APPLIED PHYSICS AND MATHEMATICS

5 2014

ISSN 2307-1621

: , ()

: 77-50415 25.06.2012 .

: 83190 10363

: ..

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: .., . , () .., . , () .., . , () .., . , () .., . , () .., . , () .., . , () .., .-. , () .., .-. , () .., .-. , () .. - . , ., () .., - . , ., () ..,- . , () .., - . , ., () .., - . , ., () .., - . , ., () ., - , ., () .., - . , ., () .., - - , ., () .., - . , ., () .., - . , ., () .., - .-. , ., () .., - , ., () .., - -. , () .., . .-. , () ., - , ., () .., - .-. , ., () .., - . , ., () .., - .-. , ., () .., - . , () .., - .-. , ., () .., - .-. , ., () .., - . , ., () .., - . , ., () ., - , ., () .., - - , ., () .., - . , ., () ., - , ., () .., - .-. , () .., - .-. , ., ()

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107258, , ., . 17, . 2, .: 8 (985) 233-07-98, E-mail: pfim@mail.ru 17.07.2014 . 6088 1/8. . .-. . 16,4. .-. . 16,9. -110. 420 .

: , 107258, , ., . 17, . 2- 107258, , ., . 17, . 2.: 8 (499) 168-21-28

ISSN 2307-1621 5 2014

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Content

SCIENTIFIC JOURNALISSN 2307-1621 5 2014

AppLIEd phySICS ANd MAThEMATICS

APPLIED PHYSICS

Yu.L. Ratis

On the physical nature of quantization of electromagnetic field 3

V.A. Buzanovskii

Gas chemical nanosensors with sensitive elements based on nickel oxide 17

APPLIED MAtHEMAtICS

S.. Batalov

Formulation of logical level of modeling in the system analysis of oil recovery complex systems 40

V.L. Vol'fson

Quantitative indicators of solutions of Diophantine equations and systems in the domain of natural numbers 50

HIStORY OF PHYSICS

AND MAtHEMAtICS

B.S. Gorobets

On the first test of the Soviet H-bomb 65

Rules of consideration,publication andreview articles 76

3

, - , , - - . , - , , , .. , - -, , , -, .

- , -

- . , - , , , :

1. ?

2. ?

3. ?

4. ?

5. ?

6. ?

7. - ?

8. ?

9. , ( )?

.. .-. , , , E-mail: ratis@rambler.ru

, , - - . -, - -. , , () -

, . , (- -) 4- , , - . - . : , , , , .

YU.L. RAtiS Doctor of Phys.-Math. Sciences, Director of Science Institute of the power engineering for the special applications Samara, Russian Federation, E-mail: ratis@rambler.ru

ON tHE PHYSICAL NAtuRE OF quANtIzAtION OF ELECtROMAgNEtIC FIELD

It is postulated that general properties of material objects are finite size, inertness and ability to interact with other material objects. On the basis of the listed postulates it is shown that quantization of electromagnetic field is the results of interaction of the polarized physical vacuum and electromagnetic wave polarizing it. It is shown that the quanta of electromagnetic field (photon) have a time-dependent (oscillating) electric charge and

nonzero rest mass. A direct consequence of all aforesaid are the uncertainty principle, a complementarity principle (corpuscular-wave dualism) and observability of the electromagnetic field of 4-potentials, in particular, in the Aharonov Bohm effect and other similar effects. The hypothesis on the physical nature of the dark matter is formulated.Keywords: quanta, photon, charge, mass, uncertainty principle.

5 20144

. - . - , , , .

1. , , , [1], - , -. , - , - - . . : 1) - - ; 2) ; 3) - - , , , (); 4) - .

, - - . - [2]. - . , , , , , - .

- . , - . - .. . - , , , - , - , . -

. , - - . - , .

- , - .. . XIX .. . - . . . , , . - - , . - , - , - . - .

- . - , - .

2. 2.1.

. , , , - . , - [2]. , - (), . , - . . - - (., , [3])

5 2014 5

. , [3] , - . - - . - - - (., , [4]).

2.2.

-. , , , .

2.3.

, .., , - . - , , , - , -, , . - - . .

, , , - . - , , , - [2]. , -. -. - , -, (- )

[5]. , - ( ), - . d el , , - . - , .. , - m = 0 [68].

, - - -. ( ). (- ) , - , - -. .

- . , - , - , , .

. - , () , - . , , , , , - .

2.4.

- . . - . , - : () , - [9]. , - , , - , . ,

5 20146

( - ) -, ( ) , . -- [2].

- : - - ? , - , , .

- , , . - , , . . - . - . , , - .

, -, ( ) , () - . . , . , - , . , , .

- , . . - , . , .

. , . , - - , .

, - - , .

- , -. , -- . , , , [68] ( , (, ) , .. e

l t ,= 0 ).

2.5. ,

, - , -- , - [10]. , - , - = =1 / h [2]. , - - , . , -- , - , .

-, - , .. [11]. ( ) - - .

2.6. -, - - . [12]. -

5 2014 7

E t ~ , E , t - , - .

- U years~1010 , R cU U . , ( ) - . - - [12]:

m c m c R RU U UC

pi pi2 2 2 = = ~ ,

m , pi C m c= 2 1 ( ) - .

- , - -. , , () , , - . .

, .. - , ,

CUR~

.

[12] m c eV 2 3310>

(m kg > 10 69 ),

m , - . - , - m kg < 10 57 (m c eV 2 2110< ), .., -, , 12 , .

[12] m , e (, - -- ). , , , , , - -

, , .. e

l t ,= 0 ). -

,

e cR RU ~ / ,

- R m ~10 6 . ~1010 R mU ~1026 , , -

e ee ~10 15 ,

ee . -

, - - . - , , , , , .., - - [58]. -, , , - -. , , . , - e e e ee e e/ / ~= 10 15 , - . , , - , -, (, ). , , . - , - - , - .

, - - , ( -), .

3. - - [1315]. ,

5 20148

- -

= pi

2 4A

cj x( ) , (1)

=m c 1 , (2)

m , j x ( ) .

, - , -, , (- ):

j x j x j xvac ( ) ( ) ( )= + . (3)

. . - , (, - ),

j x A xvac ( ) ( )= v j x A xvac ( ) ( )= , (4)

- v

j x K x x g T x x A x dvac ( ) ( ) ( ) ( )= v j x K x x g T x x A x dvac ( ) ( ) ( ) ( )= . (5)

- -, - , - j x ( ) (. (10)). (5) T - , g - :

g g = =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

. (6)

-

v ( ) ( ) ( )K x x K x x K x xc q x = + , (7)

(7) - - .

, , , [7]:

=Ax

0 . (8)

(8) - :

+ =

=

c t

A

A

0

0. (9)

, , - . .

, (1) . . , - :

j x c T g A x pi

( ) ( )=2

4. (10)

2 - : [ ] [ ]2 1= LE .

, 2 2~ e , .. -

. - 2 . - , :

= T dV00 , (11)

, (), - = m c2 .

(10) , , . (11) , . , (1) - - j x( ) ( m = 0 ).

5 2014 9

4. (1) - :

A = 0 . (12)

- (1) -:

=2

0A , (13)

.., 4- A -.

, (1) - .

5. , (1) - , - [16]. (3) (1). :

=

224

4A

cc T g A x

K x x g T x x

( )

( ) ( )) ( ) ,g A x d (14)

v ( ) ( )K x x c g x x x = 4 02 . (15)

(15), - ( [ ] =

L E3 1 ), 02 , - ( 0

2 2 = L ), -

, -, , , - . - .

(15) (14):

=

pi

pi

2

2

0

2

4

4

4

Ac

c T A x

c x x Tx

( )

( ) (( ) ( ) ,x x A x d (16)

,

=

=

pi pi

2

2

0

24

44

A

cc T A x c T A x( ) ( )

.

(17)

pi( ) ( )x T x=16 , (18)

:

=

= +

pi

2

2

0

2

16

A

A x x A x ( ) ( ) ( ) . (19)

-

pi

2

2

16= . (20)

(19) :

= +

+

2 2

0

2

A A x

x A x

( )

( ) ( ) . (21)

, -

+ =

= +

2

2

0

2

g A x x A x

A x x

( ) ( ) ( )

( ) ( )

AA x ( ), (22)

,

+ +

+ =

2

2

0

20

g A x x A x

A x

( ) ( ) ( )

( ) ( ) .

(23)

, -

g x A x

g A x

+ { } ++ =

( ) ( )

( ) ( ) .2

0

2 20

(24)

(24) :

1

0

00

2

0

2 00 2

+ { } ++ =

( ) ( )

( ) ( ) ;

x x

x (25)

( ) ( )x A x 0 . -

( , ) - , ( ).

5 201410

- . , - , (24)

( , ) exp[ ( )] ( , ) r t i k z t r tz= 0 . (26)

H = 0 ,

E = ., 0 ( , )

r t - , :

E k ik r tz

r t r tz= +

+

( , ) ( , ) ( , )

exp[

0 0 0

ii k z tz( )], (27)

,

E ik r tz

r t r tz2

0 0

2

0

2= +

+ ( , ) ( , ) ( , ) , (28)

-:

T WE

ik r tz

r t r tz

00

2

0 0

2

0

2

16

1

16

= = =

= +

+

pi

pi ( , ) ( , ) ( , )

.

(29)

- :

m c T dV 2 00= . (30)

, ,

( , ) exp[ ( )] ( , ) ( , ) r t i k z t f z t zz= . (31)

- , - - , - .

(24) - ( ). exp[ ( )]i k z tz - , f z t( , ) -

, ( , )

z -

, z . - - () () - ( , )

z (.., ).

f z t f const( , ) = =0 , (29) (19) :

00 2 2 2( ) ( , ) ( , )x k r t r tz= + ( ) . (32) ,

f z t( , )

000

0

2 2 2 2( ) ( ) ( )x f kz= [ ] + [ ]

. (33)

(25) -- (33). , (25) - , -

n f kz2 02 2 2 2

1 1( ) ( ) ( ) ( )

= + [ ] + [ ] = + (34)

( )

-.

(34) (24), :

[ ( )] ( ) [( ) ( ) ] ( ) .1 00 2 02 2 0+[ ] + = x x (35)

, - :

( ) ( , )

= x y (36)

- .

= +2 . (37)

(35) :

+ +[ ] +

+ =

2

2

0

2 2

1

0

[ ( )] ( )

[( ) ( ) ] ( ) .

x

x (38)

(38) :

[ ( )] ( ) [ ( )] ( )

[( ) ( ) ]

1 12

2

0

2 2

+[ ] + +[ ] ++

x x

( ) .x = 0 (39)

(39) -

5 2014 11

0 0( , ) exp[ ( )] ( ) r t i k z t fz= . (40)

:

+[ ] + + + ==

2 2 2 2

2

0

2

1 1[ ( )] ( )[ ( )]

[( ) ( )

k cz

2] .

(41)

(41) :

+ +

+ [ ] + +

2 2 2 2 2

2 2

0

2 2 2 2

( )

( ) ( ) (

c k

k cz

z

)) .[ ] = 0 (42)

, - ( ) (.., ( ) = 0 ) - (42) - :

2

0

2 2 2

0

1

=

=

k cz

. (43)

(42) :

+ =2

0

2

0

2

0 0 ( ) ( ) ( )

. (44)

, - ( , ) :

02 2> (45)

(44) :

0 00( , ) exp( )x y x yx y= (46)

x y2 2

0

2 2+ = (47)

, -

f k

x yz x y

x y

0

2 2 2 2

1

2

00

4+ + =

=

( ) /

exp( )

(48)

1 2 2( , ) exp( / / )x y x yx y= . (49)

n2 1> - , kz :

k k kz z z2 2 2 + . (50)

, :

2 2 2 2c kz = + . , ( ) - . , , - . - - (50).

, - (41) - , , . - (41).

( ) [ ( )] ( ) = +1 . (51)

(41) :

+ + =

=

2 2 2 2 2

0

2 2 2

( ) ( ) ( )

( ) ( ) ( ).

k cz (52)

(52) (51) (50). (52) :

+ =

= + +

2 2 2 2 2 2

2 2

0

2

1

( ) ( ) ( )

( ) ( )

( )

c k kz z

( ). (53)

(53) -

+ =2

2

20

m E V

( ) (54)

- (54), -, V -. , E V< . (56)

5 201412

(56) (55),

+ =2 2

0 ( ) ( ( )) ( ) k vz , (57)

v( )

v( ) ( )( )

= +

2

1. (58)

- , [5]. - :

( )

( )

0 1

0

= =

. (59)

, ,

[ ]

[ ] + [ ] =

=

=

( )

( ) ( ) ( )

( )

0

0

2 2 2 2

0

0

0

0

f kz

. (60)

2 in v( ) ( ) . (61)

(58) - :

=2 2

0 out z outk( ) ( ) . (62)

. , - [5]

2 in v( ) ( ) . (63)

1

= in v( ) ( ) . (64)

(64) ,

=

in v d( ) ( )1

0

. (65)

:

out zH i k( ) ( )( ) = const 01 . (66)

-

1 1

0 0 0 0

ln( )

( )k

v dz

. (67)

k v dz20

2

0

1

12

exp ( ) , (68)

0 -

v kz( ) 0 2= . (69)

(59) (69),

kz2 2 001

=+( )

( ), (70)

, - (46) = ().

(68) , - . - - , -, -, . , - , - - , , .

, (56) 0

2 2= , 2 2= .

, ~ e1 , 2 1~ e . , kz2 v( ) - , - [5]. ( ). , kz2 - v( ) - , (68) e , - 1. , - 1 = qr rD1 1exp( ) D. -

D kT ne= [ / ( )] /8 2 1 2pi .

5 2014 13

- , - .

, - ( ) - kz2 - v( ) . ( ) ( ) - - ( - v( ) ).

, - . - , ( ) :

( ) exp( )= 0 kz . (71)

- (30) :

m c dV pi 2

1

16= ( ) . (72)

, (34)

( ) ( ) ( )

= [ ] + [ ] f kz02 2 2 2 , (73)

, ,

( ) exp( / ) = kz 2 , (74)

( ) / exp( )

= + f k k kz z z02 2 2

4 , (75)

(75) (72), , :

m cL

f kkz

z

2

0

22

232

1= +

, (76)

L 0 . , L 0 - k Lz 1 .

- (69):

( ) / exp( )0 02 2 2

04= + f k k kz z z , (77)

,

k ez2 2 00

2

0

1 2

1=

+

( )

( )( ) , (78)

k cz2 2 2 2= -

0

2 2

2 2

2 2

2 2

=

=

k kk

z z

z

. (79)

- - . -, - .

Lm cC

pi

~ =2

, (80)

- , - . - , ( ), , - , - . , ( ) exp( / ) = kz 2 , ( )

-

, - ( ) ~ exp( ) akz , kz - , . , , , - , - .

-

- .

- , , - . , - , -, .

, - , , ,

5 201414

-, , .

. - , , , , - n , ( n >1). , - . n n= ( ) - [16]:

Re ( ) ( )Im ( )

( )n n P

nd

pi [ ] =

( ) ( )

02 2

2 2

0

, (81)

, n( )0 1= . - (81) -. , - , . - - , , 2 - [1315]. - -, , -, . , - , - , :1) T

.2)

.3) CP .4) -

. -

, , , . , .

, , - , - , - . .

, = .

, - ( ) () - (.. -)

m c 2 = , (82)

. , () , . , - - , , , . - .

,

= ( )m (83)

. - , - (83) . - (82) . , m m , .

(82) -- , - , - , - . ,

p kz z= (84)

(70), ,

2 2 2 0 2

2

2 2 2 0 2= +

= +c k c k kz z z

( ) ( )( ) ( ) , (85)

k cz( ) /0 = . ( ) = 0 5 10 6. m . pi = 2 4 1015 1c s/ ~ - ~ 2 eV .

[1215] , - m c eV 2 3210 . -

/ ~ .0 5 10 32 . (86)

5 2014 15

, [1315] , , . , - T * , T .

d m cdt

T sSt B ph( )

2

4= , (87)

St B -, sph .

c dcdt

k sSt B ph= ( )1 4 , (88)

sph prf2 . (88) ,

c ddl

kSt B= ( )1 4 2 (89)

,

ddl

k cSt B

=

4 2 2 (90)

-, (90) :

00

4 2

0

=

St Bk c l . (91)

- :

pi

St Bkc

=2 4

2 360

. (92)

(92) (91),

pi 00

2

0

60

=

cl , (93)

0

0

=

Hc

ltheor , (94)

-

H stheor = pi

2

0

17 1

600 3 10. , (95)

- [2]

H sexp ~ .0 25 10 17 1 , (96)

- .

6. .

1. - .

2. - , , - .

3. , (.., ),

10 10

33 2 21 <

5 201416

1. .. -

. T. 2. .: . 1989. 437 . 2. .., .., . -

. T. 2. .: . 1973. 504 .3. .. -

. .: . 1977. 367 .4. Nakamura K. et al. (Particle Data Group), J. Phys. G 37,

075021 (2010).5. .., .., . -

. . T. 3. .: . 1989. 768 .

6. .., .., . . T. 4. .: . 1989. 728 .

7. .., .. -. .: . 1981. 428 .

8. .., .. . .: (1984) 600 .

9. . -. .: . 1989. 687 .

10. . - . Z. Math. und Phys., 1913, 62. PP. 225261 (Mit M. Grossman). . T. 1. .: . 1965. 700 .

11. .. , . , 2006. 350 .

12. Spavieri G., Quintero J., Gillies G.T., and Rodrguez M.. A survey of existing and proposed classical and quantum approaches to the photon mass. Eur. Phys. J. D 61, 531-550 (2011) DOI: 10.1140/epjd/e2011-10508-7.

13. .. , - , T. 9. 3. , 2007. C. 570583

14. .. - , T. 31, . 3, -, 2007. C. 93104

15. .. -, , , 2008, 124 .

16. .., .., - . . . 8. (, -, 1992), 664 .

17. CERN experiments observe particle consistent with long-sought Higgs boson. Press release CERN. 04.07.2012.

References1. Maxwell J.C. Traktat ob `elektrichestve i magnetizme

[A treatise on electricity and magnetism]. Vol. 2. M.: Nauka [Moscow: Publishing house Sciences], 1989. 437 p.

2. Landau L.D., Lifshits E.M. Nelokal'nye vzaimodejstviya kvantovannyh polej [The classical theory of fields. Course of theoretical physics]. Vol. 2. M.: Nauka [Moscow: Pub-lishing house Sciences], 1973. 504 p.

3. Efimov G.V. Nelokal'nye vzaimodejstviya kvantovannyh polej [Nonlocal interactions of quantum fields]. M.: Nauka [Moscow: Publishing house Sciences], 1977. 367 p.

4. Nakamura K. et al. (Particle Data Group), J. Phys. G 37, 075021 (2010).

5. Landau L.D., Lifshits E.M. Kvantovaya mehanika. Nere-lyativistskaya teoriya. Kurs teoreticheskoj fiziki [Quantum mechanics. Nonrelativistic theory. Course of theoretical physics]. Vol. 3. M.: Nauka [Moscow: Publishing house Sciences], 1989. 768 p.

