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Using the same extremely noised data, two correlation methods for finding the maxima of Brillouin spectra are compared. The first method is a well-known method of correlating the received signal with the ideal Lorentzian function. In the second method, developed by the authors earlier, instead of the Lorentzian function, the same spectrum under study, but inverted along the frequency axis, is used (Backward correlation method, BWC). In addition to evaluating the accuracy of both methods, they are compared with the classical method of Lorentzian curve fitting. The accuracy of the considered methods is estimated depending on the probability of artefacts appearing in the Brillouin scattering spectra. It is shown that when the 9% probability of the artifact occurrence is exceeded, the BWC method shows better results than the other methods considered.

In this work, a new approach to solving problems of optimal control of manufacture procedures for the production of silica optical fiber are proposed. The procedure of silica tubes alloying by the Modified Chemical Vapor Deposition (MCVD) method and optical fiber drawing from a preform are considered. The problems of optimal control are presented as problems of controlling distributed systems with objective functionals and controls of different types. Two problems are formulated and solved. The first of them is the problem of the temperature field optimizing in the silica tubes alloying process in controlling the consumption of the oxygen–hydrogen gas mixture (in the one- and two-dimensional statements), the second problem is the geometric optimization of fiber shape in controlling the drawing velocity of the finished fiber. In both problems, while using an analog to the method of Lagrange, the optimality systems in the form of differential problems in partial derivatives are obtained, as well as formulas for finding the optimal control functions in an explicit form. To acquire optimality systems, the qualities of lower semicontinuity, convexity, and objective functional coercivity are applied. The numerical realization of the obtained systems is conducted by using Comsol Multiphysics.

In an era of change, business management should rely more on rigorous mathematical calculations and forecasts, rather than expert estimates, especially when it comes to distributed indicators of the company. This paper focuses on a mathematical model for the distribution of sales by their value, which is an initial-boundary-value problem for a partial differential equation of parabolic type and taking into account the stochastic nature of the sales process. The purpose of the study is to provide a mathematical justification for the methodology for calculating such a distributed sales plan, which will bring the company as close as possible to the desired distribution of sales. The methods of the theory of optimal control of distributed systems, the theory of partial differential equations, mathematical, and computer modeling are applied. Two problems of optimal control of the distributed system under consideration are formulated and interpreted in terms of the model. In the first task, the share of sales for the minimum amount plays the role of the control function. In the second task, we focus on the distribution of sales to new customers. Optimal laws are given for these functions, as well as optimality systems in strong form. An example of a numerical calculation of the model performed by Comsol Multiphysics is considered in the case when it is necessary to switch from a sales structure with dominant sales in the economy price segment to a structure in which sales in the expensive price segment prevail. Sales calculations based on the methodology proposed by the authors can be used in strategic planning of the company’s development.

