br Acknowledgments The study has been conducted with the

Acknowledgments The study has been conducted with the financial support of the Russian Foundation for Basic Research (Grant no. 15-01-07923).
Introduction Studying the scattering processes of charged and neutral particles of various natures is one of the most important problems in p-Cresyl sulfate manufacturer physics. One of these processes is electron scattering by atoms and determining the respective scattering cross-sections [1–3]. The following problems should be highlighted here:
We have explored this problem in our studies [4–10], where we suggested a method of presenting the aggregate information on the studied matter. This approach was tested on a representation of the cross-sections of electron excitation of subjects such as helium [4–7] and argon [8] atoms from normal and metastable states, and of hydrogen [9] and krypton [10] from a normal state.
The theoretical base of the representation We regard the totality of all studies from which we can extract the quantity pairs v=(S, E), where is the cross-section (or some cross-section function, e.g., S=lnQ, which does not matter in view of the below, but is useful for analyzing the quantities varying in a wide range), and E is the energy, as an aggregate source of information W. Let us note that this source is an abstraction and contains the data from all possible studies, including both the published and the future ones. What we mean is that the proposed generalization method allows to easily introduce all new appearing literary data. The publication data currently existing and available to us are regarded as a sample from the aggregate source. Our task is to establish the dependence based on the extracted information. Since for real systems, there is some error in calculations and measurements, for a statistical approach, it makes sense to treat v as a random variable related to some distribution function describing the aggregate source of information on the problem in question. It would be logical to assume that the totality of all information sources should reproduce the true relationship (1). This leads us to the fact peristalsis the relationship between the random variables (1), i.e., the so-called regression of S on E, must correspond to the expectation S with the constraint E: where is the density of the conditional probability distribution. Let us examine the deviation (the residual) δ for the regression (1) with an arbitrary function σ:
The conditional expectation of the deviation (3) with the regression in the form (2) for each value of E for the aggregate information source is equal to zero:
Therefore, the total mathematical expectation of the deviation with the regression of the form (2) for the aggregate information source is also equal to zero:
The quantity of the expected squared deviation (3), namely, with the regression of the form (2) due to (4) is equal to the squared deviation variance (3):
The regression of the form (2) is known to be characterized by the minimum of the expected squared deviation (5) among all kinds of functions σreached on the function (2) [11,12], i.e.,
As a result, we arrive at the variational principle that allows finding the best approximation to the sought-for dependence (1) in the selected class of functions. Let us now examine the aggregate information source as a composite one. Let us assume that each single information source w ∈ W is described by its own distribution function of the quantities v, and denote their joint distribution by The aggregate information source is described by the marginal distribution In addition, each information source is related to a corresponding conditional distribution of the quantity v: where is the marginal distribution density of the information sources with the distribution If we assume (which we are going to do further on) that the information sources w ∈ W form a discrete (countable or finite) set, the integral should be understood as a sum