Bán István: Csodabogarak - Anekdoták a matematikusokról & Light Biomathematics (2010)
6. Függelék (Selected Papers of István Bán)
Fe Rt (^j,7n )] j ^ 17 2, .... The most advantageous specific feature should be Fea [®j,m i F-t (^j,m)] i ÍS fixed ^ 1. Statement. The optimum wanted entity m 16o is that of the existing wanted entities which has the most advantageous specific feature. Proof. As consequence of the conditions, the following wanted elements Cj,m,t,e = #t(ai,m); Fe[aiim;Ät(aii7n)]}. From the specific features Fe[aJim; fít(aJim)], having been determined on the basis of the known state characteristic values aJim and the relations interpreted on their basis, there is a most advantageous specific feature due to the condition which should be Feo[ajiTn] Rt(ajiTn)]. This, however, is in case of eQ = 1 the element, or they are in case of e0 > 1 elements of special feature Fe[aJim; f?t(aj,m)], where e > e0. Thus, due to the existence of the most advantageous specific feature and due to the conditions, there exist(s) the wanted element(s) ~ i FCg [uj|m, j , and this is the optimum itself, as a consequence of the definition of the optimum (in case of eQ > 1, the optima): Optj m t eo = Cjtm>t,e0, i.e. the optimum wanted entity. Consequence 3. Two optima are equal if and only if their respective state characteristic values, relations, and specific features are identical. This consequence is the theorem of the equality of the wanted entities and is valid as the consequence of the optimum being the special case of the wanted entity (consequence 2). Consequence 4. Two optima differ if their state characteristic values, relations, and specific features differ. This statement follows from the consequence of the difference of the wanted entities (consequence 1), and from the consequence of the optimum as special case of the wanted entity (consequence 2). Consequence 5. The wanted entity is not additive, i.e. the relations interpreted bn the disjoint subsets A\, A2, ... of the state characteristic values, and the wanted entities Cax, Ca2, ■ ■ ■ — formed by the specific features determined on the basis of the known subsets — are not equal to the relations interpreted on disjoint subsets of state characteristic values, and with the wanted entity Ca formed by the specific features determined on the basis of the known subsets (A — A\ U A2 U • • •). This follows from the theorem of congruence of the wanted entities. First let us form the wanted entities Cax, Ca2, ••• on the subsets A\, A2,----Now the wanted entity C = C\x UC/ijU- • • will be determined from the wanted entities Cax, Ca2,----In the course of this determination it is natural that the subset sums of the state characteristic values must be given.