СЕКЦІЯ «МАТЕМАТИКА ТА КОМП’ЮТЕРНІ ТЕХНОЛОГІЇ»
(англомовна секція)
UDC 629.78 (043.2)
Makarov I.A
National aviation university, Kyiv
DETACH THE ROCKET FROM THE BOARD OF AIRCRAFTCARRIER
The work is devoted to developing of a strategy for implementing air space launches in Ukraine on the base of the aircraft AN225 a produced in Ukraine and Ukrainian “Cyclone4” rocket. The investigation was directed on the analysis of fundamental aspects of the strategy of launch of the rocket (LV) from the aircraftcarrier. All necessary calculations and graphics were made during research work. On the basis of the work the mathematical models were made. The include graphs, which describe main parameters of rocket and aircraft trajectory. During 5 years of researching works more than 10 different modes and 5 strategies of the rocket launch from the board of aircraftcarrier was reviewed. Given mode, which is described in the research work is the most efficient and vital.
Relevance, scientific and practical novelty consist of the proposed method, of launching of the rocket from the aircraft, which includes a combination of two different approaches to launch the rocket from the aircraft. During the first part of the trajectory the rocket flies like the aircraft, using a wing and start boosters to create the lifting force and thrust correspondingly. In the second part of the trajectory the rocket overcomes by the using the braking parachute and because of it’s operation the rocket comes to the required position to launch.
The goal of this scientificresearch work is to develop mathematical models and associated software to optimize the performance of the proposed strategy for air launch.
Task of the work is to create a mathematical model of the rocket launching, as well as finding of the required characteristics to modify the launcher and aircraft. Related tasks include studying of the layout and placement of bearing surfaces on the launch vehicle and placing the rocket on the board of plane.
During the research work the author had used the method of constructing a mathematical model in the environment of visual programming "Delphi7" programming language Pascal and software MathCAD for some operations. The first mathematical model constructed by the author, describes the initial stage of the launch vehicle flight with the wing immediately after its separation from the aircraft. The second mathematical model was established to describe the flight of the launch vehicle after the shooting of bearing surfaces and braking with a spin in the necessary position for the launching. For the next time the mathematical model will be developed. As a result, the research clearly demonstrates the ability and efficiency of the proposed strategy for air launch and shows the benefits of mathematical modeling as a powerful method of optimizing design of project solutions.
Scientific supervisor – Ye.O. Shkvar, Dr. of Sci. (Engineering), Prof.
UDC 004.42.514 512.772(043.2)
D. Malinina
National Aviation University, Kyiv
THE PARAMETRIC WORLD
Mathematics is very interesting discipline for many people, but only few of them know about The World of Parametric Curves. If one asked them about this World they would call the world of trigonometric functions, such as sine or cosine. But that is not true. In mathematics, coordinates X and Y of a parametric curve are expressed through a parameter t, X=cos t; Y=sin t. Almost all the famous curves may be expressed as parametric ones, such as parabola, circle, ellipsis and others. But the most interesting and impressive ones are those that allow multivalued correspondence between X and Y [1]. The most beautiful of them are the so called Rose Curves, Hypotrochoid, Hypocycloid, Lissajous Curves, Epicycloid and such unusual curves as Butterfly Curve and Hearts.
By means of MATLAB programming, I made a Graphical User Interface (GUI) program to demonstrate beauty of such curves, right Figure. It aims also to help user in getting some knowledge about all these curves and observe their change, i.e. their “behavior”, depending on varying coefficients. Usually, there are two coefficients in equations. User is prompt to set one of them as variable, so that another one will be automatically set to be a constant. All these features are explained in additional GUI window treated as the Help (left Fig.).
During this work, I significantly improved my knowledge both in programming and mathematics. You are welcome to enjoy The World of Parametric Curves, too, by means of my program!
References

