Bán István: Mathematical exploration of the enviroment (1996)
2. Mathematical fundamentals of the investigation method - 2.4 The entity of environment and the optimum - 2.4.2 Algorithmization of the wanted element and its application in local optima
The entity of environment and the optimum 41 natural processes on the basis of the possibilities ensured by the observation system. The entity of all possible state characteristic values of the states occurring in the nature, of all projections possible among them, and of all properties of state characteristics and projections should be accepted as the biomathematical definition of the natural phenomenon. The set of states perceivable by the possibilities of the living and non-living observation system, of interpretable projections and of deducible features is called the wanted entity and its elements are called the wanted elements. The notions are defined as follows: Be all the possible state characteristic values of the states occurring in the nature (ai,n) ß2,n> 03,TM, • • •, a;jn), where i = 1,2,... is the index of the state characteristic, n = 1,2,... is the indicating index of the observed value. Definition. Out of all the possible state characteristic values the state characteristic values observed within the possibilities given by the observation system should be: (^T,mi ®2,m) Ö-3,mi • • • i where j ^ 2, 771 ^ 71. Definition. Relation means the arbitrary adequacy between the state characteristic values. The following types of relations between state characteristics must be distinguished: relations which correspond theoretically to all combinatorial possibilities among the state characteristics above, relations existing in reality, and relations perceivable within the possibilities of the observing system. Definition. Be br = Rr(ai>n) (r = 1,2,...) the relations corresponding to all theoretically possible combinations of the former state characteristics Gqin. Definition. Be bs = Rs(aitU) (s = 1, 2,... N < r) the really existing relations among the former state characteristics aitH. Definition. Be bt = Rt(a,i,m) (t = 1,2,... < s) the relations perceivable on the basis of the possibilities of the observing system among the state characteristics a;im. Definition. The peculiarities of the state characteristic values and relations detectable by mathematical methods are called specific features. In all of the specific features deducible from the values and relations of the former state characteristics, one has to distinguish between all theoretically possible relations, the really existing features and those which can be interpreted among the possibilities of the observing system. Definition. Be Fc(aitn; Rr(aitn)), (c = 1,2,...) all the theoretically possible specific features of the state characteristics a;in, and of the relations br = Rr (^i,n ) • Definition. Be Td(aqn; i?r(ai,n))> (cZ = 1,2,...) the specific features existing on the basis of the state characteristics a^n, and the relations br = i?r(a;in).