Find a formula of the exponential growth of the number of scientific problems appearing in harmonically developing society.
Thus, the rate of the natural objective growth of new urgent problems dK(t)/dt, where t – is a time, is in average proportional to their current amount K(t): dK(t)/d(t) = a(t) K(t), (1)
where a(t) – is a coefficient of proportionality, characteristic for each epoch t. This coefficient defines the rate of the relative growth of the amount of problems in harmonically developing civilization:
а = а (t) = {1/К(t)} * [dK(t)/dt] (2)
If the human civilization immediately starts to solve new problems, then their general number increases by the exponential law.
Problem 1. Find a formula of the exponential growth of the number of scientific problems appearing in a harmonically developing society.
Solution. Let’s integrate the expression (1), for it let’s rewrite it in the following form: 
Let’s transfer to a definite integral over the time period from t0 (start) till t (finish): 
Hence 
what is required to prove.
From (3) let’s transfer to the periods of doubling Tп of the urgent scientific problems for 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) (4)
2. Find a formula for the period of doubling Tп of urgent scientific problems in limiting minimally developing society.
The period of doubling Тп 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.
Problem 2. Find a formula for the period of doubling of the urgent scientific problems Tп in the limiting society developing minimaly.
Solution. By the definition from the formula (3) we get: 
(5)
For the time equal to the period of doubling Tп = t0 – t the condition a(t) = a = const is true, that is why from (5): 
Hence Tп = (ln2)/а (6) or in the general case of the dependence а = a(t): Tп(t)= (ln2)/a(t), (7) what is required to prove.
3. Find a formula for the average life time τ(t) of the old problem which is being solved from the time moment t0.
Average life time τп 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 
Problem 3. Find a formula for the average lifetime τп(t) of the old problem being solved from the moment of time t0.
Time axis
The scheme of probabilities distribution
|
|
|
Integrating the expression (9), we get: w(t) = w(t0) * exp[-at] (10)
Since at t = t0 the problem still reliably existed (was not solved, does not disappear), then w(t0) = 1 (equals reliability) w(t) = exp(-at) (11)
From the expression (11) one can find the average life time of the old problem according to the definition of the mathematical average: 
so 
4. Find a relation between the doubling period Tп, life time τπ and the coefficient of problem reproduction g.
A 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.
Problem 4. Find a connection between the period of doubling Tп, life time tπ and the coefficient of reproduction of problems g.
Solution. By the definition K(t) = gK(t0), then similarly to the expression (5) one gets:
Tg = lng/а (15)
According to the definition (8): Tg =τп(t) (16)
Then, solving the equations (6), (15) and (16) jointly, one gets: Tп(t) = ln2 * tπ(t)/lng,
what is required to prove.
In the general case the equation (1) will be rewritten in the form: dK(t)/dt = gK(t) (18)
Then similarly to (3) a solution of the equation (18) will have the form:
5. Coefficient of problem reproduction “g” and main conclusions connected with it.
Let us further consider the general case, when the number of new arising urgent scientific problems is not equal to 2, but to arbitrary number g:
K(t) = gK(t0). (14)
A 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.
The general number of the urgent problems as the civilization is developing gradually increases. For timely immediate solution of new problems it is required to involve more and more number of scientific employers. Hence, it is necessary to create new research institutes systematically.
Otherwise at g = 1 the mankind or a society begin to solve infinitely qualitatively the same problems of practical character and the society does not develop intellectually.
|
|
|
At g < 1 the society for the finite time period completely solves all problems, worked by the previous generations, completely loses the intellectual interest to existence. As a result of the intellectual degeneration the society as a civilization dies.
Дата добавления: 2016-01-03; просмотров: 18; Мы поможем в написании вашей работы! |
Мы поможем в написании ваших работ!