4. CONTENTS
Mathematics
ADUKOV V.M. About Wiener–Hopf factorization of functionally commutative matrix functions.. 6
KOLYASNIKOV S.A. Units of integral group rings of finite groups with a direct multiplier of
order 3.................................................................................................................................................. 13
LESCHEVICH V.V., PENYAZKOV O.G., ROSTAING J.-C., FEDOROV A.V., SHULGIN A.V.
Experimental and mathematical simulation of auto-ignition of iron micro particles .......................... 21
MARTYUSHEV E.V. Algorithmic solution of the five-point pose problem based on the cayley
representation of rotation matrices....................................................................................................... 31
MEDVEDEV S.V. About extension of homeomorphisms over zero-dimensional homogeneous
spaces................................................................................................................................................... 39
OMELCHENKO E.A., PLEKHANOVA M.V., DAVYDOV P.N. Numerical solution of delayed
linearized quasistationary phase-field system of equations................................................................. 45
RUBINA L.I., UL’YANOV O.N. Towards the differences in behaviour of solutions of linear and
non-linear heat-conduction equations.................................................................................................. 52
SOLDATOVA N.G. About optimum on risks and regrets situations in game of two persons ........... 60
TABARINTSEVA E.V. About solving of an ill-posed problem for a nonlinear differential equa-
tion by means of the projection regularization method........................................................................ 65
TANANA V.P., ERYGINA A.A. About the evaluation of inaccuracy of approximate solution of
the inverse problem of solid state physics ........................................................................................... 72
UKHOBOTOV V.I., TROITSKY A.A. One problem of pulse pursuit............................................... 79
USHAKOV A.L. Updating iterative factorization for the numerical solution of two elliptic equa-
tions of the second order in rectangular area ....................................................................................... 88
Physics
VORONTSOV A.G. Structure amendments in heated metal clusters during its process.................... 94
GUREVICH S.Yu., GOLUBEV E.V., PETROV Yu.V. Lamb waves in ferromagnetic metals: laser
exitation and electromagnetic registration........................................................................................... 99
RIDNYI Ya.M., MIRZOEV A.A., MIRZAEV D.A. Ab-initio simulation of influence of short-
range ordering carbon impurities on the energy of their dissolution in the FCC-iron......................... 108
USHAKOV V.L., PYZIN G.P., BESKACHKO V.P. A method for monitoring small surface
movements of a sessile drop during evaporation................................................................................. 117
Short communications
BEREZIN I.Ya., PETRENKO Yu.O. Problems of vibration safety of operator of industrial tractor . 123
BOLSHAKOV M.V., GUSEVA A.V., KUNDIKOVA N.D., POPKOV I.I. Fiber and interferential
method of obtaining non-homogeneous polarized beam..................................................................... 128
BOLSHAKOV M.V., GUSEVA A.V., KUNDIKOVA N.D., POPKOV I.I. Transformation of the
spin moment into orbital moment in laser beam.................................................................................. 133
BOLSHAKOV M.V., KOMAROVA M.A., KUNDIKOVA N.D. Experimental determination of
multimode optical fiber radiation mode composition.......................................................................... 138
BRYZGALOV A.N., VOLKOV P.V. Influence of heat-treating on microhardness of silica glass
KU-1 .................................................................................................................................................... 143
GERASIMOV A.M., KUNDIKOVA N.D., MIKLYAEV Yu.V., PIKHULYA D.G.,
TERPUGOV M.V. Efficiency of second harmonic generation in one-dimentional photonic crystal
from isotropic material......................................................................................................................... 147
IVANOV S.A. Stability of two-layer recursive neural networks ........................................................ 151
KARITSKAYA S.G., KARITSKIY Ya.D. Fluorescent impurity centers for monitoring the physical
state of polymers and a computer simulation of processes of structuring of the polymer matrices .... 155
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KATKOV M.L. Existence of the fixed point in the case of evenly contractive monotonic operator.. 160
LEYVI A.Ya. Analysis of mechanisms of surface pattern formation during the high-power plasma
stream processing................................................................................................................................. 162
MATVEEVA L.V. Behavior of Gronwall polynomials outside the boundary of summability region 166
KHAYRISLAMOV K.Z. Poiseuille flow of a fluid with variable viscosity....................................... 170
KHAYRISLAMOV M.Z., HERREINSTEIN A.W. Explicit scheme for the solution of third
boundary value problem for quasi-linear heat equation....................................................................... 174
EVNIN A.Yu. Polynomial as a sum of periodic functions.................................................................. 178
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( )A t , ( )A t
- . – - ,
[6]. -
- -
– -
.
2.
1. G – G n= , e – G , [ ]G – ,
( )
g G
a g g
∈
G
( )a g . G [ ]G g
1 g⋅ . G – [ ]G . -
G , [ ]G
. [ ]G ( )a g
.
1 2{ }, , , sK e K K= – G , j jh K= – -
jK , 1, , sg g – .
( [ ])Z G G , ( )a g , -
.
j
j
g K
C g
∈
= , 1, ,j s= ( [ ])Z G .
1
s
k
i j ij k
k
C C c C
=
= ,
k
ijc – ( [ ])Z G .
A –
1
( [ ]), ( ),
s
i i i i
i
a a C Z G a a g
=
= ∈ =
( [ ])Z G . A 1, , sC C .
1 , 1
s s
k
j i i j i ij k
i i k
aC a C C a c C
= =
= = ,
kjA A
1
s
k
kj i ij
i
A a c
=
=
1 , 1
.
ss
k
i ij
i k j
A a c
= =
= (2)
, G – , s n= , ( [ ]) [ ]Z G G= , A
( ) [ ]
g G
a a g g G
∈
= ∈ { }1 2, , , ng e g g=
1 1
1 1 2 1
1 1
2 2 2 2
1 1
2
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
n
n
n n n n
a g a g g a g g
a g a g g a g g
A
a g a g g a g g
− −
− −
− −
= . (3)
, G .
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, G n a , ,
{ }1
, , , n
e a a −
, .
[ ]G ( ( [ ])Z G ) -
(3) ( (2)). [ ]G , ( [ ])Z G -
G .
– I .
[ ]G⊗ (3) ( )ka g ∈ . -
, ( [ ])Z G⊗ (2) ia ∈ . sI -
s I .
1. 1, , sχ χ – -
G 1, , sn n – .
1 1 1 2 1 2 1
1 2 1 2 2 2 2
1 1 2 2
( ) ( ) ( )
( ) ( ) ( )1
( ) ( ) ( )
s s
s n
s s s s s
h g I h g I h g I
h g I h g I h g I
n
h g I h g I h g I
χ χ χ
χ χ χ
χ χ χ
= .
– ,
1 1 2 1 1
1 1 2 2 2 2
1 2
( ) ( ) ( )
( ) ( ) ( )1
( ) ( ) ( )
s
s
s s s s
g I g I g I
g I g I g I
n
g I g I g I
χ χ χ
χ χ χ
χ χ χ
−
= ,
( [ ])A Z G∈ ⊗ :
1
A −
= Λ ,
1diag [ , ]sλ λΛ = , ,
1
( ) ( ), 1, ,j j
j g G
a g g j s
n
λ χ
∈
= = .
. 1−
:
( )1
1
,1 1
( ) ( ) ( ) ( )
0,| |
s
k i k j k i j ij
ij
k g G
I i j
h g g I g g I I
i jn G
χ χ χ χ δ−
= ∈
=
= = = =
≠
. (4)
( . [7,
.3, § 4, 2]) , G . -
(4) , 1
sI−
= .
1
A −
, (2)
, 1−
:
( )1 1
, 1 , , 1
1 1
( ) ( ) ( ).
s s
k
ik kl lj k m ml i k j l
ij
l k l k m
A A h a c g g
n n
χ χ− −
= =
= =
( . [8, §109, (8)]), -
:
1
( ) ( ) ( )
s
k m l
k ml i k i m i l
ik
h h
h c g g g
n
χ χ χ
=
= .
, – .
,
( )1
, 1 1 1
1 1
( ) ( ) ( ) ( ) ( ) ( ).
s s s
m l m
m j l i m i l m i m l i l j lij
i il m m l
h h h
A a g g g a g h g g
n n n n
χ χ χ χ χ χ−
= = =
= =
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, (4),
:
( )1
1
( ).
s
ij
m m i m
ij
i m
A a h g
n
δ
χ−
=
=
– , -
G . , -
( )1
ij i
ij
A δ λ−
= 1−
= Λ . .
, [ ]A G∈ ⊗ , G – , 1
1 1 1s sh h n n= = = = = = | |s G= .
, A , 1, , sλ λ –
.
2. (1). – -
Γ , – . -
, ( )W
( )H Γ Γ ( . [1]).
( )a g ( )ga t t∈ ∈Γ . - ( )A t
,
1
( ) ( ) ( ), 1, ,j g j
j g G
t a t g j s
n
λ χ
∈
= = Γ .
ind ( )j j tρ λΓ= – Γ ( )j tλ , -
2π , t Γ .
( ) ( ) ( )j
j j jt t t t
ρ
λ λ λ− +
= – – ( )j tλ .
1 1( ) diag [ ( ) , ( )] diag [ ( ) , ( )]s st t t t tλ λ λ λ− − + +
Λ = , ⋅ ,
– – - ( )tΛ 1
.
2. ( ) ( [ ])A t Z G∈ ⊗ – Γ -
(2). – ( ) ( ) ( ) ( )A t A t d t A t− += :
1 1 1 2 2 1 1
1 1 2 2 2 2 2
1 1 2 2
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( )
s s
s s
s s s s s
t g t g t g
t g t g t g
A t
t g t g t g
λ χ λ χ λ χ
λ χ λ χ λ χ
λ χ λ χ λ χ
− − −
− − −
−
− − −
= ,
1( ) diag [ ] ind ( ),s
j jd t t t tρρ
ρ λΓ= , , , =
1 1 1 1 2 1 1 2 1 1
1 2 2 1 2 2 2 2 2 2
1 1 2 2
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( )
s s
s s
s s s s s s s s
h t g h t g h t g
h t g h t g h t g
A t
h t g h t g h t g
λ χ λ χ λ χ
λ χ λ χ λ χ
λ χ λ χ λ χ
+ + +
+ + +
+
+ + +
= .
3.
1. 4G V= – . -
4S :
{ }4 ,(12)(3 4),(13)(2 4),(14)(2 3)V e= ,
2 2C C× . ( )A t
2- , 2 2× - ,
2 2×
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3 41 2
4 32 1
3 4 1 2
4 3 2 1
( ) ( )( ) ( )
( ) ( )( ) ( )
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
a t a ta t a t
a t a ta t a t
A t
a t a t a t a t
a t a t a t a t
= .
4V ( . [7, .3, § 5])
e (12)(3 4) (13)(2 4) (14)(2 3)
1χ 1 1 1 1
2χ 1 1− 1 1−
3χ 1 1 1− 1−
3χ 1 1− 1− 1
, ( )A t
:
1 1 2 3 4( ) ( ) ( ) ( ) ( )t a t a t a t a tλ = + + + , 2 1 2 3 4( ) ( ) ( ) ( ) ( )t a t a t a t a tλ = − + − ,
3 1 2 3 4( ) ( ) ( ) ( ) ( )t a t a t a t a tλ = + − − , 4 1 2 3 4( ) ( ) ( ) ( ) ( )t a t a t a t a tλ = − − + .
( )A t .
2. 3G S= – 3. -
3 :
{ } { } { }1 2 3, (12),(13), (2 3) , (12 3),(13 2 .K e K K= = =
jC 3( [ ])Z S :
1 2 1C 2C 3C
1C 1C 2C 3C
2C 2C 1 33 3C C+ 22C
3C 3C 22C 1 32C +
(1) - ( )A t :
1 2 3
2 1 3 2
3 2 1 3
( ) 3 ( ) 2 ( )
( ) ( ) ( ) 2 ( ) 2 ( ) .
( ) 3 ( ) ( ) ( )
a t a t a t
A t a t a t a t a t
a t a t a t a t
= +
+
3S ( . [7, .3, § 5])
e (12) (12 3)
1χ 1 1 1
2χ 1 1− 1
3χ 2 0 1−
, 1 1 2 3( ) ( ) 3 ( ) 2 ( )t a t a t a tλ = + + , 2 1 2 3( ) ( ) 3 ( ) 2 ( )t a t a t a tλ = − + , 3 1 3( ) ( ) ( )t a t a tλ = − ,
1
1 3 2 1 1 2
1 1
1 3 2 , 1 1 0
6 6
2 0 2 1 1 1
−
= − = −
− −
.
2
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1 2 3 1 1 1
1 2 2 2 2
1 2 3 3 3
( ) ( ) 2 ( ) ( ) 3 ( ) 2 ( )
( ) ( ) ( ) 0 , ( ) ( ) 3 ( ) 2 ( )
( ) ( ) ( ) 2 ( ) 0 2 ( )
t t t t t t
A t t t A t t t t
t t t t t
λ λ λ λ λ λ
λ λ λ λ λ
λ λ λ λ λ
− − − + + +
− − + + +
− +
− − − + +
= − = −
− −
,
1 1 2 3ind ( ( ) 3 ( ) ( ))a t a t a tρ Γ= + + , 2 1 2 3ind ( ( ) 3 ( ) ( ))a t a t a tρ Γ= − + , 3 1 3ind ( ( ) ( ))a t a tρ Γ= − .
3. { }8 1, i, j, kG Q= = ± ± ± ± – ,
2 2 2
i j k ijk 1= = = = − . 5 :
{ } { } { } { } { }1 2 3 4 51 , 1 , i , j , k .K K K K K= = − = ± = ± = ±
jC 8( [ ])Z Q
1 2 1C 2C 3C 4C 5C
1C 1C 2C 3C 4C 5C
2C 2C 1C 3C 4C 5C
3C 3C 3C 1 22 2C + 52C 42C
4C 4C 4C 52C 1 22 2C + 32C
5C 5C 5C 42C 32C 1 22 2C +
1 2 3 4 5
2 1 3 4 5
3 3 1 2 5 4
4 4 5 1 2 3
5 5 4 3 1 2
( ) ( ) 2 ( ) 2 ( ) 2 ( )
( ) ( ) 2 ( ) 2 ( ) 2 ( )
( ) ( ) ( ) ( ) 2 ( ) 2 ( )( )
( ) ( ) 2 ( ) ( ) ( ) 2 ( )
( ) ( ) 2 ( ) 2 ( ) ( ) ( )
a t a t a t a t a t
a t a t a t a t a t
a t a t a t a t a t a tA t
a t a t a t a t a t a t
a t a t a t a t a t a t
+=
+
+
.
8Q :
1 1− i± j± k±
1χ 1 1 1 1 1
2χ 1 1 1 –1 –1
3χ 1 1 –1 1 –1
4χ 1 1 –1 –1 1
5χ 2 –2 0 0 0
( )A t . -
.
1
( ) ( ) ( ), 1, ,5j g j
j g G
t a t g j
n
λ χ
∈
= = , :
1 1 2 3 4 5( ) ( ) ( ) 2 ( ) 2 ( ) 2 ( )t a t a t a t a t a tλ = + + + + ,
2 1 2 3 4 5( ) ( ) ( ) 2 ( ) 2 ( ) 2 ( )t a t a t a t a t a tλ = + + − − ,
3 1 2 3 4 5( ) ( ) ( ) 2 ( ) 2 ( ) 2 ( )t a t a t a t a t a tλ = + − + − ,
4 1 2 3 4 5( ) ( ) ( ) 2 ( ) 2 ( ) ( )t a t a t a t a t a tλ = + − − + , 5 1 2( ) ( ) ( )t a t a tλ = − .
( )A t .
1. , . . / . . ,
. . . – M.: , 1971. – 352 .
2. Adukov, V.M. On Wiener–Hopf factorization of meromorphic matrix functions / V.M. Adukov //
Integral Equations and Operator Theory. – 1991.– V. 14. – P. 767–774.
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3. , . . – - / . .
// . – 1992. – . 4. – . 1. – . 54–74.
4. , . . - / . . // . -
. . – 1999. – . 118, 3. – . 324–336.
5. , . . – - / . .
// . – 2009. – . 200, 8. – . 3–24.
6. , . . n / . . // -
. . – 1952. – . 7. – . 4(50). – . 3–54.
7. , . . . III / . . . – M.: , 2000. –
272 .
8. , . . / . . . – .: « », 2004.– 624 .
ABOUT WIENER–HOPF FACTORIZATION OF
FUNCTIONALLY COMMUTATIVE MATRIX FUNCTIONS
V.M. Adukov
1
An algorithm of an explicit solution of the Wiener–Hopf factorization problem is proposed for func-
tionally commutative matrix functions of a special kind. Elementary facts of the representation theory of
finite groups are used. Symmetry of the matrix function that is factored out allows to diagonalize it by a
constant linear transformation. Thus, the problem is reduced to the scalar case.
Keywords: Wiener–Hopf factorization, special indexes, finite groups.
References
1. Gokhberg I.Ts., Fel'dman I.A. Uravneniya v svertkakh i proektsionnye metody ikh resheniya
(Convolution equations and projection methods for their solution). Moscow: Nauka, 1971. 352 p. (in
Russ.). [Gohberg I.C., Fel dman I.A. Convolution equations and projection methods for their solution.
American Mathematical Society, Providence, R.I., 1974. Translations of Mathematical Monographs,
Vol. 41. MR 0355675 (50 #8149) (in Eng.).]
2. Adukov V.M. On Wiener–Hopf factorization of meromorphic matrix functions. Integral Equa-
tions and Operator Theory. 1991. Vol. 14. pp. 767–774. DOI: 10.1007/BF01198935.
3. Adukov V.M. Faktorizatsiya Vinera–Khopfa meromorfnykh matrits-funktsiy (Wiener-Hopf fac-
torization of meromorphic matrix function). Algebra i analiz. 1992. Vol. 4. Issue 1. pp. 54–74. (in
Russ.). [Adukov V.M. Wiener–Hopf factorization of meromorphic matrix functions. St. Petersburg
Mathematical Journal. 1993. Vol. 4. Issue 1. pp. 51–69. (in Eng.).]
4. Adukov V.M. O faktorizatsii analiticheskikh matrits-funktsiy (About factorization of analytical
matrix functions) Teoreticheskaya i matematicheskaya fizika. 1999. Vol. 118, no. 3. pp. 324–336. [Adu-
kov V.M. Factorization of analytic matrix-valued functions. Theoretical and Mathematical Physics.
1999. Vol. 118, no. 3. pp. 255–263. DOI: 10.1007/BF02557319].
5. Adukov V.M. Faktorizatsiya Vinera–Khopfa kusochno meromorfnykh matrits-funktsiy (Piece-
wise Wiener–Hopf factorization of meromorphic matrix functions). Matematicheskiy sbornik. 2009.
Vol. 200, no. 8. pp. 3–24. (in Russ.).
6. Gakhov F.D. Kraevaya zadacha Rimana dlya sistemy n par funktsiy (Riemann boundary value
problem for system of n pairs of functions). Uspekhi matematicheskikh nauk. 1952. Vol.7. Issue 4(50).
pp. 3–54. (in Russ.).
7. Kostrikin A.I. Vvedenie v algebru. Chast III. (Introduction into algebra. Part III). Moscow: Fiz-
matlit, 2000. 272 p. (in Russ.).
8. Van der Varden B.L. Algebra. Saint Petersburg: Lan', 2004. 624 p. (in Russ.).
18 2013 .
1
Adukov Victor Mikhailovich is Dr. Sc. (Physics and Mathematics), Professor, Department of Mathematical Analysis, South Ural State Uni-
versity.
E-mail: victor.m.adukov@gmail.com
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13. 2013, 5, 2 13
512.552.7
3
. . 1
A×Z3, A 3. -
(9,3), (9,3,3) (15,3).
: , , -
.
1.
G A b= × – , b – 3, A
a 3. ( )V G -
G .
:
0 : ( ) ( ),V G V Aϕ → ( ),x x x A∀ ∈ 1b ;
1 : ( ) ( ) ( / ),V G V A V G abϕ → ≅ ( ),x x x A∀ ∈ 2
b a ;
2
2 : ( ) ( ) ( / ),V G V A V G a bϕ → ≅ ( ),x x x A∀ ∈ b a ;
3 : ( ) ( ), ( ))V G V G a g g a g Gϕ → ∀ ∈ .
