On some properties of the solutions of the linear differential-algebraic systems

It has been compared the properties of the solutions of differential-algebraic systems and the solutions of systems of differential equations. The differences between the corresponding properties of these solutions are analyzed.

Mathematical models of the evolution of classical and solitary cylindrical waves

The evolution of nonlinear elastic cylindrical displacement waves for initial profiles in the form of a Hankel and Macdonald functions is analyzed theoretically and numerically. The difference between the two waves is that the MacDonald function has no hump, decreases monotonically and has a concave downward profile, and the Hankel function is a harmonic attenuating wave. The main novelty is that the evolution of cylindrical waves is studied for two different approaches to the solution of a nonlinear equation. Some significant differences of these waves are shown. First, the features of the Hankel wave, a harmonic wave (symmetrical profile), are briefly described. Then, theoretically and numerically, a single wave with an initial profile in the form of a MacDonald function is analyzed in more detail. Distortion of the initial profile due to the nonlinear interaction of the wave itself and the increase in the maximum amplitude during wave propagation is common to these profiles. Significant features of the McDonald wave are shown - an uncharacteristic initial profile (a profile without a classical hump) evolves in an uncharacteristic way - the profile becomes much steeper and remains convex downwards. Keywords: classical and solitary cylindrical waves; five-constant Murnaghan potential; approximate methods; Hankel and Macdonald initial wave profiles; evolution.

The story of one formula

This article aims to represent the diversity of approaches applicable to a certain mathematical problem – Stirling’s approximation was chosen here to achieve the mentioned goal. The first section of the work gives a sight of how the formula appeared, from the derivation of an idea to a publication of the strict results. Further, we provide readers with six different proofs of the approximation. Two of them use methods from calculus and mathematical analysis such that properties of logarithmic function and definite integral as well as representing functions as power series. The other two apply the Gamma function due to its connection with the notion of the factorial, namely Γ(n) = n!, n ∈ N. The last two have a probabilistic idea in their core: both of them combine Poisson distributed random variables with Central Limit Theorem to yield the desired formula. Some of the given proofs are not mathematically rigorous but rather give a sketch of a strict proof. Having all the results we assert that this story can be a good example of the variety of methods that can be used to solve one mathematical problem, even though all the listed proofs use only basic knowledge from several mathematical courses. Keywords: Stirling’s formula; factorial; Taylor series

Investigation of solutions of weakly perturbed boundary value problems for systems with impulse action

The paper is devoted to the investigation of the existence and construction of solutions of weakly perturbed linear boundary value problems for systems with impulse action in the case when the generating boundary value problem with impulse action has no solutions for arbitrary right-hand sides. The relevance of this topic is due primarily to the importance of practical application of the theory of boundary value problems in various fields of science and technology - the theory of nonlinear oscillations, the theory of motion stability, control theory, a variety of geophysical problems. On the other hand, the question of the existence and construction of solutions of boundary value problems occupies one of the central and fundamentally important places in the qualitative theory of ordinary differential equations.

The obvious — the incredible. Moebius tape

The attraction to mathematics and mathematical abilities are manifested at a fairly early age.But these abilities must be developed systematically and skillfully. The path to mathematical sciencelies not through memorizing mathematical rules, but through conscious comprehension of mathemat-ical facts.If you ask any mathematician why he loves this science, he will most likely answer: “Because itis beautiful.” A little imagination and skills – and you can turn an ordinary tape into fascinatingworks of art, incredible inventions, even try to make unusual trips on it.The purpose of the article is to interest young lovers of mathematics. The article is based on aspeech to students of the Junior Academy of Sciences of Kyiv region.