## No 3 (2020)

ON ASYMPTOTIC PROPERTIES OF THE CAUCHY FUNCTION FOR AUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATION OF NEUTRAL TYPE

#### Abstract

We investigate stability of a linear autonomous functional differential equation of neutral type. The basis of the study is the well-known explicit solution representation formula including an integral operator, the kernel of which is called the Cauchy function of the equation under study. It is shown that the definitions of Lyapunov, asymptotic and exponential stabilities can be formulated without loss of generality in terms of the corresponding properties of the Cauchy function. The conclusion is drawn that stability with respect to initial data depends on the functional space which the initial function belongs to, and, as a consequence, that there is the need to indicate this space in the definition of stability. It is shown that, along with the concept of asymptotic stability, a certain stronger property should be introduced, which we call strong asymptotic stability. The main study is devoted to stability with respect to initial function from spaces of integrable functions. Special attention is paid to the study of asymptotic and exponential stability. We use the following known properties of the Cauchy function of an equation of neutral type: this function is piecewise continuous, and its jumps are determined by a Cauchy problem for a linear difference equation. We obtain that the strong asymptotic stability of the equation under consideration for initial data from the space L 1 is equivalent to an exponential estimate of the Cauchy function and; moreover, we show that these properties are equivalent to the exponential stability with respect to initial data from the spaces Lp for all p from 1 to infinity inclusive. However, we show that strong asymptotic stability with respect to the initial data from the space Lp for p greater than one may not coincide with exponential stability.

**Applied Mathematics and Control Sciences**. 2020;(3):7-31

MATHEMATICAL MODELING OF CANCELLOUS BONE TISSUE ADAPTATION APPLIED TO THE HUMAN MAXILLODENTAL SYSTEM

#### Abstract

The bone tissue in different parts of the skeleton conforms to Wolff’s law: it aims to become optimal for the loading which acts on the corresponding bone; the bone is remodelling by means of osteosynthesis and resorption mechanisms. The modern problems of biomechanics demand research on the history of formation of bone structures in the course of time at both physiological and pathological loadings. Ever changing loadings of different nature have influence on development and functioning of the trabecular bone tissue. The mandible is one of the most liable to external and internal changes bones. Very often one has to deal with pathological changes caused by incorrect loading of different regions of bone tissue due to dysfunction of a dentition, a temporomandibular joint and so on. For example, the Popov-Godon’s syndrome which connects with tooth loss is accompanied by pathological remodelling of the surrounding bone tissue. Thus, the mathematical modeling of the cancellous bone tissue behavior in the human maxillodental system is one of the most topical problems of biomechanics and medicine. Trabecular bone tissue is a heterogeneous, porous, anisotropic material. Heterogeneity of spongy structure can be described by methods of quantitative stereology. At the same time, structural features of the trabecular bone can be described by means of the fabric tensor. This is possible to implement if there is both a constitutive relation which connects the stress tensor, the fabric tensor, and the strain tensor, and kinetic equations which describe the evolution of the fabric tensor and bone density. An initial boundary value problem on the trabecular bone tissue remodelling is stated. The effective numerical algorithm allowing to solve the problem is developed. This algorithm is implemented as a complex of problem-oriented programs. Verification of the model and identification of its parameters are carried out. All numerical calculations are performed using the ANSYS software. Trabecular bone tissue evolution is demonstrated on the set of model examples when the stress-strain state is changed. The results demonstrate different character of influence of changes of loading conditions on process of structure formation which follows from Wolff’s law.

**Applied Mathematics and Control Sciences**. 2020;(3):35-48

FUNCTIONAL DATA PREPROCESSING APPLICATION TO OIL-TRANSFER PUMPS VIBRATION PARAMETERS FORECASTING

#### Abstract

This work describes functional data preprocessing algorithm. This algorithm provides a way to reduce error in forecasting problems solution. The algorithm is a part of oil-transfer pumps vibration parameters forecasting system that enables pump failures dynamics forecasting. The author analyses existing approaches to vibration monitoring and decides to solve failure-forecasting problem as a long-term forecasting problem although researchers usually solve such problems with classification methods. An insufficiency of labeled data is the main reason of such a decision. Main ideas of the problem solution are the following. Neural network model takes and calculates periodical metered values characteristics. Time is split into periods using scales with different periods. We use shorter periods for short-term forecasting and longer periods for long-term forecasting. Functional data preprocessing provides a way to increase forecasting quality. Preprocessing key idea is following. Functions sequence transforms one of model’s inputs in order to increase correlation between input and output. Metered values distributions and dependencies between values can be variant because of observed time series nonstationarity. Author decided to modify original preprocessing algorithm to solve a nonstationarity problem. Idea of the modification is to add steps that provide preprocessing robustness i.e. allow to reduce difference between preprocessing results on different datasets. Preliminary preprocessing functions selection provides robustness. There are two variants of preliminary selection. The first one is following: function with the least difference between correlations between input and output in data subsets pass the selection. The second one is following: functions that increase correlation on both subsets pass the selection. Experiments on two pumps data prove the hypothesis that data preprocessing in vast majority of cases allows to decrease forecasting error. Modified algorithm often has less test error than original one.