A -
a , H . -
: , ( ).G a G h a h h H→ ∀ ∈
1. ( )( ) ( )V G b V A= × K ,
31 2 11 12 2 2 2 2
3 3 3
1 (1 ) (1 ) (1 )K ab a b a b ab a a G
υυ υ
υ
−− −
= = + + + + + + + + + ∩ , 1 1( )ϕ υ υ= ,
2 2( ) (1 ( )) ( )I a V Aϕ υ υ= ∈ + ∩ , 3 3( ) (1 ( )) ( )I b V G aϕ υ υ= ∈ + ∩ . ( )I a – A ,
1a − , 2
1a − ; ( )I b – G a , -
1b − , 2
1b − ( . [1]).
. 1 [2] ( )( ) ( )V G b V A= × K ,
( )1 ( ) ( ) ( )K A A A b V G= + ∩ , ( ) ( )A A A b – , -
G ( 1)( 1)x b− − , 2
( 1)( 1)x b− − {1}x A∈ . -
, K .
Kυ ∈
2 2
2 2
1h ha h haha ha
h H h H h H h H h H h H
h ha ha h ha ha bυ α α α β β β
∈ ∈ ∈ ∈ ∈ ∈
= + + ⋅ + + + ⋅ +
2
2 2
h ha ha
h H h H h H
h ha ha bγ γ γ
∈ ∈ ∈
+ + + ⋅ .
1 2, ( )V Aυ υ ∈ 3 ( )V G aυ ∈
2
2
1 h ha ha
h H h H h H
h ha haυ λ λ λ
∈ ∈ ∈
= + + ,
1
– , , - -
.
E-mail: koliasnikovsa@susu.ac.ru
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2
2
2 h ha ha
h H h H h H
h ha haυ μ μ μ
∈ ∈ ∈
= + + ,
2
2
3 h hb hb
h H h H h H
h a hb a hb aυ ν ν ν
∈ ∈ ∈
= + + .
, 0 ( ) 1ϕ υ = ( . . Kυ ∈ ) 1 1( )ϕ υ υ= , 2 2( )ϕ υ υ= , 3 3( )ϕ υ υ= ,
.
2
2
2
1 0 0 1 0 0 1 0 0
0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
1 0 0 0 1 0 0 0 1
0 1 0 0 0 1 1 0 0
0 0 1 1 0 0 0 1 0
1 0 0 0 0 1 0 1 0
0 1 0 1 0 0 0 0 1
0 0 1 0 1 0 1 0 0
1 1 1 0 0 0 0 0 0
0 0 0 1 1 1 0 0 0
0 0 0 0 0 0 1 1 1
h
ha
ha
h
ha
ha
h
ha
ha
α
α
α
β
β
β
γ
γ
γ
2
2
2
0
0
h
h
ha
ha
h
ha
ha
h
hb
hb
ε
λ
λ
λ
μ
μ
μ
ν
ν
ν
= ,
1, 1;
0, 1.
h
h
h
ε
=
=
≠
, , 1υ ,
2υ , 3υ
2h ha hha
λ λ λ ε+ + = ; 2h ha hha
λ λ λ ε+ + = ; 2h ha hha
λ λ λ ε+ + = .
, 1 2, (1 ( )) ( )I a V Aυ υ ∈ + ∩ , 3 (1 ( )) ( )I b V G aυ ∈ + ∩ .
, ,
( ) 3h h h hα λ μ ν= + + ; ( ) 3ha ha ha h hα λ μ ν ε= + + − ; 2 2 2( ) 3h hha ha ha
α λ μ ν ε= + + − ;
2( ) 3h ha hbha
β λ μ ν= + + ; 2( ) 3h h hb hha
β λ μ ν ε= + + − ; 2 ( ) 3ha h hb hha
β λ μ ν ε= + + − ;
2 2( ) 3h ha ha hb
γ λ μ ν= + + ; 2 2( ) 3ha h hha hb
γ λ μ ν ε= + + − ; 2 2( ) 3h ha hha hb
γ λ μ ν ε= + + − .
, υ ,
,
31 2 11 12 2 2 2 2
3 3 3
1 (1 ) (1 ) (1 )ab a b a b ab a a
υυ υ
υ
−− −
= + + + + + + + + + . (1)
: ( ) ( ) ( )K V A V A V G aϕ → × × ,
( )1 2 3( ) ( ), ( ), ( )ϕ υ ϕ υ ϕ υ ϕ υ= .
.
,
( ) ( )1 2 3 1 1 2 2 3 3( ) ( ), ( ), ( ) ( ) ( ), ( ) ( ), ( ) ( )w w w w w w wϕ υ ϕ υ ϕ υ ϕ υ ϕ υ ϕ ϕ υ ϕ ϕ υ ϕ= = .
,
2 2 2
(1 ) 3(1 )a a a a+ + = + + ,
2 2 2 2 2
(1 ) 3(1 )ab a b ab a b+ + = + + ,
2 2 2 2 2
(1 ) 3(1 )a b ab a b ab+ + = + + ,
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2 2 2 2
(1 )(1 ) (1 )(1 )a a b b b b a a+ + + + = + + + + =
2 2 2 2 2 2 2 2
(1 )(1 ) (1 )(1 )a b ab ab a b ab a b a b ab= + + + + = + + + + =
2 2 2 2 2 2
(1 )(1 ) (1 )(1 )a a ab a b ab a b a a= + + + + = + + + + =
2 2 2 2 2 2
(1 )(1 ) (1 )(1 ).a b ab a a a a a b ab= + + + + = + + + +
, 1 21, 1 ( )w w I a− − ∈ , 2 1 ( )w I b− ∈ ,
2 2 2
1 2 3( 1)(1 ) ( 1)(1 ) ( 1)(1 ) 0w a a w a a w b b− + + = − + + = − + + = .
( )
( )
31 2
31 2
3 31 1 2 2
11 12 2 2 2 2
3 3 3
11 12 2 2 2 2
3 3 3
11 12 2 2 2 2
3 3 3
1 (1 ) (1 ) (1 )
1 (1 ) (1 ) (1 )
1 (1 ) (1 ) (1 ).
ww w
ww w
w ab a b a b ab a a
ab a b a b ab a a
ab a b a b ab a a
υυ υ
υυ υ
υ
−− −
−− −
−− −
= + + + + + + + + + ×
× + + + + + + + + + =
= + + + + + + + + +
, 3 3 3 3w wυ υ≠ , , 2 2
3 3 3 3(1 ) (1 )w a a w a aυ υ+ + = + + .
, , ϕ -
. (1), . -
.
.
2. (9,3)
9 3
| 1 | 1G a a b b= = × = . ( )V a [3].
[2].
1 2( ) ,V a a F F u u= × = × ,
8 2 7
1 1 ( ) ( )u a a a a= − + + + , 1 8 3 6 4 5
1 1 ( ) ( ) ( ),u a a a a a a−
= − − + + + + +
2 7 4 5
2 1 ( ) ( ),u a a a a= − + + + 1 2 7 3 6 8
2 1 ( ) ( ) ( )u a a a a a a−
= − − + + + + + .
