International Journal of Mechanical Engineering and Technology (IJMET)
Volume 10, Issue 04, April 2019, pp. 830–839, Article ID: IJMET_10_04_082
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=4
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication
Scopus Indexed
THE WAVE FUNCTION OF MICROPARTICLES
AS A COMPONENT OF SYSTEM RELIABILITY
Nikolay Ivanov Petrov
Institute of Metal Science, Equipment and Technologies with Hydro- and Aerodynamics
Centre ,,Acad. A. Balevski” – Bulgarian Academy of Sciences,
67, Shipchenski prohod St. 1574 Sofia, Bulgaria
ABSTRACT
This article is based on the following scientific proposal, stated by the genius
German physicist Max Planck: “No matter can exist on its own. All matter originates
and exists only by virtue of a force which brings the particle of an atom to vibration
and holds this most minute solar system of the atom together. We must assume behind
this force the existence of a conscious and intelligent mind. This mind is the matrix of
all matter“ [1].
The article discusses the wavelength function of the microcosmos, introduced by
Planck as a component of system reliability and its sustainability. In this aspect, we
are investigating ensuring reliable sustainability at the different levels of the
structural and informational organization of the phenomena in our material world.
The statement is maintained that the unified nature of the reliability of the various
phenomena is based on the unity of the "microcosmos" - its material nature and the
capability for duality and transformations.
Key words: Reliability, microcosmos, sustainability, duality and trans-formations
Cite this Article: Nikolay Ivanov Petrov, The Wave Function of Microparticles as a
Component of System Reliability, International Journal of Mechanical Engineering
and Technology 10(4), 2019, pp. 830–839.
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1. INTRODUCTION
The author of this paper proposes a study of the mechanism of the reliability of matter and its
forms at the different levels of its structural organization. The notion of reliability is
applicable in the world of the microcosm, as it reflects the physical nature of elementary
particles that we are investigating. When observing the state of the microparticles (elementary
particles), Planck's constant appears to be the absolute measure of uncertainty. In this sense, it
should be pointed out that the uncertainty principle of Werner Heisenberg states the
following: “Absolutely accurate measurement in the micro and macro world is impossible !
“[2, 3, 4].
Based on the principle of causality in physicochemistry, Max Planck poses the question in
another way: "Are we obliged to seek a constant explanation of the universal unreliability and
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inaccuracy, associated with any physicochemical observation? It may be attributed to the
peculiarities of the individual case, to the complexity of the material object under
consideration, or to the imperfection of the measuring instrument, including even our visual
organs.
Therefore, the unreliability spreads in two directions - from the individual case to the
general and vice versa, according to the laws of physicochemistry” [Planck M. The Unity of
the physical world’s image. М., 1966] [1].
Max Planck's quantum theory and its basic equation for microparticle (MP) energy, even
written on the memorial plaque, placed on his grave ( E  hv , where h is Planck’s constant,
and v is the frequency of the physicochemical process) changes our notions in regard to the
function of the error [2]. It testifies for the presence of a tendency for a dialectical
"incarnation" of the error (equivalent to the failure in the physicochemical process) and its
opposite – the reliability.
Typical of this concept is the examination of the error (failure) in the physicochemical
processes as a necessary moment in the "reliable" functioning of the systems. It (the concept)
represents a gnoseological model of the objective reality [6].
2. REGARDING THE WAVELENGTH FUNCTION OF
MICROPARTICLES
The idea of the author of this study is to consider the wave function based on the following
experiment with the light - the most important phenomenon of nature on Earth and in the
Universe. A phenomenon that ensures the existence of humanity and of all forms of a reliable
and real biological world.
Since light possesses corpuscular-wave properties, then the following experiment is
possible. Let a beam of light rays fall on the surface of a transparent plate. According to the
wave notion regarding the nature of these beams, they are reflected from the upper surface of
the plate, and another portion is refracted and passes through it. If the intensity of the falling,
reflected and refracted beam is designated with
valid:
I , I r and I l , the following inequality will be
I  Ir  Il .
(1)
The following equation is known I  A , where A is the amplitude of the falling wave of
light [3]. As a result of this ratio, the following equation can be recorded:
2
A  Ar  Al ,
2
A, A
2
2
(2)
A
r and
l are the amplitudes of the falling, reflected and refracted wave.
where
If the light is manifested by its corpuscular properties, the intensity of the falling beam of
light will be proportional to the number of photons that are contained therein. If the number of
falling, reflected, and refracted photons is designated with
N  N r  Nl .
N , N r and N l , then follows that
(3)
In order to combine the two concepts regarding the nature of the light (wave and
quantum), the assumption is made that the number of the photons in the beam of light is
proportional to the square of the module of the respective wave. From this assumption, follow
the following formulas for the squares of modules of the reflected and refracted wave [3]:
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Ar  Al  1.
2
Ar 
2
2
(4)
N
Nr
2
; Al  l .
N
N
(5)
N N
The relationship r
determines what portion of the total number of falling photons is
reflected or the probability that a single photon is being reflected from the surface of the
considered plate. The relationship Nl N determines what portion of the total number of
falling photons passes through the surface of the plate, i.e. the probability that a single photon
can pass through the surface of the plate. We do not know which of the falling photons will be
reflected, nor which one will pass through. Their behavior is described by virtue of a
probability function, by using the squares of the amplitudes of their respective waves.
Originating from similar theoretical considerations, the German physician and chemist
Max Planck has suggested that the behavior of each microparticle can be described by single
function
  x, y , z , t 
, which he named wave function.
The probability of finding the microparticle in a small volume
determined by using the wave function in the following way [1, 4]:
dW   dV ;
2
dW
2
     ,
dV
dV in the space W is
(6)
where  is a function, which is a complex conjugate function of  . From (6) follows
that the square of the module of the wave function actually determines the probability density,

i. e. the probability that a particle may be located in a particular unit volume of the space W .
  x, y , z , t 
Therefore, the physical meaning of Planck's wave function
is probabilistic
in nature. This leads to the conclusion, that is more appropriate for it to be called wave
function of the microparticle reliability (WFMPR). This is so, because the reliability of a
given microparticle is a probability of its presence at a certain time, precisely at the supposed
place in the object (space), in the presence of the respective uncertainty of measurement (even
when observation is performed by using the most exceptional microscope)
Of course, it must be taken into consideration that Max Planck introduces the term wave
function during the time period from 1900 to 1905, when the term “reliability” had not been
formally introduced to science and only the quality of this or that physico-chemical process or
object was discussed.
In order to determine the probability of a given microparticle to be located at a particular
W , then integration of the WFMPR must be performed into the
elementary volume dV within the range from  to  , as a result of which follows:
time at any point in the space
W


  x, y, z, t  .dV  1

2
(7)
.
Since, the probability is a magnitude that varies within the range of 0 to 1, the condition
(7) is called normalization condition of the WFMPR. The physical significance of (7) is
related to the fact that, under certain conditions, the microparticle must surely be located at
some point in the space W .
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Therefore, the cumulative probability of the particle being located somewhere in the space
shall be equal to one. As a result of this follows that the normalization condition, expressed in
(7) confirms the objective existence of the microparticle in space and time.
From everything that has been analyzed above, follows the conclusion:
“The wave function of the reliability of microparticles combines their wave and quantum
properties and serves in order to describe their behavior in the space of states considered.
Planck's constant appears to be the absolute measure of the uncertainty in measuring the
reliability of microparticles (elementary particles)”.
3. THEOREM REGARDING THE RELIABLE SUSTAINABILITY OF
THE MICROCOSMOS
Already scholars during the antiquity started to discuss the idea of reliability of the
microcosmos. The main idea in Democritus's teachings is : “The being is reality – the atoms,
while non-being is the empty space”.
The study of the probability-stochastic reliability phenomena at the level of the
microparticles and at the atomic levels throughout the entire twentieth century (including the
CERN experiments that have so far given the world only the “World Wide Web” and some
conjectures regarding the existence of the X-boson particle) opens up new opportunities for
the resolution of this problem and ensures the solving of problems the in man-machine
systems [13].
Philosophical form of the theorem: "In order for different systems (obje-cts - atoms,
molecules, chemical compounds, etc.) to be generated and formed from microparticles
(microvesicles), it is necessary to fulfil the following criterion of sustainability: ,, The energy
of the internal connections between the elements of the physicochemical systems must exceed
the sum of the kinetic energy of the microparticles and the energy of the external influences
(continuous or impulse) under the respective conditions of existence”
Mathematically, this is expressed by the inequality:
,
(8)
where
is energy of the connections between the elements of the system (CES),
the cumulative kinetic energy of microparticles,
- energy of the external influences (EI).
Mathematical form of the theorem - equation for the sustainability of the system of
microparticles (SMP).
We are looking at an ever-evolving system of microparticles (SMP). Its functioning
(interactions) is represented by a system of ordinary differential equations, written in their
generic form:
xi  t   fi  t , x1 ,..., xn  , i  1,..., n; xi  t0   xi 0 ,
(9)
where t is an independent variable of the ongoing time;
functions sought that determine the basic parameters of the RTS;
x1  t  ,..., xn  t 
t0 and xi 0 are the initial set
x   t   dx  t  dt
operating conditions of the system under consideration, and the symbol
denotes a derivates of the function
xi  t 
.
f  t , x ,..., x
are the
i
i

