
- •Find a formula of the exponential growth of the number of scientific problems appearing in harmonically developing society.
- •2. Find a formula for the period of doubling Tп of urgent scientific problems in limiting minimally developing society.
- •3. Find a formula for the average life time τ(t) of the old problem which is being solved from the time moment t0.
- •4. Find a relation between the doubling period Tп, life time τπ and the coefficient of problem reproduction g.
- •5. Coefficient of problem reproduction “g” and main conclusions connected with it.
- •6. Goals and objective regularities of science development.
- •7. Objectives and criteria of importance of fundamental research.
- •General laws of science development.
- •Law of accelerated motion of science, law of accelerated rate: science develops by exponential law.
- •Law of distribution of the outstanding scientists among the men of science of different time.
- •11. Law of team spirit in science and collaboration of generations
- •12. Law of planning in the modern science.
- •13. Law of international character of science.
- •Law of exponential growth of scientific information.
- •Law of increase of repeated discoveries.
- •Law of increase of period of education for science occupation.
- •Law of particular specialization of men of science and differentiation of science.
- •18. Law of growth of national income of the state, expended for science.
- •19. Law of transformation of the modern science from descriptive methods
- •20. Law of historical depth of citations in scientific works
- •21. Law of interaction and interrelation of science.
- •22. Law of interpenetrating development in a system «Science (s) - Technique
- •23. Law of qualitative and quantitative change of productivity of work of scientists in dependence on their age
- •24. Law of increase of number of doctors and candidates of science
- •25. Law of preparation of scientific staff reserve
- •26. Law of age structure of scientific employers working in research institutes
- •27. Law of distribution of geniuses among the planet’s inhabitants
- •28. Law of influence on the process of knowledge of heretics in science
- •29. Optimal structure of science.
- •30. Horizontal scheme.
- •31. Vertical scheme – is an optimal organizational structure of the science of the 21st century.
- •32. Mechanism of realization of the vertical scheme and its functioning
- •33. International system of scientific publications
- •34. About the history of scientific publications
- •35. Peer-review procedure
- •36. Primary and secondary scientific publications
- •37. Author rights in the system of international scientific publications
- •38. Bibliometric (scientometric) indicators in the system of international scientific publications: index of scientific citation and impact-factor
- •39. Organization of the text of original article for journal
- •40. Peculiarities of style of scientific publications in English
- •41. Название (Title)
- •Introduction
- •42. Materials and Methods
- •43. Choice of journal and manuscript submission
Find a formula of the exponential growth of the number of scientific problems appearing in harmonically developing society.
As the science is developing the more and more number of scientific problems K(t) is being solved. If the problem is not dead-end, then at its solution a “tree of new problems” appears, at least – two problems.
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.
1. Problem 1. Find a formula of the exponential growth of the number of appearing scientific problems 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):
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.
2. Problem 2. Find a formula for the period of doubling of the urgent scientific problems Tп in the limiting minimal developing society.
Solution. By the definition from the formula (3) we get:
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):
Tп = (ln2)/а (6)
or in the general case of the dependence а = a(t):
Tп(t)= (ln2)/a(t), (7)
3. Find a formula for the average life time τ(t) of the old problem which is being solved from the time moment t0.
Now let’s generalize the problem and find the average lifetime of the scientific problem that is the average periods of solution of the urgent problem being considered by the developing society.
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.
3. Problem 3. Find a formula for the average lifetime τп(t) of the old problem being solved from the moment of time t0.
Solution. Let w(t) – is a probability of that the problem, existing in the moment of time t0, still exist in the time moment t. Then the quantity dw – is a probability of its solution (disappearance):
dw = - awdt (9)
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:
Comparing the expression (6) with the expression (12), we get: