physics, mathematics sciences mathematics...astola and danielian built three-parametric regular...

The journal is published since 1925, since 1967 – periodically (3 numbers in year).

EDITORIAL BOARD

DANIELYAN E.A. (editor), KIRAKOSYAN A.A. (deputy editor), SHARAMBEYAN L.T. (executive secretary)

ALAVERDYAN R.B., AVAGYAN R.M., DUMANYAN V.Zh., GEVORGYAN G.G.,

HAJYAN G.S., HAKOPIAN Yu.R., KHACHATRYAN I.G., MOVSISYAN Yu.M., NERKARARYAN Kh.V., NIGIYAN S.A., PETROSYAN A.I., SARGSYAN S.V.,

SHOUKOURIAN S.K., VARTANYAN Yu.L.

Yerevan State University Press © Proceedings of the Yerevan State University, Physical and Mathematical Sciences, 2010 © ºäÐ ¶Çï³Ï³Ý ï»Õ»Ï³·Çñ, üÇ½Ù³Ã ·ÇïáõÃÛáõÝÝ»ñ, 2010

Signed to print at 21.10.2010

Format 70�100 1/16. Printed paper 4,75. Circulation 100.

Editorial address: 1, Alex Manoogian str., 0025, Yerevan, Republic of Armenia http://www.ysu.am/site/index.php?page=9&lang=2

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY

Physical and Mathematical Sciences � 3 (223), 2010

C O N T E N T S

REVIEWS J. Astola, E.A. Danielian, S.K. Arzumanyan. Frequency distributions in

bioinformatics: the development ............................................................................. 3

MATHEMATICS Kh.A. Khachatryan. On one Urysohn type nonlinear integral equation with non-

compact operator.................................................................................................... 23

B.A. Sahakian. On the solutions of some differential equations of fractional order in complex domain..................................................................................................... 29

M.I. Karakhanyan, T.M. Khudoyan. A remark on strict uniform algebras.............. 35

H.R. Rostami. Non-unitarizable groups....................................................................... 40

MECHANICS B. Yazdizadeh. Comparison of different plane models in finite element software in

structural mechanics............................................................................................... 44

PHYSICS

R.M. Avagyan, G.H. Haroutyunyan, V.V. Harutyunyan, V.S. Bagdasaryan, E.M. Boyakhchyan, V.A. Atoyan, K.I. Pyuskyulyan, M. Gerchikov. Study on physical regularities of a wide spectrum aerosols behavior at its filtration through supeer-thin materials................................................................................. 51

R.S. Asatryan, H.S. Karayan, N.R. Khachatryan, L.H. Sukoyan. On one method

of distant infrared monitoring ................................................................................ 57

Proc. of the Yerevan State Univ. Phys. and Mathem. Sci., 2010, � 2. 2

S.G. Gevorgyan, S.T. Muradyan, M.H. Azaryan, G.H. Karapetyan. Method for measuring thickness of thin objects with a nanometer resolution, based on the Single-layer Flat-Coil-Oscillator method............................................................... 63

COMMUNICATIONS

H.A. Asatryan. On one formula of traces .................................................................... 68

Annotation in Armenian................................................................................................ 71

Annotation in Russian .................................................................................................. 74


Physical and Mathematical Sciences 2010, � 3, p. 3–22

R E V I E W S

M a t h e m a t i c s

FREQUENCY DISTRIBUTIONS IN BIOINFORMATICS: THE DEVELOPMENT

J. ASTOLA1*, E. A. DANIELIAN2, S. K. ARZUMANYAN2

1 Academician of the Academy of Science of Finland 2 Chair of Probability Theory and Mathematical Statistics, YSU

The mathematical investigation of large-scale biomolecular sequences is being

carried out by analyzing the properties of events arising in such sequences. This survey is devoted to discussion of results in this field. Based on general empirical facts being fulfilled for all frequency distributions, we discuss the axiomatics suggested by J. Astola and E. Danielyan.

The axiomatics postulates the regular variation of frequency distribution with asymptotically constant slowly varying component, the form of its shape, and the stability by parameters. The verification of the axiomatics fulfillment for well-known frequency distributions is done.

The paper describes also methods of construction of new parametric families of frequency distributions. These methods are: usage of stationary distributions of birth-death process, special functions, stable densities, etc.

The problem of stability by parameters is formulated the results on stability by parameters in terms of various classical metrics are given. The conditions of regular variation for different families of frequency distributions are formulated.

Keywords: frequency distributbiomion, olecular sequence, regular variation, convexity, stability by parameters, asymptotic expansion.

1. Introduction. Discovering the evolution by investigating the variety of

large-scale biomolecular systems there is no other way but to characterize the frequency distributions (FDs), say { nP }, of events being important for systems’ functioning. The variety and diversity of such systems do not allow to figure out and suggest a universal approximation for FD, i.e. suggest a universal model, which might be suitable in all possible situations. Based on huge datasets of biomolecular systems it has been possible to extract only some common information (statistical facts) being applicable almost to all situations for empirical FDs. Those are:

1. { } has a skew to the right, >0 for all n, nP nP 1nP� � . The conception of skewness for biologist is based on intuition and on the

shapes of graphs of empirical FDs. The quantitative aspects of the skweness

* E-mail: [email protected]

Proc. of the Yerevan State Univ. Phys. and Mathem. Sci., 2010, � 3, p. 3–22. 4

conception were not even exploited. Only for Power Law and Pareto Law defined below the parameter � was declared as a measure of skewness.

2. { } exhibits Power Law behavior as (see [1–6]). nP n ��Random variable (RV) 0 has Power Law, if (P denotes a probability)

( ) ( )nP P n c n � � �� , 1< � <�� , , . (1.1) 1n � 1

1nc n ��

��

��

�

The Power Law is used for estimation of the connectivity number in metabolic networks [2], of the rates of protein synthesis in protein sets of proka-ryotic organisms [7], of the number of expressed genes in eukaryotic cells [8, 9], of DNA sequencing structures [9], etc.

The Power Law is of interest in self-organized growing biomolecular networks, because of its scale-invariant property: � � 1

mP c P��

� n sP for integers , 1n 1s , . Self-organization means, that if we know local FDs on

successive two fractals, then we may extraporate the FD for all system [10]. m n s� �

3. The log-log plot ( lo versus lo ) of most empirical FDs g nP g n � �nP systematically deviated from the straight line and show the upward/downward convexity [1–6].

That is why many new statistical FDs have been proposed. Those are: Pareto Law (generalization of Power Law)

� � ,nP c b n b �� , -1< <b �� , 1< � <�� , , 1n

� � 1

1,

nc b n b ��

��

��

� (see [9]); (1.2)

Warring Distribution

1

11 ,,

n

nk

p p kPq q k�

� � � ��

� 0< p < <q �� , 1,n 0 1 pPq

� � , etc. (1.3)

Constructing new FDs the advantage is given to parametric ones, because by changing the parameters one hopes to find out the best approximation for unknown FD.

4. The small changes in environment do not have a dramatic influence on the structure of biomolecular system.

We may call this fact the adaptivity or the robustness. We may trust or not assumptions of statistical models that lead to different

empirical FDs for the events occurrence number in biomolecular systems. But, due to the probability theory, the replacement of observations’ independence in models by various type of weak dependence cannot have essential impact on the behavior of for large n. nP

Anyway, statistical models even being important do not describe the functional mechanism of biomolecular systems.

2. On the Mechanism. The dynamic of the biomolecular large-scale systems many authors try to explain with the help of birth-death models with various types of intensities. Their stationary solutions generate skewed to the right distributions as it requires the empirical fact 1.

Proc. of the Yerevan State Univ. Phys. and Mathem. Sci., 2010, � 3, p. 3–22.

5

In general, the development of any evolutionary large-scale complex biomo-lecular system is a result of two fundamental phenomena: Darwin natural selection, random mutation. The functioning is explained with the help of standard birth-death process [11, 12], say � � �: 0t t , (2.1) which is a homogeneous Markov Process with continuous time and countable number of states 0,1,2,... Moreover, conditional probability ( ) ( ( )ijP t P s t� � �

/ ( ) )j s i� � doesn't depend on s, and for 0t �

1( ) ( )ii iP t t o t�� , 1 1( ) ( )i i iP t t o t�� , 0,1,2,...,i � ( ) ( )ijP t o t� , i j� >1,

i� >0, 1i� � >0. Among the numerous publications devoted to birth-death models we like to

point out a pioneer paper of Yule [13], paper of Simon [14] and some recent ones (see, for instance, Granzel and Schubert [15], Bornholdt and Ebel [16], Oluic-Vicovic [17], Kuznetsov [18], Astola and Danielian [12, 19]).

The Yule’s birth model is designed to describe the evolution of new species within a genus. Its stationary solution has the Power Law as a limit case.

The Kuznetsov’s birth-death model describes the expression process of the genes in the eukaryotic cells, which exhibits a strong stochastic component (SSC), a chaotic movement (CM) of mutation, and a skewed FD of the number of events. In the model coefficients of the process are linear.

The latest one, birth-death model by Astola and Danielian gives a wide generalization of previous models. Here the coefficients are non-linear and the stationary solutions present FDs of moderate growth, i.e. � 1lim / 1n nn

P P�� .

The stationary distribution of the process (2.1) exists iff

1 1

n

kn k

� �� < �� (2.2)

with � 1 / ,n n n� � �� and takes the form 1,n

01

n

n kk

P P ��

� � , 1

01 1

1n

kn k

P ��

�

� ��

�� . (2.3)

� �1n�� presents a sequence of ratios of ''birth'' and ''death'' coefficients.

Let us interpret the process (2.1) as expressed genes process in the eukaryotic cells, which exhibits a SSC, a CM. It is a discrete process with many protein coding genes in an ''off'' state. The production of the mRNA occurs in sporadic pulses with specific mRNA transcripts starts from initiation of the transcription of the mRNA molecule of the specific gene at moment zero. Then the mRNA molecule exports from nucleus to cytoplasm of the cell where the transcript is degrades. It leads to a new mRNA copies and degradation of transcripts. We indicate the gene expression level by integers n = 0,1,2,…, assuming that it is a random process, and denote its distribution at a moment t by

� � � � � � �nP t P t n� � . The process is described as a standard birth-death process

(2.1) and � t denotes the random number of mRNA transcripts per a cell in


transcripton at a moment t. This mechanism realize the way how molecules are chosen to be included into organism over time. It is a mixture of molecular sequences being before in organism and new ones, so called mutant new sequences from other organisms. In “stable” evolution process the intensities � �1n� � of “birth” and � �n� of “death” do not depend on t and lead to stationary solution (2.2), (2.3). The summarized intensities take the form 1 1n na� ��

� �� , ,n nb� �� , (2.4) 1n

where >0, >0 and a b ,n��

n�� present the intensities of CM and SSC (2.4) at state

n. Here n�� >0, n�

� >0, . lim limn nn n� ��

��

3. General Representation. The empirical fact 1 is enough for the following conclusion: any FD � �nP may be presented in the form (2.2), (2.3) (see [20]). Indeed, due to fact 1, for we have 1n

11

00

nk

nk k

PP PP

��

�� .

Denoting � 1/ ,n n nP P� �� , we come to the first equality in (2.3). The last equality in fact 1 leads to the last equality in (2.3). Now, >0 is equivalent to (2.2).

1n

0P

The reverse statement is also true: any distribution of the type (2.2), (2.3) satisfies empirical fact 1.

According to variety and diversity of biomolecular sequences new parametric FDs were needed. Kuznetsov suggested three-parametric Kolmogorov-Warring Distribution [18]. Astola and Danielian built three-parametric Regular Hypergeometric Distribution [21], which takes the form

� � � �

11 2

00

ˆ ˆ,

ˆ1

n

nk

p k p kP P

k q k

�

�

� ��

� �� 1, n 10

1 2

ˆ ˆ ˆ ˆ( ) (ˆ ˆ ˆ ˆ( )q p q pPq p p q

� ��

� ��

� �2 )

( ), (3.1)

0< 1 2ˆ ˆp p� < <q̂ �� , where denotes the Euler's Gamma-Function. � � �

Several variations of three-parametric Regular Pareto type Distribution have

been proposed in [22–24], which finally acquires the following form ( ): 0

11

m��

� � �

� � �

1

1

11

1 1

1 1, , 1 , 1,

1 1, , 1 , 0 , 1 ,1 .

n

nm

n

n m

cP C b c nn b m b

cC b c c bn b m b

� �

� �

�

� �

�

�

��

�

� � ��

� �� ! ! �� ! ! �� ! ! �� "

�

� �

(3.2)

Easily seen that from (3.1) for 1 2ˆ ˆ ˆ1, , 1,p p p q q� � � � we get the Warring Distribution (see (1.3)), and from (3.2) for c=1 we obtain the Pareto Law (see (1.2)).


7

Although the FDs (3.1), (3.2) were constructed using other principles than the stationary solutions of standard birth-death process, but it is clear that they can be presented in the form (2.2), (2.3). Note that for FDs (1.1), (1.2) and (3.2) we have to put equal to normalization factor, and find from the equality

. After this manipulation it is obvious that, for instance, 0P 0P

1nP ��

1 21

ˆ ˆ,

ˆ1np n p n

nn q n

� ��

0� � �

, in case of (3.1),

� 11 11 1 ,

1nc n

n b n b

�

��

� �� 0,� � �� in case of (3.2),

where � �n� is a sequence of coefficients in general representations (2.2), (2.3) of these FDs.

For consider m-parametric 1m � � �n mFD P c� , where 1 2( , ,..., )m mc c c c��

and , 1,ic i m� , are parameters. It is convenient to choose parameters to be independent, and ranges of their changes be also independent. It means that there are no relationships among them of equality and of inequality types respectively.

All FDs presented above have independent parameters. But for Warring Distribution and Regular Hypergeometric Distribution the independence of ranges of parameters’ changes does not take place. The situation is improved by making the linear transformations of parameters: 1 ,q p p p� � � � � and 1 2 1 1 2ˆ ˆ ˆ ˆ ˆ ˆ1q p p p p p 2p̂� � � � � � � (3.3) in the first and in the second cases correspondingly.

Definition 1. We say that � � �mnP c�

is well-defined, if the coefficients

1,..., m� 1,..., mc c of its general representation uniquely define parameters . �T h e o r e m 1 (see [25]). All above presented FDs are well-defined. 4. Regular Variation. Due to characteristic property ,nP Ln ��

� � 0, , 1L c R n� �� # � �� (see (1.1)), the empirical fact 2 has been interpreted in mathematical sense in [19, 22] as a regular variation of FD.

Definition 2 (see [26, 27]). The sequence � �nX of positive numbers varies

regularly as with exponent n �� 1 ,R$ # � �� , if for any integer 2s

lim( / ) .s n nnX X s$��

� (4.1)

The case 0$ � presents the slowly varying sequence, which is usually denoted by � � �L n .

The Definition 2 is equivalent to the representation � , 1nX n L n n$� � ,

with some arbitrary chosen � 0 0L % .

In general, the sequence � � �L n may show quite different behavior as . n ��


L e m m a 1 (see [28], p. 6–8). Let 0 L L& & & �� . Then, there is a slowly varying sequence � � � � �L n such that lim , lim

nnL L n L L n

�� .

But, according to the properties of all before known FDs, in [23] the following general property of FDs was suggested.

Property 1. FD { } varies regularly as with exponent nP n �� ,�� 1 �! ! �� , and exhibits an asymptotically constant slowly varying component (ACSVC) L, i.e. , and ( ) , 1nP L n n n�� lim ( )

nL n L R�

�� # . (4.2)

For example (see [22, 23]), the following statement holds. T h e o r e m 2 . The FDs (3.1) and (3.2) satisfy Property 1 with

� � � � �

1 21 2

1 2 1

ˆ ˆ ˆ ˆ 1ˆ ˆ ˆ1 ,ˆ ˆ ˆ ˆ ˆ

q p q pq p p L

2p p q p� �

��

� � � � � �� p�

,

� is the parameter in (3.2), and

� � 1

1, , 1n

cL C b cn b ��

� ��

� .

In [29] the following 2m-parametric FD was considered:

�

01 1

1

01 11 1

ˆ ˆ ˆ, 1, 0, 0, 1,ˆ

ˆ ˆ ˆ1 ,ˆ

n mi

n i ik i i

n m mi

i in ik i i

k pP P n p q i mk q

k pP qk q

�

� �

�

��

�� % %� ��

� �� %� �� "

��

� ��

,

1.p

�

(4.3)

T h e o r e m 3 . The FD (4.3) exhibits the asymptotic expansion

1 11 , ,n

L MP o nn n n� � ��

� ��

(4.4)

where

� � 0

1

ˆ 1ˆ 1

mi

i i

qL P R

p��

�

�

�� #

�� , (4.5)

� �

2 2

1ˆ ˆ

,02

m

i ii

q pM L

��

� �� # ��

�.

The FD (4.3) is a generalization of (3.1). Theorem 3, in particular, says that � �nP of type (4.3) varies regularly as with exponent n �� and exhibits ACSVC L (see (4.4), (4.5)), i.e. satisfies Property 1. This is the content of the first term at the right-hand-side of expansion (4.4). The second one gives additional information on “smoothness” of� �nP , which agrees with the empirical fact 4. Indeed, the “smoothness” of continuous functions comes to light, if they can be presented in the form of T�ylor’s Series. The expansion (4.4) is the analog of such “smoothness” for a discrete case.

The asymptotic expansion (4.4) is natural for all known FDs. That is why we may even postulate it as an Extended Property 1 for FDs.


9

Denote by m$ the moment of order � �nR of FD P$ �# . If � �nP varies regularly with exponent � �� , 1 �! ! �� , then m$ ! �� for 1$ �! � and

for m$ � �� 1$ �% � (see [26]). If � �nP exhibits ACSVC, then . The

Problem of asymptotic behavior of truncated moment 1m��

� 11 n

n xx n P�

��

�!

� � as

x �� arises. T h e o r e m 4 (see [22, 30]). Let FD� �nP satisfies Property 1. Then,

1( ) ( ln )(1 (1)),x L x o x�� . (4.6) For concrete FDs even more terms of asymptotic (4.6) can be obtained. For instance, for FD (1.3) (see [30] and [22], p. 140–145): if and 1q p� �

p is an integer, then � �1 21 1( ) ln ( ( )) 2 ( ) ,x p x C A p x O x x� � �� , where

� 11

11 ;

p

nA p n�

�� if and 2q p� � p is an integer, then

22 2

2 1( ) 2 ( 1) ln ( ( )) ( ) ,2px p p x C A p O x x

x� �� '� � � � � � � (

" )�� ,

where � � 1

21

3 42 1

p

n

pA pp

�

�

��

��n . Everywhere C denotes the Euler’s constant.

5. Convexity and Monotonicity. For a sequence � �nX of positive numbers we consider the following two types of convexity: for n=0,1,2,…

1) � 2 12n n nX X X� � 0� � ! % upward (downward) convexity, 2) � � � 1 1/n n n nX X X X� �! % 2/ � log-upward (log-downward) convexity. The Problem of comparison of these convexities arises. L e m m a 2 (see [24, 31]). The upward (log-downward) convexity of

� �nX implies its log-upward (downward) convexity. For � �nX the existence of a pair “upward and log-downward convexity”

contradicts Lemma 2. But the pair “downward and log-upward convexity” may exist (see [12], p. 60–61).

Let � �nX and � �nY be positive sequences. Obviously, if � �nX and � �nY are log-upward (log-downward) convex, then � �n nX Y� is of the same type.

What can we say about � �n nX Y* ? It turns out that, for instance, the following statement holds.

L e m m a 3 (see [31]). Let � �nt ={ / , be downward convex. If }, 1n nY X n

� �n nX Y� decreases and is log-downward convex, then � �n nX Y� is of the same type.

Lemma 2 is of interest in the way of constructing FDs with given properties, because of the following statement. Let (1, ), , 1L R s� �# �� # be an integer,

11 2\{0}, 1, , 0 ...i sM R i s $ $# � ! ! ! $! , are given.


Corollary 1. The sequence 1 i

si

i

MLn n� � $�

�

� '� (" )

� decreases and is log-downward

convex starting from some index . 0 1n With the help of this statement the following general result is proved. T h e o r e m 5 (see [31]). There is a decreasing and log-downward convex

� �nFD P satisfying asymptotic expansion with above a priori given constants

1

1 ,i s

si

ni

MLP on n n� � $ � $� �

�

� ��

� n � (5.1)

(compare to (4.4)). Using the method developed in [31], it is possible to build FD of type (5.1)

with any finite number of log-upward/log-downward convex pieces in its graph, the last of which decreases and is log-downward convex.

Very often it is easier to prove such kinds of statement using continuous analogs (CA) of the sequence � �nX of positive numbers. We say that the function f(t) defined on + is a CA of 0,�� nX , if f is continuous on + , and

. The “smoothness” of f allows to apply methods of Mathematical Analysis.

0,��

� ,nf n X n� 0

The linear CA (LCA) of � �nX sometimes is more preferable for other purposes among all other continuous ones. The shape of its graph is formed by possibly minimal number of convex pieces.

Definition 3 [22]. We say that � f t defined on + 0,�� is the LCA of

� �nX , if: � � � � , 0;na f n X n b f t� is continuous on + 0,�� ; is linear on each

� � c f t

+ ,, 1 , 0n n n� . It can be easily proved that � �nX and its LCA are unimodal (or not) with

the same mode simultaneously, and have the same intervals of monotonicity and convexity.

Now let us discuss the properties of monotonicity and convexity of known FDs. The famous ones (see (1.2), (1.3)) are decreasing and log-downward convex.

For FDs (3.1), (3.2) constructed at the second stage of development we combine the results from [23] and [24] in the following statement.

T h e o r e m 6 . FDs (3.1), (3.2) are unimodal. Their graphs are formed by no more than two monotone, and no more than three log-convex (convex) pieces.

6. Around Empirical Fact 3. Due to substantiated Property 1 (see (4.2)),

� � logloglog log

n L nPn

�� n

for n=1,2,…, (6.1)

where in case of Power Law � � L n c �� doesn’t depend on n (as a rule, the biologists deal with the log-log plot of � �nP ). According to empirical fact 3 one may conclude that the upward (downward) convexity of � log / lognP n is only the

result of piecewise convexity of � � �log L n , where, obviously, the last piece is

downward convex. Note that � � lim log / log 0n

L n n��

� , which follows from the


11

following property of � � �L n : for any � 0,1� # and n large enough the inequality

� n L n n� �� ! ! holds. Now, writing (6.1) in the form � log lognP n��

� log L n� we conclude that � �log nP is piecewise convex, and may affect on the type of convexity of � �nP only in initial finite interval. But in finite interval the number of log-convex (convex) pieces for � �nP cannot be more than finite. So, this number on + 0,�� is finite too. Next argument: any biomolecular system comes to a structure with minimal “energy expenditure”, which affects on FDs. The situation, when � �nP has more than one log-convex (convex) piece under “slow mutation” leads to unnecessary “energy expenditure”. (The famous FDs (1.1), (1.2), (1.3) have exactly one log-downward (downward) convex piece). But building the mathema-tical theory of FDs in bioinformatics with arbitrary speed of “mutation” one has to allow for FDs to have more than one (at least two or three) log-convex (convex) pieces. Such a situation may be explained with the help of evolution process’ functioning. Indeed, the value 1n n� ��

� � (see (2.4)) as creates nonlinear

deterministic “shift” over time n in new transcripts. If

n ��

� 1 /n n� �� 1! over time n,

then the SSC manages the situation. The deterministic “shift” of SSC leads to the log-downward (downward) convexity of � �nP . The CM doesn’t take part in stabili-

zation process. If for “large” massif of n, then the condition � (see (2.4)) stabilizes the process. The CM gets possibility to make observed affect on the process. It’s maximal influence can be easily seen on the initial massif on indices, around the mode (now intensities of SSC and CM are comparable). Just around the mode log-upward (upward) small convex piece of

* *1 / 1n n� �� - / 1a b !

� �nP appears. Above said and Theorems 5, 6 we propose for the following Property 2. FD� �nP is unimodal and it’s graph is formed by no more than three

log-convex (convex) pieces, the last of which is log-downward (downward) convex. The Property 2 with convex (log-convex) pieces has been suggested in [23]

(in [24]). For the FD (4.3) the following statement takes place [24]. T h e o r e m 7 . Let for vectors � 1ˆ ˆ,..., mp p and � 1̂ ˆ,..., mq q in (4.3) the

numbers ( ) ( )ˆ ˆand , 1,i ip q i m� , present the i-th order statistics respectively. Then the conditions. � � � � 1 1ˆ ˆ ˆ ˆ,..., m mp q p q! ! (6.2)

are sufficient for � �nP of type (4.3) to be decreasing and log-downward convexity. The following Problem stays unsolved: find necessary and sufficient condi-

tions for the fulfillment of Property 2 with one, two, three log-convex pieces for FD (4.3).

Let � �n. be increasing sequence of positive numbers with lim nn.

�� ,

� 1lim / 1.n nn. .��

� Then the sequence � �n/ , where � 1 / ,n n b/ �.� � 0, � %

, possesses the same properties. 0,b % 1n


A deep investigation on the stationary distribution (2.2)–(2.3) with , and with * *

1 1, ,n n n n n� �. � �.� �� 1 11 , ,n n n n n� �. � �.� �� , in (2.4) has

been done in [12, 19, 32]. The following FD was extracted 0

11 :

m�

� ��

1

1 1

0 011 1

1 11 , 1, 1 1n n

nnm mn m n m

c c cP P n P c/ / / /

��

� �

� ��

�� 1 (6.3)

with either 1 10 1, , or 0 ,n nc c/ /� �! ! � �� ! ! �� ! �� .

T h e o r e m 8 . Let � �0 1, n/ /� increases, lim ( / ) 0nnn /

�� . Then:

1. � �nP varies regularly with exponent � �� , iff � �n/ varies regularly with exponent � .

2. If 1 ,�! ! �� then 1n/ � ! �� .

3. � �nP and � �n/ exhibits ACSVCs L and L1 simultaneously. Moreover,

� 0 1L P� � 1 / .c L� The statements 1 and 2 in Theorem 8 are established in [12], the statement 3

is proved below. Put � 1 1 1 0 0 0 0, 0n n na P n a P/ /� � �� .P

k

Then, due to (6.3),

� � 1

1 01

1 ... 1n

n n nk

a a c P P c c P�

��

� � � � � � � � . So, we obtain the reverse equalities

(6.4) � 1

01

1 / ,n

n kk

P c c P P n/�

�

� ��

� 1.n

Since, � � 1 ,n nn L n P n L n�/ ��

1

, therefore, from (6.4) we obtain

. Letting we prove the statement 3. � � � 1

0 11

1 / ,n

kk

L n P c c P L n n�

�

� ��

� n ��

T h e o r e m 9 (see [12], p. 54–55,67–68). Let the conditions of Theorem 8 hold, and � �n/ varies regularly with exponent � 1,� # �� . If � �n/ is downward

and log-upward convex, then � �nP decreases and is log-downward convex. According to Theorems 8, 9 under the conditions of Theorem 9 the FD (6.3)

satisfies Properties 1 and 2. 7. Back to General Representation. Below let the sequence� �n� of positive

numbers be the sequence of coefficients in general representation (2.2), (2.3) of FD � �nP . So, � 1/ ,n n nP P n� �� 1 . It follows that: (a) for a given n we have ( )1n� % ! ,

iff � 1nP Pn�% ! ; (b) � �n� increases (decreases), iff � �kP is log-downward (log-

upward) convex. Thus, one may reformulate the Property 2 in terms of � �n� .

Property 2*. � �n� is formed by no more than three monotone pieces, where the last one increases and is located under the straight line y=1.


13

Now assume that the Property 2* holds for the FD � �kP . We are going to study the question of task on convergence of series (see (2.2)), i.e.

1 1

n

kn k

� �

! �� (7.1)

for FDs of moderate growth, which means that � 1lim / 1.n nnP P ��

� In terms of � �n�

it is equivalent to the existence of limit lim 1nn

��

� . (7.2)

Any regularly varying FD� �nP , due to Property 1, is a FD of moderate growth.

To obtain a sufficient condition for the validity of (7.1) one can use the Kummer’s Test (see 3.37, p. 116–117 [33]). Namely, let be a sequence of positive numbers such that the positive limit (finite or infinite) exists

nD

� 11lim n n nn

D D� ��

� . Then (7.1) holds.

Due to (7.2), one may replace the last limit by the following one � 1lim 0n n nn

D D�� % . (7.3)

It can be easily seen that (7.3) holds, if 11 , , ,n o n R

n n$ $�� # #� �

� �(0,1)$ (we take nD n$� ), (7.4)

11 , ,1n o nn n�� ! ! ��

� � (we take nD n� ). (7.5)

This test was used in [34] in order to prove (7.1) for

� � 1

1 1 11 .... 1 1 , ... ... ... ...n

K K

�� o nn nlnn nlnn lnln lnn nlnn lnln lnn

�

� � � � � � � � ��

,

where is a natural number and 0K � 1, .� # �� All considered cases of � �n� ’s asymptotic behavior are examples of FD

� �kP of moderate growth. Now let’s discuss the Property 1. T h e o r e m 1 0 (see [20]).1) The condition (7.5) implies the regular variation of � �kP with exponent

( �� ). 2) Under the condition (7.5) the existence of ACSVC for � �kP is equivalent

to the limit relation

lim 1 0kn k n k��

��

� ��

� . (7.6)

Let the following asymptotic expansion with 1 ,12

$ �#��

�� holds for � � :nP

� �1 11

1 , , , 0 ,n � � � �+L MP o n L R M R \n n n�

� �� # #� ��

(7.7)


which in particular implies that � �nP varies regularly at infinity with exponent ( �� ) and exhibits ACSVC L, i.e. the Property 1 takes place. We easily verify that

� � �

1�

1

1111 1

1 1 1 11 1 1 1 ,

�

n �

� �

Rn o n�

n R n o n

� R R �o o o o nn n n n n nn n

� �

� �

� ��

� ��

,

where � /R M L� . Various forms of � �kP ’s asymptotic expansions, similar to (7.7), may lead to the form (7.4) for � �n� , which allows us to formulate the reverse to statement (1) in Theorem 10.

In particular, let’s replace the Property 1 by more strong but also natural Improved Property 1. For FD � �kP the asymptotic expansion (4.4) holds. The Improved Property 1 implies the asymptotic expansion (7.5). Note that

for all presented above FDs the Improved Property 1 takes place. Several relationships between � �kP ’s and � �n� ’s asymptotic expansions

are obtained in [31, 35]. For investigation of the properties of FDs � �kP it is natural to formulate them

in terms of � �n� . Property 1*. � �n� satisfies asymptotic expansion (7.5). 8. Continuity by Parameters. The simplest form of empirical fact 4 with

respect to parameters of FDs in mathematical sense is the continuity of � �nP by parameters. Sometimes FDs are given in the form of their Generating Functions (GFs).

Let us consider the finite-parametric FD � � �n mP c� . The parameters are

and 1,..., mc c � 1,...,mc = c cm� . The GF of FD � � �mnP c

� is defined as follows: for

any + ,0,1x#

� � 0

, nm mn

nP x c P c x

� �

� �. (8.1)

Having GF (8.1), it is possible to establish the continuity of � � �n mP c� by parameters with the help of Continuity Theorem for GF (see XI.6, p. 262 [36]).

1,..., mc c

Continuity Theorem. Let � � �knP be a sequence of FDs. Then, in order

as for fixed n it is necessary and sufficient the following convergence:

( )kn nP P� n ��

� � � 0 0

ask n nk n n

n nP x = P x P x = P x k +

� � � + ,0,1x# . � � for any

This idea has been developed in [22], p. 33–36, for famous FDs. For instance, in case of FD (1.3) with the help of hypergeometric series (see 9.100,


15

p.1040, [37]) and its integral representation (see 9.111, p. 1040, [37]) we have for GF of (1.3):

. (8.2) � � � � 1

1

01 1q- pP x, p,q = q - p - t - tx dt�

0Due to Continuity Theorem, if , p p' q q'� � , then the GF (8.2) with

parameters p and q tends to GF (8.2) with parameters p' and q' , if it is possible to pass to the limit under the sign of integral, which is the case in this situation.

In some cases the integral in (8.2) may be evaluated in a closed form. Due to 9.121.24, p. 1041 [37], for 1/ 2, 1p q� � one may obtain

+1 1, ,1 0,12 1 1

P x = , x+ x

� � #� � �� , (see also [38]).

9. Stability by Parameters. Let � � �n mP c� be m-parametric FD with

In general, the stability property by parameters is formulated as follows: .mmc 1# 2� R

� � �n mP c� is stable with respect to parameters 1,..., mc c in terms of some classical metric, say � . (9.1)

Here explanations are needed. All non-trivial metrics in Rm are equi-

valent to the following one '

1

m

k k m mk

c c c c�

� � �� '� � , where � 1,..., ,m mc c c 1� #�

� ' ' '1,...,m mc c c 1�

�# . In the set of sequences � � �n mP c� with different collections of

parameters one has to introduce some classical metric mc� � � � � � �� ',n m n mP c P c� � �

“suitable” to � � �n mP c� . For simplicity we write � ',m mc c� � � instead of

� � � � � �� ',n m n mP c P c� � � . Below K is a convex compact in 1 .

Definition 4. We say that FD � � �n mP c� is � -stable with respect to , if for any

1,..., mc cK 12

� '

'

| | 0lim , 0

m mm m

c c� c c

� ��

� � (9.2)

uniformly on ,mc� . 'mc K#�

Then, the empirical fact 4 for finite-parametric FD with respect to parameters takes the mathematical form (9.2).

The form of K may be chosen simple, if parameters and ranges of their changes are independent. Parameters (ranges of their changes) are independent, if there are no relations of equality type (of inequality type) among them.

All presented above FDs have independent parameters. But for FDs (1.3) and (3.1) the independence of parameters’ changes ranges doesn’t take place. The situation can be improved with the help of linear transformations:

� �1 2 1 1 2 2ˆ ˆ ˆ ˆ ˆ ˆ1 , for FD 1.3 ; 1 , , for FD 3.1� q p p p � q p p p p p p� � � � � � � � � � .

Now, for the FD � � �n mP c� with independent parameters and independent ranges of their changes one may choose K in the form


1

,m

ii

iK c c

�

3 4� 5 6� , (9.3)

where � 1,..., mc c 1# , � 1,..., mc c 17 7 # , ic ci& for 1,i m� .

The following metrics for � � �n mP c� in bioinformatics are usually used [12], [22]:

� � � � �

� � � �

' '

0 0

' '

0

, sup , Uniform Metric

, Metric in Variation .

n

m m k m mn k

m m n m n mn

c c P c c

� c c P c P c

8 �

� �

� �

�

�

� � � �

� � � �

kP

The last one is the particular case with p=1 of lp-metrics, and, obviously, . Thus, if � �',m m m mc c � c c8 &

� � � � ', � � �n mP c� is -stable, then it is � 8 -stable too.

Generally speaking, the reverse statement to the last one is not true. But for the FD (6.3) under the conditions of Theorem 8 it can be proved that � � , ' 1/ 2 , '� c c � c c� (see [12], Chapter 4).

The stability problems for introduced FDs in terms of �- and 8 -metrics are in the center of attention of several publications [39–43].

It is of interest the following Stability Criterion for � � �n mP c� in terms of lp-metrics (see [22, 43]). We assume the following conditions hold:

1. The FD � � �n mP c� allows a representation in the form

� � � � � /n m n m m n mP c g c g c g c� � � � �, 0, 0.n

2. There is mc K� #� such that for all 0n we have � � max

mn m n mc K

g c g c�#

� �� .

3. There is mc K� #� such that � min

mm mc K

c g c#� � �

� � .

4. The FD � � �n mP c� satisfies Property 1.

Let � m� � c� � �� be the exponent of � � �n mP c� ’s regular variation and

. � � 1/ � mp > c ��

T h e o r e m 1 1 . � � �n mP c� is lp-stable on K, iff

� � '

'

0lim 0

m mn m n m

c cg c g c

� ��

� �� uniformly on mc� , '

mc K#� for every . 0n

Theorem 11 was applied to FDs (3.1) and (3.2) in order to prove for them the lp-stability by parameters.

Another approach to stability of finite-parametric FDs has been suggested in [44]. It is based on monotonicity property of functions in case, when FD may be presented in the form of such functions combined by finite number of operations of finite or infinite sums, product, ratio, convolution.

10. Semi-Group Property. We already mentioned that the Power Law (1.1) is of interest in self-organized growing biomolecular networks, because of its scale-invariant property. Trying to figure out other FDs for application here one has to analyze the properties of such networks. Together with the self-organization


17

there is the second peculiarity. The FD must be of the same type in united interval as it is in each fractal forming the interval in order to extrapolate the FD in united interval and in whole system. The fractals may be chosen with approximately equal lengths in the way, which allows to postulate either the independence or some type of “weak” dependence between the numbers of event’s occurrences on each fractal. These random numbers are characterized by local FDs on fractals. Now, instead of scale-invariance the semi-group property has to take place. In contradiction to scale-invariance property, where the operation of multiplication is used, the semi-group property implies that the convolution of FDs of the same type equals to FD of exactly this type. Such semi-group property is intrinsic for normal, Cauchy’s, Levy’s distribution functions and for many other very useful ones. The semi-group property holds, for instance, for the four-parametric family of Stable Laws (see [45, 46]). Moreover, the conception of regular variation and the semi-group pro-perty for empirical FDs` continuous analogs are closely connected and supplement each other from the point of view of Probability Theory. It is just the time to notice that Stable Laws not only satisfy semi-group property, but also Property 1.

Below we introduce more powerful than the semi-group property, and, obviously, more restrictable property, which extracts the family of Stable Laws.

Definition 5. We say that distribution function S is stable, if for any , there are numbers

1,ia R#

, 1,2ib R i�# � 1,a R# b R�# such that

11 2

1 2

x a x a x aS S = S , xb b b

� � � �� R� #� � � � � ��

,

where * denotes the sing of convolution. Let us describe the parameters of Stable Laws. The first essential parameter

is the exponent, which defines the exponent � ,0,2$ # � �� of Stable Law density’s regular variation 1� �� .

Excluding Normal Law 2$ � any Stable Law � 0,2$ # has infinite variance.

Denoting by the Stable Law with exponent �S � 0,2$ # consider its two

tails: � S x$ � (left tail) and � 1 S x$� (right tail) for x R�# . The second essential parameter for S$ is asymmetry, i.e. the value of limit

� � � � + ,1

lim 1,11x

S x S xS x S x$ $

$ $

9��

� � �� #

� � ��

(the ratio of the tails difference and sum), which always exist. In other words, the asymmetry is nothing else, but the measure of skewness for � �S x . Due to empirical fact 1, we are interested in Stable Laws with maximal skewness to the right, i.e. in � S x with 19 � � .

The remained two parameters (shifting parameter and scale factor) are non-essential.

The next condition, which has to be fulfilled, if we want to use Stable Laws in bioinformatics, consists in following. The extracted densities of Stable Laws, which assumed to be continuous analogs of FDs, must be concentrated in . + 0,��


Denote by � ; ,S x $ 9 a stable density with exponent $ and asymmetry 9 . For our purposes we may use not only � ; , , 0 1S x $ 9 $! ! (onl in his y t

case � ; ,1S x $ is concentrated on + 0,�� ), but also � 2 ; ,0 , 0 2S x $ $ ,� ! ! for x#

The den+ 0,�� .

sity � ; ,0S x $ for 1 � 2 ; ,0S x $�x R# is symmetric, so, for

is+ 0,x# �� concentrated on + 0,x# �� and has skewness to the right. Now, the following families of two-parametric

densities

� � � �� 1/ 1/

, 2 ; ,0 , 0 2 �1/ 1/ ; ,,

ˆ 1 , 0 1 ,f x S x , R

f x S x , $ $$ : : : $ $ : R

: :� �$ $$ : $ $ : �

� �� ! ! #�"�

� �(10.1)

are condidates to be continuous analogs of FDs (see [22]). that the densities (10.1) are formed by no mo

convex pieces and are unimodal (see Property 2). FDs construction

� ! ! #�

Finally, note re than three

11. Disc retization of Densities. Besides the way of newbased on standard birth-death process with various forms of intensities, there is a couple of other known ways. The first one consists on construction based on dis-cretization of densities, which are concentrated on + 0,�� and satisfy Properties 1 and 2. We already have such an example: two-parametric Stable Densities (10.1).

Let � + , 0,f t t# �� , be a continuous density satisfying Properties 1 and 2. Definition 6 (see [22]). We say that FD � �nP of the type

1n� ( ) , 0,nP f t dt n

n� 0 (11.1)

is the the discretization conserves the properties of imodality of the d

discretization of f. It is very important that

monotonicity, convexity, un ensity ( )f t , i.e. at least the Properties 1 and 2 hold. It remains only to verify the Property 3.

T h e o r e m 1 2 (see [41]). The discretizations of type (11.1) of densities (10.1) are stable with respect to parameters $ and : n erms of Metric in Variation.

i t

A slightly different form of discretization of Stable Densities was used in publications [48–50].

0

( )f n , 0,P n( )n

Kf K

� (11.2) �

where ( )f t presents the corresponding Stable Density. rm of Stable Densiti

for normal, Cauchy and Levy Laws. For others there are only representations in the ntroduced types of discre-

Unfortunately, the closed fo es is possible to obtain only

form of convergent series [45, 46]. That is why above itizations for Stable Densities lead to complex expressions. At the same time, the Laplace Transform of Stable Density with asymmetry 1,9 � due to Theorem 3.1, p. 43 [46], always exists


19

� � � exp for 0 2, 1, ,�sx

� �

s � s R� s e dS x

$ ��

��

� � ! ! ; #�� 0 (11.3) � exp log for 1, .s s s � s R� � � #�"

Here only one representative with given parameters of shifting and scaling . Also for any Stable Density the right-side Laplace Trans

closed form. We may demonstrate how it is possible to build FDs with the help of Lapla

are taken form has a

ce Transform. Let

� � 0

, 0,sxs e f x dx s��

�� 0 (11.4)

be the Laplace Transform + 0,�� density � > 0f x of continuous on , which is asy to prove the following t. concentrated on + 0,�� . It is e statemen

L e m m a 4 . The function � 1 , 0z � z 1� & presents th some FD. Note that Lemma 4 is true, even if the lower limit in integral (1

some n

! , e GF of 1.4) equals to

umber $� , where R$ �# , and d on ( ) 0f x % is concentrate + ,$� �� . Due to Lemma 4, one may s e of discretization, in particular,uggest a new typ

on Laplace Transform of Stable Densities (see (11.3) too). From 1960s, after the appearance of a series of papers by Ma

basedndelbrot and his

successors, who sketched the use of Stable Laws in Economics and Biology, it comes out that Stable Laws have to be attached to Special Functions of Mathematical Analysis. Several Special Functions are connected with Stable Densities [51]. For instance, let

0( ) ,

( 1)

n

n

XE X R� n: :

:�

� #

�� , (see [45])

be the Mittag-Leffler function, where � � � denotes the Euler’s Gamma-Function. Then (see, for instance, [46], p. 169)

(�S�

The way of FDs construction based on different forms of discretizations of either Stable Densities or their Laplace Transforms may be referred as a variation of Method of Special Functions.

1 1 1/

0( ) ; ,1) , 0sx �

��E s e X X � dX s� � �� 0 .

12. Method of Special Functions.

/

,

There are other ideas, whose realizations can be interpreted as variations of Method of Special Functions. For instance, we search various Special Functions of Mathematical Analysis, which have representations in the form of positive conver-gent series and also Integral Representations. Then, forming the ratios of the n-th term and the sum we construct the probability nP for the FD� �nP . In this way the Warring, Hypergeometric, Pareto FDs and many other useful ones may be obtai-ned. Let us illustrate this way on simple example of Warring Distribution (1.3).

Consider the hypergeometric series (see 9.100, p. 103 37]), which is a Special Function:

�

9, [

� � � � 0

, , , 1 nn n

1n n

F z z$ 9

$ 9 < � � � (12.1) n< �


for positive values of arguments, where � ( 1) ( ), 0nx x x x n n .� � � � The series (12.1) is convergent in the following cases (see 9.102, p. 1040, [37 ]):

b z � c � & ! �� ! �

The following integral representation holds for ( ) 0 1; ( = 1; - 1a z! ! . ) 1 , - 0; ( ) z

0< 9% % (see 9.111, p.1040, [37]):

� � 1

1 11, , , (1 ) (1 ) ,,

�F � z t t tz dtB

� � � �� 0

0 (12.2)

where ( , )B x y denotes the . P , 1n , of the form (1.3) means ue to 1P

Beta Function that, d n

nn

0�� , we have

11 1 1

00(1, )B q 0

1( ) ( 1,1) (1 ) (1 ) , i.e. 1 ( / ),q p pP F p q t dt P p qq

� � � �� 0 where

(12.1) an 12.2)

,1,

d ( were used. In our case the conditions (b) and < 9% hOne more way for FDs construction, which may be characterized as an

addition to Method of Special Functions, consists in following. Let the FD

old.

� �nP

(2.3) in the form satisfies the general representation (2.3), (2.2) and Properties 1 and 2. We present

01

exp log(1 ) , 1 , 1n

n n n nk

P P � � � n�

� '� � � � (" )� , (12.3)

where, due to Properties 1, 2 and their various improvements in terms of � �n� , we

have � �n8� �lim 1, n n8 is monotone starting from some index n0, etc. So, 8 �n��

, in

varying sequence possessing “good” prop ies. W

particular, is a slowly ert e use the way of replacement of sums in (12.3) by integrals

+ 0log(1 ( )) , 1,

t� u du t� # ��0 , (12.4)

which doesn’t change the qualitative behavior of distributions. In this way one has to choose the interpolation � t8 for the sequence� �n8 . For instance, we may use

heorem 1, p. 55, [28]). wly varying function

the following statement (see TL e m m a 5 . There is a slo � t such that: 8

(a) � , 1;nn n8 8� (b) is infinite di ntiable;(t) ffere8 (c) + ,lim ( ( ) / ( )) 1 for 1 ;t n t n,n8 8 � # �

t��

� �� is monotone is monotone;nt 8 8 , if (d)

� � �is log - downward convex, if is log - downward convex.nt 8 8 is operation, which is called a dediscretization,

(e) By th from (12.3) we come to

a “smooth” probability density, defined on + 0,�� :

0

( ) (0) exp log(1 ( ))t

f t f � u du� '

� � � (" )0 , (12.5)


21

where (0)f may be obtained from the following equality

� ( ) 1, or 0 ( (f t dt f� � t

1

0 0exp log 1 ( )) ) .

0� u dt �� 0 0 0

ms of � t8 , for egral At the next step we choose various for which the ints po r it.

Finally, any of many variations of the reverse operation, i.e. discretization leads to new FDs.

The manner of dediscretization has been introduced and developed in [22, 5

. 301, � 2, p. 311–325.

2. Jeong H., Tombor B., Albert R., Ottval Z.N. Nature, 2000, v. 407, � 6. Rzhetsky A., Gomes S.M. Bioinformatics, 2001, v. 17, � 10, p. 988–996.

4. Wagner A., Fell D.A. Proc. Roy. Soc. London, B 268, 2001, � 1478, p. 1803–1810. 5. Wolf Y.I., Kalev G., Koonin E.V. Bio , � 2, p. 105–109.

Wuchty S. Mol. Biol. Evol., 2001, 1702.

in enomics.

tion. New York:

–880.

419. 8, � 4(4), p. 17–23 (in Russian).

odels: The abstract

ncy Distributions:

rmenii, 2009, � 109 (1), p. 21–31.

(12.4) i ssible to evaluate and get closed expressions fo

2] on example of distributions of moderate growth. The general approach for FDs of the form (2.3), (2.2) is presented in [53].

Received 02.06.2009

R E F E R E N C E S

1. Apic G., Gough J., Teichmann S.A. J. Mol. Biol., 2001, v

804, p. 651–654. 3

Essay 002, v. 24 v. 18, � 9, p. 1694–

s, 26.7. Ramsden J.J., Vohradsky J. Phys.Rev., E 56, 1998, � 6, p. 7777–7780. 8. Kuznetsov V.A. EURASIP, J. Apll. Signal Process., 2001, � 4, p. 285–296.

ences encoded9. Kuznetsov V.A. Statistics of the Number of Transcripts and Protein SequGenome. In: W.Zhang, I.Shmulevich (Eds.) Computational and Statistical Methods to GBoston: Kluwer, 2002, p. 125–171.

lection in Evolu10. Kauffman S.A. The Origins of Order: Self-Organization and SeOxford Univ. Press, 1993.

11. Saaty T. Elements of Queuing Theory. Dower Publications, 1983. 12. Astola J., Danielian E. Tampere: TICSP Series, � 27, 2004, p. 1–94. 13. Yule J.U. Trans. Roy. Soc. London, B 213, 1924, p. 21–87. 14. Simon H.A. Biometrica, 1955, v. 42, p. 425–440. 15. Glanzel W., Schubert A. Inform Process. Manager, 1995, v. 31, � 1, p. 69–80. 16. Bornholdt S., Ebel H. Phys. Rev., E 64 (3–2), 2001, p. 035104(4). 17. Oluic-Vicovic V. J. Am. Soc. Inform. Sci., 1998, v. 49, � 10, p. 86718. Kuznetsov V.A. Signal Process., 2003, v. 83, � 4, p. 889–910.

7, p. 405–19. Danielian E., Astola J. Facta Universitatis (NiŠ), 2004, v. 1Shcole, 20020. Danielian E.A., Avagyan G.P. Matem. v Visshey

21. Danielian E., Astola J. Tampere: TICSP Series � 34, p. 127–132. 1. 22. Astola J., Danielian E. Tampere: TICSP Series � 31, 2006, p. 1–25

23. Arakelian A.G. The Stability of Frequency Distributions in Biomolecular Mof Ph.D. Thesis. Yerevan: YSU, 2007 (in Russian).

24. Yakovlev S.P. The Analysis of Analytic Properties Multi-dimensional FrequeThe abstract of Ph.D. Thesis. Yerevan: SIUA, 2008 (in Russian).

tributions: The abstract of 25. Avagyan G.P. The Analysis of Properties of Regularly Varying DisPh.D. Thesis. Yerevan: SIUA, 2009 (in Russian).

26. Seneta E. Regularly Varying Functions. Lecture Notes in Mathematics. Springer–Verlag, 1976. 27. Bingham N.H., Goldie C.M., Tiegels J.L. Regular Variation. Cambridge Univ. Press, 1986. 28. Danielian E. Tampere: TICSP Series � 12, 2001, p. 1–80. 29. Arutyunian G.S., Yakovlev S.P. Matem. v Visshey Shcole, 2008, � 4 (2, 3), p. 60–63

(in Russian). 30. Arakelyan A.H., Mehrdy K.A. Matem. v Visshey Shcole, 2007, � 3 (1), p. 5–13 (in Russian). 31. Danielian E.A., Avagyan G.P. Doklady NAN A


32. Astola J. Danielian E. Facta Universitatis (NiS), 2006, v. 19, p. 109–131. 33. Thomson B.S., Bruckner J.B., Bruckner A.M. Elementary Real Analysis. Upper Saddle

. Inform. Techn. And Control, 2009, � 1, p. 8–18 (in Russian).

York and London:

ons of Pareto and Warring Distributions. M.: In and Multirate Signal

. v Visshey Shcole, 2006, � 2(2), p. 70–75 (in Russian).

� 2, p. 265–

6. 65.

al of Theoretical Statistic. India, 2008, v. 26(1), p. 121–128.

d D., Gasparian K.V. Statistica Bologna, 2008, v. 68, � 3-4, p. 134–140 (in Russian).

Facta Universitatis (NiS), 2007, v. 20, � 2, p. 119–146. y. Part 1, 2010,

River–New Jersey: Pentice–Hall, 2001, 677 p. 34. Arzumanyan S.K. Vestnik GIUA, 2009, � 12/2, p. 34–40 (in Russian). 35. Avagyan G.P36. Feller W. An Introduction to Probability Theory and its Applications. V. 1. 1st edition. John

Wiley and Sons, 1957. 37. Gransteyn I.S., Ryznik I.M. Table of Integrals, Series and Products. New

Academic Press, 1965. 38. Danielian E., Astola J. On Generating Functi

Bregovic R., Gotchev A (eds.) TICSP Workshop on Spectral Methods Proc., 2007, v. 37, p. 235–237.

39. Arakelyan A.H. Matem. v Visshey Shcole, 2006, � 2(1), p. 5–10 (in Russian). 40. Arakelyan A.H. Matem41. Astola J., Danielian E.A., Arakelyan A.H. Doklady NAN RA, 2008, 108(2), p. 99–109

(in Russian). 42. Yakovlev S.P. Proc. of Engineer. Ac. of Armenia, 2007, � 4(3), p. 26–33. 43. Yakovlev S.P. Modelirovanie, Optimizatsia, Upravlenie. Yerevan: SIUA, 2008, � 11(1),

p. 139–144 (in Russian). 44. Astola J., Danielian E.A., Yakovlev S. Proc. of Engineer. Ac. of Armenia, 2007,

272. 45. Feller W. Introduction to Probability Theory and its Applications. V. 2. 1st edition. John Wiley

and Sons, 19646. Zolotarev V.M. Transl. of Math. Monographs., Amer. Math. Soc., 1980, v. 47. Astola J., Danielian E.A., Arakelyan A.H. Doklady NAN RA, 2007, � 107(1), p. 26–36. 48. Farbod D. Far East Journ49. Farbod D., Gasparian K.V. Matem. v Visshey Shcole, 2009, v. 5(1), p. 50–54 (in Russian). 50. Farbo51. Zolotarev B.M. TVP. �., 1995, � 39, p. 354–362 (in Russian). 52. Astola J., Danielian E.53. Arzumanyan S.K. Vestnik-77 GIUA (Politechnik): Sb. Nauchn. i Metod. State

v. 1 (in Press).




ON ONE URYSOHN TYPE NONLINEAR INTEGRAL EQUATION WITH NONCOMPACT OPERATOR

Kh. A. KHACHATRYAN*

Institute of Mathematics of NAS Armenia

In the present paper the Urysohn type nonlinear integral equation with noncompact operator on the half-line is considered. It is assumed that the Wiener––Hopf–Hankel type operator is a local minorant for the initial Urysohn operator. The existence of a positive and bounded solution is proved. The limit of constructed solution at infinity is calculated. At the end of the work a list of examples is given.

Keywords: nonlinearity, iterative methods, Urysohn operator, Caratheodory condition.

§ 1. Introduction and Formulation of Theorem. In the present work the

following nonlinear integral equation

, (1) 0

( ) ( , , ( )) , (0, )f x K x t f t dt x R�

�� #0 � ��

is considered. Here ( )f x is an unknown real measurable function, satisfying equation (1) almost everywhere, ( , , )K x t = is defined on R R R� �� and satisfies the following conditions: there exists a number 0> % , such that

a) � ( , , ) 0, , , [0, ]K x t x t R R >= = >� � # � � 1� .

b) in K ? = on interval [0, ]> for each fixed ( , )x t R R� �# � . c) ( , , ) ( )K x t Carat >= 1# , i.e. function ( , , )K x t = satisfies the Caratheodory

condition in = on >1 . The latter means that for each fixed [0, ]= ># the function

( , , )K x t = is measurable in ( , )x t R R� �# � and for almost all ( , )x t R R� �# � the function ( , , )K x t = is continuous in = on the interval [0, ]> (concerning this condition we refer to [1]).

d) 0

( , , ) ,K x t dt x R> >�

�& #0 . (2)

Let and are given measurable functions on sets 0 ( )K x ( )K x� R and R� respectively, satisfying * E-mail: [email protected]


24

@ 0 1 0 00 ( ), ( ) ( ),K L R K x K x x R ,�& # � % # (3)

@ v.p. (4) 0 0( ) 1, ( )K d K= = A��

��

� �0 0 ( ) ,K d= = =��

��

% ��0

@ in 00 ( ) ( ), , ( )K x K x x R K x� � � B x on R� , (5) & ! #

@ (6) 22

0( ) .m x K x dx

�� 0 ! ��

We assume that e) � 0( , , ) ( )( ( ) ( )) , , , ,K x t x K x t K x t x t >= � =� � � � #= 1 (7)

where ( )x� is a measurable function on R� , and besides 0 ( ) 1,x� �& & ? in (8) 1, (1 ( )) ( ), 0,1.jx x x L R j� �� # �

f) We also assume that for each measurable function ( ),x. 0 ( )x ,. >& &

the functions 0,x % ( , , ( ))K x t t. and are measurable with respect

to and . 0

( , , ( ))K x t t dt.�

00t % 0x %Remark 1. It is easy to check that, if ( , , )K x t = is continuous in the totality of

all arguments on set >1 , then the conditions c) and f) are fulfilled automatically. It should be noted, that the equation (1) has been recently investigated by the

author in [2] for the particular case ( ) 1, ( ) 0x K x� �� . In the present paper the following result is proved. T h e o r e m . Let the conditions a)–f) are fulfilled. Then the equation (1) has

a nonnegative and bounded solution ( ) , ,f x x R> �& # and besides lim ( )

xf x >

�� . (9)

Moreover, if 0inf ( ) 0x R

x� ��#

� % , then ( ) 0.f x %

§ 2. The Proof of Theorem. Step I. First let us consider the following homogeneous linear equation with

sum – difference kernel

, (10) 00

( ) [ ( ) ( )] ( ) , 0S x K x t K x t S t dt x��

�� 0 %

with respect to an unknown function . From results of [3] it follows that the equation (10) has nontrivial (possessing both positive and negative values) and bounded solution . Below it will be proved that besides the solution , equation (10) has a positive non-decreasing and bounded solution with

( )S x

( )S x� ( )S x�

( )S x�

(11) inf ( ) 0.x R

S x�

�

#%

First we show that 0 ( ) ( ),K x t K x t�� % � ( , ) .x t R R� �# � (12) Indeed, let x t , then from (5) it follows that 0 ( ) ( ) (K x t K x t K x t� � ).� % � �

x t!If we assume that , then taking into account (3)–(5) we’ll have


25

0 0( ) ( ) ( ) ( ).x t K t x K t x K x t� �� % � % � � KThus, the inequality (12) is established. Now consider the following iteration:

0 ��

�� % , (13) 1 00

( ) [ ( ) ( )] ( ) , 0n nS x K x t K x t S t dt x�

0 ( ) sup ( ) , 0,1,2,...x R

S x c S x n�#

� � ��

It is easy to check by induction that 3) ( ) ( ) , 0,1,2...ni S x S x n �� 1) ( )ni S x B in ;n 2 ) ( )ni S x ? in ;x (14)

xample, let’s prove oFor e r 0n 3) :i f � it follows from (13). A

that

ssuming

( ) ( )n x S x � for any n N# , and taking i o account (12) we have S nt

1 0 00 0

( ) [ ( ) ( )] ( ) [ ( ) ( )] ( ) ( )nS x K x t K x t S t dt K x t K x t S t dt S x�

��

.� � � � � � � � �0 0� � �

The statement is proved in the same way. Now let us consider the statement.

��

1)iThe monotonicity of sequences 0{ ( )}n nS x �

� in x is easy to check, if the iterati form

nx

t S t x dt x��

��

� %0 0 (15)

on (13) is rewritten in the following

x

S x K t S x t dt K��

� � �1 0( ) ( ) ( ) ( ) ( ) , 0,n n

0 ( ) , 0,1,2...S x c n� �

0{ ( )}n nS x ��Thus, the sequence of functions has a pointwise limit:

) (n

x S x��

lim ( )nS c*� & . (16)

ce with the B. Levis t

Besides that, in accordan heorem [4], the limit function *( )S x satisfies equation (10). From (14) it follows that

*( ) ( ),S x S x S�& ?� by ,x on R� . (17)

( ) 0,S xNow prove formulae (11). As *( )S x* ; 0, then there exists

0 0x such that

*0 0( ) 0S x$ � % . (18)

), (17), (18), we obtain from

0 0 0 ( ) )x x x

S x d

Then taking into account (12 (10) 0

* *( ) [ ( ) ( )] ( ) ( ( )x x

K x t K x t S t dt K d K$ = =��

� � � � � �0 0

= =��

0 0 0

% .

Therefore, the formulae (11) is true. ing more general linear homogenous

equati

.��

�� %0 (19)

0

0 0( ( ) ( ) ) 0x

K t K t dt$�

� � �0

Step II. Now we consider the followon:

. � 00

( ) ( ) [ ( ) ( )] ( ) , 0,x x K x t K x t t dt x


26

with respect to an unknown real function ( )x. . g with equation (19) we consider the following n

Now alon on-homogenous integral equation

*0( ) (1 ( )) ( ) ( ) [ ( ) ( )] ( ) , .x x S x x K x t K x t t dt x R/ � � /

��

0

� �� #0 (20)

As solution ( )

it follows from [5], equation (20) has a nonnegative nontrivial ( ) ( )0 1x L R M R� �C and besides / # *

0 ( ) ( )x S x/ & . It can be easily shothe f

wn that unction *

1( ) ( )x S x/ � also satisfies the equation (20). Note that

0 ( )x/ ; 1( )x/ as 0 1( )( ) ( )x L R M R/ � � and # C 1 0x Rinf /

�#% .

It is obvious that the function 1 0( ) ( ) ( ) 0x x x. / /� � ( ; 0) will satisfy the equation (19).

pproach to the solution5].

) ( )] ( ) ,n

It is noteworthy that such an a of equation (19) in case when ( ) 0K x� � was suggested in [

Now we consider the following iteration:

1 0( ) ( ) [ (n 0

x x K x t��

��

� �0 K x t t dt. .�� (21)

0 ( ) sup ( ), 0,1,2..., 0.x R

x x n x. .�#

� � %

By the analogy of Step I the following facts can be established by induction: 1) ( )nj x. B in ;n 2 ) ( )nj x. ? in ;x 3) ( ) ( ), 0,1,2...nj x x n. . � (22)

sup ( )n

x R

Therefore, there exists the limit )lim ( ) (x x� (23) x. . .

�

�

�� #& ,

n

and in addition ( )x.� satisfies the equation (19) and ( )x.� ? xin . By analogywn, that if

with formulae (11), it can be sho inf ( ) 0

x Rx�

�#% , then

# 0 inf ( ) 0

x Rx9 .�

�� % . (24)

Step III. At this last stage a nontrivial solution of basic equation (1) will be d monotonicity of tioconstructed using formulae (24) an func ( )x.� . n

Let us consider the following iteration

1 0( ) ( , , ( )) , ( ) , 0,1,2...,n nf x K x t f t dt f x n>�

�

x R0

�� 0 . # (25)

ion f) it follows that each function ( )nf xFrom condit is measurable. we prove that Below

1) ( )np f x B in 2; ) ( ) ), 0,1,2.....,nn p f x n x> ( .sup ( )x R

x Rx.

.�

� �

#

#

First let us prove

� �

1) :p in the light of the properties a) and b) we have

( )1 0

0( , , ) ( ).f x K x t dt f x0 > >

�

� & �


27

1( ) ( )n nf x f x�&Assuming that and taking into account (25) we obtain

1 ( ) ( )n nf x f x� & . he inequalityNow we prove t 2 ) :p for 0n � it is obvious. Assume that 2 )p is

true for and prove it in can m N� # se of 1n m� � . In consequence of inequality (7) we obtain

1

0

( ) ( )( ) ( )) ( ) ( )) ( ) .m 00

( , , (sup sup sup

x R x R x R

x xf x t dt K x t t dt> >� >.. .K x t K x t. . .� � �# # #

�0

Using on

�

��

� � � � � �0

1 2), )p p we conclude that the sequence of functions has a limit

0{ ( )}n nf x ��

lim ( ) ( ),nn

f x f x x R�

�� # ,

satisfying

(26)

( ) ( ) , .x f

supx R

x x R>. >�

.�

��

#

& & # (27)

by ,n K CAs nf B ( )arat >1# ( )) and nf0

( ) ( , ,x K x t f�

& 0

B. Levis theorem mit function

t dt , then from

we get that li ( )f x satisfies the equation (1) 0x

). If f (

x R0 in� �

#( ) 0x

�� % , then from (27) immediately follows that / % . As

( ) sup ( )x R

x x. .� �? , then from (27) we also obtain lim ( )f x >� . x��#

Thus, the Theorem is proved. § 3. Below s of equation (1):

x t G f t dt x

we give some particular example

@ ( ) ( ) [ ( )f x x K x t K��

�0

0� � �0 ( )] ( ( )) , 0,� % (28)

where C G x x x G [0, ], ( ) , [0, ]> ># # , G ? by x on [0, ], ( ) .G> > >� (29)

00

( ) ( , ( ))[ ( ) ( )] ( ( )) , 0,f x R x f t K x t K x t G f t dt x��

�� %0 @ (30)

where ( , )R x t is a measurable function defined on R R� �� with 1) ( , ) ( [0, ])R x t Carat R >�# � , 2) )( ,R x t ? in t on [0, ]> for each fixed ,x R�#

3)

0

1( ) ( , ) , ( , ) [0, ].(

x

x R x t x t RK

�) ( )

xd K d

>=

�

��

& & # �

particular examples of such and

= = =�

��0 0G RAs we can take the following

functions: I. ( ) , (0,1), 1, .G x x R$ $ > x �� # � #

II

. ( ) sin , 1, .G x x x x R> �� #


28

1( ) , 1, .xG x xe x R>� �� III. #( ) ( ) ( ) ( )( , ) ( ) ,

2 2F x x F x xR x t u t� ��

� �

where 10( ) ( ( ) ( ) )

x

xF x K t dt K t dt

��

��

� �0 0 , and t t[0, ], 0 ( ) 1, [0, ],u C u># & & >#

in on u ? t [0, ]> . Remark 2. )

I thank the referee for usef

We note that it can be proved that the solution of equation (28is increasing.

ul remarks.

Received 16.04.2010

R E F E R E N C E S

1. Kufner A. and Fuchik S. Nonlinear Differential Equations. M.: Nauka, 1988, 304 p. 2. Khachatryan Kh.A. Doklady Rossiyskoy A uk. Matematica, 2009, v. 425, � 2,

p. 462–465 (in Russian). ussian).

kademii Na

3. Yengibaryan B.N. Izv. NAN Armenii, Matematica, 1997, v. 32, � 1, p. 38–48 (in R4. Kolmogorov A.N. and Fomin V.S. Elements of the Theory of Functions and Functional

Analysis. M.: Nauka, 1981, 544 p. 5. Arabadjyan L.G. Differential Equations, 1987, v. 23, � 9, p. 1618–1622.




ON THE SOLUTIONS OF SOME DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER IN COMPLEX DOMAIN

B. A. SAHAKIAN�

Chair of Mathematical Analysis, YSU

The paper studies differential equations of fractional order of the form � � � 1/D y z y z f z� �� in the complex domain, where 1,� � is an arbitrary

parameter, 1/D � is the Riemann–Liouville differential operator. For functions of some classes Cauchy type problems are considered.

Keywords: Riemann–Liouville operators, differential equations of fractional order, complex domain.

§ 1. Preliminaries. Let � f x be an arbitrary function from the class

. The function � 0, ,L l 0 l! ! �� 1

0

1 xD f x x t f t dt

�$$

$�� 0 is called

the Riemann–Liouville integral of order � , 0,$ $ # �� , of function � f x , and for

the function � ,0, 1$ # � � (1 )dD f x D f xdx

$ � �� $ is called the Riemann–Liouville

derivative of order $ of function � f x .

It is known that in all Lebesgue points of � f x , � � 0

lim D f x f x$

$

�

��

(and hence, almost everywhere) and, therefore, and � � 0

D f x f x$$

�

�3 4 �5 6

� � 1D f x f x7� .

Let + 10, 1 , 1$ $�

# � � � � 1 , 0, .x l� # The operators � � 0 ,D f x f x�

� � � 1/ /,..., ndD f x D f x D f xdx

� $ �� 1/ ( 1)/nD D f x� �� , , are called

Riemann–Liouville operators of successive differentiation of order

2n

/n � of function � f x . For more information on Riemann–Liouville operators see [1] (ch. IX) and [2] (§ 2). � E-mail: [email protected]


We introduce some notations. Following M.M. Djrbashian we denote

; Arg , 0 | | ,2

z z z�D�

�E � ! ! ! �

" )

'(

2

this domain is evidently many-sheeted for

0 1/�! ! and is arranged on the Riemann supface G� of the function Lnz. ( )H � is the class of functions ( )f z that are analytic in the domain �E . Let us

agree to denote � �(0; ( )) ; arg , 0 | | , .l z z z l. . D� � ! ! ! �� &. D!

Let + 0,1 ,$ # � 1 1 1 , � f$ �� z be an arbitrary function of a

complex variable and � � � 0; , | | , 0 .if re L l r. . . D# & ! ! � The function

� � � 1

0

1 ,( )

zD f z z f d

�$$

$�� 0 where integration is taken along the

intercept connecting points 0 and z, is called the Riemann–Liouville integral of order

1arg( ) ( 1)arg ,z z$ $�� $ of function ( )f z , and the function

� � 1/ dD f z D f zdz

� �� $ is called the Riemann–Liouville derivative of order 1/ �

of function � f z .

The operators 0/ ( ) ( ),D f z f z� � 1/ /( ) ( ),..., ( )ndD f z D f z D f zdz

� $ ��

1/ ( 1)/ ( ),nD D f z� �� , are called Riemann–Liouville operators of successive differentiation of order

2n /n � of the function ( )f z .

Function of Mittag–Leffler type is an entire function of the form

10

( ; )( )

n

n

zE z� n� �

� �

�

��

��

� ( 0)� % of order � with arbitrary value of parameter

� ([1], chap. VI, § 1). For any 0, 0� $% % the following formula holds (see [1], ch. III (1.16)):

1 1/ 1 1 1/

0

1 ( ) ( ; ) ( ; ),( )

, 0 , ,

zi

i

z E d z E z z re�

e r l

,$ � � � $ � .� �

.

� � � � $$

= = D . D

� � � ��

� ! ! ! ! �� & !

0 � (1.1)

� is an arbitrary parameter. § 2. Main Results. T h e o r e m 2 . 1 . Let + 0,1 ,$ # 1 / 1� $� � ( 1),� � is an arbitrary

parameter. Then in the class of functions ( ) (0; ( )),iy re L l. .#

( ) (0; ( )iD y re L l$ . ).� # the following problem of Cauchy type

1/ ( ) ( ) 0,D y z y z� �� (2.1) 0( ) | 0zD y z$�

� � (2.2) has a unique solution ( ) 0.y z � (2.3)


31

Proof. We note that for � 0;z x� # �� the similar problem is considered in the work [3] (see (3.4')– (3.5')). It is obvious that ( ) 0y z � is a solution of problem (2.1)–(2.2). We show that the solution (2.3) is unique. We note that

1/ ( ) ( ) ( ),dD y z D y z y zdz

� $ �� since 0( ) | 0,zD y z$�� then

� � 1

0( ) ,

zD y z y d D y z$ � �� 0 ,i ie z re . . =� � . (2.4)

Using the properties of fractional integrals and derivatives, from (2.4) we get that

1

1( ) ( ) ( )y z D D y z D y z$ ��

�� (almost everywhere). (2.5) Using properties of fractional integrals from (2.5) we conclude that for any the following identities also hold:

1P

� � �

� � � �

1

1

0

1

0 0

( )( ) ( )( / )

( ) ( ) .( / ) ( / )

P

P

P P zP

i PPP Pr ri i i i i

y z D y z z y d� P

re e y e e d e r y e d� P � P

�

�

�

.. . . . .��

��

� �= = = = =� �

�

�

�

�

��

� ��

0

0 0 =

(2.6)

From (2.6) for P � we get

� � � 1 1/

0 0max .

( / )

Pl

i

r l

l ly re y e d

� P

�.

�i.= =

�

�

& && 0 (2.7)

But since � � 0; ( )iy re L l. .# as , then from (2.7) follows the

statement of Theorem, i.e.

P ��

( ) 0.y z �

T h e o r e m 2 . 2 . Let + 0,1 ,$ # 1 1 $�� 1 ,� � is an arbitrary

parameter. Then the function

� 1 1

1/( ; ) ( ; ) ;1/ ,y z e z E z z� ��

��

(2.8) is the solution of the following m: Cauchy type proble

y z y z� � 0,D1/ ( ) ( ) (2.9) � �

zD y z$�� 1.0( ) | (2.10) �

Proof. We note that for � 0;z x� # ��

1/

the Theorem 2.2 is true (see [2, 3]).

By the definition of operator D � we have 1/ ( ; ) ( ; ),dD y z D y zdz

� $� �� where

� � �

� � �

1

01 11 1/

0( )� �$

1; ;( )

1 ;1/ , .

z

zi i

D y z z y d�

z E d z re e

$$

$ � . .�

� � $

� � =

��

��

� � �

� � � � �

0

0 (2.11)


Using the formula (1.1), from (2.11) we get � � 1/; ;D y z E z$ �

�� 1 . (2.12)

easily proved that But it can be � � � � 1 1

1/ 1/;1 ;1/ ,d E z E z zdz

� � ��

��

and hence

� � � 1 1

1/ 1/; ( ; ) ;1/dD y z D y z E z z y zdz

� $ � �� ; ,� � � � � � �

��

� � 1/ ; ; 0y z y z� � � �Di.e. � � and � 0; | 1.zD y z$ ��

Now we show that the solutio is unique. W (2.9)–(2.10) has another solution

n (2.8) e suppose that the problem� ;y z �� . We put � � � * ; ; ; .y z y z y z� � ��

Then it can be easily seen that the function � * ;y z � is a solution of the problem (2.1)– ; 0�(2.2), consequently �*y z , i.e. � � ; ;y z y z .� ��

Theorem 2.2 is proved. T h e o r e m 2 . 3 . Let + 0,1 ,$ # 1 / 1� $� � � 1 ,� � is an arbitrary

parameter, ( ) ( ),f z H �# ( ) (0; ( )).if re L l. .#

z

y z e z f d�

Then the function

� � � 0

; ; , � � �0 i , ,iz re e. . =� � (2.13) �

where � � 1 1

1/;e z ; 1/E z z� ��

�� is the solution of the following Cauchy

type problem: 1/ ( ),D ( ) ( )y z y z� ��

$��

f z (2.14) z 0( ) | 0.zD y �

e (2.15)

Proof. Let ,z r i.� e ,i 1 1 ,ie. =� . =� 10 .r l= =! ! ! ! ! �� By the definition of operator D $� we have:

� � � �

� � � �

� � � �

� � �

11 1 1

0 0

11 1

0 0( 1)

11 1 1

0 0

1( 1) 1 11 1 1

0 0

1; ;( )

1 ;( )

;( )

; *( )

z

ri i i i i i

i ri i

i ii

D y z z d e f d�

re e e d e e f e e d�

e r d e e f e d�

e e r d e f e d�

$$

�

=$. . . . . .�

. $ =$ . .

�

. $ =. $� .�

� � $

1= = = = � =$

= = = = � = =$

= = = = � = =$

��

�

��

� ��

� � � �

� � �

� � � �

� � �

0 0

0 0

0 0

0

= �

�

� � � 11 1 1

0 0; * ,

( )

r

i rie r d e f e d

�

. =$ .

�= = = = � = =$

��

0

0 0

(2.16)


33

� /* ,ie . �� 1�

1

1 1; ; * .i

ie e e e.

.� �= = � = = �

��

� �� where

� � � 1 1 ,if e U iV.. . 1= = =� � We put

� � � 11 1

0 0; * 1 1( )

i reI r d e U d�

. =$

� . , = = = = � = =$

�� 0 0 (2.17)

� � � 12 ; * .

( )

i rieI r d e V d�

. =$

1 1 10 0

� .= = = = � = =$

�� 0 (2.18) 0In the formulas (2.17), (2.18), by changing the order of integration and using

the formula (1.1), we get

� � � � � � 1

1 11 1 1 1 1 1

0 0 0; * ; *

( ) ( )

i ir re eI r d e U d U d r e d� �

. .= =$ $

� . . �=

1= = = = � = = = = = = = � =$ $

� �� 0 0 0 0 �

� � � � � � 1/ ;1 d�� 1 11 1 1 1 1 1

0 0 0; * *

)

ri r rie U d r e d e U E r

=.$ .

. � . �=(�

= = A A � A = �$

��

� � � �0 0 0

and

� = =�

(2.19)

� � � 1/ .r

iI ie V E r d�.2 1 1

0* ;1 1. �= � = =� � � 0 (2.20)

Moreover, according to (2.19), (2.20), from (2.16) we obtain

� �� 1/1 2 1 1 1

0;1 .i; *

riD y z I I e E r$ . f e d� .

�� =� � � � � � = =0 (2.21)

Now using the formula (2.21), we will have

� � � �

� � � � �

� � � � � � � �

� � � � �

1 1

1 1

1 1

1/1 1 1

0

1/1 1 1 1

0

1// /1 1 1

0

1/1 1 1

; * ;1

* * ;1/

;1/

;1/

ri i

ri i

ri i i i

i i i i i

d d drD y z e E r f e ddz dr dz

f re E r r f e d

f z e E e r r f e d e

f z E re e re e f e

�

�

�

�$ . .�

�. .�

�. � . � . .�

�. . . . .�

� � = = =

� � = � = = =

� � = � = = =

� � = � = =

�

�

�

� � ��

� �

� � � � � �

� � � � �

� � � � �

0

0

0

�

� � � �

1 �

�

� � � � �

1 1

10

1/1 1 1 1

0

1 1 10

; ; ,f z e z f dt f z y z�� 0i.e. � �

;1/

ri

z

z

d e

f z E z z f d�

.

��

=

� � � �

�

� � � � � �

�

0

0

� 1/ ; ;D y z y z f z� � � �� a � 0 0.D��; |zy z$ � �nd

tion is uNow we show, that the solu nique. We suppose that the problem (2.14)–(2.15) has another solution � ; .y z �� We put � � � * ; ; ; .y z y z y z� ��


� * ;y z �It can be easily proved that is the solution of the problem (2.1)–(2.2). 2.1, we have � * 0,y z;� i.e. � � ; ; .z y zyBut, according to Theorem � ��

Theorem 2.3 is proved.

Let + 0,1 ,$ # 1 1 $�� 1 ,� �T h e o r e m 2 . 4 . is an arbitrary

r, ( ) ( ),f z H �# ( ) (0; ( )).if re L l. .#paramete Then the function

; ,e z f d � � 0; ; ,i iy z a e z z re e� � z

0

. .� �� =� � � � , (2.22) � �0

auchy type probl is the solution of the following C em:

1/ ( ) ( ) ( ),D y z y z f z� �� (2.23) 0 0( ) | .zD y z a$�

� � Proof. We note that from of the Theorems (2.2)–(2.3) we have

(2.24)

� � ; 0,e z� �� and 1/D � z

�0

; .� � � �1/D e z f d f z�� 0

It is easy to see, that ( ; ) |

� � �

D y z$ ��0 0.z a� �

Consequently, the function (2.22) is a solution of the problem (2.23)–(2.24). Now w t the solution is unique.

Lee show tha

t the function � ;y z �� is a solution of the problem (2.23)–(2.24). We put � � � * ; ; ;y .y z y zz � � �� It is easy to see that the function

*y z� ;� is the solution of the , � * ; 0,y z � � problem (2.1)–(2.2). Consequentlyi.e. � � ; ;y z y z .� ��

Theorem (2.4) is proved.

Received 18.03.2010

R E F E R E N C E S

.M. Integral Transforms and Representations Functions in the Complex Domain.

Akad. Nauk Arm. SSR. Ser. Matem., 1968, v. 3, � 3, p. 171–248 (in Russian).

3. Djrbashian M.M., Sahakyan B.A. Izv. Ak d. Nauk SSSR. Ser. Matem., 1975, v. 39, � 1, p. 69–122 (in Russian).

1. Djrbashian M

M.: Nauka, 1966 (in Russian). 2. Djrbashian M.M. Izv.

a




A REMARK ON STRICT UNIFORM ALGEBRAS

M. I. KARAKHANYAN*, T. M. KHUDOYAN

Chair of Differential Equations, YSU

We study some properties of algebras of bounded continuous functions on a completely regular space, these algebras being equipped with the strong topology defined by of family multiplication operators (strict uniform algebras). We prove an analog of a theorem due to M. Sheinberg for strict uniform algebras (see [1–3]).

Keywords: Strict uniform algebra, amenable algebra, bimodule.

Let be a completely regular Hausdorff space, and 1 *( )C 1 be the algebra of all bounded complex-valued continuous functions on 1 . If we equip the space

with the topology induced by sup-norm *( )C 1 = sup{| ( ) |: }f f x x 1�

# , then

we obtain a commutative Banach algebra ( )bC 1 with the property that the maximal ideals space of which is ( ) =Cb

M 1 91 , where 91 is the Stone-Chekh

compactification for . Recall that we call the remainder of 1 1 in the extension 91 the space \91 1 with the topology induced from 91 (see [4–5]). Let

be the set of all compacts ( )1� \Q 91 12 and for ( )Q 1#� denote ˆ= ( ) = { ( ) : | = 0}Q Q b QC C f C f1 1# ,

where f̂ is the Gelfand transform of f . Then ( )QC 1 is Banach algebra with bounded approximative identity, and ( )bC 1 is -module. In the case when

and , we have QC

1 2, (Q Q 1#� ) 21Q Q21 2( ) ( )Q QC C1 1F .

Note that the remainder \91 1 has a rather complicated structure, because, for instance, in every point of the remainder the first axiom of countability fails to hold. For denote ( )Q 1#� = \Q Q1 91 . All the Banach algebras are proper closed ideals in the algebra

( )QC 1

( )bC 1 for every ( )Q 1#� . Every ideal defines a family of seminorms ( )QC 1 ( ){ }

Qg g CP 1# on ,

with

( )bC 1

( ) =g gP f T f�

, where : ( ) ( )g b bT C C1 1� is the multiplicative operator

��* E-mail: [email protected]


=gT f gf . The topology on ( )bC 1 , defined by this family of seminorms, we will

call the Q9 -topology, and we will denote by ( )Q

C 91 the algebra endowed

with the

( )bC 1

Q9 -topology (cf. [6–9]). It is easy to see that Q9 -topology is Hausdorff topology.

We will say that a closed in the Q9 -topology subalgebra of algebra �

( )Q

C 91 is Q9 -uniform, if it contains constants and separates the points of (i.e.

for any

Q1

1 2, Qx x 1# with 1 2x x; , there exists f #� such that 1 2( ) ( )f x f x; ).

Note that in the case of completely regular space 1 , the ideal can turn out to be unusable, because of its triviality.

0 ( )C 1

It should be noted here that 9 -topology of Buck on *( )C 1 is the inductive limit ( )QLind 9 of Q9 -topologies for ( )Q 1#� .

If , then is locally-compact Hausdorff space and in

that case one can introduce a strong topology on

( )Q 1#� = \Q Q1 91

*( QC )1 using the ideal ,

which we will denote by 0 ( )QC 1

( )QC 91 .

Since Q1 1 92 2 1 , then the space of maximal ideals

( ) = ( ) = ( )b Q

C QM 1 9 1 9 1 .

It can be easily seen, that the algebra ( )Q

C 91 is topologically isomorphic to

the algebra ( )QC 91 and hence the following assertions hold (cf. [2, 3]): T h e o r e m 1 .a) For any the algebra ( )Q 1#� ( )

QC 91 is Q9 -complete locally convex

algebra; b) is everywhere dense in 0 ( ) = (Q QC C1 )1 ( )

QC 91 ;

c) the space of all Q9 -continuous linear functionals on ( )Q

C 91 is

isomorphic to the space of all finite regular measures on ( )QM 1 Q1 . Proposition 1. a) The uniform topology and Q9 -topology on

coincide for every open set U in 0 ( )C U �

\{ ( ) : | =b Uf C f 11� # 0} Q1 such that QU 12 .

b) The linear space generated by 0{ ( )}i i IC U # , where { }i i IU # is the subset of

the set of all open subsets in Q1 such that i iU U Q12 2 , is Q9 -dense in

( ) ( )Q

QC C9 91 1 .

Let A be a Q9 -uniform algebra on 1 . Since the algebra ( )b QC 1 is comple-

te in the Q9 -topology, then A is a closed subalgebra of the algebra in the

sup-norm. Hence, we will denote the algebra A in the sup-norm of

( )b QC 1

( )b QC 1 by . ,b QA


37

Suppose that the Banach space X is -bimodule. Recall that ,b QA X is

Q9 -complete -bimodule, if from the fact that the net { in ,b QA }i i If # A

Q9 -converges to 0f it follows that for any x X# the nets { }i i If x # and { } converge to

i i Ixf #

0f x and 0xf respectively in the norm of the Banach space X . The bimodular operation on X defines a bimodular operation on the dual

space *X of X ( )( ) = ( ), ( )( ) = ( )f x xf f x fx. . . .

for all f A# , x X# , *X.# . Note also that linear functional *X.# is called weak* Q9 -continuous, if

from the Q9 -convergence in A of the net { }i i If # to 0f it follows that the net of functionals { }i i If . # and { }i i If. # converge in the weak topology to 0f . and 0f. respectively.

As in ([2, 3]) we define the abelian group 1 ( , )Q

*Z A X9 of all Q9 -continuous

in the weak* topology differentiations (i.e. if the net { in *:D A X� }i i If # A Q9 -converges to 0f , then the net of functionals { ( )}i i ID f # converges to in the weak* topology of

0( )D f*X ). We denote by 1

* ( , )*Z A X the abelian group of all continuous in the weak* topology differentiations *

,: b QD A � X*

. For every

, ( )Q 1#� 1 ( , )Q

Z A X9 is a subgroup of 1 *( , )Z A X .

Following B. Johnson [10] one calls a Banach algebra to be amenable,

if the group is trivial for every -bimodule ,b QA

1 * 1 * 1( , ) = ( , ) / ( , )H A X Z A X B A X *,b QA

X , where 1( , )*B A X is the abelian group, consisting of all inner differentiations ( ) =a a a.8 . .� . Analogously, the algebra A is called Q9 -amenable, if the

group is trivial for any 1 * 1 * 1( , ) = ( , ) / ( , )Q Q Q

H A X Z A X B A X9 9 9*

Q9 -complete -

bimodule ,b QA

X . Clearly, if A is an amenable algebra, then A is Q9 -amenable (i.e. from

the condition for any -bimodule 1 *( , ) = 0H A X ,b QA X it follows that

for any 1 *( , ) = 0Q

H A X9 Q9 -complete -bimodule ,b QA X ).

For the rest we need two Q9 -complete -bimodules. ,b QAProposition 2. Let ( )QM� 1# . Then there exists a measure ( )QMA 1#

and a function such that 0 ( Qg C 1# ) *= g� A , i.e. =fd fgd� A0 0 for all ( ( ). 0 ( )Qf C 1# )QC 1

T h e o r e m 2 . For any positive measure ( )QM� 1# the Hilbert space

is 2 ( ,QL 1 �) Q9 -complete Banach -bimodule. ,b QA


The proof can be done in the same manner as of the Lemma 4 in [3]. Let be the algebra of all bounded linear operators in

, and

2= ( ( , )Q QB BL L 1 � )

)2 ( ,QL 1 � 1,QJ be the ideal of nuclear operators, which is Banach space in

the nuclear norm 1 trT � T ([11]). 1,QJ becomes Banach -bimodule in the

case ,b QA

= f gf T g T T T� � � � for all ,, b Qf g A# and 1,QT J# .

It is easy to see (c.f. [11]), that for any 1,QT J# there exists a positive

function such that 0 ( Qg C 1# ) 1 1,QgT T J� � # .

T h e o r e m 3 . The Banach space 1,QJ is Q9 -complete -bimodule. ,b QAIt is well known, that the algebra QB is isometrically isomorphic, as a

Banach space, to the dual space *1,QJ (c.f. [9]). This leads to the following result.

T h e o r e m 4 . The Banach -bimodule ,b QA QB is isometrically isomorphic

as a -bimodule to the ,b QA Q9 -complete in the weak* topology Banach -

bimodule ,b QA

*1,QJ .

Using Lemma 7 form [3], one can analogously prove the following Proposition 3. Let A be Q9 -complete uniform algebra. If ( )

QA C 91; ,

then for some 1 *( , ) 0Q

H A X9 ; Q9 -complete Banach -bimodule ,b QA X .

From this Proposition we get the following result, which is the main result of the paper.

T h e o r e m 5 . Let A be Q9 -uniform algebra. Then the following conditions are equivalent:

a) = ( )Q

A C 91 ;

b) A is amenable algebra; c) A is Q9 -amenable algebra. Now consider the situation, when 1 is completely regular Hausdorff space.

In this case, as has been mentioned above, one can introduce 9 -topology in the algebra as the inductive limit *( )C 1 ( )LindQ Q9 of Q9 -topologies, where

, which we will denote again by ( )Q 1#� ( )C 91 . Then by 9 -uniform algebra A over we will mean (as above) a closed in the 1 9 -topology subalgebra in the algebra ( )C 91 , which contains constants and separates the points of . 1

It is easy to see, that 9 -topology on A is the inductive limit �Lind Q 9 of

Q9 -topologies of algebras Q

A9 , which are Q9 -uniform subalgebras of algebras

( )Q

C 91 respectively.

In the light of the obtained results, we can formulate the following results for completely regular Hausdorff space 1 .


39

T h e o r e m 6 . a) The algebra ( )C 91 is 9 -complete locally convex algebra;

b) the space of all 9 -continuous linear functionals on ( )C 91 is isomorphic to the space of all finite regular measures on ( )M 1 1 .

T h e o r e m 7 . Let A be 9 -uniform subalgebra of ( )C 91 . Then the following conditions are equivalent:

a) = ( )A C 91 ; b) A is amenable algebra. In the case, when is a compact, we get the Theorem of M. Sheinberg

from [1]. 1

Remark. Note that null-set is a set of the form with . Let is the set of all null-sets

1(0)f �*( )f C 1#

( )1� \Z 91 1# . If ( )Z 1#� , then \ Z91 is : -compact and locally-compact space and, in the light of Theorem 2.6 from [12], in *( \C )Z91 the strong topology coincides with the strong topology of Mackey (i.e. strong space of Mackey). It follows that (C Z\ )9 91 is ( )

ZC 1 9 . Hence all the

above idealogy works also for Z9 - iform algebras. unNote that in the algebra *( )C 1 one can introduce also the 19 -topology as

the inductive limit ( )LindZ Z9 of Z9 -topologies, where ( )Z 1#� , which we will denote by

1( )C 91 . This 19 -topology, as well as 9 -topology, is locally convex,

Hausdorff and 19 9& & � . For 19 -uniform algebras over 1 the analogues of Theorem 6 and Theorem 7 are true.

Received 05.09.2009

R E F E R E N C E S

1. Sheinberg M.V. Uspekhi Mat. Nauk, 1977, v. 32, � 5 (197), p. 203–204 (in Russian). 2. Karakhanyan M.I., Khor’kova T.A. Funktsional. Anal. i Prilozhen., 2009, v. 43, � 1, p. 85–

87. Translation in Funct. Anal. Appl., 2009, v. 43, � 1, p. 69–71 (in Russian). 3. Karakhanyan M.I., Khor'kova T.A. Sibirsk. Mat. Zh., 2009, v. 50, � 1, p. 96–106.

Translation in Sib. Math. J., 2009, v. 50, � 1, p. 77–85 (in Russian). 4. Gelfand I.M., Raikov D.A., Shilov G.E. Commutative Normed Rings. M.: Izd. Fiz.-Mat. Lit.,

1960 (in Russian). 5. Arkhangelskii A.V., Ponomarev V.I. Fundamentals of General Topology in Problems and

Exercise. M.: Nauka, 1974 (in Russian). 6. Buck R.C. Michigan Math. J., 1958, v. 5, � 2, p. 95–104. 7. Sentilles F.D. Trans. Amer. Math. Soc., 1972, v. 168, p. 311–336. 8. Giles R. Trans. Amer. Math. Soc., 1971, v. 161, p. 467–474. 9. Glicksberg I. Proc. Amer. Math. Soc., 1963, v. 14, p. 329–333. 10. Johnson B.E. Memoirs of the American Mathematical Society, � 127. American Mathematical

Society. Providence, R.I., 1972, 96 p. 11. Reed M., Simon B. Methods of modern mathematical physics. V. 1. New York, London:

Functional Analysis Acad. Press, 1972. 12. Conway J.B. Trans. Amer. Math. Soc., 1967, v. 126, p. 474–486.




NON-UNITARIZABLE GROUPS

H. R. ROSTAMI*

Chair of Algebra and Geometry, YSU

A group is called unitarizable, if every uniformly bounded representation of on a Hilbert space is unitarizable. N. Monod and N.

Ozawa in [6] prove that free Burnside groups

G: (G B HD � ) G H

( , )B m n are non unitarizable for arbitrary composite odd number , where n . We prove that for the same the groups

1 2=n n n 1 665n (4, )B n have continuum many non-isomorphic factor-groups,

each one of which is non-unitarizable and uniformly non-amenable.

Keywords: representation of group, unitarizable group, free Burnside group, periodic group.

Let be a group, be a Hilbert space. A representation G H : (G B H )D � is

called unitarizable, if there exists an invertible operator T such that the operator 1( )gU T g TD � is a unitary operator for any element g G# . The group is

called unitarizable, if every uniformly bounded representation G

: (G B H )D � is unitarizable.

J. Dixmier in [1] and M.M. Day in [2] proved that every amenable group is unitarizable. The question of whether the converse holds has been open since then. The first example of non-unitarizable group is constructed in [3], where it is shown the non-unitarizability of the group It is known, that if all the countable subgroups of a given group are unitarizable, then the group is unitarizable itself, and if a group is unitarizable, then all its subgroups and factor groups are unitarizable (see for example [4]). Therefore, from existence of non-unitarizable group it follows that the absolutely free group

2( ).SL �G G

F� of countable rank is non- unitarizable, and therefore any group that contains a subgroup isomorphic to the free group 2F of rank 2 is non-unitarizable.

N. Monod and N. Ozawa in joint paper [5] studying the question: “does it follow from unitarizability of group its amenability” (see [1]), obtained an inte-resting criterion, according to which the non-amenability of a given group G is equivalent to non-unitarizability of group wrA G for all infinite Abelian groups A is wreath product of groups A, where wrA G and ��

.G �



41

The first example of non-amenable grou that s isp at fy a non-trivial identity relatio

ds = ( ,..., ) ,mA A a a G is non-amenable for any odd number and . The group

n was indicated by S.I. Adian. The well-known theorem of Adian (see [6], Theorem 5) asserts that the free Burnside group

1 11 2( , ) = , ,... | = 1for all worn

mB m n a a a A * *H 1

665n > 1m ( , )B m n does

e it sati he idabout non-

amena

not contain absolutely free groups, sinc sfies t entity = 1.nxBearing on the mentioned criterion and on Adian’s Theorem

bility of groups ( , )B m n , N. Monod and N. Ozawa proved (see [5], Theorem 2) that the free Burnside groups ( , )

B m n are non-unitarizable for all

composite odd numbers 1 2=n n n , where 1 665n 2.m V.Atabekyan in pa has stre

andper [7] ngthened this result. He proved, that for

any composite odd number 1 2=n n n , where 1 665n and 1m % , any non-cyclic subgroup of free Burnside gro , )up (B m n is non zable.

This result of paper [7] indicates the new examples-unitari

of non-unitarizable period

ic groups different from Burnside groups. Actually, comparing it with result

by A.Yu. Olshanskii [8] (for odd 78> 10n ) and the result by V. Atabekyan [9] (for odd 1003n ), we obtain that an r normal subgroup of group ( , )y prope B m n is not isomo any free Burnside group and at the same time is non-un ble group.

A

rphic to itariza

ccording to [4] (see corollary 0.11), if all the countable subgroups of a given

[7], states that for every composite odd number =n n

group G are unitarizable, then the group G is unitarizable itself. The infinite cyclic oup is amenable and, therefore, is un arizable (see [1, 2]). Hence, any absolutely free group of rank 2 appearing as non-unitarizable contains countable unitarizable subgroup.

Another result of the paper

gr it

1 2n , where 1 665n and 1m % , any infinite subgroup of group ( , )B m n is finite subgroup is unitarizable. Thus, for sub of

free Burnside groups the unitarizability is equivalent to amenability. In the current work we prove that there exist finitely g

non-unitarizable, and any groups

enerated non-unitar

se is arbitrary composite odd number, where n um man

satisfy thamen

izable periodic groups of restricted period, that are different from free Burnside groups and their non-cyclic subgroups. According to the result of Dixmier–Day, non-unitarizable groups are non-amenable. Constructed below non-unitarizable groups are not only non-amenable, but also uniformly non-amenable. We prove the following

T h e o r e m . Suppo 1 2=n n n

1 665. There are continu y non-isomorphic 4-generated groups that

e identity = 1nx , each one of which is non-unitarizable and at the same time uniformly nonable.

Proof. The well-known theorem by S.I. Adian (see [10]) states, that for > 1m and odd 665n the group ( , )B m n is infinite. As it is shown in the work

1003 there are continuum many simple 2-generated non-isomorphic groups {� he given period 1003.n Let 1 2=n n n be arbitrary [11], for arbitrary

# of t odd n

}i i I


composite odd num where 1 665.n The 03. L rm a direct product = (2, )i iG B n �� of the (2, )

ber, n n et’s fo 10group B n with each group { } .i i i I� � ##

Consider ups 1 1= (2,G B n any two gro ) �� , where hic 2-generated groups of period and

that

and 2 = (2, )G B n �

1 2, { }i i I� � � ## are non-isomorp show that

2G are non-isomorphic. Proving b ontradiction, suppose

2�,n

groups 1G and y c 1 2:G GI � is some isomorphism.

It is obvious that groups (2, )B n and i� are contained in iG as subgroups ( =1,2i ). Consider the imag 2, ))e ( (B nI of subgroup (2, )B n via isomorphism I . Since the image of normal s via isomorphi a normal subgrou , then ( (2, ))

ubgroup sm is pB nI is normal subgroup in 2.G Let us show, that the intersection of

subgr 2, ))o ( (B nup I with normal subgroup 2� of the group 2G is trivial. Actually, subgroup 2since the � is simple group, then any norm l subgroup containing a non-trivial element of subgroup 2

a� , contains all the elements of that

group. Therefore, if 2( (2, )) =B nI � JC K , the 2( (2, ))B nn I �� . It is obvious that ( (2, )) (2, )B n B nI L . aper [9 roup 2� of group By Theorem 1 of p ], normal subg( (2, ))B nI is not free periodic. On the other hand by Theorem 1 of paper [12],

2� contains free periodic subgroup H of rank 2. But this is impossible since only e non-cyclic subgroup H of the group 2

subgroup th � is the group 2� . Thus,

2( (2, )) =B nI �C K . It is clear that the following isomorphisms are true:

G� 1 1 1/ (2, ) ( ) / ( (2, ))B n G B nI I . Since 1 2( ) / ( (2, )) = / ( (2, ))G B n G B nI I I and

2( (2, )) =B nI �C K , then the group 2� can be embedded into the group

2 / ( (2, ))G B n 1I � . But this is again tradiction since the groups 1� and rphic infinite groups and any proper subgroup of the group

is finite. The con

con

2� are non-isomo

tradiction proves that groups and are non-isomorphic. Since our co

1�

1G 2Gnstructed groups = (2, )i iG B n �� ( i I# ) contain non-unitarizable subgroup

(2, )B n , then they are non order to finish the proof of Theorem l

n-unitarizable either. I et’s prove that each group )

is uni m oniG ( i I#

formly non-amenable. It is known, that if a group has a unifor ly n -amenable homomorphic image, then the group is uniformly non-amenable itself (see [13], Theorem 4.1). The factor-group of group iG by the normal closure of subgroup i� is isomorphic to group (2, )B n . According to the corollary 1 of the paper [14] he group (2, ), t B n is unifo n-amenable. Thus, groups iG ( i I# ) are uniformly non-ame either, since they have a uniformly non-ame e factor-group.

Corollary. For arbitrary composite odd number =n n n , where 665,n the gr

rmly nonable nabl

1 2 1

oup (4, )B n has continuum non-isomorphic factor-groups, each one of which


43

is non-unitarizable and uniformly non-amenable. It is sufficient to notice that in each group iG , constructed during the proof

of The

Received 18.03.2010

R E F E R E N C E S

Dixmier J. Acta Sci. Math. Szeged, 1950

–291. 33.

, � 248,

t.

dv. Math.,

orem 1, the identity = 1nx holds.

1. , � 12, p. 213–227. 2. Mahlon M. Day Trans. Amer. Math. Soc., 1950, � 69, p. 2763. Ehrenpreis L., Mautner F.I. Proc. Nat. Acad. Sc. USA, 1955, � 41, p. 231–24. Pisier G. Infinite Groups: Geometric, Combinatorial and Dynamical Aspects, 2005

p. 323–362. Progr. Math., Birkhäuser, Basel. 5. Monod N., Ozawa N. Journal of Functional Analysis, 2010, v. 258, � 1, p. 255–259. 6. Adyan S.I. Izv. AN SSSR. Ser. Matem., 1982, v. 46, � 6, p. 1139–1149 (in Russian). 7. Atabekyan V.S. Matem. Zametki, 2010, v. 87, � 4, p. 500–504 (in Russian).

Dordrech8. Ol’shanskii A.Yu. Math. Appl., 2003, v. 555, p. 179–187. Kluwer Acad. Publ. 9. Atabekyan V.S. Fundament. i Priklad. Matem., 2009, v. 15, � 1, p. 3–21 (in Russian). 10. Adyan S.I. Burnside Problem and Identities in Groups. M.: Nauka, 1975 (in Russian). 11. Atabekyan V.S. Matem. Zametki, 2007, v. 82, � 4, p. 495–500 (in Russian).

n). 12. Atabekyan V.S. Izv. RAN.Ser. Matem., 2009, v. 73, � 5, p. 3–36 (in Russia13. Arzhantseva G.N., Burillo J., Lustig M., Reeves L., Short H., Ventura E. A

2005, v. 197, � 2, p. 499–522. 14. Atabekyan V.S. Matem. Zametki, 2009, v. 85, � 4, p. 516–523 (in Russian).



M e c h a n i c s

COMPARISON OF DIFFERENT PLANE MODELS IN FINITE ELEMENT SOFTWARE IN STRUCTURAL MECHANICS

B. YAZDIZADEH*

Chair of Mechanics, YSU

In solution of plane problems of mechanics there are several elements used in finite element software. In ANSYS – one of the best finite element software – there are about six type of elements. We consider different Plane Models in simple bending problem and compare the various results to distinguish the type of elements, which are suitable for solving the problem under consideration. Comparing results with analytical solutions shows that the Plane 42 and 82 models are the most suitable ones among the others. Also results in Plane 42 with usual mesh is closer to the same problem solution with fine mesh than Plane 82 in the same case.

Keywords: beam, bending, frequency, finite element.

Introduction. The finite element method is a numerical method that can be used for the accurate solution of complex engineering problems. The method was first developed in 1956 for the analysis of aircraft structural problems. Thereafter, within a decade, the potentialities of the method for the solution of different types of applied science and engineering problems were recognized [1]. Over the years, the finite element technique has been so well established, that today it is considered one of the best methods for solving a wide variety of practical problems efficiently. In fact, the method has become one of the active research areas for applied mathematicians. One of the main reasons for the popularity of the method in different fields of engineering is that once a general computer program is written, it can be used for the solution of any problem simply by changing the input data [2].

Nowadays, we live a curious situation. On one hand, most structural engineers and FE codes for computational solid mechanics are decanted. On the other, the observed mesh-size and mesh-bias dependence exhibited by these models make the academic world very suspicious about this format. Hence, a lot of effort has been spent in the last 30 years to investigate and remedy the observed drawbacks of this approach [3].

In some complicated problems, such as cracks and contacts with no effective analytic solution, numerical analysis is strongly recommended. Finite element

��*�E-mail: [email protected]


45

analysis is one of the usual ways to solve this kind of problems. In ANSYS software, there are some elements to solve plane problems, so it is necessary to choose the right type of elements type complex problems to obtain correct results.

Theoretical Aspects. We assume simple plane bending as illustrated in Fig. 1.

Fig. 1.

The displacement formula is [4]:

4

0yEIx

M�

M, (1)

where E is the module of elasticity; is the moment of inertia; I y is the vertical displacement of the point in x position.

As we know for the beam of Fig. 1, solution of above differential equation

with the following boundary conditions 0x L

yx �

M�

M, 0x Ly

�� is

� 3 2 32 36

py L L x x�EI

� � .

The maximum displacement (at 0x � ) is 3

3PLEI

.

The normal and shear stresses for the beam in Fig. 1 are obtained from relation (2) and (4) respectively [5]. Relation (4) is obtained from relation (3) applying the beam conditions:

,MyI

: � (2)

xyVQIt

= � , (3)

2

23 12xy

V yA c

=� �

� � ��

� , (4)

where is the first moment of area from to natural axis; V is the shear force acting on section; t is the beam thickness;

Q y: is the normal stress; = is the shear

stress; y is the distance of a such point, where the shear stress must be calculated; is the maximum distance from beam surface to natural axis and c A is the area of

cross section of the beam. For more information see [6]. It is clear that maximum

normal stress maxMcI

: � , and maximum shear stress max32

VA

= � � .


There are several ways to calculate the beam bending frequency. Relation (5) is obtained from Rayleigh approximation method and the error of this approximated solution is less than 0.5% [7]:

41 /2

f k EIg LND

� , (5)

where f is the first mode lateral frequency; N is the weight per unit length; g is the volume coefficient that is chosen here to be equal to 1 and k is the constant that is equal to 3.53.

We proceed by solving the problem by different plane elements, and then comparing the results with the analytical solution.

Choosing plane element instead of various type of ANSYS element is discussed in [8]. The elements, that were used, are listed and explained below.

Different Plane Elements in ANSYS Software [9]. 1. Plane 42 is used for two-dimension (2-D) modeling of solid structures as

illustrated in Fig. 2. The element can be used either as a plane element (plane stress or plane strain) or as an axisymmetric element. The element is defined by four nodes, having two degrees of freedom at each node: translations in the nodal x and directions. The element has plasticity, creep, swelling, stress stiffening, large deflection and large strain capabilities.

y

Fig. 2.

2. Plane 82 is a higher order version of the 2-D, four-node element (Plane 42) as

illustrated in Fig. 3. It provides more accurate results for mixed (quadrilateral-triangular) automatic meshes and can be used for irregular shapes without much loss of accuracy. The 8-node elements have compatible displacement shapes and are well suited to model curved boundaries.

Fig. 3.

3. Plane 182 is used for 2-D modeling of solid structures like in Fig. 2. The

element can be used as either a plane element (plane stress, plane strain or generalized plane strain) or an axisymmetric element. It is defined by four nodes, having two degrees of freedom at each node: translations in the nodal x and directions. The element has plasticity, hyperelasticity, stress stiffening, large deflection and large strain capabilities. It also has mixed formulation capability for simulating deformations of nearly incompressible elastoplastic materials and fully incompressible hyperelastic materials.

y


47

4. Plane 183 is a higher order 2-D, 8-node or 6-node element like in Fig. 3. Plane183 has quadratic displacement behavior and is well suited to modeling irregular meshes.

This element is defined by 8 nodes or 6-nodes, having two degrees of freedom at each node: translations in the nodal x and y directions. The element may be used as a plane element (plane stress, plane strain and generalized plane strain) or as an axisymmetric element. This element has plasticity, hyperelasticity, creep, stress stiffening, large deflection and large strain capabilities. It also has mixed formulation capability for simulating deformations of nearly incompressible elastoplastic materials and fully incompressible hyperelastic materials. Initial stress import is supported.

5. Plane 25 is used for 2-D modeling of axisymmetric structures with non-axisymmetric loading as illustrated in Fig. 4. Examples of such loading are bending, shear or torsion. The element is defined by four nodes, having three degrees of freedom per node: translations in the nodal ,x y and direction. For unrotated nodal coordinates, these directions correspond to the radial, axial and tangential directions respectively.

z

The element is a generalization of the axisymmetric version of Plane 42 the 2-D structural solid element, where the loading need not be axisymmetric.

Fig. 4.

6. Plane 83 is used for 2-D modeling of axisymmetric structures with non-axisymmetric loading as illustrated in Fig. 5. Examples of such loading are ben-ding, shear or torsion. The element has three degrees of freedom per node: trans-lations in the nodal ,x y and direction. For unrotated nodal coordinates, these directions correspond to the radial, axial and tangential directions respectively.

z

This element is a higher order version of the 2-D, four-node element (Plane 25). It provides more accurate results for mixed (quadrilateral-triangular) automatic meshes and can tolerate irregular shapes without much loss of accuracy. The element is also a generalization of the axisymmetric version of Plane 82 the 2-D 8-node structural solid element, where the loading need not be axisymmetric.

Fig. 5.


Solution of the Problems. The following parameters are considered for the beam:

8 2 32 , 50 , 0.01 , 0.2 , 2 10 / , 1700 /L m P N b m h m E N m kg m�� ,

where and b are the beam height and thickness respectively and h � is the beam density.

Substituting this values in (1)–(4), after calculation we obtain the values of maximum displacement, normal and shear stresses and frequency: 0.2 m, 3 106 N/m2, 7.5�104 N/m2 and 2.78148 s–1, respectively. �

Modeling in ANSYS software is implemented in two ways for each element: 1. Creation of the elements directly from nodes. In this way 3 samples

(1 element, 4 elements and 10 elements) are used. 2. Creation of the region then meshing it. In this way 2 samples (usual and

fine meshing) are used. The analysis is done with three outputs: x and displacement, normal and

shear stresses and first mode frequency. y

Results.

T a b l e 1

The maximum displacement (in meters) and error results

Displacement � plane 1

elementerr., %

4 elements

err., %

10 elements

err., %

usual elements

err., %

fine mesh elements

err., %

42 1.51E-01 24 1.98E-01 1 2.01E-01 0.5 2.01E-01 0.5 2.01E-01 0.5

82 1.51E-01 24 1.98E-01 1 2.01E-01 0.5 2.01E-01 0.5 2.01E-01 0.5

182 5.09E-03 97 5.75E-02 71 1.36E-01 32 1.97E-01 1.5 2.01E-01 0.5

183 5.09E-03 97 5.75E-02 71 1.36E-01 32 1.97E-01 1.5 2.01E-01 0.5

25 8.67E-05 100 1.38E-04 100 1.43E-04 100 1.46E-04 100 1.47E-04 100

83 1.65E-05 100 9.64E-05 100 1.09E-04 100 2.67E-04 100 1.47E-04 100

T a b l e 2

The maximum normal stress and error results

Normal stress

� plane 1

element err., %

4 elements

err., %

10 elements

err., %

usual elements

err., %

fine mesh elements

err., %

42 1.50E+06 50 2.63E+06 12 2.85E+06 5 3.09E+06 3 3.05E+06 1.8

82 41667 98 8.24E+05 72 2.70E+06 10 3.33E+06 11 3.04E+06 1.7

182 41667 98 8.24E+05 72 2.11E+06 70 3.21E+06 107 3.97E+06 32

183 41667 98 8.24E+05 72 2.11E+06 70 6.12E+06 104 4.47E+06 49

25 1193.7 100 3390 100 4814.5 100 9827.9 100 31034 100

83 207.59 100 3326.8 100 5551.5 100 29352 100 60009 100


49

T a b l e 3

The maximum shear stress and error results

Shear stress �

plane 1 element

err., %

4 elements

err., %

10 elements

err., %

usual elements

err., %

fine mesh elements

err., %

42 50000 33 50000 33 50000 33 78254 4 84146 12

82 95833 27 52941 29 11111 85 70471 6 84425 13

182 95833 27 52941 29 11111 85 69484 7.3 70893 5.5

183 95833 27 52941 29 11111 85 87309 17 85604 14

25 0 100 75 100 185 99 1308 98 5166 93

83 197 100 146 100 435 99 2425 96 6930 90

T a b l e 4

First mode frequency and error results

Frequency

� plane 1

element err., %

4 elements

err., %

10 elements

err., %

usual elements

err., %

fine mesh elements

err., %

42 2.71E+00 2.5 2.77E+00 0.4 2.75E+00 1.1 2.75E+00 1.1 2.75E+00 1.1

82 3.01E-05 8.2 5.12E+00 84 3.35E+00 20 2.75E+00 1.1 2.75E+00 1.1

182 1.48E+01 47 5.12E+00 84 3.35E+00 20 2.78E+00 0.05 2.75E+00 1.1

183 1.48E+01 47 5.12E+00 84 3.35E+00 20 2.75E+00 1.1 2.75E+00 1.1

25 3.64E-06 100 0.00E+00 100 0.00E+00 100 5.17E-06 100 0.00E+00 100

83 0.00E+00 100 0.00E+00 100 0.00E+00 100 6.29E-06 100 3.01E-05 100

Discussion. The Table 1, representing the displacement, shows that Plane 25 and 83 are

not appropriate for this purpose. Plane 182 and 183 are appropriate only in usual and fine mesh, and Plane 42 and 82 are appropriate for all conditions except in one element mesh. In usual and fine mesh element the Plane 42 and 82 errors are less than 1%.

The Table 2, representing the maximum normal stress, shows that Plane25 and 83 are not appropriate. Plane 182 and 183 only at fine mesh give some results that are close to analytical solution although the error is not acceptable. Plane 42 and 82 are appropriate for usual and fine element meshes. In fine mesh element the Plane 42 and 82 errors are less than 2%.

The Table 3, representing the maximum shear stress, shows that Plane 25 and 83 are not appropriate. Other Plane element (182, 183, 42 and 82) are appropriate only in usual and fine element meshes. There is about 5% error only in Plane 42 and 82 at usual mesh.

The Table 4, representing the first mode frequency, shows that Plane 25 and 83 are not appropriate. Plane 82, 182 and 183 are appropriate only in usual and fine meshes (error is less that 2%). Plane 42 is appropriate for all conditions.


Conclusion. The results show that the analysis of the simple beam bending with Plane 25 and 83 is not recommended in all cases. For displacement in usual and fine mesh, Plane 42, 82, 182 and 183 results are close to analytical solution, but in some problems fine mesh couldn’t be applied and usual mesh is considered for use, so in this case Plane 42 and 82 are better to use. By the same reason as above Plane 42 and 82 are suggested for normal and shear stresses. Frequency results shows, that there is no difference between Plane 42, 82, 182 and 183, so each of them is suitable for use.

In some mixed problems or problems where all the data must be calculated in one procedure, Plane 42 or Plane 82 is recommended and better for use to solve plane problems.

Special thanks to my adviser and supervisor Prof. Karen Ghazarian and Dr. Haikaz Yeghiazaryan for their help.

Received 09.04.2010

R E F E R E N C E S

1. Gupta K.K. and Meek J.L. International Journal for Numerical Methods in Engineering, 1996,

v. 39, p. 14. 2. Singiresu S.R. The Finite Element Method in Engineering. Fourth ed. Elsevier Science &

Technology Books, 2004, 658 p. 3. Cervera M. Computer Methods in Applied Mechanics and Engineering, 2008, v. 197, p. 16. 4. Popov E.P. Engineering Mechanics of Solids. First ed. New Jersy: Prentice-Hall, 1990, 760 p. 5. Beer F.P. and Johnson E.R. Mechanics of Material. 2nd ed. NY: McGraw-Hill, 1925, 532 p. 6. Timoshenko S.P. and Goodier J.N. Theory of Elasticity. 3rd ed. NY: McGraw-Hill, 1970,

506 p. 7. Timoshenko S.P. Vibration Problems in Engineering. 2nd ed. NY: D. Van Nostrand Company,

INC, 1937, 465 p. 8. Yazdizadeh B. and Yeghiazaryan H. Polytechnic University Journal (submited). Yer., 2010,

p. 9. 9. Workbench A. ANSYS Workbench Help. NY, 2007.



P h y s i c s

STUDY ON PHYSICAL REGULARITIES OF A WIDE SPECTRUM AEROSOLS BEHAVIOR AT ITS FILTRATION THROUGH SUPER-THIN

MATERIALS

R. M. AVAGYAN1, G. H. HAROUTYUNYAN1, V. V. HARUTYUNYAN2*, V. S. BAGDASARYAN2, E. M. BOYAKHCHYAN2, V. A. ATOYAN3,

K. I. PYUSKYULYAN3, M. GERCHIKOV4

1 Chair of Theoretical Physics,YSU, Armenia 2 Yerevan Physics Institute after A.Alikhanian, Armenia 3 Armenian Nuclear Power Plant 4 Nuclear Safety Solutions Limited,Canada

The behavior of aerosol particles was studied at filtration through super-thin

basalt fibers. Diffusion and inertial effects were investigated at the particle sedimentation using theoretical models for fiber filters taking into account the aerosol gas capture factor.

Composition of radioactive aerosols was analyzed over the period of Armenian NPP operation.

Keywords: basalt fiber, aerosol, capture factor, filter.

Introduction. It is known that to assure high modern requirements of clearing gases from the suspended aerosol particles, fine-fibered filters are used, which are the most effective means of catching particles in comparison with all other types of filters [1, 2].

Catching (capture) of radioactive aerosols can be carried out by means of fibrous filters, which are macroporous media with very complex geometrical characteristics. The majority of known results on determination of filter operation effectiveness was obtained experimentally and is theoretically not well-examined. The process of filtration of particles depends on a lot of parameters, and it is practically impossible to consider all of them, when carrying out the experimental work. In this connection very important and actual is the mathematical modeling of filtration process to estimate efficiency of filtration at simultaneous variation of a lot of parameters in a wide interval of their variation. The basic equations necessary for mathematical modeling of filtration processes depend on both the sizes of radioactive particles and structural features of the filter [3, 4].

The improvement of these filters is due to the use of super-thin fibers, as the effect of “gas sliding” near the surface of the thin fibers characterized by Knudsen number Kn��/� (� is the length of the free path of air molecules, � is the fiber * E-mail: [email protected]


radius), is manifested by reduction of the fiber hydrodynamic flow resistance and increase in particle sedimentation on fibers [3].

Sedimentation of submicron aerosol particles from the flow on super-thin fibers at the air (gas) velocity of about several cm/s occurs as a result of Brownian displacement of particles from flow lines, and with increase of the particle radius their filter blow-off curve goes through a maximum, caused by the influence of particles “own” size on their sedimentation at the expense of “engagement” effect, when the particle centre passes along the flow lines at a fixed distance from the fiber surface (which is smaller than the particle radius).

The maximum of particles blow-off at a fixed air (gas) velocity corresponds to the worst clearing conditions. From evaluation of the penetrating size r*, the basalt fiber filter efficiency was estimated.

It is known that for modern super-thin filters the radius of the most penetrating particles is close to fiber radius [3], which in it’s turn is comparable with free-path length of air molecules �.

Theoretical study of diffusion sedimentation depending on the penetrating size of particles is insufficient for super-thin basalt fibers. When calculating the particle radius corresponding to the blow-off maximum, it is necessary to consider the “own” size of particles.

In [3–7] the capture factor is calculated for a model filter taking into account the radius of non-diffusing particles and point particles at intermediate Knudsen numbers (Kn) and an option of the joint consideration diffusion and engagement of the finite size particles is presented at Kn=0. It is known from the general theory of aerosol filtration that one of the main effects influencing sedimentation of dispersed phase is diffusion. It can be stated after detailed analysis of capture mechanisms of aerosol particles that at filtration of aerosols through macroporous dispersion, the aerosol phase is deposited on macrograins at the expense of diffusion and inertial effects.

Results. Calculation of filtration (cleaning) factor of aerosols by macro-granular filters was made without taking into account mutual influence of the above mentioned effects.

For the coefficient factor the following formula was obtained: (1) ( ) ,d in hK l � ��where d� is the coefficient of diffusion sedimentation of aerosols; in� is the coefficient of inertial sedimentation; h is the capacity of filter layer.

Calculation of coefficients of sedimentation d� and in� was carried out using “capillary” model on the assumption that diameter of pore channels is defined by the following expression: 2 /d K l� . (2)

Under Prandtl–Taylor hypothesis, according to which the mass transfer of substances in the turbulent flow up to the laminar interlayer is carried out at the expense of turbulent diffusion, and in the laminar interlayer near to the walls of pore channel the mass transfer is carried out at the expense of the molecular diffusion, the following formula was obtained for the coefficient of diffusion sedimentation of aerosols on macrogranular filters:

f

mlK vU:� � , (3)


53

where : is the is coefficient of molecular diffusion of particles; v is the kinematic viscosity of gas; fU is the velocity in filter.

During the investigation of the diffusion effect an important factor is nuclide composition of the investigated aerosol fractions.

To determine radionuclide composition of aerosols presenting in air of pre-mises and emissions, an historical database characterizing sizes and radionuclide composition of aerosols in air emissions and ventilation systems (see Tables 1 and 2) was created at the Armenian NPP.

T a b l e 1

Nuclide composition and size of emissions during ANPP operation period (107 Bq/year)

Basic radionuclides Year of

oper. LLN* 131I 137Cs 134Cs 60Co 110mAg 90Sr 54Mn 51Cr 1978 16.9 276 0.10 – 7.80 6.40 0.03 5.60 20.0 1979 633 579 17.2 5.10 31.0 – 1.40 19.1 313 1980 428 777 48.6 46.7 18.1 – 0.30 13.6 70.0 1981 214 735 22.4 15.5 26.9 – 0.60 10.7 11.6 1982 341 230 9.5 10.0 62.8 48.5 0.37 22.3 11.5 1983 884 70 5.0 1.70 20.6 4.80 0.06 5.80 0.70 1984 1785 228 66.3 51.0 28.2 37.0 0.04 4.90 4.60 1985 754 151 60.6 33.1 17.2 71.6 0.11 4.0 16.2 1986 794 44 25.0 12.8 21.7 73.4 0.25 8.40 – 1987 259 103 13.4 5.60 34.7 122.0 0.08 7.20 17.2 1988 338 602 14.9 24.0 128 142.0 0.06 26.8 10.2 1989 181 108 10.1 – 29.4 56.60 – – – 1990 113 – 8.8 – 12.3 16.10 0.09 – – 1991 46.0 – 6.2 4.0 8.9 11.80 0 1.30 – 1994 82.0 – – – 60.1 – – – – 1995 193.0 9.70 23.3 – 83.7 – 0.15 – – 1996 121.0 23.5 15.4 0.80 22.4 25.80 0.12 0.80 11.6 1997 278.0 36.7 11.6 1.27 9.0 7.24 0.36 0.33 0 1998 238.4 28.8 9.35 1.32 18.4 7.72 0.29 1.89 21.6 1999 44.43 25.8 10.2 0.89 11.94 10.8 0.44 1.22 – 2000 30.7 26.0 4.20 5.97 17.7 22.60 0.38 8.78 – 2001 31.1 18.8 16.5 5.36 23.5 18.70 0.49 3.42 2.24 2002 9.9 59.6 7.90 2.28 6.6 2.50 0.2 0.16 – 2003 29.3 38.1 26.7 5.0 22.1 25.0 0.23 3.23 – 2004 28.5 97.1 5.59 0.38 14.6 11.3 0.04 1.35 2.53 2005 20.9 3.04** 7.0 0.83 5.10 1.35 0.03 – – 2006 18.3 3.65 5.12 0.54 9.45 1.77 0.03 – – 2007 46.0 1.90 4.88 1.60 7.15 1.17 0.05 0.10 – 2008 7.0 0.47 3.82 1.20 23.7 8.0 0.04 1.45 – % composition excluding 131I and LLN 20.59 1.4 35.08 32.89 0.28 6.83 2.9

% composition excluding LLN 59.7 6.40 3.30 10.9 10.25 0.087 2.14 7.16

* LLN are radionuclides with half-life period more than 24 hours; ** starting from 2005, new and more sensitive equipment and technique of measurement of 131I

emission were introduced at ANPP.


It is seen from the Table 1 that the greatest percentage contribution to aerosols activity is made by the radionuclides of corrosion origin (60Co and 110mAg) as well as uranium fission product 137Cs.

T a b l e 2

Concentration of radionuclides in ventilation systems (10-7 Bq/l)

Max Min Ventilation systems 137Cs 60Co 110mAg 54Mn 134Cs 137Cs 60Co 110mAg 54Mn 134Cs

2007�1� 215.3 26.1 – – – 6.7 – – – – 2�2 85000 8000 182 – 90000 28.2 27.3 37.0 – 73.8 �3 12.9 98.0 111.0 – – 7.4 55.9 74.4 – – 2�4 31005 1260 – – 72005 185 – – – 111

2008�1� 127.1 109.4 – – – 52.7 54 – – – 2�2 117.8 236.9 70.9 – – 95.7 – – – – �3 60.6 380.5 – – – – – – – – 2�4 81.9 1619 475.6 209.5 41.5 110 – – –

In 2009 the measured concentration of 58Co in 2�4 was max=193·10-7 Bq/l. When analyzing the contents of radioactive aerosols, it is necessary to take

count on the diffusion effect caused by impurity transport in direction of smaller concentrations. When taking samples from flow, it is necessary to follow a series of conditions. The sampling tube axis should be parallel to flow lines in air duct (the coaxiality condition). Otherwise, when following correct sampling speed, concentration of aerosol in sample will be lower than in investigated aerosol iC

eC e, and the aspiration coefficient ' /ia C C� will be below 1. The average flow velo-city in nozzle should be equal to low velocity in corresponding flow line iU eU(which passes the tube axis) in the gas duct (the isokinetic condition, Fig. 1).

Fig. 1. Aspiration of aerosol. a – violation of coaxiality (at correct sampling velocity); b – increased sampling velocity; c – decreased sampling velocity; d – correct sampling

velocity.

a b c d


55

At /i eU U 1N � ! flow lines diverge prior to the tube input; the particles -displace from the flow line to the tube axis under the influence of inertia, and as a result the concentration in sample becomes higher than the real one( ). ' 1a %

At 1N % the situation is vice versa and ' 1a ! (Fig. 1, c). At last, to the two above conditions it is necessary to add the third one: the

sampling tube walls (nozzle) should be infinitely thin. Errors due to non-coaxiality of sampling of particles with diameter at d

small angles v between the flow direction in air duct ( 1N � ) can be taken into account by formula (3): ' 1 4 sin /ta v S D- � , (4)

where /t eS U Dt=� is the Stokes number; 2 /18chp d= �� is the particle relaxation time; chp is the particle density; � is the dynamic viscosity of air; is the tDinternal diameter of the sampling tube.

The errors due to violation of isokinetic conditions of sampling can be estimated using empirical Beljaev-Levin formula:

� �

�

1 1 2 0.62' 1

1 2 0.62t

t

Sa

SN N

N

� � ��

� �, (5)

where . 10.2 5N�� ! !Diffusion sedimentation of aerosol particles on super-thin basalt fibers was

studied using a model of the fiber filter presenting a system of the parallel fibers located normally to the flow direction. The flow field in the system is presented as a cell model with packing density , where a is a fiber radius, is a cell radius [3, 5–7]. Analytical solution of integral equations is a large amount of work, but as a result we obtain for the total particle flow a relationship reflecting the fact of particle engagement on fibers.

� 2/a b$ � b

It is determined that for large 0 /R 8 capture factors > is equal to

� � 1/20 / 1 /R R Pe> 8� � , (6)

where Pe is the Peclet number ( 2 /Pe aU D� , where is the diffusion, U is the gas flow velocity);

DR is the dimensionless particle radius;8 �is small

( ). 1/21/ Pe8 �Thus, pure engagement effect can be expressed as

� 0 1 ,R R> � � where . (7) 0 ~ 1R

Fig. 2 (a, b) presents the results of calculations of the capture factor curves for pure engagement and for total capture factor. It is seen from the figure that curves are almost parallel, i.e. sedimentation mechanisms are additive. The effect of aerosol gas sliding near the surface promotes the accelerated movement of particles along the fiber surface that, in turn, leads to reduction of diffusion sedimentation of particles. But, on the other hand, with the increase of sliding velocity, the increase in velocity of the particle radial transport to fiber is possible, which is equivalent to reduction of the particle diffusion sedimentation. The change in the capture factor of point particles depends on these effects ratio.


a b

Fig. 2. Capture factor � dependence on the engagement parameter R for a filter with $�(1/6)2 at different Knudsen numbers. a) 8�0.05(Pe=400); b) 0.3 (Pe=12). Curve 1 corresponds to Kn�10;

curve 2 – to Kn�1; curve 3 – to Kn�0.1.

Conclusion. Analysis of the filtration process in real super-thin basalt filters has shown that it is appropriate to express the existing sedimentation mechanisms of aerosol particles in terms of capture coefficient, depending on the filter

ditions, such

nanofiber radius by fibe or example, at different ute pressure of gases.

Processes in the Environment. London, 2000, p. 597. 3. Kir

of Aerosol Sci. Roldugin V.I, Kirsh A.A. J. Colloid Sci., 2001, v. 63, � 5, p. 679.

. Kirsh V.A. Colloid Journal, 2007, v. 69, � 5, p. 649–654.

. Kirsh V.A. Colloid Journal, 2007, v. 69, � p. 655–660. 7. Lee K.W, Liu B.Y. Aerosol Science and Technology, 2010, v. 1, � 1, p. 35.

parameters and dimensionless parameters characterizing filtration conas Knudsen number and diffusion Peclet parameter.

The obtained results on aerosol capture modeling can be used to estimate the r layer gas permeability method, f

absol

The work has been executed within the frameworks of ISTC project A-1605.

Received 03.05.2010

R E F E R E N C E S

1. Reist P. Introduction to Aerosol Science. New York, 1987, p 284. 2. Spurny K. Aerosol Chemical

sh A.A., Stechkina I.B. The Theory of Aerosols Filtration with Fibrous Filters. FundamentalsN. Y.: J. Wiley & Sons. Ins., 1978, p. 165.

4.56 5,



P h y s i c s

ON ONE METHOD OF DISTANT INFRARED MONITORING

R. S. ASATRYAN1*, H. S. KARAYAN2, N. R. KHACHATRYAN1, L. H. SUKOYAN1

1 Institute of Radiophysics Researches 2 Chair of Optics, YSU

We present the development results of a new method of aerial (on a helicopter

or airplane) infrared (IR) scanning of extensive spaces with the purpose of detecting weak heat sources (fire centers at an early stage of their development) to prevent the occurrence of large-scale forest fires.

The paper presents the description of the IR radiometer as well as the measurement method of point and extended thermal sources with wavelength range of 2.5 to 5.5 microns.

Keywords: infrared sounding, fire sources, forest space, infrared radiometer.

Introduction. The monitoring of environment, the investigation and control of ecological conditions attract a great attention of the mankind, especially at the present stage of development of industry, energetics and urban building. Optoelectronic systems and devices designed for application in ecological studies and in arising extremal situations are always in the center of the scientists’ and engineers’ attention. In particular, research complexes for early detection of fire sources arising during natural calamities are indispensable. Therefore, the development and creation of infrared devices and systems of thermal monito-ring of environment, in particular large forest spaces, is a rather important problem.

The development of modern distant and effective methods of ecological monitoring of large forest spaces is more than actual. In such a situation the only method is remote monitoring from an aircraft (e.g. from a helicopter) while flying over large forests at the altitude up to 1000 m.

Brief Technical Description of a Measuring System. Structurally the measuring complex consists of two basic units: an optico-mechanical unit of the IR radiometer and an electronic control unit connected to a personal computer. It is designed to measure spectral radiance and radiation temperature (or it’s falls) of point and extended sources of infrared radiation under laboratory and field condi-tions [1–3]. To automate data acquisition and processing, the spectroradiometer is connected to a computer via a series port RS 232. Optical scheme of the optico-mechanical unit (OMU) is shown in Fig. 1. ��



Fig. 1. Optical scheme of OMU.

1 – Primary mirror of the objective; 2 – secondary mirror of the objective; 3 – radiation from an object; 4 – removable plane mirror; 5 – a sight; 6 – a modu-lator; 7 – a reference cavity; 8 – a field diaphragm; 9, 10 – projection objective; 11 – a disk with interfe-rential light filters; 12 – a sensing site of the photode-tector; 13 – a thermos for liquid nitrogen; 14 – a tele-scope; 15 – a deflection mirror.

The OMU consists of the following ma mirror objective lens of Cassegrain type.

under test, equipped with a sightin

ter to an area to be me

hich serve for refocusing the radiation from a field d

total wepresents a removable block with a

photo

Value

in parts: @ Input@ A telescope for operative pointing to an object g grid visible through an eyepiece on the OMU back panel. @ Parallax free sight for accurate pointing the spectroradiomeasured. The sight has a sighting grid with a cross and a circle, which defines

visual field boundaries of the device. @ Projection objective lenses, wiaphragm to the plane with light filters and to a sensing site of the photodetector.

They represent pairs of spherical mirrors, which are used to avoid achromatic aberrations. @ A block of removable ring wedge variable light filters, which provide a orking spectral range of 0.4 to 14 �m. @ A photodetector, which structurally r

detector placed inside it in accordance with the spectral range, a preamplifier and an adjusting mechanism.

Parameter Name Input objective 180 mm diameter Focal distance 200 mm Distances to be focused tofrom 5 m �Working spectral range I subband (spectral resolution of 10 %)

4 m

II subband (spectral resolution of 3 %) III subband (spectral resolution of 8 %)

from 0.4 to 1 from 0.4 to 1.1 m from 2.5 to 5.5 m from 7.9 to 13.5 m

Photodetectors: I subband II subband III subband

Si – photodiode InSb – photoresist

ist CdHgTe – photoresField of vision 3 mrad Noise equivalent difference of the radiation temperatures (at 295 K)

0.05 K

Continuous work time 8 hours Time of preparation to work 15 min Dimensional size of spectroradiometer: OMU 8 x 254 mm

ECU 415 x 27500 x 420 x 210 mm

Weight: OMU ECU

not more than 12 kg not more than 15 kg

Clima f operation: om –35 C to +45 C

630 to 800 mm Hg)

tic conditions oAmbient temperature Atmospheric pressure Air relative humidity

o ofr

from 84 to 107 kPa (fromup to 98% at 35oC

Powe

r voltage frequency

(220±22) V (50±1) Hz

Power consumed not more than 200 W


59

Full working spectral range of the d th the help of three sets o

g operation the OMU, by means of the wedge guide, is placed on a rotary

nic control unit (ECU) is structurally of desktop variant. All indica

utdoor

sists in the follow

evice is covered wif removable light filters and photodetectors in the subbands of 0.4 to 1.1 �m,

2.5 to 5.5 �m and 8 to 14 �m. Main technical parameters of the device are given in the Table.

Durin mechanism, which is fastened to the horizontal platform of a specially

prepared tripod. The electrotion and control elements are mounted on the front panel of the ECU.

Under laboratory conditions the ECU is placed on the table, and under oconditions it can be mounted in a helicopter with the help of dampers.

In brief, the principle of operation of the spectroradiometer coning. Inside the OMU the radiation flow from the test object is collected with

the use of an optical system (see Fig. 1) and focused into a sensing site of the photodetector. Further, a preamplifier amplifies an electric signal and transmits it to the ECU. In the ECU the electronic schemes amplify, demodulate and filter the signal from the photodetector output, and as a result of this signal appears at the output, the amplitude of which is a measure of the radiation temperature of the object. Knowing the value of the collected radiation power (from the data of preliminarily conducted energetic calibration of the device), spectral filter features of the system and amplification degree, the output signal can be exactly transformed into an absolute measurement of radiation temperatures of the objects under test.

Fig. 2. Helicopter IR s of large forests.

e note some advantages of the IR radiometer developed by us [4] compa

canning

Wred to the existing close analogs. To widen functional capabilities in the

sphere of spectral investigations of thermal objects, besides wideband interferential light filters for spectrum parts of 0.4 to 1.1, 2.5 to 5.5 and 8 to14 �m, the device is also provided with ring readjustable light filters. To eliminate chromatic aberra-


tions, the device optical scheme includes two pairs (see Fig. 1) of mirror projection objectives with light filters and the receiving site of photodetectors in the focuses.

The IR ra

diometer is mounted in the helicopter and with the help of a deflec

region (withi

ed and Point Thermal Sourc

ting that mirror, by its field of view scans (through the bottom hatch, along the helicopter motion routing) terrestrial surface of large forests (see Fig. 2).

In the presence of fire sources the radiation temperature in that n the wavelength range of 2.5 to 5.5 �m) considerably increases and the

electronic control unit registers this event. At the helicopter flight altitudes of 200, 500 and 700 m the radiometer covers, with its field of vision, surface areas of about 120, 750 and 1500 sq.m, correspondingly. With the helicopter speed of 150–200 km/hr the time of one measurement cycle is 0.1 s.

Measurment Technique of IR Flows From Extendes. Before carrying out quantitative measurements of IR radiation emitted by

an unknown source, it is necessary to fulfill energetic calibration of the spectro-radiometer, the aim of which is the measurement of the device response to the known standard source (usually a black body with known temperature). By definition, the device calibration means obtaining an electrical signal at the output, which corresponds to a radiation flow unit incident into the radiometer inlet. The calibration is expressed by some function k(�) called spectral calibration characte-ristic of the device, which includes combined effect of optical elements and ele-ctronic amplification of the whole system. k(�) is expressed in V/radiation unit with standard level of amplification degree. An output signal of the device is propor-tional to the difference between the IR radiation flows coming to the photodetector from an external source and from the internal modulated reference black body. During the calibration the radiation from the calibration black body (with known temperature) entirely fills the device field of vision. An output signal S(�) is expressed by the following ratio:

� � � �� 0, , , 1 ,Bl r T r T l� � = �� 3 � 4,S k r T� � � = �� 5 6 , (1)

is the Plunk function at the temperature and the wavelength where � ,r T� T � ; T is t erature of the calibration black body; he temp � ,l= � is the atmospheric

nsparency over the path l between the calibration source and device; 0T is the temperature of the internal reference black body; BT is the temperature of the air during the experiment.

In the windows

tra

of the atmosphere transparency (e.g. for the wavelength range of 2.5 to 5.5 m), where the transmission is high, � ,l= � can be taken to be 1, if the calibration is carried out from the distance l to several meters. Therefore, in this approximation for

equal� S � we can write

� � � � , ,S k r T� � � � 0r T3 4� �5 6 (2)

oefficient equal to 1. And amplification coefficient different from 1 the with the amplification c in measuring with the

� S � value decreases by the same factor. The Plunk function value is calculated according to the ratio


61

� � 1125, exp / 1cr T c� �T

��

� 3 � 45 6 ,

where 41 3.74 10c � � W m4/cm2 ; 4

2 1.438 10c � � m deg. The test objects, the radiation flow of which comp ely fills the device field

of vision, are extent in these measurements. In this case radiance spectral density W/cm2·stera is measured. The ratio (1) can be re-

writte

let

(W � ,T� d· m) of the objectn as

� � � � � � � � �0, , , , 1 ,�S k W T l r T r T l� � � = � � � = � 9� � � 3 � 45 6 , (3)

where � W ,T� is the radiance spectral density of the test object, 9 is the amplifi-oefficient of the whole system, and the rest of the symbols are

osphere transparencycation c the same as above. The atm � ,l= � is either measured simulta

ed with oneously or

calculat the help of data from literature [5, 6]. From the rati (3) we can get for � ,W T� :

� � � � � � � 0/ , , 1 ,

,,

BS k r T r T lW T

r l� � 9 � � = �

��

� � 3 � 45 6� . (4)

Usually the radiation of point sources does not fill the field of vision of the area A of a radiating object is known, we can measur

to the above-stated technique, that is device. If the e its spectral radiance according

� � 2

, ,plW T W TA

� � N� , (5)

where N is the solid angle of the spectroradiometer’s field of vision; is easured according to (4

object under test to the spectroradiometer. While measuring point sources, spectral

� ,W T�the total spectral radiance m ); l is the distance from the

contrast of a radiation source is also of interest, when the background radiance is comparable to the object radiation. In this case it is necessary to separate the background signal � bS � from the signal “source+background” � S � . For the spectral radiation contrast of the source we get the ratio

� � � �

2

,S l

Wk l A

� N�

9 � = �E

� , (6)

� � �

where bS S S� � �E � � . If is unknown, we may define the contrast of the he source (in W/sterad·

Aspectral luminous intensity of t m):

� � � � � 2

,W A l

k lS

I�

� � N9 � = �

� � . (7)

Calculation of the radiation temperatures of the test ob

E

jects is carried out in eveloped algorithms and programs

method of remote ecological ring of vast forest spaces will undoubtedly bring to the considerable technical-

accordance with specially d . Conclusion. Application of the given monito-


economical effectiveness and will also have a great importance in the problem of preventing the fire occurrences, especially of large-scale ones.

R E F E R E N C E S

. Asatryan R.S. Izvestia NAN RA, Tekhnitcheskie nauki, 2007, v. LX, � 2, p. 307–316 (in Russian).

2. Asatryan R.S., Asatryan S.R. et temy i Pribory, 2008, � 1, p. 17–19 (in Russian). Asatryan R.S. et al. Intern. Journal of IR and mm Waves, 2003, v. 24, � 6, p. 1035–1046.

0042. 11.04.2005 (in Armenian).

4 p. (in Russian).

Received 03.06.2010

1

al. Ecologicheskie Sis

3.4. Asatryan R.S., Gevorgyan H. G. et al. The Spectroradiometer, Patent of RA, � 1678A2,

Appl. � P20055. Kruz P., Macgloulin L., Maccvistan P. Bases of Infrared Technique. M.: Voenizdat, USSR

DD, 1964, 466. Pavlov A.V. Optoelectronic Devices. M.: Energia, 1974, 360 p. (in Russian).



P h y s i c s

METHOD FOR MEASURING THICKNESS OF THIN OBJECTS WITH A NANOMETER RESOLUTION, BASED ON THE SINGLE-LAYER

FLAT-COIL-OSCILLATOR METHOD

S. G. GEVORGYAN1, 2�, S. T. MURADYAN1, 3, M. H. AZARYAN1, G. H. KARAPETYAN1, 2

1 Center on Superconductivity and Scientific Instrumentation, Chair of Solid State Physics, YSU 2 Institute for Physical Research, Nat. Academy of Sci., Armenia 3 Russian-Armenian (Slavonic) State University

A method for measuring of any composition films and tapes thickness with a

nanometer resolution is suggested and validated experimentally. That operates on the base of a single-layer flat-coil-oscillator technique. A laboratory prototype of a device is designed and created, based on this method. Besides, PC operation in a “NI LabVIEW” software environment, as well as preliminary tests and calibration of the created device is implemented. It may find variety of applications in a research and in high-tech technology.

Keywords: Single-layer Flat-Coil-Oscillator method, a nanometer resolution thickness measuring and controlling technique, high-Tc superconductive films and tapes.

Introduction. In a micro- and, especially, in a nanotechnology there is a

great need of measurement methods (and devices on that base) that may enable measuring of the thicknesses of thin-film nanostructures, created during a tech-nological processes with a few nanometer precision [1].

Presently, optical devices are used for that purposes. They operate on the basis of interference, therefore, their resolution is close to the order of the measu-ring light’s wavelength �opt/2 O250 nm. There are also more complicated “ellipsometric” optical devices [2], based on the polarization of light, which allow to enhance the resolution of measurements by one more order of magnitude (almost 10 times), approaching it to the few tens of nanometer. However, unfortunately, they also do not permit to reach the nanometer level of resolution, and besides, they are enough expensive and also complex devices.

As opposed to the corpuscular and wave microscopes (which, in principle, can’t solve the problem of measuring/detection of a nanometer thicknesses and depths in a perpendicular of the sample face direction [3]), such resolution (even, much better) show probe microscopes (tunneling [4–5] and atomic force [6]), however, they are too much expensive, and besides, incomparably complicated � E-mail: [email protected]


devices. Therefore, it is not reasonable to use them at standard (everyday) technological processes. In addition, technical staff should pass special courses before getting permission to exploit such a complex technique. That’s why for a long time one needs to get simple and cheap ways to solve this problem.

Taking into account all these, in this work we could elaborate and create a fully new method for measuring of the thickness of thin films and tapes. It enables to measure the thickness of any composition film � with about 1–2 nm resolution in about 1 mm range. Alongside with its high resolution and wide dynamic range, as well as alongside with its simplicity and cheapness, the method has also other advantage: it doesn’t require additional processing of the specimen under study. That is especially comfortable in case of biological objects.

Description of the Method. On the basis of operation of our new method for detection of thicknesses of the films and tapes underlies a single-layer flat pick-up coil based oscillator, activated by a low power (backward) tunnel diode (TD) � a single-layer flat-coil-oscillator (SFCO) technique [7–8]. Its principle of operation is shown in Fig.1. Radiofrequency (RF, MHz-range) testing magnetic field distribution near the coil is illustrated in Fig. 1, a. When normal-metallic plate approaches to the coil (Fig. 1, b), it shields the said field distribution � due to neg-lectablly-small depth of the “skin”-layer in such a plate at the MHz-range � resul-ting in a strong distortion of the coil testing field distribution. This leads to the corresponding changes in a measuring TD-oscillator frequency � just this is the measuring parameter in the SFCO-technique.

Open Flat Coil a Metallic Plate b

Fig. 1. Principle of operation of the single-layer flat pick-up (detecting) coil based

oscillator (SFCO) technique [7–8].

So, our new method for measuring the thicknesses of the films and tapes is based on the SFCO unique technique. Schematics of such a way created laboratory device is shown in Fig. 2. It operates as follows.

Optically polished sample stage and cantilever enable springy compression of the measuring spherical or needle-type probe to the specimen. That is done due to elastic properties of the used cantilever (see Fig. 2). A normal-metallic plate (made of, for example, printed copper circuit board) is attached to its free end on the back of probe. Displacements of this plate lead to the positioned near flat-coil’s testing RF-field’s distortion (shielding). That results in the frequency shift of the flat-coil based TD-oscillator. During the measurements the sample is placed on the sample stage, under the probe (see Fig. 2), and moves. During such a scan, probe


65

goes up and down by the thickness and roughness of the sample surface. That leads to changes of mutual distance between the detecting flat coil and shielding metallic plate attached to the cantilever, which results in corresponding changes of

the measuring TD-oscilla-tor harmonic oscillations frequency. Therefore, trac-king frequency shifts, one may detect the thickness of the sample and extract the scope of its surface rough-ness. So, in a suggested by us method changes of the sample thickness and/or its surface roughness during the linear scan of the sample lead to the testing

TD-oscillator frequency shifts � this is just the measuring signal.

Cantilever Flat coil Normal metallic plate

Probe

Object under study Sample (object) stage Stand

Fig. 2. Schematics of the created prototype device, based on our new idea for measuring the thicknesses of the films

and tapes.

Actually, in a tested device two TD-oscillators are used with almost identical characteristics (with close frequencies). Due to this, influence of external and internal factors (such as temperature’s and tunnel diode bias-source voltage’s in-stabilities, drifts and other factors) on frequencies of the said oscillators, both are similar. That is why difference between the frequencies of oscillators, designed in such a “compensation” (balanced) scheme, depends on mentioned factors negligibly slow [7–8]. Therefore, use of such a balanced couple of “similar” oscillators enables to avoid experimentally consideration of the influence of mentioned (as well as some other, non-mentioned) external and internal factors, which are usually out of control during the tests [7]. In a tested coilK O7 mm flat coils based prototype TD-oscillators were activated at the frequencies close to the 25 MHz, while the difference between their frequencies was about 500 kHz. Their stability was better than *10 Hz at the room temperatures.

Calibration of the Prototype. Before the tests of the created device it was calibrated. For that the shielding metallic plate was approached (and moved away) to the measuring coil face by means of the «X-Y» stage, driven by stepping motors (with 1 �m step, on 0.2–0.7 mm distance from the coil), and the oscillator frequency shift was detected. Results of such an experiment we bring in Fig. 3.

The horizontal axis in Fig. 3 shows changes of mutual distance between the flat coil and shielding plate during calibration experiments (at the real experiments that corresponds to changes of the thickness of object under test), while the vertical axis corresponds to the related changes of the measuring TD-oscillator frequency (changes of difference frequency). The solid line is the measuring data (test data with a too small step of about 1 �m are densely merged into the shown solid line), while dashed line is the linear approximation of experimental data. As is seen from the Fig. 3, in enough wide region (more than 0.5 mm) readout of the tested prototype is linear vs. position of the shielding plate (vs. the thickness of the object under test). Starting from the slope of the curve under discussion one may estimate that 1 nm step of the metallic plate brings to the 6 Hz shift of the measuring oscillator frequency. Taking into account the said stability of the created technique


at room temperatures of about *10 Hz, we may state that the resolution of our thickness measuring prototype is around 1–2 nm.

Starting from the obtained calibration data, we organized also PC control of the created prototype device in the “NI LabVIEW” software environ-ment. In other words, developed software enables to register frequency signal-outputs from the prototype and converts them

to the values corresponding to the thickness of the object under test, using calibration data. Created device also permits to study the thickness variations of thin objects, placed on the «X-Y» sample stage, and estimate the roughness of samples’ surface along (schematics see in Fig. 2). That is possible to do in our pro-totype device with 1 �m step, by means of the sample movement «X-Y» stage, driven by the stepping motors along both of axis, and by use of the corresponding PC software control.

F, k

Hz

d, �m Fig. 3. Calibration curve of the thickness measuring prototype.

Test Data of the Prototype. Preliminary test of the created device was done by use of mylar tapes with various (but known) thicknesses. That enabled to make one more (independent) calibration of this technique, which agrees well with the results above (see Fig. 3). Such tests were done as follows:

1) before the start of measurements, iron probe was pressed to the optically polished sapphire sample stage (see Fig. 2). In that “zero” position of the probe the frequency of the testing flat-coil oscillator was measured. Its value depends on the mutual distance between the coil and normal-metallic (copper) plate, fixed on the back of the probe (attached to the free end of the cantilever (see Fig. 2).

2) then the same value was measured, but in this case mylar tape was placed between the spherical probe and the sap-phire sample stage. Five diffe-rent measurements were done during this set of tests, using different depth tapes: d = 12, 24, 36, 48 and 60 �m.

As a result of our tests the dependence of the measuring TD-oscillator frequency vs. the thickness d of the tested tape was plotted in Fig. 4. Taking into account the accuracy of our measurements (*10 Hz), one may estimate the resolution of the created by us thickness measuring prototype, using the slope

Fig. 4. Test results by use of different depth mylar tapes: d = 12, 24, 36, 48 and 60 �m.


67

of the curve, presented in this figure. And again, we came to the same resolution (1–2 nm) – from Fig. 4 we can receive: 360000 Hz/60 �m=6 Hz/1 nm=12 Hz/2 nm L L *10 Hz/2 nm. That is because of the reliabile mesurements conducted in this work.

Conclusion. Taking into account so high resolution (1–2 nm, in a dynamic region of about 1 mm) achieved in the flat-coil method based our thickness mea-surement prototype during its preliminary tests, we suppose that a thickness (depth) measurement this new method (and devices, created on that base) may find variety of applications in the research work, as well as in technology. In particular, one may use such a technique in micro- and nanoelectronics, as well as in high-temperature superconductive (HTS) film production technology. It can be used also in production process of HTS-tapes (containing inside the said HTS-films). Such a precision research instrument might be used also in various types of technological equipments, making characterization of the said tapes.

This study was supported by the Armenian National Foundation of Science & Advanced Technologies and U.S. Civilian Research & Development Foundation under Grants #ISIPA 01-04, #ARP2-3229-YE-04 and #UCEP 07/07. The study was supported also by state sources of Armenia in frames of the R&D project #72-103.

Received 16.09.2010

R E F E R E N C E S

1. Cole D.A., Shallenberger J.R. et al. J. Vacuum Science & Technol. B, 2000, v. 18, p. 440–444. 2. Franquet A., De Laet J., Schram T. et al. Thin Solid Films, 2001, v. 384, Issue 1, p. 37–45. 3. Bikov V.A. Topical review on Probe Microscopes: Russian Doctoral Degree Dissertation,

M., 2000 (in Russian). 4. Binning G. and Rohrer H. Scanning Tunneling Microscope. U.S. Patent 4,343,993. Aug. 10,

1982. Field: Sep.12, 1980. 5. Binning G. et al. Appl. Phys. Lett., 1982, v. 40, p. 178. 6. Binning R., Quate C. et al. Phys. Rev. Lett., 1986, v. 56, p. 930. 7. Gevorgyan S., Kiss T., Movsisyan A., Shirinyan H., Hanayama Y., Katsube H., Ohyama T.,

Takeo M., Matsushita T., Funaki K. Rev. Sci. Instr., 2000, v. 71, � 3, p. 1488–1994. 8. Gevorgyan S., Kiss T., Ohyama T., Movsisyan A., Shirinyan H., Gevorgyan V.,

Matsushita T., Takeo M. and Funaki K. Physica C, 2001, v. 366, p. 6–12.



C O M M U N I C A T I O N S


ON ONE FORMULA OF TRACES

H. A. ASATRYAN*

Chair of the differential equations, YSU

The paper generalizes one formula of traces, established by J. Neumann for square matrices, for nuclear operators.

Keywords: singular number, trace formula.

Preliminaries. Let be a separable Hilbert space. We will denote by H� BL H the Banach algebra of all bounded linear operators acting in the space ,

and let H

J� be the ideal of all compact operators acting in . For operator we will denote by

H� A BL H# A the unique nonnegative square root of the opera-

tor *A A . It is obvious that the compactness of one of the operators *, ,A A A A implies the compactness of remaining two operators. Let A J�# . As A is com-pact, self-adjoint and nonnegative, its nonzero eigenvalues can be rearranged in decreasing order. We will denote by � js A the j -th eigenvalue of the operator A

(note that each eigenvalue is counted with multiplicity). Numbers � js A are called

the singular numbers of the operator A (see [1, 2]). For + 1,p# � we will denote

by pJ the set of all operators A J�# , satisfying to the condition .

Then the formula

� 1

pj

js A

�

�! ��

� 1

1

ppjp

jA s A

�

�

3 4� P Q5 6� (1)

defines a norm in pJ and with respect to this norm pJ is a separable Banach

space. pJ is also two-sided ideal of � BL H and has the following property of

symmetry: for we have and pA J# *pA J# *

ppA A� (see [1]). Clearly, all the

propositions about � 1pJ p& ! � are true also for J� . Elements of 1J are called nuclear operators. * E-mail: [email protected]


69

Let’s note also the following property of ideals pJ : if � 1pA J p# & &�

and qB J# , where 1 1 1p q� � , then 1,AB BA J# .

Now we will give the definition and some properties of the trace of a nuclear operator. Let , and 1A J# � �ne be an orthonormal (finite or countable) base of the

space . Then the series H � ,n nn

Ae e� converges, and its sum does not depend on a

choice of the orthonormal base � �ne . This sum is called (matrix) trace of operator A and is denoted by � tr A . If A J�# , � B BL H# and 1,AB BA J# , then

� � tr trAB BA� . (2) The norm of 2J and the trace are connected by the following relation (see

[3]): � �2 *

22 tr A A A A J� # . (3)

The Formula of Traces. We will establish a relation, connecting norms of commutators AB BA� and * *AB B A� .

T h e o r e m . If one of the bounded linear operators ,A B , acting in Hilbert space , is nuclear, then the following equality holds: H

� � 22 * * * * * *2 2

trAB BA AB B A A A AA B B BB3 4� � � � � � �5 6 . (4)

Proof. First notice, that by the conditions of the Theorem operators AB BA� and * *AB B� A are nuclear, and since for 1 r s& & & � the following inclusion s rJ J2 is true, the left part of (4) is well-defined. We will denote

� � 22 * * * * * *2 2

trAB BA AB B A A A AA B B BB� 3 4� � � � � � �5 6 .

In view of (3), we have

� � � � ** * * * *tr trAB BA AB BA AB B A AB B A� 3 43 4� � � � � � �P Q5 6 5 6

� � � � * * * * * * * *tr tr trA A AA B B BB B A AB B A BA3 4� � � � �5 6 �

� � � � � * * * * * * * * * *tr tr tr tr trA B AB A B BA BA AB BA B A A BAB� � � � � �

� � � � � * * * * * * * * * *tr tr tr tr trA BB A A AB B A ABB AA B B AA BB� � � � �

or

� � � � * * * * * * * *tr tr tr trB A AB A ABB A B BA AA B B� 3 4 3� � � �5 6 54 �6

� � � � * * * * * * * *tr tr tr trBA B A A B AB A BAB B A BA3 4 3� � � �5 6 54 �6

� � � � * * * * * * * *tr tr tr trA AB B BA AB AA BB A BB A3 4 3� � � �5 6 546 .

According to (2), the expressions standing in the square brackets, are equal to zero, and consequently 0� � , i.e. (4) is true.

The Theorem is proved.


Remark 1. The steps in the proof of the Theorem show, that the state-ment of the Theorem remains true, if instead of nuclearity of operators A or B we assume, that one of these operators belongs to pJ , and another to qJ ,

where 1 and p& & �1 1 1p q� � .

Remark 2. The formula (4) generalises the similar formula for the square ma-trices, established by J. Neumann (see [4, 5]). If in the conditions of the Theorem one of operators ,A B is normal, then the commutability of ,A B implies the com-mutability of . This fact for square matrices has been noticed by Neumann, who has raised the question about its possible generalisations. In 1950 Fuglede, Putnam and Rosenblum have shown that this statement is true for normal operators from algebra

*,A B

� BL H (see [6]). This result of Fuglede, Putnam and Rosenblum, which goes back to Neumann, has far-reaching generalizations, which can be found in works [7, 8].

The author thanks professor M. I. Karakhanyan for the formulation of the problem and useful discussions of results.

Received 21.04.2010

R E F E R E N C E S

1. Gohberg I.C., Krein M.G. Introduction to the Theory of Linear Nonself-Adjoint Operators.

�.: Nauka, 1965 (in Russian). 2. Gohberg I., Goldberg S., Kaashoek M. Classes of Linear Operators. V.1. Birkhauser, 1990. 3. Reed M., Simon B. Methods of Modern Mathematical Physics. V. 1. Functional Analysis.

New York: Academic Press, 1972. 4. Neumann J. Port. Math., 1942, v. 3, p. 1–62. 5. Neumann J. Selected Works on Functional Analysis. �.: Nauka, 1987 (in Russian). 6. Rudin W. Functional Analysis. New York: McGraw-Hill Book Company, 1973. 7. Karakhanyan M.I. Journal of Contemporary Mathematical Analysis, 2007, � 3, p. 44–50. 8. Karakhanyan M.I. Functional Analysis and Its Applications, 2005, v. 39, � 4, p. 311–313.


Physical and Mathematical Sciences � 3, 2010

Ð ² Ø ² è à î ² ¶ ð à ô Â Ú à ô Ü Ü º ð

Ú³. ²ëÃáÉ³, ¾.². ¸³ÝÇ»ÉÛ³Ý, ê. Î. ²ñ½áõÙ³ÝÛ³Ý. Ð³×³Ë³Ï³Ý³ÛÇÝ �³ßËáõÙ-Ý»ñÁ Ï»Ýë³ÇÝýáñÙ³ïÇÏ³ÛáõÙ. ½³ñ·³óáõÙÁ ¿ç. 3–22

Ø»Í ã³÷»ñÇ Ï»Ýë³ÙáÉ»ÏáõÉ³ÛÇÝ Ñ³çáñ¹³Ï³ÝáõÃÛáõÝÝ»ñÇ Ù³Ã»-

Ù³ïÇÏ³Ï³Ý áõëáõÙÝ³ëÇñáõÃÛáõÝÁ Ï³ï³ñíáõÙ ¿ ³Û¹ Ñ³çáñ¹³Ï³ÝáõÃÛáõÝ-Ý»ñáõÙ ³é³ç³óáÕ å³ï³ÑáõÛÃÝ»ñÇ í»ñÉáõÍáõÃÛ³Ý û·ÝáõÃÛ³Ù�: ²ÏÝ³ñÏÁ ÝíÇñí³Í ¿ ³Û¹ �Ý³·³í³éáõÙ ëï³óí³Í ³ñ¹ÛáõÝùÝ»ñÇ ùÝÝ³ñÏÙ³ÝÁ: ´áÉáñ Ñ³×³Ë³Ï³Ý³ÛÇÝ �³ßËáõÙÝ»ñÇ Ñ³Ù³ñ ×ßÙ³ñÇï ¿ÙåÇñÇÏ ÷³ëï»ñÇ ÑÇÙ³Ý íñ³ ùÝÝ³ñÏíáõÙ ¿ Ú³. ²ëÃáÉ³ÛÇ ¨ ¾. ¸³ÝÇ»ÉÛ³ÝÇ ÏáÕÙÇó ³é³ç³ñÏí³Í ³ùëÇáÙ³ïÇÏ³Ý: ì»ñçÇÝë ÁÝ¹áõÝáõÙ ¿ ³ëÇÙåïáïÇÏáñ»Ý Ñ³ëï³ïáõÝ (¹³Ý¹³Õ ÷á÷áËíáÕ) �³Õ³¹ñÇãáí Ñ³×³Ë³Ï³Ý³ÛÇÝ �³ßËÙ³Ý Ï³ÝáÝ³íáñ ÷á÷áËáõÙÁ, ³Û¹ �³ßËÙ³Ý ·ñ³ýÇÏÇ Ï³éáõóí³ÍùÁ ¨ Ï³ÛáõÝáõÃÛáõÝÝ Áëï å³ñ³Ù»ïñ»ñÇ:

êïáõ·íáõÙ ¿ ³ùëÇáÙ³ïÇÏ³ÛÇ Ï³ï³ñáõÙÁ åñ³ÏïÇÏ³ÛáõÙ û·ï³-·áñÍíáÕ Ñ³×³Ë³Ï³Ý³ÛÇÝ �³ßËáõÙÝ»ñÇ Ñ³Ù³ñ: ä³ñ³Ù»ïñ³Ï³Ý Ñ³×³-Ë³Ï³Ý³ÛÇÝ �³ßËáõÙÝ»ñÇ Ýáñ ÁÝï³ÝÇùÝ»ñÇ Ñ³Ù³ñ ÝÏ³ñ³·ñí³Í »Ý Ýñ³Ýó Ï³éáõóÙ³Ý Ù»Ãá¹Ý»ñÇÝ í»ñ³�»ñáÕ ³ñ¹ÛáõÝùÝ»ñ. ÍÝÙ³Ý ¨ í³Ë×³Ý-Ù³Ý åñáó»ëÇ ëï³óÇáÝ³ñ �³ßËáõÙÝ»ñÇ, Ñ³ïáõÏ ýáõÝÏóÇ³Ý»ñÇ, Ï³ÛáõÝ ËïáõÃÛáõÝÝ»ñÇ ¨ ³ÛÉ »ñ¨áõÛÃÝ»ñÇ û·ï³·áñÍÙ³Ù�: Ò¨³Ï»ñåí³Í ¿ Áëï å³ñ³Ù»ïñ»ñÇ Ï³ÛáõÝáõÃÛ³Ý ËÝ¹ÇñÁ, �»ñí³Í »Ý ï³ñ�»ñ ¹³ë³Ï³Ý Ù»ïñÇÏ³Ý»ñÇ ï»ñÙÇÝÝ»ñáí Ï³ÛáõÝáõÃÛáõÝÁ Ñ³ëï³ïáÕ ³ñ¹ÛáõÝùÝ»ñ: îñí³Í »Ý Ñ³×³Ë³Ï³Ý³ÛÇÝ �³ßËáõÙÝ»ñÇ ï³ñ�»ñ ÁÝï³ÝÇùÝ»ñÇ Ï³ÝáÝ³-íáñ ÷á÷áËÙ³Ý å³ÛÙ³ÝÝ»ñ:

Ê. ². Ê³ã³ïñÛ³Ý. àã ÏáÙå³Ïï ûå»ñ³ïáñáí àõñÇëáÝÇ ïÇåÇ ÙÇ áã ·Í³ÛÇÝ ÇÝ-ï»·ñ³É Ñ³í³ë³ñÙ³Ý Ù³ëÇÝ ¿ç. 23–28

²ßË³ï³ÝùáõÙ Ñ»ï³½áïíáõÙ ¿ áã ÏáÙå³Ïï ûå»ñ³ïáñáí àõñÇëáÝÇ

ïÇåÇ áã ·Í³ÛÇÝ ÇÝï»·ñ³É Ñ³í³ë³ñáõÙ ÏÇë³é³ÝóùÇ íñ³: ºÝÃ³¹ñíáõÙ ¿, áñ ìÇÝ»ñ–Ðáåý–Ð³ÝÏ»ÉÇ ûå»ñ³ïáñÁ Í³é³ÛáõÙ ¿ áñå»ë ÉáÏ³É ÙÇÝáñ³Ýï àõñÇëáÝÇ ëÏ½�Ý³Ï³Ý ûå»ñ³ïáñÇ Ñ³Ù³ñ: ²å³óáõóí»É ¿ ¹ñ³Ï³Ý ¨ ë³ÑÙ³Ý³÷³Ï ÉáõÍÙ³Ý ·áÛáõÃÛáõÝÁ: ¶ïÝí»É ¿ Ï³éáõóí³Í ÉáõÍÙ³Ý ë³ÑÙ³ÝÝ ³Ýí»ñçáõÃÛáõÝáõÙ: ²ßË³ï³ÝùÇ í»ñçáõÙ �»ñí»É »Ý ûñÇÝ³ÏÝ»ñ: ´. Ð. ê³Ñ³ÏÛ³Ý. ÎáÙåÉ»ùë ïÇñáõÛÃáõÙ Ïáïáñ³Ï³ÛÇÝ Ï³ñ·Ç áñáß ¹Çý»ñ»ÝóÇ³É

Ñ³í³ë³ñáõÙÝ»ñÇ ÉáõÍáõÙÝ»ñÇ Ù³ëÇÝ ¿ç. 29–34

²Ûë ³ßË³ï³ÝùáõÙ ¹Çï³ñÏíáõÙ »Ý ÏáÙåÉ»ùë ïÇñáõÛÃáõÙ

� � � 1/D y z y z f z� �� ï»ëùÇ Ïáïáñ³Ï³ÛÇÝ Ï³ñ·Ç ¹Çý»ñ»ÝóÇ³É Ñ³í³-


ë³ñáõÙÝ»ñ, áñï»Õ � -Ý Ï³Ù³Û³Ï³Ý å³ñ³Ù»ïñ ¿, 1/D � -Ý èÇÙ³Ý–ÈÇáõíÇÉÇ ¹Çý»ñ»ÝóÇ³É ûå»ñ³ïáñÝ ¿: àñáß ¹³ëÇ ýáõÝÏóÇ³Ý»ñÇ Ñ³Ù³ñ ¹Çï³ñÏíáõÙ ¨ ÉáõÍíáõÙ »Ý ÎáßÇÇ ïÇåÇ ËÝ¹ÇñÝ»ñ: Ø. Æ. Î³ñ³Ë³ÝÛ³Ý, î. Ø. Êáõ¹áÛ³Ý. ¸ÇïáÕáõÃÛáõÝ ËÇëï Ñ³í³ë³ñ³ã³÷

Ñ³Ýñ³Ñ³ßÇíÝ»ñÇ í»ñ³�»ñÛ³É ¿ç. 35–39 ²ßË³ï³ÝùáõÙ áõëáõÙÝ³ëÇñíáõÙ »Ý áñáß Ñ³ïÏáõÃÛáõÝÝ»ñ ÉÇáíÇÝ

é»·áõÉÛ³ñ (Ï³ÝáÝ³íáñ) ï³ñ³ÍáõÃÛ³Ý íñ³ áñáßí³Í ë³ÑÙ³Ý³÷³Ï ³ÝÁÝ¹-Ñ³ï ýáõÝÏóÇ³Ý»ñÇ ÙÇ Ñ³Ýñ³Ñ³ßíÇ, áñáõÙ Ùïóí³Í ¿ �³½Ù³å³ïÏÙ³Ý ûå»ñ³ïáñÝ»ñÇ ÁÝï³ÝÇùáí ³é³ç³ó³Í ËÇëï Ñ³í³ë³ñ³ã³÷ ïáåáÉá-·Ç³: ²å³óáõóíáõÙ ¿ Ø. Þ»ÛÝ�»ñ·Ç Ã»áñ»ÙÁ ËÇëï Ñ³í³ë³ñ³ã³÷ Ñ³Ýñ³-Ñ³ßÇíÝ»ñÇ Ñ³Ù³ñ:

Ð. è. èáëï³ÙÇ. àã áõÝÇï³ñ³óíáÕ ËÙ�»ñ ¿ç. 40–43

ÊáõÙ�Á ÏáãíáõÙ ¿ áõÝÇï³ñ³óíáÕ, »Ã» ³Û¹ ËÙ�Ç Ûáõñ³ù³ÝãÛáõñ Ñ³í³ë³ñ³ã³÷ ë³ÑÙ³Ý³÷³Ï : (G B H )D � Ý»ñÏ³Û³óáõÙ ÑÇÉ�»ñïÛ³Ý ï³ñ³ÍáõÃÛáõÝáõÙ áõÝÇï³ñ³óíáÕ ¿: Ü. ØáÝá¹Ç ¨ Ü. ú½³í³ÛÇ ÏáÕÙÇó ³å³óáõóí»É ¿, áñ ³½³ï �»éÝë³Û¹Û³Ý

H

( , )B m n ËÙ�»ñÁ Ï»Ýï �³Õ³¹ñÛ³É

Ãí»ñÇ ¹»åùáõÙ áõÝÇï³ñ³óíáÕ ã»Ý, áñï»Õ : ²ßË³ï³ÝùáõÙ óáõÛó ¿ ïñíáõÙ, áñ ÝáõÛÝ Ãí»ñÇ ¹»åùáõÙ

1 2=n n n 1 665n n (4, )B n ËáõÙ�Á áõÝÇ ÏáÝïÇÝáõáõÙ

áã Ç½áÙáñý ý³Ïïáñ ËÙ�»ñ, áñáÝóÇó Ûáõñ³ù³ÝãÛáõñÁ ÇÝãå»ë áã áõÝÇï³ñ³óíáÕ ¿, ³ÛÝå»ë ¿É Ñ³í³ë³ñ³ã³÷ áã ³Ù»Ý³�»É:

´. Ú³½¹Ç½³¹». Î³éáõóí³ÍùÝ»ñÇ Ù»Ë³ÝÇÏ³ÛÇ í»ñç³íáñ ï³ññ»ñÇ Íñ³·ñáõÙ

û·ï³·áñÍíáÕ ï³ñ�»ñ Ñ³ñÃ Ùá¹»ÉÝ»ñÇ Ñ³Ù»Ù³ïáõÃÛáõÝÁ ¿ç. 44–50

ì»ñç³íáñ ï³ññ»ñÇ Íñ³·ñáõÙ Ù»Ë³ÝÇÏ³ÛÇ Ñ³ñÃ ËÝ¹ÇñÝ»ñÇ ÉáõÍ-Ù³Ý ÁÝÃ³óùáõÙ ÏÇñ³éíáõÙ »Ý ÙÇ ù³ÝÇ ï³ñ�»ñ �³½³ÛÇÝ ï³ññ»ñ: ANSYS Íñ³·ñáõÙ, áñÁ í»ñç³íáñ ï³ññ»ñÇ É³í³·áõÛÝ Íñ³·ñ»ñÇó Ù»ÏÝ ¿, ßáõñç í»ó ï»ë³ÏÇ ï³ññ Ï³: ÌéÙ³Ý å³ñ½ ËÝ¹ñáõÙ Ù»Ýù ¹Çï³ñÏáõÙ »Ýù ï³ñ�»ñ Ñ³ñÃ Ùá¹»ÉÝ»ñ ¨ Ý»ñÏ³Û³óÝáõÙ ëï³óí³Í ³ñ¹ÛáõÝùÝ»ñÇ Ñ³Ù»Ù³ïáõÃÛáõÝÁ` ³ÛÝ ï³ññ»ñÇ ï»ë³ÏÁ �³ó³Ñ³Ûï»Éáõ Ýå³ï³Ïáí, áñáÝù ÏÇñ³é»ÉÇ »Ý ïíÛ³É ËÝ¹ñÇ ÉáõÍÙ³Ý Ñ³Ù³ñ: ²ñ¹ÛáõÝùÝ»ñÇ Ñ³Ù»Ù³ïáõÃÛáõÝÁ óáõÛó ¿ ï³ÉÇë, áñ 42 ¨ 82 Ñ³ñÃ Ùá¹»ÉÝ»ñÝ ³í»ÉÇ ÏÇñ³é»ÉÇ »Ý, �³óÇ ³Û¹, ëáíáñ³Ï³Ý �çÇçÝ»ñáí 42-ñ¹ Ñ³ñÃ Ùá¹»ÉÇ ³ñ¹ÛáõÝùÝ»ñÝ ³í»ÉÇ Ùáï »Ý ïíÛ³É ËÝ¹ñÇÝ, ù³Ý Ù³Ýñ �çÇçÝ»ñáí 82-ñ¹ Ùá¹»ÉÇÝÁ:

è. Ø. ²í³·Û³Ý, ¶. Ð. Ð³ñáõÃÛáõÝÛ³Ý, ì. ì. Ð³ñáõÃÛáõÝÛ³Ý, ì. ê. ´³Õ¹³ë³ñÛ³Ý,

º. Ø. ´áÛ³ËãÛ³Ý, ì. ². ²ÃáÛ³Ý, Î. Æ. öÛáõëùáõÉÛ³Ý, Ø. ¶»ñãÇÏáí. È³ÛÝ ïÇñáõÛÃÇ ³»ñá½áÉÝ»ñÇ í³ñùÇ ýÇ½ÇÏ³Ï³Ý ûñÇÝ³ã³÷áõÃÛáõÝÝ»ñÇ Ñ»ï³½á-ïáõÙÁ, »ñ� ¹ñ³Ýù ³ÝóÝáõÙ »Ý ·»ñ�³ñ³Ï ÝÛáõÃ»ñáí ¿ç. 51–56

Ð»ï³½áïí»É ¿ ³»ñá½áÉ³ÛÇÝ Ù³ëÝÇÏÝ»ñÇ í³ñùÁ �³½³Éï» ·»ñ�³ñ³Ï

Ù³Ýñ³Ã»É»ñáí ³ÝóÝ»ÉÇë: î»ë³Ï³Ý Ùá¹»ÉÝ»ñÇ ÙÇçáóáí áõëáõÙÝ³ëÇñí»É »Ý

Proc. of the Yerevan State Univ. Phys. and Mathem. Sci., 2010, � 3.

73

¹Çýáõ½ÇáÝ ¨ ÇÝ»ñóÇáÝ »ñ¨áõÛÃÝ»ñÁ Ù³ëÝÇÏÝ»ñÇ Ýëï»óÙ³Ý ÁÝÃ³óùáõÙ ·³½-³»ñá½áÉÇ ·ñ³íÙ³Ý ·áñÍ³ÏóÇ û·ÝáõÃÛ³Ù�:

Î³ï³ñí»É ¿ Ð³ÛÏ³Ï³Ý ³ïáÙ³Ï³Û³ÝÇ ß³Ñ³·áñÍÙ³Ý ÁÝÃ³óùáõÙ é³¹Çá³ÏïÇí ³»ñá½áÉÝ»ñÇ å³ñáõÝ³ÏáõÃÛ³Ý í»ñÉáõÍáõÃÛáõÝÁ:

è. ê. ²ë³ïñÛ³Ý, Ð. ê. Î³ñ³Û³Ý, Ü. è. Ê³ã³ïñÛ³Ý, È. Ð. êáõùáÛ³Ý. Ð»é³Ñ³ñ ÇÝýñ³Ï³ñÙÇñ ÙáÝÇïáñÇÝ·Ç ÙÇ Ù»Ãá¹Ç Ù³ëÇÝ ¿ç. 57–62

´»ñí³Í »Ý ÃáõÛÉ ç»ñÙ³ÛÇÝ ×³é³·³ÛÃÙ³Ý ³Õ�ÛáõñÝ»ñÇ (Ññ¹»ÑÇ

ûç³ËÝ»ñÇ ½³ñ·³óÙ³Ý Ý³ËÝ³Ï³Ý ÷áõÉáõÙ) Ñ³ÛïÝ³�»ñÙ³Ý Ýå³ï³Ïáí û¹³ÛÇÝ ï³ñ³ÍùÝ»ñÇ (áõÕÕ³ÃÇéáí Ï³Ù ÇÝùÝ³ÃÇéáí) ÇÝýñ³Ï³ñÙÇñ ÙáÝÇ-ïáñÇÝ·Ç Ýáñ Ù»Ãá¹Ç Ùß³ÏÙ³Ý ³ñ¹ÛáõÝùÝ»ñÁ:

Ü»ñÏ³Û³óí³Í »Ý ÆÎ ×³é³·³ÛÃ³ã³÷Ç ÝÏ³ñ³·ñáõÃÛáõÝÁ, ÇÝãå»ë Ý³¨ 2.5-Çó ÙÇÝã¨ 5.5 ÙÏÙ ³ÉÇù³ÛÇÝ ïÇñáõÛÃáõÙ Ï»ï³ÛÇÝ ¨ ï³ñ³Í³Ï³Ý ç»ñÙ³ÛÇÝ ³Õ�ÛáõñÝ»ñÇ ã³÷Ù³Ý Ù»Ãá¹ÇÏ³Ý»ñÁ:

ê. ¶. ¶¨áñ·Û³Ý, ê. î. Øáõñ³¹Û³Ý, Ø. Ð. ²½³ñÛ³Ý, ¶. Ð. Î³ñ³å»ïÛ³Ý. Â³Õ³ÝÃ-Ý»ñÇ Ñ³ëïáõÃÛáõÝÁ Ý³ÝáÙ»ïñ³Ï³Ý ÉáõÍáõÝ³ÏáõÃÛ³Ù� ã³÷áÕ Ù»Ãá¹` ÑÇÙÝÁ-í³Í ÙÇ³ß»ñï Ñ³ñÃ Ïá×áí ÇÝùÝ³·»Ý»ñ³ïáñÇ íñ³ ¿ç. 63–67

²é³ç³ñÏí»É áõ ÷áñÓÝ³Ï³Ýáñ»Ý ÑÇÙÝ³íáñí»É ¿ Ï³Ù³Û³Ï³Ý �³-

Õ³¹ñáõÃÛ³Ù� Ã³Õ³ÝÃÝ»ñÇ ¨ Å³å³í»ÝÝ»ñÇ Ñ³ëïáõÃÛáõÝÁ Ý³ÝáÙ»ïñ³Ï³Ý ÉáõÍáõÝ³ÏáõÃÛ³Ù� ã³÷áÕ áõ ·ñ³ÝóáÕ, ÙÇ³ß»ñï Ñ³ñÃ Ïá×áí ÇÝùÝ³-·»Ý»ñ³ïáñÇ íñ³ ÑÇÙÝí³Í Ù»Ãá¹: Æñ³Ï³Ý³óí»É ¿ ³Û¹ Ù»Ãá¹áí ·áñÍáÕ ë³ñùÇ É³�áñ³ïáñ Ù³Ýñ³Ï»ñïÇ Ùß³ÏáõÙ, ëï»ÕÍáõÙ, “NI LabVIEW” Íñ³·ñ³ÛÇÝ ÙÇç³í³ÛñáõÙ ¹ñ³ ³ßË³ï³ÝùÇ Ñ³Ù³Ï³ñ·ã³ÛÇÝ Õ»Ï³í³ñáõÙ, ÇÝãå»ë Ý³¨ ëï»ÕÍí³Í Ñ³Ù³Ï³ñ·Ç Ý³ËÝ³Ï³Ý ÷áñÓ³ñÏáõÙ áõ ë³Ý¹Õ³-Ï³íáñáõÙ: ²ÛÝ Ï³ñáÕ ¿ É³ÛÝ ÏÇñ³éáõÃÛáõÝ ·ïÝ»É ÇÝãå»ë Ñ»ï³½áï³Ï³Ý ËÝ¹ÇñÝ»ñ ÉáõÍ»ÉÇë, ³ÛÝå»ë ¿É ÙÇÏñá- ¨ Ý³Ýáï»ËÝáÉá·Ç³ÛáõÙ:

Ð. ². ²ë³ïñÛ³Ý. Ð»ïù»ñÇ ÙÇ �³Ý³Ó¨Ç Ù³ëÇÝ ¿ç. 68–70 æ. Ü»ÛÙ³ÝÇ ÏáÕÙÇó ù³é³Ïáõë³ÛÇÝ Ù³ïñÇóÝ»ñÇ Ñ³Ù³ñ ëï³óí³Í

Ñ»ïù»ñÇ ÙÇ �³Ý³Ó¨ ÁÝ¹Ñ³Ýñ³óíáõÙ ¿ ÙÇçáõÏ³ÛÇÝ ûå»ñ³ïáñÝ»ñÇ Ñ³Ù³ñ:


Physical and Mathematical Sciences � 3, 2010

� � � � � � � � �

�. ��, . �. �� , �. �. ��. �� : �� . 3–22

� �� !��"

��!#�� $�� "�� -��" ��", �� ^$�� !��. �� $�� ^ ��!� �� " �� . � �� $�� , �� ", �� -�� . ��" � �.�. � �� .

�� !�� -�� " �� ^$�"� ��", �� "��! �� . �� -��! �� ".

�� !� �� "�� " ��^$�� : ��!�� -�� " �� , �� !�� ", ��"�� " � �.�.

�� "�� , �� !� �� "�� . �� "�� " � �� !�� . �. �. ��. �� !�� "��

�� . 23–28

� � �� "�� !�� -� �� . �� , �� –�� –� �� !��" �� " �� -�� . �� $�� !�� -�� #��. �� #�� . � �� . �. �. ��. � ��#��$ �� $ ��%��$ !��

� �� . 29–34

� � �� " �� ^�� 1/ ,D y z y z f z� �� 1,� � – ��!��" � � -

, 1/�� D � – �� !��" �� –��. �� " �� ^�� # ^�� #�.

Proc. of the Yerevan State Univ. Phys. and Mathem. Sci., 2010, � 3.

75

�. �. ��, �. �. ��. &��'�� $ ��$ ��. 35–39

� � �!� �� ^�� "�� -

�� " � �� , � ��" �� , �� "�� -��. �� . ��"�� .

�. �. ��. ��!��!�� !�� . 40–43

�� ", ��

G: (G B H )D � � � ��!��

�� . �. �� . �� , �� "-�� H

( , )B m n �� , �� , ��-�� . � � �� , �� " ��

1 2=n n n 1 665n n (4, )B n

��^� �� -��, � �� -�� , � � � � �� !� . �. �� . *�� '��$ �� $ � ��, �� !��$ � �� -

�� '��$ +�� $�� !�%�� . 44–50

�� #�� -�� !��^�� !�� . � �� ANSYS, � ��" �� #�� , �� #�� . � ��" � � �� -�� !� �� !^ �� , �� #��^ � ��" � � ��. �� !� �� , �� 42 � 82 �� , � �� ! 42 ��" ��"��" � �� #�� !� �� " � � ��, �� ! 82 ��" ��"��". �. �. ��, !. !. ��"��, #. #. ��"��, #. �. ��$��, %. �. ��,

#. �. ��, �. �. &"��"��, �. ! ��. �� '��$ �� -� �� +� � �� #�� $ ��%�� '�� $� �� . 51–56 �� !�� !��

�� !�� . �� $!^ �� " �� !�� - ��.

�� " � �� " ��.

�. �. ��, !. �. ��, '. �. ��, (. �. ��. �� -��%� �� . 57–62

�� !� �� #�� (� ��-

�� ) �� #�� "


��!^ �� (�� " � �� ) �� $�� -� #� �� .

�� -� �� , � �� " �� 2.5 �� 5.5 ��.

�. !. ! ��$��, �. �. ��, �. !. ��, !. �. ��) ��. /�� 4�� '�� #��, �� -�� !#� � ��. 63–67

�� !�� $��

�� #��. �� -� �� " ��"��" ��" � ��#��". ��$�� -�� , �� , � �� !�� "�� , � �� !^�� " �� «NI LabVIEW». �� "�� #�� #�� , � � � � ��- � � ��. !. �. ��. �� !�� . 68–70

�� , �� . ��"� �� , ��$ �� .

ÊÙ�³·ñáõÃÛ³Ý Ñ³ëó»Ý. ºñ¨³Ý, 0025, ²É. Ø ³ÝáõÏÛ³Ý, 1

ÂáÕ³ñÏÙ³Ý å³ï³ëË³Ý³ïáõ` ¸³íÇ¹Û³Ý Æ.¶.

Èñ³ïíáõÃÛ³Ý ·áñÍáõÝ»áõÃÛáõÝ Çñ³Ï³Ý³óÝáÕ` ºäÐ äà²Î,

ºñ¨³Ý-25, ²É. Ø³ÝáõÏÛ³Ý, 1, íÏ³Û³Ï³Ý 03²053345` ïñí³Í 31.01.02:

INSTRUCTION FOR AUTHORS

Aims and Scope. “Proceedings of the YSU, Physical and Mathematical Sciences” is a peer-revi ewed journal, which is published thrice a year in English. The journal presents original articles and short communications, containing new results of theoretical, experimental and applied investigations carried out in YSU or in other scientific cent ers of Armenia. The journal also presents information about new scienti fic books and conferences on subject organized or coorganized by YSU.

Submission of the Manuscript. The authors are requested to submit one hard copy (printed only on one side of A4 paper) and the electronic version of the manuscript both in English and Russian. The Russian language version will be used only for review processing.

Only papers not previously published will be accepted and authors must agree not to publish elsewhere a paper submitted to and accepted by the journal. By submitting a paper the author confirms that its submission for publication has been approved by all of the authors.

Preparation of the Manuscript for Submission. The maximum number of pages, including figures and tables, is limited to 10 for scienti fic articles and 4 for short communications. The papers should be prepared by Microsoft Word 2003 editor for Windows OS, in accordance with the following styles: font – Times New Roman; font size – 12, line spacing – 1,5; paper – A4 with the margins of 3 cm. All formulas should be typesetted using MathType editor or the standard Equation editor embedded in MS Word package.

The number of figures should not exceed 5. The figures should be located within the text using figure captions. Only grayscale figures should be included in the paper.

The first page of articl e should contain the article title, the name and complete institutional affiliation of each author, and a short abstract (the abovementioned would be present ed also in Armenian and Russian). The abstract should be followed by Keywords. The first footnote on the fi rst page should point to the e-mail address of corresponding author.

The article structure typically consists of the following sections: Introduction, Results and Discussions, Conclusion and References.

The references should be presented in English in following style: – for articles: authors’ names, journal name (using standard abbrevi ations),

year, volume and issue number, page numbers; – for books: authors’ names, full title, publisher, year, total number of

pages. Pre- and Post-Printing Process. In case of acceptance of the manuscript for

publishing the corresponding author will receive the galley proofs for checking the corrections.

After publication of the article the authors will receive one copy of journal (for each one) and 5 reprints for free.

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY, 2010, � 3, 1–76.

physics, mathematics sciences mathematics...astola and danielian built three-parametric regular...

Documents