Вирченко Н. А., Ляшко И. И., Швецов К. И. Графики функций. — К., 1979.
Scientific supervisor – Ye.O.Gayev, Dr., Prof.
UDC 511.315: 517 (043.2)
Mazur M.D.
National aviation university, Kyiv
FIBONACCI NUMBERS
Mathematics is seems to be the science of indisputable facts, formulas and strict laws. However, in this conciseness sometimes hides the amazing beauty. Mathematics can describe the hidden order in familiar to us things. One of the best variants of nature and mathematics relationship is the Fibonacci sequence.
Leonardo Bonacci—known as Fibonacci—was an Italian mathematician. He is considered to be "the most talented Western mathematician of the Middle Ages."
The Fibonacci sequence is a series of numbers where the total of two consecutive numbers equals the next number in the series. For instance, 13, 21, 34 is a part of the Fibonacci sequence because 13 plus 21 equals 34. The sequence begins 1, 1, 2, 3, 5. In some instances, the sequence begins with zero.
The Fibonacci sequence was developed in response to one of the mathematical contests that he frequently participated in. One particular contest asked the question, “Beginning with a single pair of rabbits, if every month each productive pair bears a new pair, which becomes productive when they are 1 month old, how many rabbits will there be after n months?” In order to solve this problem, Fibonacci devised a series of numbers in which each number is the sum of the previous two (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...). This proved to be the first recorded instance in European scholarship of a recursive number sequence, or one in which the relation between two or more successive terms can be expressed by a formula.
This simple row can characterize many natural phenomena, such as:

arrangement in plants;

the spirals of seeds in a seed head;

the bracts of a pinecone;

the scales of a pineapple;

the growth of every living thing;

a grain of wheat;

a hive of bees;

