В 1954 г. в соавторстве с О. В. Сармановым и Р. Э. Соловейчиком автором была поставлена следующая стохастическая задача.
Во многих исследованиях оказывается необходимым располагать сведениями о законах распределения геометрических характеристик (размер, форма и т. п.) вкраплений, содержащихся в массе твердого вещества. Однако часто непосредственно измерение зерен, представляющих собой вкрапления, не осуществимо, так как механическое отделение их от связывающей среды без существенных повреждений оказывается невозможным (например, для электрокорунда. Между тем можно высказать косвенное суждение о распределении геометрического параметра зерна гю экспериментально найденным распределениям некоторых параметров плоских сечений вкраплений, наблюдаемых в шлифе.
В настоящей статье исследуются элементарные свойства введенной нами теоретико-числовой функции х(га). Функция х(га) (определенная ниже) связана с вариационной трактовкой различных задач аддитивной теории чисел. С помощью х(га) строятся неравенства, которые выполняются для всего множества М целых рациональных положительных чисел, но переходят в равенства для некоторых подмножеств множества М. Так осуществляется экстремальная аттрибутация простых чисел, простых пар и т. п.
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.