Monetary savings of the population are the subject of research of many specialists, since their detailed study in total with other various economic variables allows for a qualitative analysis of the current social and financial situation and make predictions. The definition of such an economic characteristic of society as the distribution of the population by money savings in practice is a very non-trivial task because of the lack of reliable information on the accumulation of citizens. Often, rather than a distributed value, a constant indicator is used – the average level of household savings. In the case when the savings are distributed according to the normal law in society (as in the developed countries), the replacement of the distributed quantity by a constant is possible to simplify the tasks. It is shown that Perm krai has bimodal structure of society savings, what essentially differs from the normal distribution law and therefore from the mathematical point of view it is unacceptable to replace one with a constant value, but it is necessary to take it into account as a distributed one. This ensures the relevance of this study, whose goal is to identify and analyze the numerical characteristics of the economic structure of Perm krai. The main idea of the work is to apply the mathematical model of the accumulation spectrum of the society of D.S. Chernavskii for Perm krai, the numerical calculation of the model and the economic analysis of the obtained characteristics. Similar numerical calculations of the distribution of the population of Perm krai on savings based on official statistics have not been conducted previously. Methods of economic analysis, mathematical and computer modeling, methods of theories of ordinary differential equations, partial differential equations and stochastic differential equations, probability theory and mathematical statistics are applied. A numerical study of the mathematical model of Perm krai economic society structure was carried out, the most probable stationary levels of population savings in Perm krai were calculated, which approximately amounted to 10 and 63 subsistence minimums for 2016 year. Families seem to accumulate in the vicinity of these savings values, like elementary particles under Brownian motion (this is not a coincidence, since both processes are described by the stochastic differential Fokker-Planck equation in partial derivatives of the parabolic type). Stationary savings levels numerically characterize the consumption standards and the standard of living of a society and are formed on the basis of the value of the consumer basket. In future, it is planned to conduct research on the optimal control of the economic structure of society. Keywords the economic society structure, population distribution density by savings, stationary levels of savings, spectrum of the savings, savings in the Perm Region, economic groups of the population, social stratification, Fokker – Planck equation, money balance equation, mathematical model of D.S. Chernavskii. References 1. Chernavskii D.S., Pirogov G.G. et. al. Dinamika ekonomicheskoi struktury obshchestva [Dynamics of society economic structure]. Izvestiya vuzov. Prikladnaya nelineinaya dinamika [News of Universities. Applied Nonlinear Dynamics], 1996, vol. 4, no. 3, pp. 67–75. (In Russian).2. Igonina L.L. Investitsionnaya znachimost' sberezhenii naseleniya v Rossiiskoi ekonomike [Significance of population’s savings as investment for Russian economy]. Daidzhest-Finansy [Digest Finance], 2017, vol. 22, iss. 1 (241), pp. 4–18. (In Russian).3. Chernavskii D.S., Popkov Yu.S., Rakhimov A.Kh. Matematicheskie modeli tipologii semeinykh nakoplenii [Mathematical modelling of household savings classification]. Ekonomika i matematicheskie metody [Economics and Mathematical Methods], 1994, vol. 30, no. 2, pp. 98–106. (In Russian).4. Chernavskii D.S., Starkov N.I., Shcherbakov A.V. O problemakh fizicheskoi ekonomiki [On some problems of physical economics]. Uspekhi fizicheskikh nauk [Success of Physical Sciences], 2002, no. 9 (172). (In Russian). doi: 10.3367/UFNr.0172.200209c.1045.5. Ivanov Yu.N., Khomenko T.A. Problemy i metody statistiki sberezheniya naseleniya v sootvetstvii s kontseptsiyami SNS [Problems and methods of statistics of population’s savings according to the system of national accounts statisctics]. Ekonomicheskii zhurnal VSHE [Higher School of Economics Economics Journal], 1998, no. 4, pp. 508–515. (In Russian).6. Ruggles R. Accounting for saving and capital formation in the United States, 1947–1991. Journal of Economic Perspectives, 1993, vol. 7, no. 2, pp. 3–17. doi: 10.1257/jep.7.2.3.7. Spath J., Schmid K.D. The Distribution of household savings in Germany. IMK Studies, 2016, no. 50. 