the Univerce.
The Fibonacci numbers are also applicable in finance. Traders and investors in all markets can benefit from this timeless analysis technique, which is highly useful in determining entry and stop levels in multiple time frames and all market conditions.
In addition, if we take the ratio of two successive numbers in Fibonacci's series, dividing each by the number before it, we will find the following series of numbers:
1/1 = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.666..., 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.61538...
They seem to be tending to a limit, which is called the golden ratio.
I investigate some of the interesting applications of Fibonacci numbers.
Scientific supervisor – T. Oleshko, Assoc. Prof.
UDC 536.24:532.55 (043.2)
Meyris А.
National Technical University of Ukraine «Kiev Polytechnic Institute», Kyiv
Institute of Engineering Thermophysics, Ukrainian National Academy of Science, Kyiv
Computational modeling of heat transfer
and hydrodynamics of round pipes bunch
with grooves on their external surface
The intensification of heat transfer in a circular pipe continues to attract the attention of researchers and engineers in development of new power plants and efficient methods of thermal energy recycling. Different methods of pipes profiling (oval, elliptical, droplike, flatoval tube, etc.) and application of artificial roughness (ribs, ridges, sand roughness, etc.) have been recently used for the external heat transfer enhancement.
Nowdays the intensification of heat transfer through the use of different types of roughness, particularly the use of spherical dimples on the pipe surface causes the special interest. This type of roughness is characterized by increase of heat exchange with a slight increase of hydrodynamic drag, while providing portability and efficiency of heat exchange equipment. The use of grooves on the outer side of the tube bundles regenerators and associated projections on their inner surface intensifies the heat transfer on both sides.
During the experiment two bundles were studied: bundle of smooth pipes and bundle of pipes with spherical indentations on the surface formed by stamping staggered spherical recess with relative depth h/d = 0,3. The outer tube diameter was D = 22 mm, the relative lateral spacing was S_{1}/D = 1,7, the relative longitudinal spacing was S_{2}/D = 1,2. The experiments were conducted in the range of Reynolds numbers from 5000 to 16000, which corresponds to the transition flow regime for smooth pipes.
In theoretical part, threedimensional numerical fluid flow and heat transfer analysis were performed using commercial CFD code ANSYSCFX 14.0. The 3D model included upstream section, round pipes bunch with grooves and downstream section. The total number of grid points was around 27.4 million. The boundary conditions at the inlet and outlet were set close to experimental data. Overall 4 RANS turbulence models were tested; the models of Menter SST group were presented by standard SST model and SST γθ model. Also, two Reynolds Stress models (RSM) were tested, SSG and LRR models in particular. Comparison showed a good correspondence of experiemental data obtained with calculated results using Menter SST γθ turbulence model.
Scientific supervisor – A.A. Khalatov, Academician of the NAS of Ukraine
UDC – 519.866(043.2)
Mironov Yuriy
National Aviation University, Kyiv
STANDARD DEVIATION AND PARETO PRINCIPLE IN ECONOMICS
In this research we consider the Pareto princople and mathematic models in ABCXYZanalysis.
Pareto principle is a concept implemented in a certain relation between efforts and attained result. The common definition of the Pareto principle is: “20% of efforts result into 80% of the result”. It is widely spread in the majority of spheres of the economics and manufacturing. Now we are going to consider this implementation of the Pareto principle: “20% of the goods bring 80% of the profit”.
ABCXYZanalysis is a method of goods sorting according to their profitability and purchase frequency. After the revision, the goods are grouped by two attributes: АВС by profitability, where А are the most profitable and С are the least profitable goods, and XYZ by the demand, where X are the most predictable demand, and Z – is the least predictable demand.
Attachment to the groups A, В or С is defined according to the difference between buying and selling price. We consider every profitmaking item and calculate its net income. Then we derive a percentage worth of every profitmaking item. That is the way how the most profitable items are defined.
Attachment to the groups X, Y or Z are defined according to the standard deviation formula. Initially we derive an average quantity of goods sold in some time period. After this, standard deviation formula is applied to this mean value. RMS difference divided by an average value represents a variation coefficient, by which goods are sorted by XYZ attributes.
де:
Using the results of the research a company is able to optimize its assortment and price policy, that is to increase its income. This method is splendid to increase company's efficiency because it is based on mathematical calculations and lets analysts come to an exact reasonable solution.
Scientific supervisor – T.A. Oleshko, Assoc. Prof.
UDC 519.62(043.2)
Negodenko O., graduate student
National Aviation University, Kyiv
THE SMOOTHNESS EFFECT OF TRIGONOMETRIC INTERPOLATION
SPLINES ON THE INTERPOLATION ERROR
Recently splines are increasingly used in various fields of science and technology. Most often they are used to describe complex dependences, in computer graphics and more. Also splines are known for good approximating properties as well as for simplicity and efficiency of calculations.
The advantages of polynomial splines over other linear approximation devices showed up in the problems of interpolation functions. In many important cases interpolation splines provide the lowest possible error of on certain classes of functions.
However, polynomial splines have a number of disadvantages which to some extent constrain their use in various problems of science and technology. These disadvantages include the difficulty of constructing splines of higher degrees. Therefore, new classes of functions make an interest that have the advantages of polynomial splines and are free from the disadvantages of these splines. These new classes of functions include trigonometric splines classes.
The aim of the research was the following: to investigate the influence of differential properties of trigonometric interpolation splines on interpolation error.
In the project there was chosen function on . There were given interpolation nodes,, where , step of analytical grid , where . There was calculated the value of function in interpolation nodes. Next, there was built trigonometric interpolation spline on these nodes.
Parameter determines the smoothness of splines, as trigonometric splines of order have an absolutely continuous derivative of () order (). That’s why the practical use of these splines faces the problem of choosing the order of the spline and its influence on the interpolation error.
