Bán István: Mathematical exploration of the enviroment (1996)

2. Mathematical fundamentals of the investigation method - 2.4 The entity of environment and the optimum - 2.4.3 Optima

The entity of environment and the optimum 47 the nature is the natural phenomenon, and its part observed by its living or non-living observing systems is the wanted entity, and its element is the wanted element. (Defined in Section 2.4.2.) Definition. The wanted entity is constant if any arbitrary two wanted elements are equal. The wanted entity is varying if there exist at least two wanted elements that are not equal. Remark: In the practice one meets mostly varying entities. Consequence. From the porosity of the universe concept V follows that a varying wanted entity exists. Definition. The natural phenomenon is constant if any of its wanted entities is constant. The natural phenomenon is varying if any of its wanted entities is varying. Definition. Be CWl one wanted entity of the given natural phenomenon T, be o\ its local optimum; and be CW2 another entity and 02 its local optimum. The 02 local optimum is better than the local optimum 01, i.e. cq —> o\, if and only if it is more advantageous for the user. Consequence. If the local optima 01 and 02 of two arbitrary CWl and CW2 wanted entities of a given natural phenomenon T are not equal, then one is better than the other. This follows from the former definition and from consequence 4. Definition. Be the wanted entities of a given T natural phenomenon CWl, CW2, ..., CWn and their local optima o 1, 02, ..., on. The local optima 01, 02, ..., on are improving monotonously, if they can be ordered so that in case of two arbitrary local optima o*, and (k < k + 1 < • • • < n), the optimum Ofc+1 is better than Ofc, i.e. Ok+1 —► Ok-Consequence. If the local optima eq, 02, ..., on of the wanted entities CWl, CW2, ..., CWn of the given natural phenomenon T are not identical, then the local optima are improving monotonously. This follows from the former consequence and definition. Definition. If the local optima o\ and 02 of the wanted entities CWl and CW2 of the given natural phenomenon T are identical, then they can be substituted by the local optimum o\^ being identical with both local optima o\ and 02. Consequence. If the local optima o\, 02, ..., o„ of the wanted entities CWl, CW2, ..., CWn of the given natural phenomenon T are identical, then the local optima can be substituted by the local optimum oi^,...,ni which is identical with them. This follows from the former definition and from the possibility to form a finite number of local optimum pairs. Theorem of the existence of local optima. Be the natural phenomenon T varying, and be its wanted entities CWl, CW2, ..., CWn. Statement. In the natural phenomenon T, there exist the local optima cq, 02 5 ... ? Oji .

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