6. Landau L.D., Lifshits E.M. Kvantovaya `elektrodinami-ka. Kurs teoreticheskoj fiziki [Quantum electrodynamics. Course of theoretical physics]. Vol. 4. M.: Nauka [Moscow: Publishing house Sciences], 1989. 728 p.

7. Ahiezer A.I., Berestetsky V.B. Kvantovaya `elektrodin-amika [Quantum electrodynamics]. M.: Nauka [Moscow: Publishing house Sciences], 1981. 428 p.

8. Bogoljubov N.N., Shirkov D.V. Vvedenie v teoriyu kvan-tovannyh polej [Quantum fields theory]. M.: Nauka [Mos-cow: Publishing house Sciences], 1984. 600 p.

9. Newton I. Matematicheskie nachala natural'noj filosofii [Philosophiae naturais principia mathematica]. M.: Nauka [Moscow: Publishing house Sciences], 1989. 687 p.

10. Einstein A. Proekt obobschennoj teorii otnositel'nosti i teorii tyagoteniya. Z. Math. und Phys [Entwurf einer verallgemein-erten Relativittstheorie und Theorie der Gravitation. Z. Math. und Phys.], 1913. 62. PP. 225261 (Mit M. Grossman). M.: Nauka [Moscow: Publishing house Sciences], 1965. 700 p.

11. Logunov A.A. Relyativistskaya teoriya gravitacii [Relativ-istic theory of gravitation]. M.: Nauka [Moscow: Publish-ing house Sciences], 2006. 350 p.

12. Spavieri G., Quintero J., Gillies G.T., and Rodrguez M. A survey of existing and proposed classical and quantum approaches to the photon mass. Eur. Phys. J. D 61, 531-550 (2011) DOI: 10.1140/epjd/e2011-10508-7.

13. Ratis Yu.L. Dispersiya, pogloschenie i generaciya `elektro-magnitnyh voln v vakuume Izvestiya SNC RAN [Disper-sion, absorption and generation of electromagnetic waves in vacuum. Izvestiya SSC RAS], Vol. 9. 3. Samara, 2007. PP. 570583.

14. Ratis Yu.L. Dispersiya i pogloschenie `elektromagnitnyh voln v fizicheskom vakuume Komp'yuternaya optika [Dis-persion and absorption of electromagnetic waves in physi-cal vacuum Computer optics], Vol. 31, 3, Samara-Mos-cow, 2007. PP. 93104.

15. Ratis Yu.L. Osnovy nelinejnoj spinornoj `elektrodinamiki, Izdatel'stvo SNC RAN [The basic principles of the non-linear spinor electrodynamics]. Issue SSC RAS, Samara, 2008. 124 p.

16. Landau L.D., Lifshits E.M., `Elektrodinamika sploshnyh sred. Kurs teoreticheskoj fiziki [Electrodynamics of Contin-uous Media. Course of theoretical physics]. Vol. 8. M.: Nau-ka [Moscow: Publishing house Sciences], 1992. 664 p.

17. CERN experiments observe particle consistent with long-sought Higgs boson. Press release CERN. 7/4/2012.

Information about the author

.-.

443071, , , . 3387

E-mail: ratis@rambler.ru

Ratis Yurij LeonidovichDoctor of Phys.-Math. SciencesDirector of Science Institute of the power engineering for the special applications 443071, Samara, Russian Federation, pr. Volzhski, 3387E-mail: ratis@rambler.ru

5 2014 17

.. . , , , E-mail: vab1960@rambler.ru

- . - . . 1 , - . - 2 , - . 3 , - .

. . , - - . , , .

: , , , , -, , .

V.A. BUzAnoVSkii Doctor of techn. Sciences, Leading scientific editor LLC Nauchtehlitizdat Moscow, Russian Federation, E-mail: vab1960@rambler.ru

gAS CHEMICAL NANOSENSORS WItH SENSItIVE ELEMENtS BASED ON NICKEL OXIDE

Results of development of gas chemical nanosensors with the sensitive elements based on nickel oxide are submitted in the article. The developed devices are systematized ac-cording to construction features of their sensitive elements. Thus, three nanosensor groups are defined. Group 1 is de-vices sensitive elements of which have a covering only from nickel oxide. Group 2 is nanosensors with the sensitive ele-ments containing a covering from nickel oxide and an exter-nal layer of other material. Group 3 is devices sensitive ele-ments of which have a covering from the composite based on nickel oxide. Analytical abilities of the listed nanosensor

groups are demonstrated. Typical regularities are marked. In particular, ability of these devices to determination of rather big number of inorganic and organic chemical compounds is presented. Influence of morphology and composition of a covering and also temperature of sensitive element on sen-sitivity and selectivity of measurements, response time and recovery time is shown.

Key words: nanotechnology, nanomaterial, gas chemical nano-sensor, sensitive element, covering, nickel oxide, metrological characteristic.

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5 201418

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10 . - . 300 - 0,02 % - ~1,8 , 400 ~1,4 [1].

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5 2014 19

. - 250 400 - 0,01 % - ~1,2 [3].

- 0,02 % , - . - ~800 , - ~20 . 10 . - . - 300, 400 0,02 % - - ~1,2 [1].

, - -, 0,01 %. - ~12 . . 250 - 0,01 % - ~1,5 [3].

, - , - 0,08 %. ~15 . - . - 260 . - 0,0005 % - ~3 , 0,001 % ~4,5 , 0,002 % ~6,5 , 0,005 % ~9 , - 0,01 % ~13 , 0,02 % ~15 , 0,05 % ~16 , - 0,08 % ~17 . 3 , 15 [4].

- 0,003 % , - - . 0,0005 % -

~1,2 , 0,00075 % ~1,35 , - 0,001 % ~1,65 , 0,00125 % ~1,9 , 0,0015 % ~1,95 , 0,00175 % ~1,7 , 0,002 % ~1,8 , - 0,0025 % ~1,55 , 0,003 % ~1,4 . - 512 ( - ). - [5].

-, , 0,003 %. 0,0005 % - ~1,2 , 0,00075 % ~1,35 , 0,001 % ~1,55 , - 0,00125 % ~1,75 , 0,0015 % ~1,85 , 0,00175 % ~1,7 , 0,002 % ~1,95 , 0,0025 % ~1,65 , - 0,003 % ~1,75 . - 87,5 0,001 % ~1,75 [5].

, - , 0,003 %. - 0,0005 % - ~1,1 , 0,00075 % ~1,25 , - 0,001 % ~1,45 , 0,00125 % ~1,55 , 0,0015 % ~1,65 , 0,00175 0,002 % ~1,55 , 0,0025 % ~1,35 , 0,003 % ~1,3 . - 100 0,0005 % - ~1,6 , 0,001 0,0015 % ~1,8 , 0,002 % ~2 , 0,0025 0,003 % ~1,8 [5].

0,003 % , . -

5 201420

- 0,0005 % ~1,2 , 0,00075 % ~1,4 , - 0,001 % ~1,65 , 0,00125 % ~1,7 , 0,0015 % ~1,8 , 0,00175 % ~1,7 , 0,002, 0,0025 0,003 % ~1,75 . 125 0,0005 % - ~1,5 , 0,001 % ~2,5 , 0,0015 ~2,3 , - 0,002 % ~3,1 , 0,0025 0,003 % ~2,25 [5].

, - , 0,003 %. - 0,0005 % - ~1,6 , 0,00075 % ~2,1 , - 0,001 % ~2,35 , 0,00125 % ~2,15 , 0,0015 % ~2,6 , 0,00175 % ~2,55 , 0,002 % ~2,5 , - 0,0025 % ~2,15 , 0,003 % ~2 . - 112,5 0,0005 % ~2,25 , - 0,001 0,0015 % ~3,5 , - 0,002 % ~2,75 , 0,0025 % ~3 , 0,003 % ~2,75 [5].

, - , - 0,001 %. - ~800 , ~20 . 10 . - . - 300 0,001 % - ~1,3 , - 400 ~1,1 [1].

0,0006 % , -

. 4050 . - . - 200 . 0,0001 % - ~1,13 , 0,0002 % ~1,15 , - 0,0004 % ~1,18 , 0,0006 % ~1,2 [2].

, - -, 0,002 %. ~15 . - . 260 - 0,002 % - ~1,3 [4].

, , - 0,0035 %. - 0,0005 % ~1,9 , - 0,001 % ~3,5 , 0,0015 % ~5,5 , 0,002 % ~10 , 0,0025, 0,003 0,0035 % ~13 . - 10 , - - [5].

0,0035 % , - . 0,0005 % - ~2 , 0,001 % ~4 , 0,0015 % ~13 , - 0,002 0,0025 % ~15 , 0,003 % ~14 , 0,0035 % ~13 . - 75 0,001 % - ~5 [5].

, -, - 0,0035 %.

5 2014 21

0,0005 % - ~2,5 , 0,001 % ~5,5 , 0,0015 % ~13 , - 0,002 % ~15 , 0,0025 % ~16 , 0,003 0,0035 % ~20 . 162,5 0,001 % ~25 , 0,0015 % ~100 , 0,002 % ~300 , 0,0025 % ~600 , - 0,003 % ~1800 [5].

-, - , 0,0035 %. 0,0005 % - ~3 , 0,001 % ~5 , 0,0015 % ~15 , - 0,002 % ~25 , 0,0025 % ~24 , 0,003 % ~22 , 0,0035 % ~20 . - 150 0,001 % - ~35 , 0,0015 % ~150 , - 0,002 % ~400 , 0,0025 % ~600 , 0,003 % ~1500 [5].

0,0035 % , - - . 0,0005 % - ~3,5 , 0,001 % ~7 , - 0,0015 % ~8 , 0,002 % ~12 , 0,0025, 0,003 0,0035 % ~10 . - 150 0,001 % ~75 , 0,0015 % ~200 , 0,002 % ~600 , 0,0025 % ~1500 , - 0,003 % ~2000 [5].

, - , 0,02 %. -

~800 , ~20 . 10 . - . 300 - 0,02 % - ~1,4 , 400 ~1,2 [1].

, , 0,002 %. ~15 . . - 260 . 0,002 % - ~1,3 [4].

0,02 % , - . ~800 , ~20 . 10 . - -. 300 0,02 % - ~1,4 , - 400 ~1,3 [1].

, , 0,002 %. ~15 . - -. 260 0,002 % - ~1,3 [4].

, - , 0,02 %. - - . 300 . 0,02 % - - ~3 [1].

5 201422

- 0,02 % , . ~800 , ~20 . 10 . - . 300 0,02 % - ~5 , 400 ~4,5 [1].

, - , . . - 350 [6].

, - , - . - . 350 . , , ( ~2 ) - [6].

, - . - . 350 . , - , - ~5 [6].

, , 0,01 %. - ~12 . - . - 350 .

0,01 % ~5,5 [3].

, - , 0,002 %. - ~15 . - - . 260 0,002 % - - ~1,3 [4].

, - . - . - 350 [6].

, - , - . - . 350 . , , - ( ~4 ) [6].

, - - , . - - . 350 . , , - - [6].

- 0,005 % , - . -. ~10 . . 240 -

5 2014 23

0,005 % ~2 [7].

, , - . - . - 350 [6].

, - -, . - . 350 . -, - , ( ~2 ) - [6].

, - . - . - 350 . , , ~3 [6].

, - , 0,2 %. . ~10 . - . - 200 260 . - 0,005 % - ~1,5 , 0,2 % ~2,5 . - 21 , 28 ( 0,001 %) [7].

, , . - . 350 [6].

, - . - . - 350 . , - , ( ~2 ) [6].

, - , - . - . - 350 . -, , - ~1,5 [6].

, - , - 0,005 %. - . - ~10 . . 240 0,005 % - ~1,3 [7].

- 0,005 % , - . . - ~10 . - . 240 - 0,005 % - ~1,3 [7].

, - , 0,005 %. . ~10 . - . -

5 201424

240 . 0,005 % - - ~1,4 [7].

, , - 0,005 %. -. ~10 . . 240 - 0,005 % - ~1,5 [7].

- , - . - . - 350 [6].

, , . - . - 350 . , - , - ( ~3 ) [6].

, - , . - . 350 . , - , ~1,5 [6].

1 , - .

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, 3 (. 2): ,

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, - 3.1 - .

, - , 5 % . - . 4050 . - , , -

5 2014 25

1. ,

,

, %

1 2 3 4 5 6 7 8 9

NiO- 300 0,02 [1]

NiO- 400 0,02 [1]

4050 NiO- 200 0,03 5 7 [2]

NiO- 250-400 0,01 [3]

NiO- 300 0,02 [1]

NiO- 400 0,02 [1]

NiO- 250 0,01 [3]

NiO- 260 0,08 3 15 [4]

NiO- 300 0,001 [1]

NiO- 400 0,001 [1]

4050 NiO- 200 0,0006 [2]

NiO- 25 0,003 512 [5]

NiO- 87,5 0,003 [5]

NiO- 100 0,003 [5]

NiO- 112,5 0,003 [5]

NiO- 125 0,003 [5]

NiO- 260 0,002 [4]

NiO- 25 0,0035 10 [5]

NiO- 75 0,0035 [5]

NiO- 150 0,0035 [5]

NiO- 162,5 0,0035 [5]

NiO- 300 0,02 [1]

NiO- 400 0,02 [1]

NiO- 260 0,002 [4]

NiO- 300 0,02 [1]

NiO- 400 0,02 [1]

NiO- 260 0,002 [4]

NiO- 300 0,02 [1]

NiO- 300 0,02 [1]

NiO- 400 0,02 [1]

NiO- 350 0,01 [3]

NiO- 260 0,002 [4]

NiO- 350 [6]

NiO- 350 [6]

NiO- 350 [6]

NiO- 350 [6]

NiO- 240 0,005 [7]

5 201426

1 2 3 4 5 6 7 8 9

NiO- 350 [6]

NiO- 350 [6]

NiO- 200260 0,2 21 28 [7]

NiO- 350 [6]

NiO- 350 [6]

NiO- 240 0,005 [7]

NiO- 240 0,005 [7]

NiO- 240 0,005 [7]

NiO- 240 0,005 [7]

NiO- 350 [6]

NiO- 350 [6]

1

. - 450 . - 300 [2].

-, - 5 % , 0,03 %. - 4050 . - - . 200 - 0,03 % - ~1,35 . - 57 . , , ( ~15 %) - [2].

, - 5 % - , 0,05 %. - . 4050 . , , - . 450 . - 300 [2].

- 0,05 % -, - 5 % . 4050 . - . 200 - 0,05 % - ~1,05 [2].

, - 5 % , - 0,0006 %. - 4050 . . 200 - 0,0001 % - ~1,23 , 0,0002 % ~1,24 , 0,0004 % ~1,27 , 0,0006 % ~1,3 . , , ~10 % [2].

, - 1,2 % (.) - , 0,005 %. -

5 2014 27

. ~20 . ~10 . . - 240 . 0,005 % - ~3 . , - , ( ~1,5 ) [7].

0,2 % - , - 0,6, 1,2 1,8 % (.) . - . ~20 . - ~10 . - . ,

1,2 % 240 , - . 0,0005 % - ~4,1 , 0,001 % ~5,6 , 0,002 % ~7,8 , 0,005 % ~10,2 , 0,01 % ~15,3 , - 0,02 % ~21 , 0,05 % ~24 , 0,1 % ~30 , - 0,2 % ~32 . 1,7 , 11 . -, , , , 0,005 % -, 2,8 : 1,3 : 1,5 : 1,6 : 3 : 10,2. -. -, - , ~13 -

. 2 ,

19

~1,35 . 5-7 . , - , ( ~15 %) [2].

. 2. ,

, 5 % , 0,05 %. . 40-50 . - , - , . 450 . 300 [2].

0,05 % , 5 % . 40-50 . -

,

,

,

,

,

,

5 201428

[7].

, 1,2 % (.) , - 0,005 %. - . ~20 . ~10 . - . - 240 0,005 % - - ~1,5 . , - , ( ~15 %) - [7].

, 1,2 % (.) , - 0,005 %. - . ~20 . ~10 . - - . - 240 0,005 % ~2,8 . , - , - ~2,15 [7].

- 0,005 , - - 1,2 % (.) . -. ~20 . ~10 . - . - 240 . 0,005 % - ~1,3 . - , -

, ( ~8 %) [7].

, 1,2 % (.) , - 0,005 %. - . - ~20 . ~10 . - -. 240 - 0,005 % - ~1,6 . -, , - ~7 % [7].

, - 3.2 - : ; ; ; ; .

, , - 3 % (.) , 0,005 %. - . 330 - 0,005 % - ~2,8 [9].

- 0,005 % , 3 % (.) - . - . -

5 2014 29

330 . - 0,005 % ~1,2 [9].

, - - , 0,005 %. - - . - 200 0,001 % - ~1,5 , 0,005 % ~2,2 [10].

, - 3 % (.) - , 0,005 %. - . 330 - 0,005 % - ~1,5 [9].

0,01 % , - . . - . - 300 . - 0,0005 % ~2 , - 0,001 % ~4 , 0,002 % ~8 , 0,005 % ~15 , - 0,01 % ~24 . 5 , 21 ( - 0,001 %). , -, - 0,005 % , 4 : 15 : 2 : 7 [11].

-, - - , 0,01 %. -

- 200300 . ~50 . - . - 300 . - 0,0005 % - ~10 , 0,001 % ~22 , 0,002 % ~35 , - 0,005 % ~47 , 0,01 % ~60 . 0,5 , - 4 ( 0,001 %). , , - 0,005 % - , - 10 : 47 : 4 : 13 [11].

, - 3 % (.) - , 0,005 %. - . 330 0,005 % - ~1,3 [9].

0,005 % - , - 3 % (.) . - - . 330 - 0,005 % - ~1,4 [9].

, - , - 0,15 %. - . - . - - 300 . 0,0005 % -

5 201430

- ~5,3 , 0,005 % ~15 , 0,02 % ~45 , - 0,05 % ~75 , 0,15 % ~90 . 30 ( - 0,0002 %), 200 ( 0,15 %). - , , - ( ~12 ) , - [12].

, - - , 0,005 %. - . 200 - 0,001 % - ~1,6 , - 0,005 % ~3,5 [10].

- 0,005 % , 3 % (.) - . - -. 330 - 0,005 % - ~3,4 [9].

, - , 0,005 %. - . - . - 300 . 0,005 % - - ~7 [11].

-, -

- , 0,005 %. 200300 . ~50 . - -. 300 - 0,005 % - ~13 . - 2 [11].

- 0,01 % -, . . - - . 280 - 0,0005 % ~7 , - 0,001 % ~10 , 0,002 % ~16 , 0,003 % ~18 , - 0,005 % ~33 , 0,007 % ~45 , 0,01 % ~59 . 0,0001 %. , - , ~2,4 [13].

, - - , 0,005 %. - . . 300 . - 0,005 % ~3 . , - , ( ~20 %) [12].

5 2014 31

, , - 0,005 %. . 200 0,001 % - - ~1,4 , 0,005 % ~1,5 [10].

- 0,005 % , . - . - . 300 0,005 % - ~29 . , - - , - ~10 [12].

, - , - 0,005 %. . - . 300 . 0,005 % - ~5,5 . -, - , ( ~1,5 ) [12].

, - - , 0,005 %. . 200 0,001 % -

~1,5 , 0,005 % ~2 [10].

0,005 % - , - 3 % (.) - . . 330 0,005 % - ~3,7 [9].

, - , - 0,005 %. . - . 300 . - 0,005 % - ~2 [11].

, , - 0,005 %. - 200300 . - ~50 . - . 300 - 0,005 % - ~4 . - 2 [11].

0,005 % , - - . - . 200 - 0,001 % - ~1,3 , 0,005 % ~1,5 [10].

5 201432

, - 2, 3 4 % (.) , - 4 %. - . , 3 % 330 , . 0,005 % ~11 , 0,05 % ~35 , 0,1 % ~45 , - 0,2 % ~65 , 0,5 % ~110 , 1 % ~170 , - 1,5 % ~220 , 2 % ~230 , 2,5 % ~240 , - 3 % ~245 , 3,5 % ~250 , 4 % ~255 . - 11,2 , 4 ( 0,005 %). , , , -, , , 0,005 % - , 2,8 : 1,2 : 1,5 : 1,3 : 1,4 : 3,4 : 3,7 : 11. - 30 . , - , ( ~3,3 ) - 20 [9].

, , - 0,001 %. - . - 200 . - 0,001 % - ~3,5 [10].

- 0,001 % , . -

- . - 300 . 0,001 % - ~4,8 [10].

, - - , - 0,01 %. - . - 200 . - 0,0005 % ~3,5 , 0,001 % ~6 , 0,002 % ~9,5 , - 0,003 % ~11,5 , - 0,004 % ~13 , 0,005 % ~14 , 0,01 % ~21 . 8106 %. - 50 , - 80 ( 0,001 %). , , , -, - 0,005 % , 2,2 : 14 : 2 : 1,5 : 3,5 : 1,5. 90 - [10].

, - - , - 0,005 %. . - -. 300 0,005 % - ~4 [11].

- 0,005 % , . -

5 2014 33

200300 . - ~50 . - . 300 - 0,005 % - ~10 . 2 , - 5 [11].

, - , 0,005 %. - . . - 300 . 0,005 % - ~2,5 . , -, ~4 % [12].

, - 3 % (.) - , 0,00005 %. - . 350 0,00005 % - ~45 %. 500 , . , , - ( ~1,25 ) [14].

- 0,0034 % - , . - . - 250 [15].

, -

, 0,01 %. - 180 0,01 % - 1,54 % [16].

, - , 0,01 %. 180 0,01 % - 1,53 % [16].

- 0,5 % -, . - 180 . 0,01 % - ~80 %, - 0,05 % ~350 %, 0,1 % ~520 %, 0,2 % ~800 %, - 0,5 % ~900 %. 3 , - 2 . , - , - ~2,3 [16].

, - , 0,01 %. 180 - 0,01 % - 17,4 % [16].

, - - - , 0,01 %. 180 0,01 % - 14,3 % [16].

0,01 % , -

5 201434

- - . 180 0,01 % - 1,37 % [16].

, - - , - 0,01 %. - 180 0,01 % - 2,54 % [16].

, - , - 0,01 %. . ~12 . - -. - 350 . 0,01 % ~15 . , - - , ( ~12,5 ) [3].

- 0,01 % , . - . - ~12 . . 250 400 - 0,01 % ~5 . , , - ~3,3 [3].

, -

, 0,01 %. - . - ~12 . - . - 350 . 0,0005 % ~25 , 0,001 % ~40 , - 0,002 % ~60 , 0,003 % ~80 , 0,004 % ~100 , - 0,005 % ~120 , 0,006 % ~140 , 0,007 % ~150 , 0,008 % ~160 , 0,009 % ~167 , - 0,01 % ~172,5 . , , ( ~31,4 ) - [3].

, - , 0,08 %. - ~15 . . - 180 . - 0,0005 % - ~7 , 0,001 % ~14 , 0,002 % ~27,5 , 0,005 % ~47 , 0,01 % ~70 , - 0,02 % ~90 , 0,05 % ~125 , 0,08 % ~132 . 3 , - 5 ( 0,005 %). , , , 0,002 % - , 1,8 : 2 : 2,9 : 27,5 : 2,2. - . - , - ,

5 2014 35

~7,8 - 80 [4].

0,002 % , - - . - ~15 . . 180 0,002 % - - ~1,8 . , , ( ~1,4 ) - 80 [4].

, - - , 0,002 %. - ~15 . . 180 0,002 % - - ~2 . - , , ~1,5 80 [4].

, - , 0,002 %. - ~15 . . 180 . 0,002 % - ~2,9 .

, , ( ~2,2 ) - - 80 [4].

- 0,002 % , - . - - ~15 . . 180 0,002 % - ~2,2 . , - -, ~1,7 - 80 [4].

, , 3.3 - , - , . 1 %. . - ~78 . , - , - . 300 - 0,001, 0,01, 0,1 1 % , 16 : 33 : 52 : 78. - 20 , 3 . 0,0001 %. , - , - ( ~1,25 ) [17].

5 201436

, 3.4 -, 2, 3 5 % (.) , 3 % (.) . 0,00008 % . , 3 % - 350 , . - 0,00001 % ~30 %, 0,00002 % ~35 %, 0,00005 % ~50 %, - 0,00008 % ~60 %. - 200 , 500 ( 0,00005 %). - . -, 3 % , - ~10 % [14].

2 , - .

: ; ; -

; - ; - , , .

- (. 1 2) :

; , ; , , ; , ; , ; , ,

, . -

-, - -.

- ( 400 ) .

- - .

, - , . , - .

- - (. 1): ,

(- 1);

, ( 2);

, ( 3). 3 -

(. 2). , -

- .

- .

, - -, : -

- ;

5 2014 37

2. ,

-

,

, %

-

1 2 3 4 5 6 7 8 9

4050 NiO- + Au- 300 [2]

4050 NiO- + Au- 200 0,03 5-7 5-7 [2]

NiO- + FeO- 350 0,01 [3]

NiO- + SnO2- 330 0,005 [9]

4050 NiO- + Au- 300 0,05 [2]

4050 NiO- + Au- 200 0,05 [2]

NiO- + FeO- 250-400 0,01 [3]

NiO- + PdO- 180 0,08 3 5 [4]

NiO- + SnO2- 330 0,005 [9]

NiO- + ZnO- 250 0,0034 [15]

NiO- + In2O3- 180 0,01 [16]

78 NiO- + ZnO- + Au- 300 1 20 3 [17]

NiO- + PdO- 180 0,002 [4]

NiO- + SnO2- 330 0,005 [9]

NiO- + SnO2- 200 0,005 [10]

NiO- + SnO2- 300 0,01 5 21 [11]

NiO- + SnO2- 300 0,01 0,5 4 [11]

4050 NiO- + Au- 200 0,0006 [2]

NiO- + PdO- 180 0,002 [4]

NiO- + SnO2- 330 0,005 [9]

NiO- + In2O3- 180 0,01 [16]

NiO- + SnO2- 330 0,005 [9]

NiO- + PdO- 180 0,002 [4]

NiO- + FeO- 350 0,01 [3]

NiO- + PdO- 180 0,002 [4]

NiO- + SnO2- 330 0,005 [9]

NiO- + SnO2- 200 0,005 [10]

NiO- + SnO2- 300 0,005 [11]

NiO- + SnO2- 300 0,005 2 2 [11]

NiO- + SnO2- 300 0,15 30 200 [12]

NiO- + SnO2- 280 0,01 [13]

NiO- + In2O3- 180 0,5 3 2 [16]

10 NiO- + Au- 240 0,005 [7]

NiO- + SnO2- 200 0,005 [10]

NiO- + SnO2- 300 0,005 [12]

NiO- + In2O3- 180 0,01 [16]

NiO- + SnO2- 300 0,005 [12]

5 201438

- ;

- ;

- - ;

, , - , , - , - (, - , );

, -, -:

- - (, , , , , - ..),

- ( , );

, , - -:

(, , , ..),

- ( );

, - , - n- ( , ), -

1 2 3 4 5 6 7 8 9

10 NiO- + Au- 240 0,2 1,7 11 [7]

NiO- + SnO2- 330 0,005 [9]

NiO- + SnO2- 200 0,005 [10]

NiO- + SnO2- 300 0,005 [11]

NiO- + SnO2- 300 0,005 2 2 [11]

NiO- + SnO2- 300 0,005 [12]

NiO- + In2O3- 180 0,01 [16]

NiO- + SnO2- 200 0,005 [10]

NiO- + In2O3- 180 0,01 [16]

10 NiO- + Au- 240 0,005 [7]

NiO- + SnO2- 330 4 11,2 4 [9]

10 NiO- + Au- 240 0,005 [7]

NiO- + SnO2- 200 0,001 [10]

NiO- + SnO2- 300 0,001 [10]

NiO- + SnO2- 200 0,01 50 80 [10]

NiO- + SnO2- 300 0,005 [11]

NiO- + SnO2- 300 0,005 2 5 [11]

NiO- + In2O3- 180 0,01 [16]

10 NiO- + Au- 240 0,005 [7]

NiO- + SnO2- 300 0,005 [12]

10 NiO- + Au- 240 0,005 [7]

NiO- + SnO2- 350 0,00005 500 [14]

NiO- + MoO3- + SnO2- 350 0,00008 200 500 [14]

2

5 2014 39

- -, - n-:

- -,

- - .

, - - - - : -

, - -;

- - , , -- ;

- - ;

-. -

, - - - - - .

1. Cho N.G., Hwang I.-S., Kim H.-G., Lee J.-H., Kim I.-D. Gas sensing properties

of p-type hollow NiO hemispheres prepared by polymeric colloidal templating method. Sensors and Actuators B: Chemical. 2011. Vol. 155. 1. PP. 366371.

2. Della Gaspera E., Guglielmi M., Martucci A., Giancaterini L., Cantalini C. En-hanced optical and electrical gas sensing response of sol-gel based NiO-Au and ZnO-Au nanostructured thin films. Sensors and Actuators B: Chemical. 2012. Vol. 164. 1. PP. 5463.

3. Kim H.-J., Choi K.-I., Kim K.-M., Na C.W., Lee J.-H. Highly sensitive C2H5OH sensors using Fe-doped NiO hollow spheres. Sensors and Actuators B: Chemi-cal. 2012. Vol. 171172. PP. 10291037.

4. Wang L., Lou Z., Wang R., Fei T., Zhang T. Ring-like PdO-decorated NiO with lamellar structures and their application in gas sensor. Sensors and Actuators B: Chemical. 2012. Vol. 171172. PP. 11801185.

5. Luyo C., Ionescu R., Reyes L.F., Topalian Z., Estrada W., Llobet E., Granqvist C.G., Heszler P. Gas sensing response of NiO nanoparticle films made by reac-tive gas deposition. Sensors and Actuators B: Chemical. 2009. Vol. 138. 1. PP. 1420.

6. Liu B., Yang H., Zhao H., An L., Zhang L., Shi R., Wang L., Bao L., Chen Y. Synthesis and enhanced gas-sensing properties of ultralong NiO nanowires assembled with NiO nanocrystals. Sensors and Actuators B: Chemical. 2011. Vol. 156. 1. PP. 251262.

7. Wang L., Lou Z., Fei T., Zhang T. Enhanced acetone sensing performances of hierarchical hollow Au-loaded NiO hybrid structures. Sensors and Actuators B: Chemical. 2012. Vol. 161. 1. PP. 178183.

8. Hotovy I., Huran J., Spiess L., Romanus H., Capone S., Rehacek V., Taurino A.M., Donoval D., Siciliano P. Au-NiO nanocrystalline thin films for sensor application. Journal of Physics: Conference Series. 2007. Vol. 61. PP. 435439.

9. Liu L., Zhang Y., Wang G., Li S., Wang L., Han Y., Jiang X., Wei A. High tolu-ene sensing properties of NiO-SnO2 composite nanofiber sensors operating at 330 C. Sensors and Actuators B: Chemical. 2011. Vol. 160. 1. PP. 448454.

10. Zheng Y., Wang J., Yao P. Formaldehyde sensing properties of electrospun NiO-doped SnO2 nanofibers. Sensors and Actuators B: Chemical. 2011. Vol. 156. 2. PP. 723730.

11. Wang L., Deng J., Fei T., Zhang T. Template-free synthesized hollow NiO-SnO2 nanospheres with high gas-sensing performance. Sensors and Actuators B: Chemical. 2012. Vol. 164. 1. PP. 9095.

12. Liu X., Zhang J., Guo X., Wu S., Wang S. Enhanced sensor response of Ni-doped SnO2 hollow spheres. Sensors and Actuators B: Chemical. 2011. Vol. 152. 2. PP. 162167.

13. Chen Y., Yu L., Feng D., Zhuo M., Zhang M., Zhang E., Xu Z., Li Q., Wang T. Superior ethanol-sensing properties based on Ni-doped SnO2 p-n heterojunc-tion hollow spheres. Sensors and Actuators B: Chemical. 2012. Vol. 166167. PP. 6167.

14. Lee S.C., Choi H.Y., Lee S.J., Lee W.S., Huh J.S., Lee D.D., Kim J.C. Novel SnO2-based gas sensors promoted with metal oxides for the detection of di-chloromethane. Sensors and Actuators B: Chemical. 2009. Vol. 138. 2. PP. 446452.

15. Moon J., Park J.-A., Lee S.-J., Chu H.Y., Zyung T., Kim I.-D. Gas sensitivity modulation of oxide thin films by means of an electrical method. Sensors and Actuators B: Chemical. 2010. Vol. 148. 2. PP. 539543.

16. Feng C., Li W., Li C., Zhu L., Zhang H., Zhang Y., Ruan S., Chen W., Yu L. Highly efficient rapid ethanol sensing based on In2XNiXO3 nanofibers. Sensors and Actuators B: Chemical. 2012. Vol. 166167. PP. 8388.

17. Della Gaspera E., Guglielmi M., Perotto G., Agnoli S., Granozzi G., Post M.L., Martucci A. CO optical sensing properties of nanocrystalline ZnO-Au films: Effect of doping with transition metal ions. Sensors and Actuators B: Chemical. 2012. Vol. 161. 1. PP. 675683.

Information about the author

.

107258, ,

, 17, . 2E-mail: vab1960@rambler.ru

Buzanovskii Vladimir Adamovich Doctor of Techn. Sciences Leading Scientific Editor LLC Nauchtehlitizdat107258, Moscow, Russian Federation Alymov Lane,17, Building 2E-mail: vab1960@rambler.ru

5 201440

.. . . , , , E-mail: geoavtsyst@mail.ru

, - -. . , -

- .

: , - , , - , .

S.. BAtALoV Cand. of techn. Sciences, Associate Professor ufa State Academy of Economics and Service ufa, Russian Federation, E-mail: geoavtsyst@mail.ru

FORMuLAtION OF LOgICAL LEVEL OF MODELINg IN tHE SYStEM ANALYSIS OF OIL RECOVERY COMPLEX SYStEMS

The article describes the approach to the analysis of complex technical control systems behavior to achieve required goals, functions and normative-valuable factors. It contains compre-hensive semigraphical description of the model with multilevel and multistage integration. The article gives a detailed descrip-tion of the interaction mechanism and formulated description of complex subsystems identifying the structure of the system

complex with determination of required and sufficient condi-tions of its realization.

Key words: complex control system, model of multilevel and multistage integration of subsystems, system com-plex, normative-valuable factors, vector of technical means parameters.

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5 2014 41

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xy0(t) = x/lna0; xy1(t) = x/lna1. - xy0 xy1 xy0(t) + xy1(t) = 0 xy0(t) = xy1(t).

5 2014 45

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5 2014 47

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. .: , 1977. 341 .

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5. .., .. . II. // . . 2006. 2. . 8999.

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10. .., .. - - // -: . . . . : . 1989. . 710.

11. .. // , . 2008. 1. . 1823.

12. RU 2230895. 21 43/20. - / .. // . -. 2004. 17.

13. .. . .. - // . , , . 2008. 11. . 5862.

14. .., .., .., .. // - : . . . . : . 1998. .186197.

15. .. - - // , -. 2010. 7. . 1321.

16. .. // , , -. 2008. 6. . 3640.

17. .. - - . // , . 2008. 7. . 2735.

18. .. . : , 2007. 300 .

19. .. // . , , . 2008. 11. . 3539.

5 2014 49

20. .. // - . , , 2010. 7. . 4145.

References

1. Solodovnikov V.V., Biruakov V.F., Tymarkin B.I. [The principle of complexity in control theory]. M.: Nauka [Moscow: Publish-ing house Science], 1977. 341 p.

2. Batalov S.A. Informatsionnie sistemi i technologii [An informa-tion systems and technologies]. Ufa: UGAES [Ufa: Publishing house UGAES], 2006. 480 p.

3. Vavilov A.A. strukturnyy I parametricheskiy sintez slozhnykh system [Structural and parametric synthesis of complex sys-tems.] L.: LETI [Leningrad: Publishing house LETI], 1979. 236 p.

4. Kafarov V.V., Vetohin V.N. Osnovy avtomatizirovannogo pro-ektirovaniya himicheskih proizvodstv [Fundamentals of com-puter-aided design of chemical plants]. M.: Himiya [Moscow: Publishing house hemistry], 1987. 536 p.

5. Biruak UI. I. Insarov V.V. Upravlenie v mnogoobyektnih or-ganizatsionnih sistemah. II. Printsipi realizatsii informatsionnoi podderrzhki upravlencheskih recheniy. [Multi-Site Manage-ment in organizational systems. II. Principles for the implemen-tation of information support management decisions]. Izvestiya RAN. Teoriya i sistemi upravleniya [Izvestiya RAN. Theoru and control system], 2006. 2. PP. 8999.

6. Batalov S.A. Realizuemostj pointervaljno-kombinatcionnogo nelineinogo zakona upravlenij objektov s raspredelennimi parametrami. [Realizability interval standardized-Raman non-linear control law objects with distributed parameters]. M., Dep v VINITI RAN 13.04.05., 503-V2005 [Moscow, Pub-lishing house Dep v VINITI RAN 13.04.05, 503-V2005], 2005. 19 p.

7. Batalov S.A. Razrabotka metodologii issledovanij [Develop-ment of methodology for study of the complex system of oil production in the genesis // History of Science and Technology]. Istoriya nauki i tehniki [History of science and engineering], 2011. Vol. 2. 2. PP. 2537.

8. Spravochnik po teorii avtomaticheskogo upravlenya. Pod red. A.A. Krasovskogo. [A Guide to the Theory of Automatic Con-trol]. M.: Nauka [Moscow: Publishing house Science], 1987. 712 p.

9. Meerov M.V. Issledovanie i optimizatsiya mnogosvyaznyh sys-tem upravleniy. [Research and optimization of multivariable systems management]. Otv. redactor akademik A/A/ Voronov.

10. Batalov S.A., Kolovertnov Y.D. Princip invariantnosti mnogo-svyaznosti system ypravleniya protcessom nefteizvlecheniya iz prodyctivnyh plastov. [The invariance principle multivari-able systems process control oil recovery from the reservoir // Automatics for the oil and gas industry:]. Pribory I ystroystva avtomatiki dlya neftyanoey promychlennosti: Mezhvuz. Sb. Nauchn. Trudov. Ufa: UNI. [Ufa. Publishing house UNI], 1989. PP. 710.

11. Batalov S.A. Sistemnyy podhod k postroeniyu modeli polnoy vyrabotki neftyanyh plastov. [Systematic approach to the con-struction of a model running out of oil reservoirs // automation, remote control and communication in the oil industry]. Avtoma-tizatciy, telemehanizatciy i svyazj v neftynoy promuchlennosti.

[Avtomatization, telemechanization and link in oil industrial], 2008. 1. PP. 1823.

12. Batalov S.A. Patent RU 2230895. 21 43/20. Sposob optimizatcii neftedobychi [A method for optimizing oil produc-tion]. Batalov S.A. Otkrytij. Izobreteniy. 2004. 17.

13. YUditckiy S.A., Toch D.S. Razvitie metodov analiza povedeniy organizatcionnyh system na osnove triadnoy struktury. [Devel-opment of methods for analyzing the behavior of organizational systems on the basis of a triadic structure // devices and systems. Management, monitoring, diagnostics]. Pribory I sistemy. Up-ravlenie< kontrolj, diagnostika. 2008. 11. PP. 5862.

14. Abyzgiljdin A.YU., Aljmuhametov A.A, Kanavin YU.A., Pudnev N.A. Informatcionnye tehnologii protcessov pererabot-ki uglevodorodnogo syrjy. [Information technology processes hydrocarbon processing]. Problemy I perspectivy sovremennyh tehnologiy servisa. [Problems and prospects of modern technol-ogy service]. Mezhvuz. Sb. Nauchn. Trudov. Ufa: UTIS. [Ufa. Publishing house UTIS], 1998. PP.186187.

15. Batalov S.A. Avtomatizatciy sistemnogo kompleksa neftepro-mysla po tehnico-economichescim pokazatelym. [Automation system for oilfield complex technical and economic indica-tors]. Avtomatizatciy, telemehanizatciy i svyazj v neftynoy promuchlennosti. [Automation and remote control of the connection to the oil industry.], 2010. 7. PP. 1321.

16. Batalov S.A. Sintez sistemy upravleniy neftedobychej i controlya neftenasychennosti plasta na ranney stadii ego razrabotki. [Synthesis of control oil production and oil saturation control reservoir at an early stage of its development]. Mechatronica, Avtomatizatciy, Upravle-nie. [Mechatronics, Automation, Control.]. 2008. 6. PP. 3640.

17. Batalov S.A. Sintez mnogointervaljnoy struktury mnogo-svyaznyh systemy nefteizvlecheniy dly polnoy vyrabotki likvidiruemyh mestorogdeniy. [Synthesis of multi-hop structure of multiply oil recovery system for the full de-velopment of the liquidated deposits]. Avtomatizatciy, telemehanizatciy i svyazj v neftynoy promuchlennosti. [Automation and remote control of the connection to the oil industry]. 2008. 7. PP. 2735.

18. Batalov S.A. Avtomaticheskoe upravlenie tehnicheskimi sistemami. [Automatic control of technical systems]. Ufa: UGAES [Ufa: Publishing house UGAES], 2007. 300 p.

19. Batalov S.A. Mnogofunktsionaljny izmeriteljnyj com-plex. [Professional measuring station]. Pribory I sistemy. Upravleniy, kontrolj, diagnostika. [Devices and systems. Management, monitoring, diagnostics]. 2008. 11. PP. 3539

20. Batalov S.A. Telesistema registratcii i diagnosticheskogo analiza skvaginnyh objektov na neftepromyslah. [Telem-etry reception and diagnostic analysis of borehole facili-ties in the oil industry] [Pribory and systems. Control, Diagnostics]. Pribory i sistemy. Upravlenie, kontrolj, di-agnostika. [Devices and systems. Management, monitor-ing, diagnostics]. 2010. 7. PP. 4145.

. . ,

,

E-mail: geoavtsyst@mail.ru

Batalov Sergej AlekseevichCand. of Techn. Sciences, Associate ProfessorUfa State Academy of Economics and Service Ufa, Russian FederationE-mail: geoavtsyst@mail.ru

Information about the author

5 201450

.. . , , , E-mail: znakvicvolf@mail.ru

, 0. -, k- - , , - - . - , . -

. - .

: , , , - , , , - , -, , k-.

V.L. VoL'FSon Cand. of tech. Sciences, JSC Lanit Moscow, Russian Federation, E-mail: znakvicvolf@mail.ru

quANtItAtIVE INDICAtORS OF SOLutIONS OF DIOPHANtINE EquAtIONS AND SYStEMS IN tHE DOMAIN OF NAtuRAL NuMBERS

The paper shows that the asymptotic density of solutions of Diophantine equations or systems of the natural numbers is 0. The author gives the methods and studied estimates of the number, density and probability k-tuples to be the solution of algebraic equations for the first, second and higher orders of two or more variables, non-algebraic Diophantine equations and systems of Diophantine equations in the do-main of natural numbers. The paper contains the geometric proof of the estimate the number of solutions of Diophantine equations for two, three or more variables in the natural num-bers. The author proves the assertion about the number of

solutions of algebraic Diophantine equations of higher orders in the domain of natural numbers. The author provides esti-mates for the asymptotic behavior of quantitative solutions of Diophantine equations and systems in the domain of natural numbers.

Keywords: Diophantine equation, algebraic equation, the order of an algebraic equation, a non-algebraic equation, system, the number of solutions, the domain of natural numbers, the asymptotic density of the solutions, the prob-ability, k-tuples.

1. () . - , , , , - [1].

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5 2014 51

1

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. 2

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=

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= =

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5 201452

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5

f - , - x1 f N N

5 2014 53

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n n nk1 2, ,.. . (14) : x x xk10 20 0, ,... , 2 1k - :

x x xk10 20 0, ,... , x x xk10 20 0, ,... , x x xk10 20 0, ,... ,

,... x x x xk k10 20 10 0, ,... , , ... x x xk10 20 0, ,... .

. , x x12 22 1+ = - :

x x10 200 1= =, ; x x11 210 1= = , ;

x x13 231 0= =, ; x x14 241 0= =, ,

. -

- .

[6] -: Ax Bxy Cy N2 2+ + = , - -. - .

5 201454

, - .

, () () :

F x x a x a x aa x a x a( , )

,

1 2 11 1

2

22 2

2

12

13 1 23 2 3

2

2 2 3 0

= + + ++ + + =

(15)

aij - . -

2- . 2-

: ( ), , - [7].

, , :

D a a a= 11 22 122 . (16)

6

, F x x( , )1 2 0= - (D a a a= >11 22 122 0 ), - () ( 1 - ).

- , x x1 2, . , - x1 x2 - , - x1 . x2 , , - , - .

. , , - F x x( , )1 2 0= , () .

.

x x x x12 22 1 22 4 4 0+ + + = . (17)

(-) (17).

. (17) - D = = >1 1 0 1 0 , - . -

4 : ( , ), ( , ), ( , ), ( , )1 1 2 2 1 3 0 2 .

7

, F x x( , )1 2 0= - , () .

- , x x1 2, . , - x1 x2 - , x1 . , x2 , (15) () .

,

D a a a=

5 2014 55

. - , () , -, . , , , ( 1 ).

, - , D > 0 , - .

, - -, (15), - :

( / , / ) a a a a13 11 23 22 . (23)

, , () .

. :

2 3 4 12 14 012

2

2

1 2x x x x+ + + = . (24).

, (24) () .

. (24) :

2 1 3 2 012

2

2( ) ( )x x + + = . (25)

(25) , : x x1 21 2= = , . (25) .

, (15) F x x( , )1 2 0= . - , -. , - x1 .

, - (15) . - , A2 , A N=1 2, ... , - N .

- (15), , A2 2N ( 2 ). A2 2N , .

. :

x x x x12 22 1 22 2 0 + = . (26)

- (26) .

. (26) :

( ) ( )x x1 2 2 21 1 0 = . (27)

, (27) - , :

x x2 1= , (28)

x x2 12= . (29)

(28) - N - A2 . (29) - ( , )1 1 . , (29) A2 N +1 ( 2 - ).

, (15) x x x x12 22 1 22 2 3 0+ + = F x x( , )1 2 0= - . .

1. . x1 , x2 , : x b x b1 1 1 2= =, x c x c2 1 2 2= =, . b b1 2, c c1 2, , (15) 2N A2 ( 2 - ).

. :

x x12 13 2 0 + = . (30)

- (30) .

. (30) :( )( )x x1 11 2 0 = , : x x1 11 2= =, . , (30) - 2N A2 ( 2 ).

2. , x1 , x2 , : x a1 = x b2 = . a b, , (15) - N A2 ( 2 ).

. :

x x12 12 1 0 + = . (31)

.

5 201456

. (31) : ( )x1 21 0 = . (31) - x1 1= . , (31) - N A2 ( 2 ).

3. . (15) () ( 1 ).

. :

x x12 12 2 0 + = . (32)

- (32) .

. (32) :( )x1 21 1 0 + = , - () ( 1 ).

- . - (1) - :

F x x x a x a x a x

a x x a x x a x x

( , , )1 2 3 11 12

22 2

2

33 3

2

12 1 2 13 1 3 23 2 3

= + + +

+ + + +

++ + + + =2 2 2 014 1 24 2 34 3 44a x a x a x a .

(33)

(33) -. 17 14 , - , - [7]. , , - :

I a a a= + +11 22 33 , (34)

= + + +A A A A A11 22 33 44 , (35)

Aij aij D1

| |Ja aa a

a aa a

a aa a

=

+

+

11 1212 22

22 23

23 33

33 13

13 11 , (36)

| |Da a aa a aa a a

=

11 12 13

12 22 23

13 23 33

, (37)

| |A

a a a aa a a aa a a aa a a a

=

11 12 13 14

12 22 23 24

13 23 33 34

14 24 34 44

. (38)

, (38) | | A 0 .

8

(33) , F x x x( , , )1 2 3 0= (34) (38) (| | ,| | , | | )A D I D< >0 0 0 (- 1 ).

. - x x1 20, , . () - . - x x1 20, , , -. x3 , () .

1. (33), F x x x( , , )1 2 3 0=

(| | ,| | , | | )A D I D>

5 2014 57

9

(33), F x x x( , , )1 2 3 0= - : (| | ,| | , | | )A D I D< =0 0 - () .

. , -, , x x1 20, , , , . - x x1 20, , . - ( , )x x1 2 - () . - x x1 20, , ( , )x x1 2 . -, - () x3 . () (33) .

. :

2 1 012

2

2

3

2x x x+ = . (39)

- (39).

. (39) - , , - (39) :

x n n x n x n n1 2 2 3 21 2 1 1= + = + = + +, , ,,

n . ,

n 2 :

2 1 1 12 2n n n n n+ + < + + .

(39) A3 :

n n N2 1+ +

pi( ) [ / / ]B N NN = 0 0 .

10

(33), F x x x( , , )1 2 3 0= - (-) .

, , - , x x1 20, , , - . x x1 20, , . - ( , )x x1 2 () . x x1 20, , ( , )x x1 2 . , () x3 . (-) (33) .

3. k -, F x x xk( , ,... )1 2 0= - k - - .

. :

2 012

2

2

3

2x x x+ = . (40)

- (40).

. (40) - . , (40) - :

x m x m x m1 2 2 3 22 1 2 2 1 2= = = +, ( ) / , ( ) / ,

m . - (40) A3 , A N=1 2, ,... .

( ) / ( ) /2 1 2 2 1 2 22 2m m m+ > > > ,

m N N

5 201458

, (33) , .. . , (33) - :

a a a a33 13 23 34 0= = = = .

6 : , , , - , , - [7]. - (33) .

11

(33), - A A N3 1 2, , ...= pi( )B kNN = , - < >x x x1 2 3, ,

Pr B k NN( ) /= 2 , (41)

k ( 2 ).

, x x1 20, , - . , F x x( , )1 2 0= , - - - k . x x1 20, , , F x x( , )1 2 0= . A3 - N (33), k kN -. , pi( )B kNN = , < >x x x1 2 3, , Pr B kN N k NN( ) / /= =3 2 .

. - :

x x x x12 22 1 22 4 4 0+ + + = . (42)

- (42).

. (42) x3 , -. (42) :( ) ( )x x1 2 2 21 2 1 + + = , . : x x1 21 3= =, . , k =1 . - pi( )B NN = ,

Pr B N N NN( ) / /= =3 21 ( 2 ).

12

, (33) , A3 - pi( )B NN 2 , , < >x x x1 2 3, ,

Pr B NN( ) /1 (43)

( 2 ).

, x x1 20, , -. F x x( , )1 2 0= , - -. - A2 N . x x1 20, , , F x x( , )1 2 0= . - A3 N (33), N - N 2 . , pi( )B NN 2 , - < >x x x1 2 3, , -

Pr B N N NN( ) / / =2 3 1 .

. -

x x x12 1 222 0+ = . (44)

- (44).

. , - . (1,1). :

x x x x x xn n n n n n1 1 1 2 2 1 1 23 4 1 2 3 1+ += + + = + +, , (45)

.. . , - (45) 1, A2 , A N=1 2, ,... , N . ,

(44) N 2 - A3 , < >x x x1 2 3, , 1 / N( 2 ).

5 2014 59

(33) - , , , .

. :

x x x x12 22 1 22 2 3 0+ + = . (46)

- (46) .

. (46) :

( ) ( )x x1 2 2 21 1 1 + = . (47)

(47) , - .

, (33) - , : x x2 30, , . - A3 N 2 < >x x x1 2 3, , Pr B N N NN( ) / /= =2 3 1 .

. :

x x12 12 1 0 + = . (48)

- (48) .

. (48) :

( )x1 21 0 = . (49)

(49) x1 1= , N 2 A3 .

13

, (33), F x x x( , , )1 2 3 0= - , A3 - pi( )B N kNN = +2 -, k . < >x x x1 2 3, ,

Pr B N k NN( ) / / +1 2 (50).

( 2 ).

, , - x x1 20, , , . , , F x x( , )1 2 0= -. N k+ A2 . - x x1 20, , , - F x x( , )1 2 0= . -

N, A3 (33) pi( )B N kNN +2 , < >x x x1 2 3, ,

Pr B N kN N N k NN( ) ( ) / / / + = +2 3 21 .

. :

x x x x12 22 1 22 2 0 + = (51).

- (51) A3 .

. (51) :

( ) ( )x x1 2 2 21 1 0 = . (52)

, (52) :

x x2 1= , (53)

x x2 12= . (54) (53) N -

A2 . (54) - ( , )1 1 . , (51) A3 N N( )+1 ( 2 - ).

14

, (33),

F x x x( , , )1 2 3 0= (55)

, A3 pi( )B NN 2 2 . < >x x x1 2 3, ,

Pr B NN( ) /2 . (56)

( 2 ).

, , - x x1 20, , , . , , F x x( , )1 2 0= , 2N A2 .

x x1 20, , , F x x( , )1 2 0= . - N , - A3 (55) pi( )B NN 2 2 , < >x x x1 2 3, ,

5 201460

Pr B N N NN( ) ( ) / / =2 22 3 , (56).

. :

x x12 13 2 0 + = . (57)

- (57) .

. (57) : ( )x1 21 1 0 + = ( )( )x x1 11 2 0 = , - : x x1 11 2= =, . , (57) 2 2N A3 ( 2 - ).

, (33), F x x x( , , )1 2 3 0= , (33) , - , .

. :

x x12 12 2 0 + = . (58)

- (58) .

. (58) :( )x1 21 1 0 + = , - (- 1 ).

- F x xk( ,... )1 0= F x xk( ,... )1 0= . - - - Ak , A N=1 2, ,... , .

-, : ( )x x x2 2 13 121 2 = . : x x x x13 12 12 12 2 0 = ( ) . - x1 2 , - : x x1 20 1= =, A2 .

15

(1) Ak Nk1 ,

x f x x x x i ki i i i k= +( ,... , ,... ),1 1 1 1 ,

(1).( 2 ).

, -

x f x x x xi i i i k= +( ,... , ,... )1 1 1 ,

- x x x xi i k1 1 1,... , ,... + Ak - xi .

x x x xi i k1 1 1,... , ,... + F (1), k 1 N Ak . xi -, Nk1 .

-

x x x xi i k1 1 1,... , ,... +

F - (1), - N Ak , .. Nk1 .

(7) , , , , - (1) Ak .

,

x f x x x x i ki i i i k= +( ,... , ,... ),1 1 1 1 ,

n (1) F x xk( ,... )1 0= , - , - (1) Ak , A N=1 2, ,... , - Nk1 .

16

- n (1) Ak , A N=1 2, ,... , :pi( )B nNN k 1 , P B n NN( ) / , 0 .

15 , -. , - x f x x x x i ki i i i k= +( ,... , ,... ),1 1 1 1 , - n F x xk( ,... )1 0= , .

k - n -, , n . , , - , -, n . Ak Nk1 ,

5 2014 61

nN k1 . n F x xk( ,... )1 0= Ak , A N=1 2, ,... , nN k1 .

, - k - , F x xk( ,... )1 0= , -, , - k 1 , . - Ak ( )k N k 1 1 . n k> 1 , ( )k N nNk k . - x xk1,... F x xk( ,... )1 , - F x xk( ,... )1 0> - (59) . ) 0 , F ( ,... )0 0 0= , .. (59) x xk1 0 0= =,... . x xk1,... F x xk( ,... )1 , - F x xk( ,... )1 0> , (59) - .

2. , F ( ,... )0 0 0< . F x xk( ,... )1 0 - . - - (59) .

(59) - :

x x x12 22 32 3 0+ + = . (60)

(60) - x x x1 2 3 1= = = .

3. . :

x xm n2 1 0 = , (61)

m n, m n> >1. A2 .

. (61) - :x t x tl n l m1 2= =/ /, , l n m, . m n> , t 2 x x1 2> . , - (61) A2 : t Nl n/ < .

pi( ) [ ]/B NN n l< , (62)

[ ] . -

, .

, , F x x( , )1 2 0= , -

5 201462

A2 , , - ?

. - . - N - A2 . n , A2

n N . (63)

, . - n - n - - n . - F x x( , )1 2 0= - A2 (63).

.

F x x x x x( , )1 2 13 12 26 11 6 0= + = , (64)

3- -. - (64) A2 .

. (64) :

( )( )( )x x x1 2 31 2 3 0 = . (65)

- : x x x1 1 11 2 3= = =, , . , (65) - A2 pi( )B NN = 3 .

F x xk( ,... )1 0= - k - , - -. Nk1 - Ak . - n , - F x xk( ,... )1 0= Ak

n N k 1 . (66)

, ,