1
n
In (9) it is presumed that the functions i
are real. Under these conditions, it
is possible to record the system of differential equations (9) in the following vector form:
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The Wave Function of Microparticles as a Component of System Reliability
xi  t   fi  t , x  , x  colon  x1 ,..., xn   Rn , x  t0   x0
(10)
We can denote with SH a sphere with a radius
H and center at the beginning of the
n
x  i 1 xi2
R
n
coordinates of the Euclidean space
with norm
. With T is designated an
interval from the real axis of the space, as the following condition is fulfilled
T  a  t  
 or some number. It shall be considered that the function
, where a is
f  t , x  , f : T  S H  Rn
is continuous with respect to two arguments and fulfills the
conditions of Lipschitz (L) with respect to the second argument, i.e. the following inequality
is valid
f  t , x   f  t , y   L  x  y  , L  const  0
(11)
Under these conditions, the theorem for local existence is valid, as well as the uniqueness
and continuity of the solution
x  t , t0 , x0 
of the task from (9) regarding the sustainability of
functioning of the SMP under the conditions
t0  T and x0  Sn [6].
From the famous axiom of Andrey Lyapunov follows, that the solution
x  t , t0 , x0 
x t 
is
called undisturbed (stable) solution, and
- disturbed solution of the system of ordinary
differential equations (9) in the presence of impulse interferences [7, 8, 9, 11].
In some of the earliest scientific papers [3, 5], it is proposed to use impulse differential
equations in order to describe aggregate disturbances on the system of microparticles (SMP).
In the following works [7], it is considered that the investigation of the process of studying of
the functioning of a SMP, which is influenced by aggregate impulse disturbances, is
determined by a simplified version of the system of ordinary differential equations (9), which
are expressed as follows:
x  t   f  t , x(t )  , x  t0   x0 ,  x, f  R n  ,
and the jumps of the argument
of the first order
(12)
i x  i  N 
for which are set only
t
their points in time i , where t0  t1  ...  tk  ...   , and the respective estimate of the
x
jumps i , which are caused by the impulse disturbances.
The system of differential equations (12) including taking into consideration the
impulse disturbances, causing jumps
, is recorded as follows:
( )⁄
(
( ))
∑
(
)
(
)
(13)
).
where
is the delta function of Dirac during the time interval (
A natural transformation is performed, by replacing the second member on the right side
of (13) with the function
gt 
, where ( ) ,
)
is a defined continuous left-hand
gt 
function, locally limited by variation. The function
is assumed to be an impulse
disturbance of the equations (12). The result is a new look of the system of differential
equations (13):
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( )⁄
( ))
(
gt  ,
(14)
The transformation thus performed, preserves the simplicity and clarity that are
characteristic of the "classical" impulse differential equations. It allows to be encompassed
some "non-classical" cases of resolving the problem of impulse disturbances. Such is the case,
where during the observed interval with fixed duration is being limited not the number of the
impulses, but their cumulative impulse.
When stating the problem under investigation, it should be taken into consideration that:
there is no general definition of sustainability of the functioning of the SMP, as related to the
reliability of this system [9]. In ad-dition, it is necessary to know the properties of functions
with locally limited variations, including the Lebesgue theorem for decomposition of these
functions into a sum of continuous set functions and functions of the jumps [6, 7]. On the
other hand, in every reliability standard of a system of elements (particles), which form
objects (tangible and / or intangible) in the countries in the EU and globally, the following is
emphasized: “Reliability is a complex property of a system of objects which, depending on the
purpose of the object and the conditions of its operation, includes faultlessness, repairability,
durability and storage during transportation, either individually or in combination with these
properties, in the presence of external disturbances” [15, 16, 20, 22, 23].
4. DEFINITION OF SUSTAINABILITY OF SMP
Let us examine Cauchy’s problem in regard to SMP, which is represented by a system of
equations and fulfillment of the condition ( )
:
( )⁄
* ( ) +
( )⁄ , ( )
,
(15)
In (15) the functions
x, f , g assume values in the field of the real numbers
(
), as the function f satisfies the conditions of Caratheodory [8]. The functions ( ) g
and ( ) have a locally limited variation and continuity to the left. Thus, the accepted
t , t
 h)
conditions provide for the resolution of the problem (15) in the interval 0 0
, where
( ) is the function of the jumps of the impulse disturbances. Uniqueness of the solutions of
(15) is not supposed.
It is not difficult to formulate a problem (11) in such a way, so that the equality in the
basic equation can be understood in the ordinary sense. For this, it is sufficient for the system
) for each t  t0 . The assumption is
of equations (15) to be integrated in an interval (
accepted, that in this interval the solution of the problem (15) exists and this results in the
following integral equation:
t
x  t   x0   f  x   ,  d  g  t   g  t0 
t0
,
(16)
equivalent in the event of t  t0 as in problem (12). In (16) the time interval
is
determined by
.
By ( ) is denoted the function of the jumps (i.e. the discontinuous summand in the
g t 
decomposition of Lebesgue) of the main function of the risk
. The function h is
determined with accuracy up to a random constant addend; therefore, as certainty is
h t   x
0
0
considered that
. It is noticed that the second addend on the right side of the
formula (13) is continuously dependent on the argument t .
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An “impulse transformation” is performed by carrying out the following substitution
( )
( )
( ), ( )
( )
( )
(17)
From (17) follows that
( )
( )
( ) and ( )
( )
( ),
(18)
The so performed impulse transformations (17) and (18) result in: ,,The equation of
y t 
function continuity
in the event of locally limited variation”. It looks the following
t
y  t    f  y    h   ,   d    t     t0 
(19)
t0
It is not difficult to carry out the reverse transition from equation (19) to equation (16), i.
e. these two equations are equivalent. However, unlike (16), in equation (19) all summands
are continuous functions. This means that equation (19) has been defined in an "elementary
sense".
In the assumptions stated above, equations (19) and (16) have one solution that is located
t , T   t
 T   
0
within the interval 0
. An example of a local variation in the equation of
function continuity (19) is shown in [15,21], where by using the method of nonlinear
mathematical programming, have been solved in parallel two multifactor iterative oneparameter technogenic optimization problems, related to the reservation of technical and
economic systems.
Let us investigate the question regarding the sustainability (stability) of the solution of the
equation (15), defining Cauchy’s problem for a system of microparticles (SMP). Above all, it
is clear that the problem is reduced to the study of the stability of the null solution
i.e. pertaining to the case
f  0, t   0
x0  t   0
,
at x0  0 . We shall consider that, in equation (15), the
gt 
external influences appear to be a function of
, i.e. this concerns researching the
stability of SMP in the event of permanently acting external disturbances.
The null solution of equation (15) is called stable in the event of aggregate impulse
external influences, if for each
satisfied
 0
k

exists such
  0 , whereby the integral inequality is
g   t  .dt   , k  N ,
(20)
k 1
the inequality is valid, that looks like this: x  t    , t   0,  
(21)
In the event that (20) and (21) are not fulfilled, the null solution of (9) is unstable due to
the aggregate impulse external influences. As a result of this stability follows the analogous
sustainability, in the event of which the initial conditions are set for each
t0  0 .
If  does not depend on 0 , then the null solution of (9) is referred to as uniformly stable
in the event of aggregate impulse influences on the SMP. The null solution of (9) is
asymptotically stable in the event of aggregate impulse disturbances[12], sufficiently small
t
 k  1, k   k  N  and
0,   , resulting from x  t   0 at t   .
limitation of this variation in the interval
variation of the function
g t 
in the research (study) interval
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5. INVESTIGATION OF CAUCHY’S PROBLEM IN REGARD TO SMP
The normal functioning of the SMP is represented by the differential equation of the type
( )⁄
( )
( )⁄ , ( )
(22)
x t 
g t 
where
and
satisfy the assumption of locally limited variation and A is a
square matrix of the n-th order with real constant coefficients.
The investigation of Cauchy’s problem in regard to SMP is related to postulating the
"Theorem on the stability of the functioning of SMP (the reliability of the microparticle
system).” It is represented by the following definition: If all values of the matrix A from (22)
have a negative real part, then the solution of the equation will be uniform and asymptotically
stable. However, if the condition so formulated is violated, then this solution is unstable,
which results as well in the instability in the functioning of the SMP, i.e. its unreliability.
Proof. The solution of Cauchy’s problem, represented by (22) exists only in the following
form
t
x  t   g  t   etA g  0   A et   g   d .
(23)
0
In equation (23) all summands appear as common functions and their sum is understood in
the “elementary” sense.
e sA  ce bs
s   0,  
c

0
If it is possible to choose a number
such that
for all
, then
formula (23) through integration by parts is represented in the form of:
t
x  t    et   g    d ,
(24)
0
g   
where in the right-hand side of (24) the expression
represents a generalized
function. When implementing the inequality (21) from equation (24), it follows:
t 1 mink ,t
x t   
k 1

e( t  ) A  g    d  M ,
(25)
k 1
where the designation:
t
 t  k

bt
M  ce    g    d  e  g    d 
t 
 k 1 k 1

 bt
(26)
After entering intermediate designation, the result is:
 1

M  c 
 1 ,
b
1 e

whereby

denotes the expression

k

(27)
g    d  0 at k  
k 1
(28)
t 
In inequalities (25) and (26), the designation
(located above the sign for the
mathematical sum) represents the whole part of the number t . As a result of the so performed
mathematical analysis in order to investigate the stability (reliability) of the system of
microparticles (SMP), it becomes evident that if in (20) and (21) is assigned a number
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The Wave Function of Microparticles as a Component of System Reliability
then such a value of
 0
can be found, for which the right-hand side of (25) can turn out to
 . This demonstrates the stability of the solution to Cauchy’s problem for SMP
be less than
as represented by the differential equation (22). The uniformity of this sustainability follows
as a result of the autonomy of the assigned problem [12].
6. CONCLUSIONS
From the author’s work presented in this article follow the subsequent scientific results:
1. The problem of Max Planck's wave function has been investigated, maintaining the opinion
that at the modern stage of global science, it is more appropriate for it to be called a wave
function of the reliability of the microcosmos.
2. A philosophical form of the theorem regarding the reliability of the microcosmos and its
microparticle (elements) has been proposed.
3. The stability of the solution of a system of differential equations, describing the reliable
functioning of the microparticles, has been investigated and the corresponding theorem has
been proven.
REFERENCES
[1]
Планк, М. Единство физической картины мира. Москва, ,,Наука“, 1966, с. 50, изд.
№809/65.
[2]
Fuller, H., R. Fuller, R. Fuller. Physics Including Human Applications. Harper &
Row, Publishers, N.Y.-Hagerstown-San Francisco-London, 1978, p. 724, p. 735 [Фулър,
Х., Р. Фулър, Р. Фулър. Физиката в живота на човека. Изд. ,,Наука и изкуство“,
София, р. 735, Изд. № 28688].
[3]
Михайлова, В. Основи на физиката. Част 1 и 2. ,,Сиела”, С., 2005, с. 287-289, ISBN
954-649-316-3.
[4]
Борн, М. Физика в жизни моего поколения. ,,ИИЛ”, Москва, 1963, р. 428,
Изд. №9/0971.
[5]
Брилюен, Л. Научная неопределенность и информация. ,,Мир”, Москва, 1986, с. 28,
изд. №20/3570.
[6]
Beer, St. Cybernetics and Management. The English Universities Press LTD, London,
E.C. 1. [превод на руски: Кибернетика и управление производством. С
предисловием акад. Ал. Берг. ГИФМЛ, М., УДК 519.95: 65, 1963], p. 133.
[7]
Понтрягин, Л.С. Обыкновенные дифференциальные уравнения. Москва, Журнал
,,Наука”, 1965, с. 55-60.
[8]
Мильман, В.Д., А.Д. Мышкис. Об устойчивости движения при наличие толь-чков.
,,Сиб. Мат. Журнал”, 1960, № 2, с. 233-237.
[9]
Bird,
J.
Engineering
Mathematics.
Pocket
Books.
J. ,,Newnes”, N.Y., 2008, с. 544, ISBN: 978-0-7506-8153-7.
[10]
Мышкис, А.Д., А.М. Самойленко. Системы с толчками в заданные моменты
времени. Журнал ,,Математический сборник”,1987, №2, с. 202-208.
[11]
Popchev, Iv. Decentralized Systems. Publishing House of the Bulgarian Academy of Science. Monograph, Sofia, UDK 681.3: 62 52, 1989, р.p. 10.
[12]
Popchev, I., R. Tsoneva. Two-layer control of interconnected systems with time delays
and a condition for asymptotic stability of the overall system. - Proceedings of XI-th
European Meeting on Cybernetics and Systems Research, 21-24 April 1992, Vienna, vol.
http://www.iaeme.com/IJMET/index.asp
838
Third
Edition.
editor@iaeme.com
Nikolay Ivanov Petrov
l (editor Robert Trappl), World Scientific Publishing Co., 1992, 229-236, ISBN: 981-021991.
[13]
Popchev, I., N. Zlatareva. How to provide new problem-solving for manmachine systems. - Automatic Control-World Congress 1990 "In the service of mankind". Proceedings of the 11-th Triennial World Congress of the International Federation
of Automatic Control, Tallinn, Estonia, 13-17 August, 1990. U.Jaaksoo and V.Utkin
(ed.). V.V, Pergamon Press pic., Headington Hill Hall, Oxford, UK, 1990, p. 182.
[14]
Petrov, N. Probability, Independence, Information Society. Monograf, First pu-blication,
2012, Lodz, Poland: ,,Wydawnictwo Astra”, p. 98, ISBN 978-83-63158-20-0.
[15]
Петров, Н. Изследване на операциите пo осигуряване на надеждността на рискови
технико-икономически системи. Монография. София: Акад. изд. ,,Проф. Марин
Дринов“, 2014, p. 115, ISBN 978-954-322-790-7.
[16]
Петров, Н. Неопределеност и риск при измервания на обекти. Монография. Тракийски университет - Ст. Загора, Ямбол: PH ,,Zh.Uchkov”, с. 25, ISBN 95499-78-50-8.
[17]
Georgiev, N.L. Quality indicators for protection systems", International Journal of
Economics, Commerce and Management, vol 6, issue 4, p. 836, ISSN 2348-0386, 2018,
SJR:5.313.
[18]
Georgiev, N.L. Quality criteria for critical infrastructure protection systems. Inter-national
Journal of Economics, Commerce and Management, vol 6, issue 4, p. 265-271, ISSN
2348-0386, 2018. International Journal of Economics, Commerce
and Management,
vol 6, issue 4, 2018, ISSN:2348-0386, p. 270, SJR:5.313.
[19]
Dimitrov, V., V. Dimitrova, Optimization of machining conditions for high-speed milling
with single flute end mill cutters of element from aluminium sheets, HPL panеls and
Aluminum composite panels, International Journal of Мechanical Engineering and
Technology (IJMET), Volume 5, Issue 11, November (2014), pp. 01-09, ISSN 0976 –
6359, 2014.
[20]
БДС 27.002-86. Надеждност в техниката. Основни термини и определения. Комитет
по качество към МС на Република България, София, 1987.
[21]
D. Panayotov, V. Dimitrov, Kr. Kalev, “Comparative analysis of the processes for
machining of mold element with using TopSolid'CAM and ESPRIT,” International
Journal of Mechanical Engineering and Technology, vol. 7, issue 2, pp. 01-10, 2016.
[22]
Panayotova, G. Geometrical representation of the simple waves and blow-up. Tensor,
N.S.,Vol.68, No.1(2007), pp. 39-43, #6. ISSN 0040-3504
[23]
Dimitrov, G., G. Panayotova, Aspects of Website optimization. Proceding of the Union
os Scientist - Ruse, Book 5, Mathematics, Informatics and Physics, Vol. 12, 2015, pp.
106-114, ISSN 1314-30-77.
http://www.iaeme.com/IJMET/index.asp
839
editor@iaeme.com