PROCEEDINGS - ХТМУ · Proceedings of the Anniversary Scientific Conference with International...

UNIVERSITY OF CHEMICAL TECHNOLOGY AND METALLURGY

SOFIA

PROCEEDINGS

ANNIVERSARY SCIENTIFIC CONFERENCE

WITH INTERNATIONAL PARTICIPATION

60 Years UCTM

June 4-5, 2013

Sofia, Bulgaria

Sections: Physics, Mathematics and Technical Sciences

Automatics and Informatics

Human & Economic Sciences

Editors

Eduard Klein

Bogdana Koumanova Svetlozar Nedev Georgi Elenkov Dinka Milcheva Prepress and design Anna Staneva Printing S Print Ltd., Sofia UCTM, Sofia, 2014 www.uctm.edu ISBN: 978-954-465-074-2

http://www.uctm.edu/

Galin Milchev, Radoslav Miltchev

123

Proceedings of the Anniversary Scientific Conference with International Participation “60 Years UCTM”

Physics, Mathematics and Technical Sciences

Thermal study of the cylinders of liquid-cooled internal combustion engines using the FEA methodSorin Raţiu, Vasile Alexa, Imre Kiss...........................................................................................................................5

Heat formation under cyclic viscoelastic deformation of rubber-metal technical productsAlexander Alexandrov, Kliment Hadjov, Mihail Kolev, Milena Milenova.........................................................................11

Study on the characteristics of metallic material of the resistance structure of a workshop with traveling cranesCristina Carmen Miklos, Imre Zsolt Miklos, Carmen Inge Alic.....................................................................................14

Time-vibration superposition in a dry and sorbed liquid medium polyurethaneAlexander Alexandrov, Mihail Kolev, Milena Milenova, Aneliya Mazhdrakova.................................................................21

Integrability of fluid dynamics modelsValentina Radeva, Valery Todorov, Dimitar Kolev, Adem Zeinev, Nikolay Kitanov..........................................................25

Practical aspects of obtaining and using orthogonal modelsSvetla Lekova.................................................................................................................................................................28

Commutational properties of operators of mixed type preserving the powers - IIMiryana S. Hristova.......................................................................................................................................................36

A class of functional differential equations with “Maxima”Adem Zeinev, Valentina Radeva, Dimitar Kolev, Valery Todorov, Nikolay Kitanov.............................................................40

Automatics and Informatics

Benchmarking of industrial control systems via case-based reasoning Mincho Hadjiiski, Kosta Boshnakov, Zdravko Georgiev..................................................................................................46

Combined object-oriented approach for development of process control systemsIskra Antonova, Idilia Batchkova......................................................................................................................................51

The RSFQ electronics – alternative to the modern semiconductor electronics for development of an ultra-fast digital applications Valery Todorov..............................................................................................................................................................58

Ontology-based data integration in manufacturing execution systemsDaniela Gocheva, Dimitar Georgiev, Idilia Batchkova.................................................................................................64

TABLE OF CONTENTS


124

Model transformation between UML activity diagram and Signal Interpreted Petri NetsHabibe Eminova, Iskra Antonova, Idilia Batchkova...........................................................................................................70

Investigation and comparative analysis of the temporal database modelDimitar Pilev, Aneta Georgieva....................................................................................................................................76

Module of dynamic information system for control and organization of information at the career center of University of Chemical Technology and Metallurgy Borislav Slavov, Senya Terzieva, Rayko Stanev.......................................................................................................85

DISCO - Dynamic information system for monitoring and organization of information to the website of University of Chemical Technology and Metallurgy Borislav Slavov...............................................................................................................................................................89

Human & Economic Sciences

Cognitive strategies in the process of self-regulation of learning Ivana Radonova.........................................................................................................................................................95

Interdisciplinary approach to foreign language education in the higher technological education Senya Terzieva, Tanya Pancheva................................................................................................................................98

An effective environment for specialized education of young researchers – a satisfaction feedback Andriana Surleva, Senya Terzieva, Nina Penkova....................................................................................................101

Optimizing performance and technique in the football training of students at the University of Chemical Technology and MetallurgyVentsislav Gavrilov, Georgi Ignatov..........................................................................................................................106

Methodological platform for the preparation of certain documents on safety management in the field of railway transportLiliya Damyanova, Maria Stefanova...........................................................................................................................109

Effectiveness in retail marketing through loyaltyIrina Yovcheva..........................................................................................................................................................115

