Vol 36 Iss. 3

In April 1957, the Academy of Sciences of the USSR together with the Academy of Sciences in Berlin celebrated the 250th anniversary of the birth of its famous fellow member Leonhard Euler. The birthday of the great scientist was remembered by mathematicians all over the world and more than one of them dwelt on his works. The Leningrad Mining Institute, within whose walls, according to legend, the great mathematician used to visit, also remembered him.

In the present paper we study elementary properties of the number-theoretic function x (n) introduced by us.The function x (n) (defined below) is related to the variational treatment of various problems of additive number theory. With the help of x (n) we construct inequalities which are satisfied for the whole set M of rational positive integers, but turn into equalities for some subsets of the set M. Thus the extreme attribution of prime numbers, prime pairs, etc. is carried out.

Recently, the formal apparatus of delta functions, i.e., discontinuous functions defined by equalities, has been increasingly applied to the solution of various problems of mathematical physics ...

In a number of branches of engineering - in gravitational concentration of minerals, some processes of chemical technology and others, we meet with the constricted motion of bodies in a medium with resistance. At constricted motion conditions of streamline and resistance forces depend on such factors as body dimensions, fluid properties, proximity of neighboring bodies or vessel walls, etc.

As follows from the theoretical analysis of the process of motion of a ball in a viscous fluid in constricted conditions, the regularity of the constricted motion of a ball along the axis of a cylindrical tube is described by the relation ...

The study of the mechanism of plastic deformation is a very important problem. The mechanism of plastic deformation is influenced by a number of factors. In the present work, an attempt has been made to establish the effect of strain rate on plastic deformation. Asbestos fibers were taken as the material under test. Asbestos fibers have bonds much stronger within the fiber than between the fibers. Hence, such a fiber approaches a one-dimensional chain. The study of the mechanism on such material is of undoubted interest.

Methods for calculating yields and compositions of functions in gravity concentration have been developed for fine and coarse crushing. Calculation of coal crushing and its subsequent gravity concentration was carried out for coarse and fine crushing by different methods, which is a shortcoming of the theory. In addition, for some intermediate cases it is sometimes unclear which method should be used, and calculations by the coarse and fine crushing method give drastically different results.

The layered discharge is still a mystery to physicists. Experimental data obtained by various investigators in recent years have contributed greatly to clarifying the question of the origin of strata. However, there is still no unified theory explaining all the bizarre forms that the stratified discharge takes under different discharge conditions. Our hypothesis of the connection of strata with the group of waves of fluctuations of the plasma potential has met with a number of objections printed in the discussion paper by V. V. Potemkin and M. E. Herzenstein. The objections are basically summarized as follows: 1) the wave group of high-frequency plasma oscillations is strongly attenuated in space and cannot give a stable stratum pattern over the length of the gas-discharge tube; 2) the wave packet is blurred and cannot form a stable stratum pattern in time; 3) high-frequency plasma oscillations are very weak, detected only in the cathode region of the discharge and not always.

The problem of bounded plasma oscillations was solved by D. Bohm and E. Gross and R. L. Stratonovich. Gross and R. L. Stratonovich. D. Bohm and E. Gross did not use the kinetic equation, but proceeded from the equation of motion of the electron. In addition, they assumed that the boundary conditions at both electrodes were the same. R. L. Stratonovich took the kinetic equation for electrons as a starting point, but the boundary conditions at the electrodes were also assumed to be the same. Therefore, he met an insurmountable difficulty in solving the kinetic equation.

Passivation of metals in the process of corrosion occurs by adsorption of oxygen and other elements and compounds (adsorption passivation) and by the formation of surface films representing chemical compounds of metals (phase passivation). Adsorption passivation can be considered as the first stage of phase passivation.

Application of Poisson's solution. The problem under consideration has been the subject of study by many authors, but, despite the results achieved, is still quite far from its complete resolution, especially from the point of view of the efficiency of the applied methods. Meanwhile, for practical applications this side of the matter is very significant. Therefore, the work of V. A. Staroverova, devoted to this question, should be recognized as very relevant.

In the present work, the question of stress distribution from own weight in a rock massif with an unsupported horizontal excavation of circular cross-section and a free day surface is considered.

In this paper we consider the motion of a system consisting of three solid bodies rotating around a fixed point and having a common axis of self-rotation.The chosen system gives a schematic diagram of the working part of a crusher of the Mekhanobr type.Differential equations of motion of a solid body rotating around a fixed point were derived for the first time, as is known, by L. Euler, who also pointed out one of the cases when the solution under any initial conditions is reduced to quadrature. Further fundamental results in this matter were obtained by Lagrange, S. V. Kovalevskaya, A. M. Lyapunov and others.

When considering corrugated couplings, along with the study of their elasticity, it is also important to find out the significance of various displacements arising in them under the action of bending moments applied to the coupling flanges. The most significant among these displacements is the largest deflection of the coupling. The greater the total length of the coupling and the distance between the corrugations, the greater the coupling deflection. Excessive coupling deflections are undesirable because they contribute to misalignment and have a detrimental effect on the bearings of the shafts to be coupled.

Due to inaccuracies in the alignment of the shafts to be coupled, as well as due to misalignment during operation, it is necessary to consider the possibility of significant misalignments that are detrimental to the operation of the machine. To reduce the harmful effect of these misalignments, the cylindrical part of the coupling is provided with one or more corrugations that increase the elasticity of the coupling. The higher the elasticity of the coupling, the lower the force required to deform it, and consequently, the easier it is to ensure normal operation of the machine in the presence of misalignments.

Suppose we have a moving medium - a liquid or a gas under conditions when the latter can be considered an incompressible liquid. In this medium there are a large number of small particles, which we consider as solid bodies - material points, the position of which we will define by their coordinates, attributing to each particle some small mass and some small volume common to all particles. For definiteness we will assume that all particles have a spherical shape and motion different from the motion of the above-mentioned medium. The mass distribution in the medium will be continuous, while the mass distribution of the particles will be clearly discrete.

As is known, ductile strength is characterized by the shear resistance of a material Sshf, and brittle strength is characterized by the breakaway resistance of a material Sref. Under a complex stress state, the ductile and brittle strength criteria of a material are functions of all three true stresses S1, S2 and S3. If the functions from S1, S2, S3, expressing shear and tear resistance, are chosen correctly, then the mentioned strength criteria for a given material and its physical state before loading will be constant, independent of the type of stress state ...

In the author's paper “On One Regularity in the Location of Central Projections on the General Plane of a Picture", the proof of a theorem expressing the regularity of the relationship between the images of the elements of space obtained by projecting them from two different centers onto the general plane of the picture is given. The proved theorem is formulated as follows.

In the works of the author “Application of the method of rectangular coordinates to the solution of some problems of photogrammetry” and “About one regularity in the location of center projections on the common picture plane” the essential sides of the question of establishing regularities of connection between different in nature methods of projection adopted in descriptive geometry are stated.

Yield strengths in tension and compression are very important strength characteristics of plastically deformable materials with both large and very small plasticity. These characteristics of mechanical properties are necessary for strength calculations in bending, torsion and other types of complex stress states of structural elements and machines, the material of which resists tension and compression unequally.