Bán István: Csodabogarak - Anekdoták a matematikusokról & Light Biomathematics (2010)
6. Függelék (Selected Papers of István Bán)
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(a;jn; i?r(agn)), (c = 1,2,...) all the theoretically possible specific features of the state characteristics a;iTl, and of the relations br = Dr (ii{ ,n). Definition. Be Fd(agn; Rr(aitn)), (d = 1,2,...) the specific features existing on the basis of the state characteristics agn, and the relations br = i?r(a,-,n). Definition. Be Fe(ajtTn] Rt{ajtm)), (e = 1,2,...) the specific features interpretable on the basis of the state characteristics aym and the relations bt — Rt(a,jirn) within the possibilities of the observing system. __ Definition. On the basis of the foregoings, the natural phenomenon T is the set of the state characteristic values agn Vi and Vn, of all the possible relations i?r(ai>n) Vr, as well as of all specific features Fc(ai}n-, RT(ai>n)) Vc, i.e.: T = {(ag-„; Vi, Vn) U [^(ag«); Vr] U [Fc(ai>n; Är(a»,„)); Vc]}. Definition. From this natural phenomenon the set of states perceivable by living and non-living observation systems, and of interpretable relations and deducible specific features, i.e. the wanted entity Ce is: Ce = {(aj,m; Vj.Vm) U Vi] U [Fe(aym; i?t(aym)); Ve]}. One element of the wanted entity Ce is the wanted element C, i.e. = Fe R-t (®j,m)] } as given for a fixed, arbitrary subscript j, m, t, e. From the definition of the state characteristic values observed within the limits of the observing system, and of relations perceivable among them, as well as from that of the interpretable specific features follows that is a mathematical construction, which is a notion of local dependence. On this basis the theorem of the so-called local wanted entity can be formulated as: Theorem. Two wanted entities are equal if and only if their respective state •characteristic values, relations and specific features are identical. Be Ch,r*x.tl,ei — (ajl,mii D-tl (aji .m, )i Fei [aji ,mi > -Rtj (aji ,mi )] } and Cj2,rn.2,t2,e2 = |aj2iV»2i ^2 (aÍ2iTO2 )i Rt2(aj2rrl2)}^. The statement Cjlimi,tltei ~ ^.72,m2,*2^2 *s valid if and only if: