Applied Mathematics and Control Sciences

The idea of constructing a special tensor to describe the parameters of structurally inhomogeneous materials arose from a number of attempts to quantitatively characterize the microstructure of an elastic porous material. The using of special tensor quantities to describe the stereometric characteristics of structurally anisotropic materials makes it possible to express in a compact form the significant structural parameters of the studying objects. The predominant orientation of pores inside the sample is well described by means of the fabric tensor and the anisotropy tensor which algebraically related to the fabric tensor. The mathematical calculations presented in the work made it possible to formalize the process of calculating all the parameters necessary for constructing the structure tensor. Algorithmization of the method for determining the mean intercept length formed the basis for the developed specialized software for calculating the components of the fabric tensor. Verification of the software module was carried out by conducting a stereological study of a number of idealized test structures and a sample of porous material for which the fabric tensor was known in advance. The obtained results did not contradict natural reality, coincided with previously data and described the anisotropy degree of the studied structures with a high degree of accuracy. To demonstrate the practical using of the developed software package, the work presents the results of studying of the human femoral neck trabecular bone tissue sample and the autoclaved aerated concrete sample. Calculations of all necessary parameters were carried out and images of the fabric ellipse of the studied porous materials were presented. From the results it is clear that the structure tensor is capable to describe the stereometric characteristics of natural and artificial porous structures, and the software package allows us to automate the process of determining all necessary parameters.

The paper develops the theory of robust parameter estimation of statistical models using the apparatus of information theory. The approach of A. M. Shurygin based on the model of a series of samples with random point contamination (point Bayesian contamination model) is considered. Perhaps, stoyki (steady) estimators have the most interesting properties among the estimators proposed by A. M. Shurygin. Although this approach can be associated with F. Hampel's approach to robust estimation, the need to postulate a parametric form of the contamination point distribution when finding stoyki estimators does not allow this to be considered a robust procedure. In the first part of our work, a non-parametric method of selecting said distribution is proposed - by maximizing Shannon entropy in the neighborhood of the model distribution limited by the value of Kulbak - Leibler divergence. This way of finding the distribution density of the contamination point allows us to consider the resulting estimators as robust, and, moreover, having the optimality property. We call the obtained estimators generalized radical, since their special case is the radical estimators of A. M. Shurygin. Generalized radical estimators are widely known in foreign publications as estimators of the minimum logarithmic density power divergence (gamma-divergence), while the question of their optimality is not investigated there. Generalized radical estimators include some popular estimators of the location: the estimators of Meshalkin (Welsh), Andrews, Smith, Bernoulli, Tukey's biweight estimator, Huber-type skipped mean and the generalized Charbonnier estimators. Also in the first part of the work it is proposed to use the cross entropy functional. Cross entropy, used as an optimized functional instead of Shannon entropy, allows us to get a family of estimators with the widest range of values of the parameter specifying this family. Since the problem of maximizing the expectation of the loss function of the maximum likelihood estimators in the point Bayesian contamination model is reduced to the problem of maximizing the cross entropy, the generalized radical estimators can be interpreted as being defended against malicious modifying of the maximum likelihood estimates. In the second part of the work, another optimal solution is obtained on the basis of the formalism of A. Rényi (or the formalism of C. Tsallis equivalent in terms of our problem) that gives a new family of estimators, the special cases of which are also some well-known estimators. To select one estimate from a family defined by different divergence constraints, an optimization approach is proposed that is similar to the approach leading to stoyki estimators, but, unlike the latter, remaining non-parametric. The main theoretical results obtained in the paper are illustrated in its second part by the example of location estimating for the cosine distribution.

