When studying the geology of some area often use a variety of methods. Each of them from the point of view of a particular task brings not only information of interest to the researcher at the moment, but also information of much less interest, or not essential at this stage of research ...
Let there be a function f(x), which is a realization of some stationary random function. It is required to find an approximate value of the integral ...
ПОСТАНОВКА ЗАДАЧИВ связи с созданием высокочувствительной измерительной аппаратуры стало возможным определение малых концентраций и потоков радиоактивной примеси в атмосфере. Благодаря этому возникла важная с практической точки зрения задача определения положения и мощности источников примеси, расположенных под землей, по наблюдениям в приземном слое атмосферы. В основу решения этой задачи должна быть положена теория, объясняющая распространение радиоактивной примеси в двухслойной среде земля—атмосфера.
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 ...
Let’s consider the following problem in the theory of heat conduction. An initial temperature distribution is given in a space consisting of two media separated by a flat interface. It is required to find the temperature at any point in the space, at any moment of time. The thermal characteristics of each of the two media are assumed to be constant. The formulated problem has been considered by a number of authors for the one-dimensional case. The possibility of solving the multidimensional case using integral equations was pointed out by Münz [4]. In [5], a two-dimensional problem was solved by the method of successive approximations. In the present paper, a closed-form solution of the problem under consideration is given for the two- and three-dimensional case. The solution method can be applied to a number of similar problems.
In many experimental studies, the presence of some phenomenon is established by observing a "signal", i.e. by establishing that some, as we will say, elementary event A is realized. However, this elementary event can also be caused by secondary causes not related to the phenomenon under study. In such a case, only repeated occurrences of event A will allow us to judge with sufficient certainty the presence of the phenomenon under study. Let us conduct a series of n experiments, where event A can appear with some probability p. The phenomenon of interest to us (we will call the implementation of this phenomenon event M) can be connected to varying degrees with elementary event A. It is natural to assume that there is a rather weak dependence between event M and elementary event A if, in order to obtain certainty of the realization of M, it is necessary to observe A sufficiently often in n trials (for example, at least once t times in a row). In this paper, we limit ourselves to considering only three types of dependence of events M and A, the most important for applications (see the article).
The problem considered in this paper arose in the study of the radiation properties of smoke and fog. In the first approximation, these are sols consisting of absolutely black particles. The transparency of the layer can be characterized by the value of the average area in the flow cross-section perpendicular to the line of sight, not covered by sol particles. This point of view belongs to K. S. Shifrin. The need arises to solve the following question: what is the average value of the free area of a limited piece of a plane when n "elementary" regions are located on it, closed and congruent (such an assumption means that a monodisperse sol is considered).
In this paper, we give a solution to a general problem in geometric probability theory, which is the subject of a number of issues in modern technology (aviation agricultural technology, visibility in turbid environments, etc.). Let us consider a set A of points A₀, A..., An, randomly distributed in a circle K of radius R. We assume that the occurrence of each individual point of this set in parts of the circle K equal in area is equally probable (the law of equal probability). Let, further, the number of points in the set A be related to the value of the radius R such that the limit of the ratio exists and is finite. In other words, the average concentration of points in the circle K, with n and R increasing without limit, tends to a finite limiting concentration. We will study the random variable r, which is the smallest of the distances of an arbitrary point A₀ of set A, with its random position in the circle K, to the other points of set A. The random variable under consideration will obviously be equal to the radius of a circle with its center at point A₀, which does not contain other points of set A and has at least one point of this set on its boundary.
