Parameters of two-dimensional analytical model of an assessment of crossing of ellipses of distribution
at recognition of nanoparticles of colloidal silver are given in polyair fibers on multidimensional correlation
components of the Raman ranges with control according to polarizing characteristics. Reliability of recognition
of nanoparticles increased more than by 1000 times and was estimated on joint probability of normal
distributions of intensivnost of the Raman spectrograms of nanoparticles of silver on polyair fibers depending
on longitudinal and cross polarization of laser radiation on all range of a range with the analysis of
9 main peaks

Butov, K.V.

Danilova, S.A.

Dobrovolskaya, T.A.

Emelyanov, V.M.

Emelyanovа, О.V.

SocioEconomic Challenges Journal (Sumy State University)

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ 
Vol. 7 No 4, 04007(4pp) (2015) Том 7 № 4, 04007(4cc) (2015) 
 
 
2077-6772/2015/7(4)04007(4) 04007-1  2015 Sumy State University 
Development of Two-dimensional Analytical Model According to Polarizing Characteristics 
of the Raman Ranges at Recognition of Nanoparticles of Silver on Polyester Fibers 
 
V.M. Emelyanov*, T.A. Dobrovolskaya, S.A. Danilova, О.V. Emelyanovа, K.V. Butov 
 
Southwest State University, 94, 50 let Octyabrya St., 305040 Kursk, Russia 
 
(Received 28 September 2015; published online 10 December 2015) 
 
Parameters of two-dimensional analytical model of an assessment of crossing of ellipses of distribution 
at recognition of nanoparticles of colloidal silver are given in polyair fibers on multidimensional correlation 
components of the Raman ranges with control according to polarizing characteristics. Reliability of recogni-
tion of nanoparticles increased more than by 1000 times and was estimated on joint probability of normal 
distributions of intensivnost of the Raman spectrograms of nanoparticles of silver on polyair fibers depend-
ing on longitudinal and cross polarization of laser radiation on all range of a range with the analysis of 
9 main peaks. 
 
Keywords: Polyester fiber, Nanoparticles of colloidal silver, The Raman ranges, Polarizing characteristics 
of the Raman spectroscopy, Mathematical modeling of ranges, Two-dimensional correlation components of 
the Raman ranges, Reliability of recognition. 
 
 PACS numbers: 02.60.Cb, 02.70.Rr 
 
 
                                                          
* emelianov@nm.ru 
1. INTRODUCTION 
 
Considerably to increase reliability of recognition of 
the nanoparticles which are on fibers of fabrics correla-
tion polarizing characteristics of the Raman spectros-
copy due to multidimensional identification of a large 
number of peaks on the spectrogram allow. 
The one-dimensional assessment of identification of 
nanoparticles on fibers of polyair is carried out in 
works [1-7]. But to define the modes of drawing nano-
particles on fibers and especially their change at opera-
tion is complicated because of their small quantity. 
Carrying out statistical modeling of correlation pa-
rameters of intensity of ranges of fiber with silver na-
noparticles during operation on analytical expressions 
of correlation ellipses of interdependent parameters 
with determination of coordinates of crossing repre-
sents considerable interest, and also an assessment of 
reliability of recognition of nanoparticles of silver. 
 
2. DESCRIPTION OF THE SUBJECT AND 
METHODS OF RESEARCH 
 
2.1 Experimental Procedure 
 
We will carry out mathematical modeling of statis-
tical data for identification of crossings of ellipses of 
distributions of values of intensivnost of peaks of spec-
trograms by the following technique [3]. We will create 
the general correlation matrix for full generation of 
data on the basis of initial matrixes and we will receive 
a correlation matrix of RXY1 with a general size of 
38  38. 
Generation of the set amount of casual values is 
carried out in normal way to the law and a matrix of 
RXY1 for what the built-in MathCad Edition 14 [3-5] 
function is used. Further we define a vector of own 
numbers from the general correlation matrix of RXY1. 
As the generated casual values of a vector of own 
numbers possess some correlation which is negatively 
affecting modeling accuracy, it is necessary to lead 
them to an uncorrelated look for what the special pro-
gram developed by authors in the environment of 
MathCad Enterprise Edition 11 is used.  
Further transformation of uncorrelated values 
through a fundamental matrix of UR of a correlation 
matrix of RXY1 in the correlated follows. 
 
2.2 The Processing of the Experimental Data 
 
The generated 96 values of data for 1 peak on an axis 
X and Y are given in Fig. 1a, and for 4 peaks in Fig. 1b. 
 
 
a 
 
 
b 
 V.M. EMELYANOV, Т.А. DOBROVOLSKAYA, ET AL.  J. NANO- ELECTRON. PHYS. 7, 04007 (2015) 
 
 
04007-2 
 
c 
 
 
 
d 
 
Fig. 1 – Two-dimensional correlation dependences of the Ra-
man polarizing spectrograms of polyester fibers with nanopar-
ticles of silver and without nanoparticles at polarization across 
and along fibers: a, b – a general view at generation of n  96 
of data for 1 and 4 peaks; c, d – a general view at generation of 
n  1536 of data for 1 and 4 peaks 
 
As data of Fig. 1a are crossed, it is possible to find a 
point of intersection easily. According to Fig. 1b it is 
visible that ellipses of distribution aren't crossed. 
Therefore it is necessary to generate bigger number 
of data. 1536 generated data are given in Fig. 1с and 
1d. On them it is visible that ellipses of 1 peak are 
crossed with considerable overlapping, and distribution 
ellipses 4 peaks are crossed normally. 
Crossings of analytical ellipses of distribution of 
these peaks of spectrograms are given in Fig. 2. These 
ellipses are constructed on analytical mathematical 
expressions with parameters of population means, av-
erage quadratic deviations and coefficients of correla-
tion for polyair fibers without nanoparticles (a continu-
ous curve) and with nanoparticles (a curve points). For 
convenience of construction these drawings are devel-
oped on 180 round axis X in comparison with Fig. 1. 
 
3. DESCRIPTION AND ANALYSIS OF RESULTS 
 
Crossing happens in Fig. 2a, b X1  488.2 and 
Y1  485.9, and Fig. 2c, d of X4  409.79 and Y4  2094.9. 
Crossings of ellipses of distribution on Fig. 1 find 
manually with number of the generated data. On Fig. 2 
crossing are allocated with use of analytical mathemat-
ical expressions also manually at selection of the gen-
eralized R ellipse parameter – the equivalent radius of 
curvature. 
For mathematical identification of crossing it is 
necessary to create system of the analytical equations  
 
a 
 
b 
 
c 
 
d 
 
Fig. 2 – Crossing of ellipses of distribution at a task on analyt-
ical expressions (a continuous curve – without nanosilver; a 
curve with points – with nanosilver): a – a general view for 
1 peak; b – the increased fragment with crossing of ellipses of 
distribution of 1 peak; c – a general view for 4 peaks; d – the 
increased fragment with crossing of ellipses of distribution 
4 peaks 
 
 DEVELOPMENT OF TWO-DIMENSIONAL ANALYTICAL MODEL ACCORDING… J. NANO- ELECTRON. PHYS. 7, 04007 (2015) 
 
 
04007-3 
and it will give coordinates of a point of intersection.  
The analytical assessment of crossing of ellipses of 
distributions is made on systems of the equations with 
finding of coordinates of a point of intersection (1)-(2): 
iii  0…50     xiii  10iii + MENXAg90;   
x1iii  – 10iii + MENXAg90;                                            
x6iii  10iii + MENX0; x7iii  – 10iii + MENX0.     (1) 
 
y21iii  0.5 YAg90 [2rXYAg_0_80  (xiii-MENXAg90) 
/XAg90] + 0.5YAg90{[2rXYAg9_0_80(xiii-
MENXAg90)/XAg90]2-4[((xiii-MENXAg90)/XAg90)2 
–[1-(rXYAg_0_80)2]12.847]}0.5 + MENYAg90;               (2) 
 
The analytical equations for points are similarly re-
ceived: y31iii, y41iii, y51iii, y61iii, y71iii, y81iii, y91iii. 
Calculations on analytical expressions it is made by 
criterion of crossing of ellipses in one point (only con-
tact) for coordinates of limit values R114  2.6355 and 
Х114  488.21 on 1 peak of the spectrogram. For double 
crossing of ellipses of distribution points of intersection 
with results are revealed Z1i,1  Х1i(1)  485.21÷503 and 
Z1i,3  Х1i(2)  488.21÷473 at change 
Z1i,0  R1i  2.6355÷2.6739. 
For longitudinal polarization at R114  2.6355 limit 
value of crossing is revealed Y114  485.98. This begin-
ning of double crossing of ellipses of distribution 
Z1i,2  Y1i(1)  485.98÷517 and Z1i,4  Y1i(2)  485.98÷450 
at change of equivalent radius of an ellipse Z1i,0  
R1i  2.6355÷2.6739 on Fig. 3b. 
Calculations on analytical expressions for 4 peaks it 
is shown on in Fig. 3c and 3d. Coordinates of limit val-
ues R414  3.58427 and Х414  409.81 only contacts are 
received in a point. Х3(1)  485.948874 and 
Х3(2)  485.948889 for R4  41.7321043. For double 
crossing of ellipses of distribution points are revealed 
Z4i,1  Х4i(1)  409.81÷412 and Z4i,3  Х4i(2)  409.8÷ 
400.5 at change Z4i,0  R4i  3.58427÷3.58887 on 
Fig. 3c. 
For coordinate Y414  2095.4 at R414  3.58427 limit 
value of crossing is revealed. Double crossing happens in 
points of Fig. 3g and results of the decision 
Z4i,2  Y4i(1)  2095.4÷2125 and Z4i,4  Y4i(2)  2095.3÷1960. 
 
4. CONCLUSIONS  
 
At a choice of points of intersection of the ellipses of 
distribution constructed on population means, average 
quadratic deviations and correlation coefficients with 
selection of equivalent radius of ellipses of distribution 
crossing coordinates are received. 
It is revealed that the closest compliance takes 
place at estimates XAn, Yan, Ran и XРn, YРn, RPn in 
comparison with the experimental XЭn, YЭn, where to 
estimate the equivalent radius of curvature of ellipses 
it wasn't presented possible in general. At an assess-
ment of reliability of identification of nanosilver on fi-
bers of polyair the following values for one-dimensional 
measurements on axis X in the cross direction of fiber 
are revealed: 
for cross polarization of X 
pXT  (0.97917  0.92208  0.97917  0.99653  0.97917  
0.99826 0.97917 0.98958 0.99479),                               (3)  
 
 
a 
 
b 
 
c 
 
            d 
 
Fig. 3 – Images of crossings of ellipses of distribution on analyti-
cal expressions: a, b – a look when crossing X1i(1), X1i(2) and Y1i(1), 
Y1i(2); c, d – a look when crossing X4i(1), X4i(2) and Y4i(1), Y4i(2) 
 
for longitudinal polarization of Y 
pYT  (0.78125  0.70833  0.77083  0.81250  0.77083  
0.72917   0.79167  0.72917  0.87500),                           (4) 
 V.M. EMELYANOV, Т.А. DOBROVOLSKAYA, ET AL.  J. NANO- ELECTRON. PHYS. 7, 04007 (2015) 
 
 
04007-4 
and at two-dimensional measurement in cross X and 
longitudinal Y directions taking into account correla-
tion coefficients: 
pXYT  (0.9896  0.92708  0.99306  0.99935  0.98611  
0.99826 0.97917  0.99653  0.99479).                             (5)  
Here it is visible that the offered method gives an 
essential prize in an assessment of reliability of defini-
tion of the modes of drawing nanoparticles of silver on 
fibers. For example, for 4 peaks of i=3 the increase oc-
curred more than by 1000 times. 
 
 
REFERENCES  
 
1. V.M. Emelyanov, T.A. Dobrovolskaya, V.V. Emelyanov, 
E.J. Orlov, Nanotechnics 2, 81 (2013). 
2. V.M. Emeljanov, T. Dobrovol’skaja, S. Danilova, 
V. Emeljanov, E. Orlov, Open J. Metal. 3 No 3, 29 (2013). 
3. V.M. Emelyanov, T.A. Dobrovolskaya, S.A. Danilova, 
V.V. Emelyanov, K.V. Butov, E.J. Orlov, J. Nano- Elec-
tron. Phys. 5 No 4, 04001 (2013). 
4. V.M. Emelyanov, T.A. Dobrovolskaya, V.V. Emelyanov, 
E.J. Orlov, Modern Problem. Sci. Educ. 6 (2013). 
5. V.M. Emelyanov, T.A. Dobrovolskaya, V.V. Emelyanov, 
E.J. Orlov, Fundamental Res. 10, 3310 (2013). 
6. V.M. Emelyanov, T.A. Dobrovolskaya, S.A. Danilova, 
V.V. Emelyanov, Nat. Eng. Sci. 6, 415 (2013). 
7. V.M. Emelyanov, T.A. Dobrovolskaya, S.A. Danilova, 
V.V. Emelyanov, K.V. Butov, E.J. Orlov, Science of Science 
6 (2013). 
 
 


Development of Two-dimensional Analytical Model According to Polarizing Characteristics of the Raman Ranges at Recognition of Nanoparticles of Silver on Polyester Fibers

Electronic Sumy State University Institutional Repository

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ Vol. 7 No 4, 04007(4pp) (2015) Том 7 № 4, 04007(4cc) (2015)   2077-6772/2015/7(4)04007(4) 04007-1  2015 Sumy State University Development of Two-dimensional Analytical Model According to Polarizing Characteristics of the Raman Ranges at Recognition of Nanoparticles of Silver on Polyester Fibers  V.M. Emelyanov*, T.A. Dobrovolskaya, S.A. Danilova, О.V. Emelyanovа, K.V. Butov  Southwest State University, 94, 50 let Octyabrya St., 305040 Kursk, Russia  (Received 28 September 2015; published online 10 December 2015)  Parameters of two-dimensional analytical model of an assessment of crossing of ellipses of distribution at recognition of nanoparticles of colloidal silver are given in polyair fibers on multidimensional correlation components of the Raman ranges with control according to polarizing characteristics. Reliability of recogni-tion of nanoparticles increased more than by 1000 times and was estimated on joint probability of normal distributions of intensivnost of the Raman spectrograms of nanoparticles of silver on polyair fibers depend-ing on longitudinal and cross polarization of laser radiation on all range of a range with the analysis of 9 main peaks.  Keywords: Polyester fiber, Nanoparticles of colloidal silver, The Raman ranges, Polarizing characteristics of the Raman spectroscopy, Mathematical modeling of ranges, Two-dimensional correlation components of the Raman ranges, Reliability of recognition.   PACS numbers: 02.60.Cb, 02.70.Rr                                                             * emelianov@nm.ru 1. INTRODUCTION  Considerably to increase reliability of recognition of the nanoparticles which are on fibers of fabrics correla-tion polarizing characteristics of the Raman spectros-copy due to multidimensional identification of a large number of peaks on the spectrogram allow. The one-dimensional assessment of identification of nanoparticles on fibers of polyair is carried out in works [1-7]. But to define the modes of drawing nano-particles on fibers and especially their change at opera-tion is complicated because of their small quantity. Carrying out statistical modeling of correlation pa-rameters of intensity of ranges of fiber with silver na-noparticles during operation on analytical expressions of correlation ellipses of interdependent parameters with determination of coordinates of crossing repre-sents considerable interest, and also an assessment of reliability of recognition of nanoparticles of silver.  2. DESCRIPTION OF THE SUBJECT AND METHODS OF RESEARCH  2.1 Experimental Procedure  We will carry out mathematical modeling of statis-tical data for identification of crossings of ellipses of distributions of values of intensivnost of peaks of spec-trograms by the following technique [3]. We will create the general correlation matrix for full generation of data on the basis of initial matrixes and we will receive a correlation matrix of RXY1 with a general size of 38  38. Generation of the set amount of casual values is carried out in normal way to the law and a matrix of RXY1 for what the built-in MathCad Edition 14 [3-5] function is used. Further we define a vector of own numbers from the general correlation matrix of RXY1. As the generated casual values of a vector of own numbers possess some correlation which is negatively affecting modeling accuracy, it is necessary to lead them to an uncorrelated look for what the special pro-gram developed by authors in the environment of MathCad Enterprise Edition 11 is used.  Further transformation of uncorrelated values through a fundamental matrix of UR of a correlation matrix of RXY1 in the correlated follows.  