PSI - Issue 65

Dumanskaya E.I. et al. / Procedia Structural Integrity 65 (2024) 75–82 Dumanskaya E.I., Emelyanov I.G. / Structural Integrity Procedia 00 (2024) 000–000

78

4

differential and integral equations, the object of study is usually divided into a number of subdomains, which constant mathematical dependencies or dependencies according to set laws are accepted. This technique allows solving many scientific and applied problems. It is known that an integral equation can be reduced to a system of linear algebraic equations. Let us divide the period of a person's life � � into N time periods (day, month, year, ...). Let's assume that in each time period, the functions entered above have a constant signification. In the shorter the time period, the mathematical model describes more accurately the phenomenon under study. Thus, the time period is equal to

1    t T N i L

, ( 1.... ) i N 

(9)

For each time step of equation (7) is

1 t D C sS d c     1 2 [( 1 ]

(10)

f t

1 1 1 1

.  On the right-hand side of the system, the multiplication � � � � is the probabilities of committing a crime in each time period. Since the statistical information on various social phenomena required for the solution of system (10) is represented annually, let us assume a constant t and dimensionless time period of person's supervision is one year, and then the system (10) takes the form ] 2 [( 1 dN N N N cN C sS  N f t t D  

1 f D C sS d c     1 1 2 [( 1 ]

(11)

1

2 [( 1 

D

cN C sS

f

]   dN

.



N

N

Let us assume that in the system (11) in brackets there is known quantitative information on various negative social phenomena in society, and a quantitative negative characteristic of the person. On the right side is an unknown discrete signification of the probability density of the crime under the current conditions, which depends on time. Since each individual has a unique set of both positive and negative personality traits, which are difficult to formalize, and which can be recognized only by their manifestations in social life, let us accept the following hypothesis of the crime's procedure. In the case in which the person committed a crime, the prerequisites for that are accumulated in the life according to normal law, Piskunov (1985). This law, which is quite universal, is often used in the study of technical objects (wear of parts in many mechanisms, etc.). It can describe various processes, including humanitarian, by selecting appropriate parameters. Let us assume that the density of the probability of crime is expressed by the formula (12) In equation (12), parameter a is the maximum point and at the same time the center of symmetry, the parameter � is the distance from this center to the inflection point. If � is small, then the curve is high, if � is large, then it is wide. Thus, by choosing the appropriate parameters a and � it is possible to describe various variants of the probability of crime, from those that have been slowly increasing in recent years to rapidly developing - synergetic. Since the mathematical model is designed to predict of personality criminal behavior for persons who committed a crime in the last ( N ) year of observation, for the convenience of calculations and clarity, we take a = 0 in formula . 2   ( ) 1 2 2 ( )  t a  2 e f t  