Mathematics in Modern Technical University

Finding finite sums, products, and limits of some numerical sequences. Part 2. Application of methods of higher mathematics. Finding sums of series

2023-12-29T15:08:19+02:00

Compared to the methods of elementary mathematics, the methods of higher mathematics significantly expand the range of possibilities when finding sums and products of elements of some numerical sequences and sums of numerical or functional series. This article examines the application of de Moivre’s, Euler’s formulas and Newton's binomial, examples are given. Considering a suitable function on the segment $[a,b]$ and forming the corresponding integral Riemann sum for it, we find its value by integration. Using the Taylor & Maclaurin series of some known functions in $R$ or $C$, integrating and differentiating the corresponding power and functional series, we obtain the sums we are looking for. An example of the application of the Wallis formula is given. Expanding some functions into a Fourier series, we find for certain values of the argument the sum of many interesting numerical series. An example of using Lyapunov's equality to calculate the sum of trigonometric series is given. The use of operational calculus methods for finding sums of numerical, functional and trigonometric series is considered, an example of the use of Dirac's $\delta $-function and its properties is given.

Comparison of the results of calculating sin1 ° using the al-Kashi method and modern technologies

2023-12-29T19:44:28+02:00

The article discusses methods for calculating the value of sin1 ° using the al-Kashi method and interpreted object-oriented programming language Python. The calculation of the value by each method is presented and the comparison of the obtained results is presented. The convergence of the al-Kashi method using Banach's theorem has been established.

The philosophy of music of Pythagoras

2023-12-29T20:10:39+02:00

The article reproduces the image of the great thinker Pythagoras - one of the most popular scientists and the most mysterious personality, the philosopher.

Pythagoras created the brightest and most modern "religion": he nurtured in humanity a belief in the power of reason, the belief that the key to the mysteries of the worldview is mathematics. Music for Pythagoras became not only a means of inspiration but also a subject of scientific research, it was in music that Pythagoras found direct proof of his statement: "Everything is a number." 2,500 years ago, Pythagoras guided people on the path of triumph of the Mind. The whole world, Pythagoras argued, is a harmonious number. And these numbers form the ratio as well as the intervals between different degrees of scale. From time immemorial, numbers seemed to people to be something mysterious. Any object could be seen. The number cannot be touched and, at the same time, numbers really exist, because all objects can be counted... Pythagoras and his followers believed that everything in nature is measured, everything is subject to numbers, and to know the world means to know the numbers that control them.

If before Pythagoras, music was understood magically and understood as the embodiment of the forces of nature, used mainly in ritual and religious rites, it is Pythagoras who became the progenitor of the mathematization of the musical phenomenon.

The main grain of Pythagorean world harmony is the idea of harmony in a mathematically ordered whole. Pythagoras came to this idea when he discovered that the basic harmonic intervals: octave, pure fifth and pure fourth - occur when the lengths of the strings are 2:1, 3:2 and 4:3. Drawing analogies between the orderliness of the material world and the orderly mathematical relationship in music, he suggested that each planet in its rotation around the Earth emits a tone of a certain height, passing through the clean upper air - the ether. All the celestial sounds of all the planets, merging, form what is called "harmony of the spheres" or "music of the spheres."

The laws of music and mathematics are the basic essence of natural existence, according to which the universe is not only built, but also moves and develops.
The teachings of Pythagoras showed the unity of everything in the set, and the main purpose of man was expressed in the fact that through self-development man must achieve a connection with the cosmos.

Open Mathematical Olympiad of Igor Sikorsky Kyiv Polytechnic Institute, 2021. Round I

2023-12-29T20:33:55+02:00

This article provides the problems and their solutions for categories A and B of the first round of the Open Mathematical Olympiad conducted at Igor Sikorsky Kyiv Polytechnic Institute in 2020/2021 academic year. Due to the quarantine restrictions, the Olympiad was held remotely, which, on the one hand, affected the number of participants, and on the other, allowed students from other educational institutions to participate in the Olympiad.

The boundary value problems for the linear singularly perturbed systems of differential-algebraic equations

2023-12-27T20:51:30+02:00

The article considers the main stages of the development of the theory of asymptotic integration of boundary-value problems for linear singularly perturbed differential-algebraic systems. The need of developing constructive methods of approximate integration of boundary-value problems for differential-algebraic systems is due to the importance of their practical application in the theory of nonlinear oscillations, stability of motion, control theory, radio engineering, and biology.

In the present paper the authors offer a review of literary sources, which consider the methods of constructing asymptotic solutions of singularly perturbed systems of differential equations with a degenerate matrix with derivatives under the condition of stability of the spectrum of the limit pencil of matrices. It is noted that the problem of constructing asymptotic solutions of boundary-value problems for systems of this type is poorly studied, and therefore relevant. In particular, the question of the conditions for the existence and uniqueness of the solutions of these problems and the development of methods for constructing their asymptotics in various cases related to the behavior of the spectrum of the limit pencil of matrices has been poorly researched.

In memory of Victor Oleksandrovych Haidey

2023-12-27T20:24:01+02:00