The focus was on cases where the functions and , were selected in the role of testing examples. In the interval there were given 9 interpolation nodes and there was found the value of functions in these nodes. Trigonometric interpolation spline was built on these nodes, and there was calculated interpolation error having different values of parameter. While comparing these errors it appeared that for two testing examples with the increase of the order of spline the interpolation error increased.
This is explained by the fact, when using trigonometric splines to approximate functions generally at the ends of interpolation segment the wellknown Gibbs phenomenon takes place that significantly degrades approximation quality.
Thus, we can conclude that to reduce errors it is appropriate to use special methods in order to diminish the harmful effects of Gibbs phenomenon.
Scientific supervisor – V.P. Denisiuk, Doc. phys. math. sc., Prof.
UDC 621.45.038(043.2)
Panchenko N.А.
National Technical University of Ukraine «Kiev Polytechnic Institute», Kyiv
Institute of Engineering Thermophysics, Ukrainian National Academy of Science, Kyiv
COMPUTER SIMULATION OF THE FLAT PLATE FILM COOLING VIA COOLANT BLOW THROUGH THE DOUBLE JET HOLES
T
Fig. 1 – The double jet holes configuration.
he film cooling is one of the primary techniques to protect blades of highpreformanced gas turbines. The improvement of thermal efficiency of gas turbines can be achieved by increase of the cooling efficiency via application of advanced cooling technologies. Traditional injection of cooling jets through inwall inclined cylindrical holes forms the «kidney pair vortices» transporting a hot external flow underneath the cooling jet and leading to the cooling jet separation from cooled surface. As known, secondary flow is not destroyed by means of the shaped holes instead of cylindrical ones. Moreover, the complex hole shape leads to some technology constraints and high production cost. In all novel film cooling technologies is very important to reduce the influence of secondary flow, to distribute a coolant more uniformly in the spanwise direction, as well as to minimize the coolant mixing with external hot flow. One of the promising technologies is the double jet film cooling configuration with complex angles of coolant supply, which prevents the cooling jets separation and improves the coolant spanwise distribution. As a result, more uniform film cooling effectiveness can be expected in this case.
The main relative geometric parameters of the investigated double jet cooling configuration (Fig. 1) are: t/d = 4, t_{1}/d = 3.125, Δ/d = 1, α = 30º, β_{1} =β_{2} = 30º.
Results of the film cooling experimental studies over a flat plate using double jet holes configuration at the inlet have shown that at low (m = 0,50) and medium (m = 1,0) blowing ratio the cooling effectiveness of this configuration is greater of the traditional twoarray inclined cylindrical holes configuration, however corresponds it at m = 1,5.
In theoretical part, threedimensional numerical fluid flow and heat transfer analysis was performed using commercial CFD code ANSYSCFX 14.0. The 3D model includes plenum, double jet cooling holes and main flow duct. The total number of grid points is around 3.8 million. The boundary conditions at the inlet and outlet were set close to the experimental data. All togwther 6 RANS turbulence models were tested; the models of kε group were presented by the standard kε model and RNG kε model, while kε group models – by the standard kε model and Menter SST model. Also, two Reynolds Stress models (RSM) were tested, namely SSG and LRR models. Comparison has shown a good agreement of experiemental data obtained with calculated results using standard kε turbulence model.
Scientific supervisor – A.A. Khalatov, Academician of the NAS of Ukraine
UDC 515.126(043.2)
Rybalkina T.V., assis. prof.
National Aviation University, Kyiv
TOPOLOGICALLY EQUIVALENT ORIENTED CYCLES
OF LINEAR MAPPINGS
This is a joint work with V. Sergeichuk. We consider oriented cycles of linear mappings of the form
in which or . Let be transformed to
(with the same length t) by a system of bijections, that is,
We say that and are isomorphic if all are linear, and and are topologically equivalent if all and are continuous.
A cycle is regular if all are bijections, and singular otherwise. Each cycle possesses a regularizing decomposition
,
in which is regular and all are indecomposable singular.
Theorem. [1]. (a) Let or , and let and be topologically equivalent. Let
,
be their regularizing decompositions. Then and are topologically equivalent, , and after a suitable renumbering their indecomposable singular summands and are isomorphic for all .
(b) If and are regular, then they are topologically equivalent if and only if the linear operators and are topologically equivalent (as the cycles of length 1).
1. Rybalkina T.V., Sergeichuk V.V., Oriented cycles of linear mappings and their topological classification, UMJ. 66 (2014) 13991405.
UDC 517.938(043.2)
Romanchuk A., Rychik A.
National aviation university, Kyiv
CATASTROPHE THEORY AND UNEXPECTED BEHAVIOR
OF COMPLICATED SYSTEMS. PROBLEMS OF PREDATOR/PREY
RELATIONSHIPS AND THEIR MATHEMATICAL DESCRIPTION
Actuality of our research preconditioned of ubiquity of the theory of catastrophes [1, 2]. It can be applied on heart rhythm, geometrical and physical optics, embryology, linguistics, experimental psychology, economics, hydrodynamics, geology, theory of elementary particles. We can meet with it when we describe wars, life of predators and preys, different cataclysms such as tsunami, hurricanes and volcanic eruptions and it’s frequency. It can be useful when we modeling brain activity, psychological disturbances, rebellions of prisoners, behavior of speculators, influence of alcohol on drivers, politics of censure on the erotic literature.
The goal of this presentation is to spread an idea of necessity of studying the catastrophe theory among students and people who approach to become educated.
Results of explorations weren’t unexpected. We were assured that it is the basic theory of world organization and it is interesting to realize that there is the connection between the heart rhythm and natural disasters. Only one formula can explain terroristic acts and extinction of species of animals.
Basic outcomes. Catastrophe is the number of salutatory changes, appeared as a sudden reply of system on gradual changing ox external conditions.
There are also two terms which are connected with the theory of catastrophes. They are theory of specialties and bifurcation. The first implies grand generalization of exploration of functions on maximum and minimum, the second means ‘division on two’ and applies to determine quality reconstruction or metamorphoses of different objects during the changing of parameters from which they are dependent.
We can see the theory of catastrophes everywhere so we need to be informed about how it operates and what consequences it has. In particular, the set of complicated problems of predator/prey relationships with logically unexpected behavior under some special superpositions of conditions has been analyzed on the base of corresponding mathematical model and its realization in the form of corresponding algorithm and computer program.
References