2.2 The Processing of the Experimental Data  The generated 96 values of data for 1 peak on an axis X and Y are given in Fig. 1a, and for 4 peaks in Fig. 1b.   a   b  V.M. EMELYANOV, Т.А. DOBROVOLSKAYA, ET AL.  J. NANO- ELECTRON. PHYS. 7, 04007 (2015)   04007-2  c    d  Fig. 1 – Two-dimensional correlation dependences of the Ra-man polarizing spectrograms of polyester fibers with nanopar-ticles of silver and without nanoparticles at polarization across and along fibers: a, b – a general view at generation of n  96 of data for 1 and 4 peaks; c, d – a general view at generation of n  1536 of data for 1 and 4 peaks  As data of Fig. 1a are crossed, it is possible to find a point of intersection easily. According to Fig. 1b it is visible that ellipses of distribution aren't crossed. Therefore it is necessary to generate bigger number of data. 1536 generated data are given in Fig. 1с and 1d. On them it is visible that ellipses of 1 peak are crossed with considerable overlapping, and distribution ellipses 4 peaks are crossed normally. Crossings of analytical ellipses of distribution of these peaks of spectrograms are given in Fig. 2. These ellipses are constructed on analytical mathematical expressions with parameters of population means, av-erage quadratic deviations and coefficients of correla-tion for polyair fibers without nanoparticles (a continu-ous curve) and with nanoparticles (a curve points). For convenience of construction these drawings are devel-oped on 180 round axis X in comparison with Fig. 1.  3. DESCRIPTION AND ANALYSIS OF RESULTS  Crossing happens in Fig. 2a, b X1  488.2 and Y1  485.9, and Fig. 2c, d of X4  409.79 and Y4  2094.9. Crossings of ellipses of distribution on Fig. 1 find manually with number of the generated data. On Fig. 2 crossing are allocated with use of analytical mathemat-ical expressions also manually at selection of the gen-eralized R ellipse parameter – the equivalent radius of curvature. For mathematical identification of crossing it is necessary to create system of the analytical equations   a  b  c  d  Fig. 2 – Crossing of ellipses of distribution at a task on analyt-ical expressions (a continuous curve – without nanosilver; a curve with points – with nanosilver): a – a general view for 1 peak; b – the increased fragment with crossing of ellipses of distribution of 1 peak; c – a general view for 4 peaks; d – the increased fragment with crossing of ellipses of distribution 4 peaks   DEVELOPMENT OF TWO-DIMENSIONAL ANALYTICAL MODEL ACCORDING… J. NANO- ELECTRON. PHYS. 7, 04007 (2015)   04007-3 and it will give coordinates of a point of intersection.  The analytical assessment of crossing of ellipses of distributions is made on systems of the equations with finding of coordinates of a point of intersection (1)-(2): iii  0…50     xiii  10iii + MENXAg90;   x1iii  – 10iii + MENXAg90;                                            x6iii  10iii + MENX0; x7iii  – 10iii + MENX0.     (1)  y21iii  0.5 YAg90 [2rXYAg_0_80  (xiii-MENXAg90) /XAg90] + 0.5YAg90{[2rXYAg9_0_80(xiii-MENXAg90)/XAg90]2-4[((xiii-MENXAg90)/XAg90)2 –[1-(rXYAg_0_80)2]12.847]}0.5 + MENYAg90;               (2)  The analytical equations for points are similarly re-ceived: y31iii, y41iii, y51iii, y61iii, y71iii, y81iii, y91iii. Calculations on analytical expressions it is made by criterion of crossing of ellipses in one point (only con-tact) for coordinates of limit values R114  2.6355 and Х114  488.21 on 1 peak of the spectrogram. For double crossing of ellipses of distribution points of intersection with results are revealed Z1i,1  Х1i(1)  485.21÷503 and Z1i,3  Х1i(2)  488.21÷473 at change Z1i,0  R1i  2.6355÷2.6739. For longitudinal polarization at R114  2.6355 limit value of crossing is revealed Y114  485.98. This begin-ning of double crossing