Aviation
and Astronautics-2011 10-th International Conference ( S.S.Krylov, A.A.Puntus Moscow
Aviation Institute ( 4, Volokolamskoye shosse, Moskow, 125993, On November,
8-10-th, 2011 at the Moscow Aviation Institute (National Research University)
10-th International conference On October, 25-th, 1972 on a
building of the Moscow Aviation Institute the memorial board has been
established: The international conference This survey, prepared by
Members of conference Organizing Committee, is presenting the materials on some
papers of students and post-graduate students of faculty A.A.Krasnukhin, A.A.Puntus. (Moscow Aviation Institute, Forecasting problem of satellite flight on high elliptical
orbit. This survey covers
the problem of forecasting satellite flight on high elliptical orbit. This
orbits' specialty is in apogee altitude exceeding much perigee one. Satellites,
using high elliptical orbits have rather high speed at perigee, but it reduces
dramatically at apogee. When spacecraft is near apogee for ground observer it
may seem that the satellite do not move for several hours. That means his orbit
becomes quasi-geostationary. Quasi-geostationary can be located at any point on
the globe. This quality is used in southern and northern latitudes, distant
from the equator. Special mathematic
motion model, based on corresponding geo-potential and standard atmosphere
models, heeding moon and sun gravity and light pressure, is applied to forecast
satellite flight. Motion at high elliptical orbit is described in mathematical
model by a system of differential equations rectangular geocentric inertial
coordinate system. Speed vector and vector of coordinates are initial
conditions for integrating the system of differential equations. A.V.Naumov, S.V.Ivanov. (Moscow Aviation Institute, The problem of distribution of the investment capital
allocated for reconstruction of the aerospace complex. Nowadays
the reconstruction of the companies and structures of the aerospace branch is the
actual task. The aerospace complex is a foundation which provides the launch of
rockets, space groups control, maintenance of the infrastructure which is
necessary for development and effective use of products and systems of the
aerospace branch. We
present a mathematical model which is offered for investments of capital which
are assigned for reorganization of the structure of the aerospace complex. This
model is based on two-stage bi-level stochastic programming problem with
quantile criteria. A
lot of indefinite and random factors fluent on the quality of the aerospace
complex. In the offered model demand for products of the aerospace complex is
random factor. We use quantile criteria for adequate risk discount. The
quantile criteria is the level of loss in realization of the project which cannot be exceeded with given
level of probability. The
process of making a decision in this model has two stages. On the first stage
we determine the volume of investment capital allocated for development of
different branches of the aerospace complex. On the second stage we define the
optimal price for realized production. This price depends on the realization of
random demand. Participation
of the competitor on the market models by bi-level optimization programming
problem. The model supposes participation on the market of two players: the
leader (the state, the owner of aerospace complex) and the follower (foreign
competitors who produce the similar products). The follower chooses his own
price for products depending on the price which the leader can set. We
have made an analysis of such model. We have got analytical formulas of optimal
price for the production of the leader and the follower depending on the first
stage strategy and demand realization. We have got evident specie of the
criteria function of the second stage problem. This circumstance allowed us in
scalar case (in case of one branch) to build the deterministic equivalent to
the original problem. We show the decision of the problem in scalar case.
Possible methods for solving the problem in common case are discussed. A.I.Emelkin, A.A.Puntus, M.N.Mjakshina, V.I.Polezhaev. (Moscow
Aviation Institute, Parametrical
research of heat exchange and current structure in an inclined layer at face
heating. The convection and heat exchange in an inclined layer
in the presence of a longitudinal gradient of temperature (a case of heating of
a layer from an end face) are studied much less, than in case of an inclined
layer at a cross-section supply of heat. However, this case also is interesting
to the theory of a thermal convection and has applied value as can serve as
convection model at cultivation of crystals in special horizontal capacity, at
heating from below vertical slices in mines, at cooling of various chinks, at
research of corresponding currents in atmosphere, ocean, an asthenosphere
and in other cases. In this work the heating from an end face horizontal is
considered and having warmed up from below vertical layers, and also the
intermediate positions reached at turn of a layer. Calculations are executed
for the numbers of the Rayleigh limited to value 10*5 at different and
relations of the parties (L/H=1, 5, 10, 30) for a finding of typical modes and
current and heat exchange characteristics. As starting at increase in length of
a layer, and also for tests the case of square area was considered also at
different angles of slope. Even today here the interesting features caused by
interactions of convective mechanisms of the various natures are found out. The
basic attention is given revealing of parameters at which extreme values of
heat exchange and physical interpretation of features of structure of a current
and heat exchange which lead to extreme effects are realized. Calculations
are executed on the basis of Network Computer Laboratory of Institute of
problems of mechanics of the A.I.Kibzun, A.I.Chernobrovov. (Moscow
Aviation Institute, Minimization
of the CVaR by a stochastic quasi-gradient algorithm to
obtain upper bound of quantile. Generally,
there are plenty of random factors in the tasks from the fields of technology
and economics related to the optimization of resource allocation. In the
optimization problems with random factors quantile criterion (Value-at-Risk) is
frequently used. For example, in aircraft control terminal quantile is used as
a criterion (i.e. deviation from the target (the miss), guaranteed with some
probability). Practical
solution of the problem of finding the quantile's minimum is complicated by the
fact that the explicit form of the quantile is often impossible to find. A
stochastic quasi-gradient algorithm for solving this problem is also known.
That algorithm does not use the explicit form of quantile; but it requires the
convexity of quantile and fulfillment of some hardly verifiable conditions. Conditional
Value-at-Risk (CVaR) describes the average loss in
the worst cases (when the objective function is more than quantile). Finding
the explicit form of CVaR is often not easy. But that
form has some useful properties, particularly it is easy to check its
convexity. For example, if the objective function is convex by strategy then CVaR is also convex by strategy. We consider
the method of constructing an upper bound of quantile based on the solution of
the problem of minimizing the CVaR. Stochastic
quasi-gradient algorithm is applied to solve this problem. The convergence of
that algorithm is proved with unitary probability. This method
for approximation of quantile has the following useful properties: -
applicability to a wide class of
problems; -
does not require the convexity of
quantile; -
if
the level of reliability is close to 1, the upper bound is close to an exact
value. The paper
provides examples of applying the algorithm as well as some practical aspects
of building the numerical solution methods. V.G.Komov, O.N.Tretiyakova. (Moscow
Aviation Institute, Development of the controlling program
for kinematic system of the machine of a high-speed laser-beam cutting. Software
development for management system by multi-coordinate machine of a high-speed
laser-beam cutting was the purpose of the given operation. Any
machine of a high-speed laser-beam cutting represents the industrial
technological complex providing handling of various materials on physical properties
fulfills high-precision cutting preparations which are used in various
industries, including by the space. Machine basic elements are the coordinate
desktop on which preparation, the laser, the controller, the computer is
allocated. Simple kinematic systems where the coordinate desktop has one or two
levels of freedom, don't allow to reach high
speeds are sharp. At magnification of speed are sharp with usage of simple
systems strongly quality of performance of preparations worsens. For
magnification of speed are sharp it is necessary to increase capacity of
engines, and it conducts to growth of mass of setting and power inputs. For
the decision of this problem the projected plant considered in given operation
which includes system with parallel kinematics is offered. In it concerning the
main large coordinate desktop, the easy additional coordinate little table
which can make relocation with the big accelerations is moved to planes of its
parallel surface. It provides necessary accuracy of leading of preparation to a
demanded point since allows to accelerate and to brake an additional little
table quickly. Besides, such kinematic system of the machine of a high-speed
laser-beam cutting provides lowering of expenses of the electric power, i.e.
does setting economic while in service at its implementation in mass production
for cutting preparations. In
operation the algorithm of the synchronous control has been offered by similar
machines in which basis modelling of the relative driving of the main and
additional coordinate desktops of setting is necessary. The controlling program
implementing given algorithm is created. The program is written in C#, the controller
works with usage G-codes, transmission of controlling commands from the
computer to the controller is provided. The program is used at designing of the
machine with parallel kinematics for a laser-beam cutting in Open Company NPTS
V.N.Panovskiy. (Moscow
Aviation Institute, Realization of methods of global
optimization using interval analysis. While
designing aircraft structures it is often necessary to optimize certain
parameters. They are usually the weight, flight range, aerodynamic
characteristics, etc. Optimized parameters are specified by the designer from
some restrictions that appear depending on the physical formulation of the
problem or resource limitations. The solution of our problem is the object that
most closely matches the required specification - this shows the importance of
developing an effective optimization method. The aim of this study was to develop
an efficient algorithm for global optimization using interval analysis. During
the study of the possibility of using interval methods the main problems
hindering the development of algorithms were identified and subsequently solved.