Арнольд В. І. Теорія катастроф, третє видання, М1990

https://ru.wikipedia.org/wiki/
Scientific supervisor – Ye.O. Shkvar, Dr. of Sci. (Engineering), Prof.
UDC 532.526(043.2)
Strelyayev O.Yu.
National Technical University of Ukraine «Kyiv Polytechnic Institute», Kyiv
Modern approaches of turbulent flow control
and their mathematical models
Energy, primarily from fossil fuels, underlies modern industrial development and usage of power plants and vehicles. Meanwhile efficient energy use is the greatest challenge to scientists and engineers all around the world. The major goal of the researches in the field of energy efficiency is to reduce the amount of energy (fuel) required by contemporary devices. Modern power plants mainly use liquid or gaseous coolant, so the significant way to increase their effectiveness can be found based on general principles of fluid mechanics, associated with deliberate action on the structural features of the flow and creation of turbulence and its further evolution. There are two different and promising areas on this way: to impact on the structure of the flow in the boundary layer or effect on contact between surface and the liquid. In both cases attention should be paid on the solution of such problems as: intensification of convective heat exchange processes, efficiency of combustion processes, reducing additional losses like vibrations and noise generation, friction drag reduction caused by viscous properties of a moving fluid and loss of stability in the boundary layer. First direction can be realized by providing energy from outside, for example, by sucking or blowing gas or liquid in the boundary layer, the use of polymer additives, creating rotational motion of the rigid surface, etc. The second direction includes methods aimed at reducing the frictional resistance by changing the properties of the liquid contact with the rigid surface. This can be achieved by applying surface that can move along the stream, injection of gaseous microbubbles in the boundary layer of liquid, creation of air or steam gap between fluid flow and sleek surface. Author’s research in this area is based on the methodology of computational fluid dynamics, which involves the use of formal representations of physical laws and relationships, mathematical and numerical methods of calculation the flow parameters, and allows with the prespecified accuracy explore the effects of turbulent flow. The fundamental bases of almost all CFD problems are the Navier–Stokes equations, which define any singlephase fluid flow. In this field of researches, this approach has some significant advantages over the experimental methods of research, and often has no alternative, because it allows to define local parameters of turbulent flows with requested for practical purposes resolution and based on this data make optimization of geometrical or operational parameters of the flow control that is impossible or very uneffective in case of the experimental approach. Therefore, the numerical simulation is a modern, perspective and priority research method aimed on detailed study of the properties and effects of turbulent flows.
Scientific supervisor – Ye. O. Shkvar, Dr. of Sci., Prof.