1
1 2( )V G a b u u K= × × × × ,
Kυ ∈
3 6 2 6 3 21 21 1
1 (1 ) (1 )
3 3
a b a b a b a b
υ υ
υ
− −
= + + + + + + ,
3
1 2, (1 ( )) ( )I a V aυ υ ∈ + ∩ , 3
( )V G a – ( . [3]) 3( ) 1ϕ υ = .
8
0
i
ii
aλ=
,
8
0
i
ii
aμ=
,
( 2 3 4
0 0 1 1 2 2 3 3 4 4
5 6 7 8
5 5 6 6 7 7 8 8 6 3 7 4
2 3 4 5 6
8 5 0 6 1 7 2 8 3 0
4 1
1 3 (1 ) 1 ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( 1) ( ) ( ) ( 1)
( )
a a a a
a a a a b ab
a b a b a b a b a b
a
υ λ μ λ μ λ μ λ μ λ μ
λ μ λ μ λ μ λ μ λ μ λ μ
λ μ λ μ λ μ λ μ λ μ
λ μ
= + + ⋅ + + ⋅ + + ⋅ + + ⋅ + + ⋅ +
+ + ⋅ + + ⋅ + + ⋅ + + ⋅ + + ⋅ + + ⋅ +
+ + ⋅ + + − ⋅ + + ⋅ + + ⋅ + + − ⋅ +
+ + ⋅
)
7 8 2 2 2 2
5 2 3 6 4 7 5 8
3 2 4 2 5 2 6 2
6 0 7 1 8 2 0 3
7 2 8 2
1 4 2 5
( ) ( ) ( ) ( )
( 1) ( ) ( ) ( 1)
( ) ( ) .
b a b b ab a b
a b a b a b a b
a b a b
λ μ λ μ λ μ λ μ
λ μ λ μ λ μ λ μ
λ μ λ μ
+ + ⋅ + + ⋅ + + ⋅ + + ⋅ +
+ + − ⋅ + + ⋅ + + ⋅ + + − ⋅ +
+ + ⋅ + + ⋅
(2)
, 3
1 2, (1 ( ))u u I a∈ + ,
2 6 6
1 1 ( 1) ( 1)u a a a a= − − + − , 6 3 2 3
2 1 ( 1) ( 1) ( 1)u a a a a a a= − − + − + − .
: K F Fϕ → × ,
( )1 2( ) ( ), ( )ϕ υ ϕ υ ϕ υ= .
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1 2,u u
1
1 8
a a+ 2 7
a a+ 3 6
a a+ 4 5
a a+
1u 1 –1 1 0 0
2
1u 5 –4 3 –2 1
3
1u 19 –18 15 –9 3
2u 1 0 –1 0 1
2
2u 5 1 –4 –2 3
3
2u 19 3 –18 –9 15
1 2u u –1 0 1 1 –1
2
1 2u u –5 –1 5 3 –4
2
1 2u u 1 1 –2 0 –1
2 2
1 2u u 7 1 –6 –3 5
. 1 (2) ,
2 3 3 3
1 2 1 2( ,1),( ,1),(1, ),(1, ) ( )u u u u Kϕ∈ ,
1 2 1 20 , , , 2i i j j≤ ≤ ( )1 2 1 2
1 2 1 2, ( )i i j j
u u u u Kϕ∈ ,
1 2 1 21, 2, 2, 1i i j j= = = = 1 2 1 22, 1, 1, 2i i j j= = = = .
, .
2. 2 2 3 3 3
1 2 1 2 2 1 2( ) ( , ) ( ,1) (1, ) (1, )K u u u u u u uϕ = × × × . 3 3 3( )F F Kϕ× ≅ × × .
, ( )V G ,
, ( . [1, . 45]),
.
3. ( )W G – , | ( ): ( ) | 1V G W G = ,
1 2 3 4 5 6( )V G a b u u u u u u= × × × × × × × ,
8 2 7
1 1 ( ) ( )u a a a a= − + + + , 2 7 4 5
2 1 ( ) ( ),u a a a a= − + + +
8 2 2 2 7
3 1 ( ) ( )u ab a b a b a b= − + + + , 2 2 7 4 5 2
4 1 ( ) ( )u a b a b a b a b= − + + + ,
2 8 2 7 2
5 1 ( ) ( )u ab a b a b a b= − + + + , 2 7 2 4 2 5
6 1 ( ) ( )u a b a b a b a b= − + + + .
. ( )W G ( )V G , -
( )V C , C G . ( )V C
.
3 3
( )V a a= , ( )V b b= , 3 3
( )V a b a b= , 6 6
( )V a b a b= ,
1 2( )V a a u u= × × , 3 4( )V ab ab u u= × × , 2 2
5 6( )V ab ab u u= × × .
2 ,
1 1 2 2
1 2 5 6 1 2 1 2( ) ( , )u u u u u u u uϕ − −
= , 1 1 3
2 3 5 6 2( ) ( ,1)u u u u uϕ − −
= ,
1 1 3
1 4 5 6 1( ) (1, )u u u u uϕ − −
= , 1 1 3
2 3 4 5 2( ) (1, )u u u u uϕ − −
= .
.
3. (9,3,3)
9 3 3
| 1 | 1 | 1G a a b b c c= = × = × = . A A a b= × .
( )V A 3.
( )V A a b F= × × , 1 6...F u u= × × .
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1 1 6( ) ...V A a b u u K= × × × × × × ,
Kυ ∈
3 6 2 6 3 21 21 1
1 (1 ) (1 )
3 3
a b a b a b a b
υ υ
υ
− −
= + + + + + + ,
3
1 2, (1 ( )) ( )I a V aυ υ ∈ + ∩ , 3
( )V G a – ( . [3]) 3( ) 1ϕ υ = .
, ( )3
1 ( ) , 1,...,6u I a∈ + = .
: K F Fϕ → × , ( )1 2( ) ( ), ( )ϕ υ ϕ υ ϕ υ= .
2, 3 3
( )i
ab a= , 0,1,2i = , 3 3
( ,1),(1, ) ( )u u Kϕ∈ .
( ) ( ) ( )3 3 5 6 5 61 2 1 2 4 4
5 51 2 1 2 3 4 3 4 6 6, , , , , ( ), , 0,1,2i j i i j ji i j j i j
u u u u u u u u u u u u K i jϕ∈ =
,
1 3 5 2 4 6 1 3 5 2 4 61, 2, 2, 1i i i i i i j j j j j j= = = = = = = = = = = =
1 3 5 2 4 6 1 3 5 2 4 62, 1, 1, 2i i i i i i j j j j j j= = = = = = = = = = = = .
GAP ( . [4]) ,
( )6 3 5 62 4 1 2 4
52 4 6 1 2 3 4 6, ( ), , 0,1,2.i j j ji i j j j
u u u u u u u u u K i jϕ∉ =
, .
4. 1 12...K w w= × × ,
2 2
1 1 2 1 2( ) ( , )w u u u uϕ = , 2 2
2 3 4 3 4( ) ( , )w u u u uϕ = , 2 2
3 5 6 5 6( ) ( , )w u u u uϕ = ,
3
4 2( ) ( ,1)w uϕ = , 3
5 4( ) ( ,1)w uϕ = , 3
6 6( ) ( ,1)w uϕ = ,
3
7 1( ) (1, )w uϕ = , 3
8 2( ) (1, )w uϕ = , 3
9 3( ) (1, )w uϕ = ,
3
10 4( ) (1, )w uϕ = , 3
11 5( ) (1, )w uϕ = , 3
12 6( ) (1, )w uϕ = .
3 3 3 3 3 3 3 3 3( )F F Kϕ× ≅ × × × × × × × × .
, , .
5. ( )W G – , | ( ): ( ) | 1V G W G = .
4. (15,3)
15 3
| 1 | 1G a a b b= = × = . ( )V a
[2]. .
0( )V a a u F= × × , 1 2 3F u u u= × × ,
3 12
0 1 ( ),u a a= − + + 1 6 9
0 1 ( )u a a−
= − + + ,
1 14 2 13 3 12 5 10 6 9 7 8
1 3 3( ) 2( ) ( ) 2( ) 3( ) 3( )u a a a a a a a a a a a a−
= − − + − + − + + + + + + + ,
2 13 3 12 4 11 5 10 6 9 7 8
2 3 2( ) ( ) 2( ) ( ) 2( ) ( )u a a a a a a a a a a a a= − + + + + + − + − + + + ,
1 14 2 13 3 12 4 11 5 10 6 9
2 3 3( ) 3( ) 3( ) 2( ) 2( ) ( )u a a a a a a a a a a a a−
= − + + − + + + − + + + − + ,
14 3 12 4 11 5 10 6 9 7 8
3 3 ( ) 2( ) 2( ) ( ) ( ) 2( )u a a a a a a a a a a a a= + + − + − + − + + + + + ,
1 2 13 3 12 4 11 5 10 6 9 7 8
3 3 3( ) ( ) 3( ) 2( ) 3( ) 2( )u a a a a a a a a a a a a−
= + + − + − + + + + + − + .
1 0 1 2 3( )V G a b u u u u K= × × × × × × , Kυ ∈
5 10 2 10 5 2 5 1031 2 11 1
1 (1 ) (1 ) (1 )
3 3 3
a b a b a b a b a a
υυ υ
υ
−− −
= + + + + + + + + + ,
5
1 2, (1 ( )) ( )I a V aυ υ ∈ + ∩ , 5
3 (1 ( )) ( )I b V aυ ∈ + ∩ .
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ψ ( )V a 5
( )V G a ,
5
: a ab aψ → . 5 5
0( )V G a ab a u F′ ′= × × , 1 2 3F u u u′ ′ ′ ′= × × ,
( ), 0,1,2,3.i iu u iψ′ = =
, 5
1 2 3, , (1 ( ))u u u I a∈ + , 1 2 3, , (1 ( ))u u u I b′ ′ ′ ∈ +
4 5 4 3 2 10
1 1 ( 1)( 1) (2 2 2 1)( 1)u a a a a a a a a= + + − − − − + − + − ,
4 3 2 5 3 2 10
2 1 (2 2 1)( 1) (2 2 1)( 1)u a a a a a a a a a= − − − + + − − − − + − ,
4 3 2 5 4 2 10
3 1 ( 2 2 1)( 1) ( 2 2 1)( 1)u a a a a a a a a a= + + + + − − − − − + − ,
5 4 5 3 5 5 5
1 ( 2 2 )( 1)u a a a a a a a a b′ = + + − − −−
4 5 2 5 5 5 2
(2 2 )( 1)a a a a a a a b− − − + − ,
5 4 5 3 5 2 5 5
2 (2 2 )( 1)u a a a a a a a a b′ = + − + − − +
3 5 2 5 5 5 2
( 2 2 )( 1)a a a a a a a b+ − + − − ,
5 4 5 2 5 5 5
3 (2 2 )( 1)u a a a a a a a a b′ = + − − + − +
4 5 3 5 5 5 2
( 2 2 )( 1)a a a a a a a b+ + − − − .
( )5
0 1 ( )u I a∉ + , ( )0 1 ( )u I b′ ∉ + .
: K F F Fϕ ′→ × × ,
( )1 2 3( ) ( ), ( ), ( )ϕ υ ϕ υ ϕ υ ϕ υ= .
, 9 3× , ,
( ) ( ) ( )3 3 3
,1,1 , 1, ,1 , 1,1, ( ), 1,2,3i i iu u u K iϕ′ ∈ = .
GAP ( . [4]) ,
( )3 3 31 2 1 2 1 2
1 2 3 1 2 3 1 2 3, , ( ), , , 0,1,2( 1,2,3)i j ki i j j k k
u u u u u u u u u K i j kϕ′ ′ ′ ∈ = = .
, . 2.
2
1i 2i 3i 1j 2j 3j 1k 2k 3k
0 0 0 1 0 1 2 0 2
0 0 0 2 0 2 1 0 1
0 0 1 0 0 2 1 0 2
0 0 1 1 0 0 0 0 1
0 0 1 2 0 1 2 0 0
0 0 2 0 0 1 2 0 1
0 0 2 1 0 2 1 0 0
0 0 2 2 0 0 0 0 2
1i 2i 3i 1j 2j 3j 1k 2k 3k 1i 2i 3i 1j 2j 3j 1k 2k 3k
0 1 0 0 2 0 2 1 2 0 2 0 0 1 0 1 2 1
0 1 0 1 2 1 1 1 1 0 2 0 1 1 1 0 2 0
0 1 0 2 2 2 0 1 0 0 2 0 2 1 2 2 2 2
0 1 1 0 2 2 0 1 1 0 2 1 0 1 2 2 2 0
0 1 1 1 2 0 2 1 0 0 2 1 1 1 0 1 2 2
0 1 1 2 2 1 1 1 2 0 2 1 2 1 1 0 2 1
0 1 2 0 2 1 1 1 0 0 2 2 0 1 1 0 2 2
0 1 2 1 2 2 0 1 2 0 2 2 1 1 2 2 2 1
0 1 2 2 2 0 2 1 1 0 2 2 2 1 0 1 2 0
1 0 0 0 0 1 1 0 0 1 1 0 0 2 1 0 1 2
1 0 0 1 0 2 0 0 2 1 1 0 1 2 2 2 1 1
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1 0 0 2 0 0 2 0 1 1 1 0 2 2 0 1 1 0
1 0 1 0 0 0 2 0 2 1 1 1 0 2 0 1 1 1
1 0 1 1 0 1 1 0 1 1 1 1 1 2 1 0 1 0
1 0 1 2 0 2 0 0 0 1 1 1 2 2 2 2 1 2
1 0 2 0 0 2 0 0 1 1 1 2 0 2 2 2 1 0
1 0 2 1 0 0 2 0 0 1 1 2 1 2 0 1 1 2
1 0 2 2 0 1 1 0 2 1 1 2 2 2 1 0 1 1
1 2 0 0 1 1 2 2 1 2 0 0 0 0 2 2 0 0
1 2 0 1 1 2 1 2 0 2 0 0 1 0 0 1 0 2
1 2 0 2 1 0 0 2 2 2 0 0 2 0 1 0 0 1
1 2 1 0 1 0 0 2 0 2 0 1 0 0 1 0 0 2
1 2 1 1 1 1 2 2 2 2 0 1 1 0 2 2 0 1
1 2 1 2 1 2 1 2 1 2 0 1 2 0 0 1 0 0
1 2 2 0 1 2 1 2 2 2 0 2 0 0 0 1 0 1
1 2 2 1 1 0 0 2 1 2 0 2 1 0 1 0 0 0
1 2 2 2 1 1 2 2 0 2 0 2 2 0 2 2 0 2
2 1 0 0 2 2 1 1 2 2 2 0 0 1 2 0 2 1
2 1 0 1 2 0 0 1 1 2 2 0 1 1 0 2 2 0
2 1 0 2 2 1 2 1 0 2 2 0 2 1 1 1 2 2
2 1 1 0 2 1 2 1 1 2 2 1 0 1 1 1 2 0
2 1 1 1 2 2 1 1 0 2 2 1 1 1 2 0 2 2
2 1 1 2 2 0 0 1 2 2 2 1 2 1 0 2 2 1
2 1 2 0 2 0 0 1 0 2 2 2 0 1 0 2 2 2
2 1 2 1 2 1 2 1 2 2 2 2 1 1 1 1 2 1
2 1 2 2 2 2 1 1 1 2 2 2 2 1 2 0 2 0
, .
