Loe raamatut: «All sciences. №3, 2023. International Scientific Journal»

Sultonali Mukaramovich Abduraxmonov, Foziljon Oripovich Obidov, Ibratjon Aliyev, Sanjar Sadiqjonovich Odilov, Nurmuxammad Sultonaliyevich Toyirov Toira Abdusalyamova ja teised

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Authors: Aliyev Ibratjon, Abduraxmonov Sultonali Mukaramovich, Odilov Sanjar Sadiqjonovich, Toyirov Nurmuxammad Sultonaliyevich, Obidov Foziljon Oripovich, Abdusalyamova Toira

Editor-in-Chief Ibratjon Xatamovich Aliyev

Illustrator Ibratjon Xatamovich Aliyev

Illustrator Obbozjon Xokimovich Qo'ldashov

Illustrator Sultonali Mukaramovich Abduraxmonov

Cover design Ibratjon Xatamovich Aliyev

Cover design Ra'noxon Mukaramovna Aliyeva

Acting scientific supervisor Sultonali Mukaramovich Abduraxmonov

Economic Manager Farruh Murodjonovich Sharofutdinov

Economic consultant Botirali Rustamovich Jalolov

Proofreader Gulnoza Muxtarovna Sobirova

Proogreader Abdurasul Abdusoliyevich Ergashev

Proofreadfer Ekaterina Aleksandrovna Vavilova

ISBN 978-5-0059-9245-1 (т. 3)

ISBN 978-5-0059-5900-3

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PHYSICAL AND MATHEMATICAL SCIENCES

THE IMPORTANCE OF DIFFERENTIAL EQUATIONS IN THE STUDY OF GENERAL LAWS AND THE SIMPLEST CASES OF TRANSFORMATION

Aliev Ibratjon Khatamovich

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

Ferghana State University, Ferghana, Uzbekistan

Аннотация. Изучение окружающего мира непосредственно сводит к необходимости ведения тех или иных прогнозов, которые сводятся уже к важности установления для них основных законов мироздания, которые можно наблюдать в ходе изучения тех или иных явлений. При этом часто использование физических законов, возможные для описания с использованием не только обычных уравнений, но и дифференциальных уравнений, первого и вторых порядков, в том числе и большого количества уравнений в частных производных, довольно часто используемых при этом исследовании и понимании.

Ключевые слова: дифференциальные уравнений в частных производных, обыкновенные дифференциальные уравнения, математическое моделирование, аналогия, закономерности.

Annotation. The study of the surrounding world directly reduces to the need to make certain forecasts, which are already reduced to the importance of establishing for them the basic laws of the universe, which can be observed during the study of certain phenomena. At the same time, there is often the use of physical laws that are possible to describe using not only ordinary equations, but also differential equations of the first and second orders, including a large number of partial differential equations, quite often used in this study and understanding.

Keywords: partial differential equations, ordinary differential equations, mathematical modeling, analogy, regularities.

Coming to the study of the laws of the world in physical science, certain laws were most often distinguished, the initial ones among which are the mechanical laws created by Newton and developed mathematically from his side, along with other scientists, among whom the figure of Leibniz stands out vividly. For an example of this statement, we can give the differential forms of the basic equations of motion (1), which in turn are reduced to certain values in the formulas of acceleration (2), force (3), work (4), power (5) and others.

The present moments of understanding can most often be considered precisely in differential forms of meaning, for the reason that they can be numerically determined by introducing some transformations, namely, by converting (6) and taking a certain integral with the establishment of certain boundaries (7).

Such directions are developed not only in mechanical terms, but also in other branches of physics, electrostatics, electrodynamics, magnetostatics, magneto-dynamics and others can be a vivid example of this. To prove this, it is enough just to mention that the very concept of current strength is a derivative of the charge time, and voltage is a derivative of the work charge.