Supply process as a factor for achieving competitive advantageGeorgi Kondev..............................................................................................................................................................119

Contemporary trends in the e-business systems at the enterprises of mining and manufacturingGalin Milchev, Radoslav Miltchev..............................................................................................................................123


28


PRACTICAL ASPECTS OF OBTAINING AND USING ORTHOGONAL MODELS

Svetla Lekova

Department of Applied Mechanics, University of Chemical Technology and Metallurgy8 Kl. Ohridski, 1756 Sofia, Bulgaria, E-mail: [email protected]

ABSTRACT

This work considers problems related to the usage of orthogonal expansions. By comparing approximation with Kautz functions and with Laguerre functions the advantages of using Kautz functions are convincingly demonstrated. It is shown that applying Kautz functions (with different real poles) has significant advantages for processes devel-oping with high speed in the beginning but dumping relatively slow as well as for processes with time delay. When applying orthogonal Laguerre functions approximately the same precision is achieved with a much bigger number of coefficients.

Another practical aspect in the obtaining of orthogonal models, whose effect is comparable to the effect of scaling factor on the accuracy, is the numerical differentiation. То avoid the numerical differentiation it is supported to use interpolation methods by Matlab.

Keywords: mathematical model; orthogonal functions; Kautz functions.

KAUTZ FILTERS

They are proposed by Kautz in 1954. There were no computers at the time and only analytical solutions were possible. An important assumption when applying the functions of Laguerre or Kautz, is a quadratic integral value during the function to be limited (square integrable). In this case, the accuracy of the approximation is increased when the number of terms is increased and this ensures convergence.

There are three cases of Kautz framework that require different levels of complexity: with non-identical real poles, with non-identical complex poles, with a combination of real and complex poles. Relevant for the subject is the first case of distinct real poles [1]. Let −α1/2, −α2/2, ... , −αN /2 are the locations of the actual poles of Kautz framework (αi is a positive real constant). The transfer function of the N-th Kautz filter is:

( ) ( )( ) ( ) ( )

1 1

1 2 1

/ 2 ... / 2 ( ) .

/ 2 / 2 ... / 21 .

/ 2

NN N

N

N

s sK s q

s s s

s

α αα α α

α

−

−

− −=

+ + +

+

(1)

It is orthogonal to all preceding components. The normalization factor is N Nq α=

For this procedure, there is no limit if the pole is the same. Choosing the same pole the Laguerre functions are obtained. This Kautz procedure can be extended for complex poles. Like the Laguerre model, the approximation of the impulse response with model using Kautz framework is:

1( ) ( )i i i

iw t k tβ α

∞

=

= ∑ (2)

Cutting the infinite functional series (2), the Fourier coefficients are determined by

1 1 10

( ) (t) k t w dtβ α∞

= ∫

2 2 2

0

( ) (t) k t w dtβ α∞

= ∫ (3)

0

( ) (t) N N Nk t w dtβ α∞

= ∫

Svetla Lekova

29

Simulation modeling of experimental step/ im-pulse response

In the context of control systems, models of con-tinuous transfer function are preferred and widely used. Most existing methods for identification of the transfer function do not consider the time delay of the process (or insensitivity), or it is assumed that the delay time is accurately known. In the industrial processes time delay is important for obtaining a representative character-istic of control systems, hence the continuing interest in the identification of processes with a delay. In [2] were analyzed methods for approximation of unknown delay with a model based on rational transfer function (Laguerre formula), as well as with orthogonal Laguerre polynomials. Even when the system has big delay, these methods lead to an acceptable error of approximation.

Selection the location of the poles/ pole. In theory, the choice of the pole α slightly affects the existence and convergence patterns of Laguerre models. In practice, the incorrect selection of α leads to high-order models, in order to achieve the desired accuracy. The location of the pole α must be chosen so that for a given N the error in the given criteria is reduced. This approach is particularly effective if the poles of the system are in a relatively small area and the parameter α is selected in the neighborhood of the dominant pole of the system. Using Kautz functions desired precision of a lower-order model is achieved and the range is also broadened which allows a reasonable speed of convergence of the decomposition.