36 p.8. Kitamura Y., Takayama N., Arita F. Household savings and wealth distribution in Japan. In Book: Life cycle savings and public policy: A cross-national study of six countries. Ed. A. Borsch-Supan. San Diego, Academic Press, 2003. 34 p.9. Jantti M., Sierminska E., Van Kerm P. Modelling the joint distribution of income and wealth. IZA Discussion Papers, 2015, no. 9190. 31 p.10. Rimashevskaya N.M., Diskin I.E. Sberezheniya naseleniya kak istochnik investitsii [Personal saving as a source of investments]. Ekonomicheskie nauki sovremennoi Rossii [Economic Sciences of Contemporary Russia], 1998, no. 2. (In Russian) Available at: http://www.cemi.rssi.ru/ecr/1998/ 2/content2.htm (accessed 19.04.2018).11. Matovnikov M.Yu. Sberegatel'naya aktivnost' naseleniya Rossii [Savings activity of Russian population]. Den'gi i kredit [Money and Credit], 2015, no. 9, pp. 34–39. (In Russian).12. Chernavskii D.S., Shcherbakov A.V. Sotsial'no-ekonomicheskii byulleten'. 2016 [Social and economic bulletin. 2016]. Tsentr sotsial'no-ekonomicheskogo prognozirovaniya im. D.I. Mendeleeva. Moscow, Grifon Publ., 2016. 72 p. (In Russian).13. Chernavskii D.S., Suslakov B.A., Chernavskaya O.D. et.al. O sotsial'no-ekonomicheskoi strukture obshchestva [About social and economic structure of society]. Zakonodatel'stvo i ekonomika [Legistlation and Economy], 1995, no. 7-8, pp. 8–14. (In Russian).14. Gyul'mamedova G. Sravnitel'nyi analiz modelei denezhnykh i material'nykh nakoplenii [Comparative Analysis of the models of money and materials accumulation]. Aktual'nye problemy ekonomiki [Acute Issues of Economy], 2012, vol. 138, no. 12, pp. 322–327. (In Russian).15. Orudzhev E.G., Gyul'mamedova G.A. O smeshannykh zadachakh na konechnom prostranstve nakoplenii [About mixed tasks at the final space of savings]. Aktual'nye problemy ekonomiki [Acute Issues of Economy], 2011, no. 11, pp. 431–441. (In Russian).16. Krysova E.V., Shatrov A.V. Metody stokhasticheskoi dinamiki v matematicheskom modelirovanii sotsial'no-ekonomicheskikh protsessov [Stochastic dynamics methods in mathematical modelling of social and economic processes]. Matematicheskii vestnik pedvuzov i universitetov Volgo-Vyatskogo regiona [Mathematical Bulletin of Pedagogical Universities and Other Universities of Volga-Vyatka Region], 2006, no. 8, pp. 301–315. (In Russian).17. Mandel' L., Vol'f E. Opticheskaya kogerentnost' i kvantovaya optika. Per. s angl. [Optical coherence and quantum optics. Trans. from Engl.]. Moscow, Fizmatlit Publ., 2000. 896 p. (In Russian).18. Tikhonov V.I., Mironov M.A. Markovskie protsessy [Markov chain]. Moscow, Sovetskoe radio Publ., 1977. 488 p. (In Russian).19. Kurzenev V.A., Lychagina E.B. Stokhasticheskoe modelirovanie dinamiki ekonomicheskoi sistemy [Stochastic modelling of dynamics of economic system]. Upravlencheskoe konsul'tirovanie [Management Consulting], 2013, no. 5 (53), pp. 78–83. (In Russian).20. Oksendal' B. Stokhasticheskie differentsial'nye uravneniya. Vvedenie v teoriyu i prilozheniya. Per. s angl. [Stochastic differential equations. Introduction to the theory and application]. Moscow, Mir, AST Publ., 2003. 408 p. (In Russian).21. Chen Mu-Fa. Stochastic models of economic optimization. Recent Developments in Stochastic Analysis and Related Topics, 2004. doi: 10.1142/9789812702241_0007.22. Mantegna R.N., Stanley H.E. An introduction to econophysics, correlation and complexity in finance. Cambridge University Press, 2000. 162 p. doi: 10.1063/1.1341926.23. Kendrick D.A. Stochastic control for economic models. The University of Texas. 2002. Available at: https://liberalarts.utexas.edu/files/495395 (accessed 19.04.2018).24. Vladimirova D.B., Derevyankina P.O. Optimal'noe upravlenie raspredelennymi sistemami v zadachakh sotsial'no-ekonomicheskogo analiza [Optimal management of distribution systems in the tasks of social and economic analysis]. Global'nyi nauchnyi potentsial [Global Scientific Potential], 2016, no. 3 (60), pp. 40–43. (In Russian).25. Fursikov A.V. Optimal'noe upravlenie raspredelennymi sistemami. Teoriya i prilozheniya [Optimal management of distribution systems. Theory and application]. Novosibirsk, Nauchnaya kniga Publ., 1999. 350 s. (In Russian).