This statement can be given for a large number of very different understandings, but the important fact is that such an approach, unlike classical mathematical regulation, becomes the only one when it is necessary to describe the gravitational characteristics of space on the scale of the entire space. An example of this kind of phenomena, where the use of derivatives and, accordingly, differential equations becomes known quantum physics.

However, on the scale of phenomena where the classical mathematical apparatus can no longer perform its functions, it is not so much the usual classical derivatives that are reduced to ordinary differential equations that are important, unless, of course, the simplest cases are not taken into account, a vivid example of which is overcoming the potential well of a particle or describing its motion, or other similar trivial cases, only partial differential equations are more interesting.

Used literature

1. Pontryagin L. S. Ordinary differential equations. – M.: Nauka, 1974.

2. Tikhonov A. N., Samarsky A. A. Equations of mathematical physics. – M.: Nauka, 1972.

3. Tikhonov A. N., Vasilyeva A. B., Sveshnikov A. G. Differential equations. – 4th ed. – Fzimatlit, 2005.

4. Umnov A. E., Umnov E. A. Fundamentals of the theory of differential equations. – Ed. 2nd – 2007. – 240 p.

5. Charles Henry Edwards, David E. Penny. Differential Equations and the problem of eigenvalues: Modeling and calculation using Mathematica, Maple and MATLAB = Differential Equations and Boundary Value Problems: Computing and Modeling. – 3rd ed. – M.: "Williams", 2007.

6. Elsholts L. E. Differential Equations and Calculus of Variations. – M.: Science, 1969.

SOME OPERATIONS AND SPECIAL CASES OF MATHEMATICAL ANALYSIS IN THE EXPONENTIAL SET

Aliev Ibratjon Khatamovich

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

Ferghana State University, Ferghana, Uzbekistan

Аннотация. Важность определения и преобразования ингенциальных чисел и настоящего множества с каждым днём становится всё более очевидном, особенно с входом данного понятия в математическую физику, но и как чисто математический объект они представляют не малый интерес, хотя при этом имеют и практическое применение. В настоящей работе, описаны методы проведения некоторых алгебраических операций с ними, в том числе с использованием формулы Эйлера и интеграллами.

Ключевые слова: ингенциальные числа, математический анализ, алгебраические операции, формула Эйлера, интегрирование, производные.

Annotation. The importance of defining and converting exponential numbers and a real set is becoming more and more obvious every day, especially with the entry of this concept into mathematical physics, but as a purely mathematical object they are of no small interest, although they also have practical applications. In this paper, methods of performing some algebraic operations with them are described, including using Euler’s formula and integrals.

Keywords: inertial numbers, mathematical analysis, algebraic operations, Euler formula, integration, derivatives.

The very process of logarithmization of an exponential number of a general form can be seen in (1).

Thus, when logarithming, 2 parts of the expression itself are formed – the real one, as the natural logarithm of the coefficient of the ingential part and the logarithm of the ingential unit, which is defined in (2).

That is, in this case, the question arises to what degree it is necessary to raise the Euler number so that it gives an exponential unit. The answer is quite simple – it is a negative logarithm of zero (2) from this it follows that the logarithm of the exponential number is (3).

It is also interesting to solve the Euler equation with a tangential unit, and then with a general form of an exponential number, which was described further, taking the expressions as unknowns. And for this, we can initially proceed from Taylor expansions (4—6).

Which is easily proved, since when the unknown is zeroed, the sine in (5) is also zeroed, and the cosine in (6) is equal to one. And it already follows from this (7).

And the unknown in (7) can be all kinds of numbers, both complex, when substituting which the remarkable Euler equality follows, and exponential. And to begin with, let’s consider a special case with an exponential unit and perform the following transformations (8).

Based on this relation, we perform transformations in (9), leading to equation (10), while taking into account that this expression is identical, it is possible to differentiate both parts of the equation in (11) by performing the corresponding transformations.