The first problem was that the interval analysis works not with real or complex
numbers, it works with completely different structures - real intervals. From
that comes the necessity of redefining all operations of elementary algebra and
figuring out how to change the basic operations in order to get the right
result. The second problem is to establish a inclusion
function - a notion that is associated with real function. Making wrong
inclusion function causes algorithm handling a large number of values that
did not actually reached by the initial function. The
solution of two mentioned above problems has allowed to
create two global optimization algorithms: -
dichotomy of the direct image; -
imaginary values cutoff. The
result of the work is the development of algorithmic support and software for
new methods of global optimization. The developed methods have been tested on
standard tasks of conditional global optimization of functions with simple and
complex structure of isolines. In all cases, the answer was found with a given
accuracy and in an acceptable time, which confirms the possibility of using the
apparatus of interval analysis. Thus, the obtained results allow us to hope for
the successful application of the methods for the design of aircraft structures.
In the future we plan to improve the developed methods by creating new algorithms
for constructing the functions of inclusion, and develop new ones, using both
sets of interval analysis and other mathematical sciences. G.U.Shevchenko, O.N.Tretiyakova. (Moscow
Aviation Institute, Development of elements of software
for process control of laser cutting. This
work is a continuation of work on the creation of programs for automation of
technological processes and systems to create innovative laser technologies. In
[1] was described software for technology systems for the enterprise In this paper
we presented the software for controlling the laser processing equipment with
computer numerical control (CNC), which was developed for SPC Ltd. The software
is intended for general machine control laser cutting, to load into memory
controller technology programs, to run these programs, for monitoring the
treatment and diagnosis for the general state of the control system. Program interface
is a standard application window of the operating system MS Windows. For
control used standard input-output (keyboard and mouse), it is also possible to
connect additional devices, such as machine control, encoders, touch panels,
etc. This software
is designed and used on the machines for laser cutting, Ltd. established in SPC
References 1. O.N.Tretiyakova, G.U.Shevchenko.
Development of control program to install laser controlled thermo-splitting
semiconductor materials. In Sat: 9 International Conference D.V.Metlitskaya. (Moscow
Aviation Institute, Application of
the modified method of artificial immune systems to an optimal control problem
of discrete dynamic systems. The
method of artificial immune systems (AIS) concerns met heuristic methods of
search and is the representative of evolutionary methods. AIS uses ideas
borrowed from immunology, simulating work of immune system of a live organism. AIS finds the decision which can differ a little from
optimum, but thus is comprehensible from the practical point of view. Unlike
classical methods of optimization, AIS can be applied, when almost completely
there is no information on character and properties of investigated function. In the given
work application of modified AIS of search of a conditional global extremum to
an optimal control problem of the discrete determined systems is considered.
The detailed algorithm of the decision of a task in view is offered. On its
basis the complex of the software is generated, which allows to solve a task in
view for various models of control systems and various functionals. The complex
has the convenient interface with which help it is possible to change easily
problem and method parameters, and also to analyze overall performance of a
method. Some test
examples are solved by means of the developed complex. On examples it is shown
that the offered method allows to select parameters in such a way that the
received approximate solution coincide with the optimum with enough accuracy. AIS is applied to a wide range of
problems of designing of difficult aerospace systems, including optimal control
problems of trajectories of aircrafts of various types. E.A.Aleshina. (Moscow
Aviation Institute, Bacterial Foraging Optimization for optimal control of
discrete deterministic systems. Bacterial Foraging Optimization (BFO) proposed by Y.Liu and K.M.Passino is a swarm
intelligence met heuristic; search of the minimum is done by set of agents,
then the collected information is aggregated and it gives pivots for search on
the next iteration. The
main principles of BFO are the following: -
the search is done by a
colony of bacteria (particles); -
every particle tends to
reach areas with better values of criterion and avoid noxious areas where the
criterion value is worse; -
search direction is randomly
chosen; -
particles interact, attracting
and repelling to each other; -
some bacteria with the worst criterion values restart search from a random
point in the decision space or from the location of the currently best
solution. To
apply BFO to optimal control problem of discrete deterministic systems a
control vector was optimized; a control criterion was chosen as a function to
minimize. To find criterion value firstly the state vector was recalculated and
then based on its value and control vector the desired was evaluated. Based
on BFO algorithm, a program for solving the mentioned problem was created.