6. 1 9...K w w= × × ,
1 1 3 1( ) ( , , )w u u uϕ ′= , ( )2 2 2
2 2 2 1 2 3( ) , ,w u u u u uϕ ′ ′ ′= , ( )2 2
3 3 3 1 3( ) , ,w u u u uϕ ′ ′= ,
( )2 2
4 1 3 1 3( ) 1, ,w u u u uϕ ′ ′= , 3
5 2( ) (1, ,1)w uϕ = , 3
6 3( ) (1, ,1)w uϕ = ,
3
7 1( ) (1,1, )w uϕ ′= , ( )3
3 8 2( ) 1,1,w uϕ ′= , ( )3
9 3( ) 1,1,w uϕ ′= ,
3 3 3 3 3( )F F Kϕ× ≅ × × × × .
, , -
.
7. ( )W G – , | ( ): ( ) | 1V G W G = ,
0 1 12( ) ...V G a b u u u= × × × × × ,
3 12
0 1 ( )u a a= − + + , 14 2 13 3 12 4 11 5 10 6 9
( ) 3 2( ) 2( ) 2( ) ( ) ( ) ( )u x x x x x x x x x x x x x= − + + + − + + + − + + + ,
1 ( )u u a= , 2
2 ( )u u a= , 2
3 ( )u u a= , 4 ( )u u ab= , 2 2
5 ( )u u a b= , 4
6 ( )u u a b= , 2
7 ( )u a b ,
4 2
8 ( ),u u a b= 8
9 ( )u u a b= , 3
10 ( )u u a b= , 6 2
11 ( )u u a b= , 12
12 ( )u u a b= .
. ( )V C , ( )W G , C -
G .
5 5
( )V a a= , ( )V b b= , 5 5
( )V a b a b= , 10 10
( )V a b a b= , 3 3
0( )V a a u= × ,
0 1 2 3( )V a a u u u u= × × × × , 0 4 5 6( )V ab ab u u u u= × × × × ,
2 2
0 7 8 9( )V a b a b u u u u= × × × × , 3 3
0 10 11 12( )V a b a b u u u u= × × × × .
6 ,
1 1 1
10 11 12 1 3 1( ) ( , , )u u u u u uϕ − − −
′= , ( )1 1 1 2 2 2
1 2 3 4 6 7 2 2 1 2 3( ) , ,u u u u u u u u u u uϕ − − −
′ ′ ′= ,
( )1 1 1 1 1 1 2 2
1 7 8 9 10 12 3 3 1 3( ) , ,u u u u u u u u u uϕ − − − − − −
′ ′= , ( )1 1 1 1 1 1 2 2
1 3 7 9 10 12 1 3 1 3( ) 1, ,u u u u u u u u u uϕ − − − − − −
′ ′= ,
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3
2 4 6 9 12 2( ) (1, ,1)u u u u u uϕ = , ( )1 1 1 1 1 1 1 3
3 4 6 7 8 9 10 12 3( ) 1, ,1u u u u u u u u uϕ − − − − − − −
= ,
( )1 1 1 1 1 3
1 3 4 8 10 11 12 1( ) 1,1,u u u u u u u uϕ − − − − −
′= ( )3
1 3 5 9 10 2( ) 1,1,u u u u u uϕ ′= ,
( )1 1 1 1 1 3
1 3 6 7 8 9 11 3( ) 1,1,u u u u u u u uϕ − − − − −
′= .
.
1. , . . / . . . –
: . . ., 1987. – 210 . ( . 24.09.87, 2712– 87).
2. , . .
/ . . . – , . . . .,
2000. – 45 . ( . 30.03.00, 860– 00).
3. , . . 7 9 / . . , . . // -
, . – 1999. – 11(450). – . 81–84.
4. Martin Schönert et al. GAP – Groups, Algorithms, and Programming. Lehrstuhl D für Mathe-
matik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, sixth edition, 1997.
UNITS OF INTEGRAL GROUP RINGS OF FINITE GROUPS
WITH A DIRECT MULTIPLIER OF ORDER 3
S.A. Kolyasnikov
1
The description of units of integral group rings of finite groups of type A×Z3 was obtained, where A
contains a central subgroup of order 3. For example, the unit groups of integral group rings of Abelian
groups of the types (9,3), (9,3,3) and (15,3) were found.
Keywords: Abelian group, group ring, unit group of group ring.
References
1. Bovdi A.A. Mul'tiplikativnaya gruppa tselochislennogo gruppovogo kol'tsa (Multiplicative group
of integral group ring). Uzhgorod: Uzhgorodskiy gosudarstvennyy universitet, 1987. 210 p. (in Russ.).
2. Kolyasnikov S.A. O gruppe edinits tselochislennogo gruppovogo kol'tsa konechnykh grupp
razlozhimykh v pryamoe proizvedenie (About group of units of integral group ring of finite groups fac-
torable into direct composition). Novosibirsk, Red. Sibirskogo Matematicheskogo Zhurnala, 2000. 45 p.
(in Russ.).
3. Alev R.Zh., Panina G.A. Russian Mathematics (Izvestiya VUZ. Matematika). 1999. Vol. 43,
no. 11. pp. 80–83.