Since the final equality (11) can be represented as in (12), further carrying out additional differentiation, also introducing the condition that this is an identity, and in (13) the differentiation process for the right side of equality is described in detail. And for the left part there is no need for a detailed painting.

When the differentiation is made, it is enough to make elementary transformations, we get the trigonometric form of the special case (14).

Now, when the general form for the doubly differentiated case is obtained, it is necessary to return to the primordial ones, because this is the identity, resulting in the following equalities (15—16).

And indeed, this value is close to the most potential value, so this expression can be considered the second kind of writing of the exponential unit. Now, it is possible to proceed to the solution of the Euler equation for the general form of the intentional numbers, having carried out the first substitution and the usual replacement operations at stage (17) and (18) at the beginning.

When the necessary transformations come to an end, and other actions no longer take place, it is also sufficient to differentiate both parts of equality as a valid identity (19).

Differentiating the first part of the equality, we can come to the result in (20), and for the second part, the calculations will continue throughout (21).

Then, applying (22—25), one can come to the form (26).

As a result, it is enough to equalize both results in (20) and (26), since these are two parts of the identity, and then get (27) with the necessary simplification, and already in (28) with additional simplification and differentiation as an identity.

At the same time, the differentiation of the first part of equality is obvious in (29), as well as the second in (30), after which equality and the resulting transformations can be introduced into (31).

As a result, equalities are formed that need to be integrated twice, because their derivatives were taken earlier, getting (32).

Integrating the first part, a separate result is obtained in (33) and integrating the second part in (34).

Thus, it is possible to arrive at equality (35), from where it is possible to arrive at another equality in the same equation.

The result is really quite surprising, but this is equality (35), which came out after substituting the general form of an ingential number into Euler’s formula and the solution for this case is the ingential number (36). Thus, this is the first full-fledged equation, the solution of which was an intentional number.

Although the complex numbers themselves are located on the axis of numbers, this interval can also be expressed on the tangential plane. This coordinate system has an axis starting from infinity as the ordinate, and the abscissa has all real numbers. Thus, all exponential numbers can be represented on such a rectangular coordinate system, in the case of adding complex numbers – already in space.

Used literature

1. I. V. Bargatin, B. A. Grishanin, V. N. Zadkov. Entangled quantum states of atomic systems. Editorial office named after Lomonosov. 2001.

2. G. Kane. Modern elementary particle physics. Publishing house Mir. 1990.

3. S. Hawking. The theory of everything. From singularity to infinity: the origin and fate of the universe. Publishing house AST. 2006.

4. S. Hawking, L. Mlodinov. The supreme plan. A physicist's view of the creation of the world. Publishing house AST. 2010.

5. T. D'amour. The world according to Einstein. From relativity theory to string theory. Moscow Publishing House. 2016.

6. S. Hawking, L. Mlodinov. The shortest history of time. Amphora Publishing House. 2011.

ABOUT RESEARCH ON THE COLLATZ HYPOTHESIS IN THE FACE OF A MATHEMATICAL PHENOMENON

Aliev Ibratjon Khatamovich

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

Ferghana State University, Ferghana, Uzbekistan

Аннотация. Когда об этой задаче рассказывают молодым математикам – их сразу предупреждают, что не стоит браться за её решение, ибо это кажется невозможным. Простую на вид гипотезу не смогли доказать лучшие умы человечества. Для сравнения, знаменитый математик Пол Эрдеш сказал: «Математика ещё не созрела для таких вопросов». Однако, стоит подробнее изучить данную гипотезу, что и исследуется в настоящей работе.

Ключевые слова: гипотеза Коллатца, числа-градины, ряды, алгоритм, последовательности, доказательства.

Annotation. When young mathematicians are told about this problem, they are immediately warned that it is not worth taking up its solution, because it seems impossible. A simple-looking hypothesis could not be proved by the best minds of mankind. For comparison, the famous mathematician Paul Erdos said: «Mathematics is not yet ripe for such questions.» However, it is worth studying this hypothesis in more detail, which is investigated in this paper.