Graphical user interface of the utility is convenient for putting in problem
characteristics and setting up the parameters of algorithm. The program allows
user to make series of experiments with the same algorithm parameters in one
click. This is especially important since the nature of BFO is stochastic. Solving of sample problems shows that it's necessary
to examine different modifications of BFO to have the results acceptable for
practice use. This examination should take into account the experience of other
met heuristics, and especially of Particle Swarm Optimization as it's among the
most effective ones. I.E.Evdokimov, V.S.Nikolaenko,
G.S.Filippov, B.Yu.Jashenko. (Moscow Aviation Institute, Indicatrix mathematical modelling of the
difficult radiator. Objective of this
research is working out the method and on its basis the program of calculation
of difficult radiators in which the radiant stream from diffuse radiation sources is formed by
system mirror, diffuse and diffuse-mirror reflectors with the set
optical characteristics. The developed method is
received on the basis of an iterative-zonal
method and the programs used in computer graphics for definition of light
exposure of objects. It allows to count radiation of warm surfaces taking into
account various reflections and to define distribution of radiant streams in
space taking into account easing in the environment. The design
procedure is realized in the form of the program in language Borland DELPHI. It's algorithm
was checked by means of computing MathCAD environment. The developed technique
yields good results at calculation of distribution of a stream of radiant
energy from geometrically difficult figures which surfaces are set in the form
of a set of points with coordinates (X, Y, Z). Reliability of the
developed technique is confirmed by numerical and modelling experiment with use
of light model of a radiator. In numerical
experiment good coincidence of results of calculations by the developed
technique with the data of analytical indicator calculations, mentioned in
reference books is received. In modelling
experiment measurements of indicator of the ax symmetric light model consisting
from diffuse of the
radiating ring and system of formation of a radiant stream, executed in the
form of two coaxial mirror reflectors () were spent. Good
coincidence of results of calculation to the data of modelling experiment is
received also. Result of the
executed research is the package of the applied programs, allowing to spend mathematical modelling of difficult radiators. N.F.Kudimov, O.N.Tretiyakova. (Moscow Aviation Institute, Turbulence Models study for analysis of thermodynamic and gas-dynamic
processes in solar-battery cells. As well as in the
previous works [1, 2] we consider a
solar panel is installed on an object
moving at subsonic speeds on the
ground surface. The real
engineering application is the car
equipped with the solar panels used as primary power supply
replacing the internal combustion engine. In [1]
has been shown that an inefficient
heat abstraction has occurred in air flow separation zones. To
improve the cooling there are offered to turbulize somehow an
air flow in order to increase the effective thermal
conductivity, and therefore the task of turbulent
flow's analysis using
various turbulence models becomes an
important. For the purposes of analysis of heat
transfer in solar cells
there has been resolved additional dual
problem consisting of the
calculation of radiative-conductive heat
transfer in solar cells and
calculation of gas-dynamic flow near the
surface of solar cell by using various turbulence
models. (Spalart-Allmaras, k-ε, SST, RSM, Cebeci-Smith, Baldwin-Lomax). The analysis of numerical
experiments allows authors to drawing
up the conclusions about the feasibility of a
particular model. References 1. N.Vinnichenko, I.Uvarov, I.Znamenskaya. Analysis
of convective flow on solar cells cooling. ISFV14 - 14th International Symposium on Flow Visualization,
2.
