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Synergetics and architecture. (English) Zbl 1185.00032
A series of phenomena referring to language, literary criticism, economics, quantum physics and architecture is studied from the synergetics point of view as the main property of self-organizing complex systems. A whole series of mathematical formulas describing these phenomena are identical and can be related to different situations: formulas relating to the Bose-Einstein distribution of particles and the distribution of words from a given dictionary described by Zipf’s law.
In the first section we have a very interesting description of synergetics that is related to the example given by G. Hackle: “in a certain community the guests bring their own wine to weddings and all wines were mixed before drinking. One guest thought that if all the other guests would bring wine, he can bring water, because after mixing no one would notice this fact. But if the other quests did the same, as the result everyone will drink water”. Basing on this story we have the introduction of two different arithmetics; one (linear) that ensures “equal rights for everyone” (“We all live happily, divide all things equally”), which can effectively work only in extreme situations and for short periods of time (for example the war time can be given, where everyone wants to give as much as possible) – this arithmetic leads to the certain rules which ensure the establishment of equilibrium and the system becomes self-organizing (like in the living nature, where everything follows the order of things (some animals eat other ones, a third group feeds on the first on and so on)), but also the cataclysms can occur. The second arithmetic (nonlinear) can burst despite the fact that the equilibrium described above may last for a long time, because some catastrophic phenomena can occur. In the paper this arithmetic is presented basing on the Kolmogorov axioms and finally it depends on one parameter \(\beta\), which has the physical meaning of \(1/T\), where \(T\) is the temperature and this arithmetic can be related to the zeroth kind phase transition. Moreover, any model of society, primitive or complex, based on \(\beta\)-arithmetic, leads to the same phase transition.
In the second section we have the presentation of the salary model, where appears the same \(\beta\) parameter as in the new definition of arithmetic. But this model also can be related to the Allen-Jones fountain effect for liquid helium-4.
The section 3 is devoted for considerations about the dimension with its connections for the problem of self-organization of complex systems and the problem of complexity increase with its relation to the increase of entropy.
In sections 4, 5, 6 and 7 we have the presentation to the problem of linguistical statistics. In section 4 the Zipf’s law is presented, while in the fifth one we have a new viewpoint on the problem of frequencies. From Zipf’s law it can be deduced that we can have three types of word: (i) superfrequent, (ii) frequent and (iii) rare. It seems that the super frequent words are not necessary to recognize the text because they don’t play an essential role in clarifying the meaning of the text. But sometimes these words are the unless one because for example in English language they make possible to distinguish the meaning of phrasal verbs, thus it is necessary to introduce a preference function. Since the number of words occurring only once constitutes no less than 1/3 of the dictionary, it can be assumed that its dimension can be zero. Basing on this assumption we can see (section 6) that Zipf’s law can be related to the dequantization of the Bose-Einstein distribution law in dimension zero. Moreover, this can be related to the \(\beta\) parameter that was presented in first section of the paper. This quantized form of the Zipf’s law is used to the statistical description of the structure of natural language texts and this is presented in section 7 for the case of Quiet Flows the Don. Thus we have the calculations of \(\beta\) parameter (the inverse temperature \(1/T\)) in the case of some additional novels.
Section 8 is devoted for the problem of the pointillist processing of photographs.
In section 9 we have the divagations about the criterion, which determine whether or not a new building spoils the ensemble into which it is enclosed.

MSC:
00A71 General theory of mathematical modeling
82C99 Time-dependent statistical mechanics (dynamic and nonequilibrium)
00A66 Mathematics and visual arts
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