A scheme for constructing direct adaptive control systems with an implicit reference model for linear continuous objects is being developed. The key condition for the convergence of the tunable parameters to their ideal values is the constancy of the "rich" harmonic excitation of the system. In real systems, this requirement is difficult to fulfill because the setting effects must be worked out by the system over a limited time interval. The desire to obtain fast parametric convergence in adaptive control systems is the engine of research in the field of hybrid adaptive systems. In this paper, hybridity is understood in the sense that the control object is continuous, and the controller, or any part of it, is discrete. A scheme for the synthesis of a hybrid adaptation circuit is proposed, in which the continuous part of the algorithm uses as a parameter a discrete estimate of the ideal adjustment matrix of the regulator. The task is to obtain a parametric error model in the form of a linear regression equation with respect to the parameters of the ideal setting of the regulator. A quadratic criterion of parametric mismatch is introduced. The corresponding minimization problem according to the desired estimates leads to the least squares algorithm, which can be used both in one-step and recurrent form. It is shown that the recurrent least squares algorithm can be used in this part. Stability analysis is carried out by the Lyapunov function method. The conditions under which a closed system is dissipative are determined. The purpose of this work is to show the advantage of the hybrid adaptation circuit over the traditional one. To do this, the theoretical results are supplemented with examples of numerical modeling. The dynamics of the system is compared with the traditional continuous adaptation algorithm and with the hybrid algorithm. In both cases, the Lyapunov function under study is non-increasing. This reflects stability of the system. However, in a traditional system, the convergence rate is significantly lower, which leads to a larger coordinate error. In a system with a hybrid adaptation contour, there is a relatively fast both coordinate and parametric convergence into an admittedly small dissipation zone.

This article is devoted to the current problem of searching for patterns in large volumes of statistical data. The tool for data analysis is regression analysis. When constructing regression models, researchers often strive only for their high quality of approximation. But, as noted in modern scientific works, such a metric alone is not enough. Therefore, interpretable machine learning is actively developing today. Previously, the author proposed a definition of a quite interpretable linear regression, and the problem of its construction was formalized as a mixed integer 0-1 linear programming problem. Research has revealed the high efficiency of the developed mathematical apparatus in solving problems of big data processing. Therefore, it was decided to expand the proposed technology for constructing quasilinear regressions. The article gives a definition of a quite interpretable quasilinear regression, which includes 6 conditions. An algorithm has been developed for interpreting the influence in the estimated quasilinear regression of monotonically transformed explanatory variables on the dependent variable. The problem of constructing a quite interpretable quasilinear regression is formalized as a mixed integer 0-1 linear programming problem. It is shown how to select the acceptable limits of the parameter M in this problem. To demonstrate the performance of the proposed mathematical apparatus, the problem of modeling the compressive strength of concrete using data containing more than 1000 observations was solved. For this purpose, the VInter-2 program was used. The constructed model included the following transformed variables: cement-water ratio, blast furnace slag, plasticizer and concrete age. The constructed regression turned out to be better in terms of the quality of approximation and simpler in the structure of the existing model. An interpretation of the constructed quasilinear regression is given. The influence of explanatory variables on the strength of concrete in it is consistent both with the substantive meaning of the problem and with other existing mathematical models. The technology proposed in the article for constructing quite interpretable quasilinear regressions has high potential for solving problems of big data processing in various subject areas.

Non-stationary effects in the operation of a two-channel RQ system with the ex-change of requests during the service process are considered. The study is aimed to identify the features of the functioning a hybrid call center with parallel service by a human dispatcher and an intelligent voice bot. Requests that do not obtained service are transferred to orbits, from where retry calls are made. After a random time, determined by the patience of clients, requests leave the orbit. An intelligent bot has a higher service rate compared to a human dispatcher. The flow of input requests is distributed randomly between service channels. It is taken into account that some clients during the course of service may prefer to switch to an alternative dispatcher. During the operation process, the human dispatcher has short breaks for rest and a long break for lunch. At these time periods, maintenance is performed entirely by a bot. The intermittency of service modes leads to non-stationary effects in the form of a current modification in the throughput of the QS. Incoming requests are described by Poisson processe when the service is open. Numerical calculations are based on discrete event simulation modeling of the system. At each step, the model time is increased by a fixed step amount. The state of the system changes randomly over a time step, as a Markov process, i.e. the transition probability depends only on the current state of the QS and does not take into account memory effects. A simulation modeling algorithm is proposed, based on dividing the entire period of operation of the QS into small intervals, when the probabilities of changes are small. The presence of alternatives in the operation of the QS leads to branching of the process. The statistical nature of the functioning of the QS is taken into account in the ensemble of realizations of the computer model and is characterized by the dispersion of the results. As an example, a simulation of a call center for a housing management company was carried out and showed the prospects of increasing the share of intelligent bots in servicing calls.