# Formalization of the notion of technical systems evolution (Part 4):Evolutionary series of representative technical systems

### Y. B. Karasik, Thoughts Guiding Systems Corp., Ottawa, Canada. e-mail:karasik@sympatico.ca

In the previous article I considered series of systems representing majority. Now I am going to generalize this approach and consider series of systems representing something else, not neceserily majority. In what follows such series are called series of representative technical systems. What they represent is not important for this definition. It could be anything for which an evolutionary trend exists.

Consider, for example, systems representing the nascent technical systems as opposed to systems representing those, which die out. At any point in time there are systems, proportion of which decreases and proportion of which increases. If both of them belong to the same distribution then transition from the former to the latter is definitely a trend. This trend can be formalized as follows:

Definition:

Let {s: s ∈ S(t) and C(s) = k} be a subset of systems S at a moment t, which value of criteria C is k. (For example, if S are planes and C is the number of wings, then {s: s ∈ S(t) and C(s) = 2} are all planes that have 2 wings at a moment t.) In what follows {s: s ∈ S(t) and C(s) = k} is denoted as S(C, k, t) and the number of systems in it is denoted as ||S(C, k, t)||.

Evolutionary series of representatives of nascent technical systems is a series {sn} where sn ∈ S(C, k(tn), tn) and function k(t) is determined from the following equation:

 d ||S(C, k(t), t)|| d ||S(C, m, t)|| = max dt m dt
and {tn} are the points of discontinuity of function k(t).

These serise represent transitions from systems loosing the majority status (but which have not lost yet) to nascent technical systems, which will gain majority some day after. There could be series of other types of representative systems, of course.