N.F.Kudimov, O.N.Tretiyakova. Mathematical model of the radiative-conductive heat transfer in solar cells in
convective heat transfer conditions. In Sat: 9 International Conference
O.V.Ryazantseva, A.V.Panteleyev. (Moscow Aviation Institute, Application of adaptive greedy method of random search
global extremum to optimal control of discrete systems. A characteristic
feature of the solution of topical problems of the design of complex aerospace
systems is the widespread use of global optimization met heuristic methods,
allow one to obtain an approximation to the extreme To implement this method, a
program was developed. It implements the following features: input parameters
of the problem, input parameters of the method, the solution of the problem;
tabular and graphical presentation of the solutions, and view the protocol
solutions and help information. It was decided a few test cases with this
program help. The results indicate the effectiveness of the adaptive greedy
method of random search global extremum in optimal control of discrete systems. O.N.Tretiyakova. (Moscow Aviation
Institute, Experience in the development and application of
Internet technologies for learning physics students MAI. Developed
by us and used for several years at the Department of Physics, MAI system of
distance learning physics students of technical university can make to the
learning process the following elements: -
carrying out remote labs in physics, -
carrying out course work in physics and disciplines of
specialization, -
an online test at the rate of Physics and disciplines
of specialization, -
organization of independent work of students, -
an electronic library
of textbooks on physics for students of technical universities. When you create all these
developments used the concept of level differentiation in teaching students of
the university. All system components are presented in an educational website
Department of Physics 801 Moscow Aviation Institute at: http://www.kaf801.ru/ P.A.Panteleev, A.G.Badalova. (Moscow Aviation Institute, Software
package Developed a package
of programs in MathCAD allows you to define the best strategy for risk
management in the enterprise aviation industry, specializing in the production
of new equipment, as well as to select the most preferred method of risk
management, depending on the level of risk. We consider the following strategies:
self-containment, a harmonized risk management strategy, risk transfer, and
implement their methods: avoidance, transfer, distribution, funds reservation,
association, localization, diversification, limiting the compensation, warning. To solve
the problem generated by the authors applies the algorithm of the analytic
hierarchy process. Compiled by the hierarchical model, which includes four
levels of hierarchy: I - the risk of the enterprise; II - risk levels (low,
below average, average, above average, high); III - risk management techniques;
IV - Risk Management Strategy. Based on the method of paired comparison, the
corresponding matrices are composed of judgments. The method consists in
finding the largest eigenvalue of the matrix of judgments and the corresponding
eigenvector (usually normalized). Eigenvector reflects the ordering of
priorities, and the eigenvalue is a measure of the consistency of judgments.
The final vector of influence strategies at risk (priority strategies) is
obtained by finding the product of the matrix effect on the fourth level of the
third, the third level in the second, the second level to the first. Application
of the described software program showed that, in general, for the enterprise
strategy is the most preferred strategy harmonized risk management strategy
further risk transfer and risk-taking strategy. If the risk level is low, better
to adopt a strategy of risk-taking, more harmonized strategy for management and
risk transfer strategies. If the risk level below the average, it is better to
apply the harmonized management strategy, a strategy more risk taking and risk
transfer. If the risk level of average or above average, it is preferable to a
harmonized strategy management, strategy further risks transfer and adoption
strategy. If the risk level is high, it is better to adopt a strategy of risk
transfer, then the strategy of a harmonized management and risk-taking
strategy. The results obtained can be
used in the feedback loop strategic risk management systems. S.D.Makhijanova, A.A.Puntus. (Moscow Aviation Institute, About
limiting cycles in double oscillator, raised by a
nonlinear friction. In the presented
research work mathematical modelling double oscillator with a slice-nonlinear
dry friction is spent and research of corresponding model is conducted, and
also problems of modelling of phase portraits double oscillator n the personal
computer are considered. The method of research of considered mathematical
model consists in the analytical decision of the independent system of the
differential equations received as a result of modelling, modelling of the
given system on the computer further is spent, then the corresponding numerical
method gets out, the program of calculation with use of computer facilities is
made and realization of program calculation on the computer facilities is
spent. As a result modelling process on the computer of corresponding phase
portraits is spent. In the course of performance of the given scientific work
the mathematical model double oscillator with a slice-nonlinear dry friction
has been constructed, the analytical decision of the equations in two areas is
resulted. Analytical and numerical modelling of the given system on the
computer is considered, corresponding phase portraits are received as a result
of the given research. Limiting cycles are received by the following technique.
