The arithmetic-geometric mean B (a, B) is the general limit of sequences defined by recurrence relations...

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

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)...

Whenever the study of nature moves from judgments of qualitative nature to the elucidation of quantitative regularities, we turn to science, the content of which are quantitative relations and geometric forms of the real world. Mathematical methods come to the researcher and become a powerful means of work, allowing to reveal the general laws, deeply hidden by the diversity of constantly changing phenomena and the abundance of observable facts ...

The arithmetic-geometric mean algorithm introduced by Gauss is a remarkable example of approximation of a multivalued transcendental function by means of algebraic. In Gauss's works published during his lifetime and in the remaining posthumous materials, almost no attention is paid to the convergence of the algorithm and the branching of its terms is not considered at all.

In questions related to the approximate definition of a function, one encounters the problem of constructing an approximate expression of a function by its mean values given for a number of intervals. An example of this may be drawing the equation of a distribution curve or drawing the equation of a regression line of one of two random variables on the other. To the same problem is given the finding of the distribution of a mineral in a well on the basis of readings obtained from core analyses, and a number of other sampling questions.

Широкое обобщение тета-функций дается решением дифферен­циального уравнения ...

Известная теорема Гюльдена, устанавливающая зависимость между объемом тела, образуемого вращением плоской фигуры, площадью этой фигуры и длиной окружности, описанной ее центром тяжести, является частным случаем гораздо более общего положения.

The spatial position of a curved borehole is determined from measurement data. Measurements give in a number of points Mi (i=1, 2,..., n), taken along the length of the borehole, values of inclination angles θi and azimuth ϕi. According to these data and distances St of points Mi from the wellhead, the position of points Mi and the well as a whole is determined. This determination can be made in various ways, each of which gives an approximate position of the borehole. The question arises about the evaluation of possible deviation of the obtained position of the well from the actual one and the dependence of the accuracy of determining the spatial position of the well on the accuracy of angle measurements and the number of points in which such measurements are made.

The study of the process of coal destruction by water jet in order to build a rational theory of the phenomenon is a difficult theoretical and experimental task. The complexity of the study is caused by insufficiently studied process of brittle fracture, the complexity of the structure of the fractured rock, the lack of our information about the destructive agent and its action. In such a complex' environment, it seems natural at the first steps of the research to refuse from full consideration of all factors acting in the process of fracture, to simplify and schematize the phenomenon. The study carried out in a simplified scheme gives only approximate dependences between the mechanical characteristics of the rock being fractured and the parameters characterizing the jet producing the fracture. However, the obtained dependencies, having been subjected to experimental verification, can be evaluated with respect to the accuracy of the results they deliver and the acceptability of their use in the practice of technical calculation. On the basis of experience, corrections can be made to them, taking into account the complexity of the actual phenomenon and bringing the simplified scheme closer to reality.

Mathematics and mechanics are represented in the “Notes of the Leningrad Mining Institute” by studies in various fields of analysis, geometry and mechanics. Considering the works published in the pages of the Notes of the LMI for the fifty-year period of their existence, one can get a general idea of the work of the departments of mathematics and mechanics, the orientation of these works, their nature and the results achieved.

The above transformation of series expressing theta functions has been well known for a long time. It was obtained by Jacobi in 1828 and is related to his studies on the theory of elliptic functions.

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.

When calculating mineral reserves in deposits, instead of the volume of the actual ore body, the volume of a body that is sufficiently close to it and has the correct geometric shape is usually calculated. An unsuccessful choice of such a geometric body can lead to a significant decrease in the calculation accuracy or greatly complicate the calculations. In the practice of calculating reserves with such exploration data, the volume of the explored body was sometimes calculated as the volume of a cone with a base equal to the contoured area on the horizon, and with the apex at the point where the drill well exits the deposit, without taking into account the thickness of this latter. The calculation made in this way gave reserves below the minimum determined by exploration data, and the discrepancy reached a significant value of several tens of percent. Below are methods for calculating the volume of a conoidal body, which can serve to calculate the reserves of a part of an ore deposit limited by a contoured area at a certain horizon and cut by a drill hole at depth.