Найти формулу для периода удвоения Tп актуальных научных проблем в обществе, развитие минимально ограничивая.

From (6) let’s go to the periods of doubling Td of the urgent scientific problems from the moment t0 (start epoch) till the moment t (future epoch). That is let’s solve the problem for the limiting case of the minimal requirements to the developing society: K(t) = 2K(t₀)

Definition 1: The period of doubling T_d of the urgent scientific problems is a time of development of society, during which the number of new problems, being solved by the society, increases twice.

Solution. By the definition from the formula (6) one gets: 2K(t₀) = K(t₀)exp {∫ ^t _t0 a(t)dt} (8)

For the time equal to the period of doubling

Td = t₀−t the condition a(t) = a = const is true, that is why from (8) one has:

2=exp{a∫ ^t _t0 dt} = exp{a(t − t₀)} = exp {aTd } (9)

hence Td = ln 2/a (10)

or in the general case of the dependence

a = a(t): Td (t) = ln 2/a(t) (11) what was required to prove.

Find a formula for the average life time τ (t) of the old problem which is being solved from the time moment t₀.

Найти формулу для среднего времени жизни τ(t) старой задачи, которая решается с момента t₀.

Average lifetime τa of a scientific problem – is a time during which one old problem is solved and ”g” new problems appear, which are genetically connected to the old one. In general case τ = τ (t).

Solution. Let w(t) – is a probability that the problem, existing at the time moment t0, still exists at the time moment t. Then the quantity dw – is a probability of its solution (disappearance) for the period between t and (t + dt) (see fig.). It is obvious then (the minus sign means "disappearance"):

dw = −a ⋅ w ⋅ dt. (12)

Integrating the expression (12), one gets:

w(t) = w(t₀) ⋅ exp {−a ⋅ t} (13)

Since at t = t₀ the problem still reliably existed (was not solved, does not disappear), then w(t₀) = 1 (equals reliability)

w(t) = exp {−a ⋅ t} . (14)

From the expression (14) one can find the average life time of the old problem according to the definition of the mathematical average:

τ_a = ∫ ^∞₀ t(−dw) = ∣substitute (замена) (12), (14)∣ = a ∫ ^∞₀ t⋅w dt ={u=t, dv=e^-atdt, du=dt, v=e^-at/-a}=а(-te^-at/a|^∞₀ + 1/a∫^∞₀e^-atdt)= -t/e^at|^∞₀ + 1/-аe^at| ^∞₀)=(-∞/e^a∞+0/e^a0)+(1/-аe^a∞ + 1/аe^a0)=(0+0)+(1/-∞ +1/а)=1/а

So τ_a = 1/a .

Find a relation between the doubling period T _п , life time τ_π and the coefficient of problem reproduction “g”.

Найдите связь между периодом удвоения T_n, временем жизни и коэффициентом воспроизводства задачи “g”.

Definition 3:Coefficient of problem reproduction g – is a number equal to a ratio of newly arising problems to the number of the old scientific problems in this developing society.

Find a relation between the period of doubling Td, life time τ_a and the coefficient of reproduction of problems g.

By the definition K(t)=g⋅K(t₀), then similarly to the expression Td = τa ln 2 = 0.69τa. one gets: According to the definition (2): T_g=τ_a(t).