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Modern scientific research and their practical application

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Modern scientific research and their practical application. VolJ11309


CONTENTS

J11309-254

LINEAR QUADRATIC GAUSSIAN PROBLEM FOR DISCRETE-TIME MARKOVIAN JUMP LINEAR SYSTEMS

Korotkova N.N., Mozhenkov V.V., Belyakov E.V.

J11309-255

DEPOLARIZATION CURRENTS IN POLYVINYLIDENE FLUORIDE CAUSED BY RELAXATION OF CHARGE AND POLARIZATION Revenyuk T. A., Sergeeva A. E

J11309-256

PROPAGATION OF ELASTIC WAVES IN BILAYER FERRITE-PIEZOELECTRIC STRUCTURE Galichyan T.A.

J11309-257 "EXPANSION" of the UNIVERSE Klinkerman R.V.

J11309-259 - COMMUTATION OF LINEAR OPERATORS Ahramovich M.V.

J11309-260

MATHIMATICAL CALCULATION OF THE MASS FLOW OF NITROBENZENE FLOWING THROUGH THE PIPELINE

Mocretsova I.S., Rebro I.V., Mustafina J.A., Galitsyna T.A, Koroteyeva E.A, Perepechenova T.N.

J11309-261 EXTENDED OF TABLE ALGEBRA: MULTISET TABLE ALGEBRA Buy D.B.1, Glushko I.M.2



J11309-254

Korotkova N.N., Mozhenkov V.V., Belyakov E.V.

LINEAR QUADRATIC GAUSSIAN PROBLEM FOR

DISCRETE-TIME MARKOVIAN JUMP LINEAR SYSTEMS

Volzhsky polytechnical institute, Volzhsky, Engelsa 42a, 404121

In this report we describe the new proof of the theorem on the optimal control of

linear stochastic systems with Markovian switching with use of quadratic functional

cost value.

Key words: linear stochastic systems, quadratic functional cost.

Introduction

Switched systems arise naturally in many engineering fields, such as power

electronics, embedded systems, manufacturing and communication networks, etc.

Incorporating the switching behavior in the model and controller structures offers

much greater freedom and more possibilities for capturing complex system dynamics,

achieving stabilization and improving the overall performance of the feedback

systems.

Many theoretical and numerical tools have been developed for the stability

analysis of various switched systems.

These stability results have also led to some controller synthesis algorithms that

ensure stability of some simple switched systems

System control conditions:

==

++=+

.;

;)(),(),(

0

0

1

ixx

wjujBxjAx jjjjjj

θ

δθθ

with quadratic functional cost is solved in [1] using dynamic programming by

Bellman equasion, but the result of that article can also be achieved with

transformation of the functional cost.

Theorem



Fix the probability space ),,( PFΩ and consider the following class of dynamic

systems:

==

++=+

.;

;)(),(),(

0

0

1

ixx

wjujBxjAx jjjjjj

θ

δθθ

(1)

where nj Rx ∈ , jw is the sequence of vectors of the independent random values

and with average zero and covariance E (E-identity matrix) with l dimension, ju is

the sequence of the control functions with m dimensions.

jθ is the Markovian chain with stationary transition function having the range

of values *,...,2,1* NF = , where

( ) ( ) ijnnnn pijPijP ====== ++ θθθθθ 101 ,..., . (2)

Assume that the start values 0x are the random vector with the mean 0m and

covariance 0С , FjjijBijA ∈,)(),,(),,( δ are the effective matrices with the associated

dimensions, ,...2,1,0:,, =jwx jjj θ are statistically independent.

Assume that jx , jθ are observable and find the control functions ),( jjj xuu θ=

(note: ),( jjx θ is the Markovian process was used). We consider the unbounded state,

where the range of values for control functions becomes mR .

With the above assumptions we find the control from the class of feedback

controls that minimises the following functional cost value:

( )

++= ∑

−

=

1

0)(

N

kNNNkkkkkkN xMxuNuxMxMuJ , (3)

where kM are nn× symmetrical non-negative matrices, kN are mm×

positively defined matrices and [ ]M is the expected value.

In that case optimal control is defined the formulas:

jjjj xiLixu )(),(0 −= for ij =θ when 1,...,0 −= Nj ,

where

[ ][ ] ×××+=−

++

1

11 ),(),()(),()( ijBxKMijBNiL jjjjjj θθ [ ] ),(),()(),( 11 ijAxKMijB jjjj ×× ++ θθ ,



FiiK j ∈),( - are symmetrical non-negatively defined nn× matrices calculated

with the reverse recursion with the formula

[ ] −+××= ++ jjjjjj MijAxKMijAiK ),(),()(),()( 11 θθ

[ ] ×××− ++ ),(),()(),( 11 ijBxKMijA jjjj θθ

[ ][ ] ×××+×−

++

1

11 ),(),()(),( ijBxKMijBN jjjjj θθ

[ ] ),(),()(),( 11 ijAxKMijB jjjj ××× ++ θθ when 1,...,0 −= Nj , Fi∈ ;

jj MiK =)( when Nj = , Fi∈ ;

and minimal control cost is given with

[ ] [ ][ ]∑−

=++++=

1

011000000

0 )()()()()(N

jjj jKMjtrCKtrmKmuJ δθδθ ,

where tr[] trace of the matrix.

Proof

Consider

[ ] [ ]=−=− ++++++ ),(111111 jjjjjjjjjjjjjj xxKxxKxMMxKxxKxM θ

( )[ ××++= +1)(),(),( jjjjjj KwjujBxjAMM δθθ

( ) ] [ ]),(),()(),(),( jjjjjjjjjjjj xxKxMMxwjujBxjA θθδθθ −++× .

Since jx , ju , ),( jjA θ , ),( jjB θ are measurable relatively to ( )jjx θ, , then by the

properties of the mathematical expectation they can be taken out of the sign of the

mathematical expectation. Since jw are independent of ( )jjx θ, , then

[ ] [ ] 0),( == jjjj wMxwM θ . Substituting for jK , we get

[ ]=−+++ jjjjjj xKxxKxM 111

( )( )[ ][ += ++ jjjjjj xijAxKMijAxM ),(,),( 11 θθ

( )( )[ ] ++ ++ jjjjjj uijBxKMijBu ),(,),( 11 θθ

( )( )[ ] ++ ++ jjjjjj uijBxKMijAx ),(,),( 11 θθ

( )( )[ ] ]++ ++ jjjjjj xijAxKMijBu ),(,),( 11 θθ

( )( )[ ][ ]−+ ++ jjjjjj wjxKMjwM )(,)( 11 δθθδ



( )( )[ ]([ −+− ++ jjjjjj MijAxKMijAxM ),(,),( 11 θθ

( )( )[ ] ×− ++ ),(,),( 11 ijBxKMijA jjjj θθ ( )( )[ ] ×+−

++

1

11 ),(,),( ijBxKMijBN jjjjj θθ

[ ] ) ]jjjjj xijAxKMijB ),(),()(),( 11 θθ ++× .

After the addition and subtraction of [ ]jjj uNuM , and the reduction of

( )( )[ ][ ]jjjjjj xijAxKMijAxM ),(,),( 11 θθ ++ we have

[ ]jjjjjj xKxxKxM −+++ 111 [ ]++−= − jjjkjjj uNuxMxM

( )( )[ ][ ]++ ++ jjjjjj wjxKMjwM )(,)( 11 δθθδ

( )( )[ ]( )[ +++ ++ jjjjjjj uijBxKMijBNuM ),(,),( 11 θθ

( )( )[ ] ++ ++ jjjjjj uijBxKMikjAx ),(,),,( 11 θθ ( )( )[ ] +++ jjjjjj xijAxKMijBu ),(,),( 11 θθ

( )( )[ ] ×+ ++ ),(,),( 11 ijBxKMijAx jjjjj θθ ( )( )[ ] ×+−

++

1


[ ] ]jjjjj xijAxKMijB ),(),()(),( 11 θθ ++× .

Rewrite the above in the following form

[ ]=−+++ jjjjjj xKxxKxM 111 [ ]++− jjjjjj uNuxMxM

( )( )[ ][ ]++ ++ jjjjjj wjxKMjwM )(,)( 11 δθθδ

( )( )[ ][ ]([ +++ ++ jjjjjj uijBxKMijBNM ),(,),( 11 θθ ( )( )[ ] )×++ jjjjj xijAxKMijB ),(,),( 11 θθ

( )( )[ ] ×+×−

++

1


( )( )[ ][ ]( ++ ++ jjjjjj uijBxKMijBN ),(,),( 11 θθ ( )( )[ ] )]jjjjj xijAxKMijB ),(,),( 11 θθ ++ .

Sum the received equation

[ ]=−∑−

=+++

1

0111

N

jjjjjjj xKxxKxM [ ]++−∑

−

=

1

0

N

jjjjjjj uNuxMxM

( )( )[ ][ ]++∑−

=++

1

011 )(,)(

N

jjjjjjj wjxKMjwM δθθδ

( )( )[ ][ ]([∑−

=++ +++

1

011 ),(,),(

N

jjjjjjj uijBxKMijBNM θθ

( )( )[ ] )×+ ++ jjjjj xijAxKMijB ),(,),( 11 θθ

( )( )[ ] ×+×−

++

1




( )( )[ ][ ]( ++× ++ jjjjjj uijBxKMijBN ),(,),( 11 θθ

( )( )[ ] )]jjjjj xijAxKMijB ),(,),( 11 θθ +++ .

But, on the other hand, that is equivalent to

[ ]=−∑−

=+++

1

0111

N

jjjjjjj xKxxKxM [ ] [ ]NNN xKxMxKxM +− 000 [ ] [ ]NNN xMxMxKxM +−= 000 .

Taken these two equasions we get the following formula for the control cost

[ ] ( )( )[ ][ ]++= ∑−

=++

1

011000 )(,)()(

N

jjjjjjj wjxKMjwMxKxMuJ δθθδ

( )( )[ ][ ]([∑−

=++ +++

1

011 ),(,),(

N

jjjjjjj uijBxKMijBNM θθ ( )( )[ ] )×++ jjjjj xijAxKMijB ),(,),( 11 θθ

( )( )[ ] ×+×−

++

1


( )( )[ ][ ]( ++ ++ jjjjjj uijBxKMijBN ),(,),( 11 θθ ( )( )[ ] )]jjjjj xijAxKMijB ),(,),( 11 θθ ++ .

By using controls we can effect only the final sum since the matrices

( )( )[ ][ ] 1

11 ),(,),(−

+++ ijBxKMijBN jjjjj θθ , 1,...,0 −= Nj ,

are non-negatively defined, )(uJ takes the minimal value when

( )( )[ ][ ] ++ ++0

11 ),(,),( jjjjjj uijBxKMijBN θθ ( )( )[ ] 0),(,),( 11 =++ jjjjj xijAxKMijB θθ ,

1,...,0 −= Nj ,

or

[ ][ ] ×××+−=−

++

1

110 ),(),()(),(),( ijBxKMijBNixu jjjjjjj θθ

[ ] jjjjj xijAxKMijB ),(),()(),( 11 ××× ++ θθ , 1,...,0 −= Nj .

Thus we have minimal control cost is

[ ] [ ][ ]∑−

=++++=

1

011000000

0 )()()()()(N

jjj jKMjtrCKtrmKmuJ δθδθ .

Theorem is proved.

This method of proof is supposed to be used for other systems.

References:



1. Fragoso M.D. Discrete-time jump LQG problem / M.D.Fragoso //

International journal of System Science - 1989 - Issue 12. Volum 20 - pg. 2539-254

УДК 537.226.83

PACS 77.84.-s, 73.61.Ph, 72.20.jv J11309-255

Revenyuk T. A., Sergeeva A. E.

DEPOLARIZATION CURRENTS IN POLYVINYLIDENE FLUORIDE CAUSED BY RELAXATION OF CHARGE AND POLARIZATION

Department of Physics and Material Science, Odessa National Academy of Food Technologies, ul. Kanatnaya 112, 65039 Odessa, Ukraine

The procedure has been developed for extracting homocharge and heterocharge

currents from experimentally measured thermally stimulated depolarization currents

of corona poled PVDF. Application of different depolarization modes supplemented

with the isothermal currents allowed to obtain such parameters of the relaxation

processes, as activation energies, characteristic frequencies and time constants.

Key words: polymer ferroelectrics, homocharge, heterocharge, depolarization.

1. Introduction

Polyvinylidene fluoride (PVDF) and its copolymers received considerable

attention during last years because of their high piezo- and pyroelectric activity

whose origin is not fully understood [1–9]. Their specific properties are usually

attributed to the high level of the residual polarization [4–7], although some

researchers believe that the injected charge can also play an important role [8].

Despite the fact that PVDF is often considered as a polymer ferroelectric [5],

some of its electrical properties can be explained in terms of the conventional theory

of polar dielectrics and electrets. The phenomenological Gross-Swann-Gubkin model

of electrets [4-6] assumes availability of two kinds of charges in polar dielectrics,

namely the homocharge σ(t) whose sign coincides with the polarity of electrodes

during poling and the heterocharge Р(t) (internal polarization) arising from micro-

and macrodisplacement of intrinsic charges in the dielectric under action of the

electric field. In the case of PVDF one can assume that the heterocharge represents



the dipole polarization, while the homocharge is formed by charges trapped at or near

the surface [8].

Stability of the electret state in a polar dielectric depends on the interaction and

the resulting mutual relaxation of the homocharge and the heterocharge. Since the

heterocharge (polarization) is usually of the primary importance for PVDF, the role

of the homocharge was not given enough consideration so far, although the

stabilizing effect of the space charge on the residual polarization has been already

discussed [7,8].

Thermally stimulated depolarization (TSD) is a commonly used method for

identifying relaxation processes in charged polymer electrets [7]. However, it is very

difficult to separate the influence of the homocharge and the heterocharge on the TSD

currents, especially if the corresponding peaks are superimposed in a wide range of

temperatures.

In the present work, we illustrate how to extract depolarization currents caused

by relaxation of the homocharge and the heterocharge from the total TSD current by

solving the inverse problem and revealing the relaxation behavior of both

components from the experimentally measured TSD current. Moreover, it is shown

that application of different TSD modes supplemented with the isothermal

depolarization currents makes it possible to evaluate important parameters of the

relaxation processes, such as activation energies, characteristic frequencies and time

constants.

2. Experimental procedure

The study was performed on uniaxially stretched 25 µm thick PVDF films

supplied by Plastpolymer (Russia) and composed of β-form crystallites and

amorphous phase in nearly equal volume fractions. A metal electrode of 0.1 µm

thickness was deposited on one surface of each sample by thermal evaporation of Al

in a vacuum. The other side of the sample was subjected to a negative corona

discharge initiated by a pointed tungsten electrode with the automatically controlled

potential, while the metallized rear surface was grounded. The vibrating control grid

between the sample surface and the corona electrode was kept at a constant potential



of 3 kV. All samples were charged at room temperature under a constant charging

current density [8] of 90 µA/m2 for 30 min and then short-circuited and conditioned

at room temperature for 24 hours (except those intended for measuring kinetics of the

electret potential).

Four versions (modes) of depolarization were applied to study relaxation

processes, namely thermally stimulated (T) and isothermal (I) depolarization of short-

circuited (S) and open-circuited (O) samples. The modes were thus referred as TS,

TO, IS and IO modes with the first letter indicating the temperature regime

(thermally stimulated or isothermal) and the second one indicating the electric state

(short-circuited or open-circuited). Additional experiments on the thermally

stimulated kinetics of the electret potential (TP) have been also performed after 24

hours of conditioning in the open circuit configuration. The Teflon film of 10 μm

thickness was used as a dielectric gap in TO and IO modes. All thermally stimulated

experiments were performed under a constant heating rate of 3 K/min. In isothermal

experiments, temperature was maintained constant after the desired value was

achieved by fast heating. The electret potential in the TP mode was measured by the

Kelvin method and recorded continuously.

3. Results and their discussion

The following are the main features of experimental curves seen in fig. 1 and

fig. 2.

20 40 60 80 100 120

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0

100

200

3003

2

1

0

31

2

Volta

ge, V

Curre

nt, µA

/m2

Temperature, oC



Fig. 1 Thermally stimulated currents in the TS mode (1) and the TO mode (2) and the electret potential in the TP mode (3).

– The depolarization current in the TS mode shows a broad "non-classical" peak

with a maximum around 65°C;

– The inversion of the TSD current is observed in the TO mode, while the

direction of the current coincides with that in the TS mode during the initial stage of

the heating;

– The electret potential in the TP mode has a maximum at 40 oC.

– The current slowly decreases with time in the IS mode at all temperatures,

while the isothermal current changes its direction in the IO mode at elevated

temperatures.

The above mentioned features can be explained in the frames of a model

assuming existence in the samples of the homocharge and the heterocharge [1, 6-8]

with the former representing the charge trapped at the surface and the latter standing

for the polarization formed in the bulk due to the electric field created by the

homocharge. The two types of the charge are obviously interdependent.

Fig. 2 Isothermal transient currents at different temperatures in the IS mode (a) and the IO mode (b); 1 – 45 oC, 2 – 55 oC, 3 – 70 oC.

At first we examine charging and relaxation processes qualitatively. It is

reasonable to assume that the negatively charged particles (ions and/or electrons),

supplied by corona discharge are adsorbed and thermalized on the surface of the

0 5 10 150

2

4

6

0 5 10 15

-0.1

0.0

0.1

0.2

0.3(a)

3

2

1Cu

rrent

, µA/

m2

Time, min

Curre

nt, µ

A/m2

Time, min

(b)

3

2

1



sample because of their low (thermal) energy. The redundant charge localized on the

surface or in the near-to-surface layer forms the homocharge having a certain surface

density σ and producing the uniform field E in the bulk of the sample. The high

electron affinity of the fluorine atoms facilitates the charge trapping and formation of

the stable homocharge.

The uniform internal polarization P (heterocharge) appears mainly due to

alignment of CH2-CF2 dipoles in the field created by the homocharge. This is

equivalent to developing the bound surface charge P having the sign opposite to that

of the homocharge σ. Among all polarization processes in PVDF, the alignment of

CH2-CF2 dipoles is the main one, because of their large dipole moment of 2.1 D [1].

If the polarization P is zero, the field in the bulk of the sample is created by the

total surface charge σ. When polarization P starts to grow, the depolarizing field

appears, which is immediately "neutralized" by a fraction of the surface charge equal

to the neutralized (screened) polarization. Thus, the electric field in the bulk is now

created not by the total charge σ, but by the difference (σ-P) between the surface

charge and the polarization. Hence, one can consider the total surface charge σ as

consisting of two parts σ=σ1+σ2, the first one representing the compensating charge

(σ1=P) and the second one σ2=σ-P creating the electric field in the bulk of the

sample.

After short-circuiting of the charged samples (in the TS and the IS modes), the

"excessive" charge σ2 disappears. Then the equilibrium between the homocharge and

the heterocharge (σ=σ1=P), as well as zero internal field (E=0) are maintained due to

the current flowing through the external circuit, so the measured current corresponds

to the relaxation of the heterocharge.

However, if, after the short-circuiting and forming the σ=σ1=P equilibrium, a

non-conductive dielectric gap is introduced between one of the electrodes and the

surface of the sample (TO and IO modes), one can observe the relaxation currents of

both, the heterocharge and the homocharge flowing in opposite directions. The field

in the bulk is not zero any more, so the surface charge (homocharge) is either forced

to drift in its own field from one surface of the sample to another one through the



whole thickness of the sample (in the absence of the intrinsic conductivity), or it is

slowly neutralized by charge carriers responsible for the intrinsic conductivity. In any

case, the relaxation of the heterocharge takes place in non-zero field conditions and is

caused by thermal disordering of the aligned dipoles [1, 12].

It will be shown now that the two components of the total depolarization current

can be obtained from the experimental i(T) dependence in the TO mode (fig. 1, curve

2). It is known [1, 8] that the TSD current i(t) and the electret potential V(t) in

experiments with a non-conducting spacer or an air gap introduced between the

sample surface and the electrode depend not only on the interrelation between the

homocharge and the heterocharge, but also on their time derivatives, so that

−=dt

tddt

tdPsti )()()( σ, (1)

)]()([)(1

1 tPtsxtVo

−= σεε , (2)

dttdV

xti o )()(

1

1 ⋅−=εε

, (3)

where )/( 111 εεε oo xxxs += , t is the time, ε and xo the dielectric constant and

the thickness of the sample, ε1 and x1 the corresponding values of the dielectric gap,

εo is the permittivity of a vacuum.

The conductive component ic(t) of the total current can be expressed as

dttdtV

xgti

oC

)()()( σ−== , (4)

where )/exp( kTQgg o −= is the specific conductivity, k Boltzmann's constant, T

the temperature, Q the activation energy of the intrinsic conductivity, go the pre-

exponential factor. Integrating Eq.(3) and substituting time t for temperature T in Eq.

(l)–(4) according to )1( btTT o += , where b is the heating rate, To the initial

temperature, we obtain expressions for temperature dependences of the homocurrent

i1(T) and the heterocurrent i2(T), as well as for the voltage across the sample (electret

potential) V(T)



∫∞

−−==

Tooo

o dTTikTQ

xbTgx

dtdTi ')'(exp)(

1

11 εε

σ, (5)

dtd

sTi

dtdPTi σ

+==)()(2 , (6)

∫∞

=Too

dTTibT

xTV ')'()(

1

1

εε . (7)

All the values at the right hand side of Eq. (5)–(7) are known, or can be found

experimentally. Results of the calculations according to the Eq.(5)–(7) based on the

data of Fig. 1 are shown in Fig. 3. Values of Q=0,76 eV and the pre-exponential

factor go=0,18 Ω-l⋅m-1 were obtained from the steady-state values of the isothermal

charging currents and voltages.

As one can see from Fig. 3, the homocurrent and the heterocurrent form two

broad peaks with almost coinciding maxima. The heterocharge decays faster in the

low-temperature region, while the homocharge remains relatively stable. This is

probably the reason of the initial increase of the thermally stimulated potential (see

curve 3 in Fig. 1 and curve 3 in Fig. 3). The current inversion in TO and IO modes is

caused by the change of ratio between the homocurrent and the heterocurrent at high

temperatures (curves 1 and 2 in Fig. 3).

It is known that the inversion of the TSD current can be caused by the over-

polarization, i.e. by appearing of the additional heterocharge in the field of the

homocharge, the voltage in this case should be decreasing [8]. However, this has not

been observed experimentally in our case (Fig. 1). From the other side, the initial

growth of the electret potential during the heating cannot be caused by increasing of

the surface charge density σ, because the charges in this case should move in the

direction opposite to that of the electric field created by the charges, that is not

possible. Therefore, the first TSD peak and the corresponding increase of the electret

potential (Fig. 1) are caused by the faster decay of the heterocharge (polarization) in

comparison with the homocharge. It is possible that in PVDF not all polarization is

destroyed during the first stage of the heating, but only the least stable part.



Fig. 3 Temperature dependences of the homocurrent (1), the heterocurrent (2) and the voltage across the sample (3) calculated according to the model.

Thus, the heterocharge in PVDF films is not sufficiently stable, therefore its

lasting conservation is possible only in presence of the stabilizing field of the

homocharge. We believe that many specific properties of PVDF are related to a

fortunate combination of the large dipole moment of CH2–CF2 units (2.1 D) [3]

promoting formation of the important heterocharge and the high electron affinity of

the fluorine atoms (3.37 eV) promoting creation of the stable homocharge. Although

the electret state in PVDF is unstable, the self-balanced mutual relaxation of the

homocharge and the heterocharge is delayed due to the stabilizing action of the

homocharge.

It is assumed in the theory of electrets [1,4,5] that the homocharge and the

heterocharge decay exponentially with the temperature dependent time constants.

Therefore, the following expressions are valid for IO and IS modes

−−=

111 exp)(

ττσ ts

ti o, (8)

−−=

222 exp)(

ττtP

ti o, (9)

20 40 60 80 100 1200

2

4

6

8

0

10

20

30

40

2

Volta

ge, V

3

2

1

Curre

nt, µ

A/m

2

Temperature, oC



=

kTQ

gT

o

o exp)(1εε

τ , (10)

=

kTWT o exp)(2 ττ , (11)

where W is the activation energy of the heterocharge relaxation, τ1 and τ2 the

corresponding time constants.

Applying Eq.(8)–(11) to the experimental curve shown in Fig. 2, we calculated

the following parameters of the homocharge and heterocharge relaxations: the

activation energies (Q=0,76 eV and W=0,54 eV), the characteristic frequencies (f2=

1/τo =7,4 MHz and f1=(go/εoε)=1,7 GHz), the time constants at 20°C (τ1 =31 000 s

and τ2 =2 800 s). The results indicate that the homocharge is much more stable than

the heterocharge.

4. Conclusion

It is shown how to extract depolarization homocharge and heterocharge currents

from experimentally measured TSD current and reveal the relaxation behavior of the

both components. The application of different TSD modes supplemented with the

isothermal depolarization currents allowed to find important parameters of the

relaxation processes, such as activation energies, characteristic frequencies and time

constants.

The uniform field approximation assumed in this paper is justified only in the

case of high poling fields exceeding 50–60 MV/m. At lower fields, one should

consider injection of charge carriers in the bulk resulting in non-uniformity of the

field and the polarization in the thickness direction [6,8].

The proposed method allows to analyze interrelation between the homocharge

and the heterocharge not only in PVDF, but also in other dielectrics. Introduction of

polar groups with simultaneous creation of deep traps for the charge carriers might be

a promising procedure for increasing stability of the residual polarization in polar

polymer dielectrics. Therefore, if the appropriate conditions exist for creating and

trapping of the homocharge, then the high level of the residual polarization can also

be ensured for a long period of time.



References

1. G. M. Sessler (ed.), Electrets, Vol. 1, Third Edition (Laplacian Press, Morgan

Hill, 1999).

2. M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics

and Related Materials (Oxford University Press, Oxford, 1977, reprinted 1996).

3. Furukawa T. Ferroelectric Properties of Vinylidene Fluoride Copolymers

//Phase Transitions, 1989.-Vol. 18, P. 143-211.

4. B. Gross, J. Chem. Phys., 17, 866 (1949).

5. A. N. Gubkin, Electrets (Nauka, Moscow, 1987).

6. von Seggern H. and Fedosov S. N. Importance of screening charge dynamics

on polarization switching in polyvinylidene fluoride //Appl. Phys. Lett. – 2007 -

v.91.- P. 062914

7. J. van Turnhout, Thermally Stimulated Discharge of Polymer Electrets

(Elsevier, Amsterdam, 1975).

8. Sergeeva A. E. Corona poling of a ferroelectric polymer /A. E. Sergeeva, S.

N. Fedosov, J. A. Giacometti // Polymers and Liquid Crystals, Proceedings of. SPIE.-

1999.-4017.-P. 53—58.

9. Kochervinskii V. V. Mechanism of Polarization and Piezoelectric Behavior in

Crystallizable Ferroelectric Polymers from the Standpoint of Propagation of Soliton

Waves //Высокомолекулярные соединения. Сер.С.-2006.-т.48,1.-С. 38-57.

10. Малышкина И.А. Исследование процессов диэлектрической

релаксации в сополимерах винилиденфторида и гексафторпропилена / И.А.

Малышкина, Г.В. Маркин, В.В. Кочервинский //Физика твердого тела.- 2006.- т.

48, вып. 6.-С.1126 1129.

11. Fridkin V.The Switching in one Monolayer of the Ferroelectric Polymer. /

V. Fridkin, A. Ievlev, K. Verkhovskaya, G. // Ferroelectrics.-2005.- v. 314.-P. 37-46.

12. Bauer F. Ferroelectric Polymers for High Pressure and Shock Compression

Sensors //Mat. Res. Soc. Symposium (Materials Research Society).-2002.-P. 698.



UDC 537.9

J11309-256

Galichyan T.A. PROPAGATION OF ELASTIC WAVES IN BILAYER FERRITE-

PIEZOELECTRIC STRUCTURE

Novgorod State University after Yaroslav the Wise

173003 Velikiy Novgorod, Russia

Introduction

Structures based on ferrites and piezoelectrics are interesting, because there are

effects which can be absent in individual ferrite and piezoelectric components and

appears due to mechanical interaction between the ferrite and piezoelectric

subsystems. Magnetoelectric (ME) effect is one of such effects, which consists in

inducing the electric polarization in an applied magnetic field, or vice versa inducing

the magnetization in an applied electric field. ME effect in ferrite-piezoelectric

structure is caused by the elastic interaction of magnetostrictive and piezoelectric

subsystems. The frequency dependence of the effect is defined by the dispersion

relation for such type of waves, because ME effect in composites is caused by the

elastic interaction of magnetostrictive and piezoelectric phases. The distribution of

elastic waves in bilayer medium significantly differs from the distribution in

homogeneous medium. Attempts were made earlier in Ref. [1] to consider ME effect

in such structures, but in the same time, a supposition was made, that the amplitude

of oscillations was not changing its direction perpendicular to the line of the partition.

This suggestion can be used in the description of the effect for rather thin layers, with

some accuracy. At more detailed examination this method, used in Ref. [1], reduces

to the method of effective parameters, which was first used in describing the

frequency dependence of (ME), effect in Refs. [2,3]. Recently, in Ref. [4], the

dispersion relation was obtained for a structure, representing a thin film grown on a

semi-infinite substrate. In this paper, the distribution of the elastic waves in bilayer

ferrite-piezoelectric structure was investigated given the fact that the amplitude of the



wave is changed on the thickness of the sample. The dispersion relation for the elastic

waves of acoustic range was obtained. Under the limiting transition it is shown that

the dispersion relation, when one of the layers is thinner than the other, passes to a

dispersion relation of one, or the other media.

Model

As a model, we consider a structure, composed of piezoelectric and

magnetostictive phases mechanically interacting on the boundary (fig.1). The

alternating magnetic field exits elastic oscillations in the ferrite component

transferring through the boundary of the partition to the piezoelectric component,

which brings to interconnected oscillations of ferrite and piezoelectric subsystems

and an electric field is occurred.

fig. 1. 1- piezoelectric phase with 𝑡 𝑝 thickness, 2- magnetostrictive phase

with 𝑡 𝑚 thickness, 3- electrodes.

The amplitude of the vibrations will be inhomogeneous, perpendicular to the

line of the partition, as there is a sharp boundary between the ferrite and piezoelectric

layers.

Z

3

1

X

m

p

2



Dispersion relation

Take into account inhomogeneity along the Z axis, the solution of the equation

for the displacement vector of the media will be presented as planar waves, which

amplitude is changed on the thickness of the sample which is shown in (1).

𝑢(𝑥, 𝑧) = 𝑔(𝑧) 𝐴 𝛼 𝛼

𝛼 cos(𝜔𝑡 − 𝑘𝑥) + 𝐵 𝛼 sin(𝜔𝑡 − 𝑘𝑥), (1)

where the index α is accordingly m for the ferrite and p for the piezoelectric.

𝐴 𝛼 , 𝐵 𝛼 are the constants of integration.

The substitution of this expression into the expression for the motion of the

media carries to the equation for function 𝑔(𝑧) 𝛼 . Solution of this equation depends

on the ration of elastic wave velocity in ferrite and piezoelectric phases. For the sake

of definiteness, we choose a structure of ferrite and piezoelectric, namely, ferrite-

nickel spinel and lead zirconate titanate (NFO-PZT). In this case, the velocity of

propagation of elastic waves in the ferrite will be higher than that in the piezoelectric

phase.

The boundary conditions, namely, the normal components of stress tensor on

top and bottom surface are zero, the displacement of ferrite and piezoelectric media

on interface are the same, so as the shear stresses. These boundary conditions produce

a system of equations, consistency of equations give the dispersion relation in the

following form:

𝑌 𝜒 𝑚 𝑚 𝑡ℎ( 𝜅 𝑚 ) = 𝑌 𝜒 𝑝

𝑝 𝑡𝑔( 𝜅 𝑝 ). (2)

𝜅 𝛼 = 𝜒 𝛼 𝑡 𝛼 is a non-dimensional parameter, 𝑡 𝛼 is the thickness of ferrite and

piezoelectric phases accordingly 𝜒 2 = −2(1 + 𝜈) 𝜔

2

𝑉 𝑚 𝐿2 − 𝑘2

𝑚 ,

𝜒 2 = 2(1 + 𝜈) 𝜔

2

𝑉 𝑝 𝐿2 − 𝑘2

𝑝 . 1𝑉 𝑚 𝐿2 = 𝜌 𝑚

𝐸 𝑚, 1𝑉 𝑝 𝐿2 = 𝜌 𝑝

𝐸 𝑝 , 𝑉 𝑚 𝐿

, 𝑉 𝑝 𝐿 are the velocities of

longitudinal waves, respectively in the ferrite and piezoelectric media, 𝑌 𝑚 , 𝑌 𝑝 are the

Young's modulus of ferrite and piezoelectric phases accordingly.

Here, (2) defines the dependence of angle frequency ω from the wave vector k,

in an implicit form, at the distribution of elastic waves in bilayer ferrite-piezoelectric

structure.



In limited case, when the thickness of the piezoelectric phase is much less

than the thickness of the ferrite phase, we have relation in (3), which presents the

dispersion relation for the ferrite phase.

𝜔 = 𝑉𝐿 𝑘 𝑚 (3)

In case, when the thickness of ferrite phase is much less than the piezoelectric

phase, we have the (4), which presents the dispersion relation for piezoelectric phase.

𝜔 = 𝑉 𝑝 𝐿 𝑘 (4)

For thin ferrite and piezoelectric phases, when 𝜅 𝑚 = 𝜒 𝑡 ≪ 1 𝑚

𝑚 and

𝜅 = 𝜒 𝑡 𝑝 𝑝

𝑝 ≪ 1, the approximate expression is obtained, for the dispersion relation,

which is shown in (5).

𝑌 ⋅ 𝑚 ( 𝜒)

2 ⋅ 𝑚 𝑡 𝑚 = 𝑌 ⋅ (

𝑝 𝜒) 2

𝑝 ⋅ 𝑡 𝑝 (5)

Taking into account the expressions for 𝜒 𝑚 , 𝜒 𝑝 and (5), after simple

transformations, the dispersion relation is obtained in the following form, shown in

(6).

𝜔 = 𝑌⋅ 𝑚 𝑡+ 𝑌⋅ 𝑡 𝑝 𝑝 𝑚

𝜌⋅ 𝑡 𝑚 𝑚 + 𝜌⋅ 𝑡 𝑝 𝑝 (6)

Thereby, for small thicknesses of ferrite 𝑡 𝑚 and piezoelectric 𝑡 𝑝 , the linear

dependence of angle frequency from the wave vector is retained.

Conclusion

The amplitude of the wave is changed on the thickness of the sample, on

direction perpendicular to the line of the partition, in the distribution of the elastic

waves in bilayer structure. The boundary conditions produce a system of equations

which give the dispersion relation for the planar oscillations. Nonlinear relation

between the angle frequency ω and the wave vector k is defined from that expression.

In limited cases, when one of the layers is thinner than the other, this relation passes

to a dispersion relation of one, or the other media, thus the velocity of the distribution

of elastic waves is less than the velocity of the distribution of elastic waves in ferrite,

but more than in piezoelectric phase. In limited cases, this dispersion relation passes

into dispersion relation for ferrite and piezoelectric accordingly.



References:

1. M. I. Bichurin, V. M. Petrov, S. V. Averkin and A. V. Filippov,

“Electromechanical resonance in magnetoelectric layered structures,” Physics of the

Solid State, 2010, vol. 52, no. 10, pp. 1975-1980.

2. Fillipov D.A., Bichurin M.I. Petrov V.M., Laletin V.M., Poddubnaya N.N and

Srinivasan G., “Giant magnetoelectric effect in composite materials in the area

electromechanical resonance,” Technical Physics Letters, 2004, vol. 30, no. 1, pp. 15-

20.

3. Fillipov D.A., Bichurin M.I., Petrov V.M., Laletin V.M. and Srinivasan G.,

“Resonant amplification of the magnetoelectric effect in composite ferrite-

piezoelectric materials,” Physics of the Solid State, 2004, Vol. 46, no. 9, pp. 1621-

1627.

4. Fillipov D.A., “Magnetoelectric effect in thin-film magnetostrictive-

piezoelectric structures grown on a substrate,” Physics of the Solid State, 2012, vol.

54, no 6, pp 1112-1115.

J11309-257

Klinkerman R.V.

"EXPANSION" of the UNIVERSE

Branch of Siberian Federal University, Krasnojarsk region, Zheleznogorsk,

Kirova st, 12a, 662971

In this paper we show that the Fridman model of Universe expansion do not

lead to physical consequences for inner objects of Universe. We offer another ways

for explanation of Universe evolution, red shift and asymmetry of elementary

particles composition.

Key words: evolution, scale factor, red shift, neutrino, condensation, chaos,

conservation laws, charges.

Strange position is observed in cosmology regarding an explanation of red shift.

For a basis the model of Fridman-Einstein or its modifications is taken. In such



models changes a scale factor for time and spatial coordinates (further « system of

coordinates »), covering all Universe. This factor at one average density of substance

on a present instant monotonically increases from 0, at another – at first increases,

then decreases up to 0. Of such factor behaviour the conclusion is done, that the

Universe has arisen from a condition with infinite density, and then it extends and

density falls. In the second case after a while the Universe starts to be compressed up

to infinite density.

But all of physicist know that variation of scale of system of coordinates does

not influence the physical processes proceeding in it. It is possible to pass simply

from one system to another; thus the numerical description of physical process will

be another, but its essence will not change. Together with a scale factor

synchronously change all the sizes, down to elementary particles, so the result is zero.

In another way it is possible to tell, that units of measurements of space and time

have changed, but in other units process have the same character.

How then red shift is explained? In an instant of emission of light in a distant

galaxy the system of coordinates with a scale factor and appropriating frequency of

light is fixed. Arrival of light to the observer consider in system of coordinates with

the increased scale factor because of its evolution and, accordingly, with the

decreased frequency. Required red shift turns out. But in fact in new system of

coordinates the measuring equipment, substance of which it consists, atoms on which

the arrived radiation operates, all has other sizes and works in other pace. So not any

shift will be fixed.

From the logic point of view such "expansion" is fictitious: physics are invariant

to variation of scale factor for the whole of system of coordinates. Supporters of the

concept of "expansion" know this difficulty, and they have invented a subterfuge:

they say that extends only space between galaxies, but inside of galaxies all is defined

by interaction between their components, therefore inside the scale factor does not

operate. If to accept this sight the physics are really change: in ones sites there is one

scale, in others another. But in my opinion such sight carries specially fitting

character: force logic to receive the necessary result.



Here except for a mathematical reason about emptiness of the standard approach

there is still a sensation of absurd from the philosophical point of view: the Universe,

all the world cannot to be described by a certain system of the equations. For this

purpose it is necessary to leave the Universe and to look at it as on the limited

experimental object.

But red shift exists, and it is necessary to explain it somehow. I consider that the

explanation should lay in the physics, instead in superficial mathematics. By present

time I see two versions. The first – “tiredness” of light because of small space non-

ideality. The basic objection against this hypothesis – erosion of galaxies images -

can be overcome by a certain model of a structure of space.

The second - the Universe extends, incorporating surrounding chaos. Similar as

the dewdrop grows at night cooling air.

At this version I see two subversions: flat space with borders and the curve

closed boundless space.

“Tiredness” of light is detailed thus: the space will consist of the associated pairs

' neutrino - antineutrino ' with a spin 0. Such space should be very ' smooth '. These

pairs, being bozons, form boze-einstein condensate. Light, cooperating with this

pairs, translates them in overcondensate condition, losing thus energy. As the space is

homogeneous and isotropic the lateral component, by virtue of symmetry, do not

exist and the image of distant galaxies is not eroded.

The possibility of this pairs «looseness» is not exclude similar that as under

influence of light electrons of atoms pass to higher orbits. Such approach reminds the

theory of superconductivity BCS.

The given interaction concerns to family electroweak.

Interaction with chaos most natural to consider as occur in stars where greater

temperatures and intensive thermal movement take place. The last is in the big degree

chaos. But the chaos should be understood in more general sense: it is something that

does not give in to laws. Therefore in thermal movement should be more chaos, than

simple movement of particles, there should be something breaking this picture,



leaving from its frameworks. It is communication with external (to Universe)

absolute chaos.

What is such, more chaotic, than thermal movement? This is violation of the

laws of conservation, in particular, energy, but not only.

The capital question takes place: why our Universe is asymmetrical concerning

particles and antiparticles? Why there is a lot of protons in it, but there are no

antiprotons, there is a lot of electrons, but there are no positrons? If it was symmetric,

substance would annihilate and all disappeared (except for neutrinos and

antineutrinos to which is no where to annihilate). In reactions with elementary

particles the laws of conservation of lepton and barion charges take place: if the

lepton is born also the antilepton is born; if the barion is born also antibarion is born.

It turns out: on one hand symmetry take place, on other – it is not present. Here the

idea of the chaos breaking laws of conservation of charges should work: in rare cases

at reactions with elementary particles in our world remain protons and electrons, and

antiprotons and positrons are absorbed by external chaos. Why protons and electrons,

instead of their antiparticles? Probably an existing substance here influences: the born

protons and electrons enter systems of identical particles with existing, and this

rescues them from annihilation or from absorption by chaos.

Such state of affairs has very the general character. The life follows about on the

same way. Living beings absorb necessary substances from an environment and

throw out unnecessary. But in living beings the given processes well work only

certain time; further they are upset, and the being dies. Maybe the same fate waits for

Universe.

References:

1. Landau, L.D., Livshitz, E.M. Field theory. – Moscow: “Nauka”, Glavnaja

redaktzija fiziko-matematicheskoy literatury, 1988, s. 457 - 469.

2. Novikov, I.D. Evolutzia Vselennoj. - Moscow: “Nauka”, Glavnaja redaktzija

fiziko-matematicheskoy literatury, 1983, s. 23.



3. Arhangelskaja, I.V., Rozental, I.L., Chernin, A.D. Kosmologia i fizicheskij

vacuum. – Moscow: “KomKniga”, 2007, s. 14 - 22.

4. Hoyle, F. Frontiers of Astronomy. – Published by The New American Library

of World Literature, Inc., 501 Madison Avenue, New York 22, 1957, pp. 270 - 313.

UDC 517.98

J11309-259

Ahramovich M.V.

q - COMMUTATION OF LINEAR OPERATORS

Tavrida National V.I. Vernadsky University Simferopol, Vernadsky ave.

In this paper some properties of a pair of linear operators satisfying the relation

of q - commutation qBAAB = are investigated.

Key words: the commutation of operators, the "wild" problem, the measurable

operator, the von Neumann algebra.

Introduction

For arbitrary linear operators A and B in a complex vector space V consider

the quadratic relation in general form:

( ) 0, 7652

4322

12 =++++++= IcBcAcBcBAcABcAcBAP ,

where the leading coefficients satisfy the following inequality:

04321 >+++ cccc .

It is known (see [1]) that if the space V is finite then by using an affine change

of variables the given quadratic relationship can be reduced to one of the following

canonical forms:

1. 02 =A .

2. IA =2 .

3. 02 =+ BA .

4. qBAAB = .

5. qBAIAB =+ .



6. [ ] ABA =, .

7. [ ] 2, ABA = .

8. [ ] IABA += 2, .

9. [ ] BABA += 2, .

Some of these relations were considered by many authors ([1], [2]).

In this paper we investigate the relation of q -commutation of two linear

operators A and B :

qBAAB =

which generalizes the classical relation of commutation:

BAAB = .

Linear operators in finite-dimensional vector spaces

Let V is a finite-dimensional vector space and ( )VB is an algebra of all linear

operators in V .

The finite set ( )mAAA ,,, 21 of operators in ( )VB is called indercomposable

if the vector space V can not be represented as a direct sum of nontrivial subspaces

NMV ⊕= ,

each of which is invariant with respect to each operator mkAk ,,2,1, = .

Bounded operators have the following criterion of indercomposable.

Statement 1. ([3]) A finite set of operators in ( )VB is indecomposable if and

only if the following condition

=

==

RRmkRARA kk

2

,,2,1,

leads to 0=R or IR = .

We say that two finite sets of liners operators ( )mAAA ,,, 21 and

( )mBBB ,,, 21 over the finite-dimensional vector spaces V and W are similar if an

invertible operator WV →:S exists, such as:

mkBSSA kk ,,2,1,1==− .



Consider the problem of classification of arbitrary finite sets of linear operators

in a finite dimensional vector space up to a similarity transformation. This problem is

one of the oldest problems in linear algebra and is a complex ("wild") for a pair of

operators in general form. Therefore additional conditions (such as nilpotency, self-

adjoint, commutation, etc.) impose on the operators. With some restrictions the

classification problem manageable i.e. is a "tame" and other conditions leave the task

"wild". We say that the problem of classification up to similarity transform a set of

linear operators is "wild" if it contains a sub-task of classification up to a similarity

transformation a pair of operators without additional conditions.

In paper [4] has been considered the problem of the canonical form of a pair of

nilpotent operators ( )BA, with additional relations involving q -commutation

relation:

==

==

.0

0,02

32

qBAABAB

BA

It is proved that the problem of classification of the pair of operators up to a

similarity transformation is a "wild" problem.

To prove this the following constructions have been used:

Let ( )BA, is an arbitrary pair of operators in ( )VB . Denote

kjj

k

jjk ,,1,,:

1==⊕= ∑

=VVVV

Let’s consider the following operators:

1. 25 VV →C: ,

=

BAIIIIII

C0

0.



2. ( )5VBD∈ ,

=

IdId

IdId

Id

D

5

4

3

2

1

00000000000000000000

where Cdi ∈ , 0≠id ,

ji dd ≠ if 51 ,,i,jj,i =≠ .

3. ( )151 VBY ∈ ,

=

000000

00

1

IY where I is identity and 0 is zero operators in 5V

.

4. ( )152 VBY ∈ ,

=

0000000

2

DIY .

5. 2153 : VV →Y , ( )003 CY = where 520

0000000000

0 ,=

= .

6. ( )32VBY ∈ ,

=

12151515

322152

22151

00000

qYYYY

Y

,,

,,

,

where 0, ≠∈ qq C ,

( )0000 15,2 = , ( )T0000 2,15 = ,

=

0000

0 2,2 ,

=

000000000

0 15,15 .

7. ( )32VBX ∈ ,

=

15152151515

15222152

2151515

000000

00

,,,

,,,

,, IX where

=

II

II

000000

.

The operators X and Y satisfy next relations:



==

==

qYXXYXY

YX0

02

32

.

In addition, we have the following result:

Theorem 1. If ( )YX , and ( )YX ~,~ are two pairs of linear operators in 32V constructed

respectively by pairs of linear operators ( )BA, and ( )BA ~,~ in V then the pairs ( )YX , and

( )YX ~,~ are similar if and only if the pairs ( )BA, and ( )BA ~,~

are similar.

Theorem 2. The pairs of operators ( )YX , and ( )YX ~,~ are indecomposable in 32V if and

only if the pairs of operators ( )BA, и ( )BA ~,~ are indecomposable in V .

q -commutation of unbounded operators

Let H is a Hilbert space over the field of complex numbers С . Let’s consider

two self-adjoint operators HH →:, BA with the q -commutation relation:

0,, ≠∈= qqqBAAB C .

If the operators A and B are self-adjoint then the parameter q is equal to 1 or

1− that is the relation q -commutation of self-adjoint operators reduce to their

commutation and anticommutation (see [5], [6]).

Bounded self-adjoint operators A and B commute (anticommute) if and only if

they commute (anticommute) on each vector H∈ξ :

( ) H∈∀−== ξξξξξ ,BAABBAAB .

If the operators A and B are unbounded then the pointwise commutation (or

anticommutation) may not exist. For example, if ( ) ( ) 0=∩ BDAR . Therefore the

definition of a commutation (as the definition of an anticommutation) of unbounded

self-adjoint operators cannot be entered directly.

Let A and B are arbitrary self-adjoint operators in H . Denote by ( ) R∈λλAE

and ( ) R∈µµBF the spectral families of orthogonal projections of the operators A

and B , respectively. For arbitrary ∞<≤ ML,0 let’s construct the operators:



( ) ( )∫∫−−

==M

MBM

L

LAL dFBdEA µµλλ , .

We say that the self-adjoint operators

( ) ( )∫∫∞

∞−

∞

∞−

== µµλλ BA dFBdEA ,

commute (anticommute) if for any ∞<≤ ML,0 commute (anticommute) the

bounded operators

( ) ( )∫∫−−

==M

MBM

L

LAL dFBdEA µµλλ , :

( )LMMLLMML ABBAABBA −== .

Note that in the case of bounded self-adjoint operators this definition is

equivalent to the pointwise commutation (and anticommutation).

The anticommutation of self-adjoint unbounded operators have been studied in

[5], [7]. In the paper [5] has been shown that two self-adjoint (in general case are

unbounded) operators A and B anticommute if and only if they anticommute on an

invariant dense set in H of integral joint vectors of A and B .

q -commutation of measurable operators

Let’s consider q -commutation of measurable and locally measurable operators.

Let M is a von Neumann algebra i.e. is a *-algebra with a unit in ( )HB closed

in the weak operator topology.

Let ( )MS is an algebra of measurable operators affiliated with M and let

( )MLS is an algebra of locally measurable operators affiliated with M . These

algebras are closed under the operations of a strong sum STST +=+ : and a strong

product TSST =⋅ : of operators.

The algebras ( ) ( )MMSM, LS, related together by the following inclusions:

( ) ( )MMSM LS⊆⊆ .

The commutation of measurable and locally measurable self-adjoint operators

affiliated to an arbitrary von Neumann algebra was studied in [8], [9].



In [10] the anticommutation of measurable operators are investigated. For

measurable self-adjoint operators with the following results have been obtained:

Theorem 3. Let BA, are self-adjoint operators of in ( )MS . If the operators

AB and BA− are the same on any strongly dense subspace

( ) ( )BADABDDD ∩=⊆1 then the operators A and B anticommute in the

algebra ( )MS :

ABBA ⋅−=⋅ .

Theorem 4. The two self-adjoint operators ( )MSBA ∈, anticommute as

elements of the algebra if and only if they strongly anticommute.

For a locally measurable anticommuting self-adjoint operators ( )MLSBA ∈,

an invariant locally measurable subspace D , such that

( )BAHD b ,⊆

( ( )BAHb , is a set of bounded joint vectors of operators A and B ) has been

constructed. The basic properties of this subspace have been researched. By using the

properties of the subspace D the following results for the locally measurable

anticommuting self-adjoint operators have been obtained:

Theorem 5. Let A and B are locally premeasurable operators affiliated with a

von Neumann algebra M , D is a locally measurable subspace in H , such that:

1. ( ) ( )BADABDD ∩= .

2. DDBDDA →→ :,: .

3. ξξ BAAB −= for any D∈ξ .

Then ABBA ⋅−=⋅ .

Theorem 6. The two self-adjoint operators ( )MLSBA ∈, anticommute if and

only if they anticommute as elements of the algebra.

Conclusion

The main result of this work is the studying of some properties of a pair of

operators BA, satisfying q -commutation relation qBAAB = .



It is proved that the problem of classifying up to a similarity transformation of

the pair of operators satisfying the conditions:

==

==

.0

0,02

32

qBAABAB

BA

is a "wild" problem.

It has been considered the q -commutation of measurable and locally measurable

self-adjoint operators. It has been proved that two self-adjoint operators

( )MSBA ∈, ( )( )MLSBA ∈, anticommute as elements of the algebra ( )MS

( )( )MLS if and only if they strongly anticommute.

References:

1. Ostrovskyi V., Samoilenko Yu. Introduction to the theory of representations

of finitely presented *-algebras. Representations by bounded operators. Rev. Math &

Math. Phys., Vol. 11. - Gordon & Breach, London. - 1999. - 261 p.

2. Наймарк М.А. Нормированные кольца. М.: Наука. Главная редакция

физико-математической литературы. - 1968. - 664 с.

3. Ахрамович М.В., Муратов М.А. О классификации пары q -

коммутирующих операторов в конечномерном линейном пространстве //

Таврич. вестник информатики и математики. - 2010. - 2. - С. 17-26.

4. Ахрамович М.В., Муратов М.А. Задача класифікації пари q -

коммутуючих нільпотентних операторів // Наукові вісті НТУУ "КПІ". - 2011. -

1. - С. 42-47.

5. Самойленко Ю.С. Спектральная теория наборов самосопряженных

операторов. - Киев: Наукова Думка. - 1984. - 232 с.

6. Kamei E. Operators with skew commutative cartesian parts // Math.

Japonica. - Vol. 25, No. 4. - P. 431–432.

7. Vasilescu F.-H. Anticommuting self-adjoint operators // Rev. Roum. Math.

Pures Appl. - 1983. - Vol. 28, No. 1. - P. 77-91.



8. Муратов М.А. К вопросу о коммутируемости локально измеримых

операторов, присоединенных к алгебре фон Неймана // Учёные записки

Таврического университета им. В.И. Вернадского. Серия "Математика.

Механика. Информатика и Кибернетика". - 2006. - Т. 19(58), 2. - С. 52-62.

9. Муратов М.А., Самойленко Ю.С. О коммутируемости измеримых

операторов, присоединенных к алгебре фон Неймана // Учёные записки

Таврического университета им. В.И. Вернадского. Серия "Математика.

Механика. Информатика и Кибернетика". - 2007. - Т. 20(59), 1. - С. 70-79.

10. Ахрамович М.В. Об антикоммутируемости измеримых операторов,

присоединенных к алгебре фон Неймана. // Ученые записки Таврического

национального университета им. В.И. Вернадского. Серия «Физико-

математические науки». - 2012. - Т. 25(64), 2. - С. 1-14.

UDC. 51-72

J11309-260

Mocretsova I.S., Rebro I.V., Mustafina J.A.,

Galitsyna T.A, Koroteyeva E.A, Perepechenova T.N.

MATHIMATICAL CALCULATION OF THE MASS FLOW OF

NITROBENZENE FLOWING THROUGH THE PIPELINE

Volzhskiy polytechnical institute (branch)

Volgograd state technical university, www.volpi.ru

This article deals with methods of mathematical calculation in designing a

pipeline and selecting hydraulic machines.

Key words: pipeline designing, selecting hydraulic machines.

One of the most sufficient problems nowadays is controlling hydrosystems.

Thus, unstable processes in pipeline systems may result in overload and cavitation in

the equipment and, consequently, improper junctions, leakages, damage and

destruction of the system elements. The most important feature of sufficiently non-



steady process becomes the speed of impact pulse spreading which is necessary to

know for preventing a hydrosystem from dynamic overload. The study of hydraulic

impact connected with local pressurizing is based upon the research on ordinary

pipelines by N.E. Zhukovsky. [1]

The significance of this problem for the contemporary world causes us to

consider methods of mathematical calculations in designing a pipeline and selecting

hydraulic machines.

A practical task is supposed to be considered: nitrobenzene of 20 0С is draining

down the pipeline with the diameter of 25×2,5. The starting point of the pipeline is

200 mm higher than the terminal point. The length of the horizontal part of the

pipeline is 240 m. Mass flow of nitrobenzene is considered to be calculated.

The theoretical basis before starting to solve the task is as follows: the actual

environment within the hydrosystem is considered as a dynamic system where the

principle of motion and substance quantity conservation, equations of continuity and

state can be applied. As for the equation of state there exists a supposition: the

density of the liquid and the area of a pipe cross-section depend on the intrinsic

pressure, according to Hooke's law. Linear law of deformation of the pipe material

and liquid implies isothermal process. The inertia of pipe walls is not considered.

Friction stress on the pipe walls, in accordance with hypothesis of quasistandardness,

depends on the instantaneous speed of the liquid and coefficient of the hydraulic

resistance λ in Darcy-Weisbach formula. [1]

We possess the following original data: pipeline diameter: dн δ× = 25×2,5 mm

(external diameter×wall thickness); pipeline length: = 240 m; liquid: nitrobenzene t

= 20 0С.; difference between the starting point and terminal point of the pipeline h =

200 mm. We are supposed to calculate mass flow of nitrobenzene G[kg/s].



Figure 1. Diagram of the pipeline calculation.

Two methods of mathematical calculations of mass flow of nitrobenzene are to

be considered.

The 1st method: analytical.

Mass flow of the liquid can be found with the help of this formula:

ρω ⋅⋅=сеч

FG [kg/s], where ω [m/s] – the speed of nitrobenzene, it is

unknown, F [m2] – the free area of the flow, it can be calculated using this formula:

4

2

BdF π

= , ρ [kg/m3] – the density of nitrobenzene, it can be found in the reference

literature.

The unknown speed of nitrobenzene is possible to find using Reynolds’ formula

of number calculation:

µρω ⋅⋅

= BdRe ⇒ ρµω⋅⋅

=Bd

Re [m/s] (1), where µ [Pa*s] is the

coefficient of dynamic viscosity of nitrobenzene at t = 20 0С.

However, the exact value of Reynolds’ number is unknown. Let us suppose that

the condition in the case of voluntary flow is laminar. The full pressure of the flow is

accompanied by linear loss and is caused by the difference of terminal points height

of the pipeline h = 200 mm. The full pressure is to be found with the help of Darcy-

Weisbach equation:

h=2

l=

dН×δ

25×2,5m



gdh

B 2

2ωλ ⋅⋅=

[m] , where h = 200 mm = 200 310−⋅ [m]

The coefficient of the friction resistance Re64

=λ is for the liquid, flowing

through a straight round pipe in the laminar condition.

Consequently: 3

2

102002

−⋅=⋅gd

B

ωλ or 2,02Re

64 2

=⋅⋅gd

B

ω (2)

Let us place equation (1) in the equation (2), and thus we obtain the following

2,02

Re

Re64

2

=

⋅⋅

⋅⋅g

dd

B

B

ρµ

.

Brackets are opened: ⇒=⋅⋅⋅

⋅⋅ 2,02

ReRe64

2

22

gddBBρµ 2,0

2Re6423

2

=⋅⋅⋅⋅⋅gd

Bρ

µ .

µρω ⋅⋅

= BdRe is placed and we obtain

2,02

6423

2

=⋅⋅⋅⋅⋅⋅⋅⋅µρµρω

gdd

B

B or 2,0

264

2=

⋅⋅⋅⋅⋅gd

Bρ

µω .

The speed of the flow is determined by the formula: 6422.0 2

⋅⋅⋅⋅⋅

=µρω gd

,

where

][1020205,22252 3 mmmdd HB−⋅==⋅−=−= δ

The original data are placed and we obtain the following:

( ) sm /059,064240101,2

81,92120310202,03

23

=⋅⋅⋅

⋅⋅⋅⋅⋅= −

−

ω

Thus we obtain the mass flow of nitrobenzene:

( ) ]/[022,012034

102014,3059,023

skgFG сеч =⋅⋅⋅

⋅=⋅⋅=−

ρω



The 2nd method: geometrical.

When free flow of nitrobenzene through straight round pipe in the absence of

local resistance the energy loss depends on the pipeline length and is caused by the

power of viscosity hard walls limiting the flow. The difference between the heights of

the pipeline terminal points is 200 mm, which is minor as compared with the length

of the pipeline 240 m. Therefore, we can presuppose that the flow condition is

laminar that is Re1 2300≤ .

Supposing Re1 = 500. To determine the speed of nitrobenzene flow we need

Reynolds’ criterion: µρω ⋅⋅

=d1

1Re .

Consequently we obtain

]/[0457,012031020102,2500Re

3

3

1 smd

=⋅⋅⋅⋅

=⋅⋅

= −

−

ρµω .

Let us calculate friction losses by the formula of Darcy-Weisbach:

gdh

2

2

1ωλ ⋅⋅=

[m] , where Re64

=λ for laminar flow.

The data are placed and we obtain: ( ) ][16,0

81,920457,0

1020240

50064 2

31 mh =⋅

⋅⋅

⋅= − .

It should be noted that the difference between the heights of the pipeline

terminal points is to be h = 200 mm = 0,2 m.

Thus, Reynolds’ number has been chosen incorrectly as it is necessary perform

the equality of h1 = h.

Supposing Re2 = 1000. The speed of nitrobenzene flow is determined:

sm /091,012031020102,21000

3

3

2 =⋅⋅⋅⋅

= −

−

ω ,

Friction losses are calculated: ( ) mh 32,0

81,92091,0

1020240

100064 2

32 =⋅

⋅⋅

⋅= −

We can conclude again that Reynolds’ number has been chosen incorrectly as

we have obtained that h2 > h , but we are considered to obtain h2 = h.



The calculations performed allow to design a graph of h dependence on Re

number:

if Re1 = 500 h1 = 0.16 m; if Re2 = 1000 h2 = 0,32 m

Figure 2. Calculation graph.

The geometrical calculation is as follows:

1. Let us mark the points with coordinates of (Re1; h1) and (Re2; h2).

2. The two points are connected with a straight line.

3. From the point h = 0,2 [m] we draw a straight line parallel to the abscissa

axis up to the point of intersection with the straight line; from the point of intersection

we draw perpendicular to the abscissa axis. We obtain the truth value in the scale Re

= 625.

Graphically obtained result of Reynolds’ number is placed in 1024,062564

==λ

and the speed of nitrobenzene flow is calculated:

]/[057,012031020102,2625

3

3

sm=⋅⋅⋅⋅

= −

−

ω Thus we determine losses in the pipeline:

( ) ][203,081,92

057,01020

2401024,02

3 mh =⋅

⋅⋅

⋅= − .

The number of losses determined is practically equal to the given condition

h



h = 200 mm = 0,2 m.

Therefore, knowing the speed of nitrobenzene flow in the pipeline, we can

determine mass flow of nitrobenzene:

( ) ]/[0215,012034

102014,3057,023

skgFG сеч =⋅⋅⋅

⋅=⋅⋅=−

ρω .

The offered methods of mathematical calculation of nitrobenzene mass flow

have demonstrated almost similar results.

References:

1. Burayeva, L.A. Mathematical modeling of hydraulic impacts in manifold

pipeline systems / autoref. candidate’s phys.-math. Sciences thesis by Burayeva L.A.

– Stavropol, 2006.

UDC 004.655

J11309-261

Buy D.B.1, Glushko I.M.2

EXTENDED OF TABLE ALGEBRA: MULTISET TABLE ALGEBRA 1 Taras Shevchenko National University of Kyiv Kyiv, Volodymyrska Street

64/13, 01601 2Nizhyn Gogol State University Nizhyn, Kropyiv'yanskoho 2, 16600

Introduction

The relational data model is nowadays in widespread use as in database

scientific research so and in practice. In its formal definition, originally proposed by

E. Codd [1], the relational model is based on sets of tuples, i.e. it does not allow

duplicate tuples in a relation. Many database languages and systems do require a

relational data mode with multi-set semantics though. There are two major reasons

for this. In the first place, relations allowing duplicate tuples are useful in many

application domains where duplicate entities can exist. In the second place, in the



relational data model removal of duplicates after implementation of operations of

projection and union assumes merge of identical elements or realization of other

labour-intensive actions.

This problem was also considered in the works of Paul W.P.J. Grefen and Rolf

A. de By [2], G. Lamperti, M. Melchiori, M. Zanella [3], H. Garcia-Molina, J.D.

Ullman, J. Widom [4], D. Buy, S. Polyakov [5]. However this question requires

specification and extension.

Multiset: basic definitions

Let’s introduce the basic concepts of multisets in terms of monograph [5].

Multiset α with basis U is a function +→ NU:α , where U is an arbitrary set,

,...2,1=+N is the set of natural numbers without zero.

Pair α>∈< na, means that element a has 1≥n duplicates in multiset α .

Let D be the universe of element of multiset bases, then power set )(DP –

universe of multiset bases. Characteristic function of multiset α is a function

ND →:αχ , the values of which are specified by the following piecewise schema:

∈

=else;,0

,if)()(

ααχα

domddd

for all Dd ∈ .

1-multisets are multisets whose range of values is the empty set or single-

element set 1. These multisets are the analogues of ordinary sets.

The operations over multisets are defined in terms of characteristic functions in

monograph [6]. Authors define operations of multiset union 1 , intersection 1 ,

difference 1\ , which build 1-multisets, and operations of multiset union All,

intersection All, difference All\ , which build multisets of general view. The

Cartesian product of multiset ⊗ , the operation )(αDist , which build 1-multiset, and

analog of a full image for multisets are defined too.

Multiset table algebra

Among the two sets that are considered, A is the set of attributes and D is the

universal domain. An arbitrary (finite) set of attributes A⊆R is called scheme. The



tuple of scheme R is the nominal set on pair R , D . The projection of this nominal

set for the first component is equal to R . The set of all tuples on scheme R is

designated as )(RS and the set of all tuples is designated as S .

The table is pair R,ψ , where the first component ψ is a finite multiset, basis

of which )(ψΘ is the set of tuples of the same scheme and other component R is a

scheme of table. Thus, a certain scheme is ascribed to every table. The set of all table

on scheme R is designated as )(RΨ and the set of all table is designated as

RR)(Ψ=Ψ .

The notation ),( ψsOcc denotes the number of duplicate tuple s in multiset ψ .

Let's agree multiset to write down as ,..., 11

knk

n ss , where ),( ψii sOccn = , ki ,...,1= ,

,...,)( 1 kss=Θ ψ is basis of multiset ψ .

Under multiset table algebra is understood algebra ΞΩΨ ,, P , where Ψ is the

set of all tables, Ξ∈∈⊆

ΨΨΨΞ ⊗=Ω ξ

ξπσ ,,,,,,,,

,,,, 21

21~,,,,,\,, Pp

RRRXRRRRRXRpRRR

P RtAllAllAll A is

signature, P , Ξ are the sets of parameters.

Let’s define the operations. The union RAll

,Ψ (intersection R

All,Ψ

, difference RAll

,\Ψ )

of the tables on scheme R is binary operation which is derived by restriction of

analogous operations All, All

, All\ over multiset on set of all tables on scheme R .

We will consider every operation separately. Bases of multisets 1ψ and 2ψ are

designated as )( 1ψΘ , )( 2ψΘ accordingly.

Hence, )()()(:, RRRRAll Ψ→Ψ×ΨΨ

, RRR AllR

All ,,, 212,

1 ψψψψ =Ψ .

Basis of multiset 21 ψψ All of the resulting table is equal to union of bases of

multisets of input tables: )()()( 2121 ψψψψ ΘΘ=Θ All . Duplicate tuples, which

appear after implementation of operation, are not removed from the result. The

number of duplicates is given by the following formula:



ΘΘ∈+ΘΘ∈ΘΘ∈

=);()(якщо),,(),(

),(\)(якщо),,(),(\)(якщо),,(

),(

2121

122

211

21

ψψψψψψψψψψ

ψψ

ssOccsOccssOccssOcc

sOcc All

where )()( 21 ψψ ΘΘ∈ s .

)()()(:, RRRRAll Ψ→Ψ×ΨΨ

, RRR AllR

All ,,, 212,

1 ψψψψ =Ψ .

Basis of multiset 21 ψψ All of the resulting table is equal to intersection of

bases of multisets of input tables: )()()( 2121 ψψψψ ΘΘ=Θ All , and the number of

duplicates is given by the following formula:

)),(),,(min(),( 2121 ψψψψ sOccsOccsOcc All =, де )()( 21 ψψ ΘΘ∈ s .

)()()(:\ , RRRRAll Ψ→Ψ×ΨΨ , RRR All

RAll ,\,\, 212

,1 ψψψψ =Ψ , де R,1ψ ,

).(,2 RR Ψ∈ψ Basis of multiset 21 \ ψψ All of the resulting table is defined as

follows: ),()(\)()\( 212121 ψψψψψψ >ΘΘ=Θ CAll , where

),(),()()(|),( 212121 ψψψψψψ sOccsOccssC >∧ΘΘ∈=> . The number of

duplicates is given by the following formula:

∈−

ΘΘ∈=

> ),(якщо),,(),(

),(\)(якщо),,()\,(

2121

21121 ψψψψ

ψψψψψ

CssOccsOcc

ssOccsOcc All , where

),())(\)(( 2121 ψψψψ >ΘΘ∈ Cs .

Let ,~: falsetrueSp → be partial predicate on the set of all tuples. Selection

over a predicate p of table on scheme R is a unary partial parametric operator Rp,σ

which compares the table with its subtable containing tuples on which predicate p is

true.

Hence, Ψ→Ψ ~:,Rpσ , )(|,, pdomRdom Rp ⊆Θ= ψψσ ,

( ) RtruespssRRp ,~)()(||,, −∧Θ∈= ψψψσ , where )(, RR Ψ∈ψ , −~ –

generalized equality (both parts of equality are simultaneously undefined or both

defined and equal).

Basis of multiset of the resulting table is defined as follows:

~)()(|)( truespss −∧Θ∈=′Θ ψψ , де ~)()(|| truespss −∧Θ∈=′ ψψψ . Depending



on the value of a predicate p on tuple s either all the duplicates of this tuple are

selected or none of them, that is: ),,(),( ψψ sOccsOcc =′ where )'(ψΘ∈s . Note that

the selection does not generate new duplicates.

Let A⊆X be (finite) set of attributes. Projection over a set of attributes X of

table on scheme R is a unary parametric operator RX ,π . The value of such projection

is the table on scheme XR , which consists from restrictions on the X of all tuples

of input table.

Hence, )()(:, XRRRX Ψ→Ψπ , ( ) XRRRX ,,, ψψπ ′= , where )(RΨ∈ψ .

Basis of multiset ψ ′ is defined as follows: )(||)( ψψ Θ∈=′Θ sXs . Duplicate

tuples, which appear after implementation of operation, are not removed from the

result. The number of duplicates is given by the following formula:

∑′=

Θ∈

=′′

sXss

sOccsOcc|

),(),(),(

ψψψ , де )(ψ ′Θ∈′s .

Join of table on scheme 1R and table on scheme 2R is a binary operation 21 ,RR

⊗

whose value is the table on scheme 21 RR consisting of all the unions of compatible

tuples of input tables. Hence, )()()(: 21212,1

RRRRRR

Ψ→Ψ×Ψ⊗ ,

2122,11 ,,,21

RRRRRR

ψψψ ′=⊗ , де ),( 11 RΨ∈ψ )( 22 RΨ∈ψ . In other words, each

tuple of 1ψ is paired with each tuple of 2ψ , regardless of whether it is a duplicate or

not. The set of tuple )()(|)( 21221121 ssssss ≈∧Θ∈∧Θ∈=′Θ ψψψ is basis of

multiset ψ ′ . The number of duplicates is given by the following formula:

),,(),(),( 221121 ψψψ sOccsOccssOcc ⋅=′ де )'(' ψΘ∈s and 21' sss = .

Let’s define the renaming operation. Rename of table on scheme R is a unary

parametric operator RR ,ξψ , where AA→~:ξ is injective partial function on the set

of attributes, which carries out renaming of attributes of input tables corresponding

the function ξ . To rename a table means to rename the attributes of its scheme, i.e. to

rename the tuples of table. Rename of tuples we will be carries out in terms [5]. Let

AA→:η be a function renaming attributes. Rename of tuples corresponding



function renaming attributes η is a function SSRs ′→:η ,

|)(),()( 21 sAAsAsRs πηη ∈= , where \domξAidξη = . Scheme R is ξ -permissible

if ∅=)\(][ ξξ domRR . Here ][Rξ is full image of set R corresponding the

function ξ . The set of all table on scheme R , where scheme R is ξ -permissible,

designated as )(RξΨ .

Rename of table on scheme R corresponding injective partial function renaming

attributes AA→~:ξ is a unary parametric operator RR ,ξψ whose domain is )(RξΨ .

The value of such rename is defined as ][],[),(, RRsRR R ηψψψ ηξ = , )(Rξψ Ψ∈ ,

where \domξAidξη = and ][, ψη RRs is full image of multiset ψ with basis )(ψΘ

regarding of function ηRs . Basis of multiset ][ψηRs is full image of set )(ψΘ

regarding of function ηRs . The number of duplicates is given by the following

formula: ),(])[,( ψψη sOccRssOcc =′ , where )(1 sRss ′∈ −η

, )(ψ ′Θ∈′s .

Let’s define the active complement operation like to [6]. Let’s define a few

auxiliary concepts for this.

The active domain of attribute RA∈ relative to the table R,ψ is the table

),(,, RD RAA ψπψ = . The saturation of the table R,ψ is the table

),(...),(),( ,,,...,,,1121

1RRRC RAAAAAARA n

nnψπψπψ

−

⊗⊗= .

Active complement of table on scheme R is a unary operator R~ which

compares table complement in its saturation. Hence, R~ : )()( RR Ψ→Ψ ,

RRCR RAll

,\),(,~ , ψψψ Ψ= , where )(RΨ∈ψ .

Statement. Any expression over multiset table algebra can be replaced by

equivalent to him expression which uses only single-attribute, single-tuple constant

table, operations of selection, join, projection, union, difference, and renaming.



Proof Really, let R,ψ be constant table, ,..., 11

mnm

n ss=ψ , ,..., 1 pAAR = be

scheme of table, ),( ψii sOccn = , and ,,...,,11 pipii dAdAs = , mi ,...,1= . We will

replace this table by the expression

.,...,

,...,

...,...,

,...,

,,...,,1

,

1

,,...,,1,

,,2,,...,,21

,

,1,,...,,11

,

11211

11211

11211

2

11211

1

⊗⊗

=

⊗⊗

⊗⊗

⊗⊗

−

−

−

−

Ψ

=

Ψ

ΨΨΨ

ΨΨ

pppi

All

pppAllm

AllAllpppAll

AllpppAll

ipAAAAAin

Rm

i

npAAAAAnR

n

RRpAAAAA

R

n

RpAAAAA

R

n

dAdA

dAdA

dAdA

dAdA

We will show that the operation of intersection and active complement can be

replaced by the operations marked in formulation of the statement.

The operation of intersection can be replaced by difference:

( )RRRRR RRRAllAllAll

,\,\,,, 2,

1,

12,

1 ψψψψψ ΨΨΨ = [7].

The operation of active complement can be expressed with join, projection,

difference: RRCR RAll

,\),(,~ , ψψψ Ψ= , where

),(...),(),( ,,,...,,,1121

1RRRC RAAAAAARA n

nnψπψπψ

−

⊗⊗= , ,..., 1 nAAR = .

Conclusions

Multisets analog of the table algebra is constructed. The concept of the table is

specified, using concept of a multiset. A certain scheme is ascribed to every table. It

is entered the main concepts such as scheme, tuple and table. A scheme is an arbitrary

(finite) set of attributes. The tuple is the nominal set on the set of attributes and the

universal domain. The table is pair, where the first component is an arbitrary multiset,

basis of which is the set of tuples of the same scheme and the second component is a

scheme of table.

The basic operations of tables are defined. This is analogs of set-theoretic

operations (union, intersection, difference) and special operations (selection,



projection, join, active complement and renaming). For each operation are defined a

basis of the resulting table and number of duplicates of every tuple. It is shown that

among the main operations over tables is such which are expressed through others.

References

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AUTHOR INDEX

Belyakov E.V. student

Volzhski 404125

Korotkova N.N., Ph.D. docent Volzhski

404125

Mozhenkov V.V Art. St. Volzhski

404125

Revenyuk T. A Odessa

65088

Sergeeva A. E Doctor of Physical and Mathematical Sciences

professor Odessa 65088

Galichyan T.A.

graduate student der. Radishchevo Chudovsky district Novgorod region. 174210

Klinkerman R.V.

applicant Zheleznogorsk Krasnoyarsk Territory 662972

Sagan E V PhD docent Kherson

73024

Ahramovich M.V.

graduate student Simferopol ARKrym 95007

Galitsyna T.A

specialist Art. St. Volzhski Volgograd region 404104

Koroteyeva E.A

student Volzhski Volgograd region 404104

Mocretsova I.S

specialist Volzhski Volgograd region 404104

Mustafina J.A.,

PhD docent Volzhski Volgograd region 404104

Perepechenova T.N

student Volzhski Volgograd region 404104

Rebro I.V

PhD docent Volzhski Volgograd region 404104

Buy D.B Doctor of Physical and Mathematical Sciences

professor Nezhyn 16600

Glushko I.M Nezhyn

16600

Buy D.B Doctor of Physical and Mathematical Sciences

professor Kiev-680 03680

Kompan S V graduate student docent Kiev-680

03680

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