**Applied Mathematics and Control Sciences**. 2020;(3):51-72

ANALYSIS OF THE ECONOMIC FACTORS INFLUENCE IN THE OPTIMAL CONTROL PROBLEM OF HOUSEHOLD SAVINGS

#### Abstract

Household savings are an important component of the financial system. An increase in the number of households following the savings and investment financial strategies leads to economic growth. The financial behavior of the Russian population can be described as passive without a long-term strategy. Therefore, the task of its activating requires careful state regulation. Special significance it acquires in the middle class, which has sufficient income and seeks to rationalize in all areas of their lives. The article explores the problem of managing of middle-class households savings using optimal control methods, correlation and regression analysis, mathematical and computer modeling. A mathematical model of the problem of managing savings by regulating the non-savers share is presented in the form of a boundary control problem for a distributed system with a parabolic equation. The methodology for obtaining quantitative estimates of the impact of economic factors on the household savings distribution by their size is described. An example of determining the optimal reduction of the share of non-savers households in the Perm Region is given and its quantitative relationship with the volume of investment is established. The proposed methodology and the results of the calculations can be used in the development of programs to stimulate savings and socio-economic development of the region in order to increase the validity of management decisions.

**Applied Mathematics and Control Sciences**. 2020;(3):75-88

ON THE CONSTRUCTION OF PROGRAM CONTROL IN THE PROBLEM ON ATTAINABLE VALUES OF ON-TARGET FUNCTIONALS FOR DYNAMIC ECONOMIC MODELS WITH DISCRETE MEMORY

#### Abstract

Main constructions and relationships for program control actions are proposed as applied to the problem on attainable prescribed values of on-target functionals for continuous-discrete dynamic economic mathematical model with discrete memory under given polyhedral constrains with respect to control. The form of on-target functionals covers widely used kinds of functionals such as multipoint, integral ones and linear combinations of those. The feature of the control system under consideration is the presence of two kinds of the state variables, namely, a part of them depends on continues time, whereas others depend on discrete time. Aftereffect of the system is defined by its discrete memory located at a given collection of instants. The results are obtained on the basis of the principal statements from the general theory of continuous-discrete systems. In the constructive part of the research, the basic idea is the reduction of the original problem to a variant of the general moment problem with taking into account pointwise polyhedral constraints on controls. This allows us to construct estimates of the attainability set and to build program controls on the base of solutions to a series of linear programming problems. Every such a problem provides us with values of the program control on a partial segment. All these values are used while constructing the program control as a whole. The mentioned procedures use in essence the Cauchy operator to the hybrid system under consideration. The property of this operator are studied in the cited previous papers. The obtained results constitute an instrumental basis for efficient studying and constructing solutions to urgent applied problems with constrained resources of control.

**Applied Mathematics and Control Sciences**. 2020;(3):89-104

RESEARCH OF THE BANKRUPTCY PREDICTION RELIABILITY IN CASE TO INTRODUCE A NEW CATEGORY OF THE COMPANIES FINANCIAL POSITION

