Bán István: Biomathematics - PMSB Methods in Forestry (2002)
2. The environment - 2.2. Algorithmization of a wanted phenomenon and its application in local optima
2.2. Algorithmization of a wanted phenomenon Proof: 1) Statement-, from completeness of B follows the completeness of d. From the completeness of B follows that in the arbitrary environment of each of its values, there is a value differing from the former one. When taking the set of arbitrary environments of 6; within the closed interval [6,; 6, ], for each arbitrary environment there is a value differing from b] . When taking the set of the arbitrary environments 6, within the closed 2 1 interval [6,; b ], for each arbitrary environment there is a value differing 2 11 L 2 2 1 from bt . [bt; bt ] = [bt; bt ], and due to the definition of completeness, it is complete. Due to the Definition 5 the state characteristic values b] and b\ are at 1 2 the same time the sizes bt and bt , too, and due to Definition 7 the distance d 1 2 between the sizes b; and b, is the sign-correct difference between them, L 12 2 1 hence the absolute value of distance d is: |d| = [b;; bt ] = [ö£; bt], thus it is complete. From the definition of completeness and by taking also the arbitrary orientation of the absolute value of distance d, the completeness exists, thus the distance d itself is complete, too. 2) Statement: if B is not complete, then d is either complete or not. It follows from the definition of the completeness of B that if it has a value for which a value differing from the previous value does not exist in a given environment, then B is not complete. If this value of B is also an element of d, then d is not complete. If b has value(s) for which no value(s) differing from the former one(s) exist(s) in given environment(s), but this value (these values) is (are) not elementis) of d, then d is complete in consequence of Statement 1. ■ 2.2. Algorithmization of a wanted phenomenon and its application in local optima One of the greatest importance of the biological notion defined by biomathematics: process, state, state characteristic and state characteristic value is that they yield the possibility to interpret (Bán 1974, 1988), to project and to investigate the so-called continuity state of natural events in time. It arises as requirement of each member of the nature to understand the natural phenomena occurring as the entity of natural processes on the basis of the possibilities ensured by the observing system. The entity of all possible state characteristic values of state occurring in nature, of all the possible projections among them and of all state characteristics, and of all features of their projections, respec- 27