Readers are offered a deep and comprehensive study of the Pythagorean theorem, one of the most famous and important theorems in geometry. The article presents twenty-five different ways to prove this theorem, covering a wide range of mathematical methods and concepts. It actually demonstrates how the Pythagorean theorem can be proven using classical geometric approaches, as well as using modern mathematical tools. Readers will be able to get acquainted with proofs based on metric relations in a triangle, trigonometric properties of angles, elements of vector algebra, properties of a circle, tangent, chord and secant, as well as more complex proofs using differential equations and mechanics. The article will be interesting and useful for a wide range of readers, including high school students, students of mathematical specialties, mathematics teachers and everyone who is interested in the history and development of mathematical knowledge.

The purpose of this work is to construct a numerical analytical solution to the problem of plane wave scattering on a model in the form of a finite wedge-shaped object with a smoothed surface. Such an object is a simplified model of an airplane wing profile. The obtained solution will make it possible to evaluate the influence of the smoothing radius and the conductivity of the smoothing surface on the dispersion characteristics of a plane harmonic wave. This article is the first part of this work. It discusses the method of partial regions, thanks to which it is possible to construct an analytical solution. The peculiarities of applying the method of partial regions specifically to this model of the object are indicated, the application of point wise conjugation of fields on the boundaries of partial regions is considered.

Polytetrafluoroethylene (PTFE) films are deposited in parallel and perpendicular magnetic fields (MF) by electron‐enhanced vacuum deposition (EVD) and EVD + low‐temperature plasma (LTP) methods. The structure, morphology, and nanomechanical properties of the films are studied by infrared spectroscopy (IRS), atomic force microscopy (AFM), and spectroscopic ellipsometry. The structure of the thicker films is closer to that of bulk PTFE than that of thin films. The films' crystallinity and surface roughness are higher than those deposited without MF. The birefringence of the refractive index (n) of the films deposited in the MF is inverse to the anisotropy of the n of the films deposited without MF. The hardness of the films is close to that of bulk PTFE.

This paper deals with investigatson of the process of drill bit whirling on the rough plane of the well bottom. Nonholonomic kinematic models of the mechanical interaction of contacting bodies with defects at the initial stage of system self-excitation are applied. On the basis of the results of experimental studies, it is believed that one of the main factors influencing on the whirlings vibrations is the geometry of the bit. The bit is considered to be an absolutely rigid ellipsoidal body, the well bottom surface is supposed to be a plane. The resulting oscillations are associated with spontaneous bending deformations of the drill string, which are accompanied by continuous contact of the bit and the rock. The equations of motion of the bit in the linear approximation are obtained. The analysis of the solution of the linearized equations is carried out, and the frequencies of the arising periodic motions are found. The forms of oscillations under different geometrical parameters of an ellipsoidal bit are analyzed. It is shown that the bit can rotate in backward direction, moving at angular velocities that exceed the angular frequency of the drill string. The research results can be used in the development of new types of drilling rigs.

A two-dimensional periodic Stokes flow of viscous incompressible fluid in a rectangular cavity with constant velocities applied to the top and bottom walls is considered. The study of advection regimes is reduced to the sequential solving of two problems. To solve the first problem, the analytical method of superposition, which allows one to obtain any desired accuracy of the velocity field, is used. To solve the second problem, which is associated with obtaining the trajectories of individual fluid particles, numerical calculations of the Cauchy problem were performed. An analysis of the boundary conditions accuracy was performed based on the control of local integration. The advection of the selected volume of fluid in the rectangular cavity under the periodic motion of walls for a finite period time based on piecewise spline cubic interpolation is modeled. The obtained numerical results agree with well-known experimental data.