Keywords: Collatz hypothesis, hailstone numbers, series, algorithm, sequences, proofs.

In short, its essence is as follows. A certain number is selected and if it is not even, it is multiplied by 3 and 1 is added, if it is even, then divided by 2.

We can give an algorithm of this series for the number 7:

7 – 22 – 11 – 34 – 17 – 52 – 26 – 13 – 40 – 20 – 10 – 5 – 16 – 8 – 4 – 2 – 1

Next, a cycle is obtained:

1 – 4 – 2 – 1 etc.

This leads to the hypothesis that if you take any positive integer, if you follow the algorithm, it necessarily falls into the cycle 4, 2, 1. The hypothesis is named after Lothar Collatz, who is believed to have come to this hypothesis in the 30s of the last century, but this problem has many names, it is also known as the Ulam hypothesis, Kakutani's theorem, Toitz's hypothesis, Hass's algorithm, the Sikazuz sequence, or simply as "3n+1".

How did this hypothesis gain such fame? It is worth noting that in the professional environment, the fame of such a hypothesis is very bad, so the very fact that someone is working on this hypothesis may lead to the fact that this researcher will be called crazy or ignorant.

The numbers themselves that are obtained during this transformation are called hailstones, because, like hail in the clouds, the numbers then fall, then rise, but sooner or later, all fall to one, at least so it is believed. For convenience, we can make an analogy that the values entered into this algorithm are altitude above sea level. So, if you take the number 26, then it first sharply decreases, then rises to 40, after which it drops to 1 in 10 steps. Here you can give a series for 26:

26 – 13 – 40 – 20 – 10 – 5 – 16 – 8 – 4 – 2 – 1

However, if we take the neighboring number 27, it will jump at a variety of heights, reaching the mark of 9,232, which, continuing the analogy, is higher than Mount Everest, but even this number is destined to collapse to the Ground, although it will take 111 steps to reach 1 and get stuck in the same loop. The same interesting numbers can be numbers 31, 41, 47, 54, 55, 62, 63, 71, 73, 82 and others. For comparison, we can analyze the table (Table 1) and the graph (Fig. 1) for these interesting numbers.

Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Fig. 1. Graph of values for interesting numbers-granules of the algorithm

When the path of one number is so much different even from the neighboring one, how do you even approach the proof of such a hypothesis? Of course, all mathematicians were at a loss and absolutely no one could solve this problem. So Jeffrey Lagarias is a world expert on this problem, and he said that no one should take up this problem if he wants to become a mathematician. A large-scale work was carried out and a huge number of hailstones were studied, trying to find a pattern. Here it can be argued that all values come to one, however, what can be said about the path that all numbers take? The interesting thing is that this path is absolutely random.

For example, we can give a graph of all the values of this algorithm from 1 to 100 (Fig. 2).

Fig. 2. Graph of values for numbers-granules from 1 to 100

As you can see, most often growth begins initially and after a sharp decline, while the value of the number is simply not considered, however, if you make the graph logarithmic, there is a downward trend in its fluctuations. It can also be observed on the stock market on the day of the collapse, which is not accidental, because these are examples of geometric Brownian motion, that is, if you take logarithms and calculate the linear component, the fluctuations seem random, as if a coin was thrown at each step. And if we consider this function analysis as part of mathematical analysis, then there begins to be an obvious connection with probability theory. From where it turns out that when heads are obtained, the line goes up, and when tails go down, from where a special graph is obtained.

If we consider this chart when compared with the same exchange, then it is more likely in a short-term analysis, although in the long term, stocks are still growing, and "3x+1" is falling. You can also pay attention to the highest digit of the gradient numbers – this means a histogram, which is obtained by counting the number of digits from which the numbers begin in a number of granules for a particular number of the algorithm. If you add these values each time, for 1, 2, 3, etc., more and more data is obtained, while the ratio of the height of the columns becomes more and more ordered.

