Another way was also chosen. A scattering graph was constructed by taking the numbers themselves on one axis and the values on the other. If it can be proved that in any sequence of the algorithm there is a number smaller than the original one, it is possible to confirm the Collatz hypothesis. But any initial number is reduced to a smaller number, which in its own sequence will lead to a number even smaller, etc., up to 1.

That is, the only possible outcome for this particular case is the 4-2-1 cycle, but it has not yet been possible to prove this.

Although in 1976, Riho Terras showed that almost all sequences include values below the original one. In 1979, it was shown that the values would be less than the original ones by these values raised to the power of 0.869. Later, in 1994, the degree became more precise – 0.7925. Here, almost all the numbers mean that when the initial values tend to infinity, the proportion of the limiting function tends to 1. In 2019, mathematician Terry Tao was able to prove that this algorithm obeys even stricter restrictions.

He managed to show that all numbers will be less than the values of the function at any point, provided that the limit of the function, when the variable tends to infinity, will be equal to infinity. In this case, the function can grow arbitrarily slowly, the same logarithm, or logarithm logarithm, or logarithm logarithm, etc. This allows us to assert that arbitrarily small numbers exist in the series of any initial number. And as it was said in 2020, only a direct proof of the hypothesis can be better than this.

Used literature

1. Hayes, Brian. The ups and downs of hailstone numbers. American. – 1984. – No. 3. – pp. 102-107.

2. Stewart, Ian. The greatest mathematical problems. – M.: Alpina non-fiction, 2015. – 460 p.

3. Jeff Lagarias. The 3x+1 and its generalizations. American Mathematical Monthly. – 1985. – Vol. 92. – P. 3—23.

GENERAL IDEA OF THE CONCEPT AND USE OF DIFFERENTIAL EQUATIONS IN THE DESCRIPTION OF SOME DYNAMIC PHENOMENA

Aliev Ibratjon Khatamovich


2nd year student of the Faculty of Mathematics and Computer Science of Fergana State University


Ferghana State University, Ferghana, Uzbekistan

Аннотация. Как некогда сказал Стивен Строгац: «Со времён Ньютона человечество пришло к осознанию того, что физики выражаются на языке дифференциальных уравнений». Разумеется, что данный язык используется далеко за пределами физики и талант использовать его, ровно, как и воспринимать, даёт новые краски при изучении окружающего мира. В настоящей работе описывается общее представление об этом методе и сам процесс его изучения.

Ключевые слова: дифференциальные уравнения, исчисления, алгоритмы, математическая физика.

Annotation. As Stephen Strogatz once said: «Since the time of Newton, mankind has come to realize that physicists are expressed in the language of differential equations.» Of course, this language is used far beyond physics and the talent to use it, exactly as to perceive it, gives new colors when studying the surrounding world. This paper describes a general idea of this method and the process of studying it.

Keywords: differential equations, calculus, algorithms, mathematical physics.

By themselves, differential equations arise every time it is easier to describe a change than absolute values. For example, it is easier to describe the nature of an increase or decrease in the growth or decline of the population or population of a particular species than to describe certain values at a certain point in time. In physics, more precisely in Newtonian mechanics, motion is described by force, and force is determined by constant mass and changing acceleration, which is a statement of change.