Error in the impulse response. The evaluation of the Kautz coefficients by (3), in fact reduces the quadratic error:

( ( ) ( ) ) 21

10

-N

i i ii

dE h t k t dtdt

β α∞

=

= ∑∫ (4)

After substitution of the estimated coefficients in (4) is obtained

( ) 21

10

2 -( ) N

ii

dE h t dtdt

β∞

=

= ∑∫ (5) The first term in the right-hand side of (5) is

independent of αi. Therefore, the minimum of the error function in an impulse response corresponds to the maxi-mum of N 2

i 1 max,iβ=→∑ (6)

considering αi , i = 1, 2, … , N.

Error in step response. After obtaining the coeffi-cients iβ , step response of the model can be calculated. Therefore, the integral square error between the measured step response )(th and step response of orthogonal model ( )thapr is:

2step

0

[ ( ) ( ) ] -m

apr

TE h t h t dt= ∫ (7)

where Tm is time greater than the time for establishing of process Tstep. (Tstep is defined as the time required for the process to reach ± 2 % of steady state.) Based on this error function, the location of a pole/ poles can be chosen so that Estep is a minimum. Finding the best value of α (for a special case of the Laguerre functions), using criterion (6) or (7) can be solved by an unconditional one-dimensional optimization problem. But this is not applicable when using Kautz functions.

Model order. The order of the model N is also important for the accuracy of the sought-after orthogonal model. Common practice in identification is to choose the order according to the type of change of the error function, for example E1 or Estep. The order N is chosen so that with its increase the error of model decreases insignificantly.

In determining the appropriate decision the rule of thumb is to choose α such that the error between the step response of the sought-after pattern/ model and that taken from the real process [3] is minimized. In discrete form this error is:

( ) ( )( ) 2

1

1 M

s saprk

sq h kT h kTM =

= −∑ (8)

where ( )sh kT is the real output of the process in the step change of input, while the characteristic ( )saprh kT is obtained from the model with the same input. (Ts interval sampling; M - the number of discrete points in [0, Tm]).

Choice of Tm: The initial part of the step response contains frequency information about the process. There-fore it is pointless to use excessive data algorithm after steady state and so Tm must satisfy:

Tm = (1.2 ÷ 1.5) Tstep (9)Choice of M: The number of calculations becomes

a problem for many large points M. M is recommended to be about 200 [3].

Model order: In general, the order of the process is unknown. Most identification schemes assume that


30

the model order is known. The proposed here approach does not follow such a strategy. If the sq error (8) is too large, the order of the evaluated model is increased by one. Furthermore, this approach makes no assumptions about the time delay.

Estimation of Kp: Common practice in the process control is the transient coefficient Kp to be determined directly from the steady state as Kp= [ y(∞) – y(0) ] / h where h is the amplitude of input step signal. During the presentation of orthogonal functions knowledge of Kp has an important role in obtaining correction coefficients

1

1

2 ( 1) /

pN N

iii

i

KK

β α−

=

=−∑

(10)

and eliminating the static error of the model.Influence of steady state error: Common problem

for all identification methods using step test signals is

that low-frequency noise leads to errors of estimation. This error is reduced if the amplitude of the test signal or the signal / noise ratio is increased. If Kp is known the steady-state error can be removed and the proposed method can be applied to the modified curve of char-acteristic. After acceptance of N and calculation of the Fourier coefficients by (3) and KN by (10) the coefficients {βi} can be adjusted:

ˆ i N iKβ β= (11)

Comparison of orthogonal Kautz and Laguerre modelsThe first case concerns the delay objects/ processes

with self-settling and a large difference in the time constants:

exp (-0.1 )( ) (1 ) (1 0.1 ) (1 0.01 )

sW ss s s

=+ + + (12)

In case of pure delay, even when the poles in Kautz

filters are the same as the poles of the system, the Kautz model will not be identical to the output system. It is expected the model will compensate the delay in the direction of the lower poles of the filter (–αi /2). Fig. 1a, b compares the results of orthogonal Kautz model at α = [16 140 2] (fourth row of Table 1) and a Laguerre model with three coefficients (N = 3), α = 3.

On Fig. 2 are given the real (with continuous line) and the Laguerre approximated impulse response - with broken line at N = 3; with points for N = 7 (for the same α, α = 3). It can be seen that with increase of the coef-ficients up to 7 is reached approximately the same ac-

Fig. 1. Comparison of orthogonal Kautz and Laguerre models with 3 coefficients: the real characteristics is a continuous line, Kautz model - points, Laguerre model - a broken line.

Fig. 2. Orthogonal Laguerre models with 3 and 7 coefficients.

