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 Theorem-. The theorem of the optimum wanted phenomenon. Be aj m j = 1, 2, ...; m - 1, 2, ... state characteristic values. The relations Rfitij m) t = 1, 2, ... should be existent. Be the features proved on the state characteristic values aj m and on the state characteristics Rt{ajm) and relations defined by mathematical methods: Fe[ajm\ Rt(ajnz)] e = 1, 2, ... Be optimum the for us most advantageous feature Fe[ajm\ Rt(a,j m)\ e0 fixed as > 1. Statement: Optj is out of the existing wanted phenomena that one which has the feature most advantageous for us. Proof: Due to the conditions the entity of the following wanted phenomena does exist: ^j,m,t,e = [aj,m’ Rt(aj,m)]> ■ Out of the properties Fe[aJ>mi Rt(aj m)\ determined from the knowledge of the state characteristic values aj m and on the relations Rt(ajm) interpreted on them, there exists — due to the condition — the for us most advantageous property, this should be denoted by Fe [aj m-, Rt{aj m)]. This, however, is in case of e0 = 1 or they are in case of e0 > 1, the element(s) of the feature Fe[a] m\ Rfcij m)\. Due to the existence of the most advantageous feature and due to the conditions, there exist(s) the wanted phenomenon/phenomena Cj>m t eo = [czy m; Rt(ajm)]; Fe[aj m\ Rt(ajm)], which is due to the definition of the optimum Optj,m,t,e = Cj,m,t,e ün case of e0 > 1 plural has to be understood), i.e. the opti­mum wanted phenomenon. ■ Consequence 3: Two optima are then and only then equal, if their correspond­ing state characteristic values, their relations and their features are equal. This consequence is the theorem of the equality of the wanted phe­nomena, and holds in consequence of the optimum as the special case of the wanted phenomena (Consequence 2). Consequence 4: Two optima are different, if their state characteristic values, their relations or any of their peculiarities are different. This statement follows from the difference of the wanted phenomena (Consequence 1) and from the consequence of the optimum being the special case of the wanted phenomena (Consequence 2). Consequence 5: The wanted phenomenon is not additive, i.e. the relations interpreted on the disjunct subsets Alt A2, ..., and by knowing them the entity of the wanted phenomena , CAi, ... constructed from their prop­erties is not identical with the wanted phenomenon CA (A = A1 A2 ^ ...)

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