In the development of a strategic concept for the development of regional socio-economic systems, an important tool is their analysis and mathematical modeling. The paper deals with optimal control problem for a system describing the distribution of the region’s population by monetary savings. An example of a numerical study of the model for the region of the Russian Federation (Perm Region) was carried out using Comsol Multiphysics. To calculate it, clusters of the population of the region were defined by the level of average per capita income. Model parameters were determined on the basis of statistical data and expert estimates in each cluster. Then, stationary distributions were obtained for each cluster and for the entire region. It turned out that the distribution of the Perm Region’s population by monetary savings is a bimodal and highly differentiated one. The distribution with a less pronounced differentiation was chosen as a target. The controllable function was a function of migrants. The combination of socio-economic and migration policy decisions (including support of the needy population and attract migrants) according to the found optimal rule will reduce the differentiation of the population by monetary savings. Applying methodology of optimal control to mathematical models of socio-economic distributed systems can help the leadership of the region to make more accurate and balanced decisions and, consequently, to increase the effectiveness of the region’s management.

Fatigue tests of materials are characterized by long duration and high cost. In this regard, it is relevant to develop methods for modeling test results in the widest possible range of loads. Fatigue curve mathematical model includes the equation of the relationship between the amplitude of the stress and the durability of the samples, considering the random nature of the values of durability and the limit of unbounded endurance. At the first stage, the values of the model parameters are determined using the maximum likelihood function using real test data in a limited range of stress amplitudes. At the second stage, the problem solution is found considering the random value of the limit of unbounded endurance. Moreover, the mean estimate is obtained from the solution of the first part of the problem. The estimate of the unbounded endurance limit variance is obtained by calculation from the variance balance conditions. The results of modeling fatigue tests for aluminum alloy samples in a wide range of stress amplitude values are presented. Simulation results for determining the values of the fatigue curve left tolerance are considered.

In the paper the problems of optimal control of distributed systems described the process of production of quartz optical fiber are considered. This process includes two main stages: the alloying of quartz tubes and the optical fiber drawing from a preform. The following problems of optimal control of distributed systems are formulated: two-dimensional problem with boundary control and one-dimensional problem with distributed observation which describe the process of quartz tubes alloying, and one-dimensional problem with boundary control and distributed observation for problem of quartz optical fiber drawing. For these problems the optimality systems are obtained and the algorithms for their realization are proposed.

The problem of optimal control of a distributed system with a moving thermal source of action is solved. This approach is realized for the process of alloying quartz optical fibers in two-dimensional and one-dimensional formulations. The control problem is solved using variational approaches and convex analysis. As a result of the solution, optimality conditions are obtained in the form of systems of partial differential equations. Numerical studies have been carried out, and management functions have been obtained. Thus, a method for managing a distributed system with a moving source of influence has been proposed and implemented.

A one-dimensional model of quartz optical fibers extraction was considered and the problem of drawing parameters stability was solved. More over two numerical experiments demonstrating the states of stability or instability depending on the values of small harmonic oscillations introduced into the system, as well as on the furnace parameters were performed.

Sales volume is one of the most important characteristic of business efficiency. In a detailed analysis and planning of sales it is necessary to consider not only lumped, but also distributed indicators. For this purpose, it is proposed to use developed mathematical model of distribution of orders at their cost. The model constructed by stochastic dynamics methods and described by the initial-boundary value problem for a one-dimensional partial differential equation of parabolic type. The paper presents the model rationale and also numerical example made in Comsol Multiphysics.

An optimal control problem for a distributed system with compromise control is considered out of this work. The solvability is shown and an optimization system is obtained for the optimal control issue with boundary monitoring and boundary supervision described by equations of melts continuity and movement. An algorithm for implementing the optimal control problem is suggested and the outcome is analyzed. The issue is resolved for a few types of initial conditions used with the radius deviation function from its programmed determination.

One of the most important requirements for optical fibres as waveguiding media is uniformity. Polarisation-maintaining anisotropic fibres contain a special type of nonuniformity, which leads to polarisation cross-talk: optical power is transferred from one polarisation mode to the orthogonal mode. In this paper, we report a technique for detecting and locating such nonuniformities in a PANDA anisotropic single-mode fibre using polarised reflectometry.

A study was performed to determine the regimes to be used for nitriding titanium and zirconium, and a theoretical evaluation was made of the reaction of TiN and ZrN with products from the dissociation of quartz glass at temperatures of 2000 K. The results show the promise of using nitrides of titanium and zirconium as the material for the heaters of a furnace which is used to draw optical fibers of quartz glass.

Anisotropic, polarisation-maintaining fibres have been studied using a reflectometer and integrated optic polariser. Linearly polarised pulses were launched into the fibre under test at different angles between their plane of polarisation and the main optical axis of the fibre. A special procedure for the correlation analysis of these reflectograms is developed to enhance the reliability of the information about the longitudinal optical uniformity ofanisotropic fibres.

The form of the interface of polymer melts in bilaminar streams during flow in spinneret channels has been studied on a batch laboratory industrial set-up. Numerical calculations of the coordinates of the interface in bilaminar streams have been carried out using the previously proposed mathematical model. The results of experimental studies agree well with theoretical calculations, which indicates the adequacy of the theoretical model for a real process.