Svetla Lekova

31

curacy as what is obtained when using three orthogonal Kautz functions. In Table 1 are given values sq in (8) for actual and approximated impulse response as well as the correction coefficients KN in (10). These results illustrate the robustness of the process with regard to the choice of the scaling vector α in a rather wide range. Data are not presented graphically, because in all these cases the results practically coincide with the orthogonal model of Fig. 1a, b. On Fig. 3 is presented approximation of the impulse response for N = 4, α = [80 40 10 2.2]. It is important to underline that high accuracy of the approximation is achieved with a small number of coefficients and arbitrary choice of α, which does not even cover the range of time constants / poles. And it was shown that a priori information about the small time constants is not essential.

Another example of application of the Kautz func-tions with real poles is in processes / objects which may be regarded as the sum of first-order objects or the higher order with self-settling with a large difference in the time constants and a pure delay. Comparison of approxima-tions with orthogonal Kautz and Laguerre models here addresses the following test case:

101 0.5

) 501 ( ) 1 ( ) (-0.25 0.5 )(

s.s.ssexpsW

++

++= (13)

The results for the criterion (8) with different values of α are shown in Table 2. It can be seen that for higher values of Ts the accuracy decreases sharply (the third column of the Table 2). Below are shown the data for N = 161. Because of the significant delay and the error of numerical integration, even when the poles of the Kautz filters are identical with the poles of the transfer function (13), namely, α / 2 = [10 2 1], the error sq in step response

is different from zero. In approximation with orthogonal Laguerre functions, if the scaling factor α is chosen only considering the largest time – constant, then α must be in the interval [1.6, 5]. Due to the complexity of the transfer function (13), a higher value for α is preferable, which takes into account the steepness of the transition at the beginning of the characteristic. Most acceptable results have been obtained at α = 4. On Fig.4 a, b are presented the impulse response and the step response approximations for these three discussed cases: N = 4, α = [20 4 2] and the approximation with Laguerre functions, when α = 4 and N = 4, N = 8.

Models of Fig. 4 are obtained by the following Fourier coefficients:

Table 1. Results from different Kautz models.

α sq

Eq. (8)

KN

Eq. (10)

[ 200 20 2 ] 0.0031 1.030

[ 180 18 2 ] 0.0024 1.018

[160 16 2] 0.0017 1.006

[140 16 2] 0.0015 1.004

[120 16 2 ] 0.0014 1.002

[100 18 1] 0.0016 1.010

[80 40 10 2.2 ] 0.0008 1.007

Table 1. Results from different Kautz models.

Fig. 3. Approximation with four Kautz functions, α = [ 80 40 10 2.2 ].

Table 2. Influence of the number of experimental data.

α

sq

M=161,

Ts=1/20

M=81,

Ts=1/10

[20 4 2 2 ]

[20 4 2 ]

[17 5 2.4 1.82]

[17 5 2.4 ]

α = 4, N = 8

α = 4, N = 4

9.62E-05

1.54E-04

5.92E-05

2.16E-04

9.78E-05

0.0066

0.0018

0.0058

0.0013

0.0029

0.0026

0.0067

Table 2. Influence of the number of experimental data.


32

β (Laguerre) = [0.9345 0.3208 0.5017 0.2407 0.1930 0.1190 0.0840 0.0568];

β (Kautz) = [1.1262 -0.1426 0.2520].Processes with similar characteristics are the

absorption/desorption of wood fibrous sorbents for purification of water from oil contamination, and the purification of waters by heavy metals [4, 5] as well as the experimental curves of viscoelastic deformation discussed in [6, 7]. All these processes are characterized by high speed at the beginning, but relatively slowly reach the steady state.

Influence on the accuracy of numerical differentia-tion

Another practical aspect in the obtaining of orthogo-nal models, whose effect is comparable to the effect of scaling factor on the accuracy, is the numerical differ-entiation. For experimental data, we do not know the formula of the function so there are two basic approaches to numerical differentiation:

- Numerical differentiation using finite differences (or formulas for numerical differentiation in point pat-terns with equidistant nodes). Applying it to the step response, for the simplest case of finite difference of first order is obtained (forward or backward):

( ) ( ) ( 1)

s s

k k ks

h k T h kTh h w

T+ −

′ = ∆ = = (14a)

or( ) ( ) ( 1)

s s

k k ks

h kT h k Th h w

T− −

′ = ∆ = = (14b)

- Numerical differentiation by formulas of nu-merical differentiation using interpolation polynomials. The most widely used formula is based on Newton’s interpolation polynomial forward or backward. This enables efficient calculation of the derivatives, offering a posterior estimate of the error. The latter means that the addition of each following addend in the right part of the formulas makes more precise/exact the required value. Therefore, the calculations are terminated when a new addend becomes a module less than the accuracy of the data (i.e. a fatal/ineluctable error).