On pressing of button So, in the given research work
the analytical decision received as a result of mathematical modelling double
oscillator independent system of the differential equations is passed. Besides,
in the given research work the method of research of considered mathematical
model double oscillator and as a result applications of this method construction
of limiting cycles in such double oscillator, raised by a nonlinear friction is
executed is presented. For task in view realization, construction and research
of considered limiting cycles is executed, modelling of the given system on the
personal computer is spent, the choice of a corresponding numerical method is
realized, the necessary program of calculation is made, calculation on the
computer facilities is realized and phase portraits double oscillator
corresponding to various entry conditions and in various considered areas of a
phase plane are received. S.S.Leonov, A.A.Puntus (Moscow Aviation Institute, A.N.Vasilyev. ( The methods of neuron network modelling at the
solution of typical problems of hydrodynamics. The purpose of the present
work is efficiency checking of the methods of neuron network modelling at the
solution of two typical problems of hydrodynamics. In the first problem (the Couette flow) there considered flow of an incompressible
viscous fluid in horizontal layer of the given thickness. The lower bound of
the layer is motionless, and upper moves with the constant horizontal velocity
directed along the axis of abscissa. The axis of ordinates is directed
vertically upwards. The stationary solution is searched. We suppose that the
flow is flat. Moreover, the flow is horizontal. Also there is no horizontal
pressure gradient. In the second problem (the
Poiseuille flow) there considered flow of an incompressible viscous fluid in
the layer of the given thickness and the given length with solid boundaries
under the influence of the pressure difference enclosed to the edges. As well
as in the previous problem we are searching the stationary solution. The flow
is flat and horizontal. Process of solution of these
problems implies the realization of the following algorithm: -
assignment of domain in which the solution is being
searched (definition of sub-domain with the ratios changing, definition of
boundary sections with the changing of the admissible condition type); -
assignment of operators appearing in the equation,
initial-boundary conditions, other relations and constraints; -
selection of the neuron decomposition to find the
solution (the monolayer perceptrons are used at the
solution); -
assignment of the kind of generation of test points
with possibility of the assignment of different laws of allocation of these
points depending on geometry of the area; -
assignment of the error functional as a characteristic
of the model quality; -
minimization of the error
functional in enrolment of the test points. The present algorithm was
realized in MathCAD 14 system. The result of work is: -
receiving the approximate solution for Couette flow and Poiseuille flow; -
receiving inaccuracy of the approximate solutions
considered problems via meaning of the error functional; -
visual comparison of both
exact and approximate solutions. Yu.O.Khan, A.I.Fedyushkin,
A.A.Puntus. (Moscow Aviation Institute, Mathematical modelling of a natural convection at the
normal and lowered gravitation. The purpose of the
given work was construction of mathematical model for calculation of a thermal,
concentration and thermo-concentration convection in dimensionless variables at
various numbers of the Rayleigh. The problem of a natural convection of a
viscous incompressible liquid in the square closed area with heat-isolated horizontal
walls and with the set temperatures on boundary vertical walls is considered.
The mathematical model of the given problem represents the two-dimensional
equations of Navier-Stokes in dimensionless variable Boussinesq in approach. For speed on all borders the sticking
condition is laid down, and for temperature such conditions are laid down: the
bottom and top horizontal borders - heat-isolated, and the right and left
vertical borders - isothermal. As entry conditions the motionless liquid with
linear temperature distribution is set. For modelling of a considered
convective problem dimensionless record of the equations which includes
dimensionless parameters - numbers of Prandtl, the Rayleigh, by Grashof,
Schmidt was used. The system of the equations of Navier-Stokes in approach of Boussinesq was used. Calculations have been
spent with use of modern hydrodynamic program Fluent. Comparison of the
received results with The concentration model of a convection is defined by thermal number of the Rayleigh
and Schmidt's number. The field of concentration under the influence of a convection starts to change with smaller numbers of the
Rayleigh, than a temperature field as Schmidt's number is more than number of
Prandtl. The number of Prandtl isn't enough, that is conductivity of heat much
more, than viscosity, therefore a field of temperature under the influence of a
liquid starts to change only at enough great number of the Rayleigh (more than
100). And when the temperature field invariably, a concentration field already
considerably changes. The conclusion from here follows that concentration
heterogeneity under normal conditions is less, than in the conditions of the
lowered gravitation. As a result of the given work
dependence of a thermal and concentration convection, influence of number of
the Rayleigh is received and influence of changes of numbers of Prandtl and
Schmidt on liquid movement in the closed volume is investigated. Sergey Sergeevich
Krylov, the dean of faculty information systems and
technologies, information systems and technologies, program engineering,
problems of preparation of qualified professionals. Artur Agafonovich Puntus,
Professor of department of differential equations of the Moscow Aviation
Institute ( the qualitative theory and applied
problems of differential equations, problems of teaching of courses of higher
mathematics, problems of preparation of qualified professionals. |
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