#### Abstract

The motivation for this research was the result obtained earlier by the authors in the field of developing industry models for predicting bankruptcy with high prognostic ability. The article examines the prediction reliability of the financial position of companies in the case of introducing an additional category of financial position that reflects the position between financial solvency and insolvency (bankruptcy). The authors hypothesize that the reliability of models decreases if the requirements for their accuracy increase due to the introduction of an additional category of financial position. Hypothesis testing is performed using a non-entropic approach. This approach should reduce the measure of uncertainty in terms of the uncharacteristic nature of some of the identified features of financial position relative to the initial categories. At the same time, features of financial position are defined as ranges of specific weight of balance sheet items that have positive or negative information importance. Information importance is determined based on the methods of system-cognitive analysis, implemented automatically in the EIDOS X++ system, as well as by reproducing information models using MS Excel tools. Normalization of the informational importance values of features and their interpolation allowed us to obtain functions similar to the membership functions in the theory of fuzzy sets. When constructing membership functions relative to ranges of significant balance sheet items ("Fixed assets", "Inventory", "Accounts Receivable", "Short-Term financial investments", "Retained earnings (uncovered loss)", "Accounts payable"), ranges with zero or insignificant values of characteristic functions corresponding to the initial categories of financial position are identified. This actually meant a high level of uncertainty in the prediction. The authors propose to introduce additional linguistic variables and their corresponding fuzzy sets, whose carriers are the relative scales of the above balance items, this will reduce uncertainty. A total of 5 such fuzzy sets were identified, where the researchers used the concept of "gray zone" as a linguistic variable, which was actually used as a new category of financial position. All calculations are shown on the example of fixed assets. The prognostic ability of models based on an optimized sample, where the category of the position of companies that have at least 3 out of 5 features of the "gray zone" has been replaced, is reduced, as expected, but only slightly. And in the case of reproducing algorithms of system-cognitive analysis using MS Excel tools, there is even an increase in the prognostic ability of one of the models. In fact, the hypothesis that the reliability of models decreases if the requirements for their accuracy increase was not confirmed. From an economic point of view, the theoretical significance of the obtained result is that with the help of a non-entropic approach it was possible to show the need to introduce a new category of financial position. From a mathematical point of view, the theoretical significance lies in the fact that membership functions for linguistic variables are obtained based on real data on the financial position of almost two hundred Russian companies, these reduction functions can be used by specialists in the field of fuzzy set theory in the future. The results obtained are applicable at least for the construction industry, but can also be replicated relative to other sectors of the economy when forming the corresponding samples.

**Applied Mathematics and Control Sciences**. 2020;(3):105-122

STRATEGIC MANAGEMENT OF THE REGIONS IN THE CONTEXT OF DOMESTIC AND FOREIGN EXPERIENCE

#### Abstract

The results of analyzing the approaches, concepts, programs, as well as the practices of strategic management of regional development, existing in Russia and abroad, are presented. Based on the revealed trends and taking into consideration the factors of global instability in the world, a complex of scientific provisions is worded with regard to forming of a relevant management strategy for social and economic development of regions in modern-day Russia. The suggested approach includes 10 provisions. 1. The strategy of the region social and economic development should be considered as an integral structure of subsystems taking into account the whole range of its natural, economic and social components, which constitute a unified territorial complex with multiple intricate internal and external relations. 2. It is necessary to analyze the social and economic status of the region in the totality of the fields of its production-and-economic activities; and compare the relevant set of indicators, provided by the state and regional statistics, to the available data obtained as a result of monitoring the functioning of the industrial enterprises. 3. The strategic priorities in the social and economic development of industrial enterprises in the Russian Federation entity should be formed through a system of supporting the making of managerial decisions on the region’s development, which would include the earlier mentioned methods, models and program complexes of the information-and-analytical system. 4. Russian Federation entities should elaborate the strategies of their social and economic development in compliance with the Forecast of the Science and Technology Development in the Russian Federation for the Period till 2030. 5. Since Russian Federation entities, in their essence, are integral formations with a unified wide range of natural, economic and social components, and are interested in proportionate and well-balanced development of those, it means that in order to ensure the strategic industrial and economic priorities of these regions, it is necessary to perform a complex analysis of the supporting infrastructure subsystems (engineering and technical facilities and communications, resources of all kinds, etc.) for the industrial enterprises being developed. 6. The concepts and programs of the social and economic development of regions should not only be properly elaborated and informationally tied to relevant federal programs. 7. No matter the scale of the regional development plans and the ambitions within those, as formed by the regional departments of state authorities, all of that can only be fulfilled by the population of those entities. 8. The strategy of social and economic development of any region, as a rule, comprises the programs for development of the secondary and higher professional education, the forming of which should be based on the forecasts of the demographic increase of population, development of industrial enterprises and organizations located on the region’s territories, etc. 9. It is impossible to fulfill the strategies of the development of Russian Federation entities without such an important component as public healthcare. 10. Various social programs and social protection of the population in general should be one of the crucial aspects when forming the strategies of development of Russian Federation entities

**Applied Mathematics and Control Sciences**. 2020;(3):123-149

CONCEPTUAL MODEL OF HUMAN RESOURCES MANAGEMENT AND PROJECT MANAGEMENT FOR AN SOFTWARE DEVELOPMENT

#### Abstract

The article provides an analysis of digital economy level in the Russian Federation, based on I-DESI index and software development was singled out as the most important industry for digital transformation. The importance stimulated by market growth dynamic and Russian Federation IT market product structure. An analysis of possibility using workflow based on data and data mining with according to software development specific are produced in the article. After the analysis there are conclude that digital transformation in software development are possible. Then generalized model of human resources management and project management for a software development organization is produced. Human resources management and project management in software development organization often executed manually and not effective. With this in mind produced model contains intelligence decision support system as crucial element in management and assumes using current operation tools. The model using can increase software development organization effectiveness and increase the process standardization due to base the process on accumulating and analysis data.