A very important point in the numerical differen-tiation of some functions is possible instability of the problem, as small errors in input data lead to large errors in the result and sometimes is obtained “the bang of the error”. This event occurs when the derivatives increase very much, which is recognized for example by large and unrelenting values of the finite differences. To solve this case and find a satisfactory approximate value of the derivative of unstable problems very small step Ts

can be used.The two cases, examined in section 3, of applica-

bility of Kautz models (with different real poles) have significant advantages for processes developing with

Fig. 4. Comparison of orthogonal Kautz and Laguerre models: actual characteristic with a continuous line; Kautz model (N = 4) - with dots; Laguerre model - with broken line (for N = 4); and quenched with points (for N = 8).

Svetla Lekova

33

high speed in the beginning but dumping relatively slowly. This high speed leads to significant errors in numerical differentiation of finite differences of the first order, while in the second approach the members of the interpolation polynomial increase. The problem of nu-merical differentiation is very complicated and becomes comparable in volume to the stages of modeling.

This work proposes an approach in which the experi-mental step responses are approximated or interpolated (there are numerous methods), and then looks for their analytical derivatives. This task is not simpler than nu-merical differentiation but using Matlab does not require a thorough understanding of the methods.

Interpolation is a process for estimating values that lie between known data points. It is easy to create and work with piecewise-polynomial functions from Spline Toolbox. Its typical use involves the construction and subsequent use of a piecewise-polynomial approxima-tion. This construction would involve data fitting, but there is a wide range of possible data that could be fit. An exact fit would involve interpolation, an approximate fit might involve least-squares approximation or the smoothing spline. But the function to be approximated may also be described in more implicit ways, for exam-ple as the solution of a differential or integral equation. Care has been taken to make this work as painless and intuitive as possible. In particular, the user need not worry about just how splines are constructed or stored for later use, nor need the casual user worry about such items as “breaks” or “knots” or “coefficients”. It is

enough to know that each function constructed is just another variable that is freely usable as input (where appropriate) to many of the commands.

The supported interpolant methods by Matlab are shown below.

- Linear interpolation. This method fits a different linear polynomial between each pair of data points.

- Nearest neighbor interpolation. This method sets the value of an interpolated point to the value of the nearest data point. Therefore, this method does not generate any new data points.

- Cubic spline interpolation. This method fits a different cubic polynomial between each pair of data points.

- Shape-preserving Piecewise cubic Hermite interpolation (PCHIP). This method preserves monotonicity and the shape of the data.

The type of used interpolant depends on the characteristics of the data being fit, the required smoothness of the curve, speed considerations, post-fit analysis requirements, and so on.

When choosing an interpolation method, keep in mind that some require more memory or longer computation time than others. For example nearest neighbor interpolation is the fastest method. However, it provides the worst results in terms of smoothness. Linear interpolation uses more memory than the nearest neighbor method, and requires slightly more execution time. Unlike nearest neighbor interpolation its results are continuous, but the slope changes at the vertex

Table 3. Comparison between Kautz models and various interpolations.

α

sq

Linear interpolation

'linearinterp'

w2

Cubic spline interpolation 'splineinterp'

w3

Smoothing spline 'smoothingspline'

w4

Shape-preserving Piecewise interpol.

'pchipinterp'

w5

[20 4 2 2 ]

[20 4 2 ]

[17 5 2.4 1.82]

[17 5 2.4 ]

α = 4, N = 8

α = 4, N = 4

0.0092

0.0087

0.0101

0.0129

0.0095

0.0246

1.07E-04

1.39E-04

5.80E-05

2.34E-04

9.74E-05

0.0065

1.58E-04

1.42E-04

3.19E-05

0.0012

4.01E-06

0.0056

8.43E-07

2.94E-06

5.39E-05

6.15E-04

3.59E-05

0.0072

Table 3. Comparison between Kautz models and various interpolations.


