Computer Experiments in Groups of Permutations With Broken Symmetry

Abstract

В серии компьютерных экспериментов на основе программ расчета групп симметрии при изучении струк-тур с нарушенной симметрией обнаружены новые множества с «нулем», «единицей», без обратных элементов, с конвергентными (сходимостью) и дивергентными (расходимостью) свойствами орграфа структуры. В работе продемонстрированно, что существует возможность перехода от множества преобразований с конвергентны-ми свойствами орграфа структуры к структурам множеств с дивергентными свойствами заменой некласси-ческих матриц с нарушенной симметрией на обратные матрицы подстановок. А также показана конкретная корреляция этих свойств с моделями накопления и послойного роста, используемыми в нанокластерном ана-лизе, радиотехнике, при математическом описании состояний некоторых экономических систем и объектов кристаллохимии, что является наглядной демонстрацией принципа «диссимметрии» П. Кюри. A.G. and N.G. Stoletovs, Vladimir, e-mail: godograf@list.ru In computer simulations, by studying the tables, sets of binary transformations with broken symmetry we've discovered the new sets with «zero» and «one» without inverse elements, with convergent and divergent properties of structure's oriented graphs. It is possible to continue from a variety of transformations with convergent properties of oriented graphs' structure to the structure of the set with divergent properties. A correlation of these properties with the accumulation models and the layer growth used in the analysis of the nanoclusters, radio technique, as well as the description of the conditions of some economic systems and the chemistry of crystals. This is a clear demonstration of the principle of «dissymmetry» pierre Curie.

Structure Transformations in Broken Symmetry Groups—Abstraction and Visualization

“…Each modification [9] can be visualized by analyzing the substitution itself. Substitution visualization of three operations, randomly chosen from Table 1, is presented in Figure 2 [15]. Evidently, the resulting table is not a classic multiplication Cayley table, as inverse elements are absent and properties change "by rows" and "by columns".…”

“…Each modification [9] can be visualized by analyzing the substitution itself. Substitution visualization of three operations, randomly chosen from Table 1, is presented in Figure 2 [15]. [1], g [8] Table 1 in one dimensional space (а) and g[0], g [1], g [8] in two-dimensional space (b).…”

“…In one of the computer experiments [11] using a designed program of binary matrices multiplication with classic and non-classic substitutions ( Figure 3) numerical set of 34 number values made a substitution set consisting of 91 BSG operations. Out of the set, two operations subgroups can be highlighted (including g[0]), which create small finite subsets (Table 2) visualization of three operations, randomly chosen from Table 1, is presented in Figure 2 [15]. [1], g [8] Table 1 in one dimensional space (а) and g[0], g [1], g [8] in two-dimensional space (b).…”

“…A transition to the orgraph with divergent vectors ("All roads start in Rome"), formally can be seen as "reverse" operation, g[1] −1 , where upper and lower rows switch places in substitutional matrices [15]. A graphic representation of the reverse to the transformations from Table 3 operations is shown on Figure 4 as a divergent process (diverging "development spiral").…”