**Applied Mathematics and Control Sciences**. 2020;(3):150-163

COMPARATIVE ANALYSIS OF THE UNIVERSITIES CLUSTERING RESULTS FOR DIFFERENT SETS OF PARAMETERS

#### Abstract

To increase the competitiveness of domestic universities, the "5-100 Project" is being implemented, in which from 2013 to 2020 21 universities that have been selected based on the results of "Road maps", which must necessarily reflect international cooperation, as well as plans for the development of educational programs, take part. This year, it is planned to summarize the results of the program. The interpretation of results can be heterogeneous, since it is difficult to select parameters for comparison when evaluating complex multi -parameter objects. The clustering allows you to group similar objects, while the task of the researcher is to interpret the results. It is always easier to compare objects with similar parameters. As a result, two groups were obtained. This paper analyzes the problem of evaluating the effectiveness of Universities in various parameters and their totality. Clusterization was performed using the hierarchy method in the R program. This paper examines 11 universities from the list of "5-100", information about the activities of which is most fully presented on the Internet. Built according to data provided in the University self-study reports. The analysis was carried out separately for each of the sections in order to determine the most significant parameters for the University (educational, research, international, financial and economic activities, infrastructure and training of disabled people and persons with disabilities), as well as for a set of characteristics.

**Applied Mathematics and Control Sciences**. 2020;(3):164-181

ADAPTED BOX-COUNTING METHOD FOR ASSESSMENT OF THE FRACTAL DIMENSION OF FINANCIAL TIME SERIES

#### Abstract

This paper continues research within the framework of the scientific direction in econophysics at the Department of Information Systems and Mathematical Methods in Economics of the faculty of Economics of PSU. Modeling and prediction of financial time series is quite a perspective area of research, because it allows participants of financial processes to reduce risks and make effective decisions. For example, we could research financial processes with the help of fractal analysis. In the article there is studied and worked out in detail one of the methods of fractal analysis of financial time series - the box-counting method for assessment of the fractal dimension. This method is often used in studies conducted by domestic authors, but the authors do not delve into the characteristics and problems of using the box-counting method for analysis of time series, that means that the answers to the interested questions have not yet been given. The main problem is that, as a rule, the analyzed object in the tasks of applying the box-counting method to time series is a computer image of the plot of series. In the article there is proposed the procedure of adaptation of the box-counting method for assessment of the fractal dimension of time series, the procedure does not require the formation of a computer image of the plot. In the article there is considered following difficulties developed from this adaptation: 1) high sensitivity of the resulting estimation of the dimension to the input parameters of the method (the ratio of the sides of the covered by cells plane with the plot; the used range of lengths of the side of the cell; the number of partitions of the plane into cells); 2) the non-obviousness of choosing the optimal values of these parameters. In the article there are analyzed approaches to the selection of these parameters that were proposed by other authors, and there are determined the most suitable approaches for the adapted box-counting method. Also there are developed unique methods for determining the ratio of the sides of the plane with the plot. In the paper there is written the computer program that implements the developed method, and this program is tested on the generated data. The study obtained the following results. The fact of sensitivity of the adapted box-counting method to input parameters is confirmed, that indicates the high importance of the correct choice of these parameters. According to the study, there is found out inability of the proposed methods of automatic determination the ratio of the sides of the plane in relation to artificial time series. There are obtained the most precise (in a statistical sense) estimates of fractal dimension, those found by means of the adapted box-counting method, with the fixed ratio of the sides 1:1. According to comparing the adapted box-counting method and R/S analysis, there are obtained the most precise estimates by the second method (R/S analysis). Finally in the paper there are formulated the possible directions for further research: 1) comparison of the accuracy of various methods for assessment of the fractal dimension on series of different lengths; 2) comparison of the methods of fractal analysis and p -adic analysis for modeling and prediction of financial time series; 3) determination of the conditions of applicability of various methods; 4) approbation of the developed methods for determining of the ratio of the sides of the plane with the plot on real economic data.

**Applied Mathematics and Control Sciences**. 2020;(3):185-218