Vol 48 Iss. 3

The two-element arithmetic-geometric mean algorithm, which constitutes an example of approximating a multivalued transcendental function of one variable by means of an algebraic one, was introduced into consideration by C. F. Gauss.

... The function M (a, b) is a homogeneous infinitely multivalued function of its arguments. The properties of the function M (a, b) were specified by C. F. Gauss. However, the results of his research, published in posthumous editions, represent only separate notes, not connected by unity of presentation, and it is difficult to judge from them the form of a complete presentation of the subject. The latter has prompted numerous comments to reconstruct a coherent statement of the properties of the arithmetic-geometric mean, and a number of studies to improve or modify certain details of such a reconstruction. The properties of the function M (a, b) follow directly from the properties of the algorithm defining it. Meanwhile, the proposed statements of the properties of the arithmetic-geometric mean go beyond the defining algorithm and involve considerations that are superfluous to the task at hand.

Let S^(k), k=1, 2, . . . . - class of functions...

Differential equations of parabolic type describe many diverse problems from physics and engineering. Their solution, represented in the form of the classical Fourier expansion, is suitable only at large and average values of time. Meanwhile, a number of specific technical problems require consideration of short-time processes. There are already some partial results in this direction.

Consider a function F(x, y), where y = y (x), satisfying the following conditions...

It can be shown that the well-known Newton's algorithm αі = αi* — ∆, which serves to refine the root of a polynomial, is a representative of a related group of algorithms...

The problem of approximate quadrature is one of the most studied problems in analysis. Emerging from the needs of computations associated with the solution of various applied problems, it has been extensively developed and has been the subject of numerous studies. The works in this field are exhaustive, but the study of the outside world poses new problems where seemingly everything is known. An example is the approximate quadrature formula for a complex function F(y₁, . . . , y_n)...

There is no optimal (in the sense of mean square error) interval of Govt interpolation by one, two, three, etc. equal to T ordinates of the “pure” signal x (t). The smaller T, the smaller the mean square of the error. If an interference n (t) is added to the signal, then for one-point (step) interpolation Topt also does not exist at any combinations of x (t) and n (t). However, already for two-point (linear) interpolation, the interval Gopt, which gives the minimum of RMS error, exists, and linear interpolation of discrete measurements can be more accurate than continuous measurements. At the same time, linear preemptive extrapolation does not improve the accuracy.

Integrators are widely used in many modern automation and counting and solving devices. They are used, in particular, to determine the coordinates of the location of some object if its velocity is known. Often the characteristic input signal of a dynamic system, or one of its components, is a random process. Then the output is also a random process, the parameters of which must be determined. We come to the formulation of such a problem, in particular, when determining the random component of the error at the output of the integrating drive.

When processing experimental data, the problem of identification and exclusion of sharply deviating results of observations is of essential importance. The works of many authors are devoted to this issue, offering various criteria (statistics) for evaluating sharply deviating results. However, as noted by E. Gumbel and Yu. V. Linnik, most authors are limited to the case of normal distribution of the results of observations.

The modern development of rock mechanics has put forward the important task of studying the effect of strain rate on rock properties. The study of rock fracture processes under dynamic loading is of great importance for the improvement of mining technology. Mechanical properties under dynamic loading is important to know when designing mining machines, drilling wells at high speeds, destruction of rocks by explosion.

Determination of the largest dynamic forces in hoisting ropes is associated with the study of oscillatory processes in a variable length thread with a load at the end. Differential equations of longitudinal vibrations of an elastic weighted thread were first derived and solved by N. P. Neronov. These equations and their exact solution obtained by the method of characteristics are complicated and inconvenient for practical application. However, brought to a numerical example, they can be used to compare approximate methods with the exact solution of the problem .

The problem of radial vibrations of a hollow infinite cylinder, on the inner surface of which a variable pressure f(t) acts, is reduced to the solution of the differential equation...

Consider the problem of calculating the natural frequencies of axisymmetric oscillations of a thin hemisphere of radius R with mass M, rigidly fixed at the equator. The system of equations in displacements describing the oscillations of the shell element has the form...

Calculation of the integral from the square of the Lejandre function...

We consider the motion of a system consisting of two solid bodies rotating around a fixed point and having a common geometric axis of proper rotations. We study the uninterrupted rolling of the first body of the system on the inner conical surface of the stationary body (bowl). Such motion corresponds to the working and idle modes of the crusher. Due to the difficulties arising in the study of the motion of a system of solid bodies rotating around a fixed point, a number of constraints have to be introduced in the problem formulation. For the limiting case, the differential equations of motion of each body of the system separately and of the system as a whole are drawn up.

In many cases, especially for systems of high order, when solutions of differential equations and algebraic stability criteria become very cumbersome, frequency study is more convenient and illustrative, especially in those cases, when for some links of the automatic system, easily amenable to modeling, it is easier to experimentally remove the frequency characteristics than to make differential equations of dynamics.

Consider the dynamic elastic field in the explosion of an elongated charge parallel to one of the exposed surfaces of a rock massif ledge. Mathematically, such an effect can be described quite accurately by direct and reflected from the exposed surfaces perturbations caused by a source moving with detonation velocity such as a center of expansion.

An unsteady axisymmetric unspun flow of incompressible ideal fluid in the absence of mass forces is considered. The problem is solved in a cylindrical coordinate system (g, f , z), the origin of which coincides with the center of a circular outlet hole of very large diameter, the z-axis with the axis of symmetry of the flow, the position of the plane f = 0 is indifferent due to the axial symmetry.

In the study of some problems of streamline bodies with supersonic gas flow, it is more convenient to write down the system of hydromechanics equations in natural coordinates rather than in Cartesian coordinates. Similar coordinates were used in solving problems of the flow of flat and axisymmetric bodies with hypersonic gas flow by the “boundary layer” method.

Let us consider a two-component medium, for example, a mixture of gas or liquid and suspended particles. We will take into account the interaction between them according to the linear or quadratic law. The fluid motion is assumed to be one-dimensional (vertical), steady and laminar. The flow extends wide enough in horizontal directions, so that no vertical walls are considered. Of particular importance is the choice of continuity equations. In the author's works, an attempt was made to use continuity equations specially derived for the case under consideration, but the difficulties of extending them to three-dimensional space forced the author to switch to the generally accepted continuity equations.

Let us consider the issue of planning ore extraction of a given composition on the example of the Kingisepp phosphorite mine. The whole complex of open-pit mining of the Kingisepp phosphorite deposit is characterized by the complexity of interrelationships between technical, mining, geological and economic factors.