One of the stages of producing multilayer films and two-component fibers (including optical fibers) is joint extrusion of two or more molten polymers through a forming channel. Here there is layered or stratified flow [i]. Areas of steady-state stratified flow are the most completely studied where the profile of the flow rate and the position of the interface of the liquid layers is unchanged over the length of the channel [1-4]. Fewer works are devoted to studying the initial area of combined flow where, as is shown in an experiment in [5] and in theoretical finite element analysis [6], the coordinates of the interface vary considerably.

Quantitative dependences have been obtained for pressure gradients, viscosities, and position of the liquid interface as functions of the flow rates of materials in two-layer axially symmetric flow.

The isothermal flow of an anomalously viscous fluid in a screw channel is investigated with allowance for circulation flow of the fluid and the side ribs of the channel.

It has been shown that deviations in size of the capillary height exert the greatest affect on the flow rate of polymer melt through the spinneret hole at an assigned pressure drop; deviations in its diameter exert smaller effect; and deviations in the core entry angle exert the least effect.

Investigation of the formation of a stream of liquid at the entrance to a spinneret hole has shown that the size of the disturbing zone in the reservoir and the profile of liquid flow velocity at the entrance to the capillary depend on the Reynolds number and the rheological properties of the liquid.

The form of the interface in the melting of a polymer in an extruder has been studied. It has been found that it differs considerably from right-angled, and melting of the polymer takes place over its entire curvolinear surface.

Circulation currents of the polymer melt in the transverse section of the extruder screw channel, temperature fields in the polymer melt and in the solid plug, and the pressure distribution (pressure gradient) along the length of the melting zone have been investigated.

A comparison of theoretical and experimental data on pressure drop in the spinneret channel has been performed for a model spinneret channel and for standard spinnerets. Comparison of the experimental results with theoretical data has shown good correspondence of the integral flow characteristics, which indicates adequacy of the description of flow processes in channels of complex form within the framework of the mathematical model used. Comparison of the results from experimental and calculated determination of inlet pressure losses in capillaries has confirmed the agreement in character of the dependence of the Bagley inlet correction on the Reynolds number. This affords a basis for considering that the calculation method models rather well the flow of liquids in a zone of unstabilized flow in spinneret channels.

The pressure distribution of a polycaproamide melt along the length of the spinneret channel has been studied.

The effect of dissipation of mechanical energy on the temperature fields of polymer melts and solutions in the channels of spinnerets for fibre spinning has been investigated.

Nonsteady flow of a viscoelastic liquid which obeys the DeWitt rheological equation in the capillary of a spinneret for spinning man-made fibres has been studied.

The effect of polymer physical properties, rate of extruder screw rotation, and preliminary heating of the polymer on its rate of fusion have been studied.

A theoretical study has been made of the liquid pressure distribution at the entrance to spinneret holes. It has been shown that the pressure distribution ahead of the capillary and at its entry section depends on the rheological properties of the liquid and the Reynolds number.

A model of polymer melting in an extruder, based on the solution of the twodimensional nonstationary conservation equations, is proposed.

The contemporary state of the theory of melting of polymeric materials in plasticizing extruders has been discussed.

A theoretical study of the flow of a caprolactam melt in a cylindrical channel has been made.

The melting of polycaproamide flake has been studied experimentally on a 32 mm laboratory extruder under various processing regimes. The lengths of the extruder charging and melting zones have been measured, plus the cross-sectional area of the polymer plug along the length of the melting zone.

The effect of the entry cone angle of a spinneret hole on the flow characteristics of the polymer melt has been investigated. This dependence has a complex character, which is expressed in the fact that initially the pressure drop in the spinneret hole decreases with increase in the angle, then rises, and again decreases.

The form and size of the solid polymer plug in melting polymeric materials in plasticizing extruders depend on the source of heat: external, dissipation of mechanical energy, or preliminary warming of the flake.

Isothermal laminar flow of a polymer melt in a spinneret hole of complex geometric form has been examined. Liquid velocity profiles have been constructed in the directions along and transverse to the channel, as well as the distribution of the current function over the area of its cross section.

The effect of fluctuations in polycaproamide melt temperature before the spinneret on flow rate of the melt through a spinneret hole, and, consequently, on the linear density of elementary filaments, has been investigated.