of ellipses of distribution Z1i,2  Y1i(1)  485.98÷517 and Z1i,4  Y1i(2)  485.98÷450 at change of equivalent radius of an ellipse Z1i,0  R1i  2.6355÷2.6739 on Fig. 3b. Calculations on analytical expressions for 4 peaks it is shown on in Fig. 3c and 3d. Coordinates of limit val-ues R414  3.58427 and Х414  409.81 only contacts are received in a point. Х3(1)  485.948874 and Х3(2)  485.948889 for R4  41.7321043. For double crossing of ellipses of distribution points are revealed Z4i,1  Х4i(1)  409.81÷412 and Z4i,3  Х4i(2)  409.8÷ 400.5 at change Z4i,0  R4i  3.58427÷3.58887 on Fig. 3c. For coordinate Y414  2095.4 at R414  3.58427 limit value of crossing is revealed. Double crossing happens in points of Fig. 3g and results of the decision Z4i,2  Y4i(1)  2095.4÷2125 and Z4i,4  Y4i(2)  2095.3÷1960.  4. CONCLUSIONS   At a choice of points of intersection of the ellipses of distribution constructed on population means, average quadratic deviations and correlation coefficients with selection of equivalent radius of ellipses of distribution crossing coordinates are received. It is revealed that the closest compliance takes place at estimates XAn, Yan, Ran и XРn, YРn, RPn in comparison with the experimental XЭn, YЭn, where to estimate the equivalent radius of curvature of ellipses it wasn't presented possible in general. At an assess-ment of reliability of identification of nanosilver on fi-bers of polyair the following values for one-dimensional measurements on axis X in the cross direction of fiber are revealed: for cross polarization of X pXT  (0.97917  0.92208  0.97917  0.99653  0.97917  0.99826 0.97917 0.98958 0.99479),                               (3)    a  b  c              d  Fig. 3 – Images of crossings of ellipses of distribution on analyti-cal expressions: a, b – a look when crossing X1i(1), X1i(2) and Y1i(1), Y1i(2); c, d – a look when crossing X4i(1), X4i(2) and Y4i(1), Y4i(2)  for longitudinal polarization of Y pYT  (0.78125  0.70833  0.77083  0.81250  0.77083  0.72917   0.79167  0.72917  0.87500),                           (4)  V.M. EMELYANOV, Т.А. DOBROVOLSKAYA, ET AL.  J. NANO- ELECTRON. PHYS. 7, 04007 (2015)   04007-4 and at two-dimensional measurement in cross X and longitudinal Y directions taking into account correla-tion coefficients: pXYT  (0.9896  0.92708  0.99306  0.99935  0.98611  0.99826 0.97917  0.99653  0.99479).                             (5)  Here it is visible that the offered method gives an essential prize in an assessment of reliability of defini-tion of the modes of drawing nanoparticles of silver on fibers. For example, for 4 peaks of i=3 the increase oc-curred more than by 1000 times.   REFERENCES   1. V.M. Emelyanov, T.A. Dobrovolskaya, V.V. Emelyanov, E.J. Orlov, Nanotechnics 2, 81 (2013). 2. V.M. Emeljanov, T. Dobrovol’skaja, S. Danilova, V. Emeljanov, E. Orlov, Open J. Metal. 3 No 3, 29 (2013). 3. V.M. Emelyanov, T.A. Dobrovolskaya, S.A. Danilova, V.V. Emelyanov, K.V. Butov, E.J. Orlov, J. Nano- Elec-tron. Phys. 5 No 4, 04001 (2013). 4. V.M. Emelyanov, T.A. Dobrovolskaya, V.V. Emelyanov, E.J. Orlov, Modern Problem. Sci. Educ. 6 (2013). 5. V.M. Emelyanov, T.A. Dobrovolskaya, V.V. Emelyanov, E.J. Orlov, Fundamental Res. 10, 3310 (2013). 6. V.M. Emelyanov, T.A. Dobrovolskaya, S.A. Danilova, V.V. Emelyanov, Nat. Eng. Sci. 6, 415 (2013). 7. V.M. Emelyanov, T.A. Dobrovolskaya, S.A. Danilova, V.V. Emelyanov, K.V. Butov, E.J. Orlov, Science of Science 6 (2013).   

https://essuir.sumdu.edu.ua/bitstream/123456789/43194/1/Emelyanov.pdf

Development of Two-dimensional Analytical Model According to Polarizing Characteristics of the Raman Ranges at Recognition of Nanoparticles of Silver on Polyester Fibers

Abstract

Similar works

Full text

Available Versions

SocioEconomic Challenges Journal (Sumy State University)

Electronic Sumy State University Institutional Repository