So for the first billion sequences, the most frequent value is one, 29.94% of all cases, 2 – 17,47%, 3 – 12,09%, 4 – 10,63%, 5 – 7,94%, 6 – 6,16%, 7 – 5,76%, 8 – 5,31%, 9 – 4,7% and the larger the figure, the less often it turns out to be ahead.

This arrangement is typical not only for hailstone numbers, there are many examples, these are the population of countries, the value of companies, all physical constants or Fibonacci numbers, and much more. This law is called Benford's law. Surprisingly, if you trace the violation of the Benford law in tax returns, you can even determine the fact of fraud. This law also helps to identify anomalies in the counting of votes in elections or many other things.

The best effect of this law occurs when the numbers entered in it have a spread of several orders of magnitude, as in this case, but Benford's law, unfortunately, cannot tell whether all the numbers end up in the 4-2-1 cycle. To do this, you need to use a different method. Initially, it is strange that this algorithm reduces all numbers to 1, given that even and non-even numbers equally and non-even numbers increase by more than 3 times, and even numbers decrease by 2 times.

This suggests the conclusion that all sequences should, in theory, go up, not down. But it is worth paying attention to the fact that whenever an operation is performed with a non-even number, that is, when it is multiplied by 3 and 1 is added, it necessarily turns into an even number, therefore, the next step it will always be divided by 2. It turns out that non-even numbers are not tripled, but multiplied by (3x+1) /2 or more precisely by 1.5, because 0.5 for large numbers can be ignored. So the maximum growth from this is exactly 1.5.

A graph has already been given for all numbers from 1 to 100, but it is worth considering a small case for all non-even numbers. As you know, in the second step they turn into even values, and then exactly half of them are immediately reduced, after dividing again to non-even. But every 4 numbers will have to be divided by 2 twice, which means that these non-even numbers are ¾ of the previous one. Every 8th number will have to be divided by 2 three times to get an even number. Every 16 four times, etc.

So taking the geometric mean, you can see that in order to get from one non-even number to another through all even numbers, you need to multiply it by ¾, which is less than one, hence it turns out that statistically, this sequence decreases more often than it grows.

Let's give an example for a large number, for example 341. Its row looks like this:

341 – 1024 – 512 – 256 – 128 – 64 – 32 – 16 – 8 – 4 – 2 – 1.

He had only one non-even and all even numbers, which is why this series is remarkable. However, they can be depicted both in the form of graphs and in the form of trees, showing how one of the numbers is connected with the next in its sequence, creating a graph.

And if the hypothesis is correct, then any number should be in this huge graph, consisting of an infinite number of "streams" forming 4-2-1 cycles in one stream. There is an interesting visualization of such a graph, which uses an algorithm that on non—even numbers, it rotates clockwise at the selected angle, and counterclockwise on even numbers.

As a result, an interesting curved structure is obtained, more often in one direction. Resembling coral, algae or a tree in the wind. But this is only for a small number of numbers, for huge arrays, changing the angles of rotation, you can create huge and dazzlingly beautiful figures, as if generated by nature.

The hypothesis seems to be incorrect only in 2 cases:

1. If a number is found that will give infinity in the algorithm, that is, for some unknown reason, this "force of attraction" to 4-2-1 should not act on it;

2. Somewhere there is a sequence that would form its own closed loop, and all the numbers in it should be outside the main graph.

However, none of these options has been found yet, although all numbers up to 2 to the 68th power have already been checked by a simple search, which equals 295,147,905,179,352,825,856 numbers. It is known for sure that all the numbers from these values come to the 4-2-1 cycle. Moreover, based on these data, it is calculated that even if there is such a special data cycle, it should consist of at least 186 billion numbers. And it turns out that all the works indicate that the hypothesis is true, but still does not prove it.

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.

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