34

points. Cubic spline interpolation has the longest relative execution time, although it requires less memory than cubic interpolation. It produces the smoothest results of all the interpolation methods. Cubic interpolation requires more memory and execution time than either the nearest neighbor or linear methods. The cubic spline and shape-preserving methods are slower, but the resulting curves are often very smooth.

Goodness-of-fit statistics, prediction bounds, and weights are not defined for interpolants. Additionally, the fit residuals are always zero (within computer precision) because interpolants pass through the data points. Interpolants are defined as piecewise polynomials because the fitted curve is constructed from many „pieces.“ For cubic spline and PCHIP interpolation, each piece is described by four coefficients, which are calculated using a cubic (third-degree) polynomial. The piecewise polynomials described here always produce a well-behaved fit, and thus they are more flexible than parametric polynomials and can be effectively used for a wider range of data sets.

The methods discussed above for smoothing and interpolation are tested by step responses (13). On

Fig. 5 are given the corresponding derivatives and the methods of Table 3 are applied to the source data viewed as discrete points M (M = 161). Numerical differentiation by (14a) - w1, is identical to w2 and therefore not included in the table. The nearest neighbor interpolation and formula (14b) lead to unacceptable results and therefore are not included. Fig. 5 shows that impulse characteristics differ significantly in the initial section, which has an important role in the accuracy of the orthogonal model. By the values of the criterion (8) of Table 3 it is clear that the numerical differentiation, using first differences forward (the most common and affordable method), gives the most unacceptable results. In an orthogonal model with four Fourier coefficients the remaining three methods give comparable results.

Final results of step responses with α = [17 5 2.4 1.82] are shown in Fig. 6 (for N = 4) and Fig. 7 (for N = 3). It is seen that the use of numerically differentiating dif-ferences in the first four coefficients in the orthogonal model leads to greater error than using a suitable method of interpolation and the model with three coefficients. On the other hand, these results illustrate the robustness of the orthogonal Kautz model to the choice of the scaling vector α in a rather wide range.

CONCLUSIONSBy comparing the approximation with Kautz and

Laguerre functions convincingly are shown the charac-teristics and advantages of using Kautz functions. The following conclusions can be drawn:

Kautz functions are applicable in case of objects with a large difference in time constants and pure de-lay. When using Laguerre functions approximately the same accuracy is obtained with a much larger number

Fig. 5. Comparison of approximated impulse responses (N = 4) when applying the different interpolation methods.

Fig. 6. Comparison of orthogonal Kautz models (N = 4) when applying the different interpolation methods.

Fig. 7. Comparison of orthogonal Kautz models (N = 3) when applying the different interpolation methods.

Svetla Lekova

35

of coefficients.In the presence of pure delay it is necessary the poles

of Kautz filters to be selected smaller than the poles of the transfer function.

It is shown that the approximation with Kautz func-tions is robust to the choice of α in a quite wide range, even if it is not necessary to cover the whole range of time constants/ poles.

Especially when objects/processes that occur at high speed in the beginning, but relatively slowly subside, the use of numerical differentiation in first differences leads to greater error than using an appropriate method of interpolation of the step response and the corresponding analytical derivative.

REFERENCES

1. D. Deschrijver, T. Dhaene, Data-driven model order reduction using orthonormal vector fitting. Model Order Reduction: Theory, Research Aspects and Applications, 13, 3, 2008, 341-359.

2. S. Lekova, Dynamic models of objects with time delay, Journal of the Technical University Sofia, branch Plovdiv, “Fundamental Sciences and Applications”, 14, 2009, 161-168.

3. L. Wang, W. Cluett. From Plant Data to Process Control: Ideas for Process Identification and PID Design. Taylor and Francis, London, 2000.

4. S. Lekova. Modeling processes to purify water from heavy metals. J. of Ecology and Industry, 8, 1-2, 2006, 143-144.

5. S. Lekova, S. Nenkova. Optimization of wood fiber sorbents to purify water from oil pollution. J. of Ecol-ogy and Industry, 8, 1-2, 2006, 141-142.

6. S. Lekova, A. Aleksandrov, М. Milenova, Tz. Tzolov. Orthogonal dynamic model of viscoelastic deformation under variable concentration and loading. Journal of Theoretical and Applied Mechanics, Sofia, 43, 3, 2013, 51–58.

7. A. Aleksandrov, S. Lekova, М. Milenova. Generalized Model of Viscoelastic Deformation. Int. J. of Pure and Applied Mathematics, 81, 4, 2012, 635-646.

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