of which are the closest neighbors’ numbers of pairs. It has been shown that the level of order is calculated by expansion of the A matrix to matrices that correspond to the idealsymmetry compositions. The theorem has been formulated and proved as to the type of the pair matrix expansion to matrices that correspond to the ideal-order compositions. Maximum and minimum symmetry word structures have been found.

    Mathematics Subject Classification: 05A99; 62B10; 94A17; 60J10
    Keywords: Entropy; Information; Symmetry; Combinatorics