4. Martin Schönert et al. GAP – Groups, Algorithms, and Programming. Lehrstuhl D für Mathe-
matik. Rheinisch Westfälische Technische Hochschule, Aachen, Germany, sixth edition, 1997.
6 2013 .
1
Kolyasnikov Sergey Andreevich is Senior Lecturer, General Mathematics Department, South Ural State University.
E-mail: koliasnikovsa@susu.ac.ru
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, -
( (6))
20–
40 . ,
,
-
-
.
-
-
.
-
(1). . 5.
. 5 -
20–40 . , -
, -
. -
.
, -
. ,
.
. -
. , -
, , .
, , ,
.
-
-
,
2–25 600–1100 K -
.
1. Allen, C.M. Energetic-nanoparticle enhanced combustion of liquid fuels in a rapid compression
machine / C.M. Allen, T. Lee // Proc. 47th AIAA Aerospace Sciences Meeting Including The New Ho-
rizons Forum and Aerospace Exposition. – 2009. – AIAA 2009-227.
2. Trunov, M.A. Ignition of aluminum powders under different experimental conditions /
M.A. Trunov, M. Schoenitz, E.L. Dreizin // Propellants, Explosives, Pyrotechnics. – 2005. – Vol. 30. –
P. 36–43.
3. Utilization of iron additives for advanced control of NOx emissions from stationary combustion
sources / V.V. Lissianski, P.M. Maly, V.M. Zamansky, W.C. Gardiner // Ind. Eng. Chem. Res. – 2001. –
Vol. 20, 15. – P. 3287–3293.
4. / . . , . . ,
. . . // . – 1996. – T. 32, 3. – C. 24–34.
5. Cashdollar, K.L. Explosion temperatures and pressures of metals and other elemental dust clouds /
K.L. Cashdollar, I.A. Zlochower // Journal of Loss Prevention in the Process Industries. – 2007. – Vol. 20.
– P. 337–348.
. 5.
. (1, 3) –
, (2, 4) – -
(5). 1, 2 – 30 , 3, 4 – 2 , 5 –
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
29. . ., . .,
.- ., . ., . .
2013, 5, 2 29
6. Dynamics of dispersion and ignition of dust layers by a shock wave / V.M. Boiko, A.N. Papyrin,
M. Wolinski , P. Wolanski // Progress in Astronautics and Aeronautics. – 1984. – Vol. 94. – P. 293–301.
7. Ignition of dust suspensions behind shock waves / A.A. Borisov, B.E. Gel’fand, E.I. Timofeev et
al. // Progress in Astronautics and Aeronautics. – 1984. – Vol. 94. – P. 332–339.
8. Photoemission measurements of soot particles temperature at pyrolysis of ethylene /
Y.A. Baranyshyn, L.I. Belaziorava, K.N. Kasparov, O.G. Penyazkov // Nonequilibrium Phenomena:
Plasma, Combustion, Atmosphere. – Moscow: Torus Press Ltd, 2009. – P. 87–93.
9. Optical properties of gases at ultra high pressures / Yu.N. Ryabinin, N.N. Sobolev,
A.M. Markevich, I.I. Tamm // Journal of Experimental and Theoretical Physics. – 1952. – Vol. 23, 5.
– P. 564–575.
10. , . . / . .
// . – 2004. – T. 40, 1. – . 75–77.
11. , . . /
. . , . . // . – 2011. – . 47, 6. – . 98–100.
12. /
. . , . . , . . . // - . – 2012. –
. 85, 1. – . 139–144.
13. , . .
/ . . , . . // . – 2001. – . 37, 6.
– . 46–55.
14. /
. . , . . – .: , 1979. – 312 .
15. , . . / . . , . . // -
. – 2003. – . 39, 5. – . 65–68.
EXPERIMENTAL AND MATHEMATICAL SIMULATION OF AUTO-IGNITION
OF IRON MICRO PARTICLES
V.V. Leschevich
1
, O.G. Penyazkov
2
, J.-C. Rostaing
3
, A.V. Fedorov
4
, A.V. Shulgin
5
The experimental and numerical studies of autoignition and burning of iron microparticles in oxy-
gen were introduced. Parameters of high-temperature gas environment were created by rapid compres-
sion machine. Powders with a particle size 1–125 microns were studied at the oxygen pressure of 0,5–28
MPa and the temperature of 500–1100 K. Ignition delay time was measured by registration of radiation
from the side wall of the combustion chamber and by measuring the pressure on the end wall. Critical
auto-ignition conditions are determined according to size of the particles, oxygen temperature and pres-
sure. For description of ignition mechanisms a punctual semiempirical simulator was introduced. The
simulator gave the opportunity to describe the experimental data on dependence of ignition delay time of
iron particles on the ambient temperature, taking into account correlations of critical ignition tempera-
tures with pressure.
Keywords: iron micro particles, oxygen, rapid compression machine, ignition delay time, mathe-
matical simulation.
1
Leschevich Vladimir Vladimirovich is Junior Researcher, A.V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences
of Belarus.
2
Penyazkov Oleg Glebovich is Corresponding Member of the National Academy of Sciences of Belarus, director, A.V. Luikov Heat and Mass
Transfer Institute of the National Academy of Sciences of Belarus.
3
Rostaing Jean-Christophe is L’Air Liquide, Centre de Recherche Claude-Delorme, France.
4
Fedorov Alexander Vladimirovich is Doctor of Physical and Mathematical Sciences, Head of Laboratory, S.A. Khristianovich Institute of
Theoretical and Applied Mechanics of the Siberian Branch of the Russian Academy of Sciences.
5
Shulgin Alexey Valentinovich is Candidate of Physical and Mathematical Sciences, Senior Researcher, S.A. Khristianovich Institute of Theo-
retical and Applied Mechanics of the Siberian Branch of the Russian Academy of Sciences.
E-mail: shulgin@itam.nsc.ru
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References
1. Allen C.M., Lee T. Energetic-nanoparticle enhanced combustion of liquid fuels in a rapid com-
pression machine // Proc. 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum
and Aerospace Exposition. 2009. AIAA 2009-227.
2. Trunov M.A., Schoenitz M., Dreizin E.L. Ignition of aluminum powders under different experi-
mental conditions. Propellants, Explosives, Pyrotechnics. 2005. Vol. 30. pp. 36–43.
3. Lissianski V.V., Maly P.M., Zamansky V.M., Gardiner W.C. Utilization of iron additives for ad-
vanced control of NOx emissions from stationary combustion sources. Ind. Eng. Chem. Res. 2001. Vol.
20, no. 15. pp. 3287–3293.
4. Zolotko A.N., Vovchuk Ya.I., Poletaev N.I., Frolko A.V., Al'tman I.S. Fizika goreniya i vzryva. 1996.
Vol. 32, no. 3. pp. 24–34. (in Russ.).
5. Cashdollar K.L., Zlochower I.A. Explosion temperatures and pressures of metals and other elemen-
tal dust clouds. Journal of Loss Prevention in the Process Industries. 2007. Vol. 20. pp. 337–348.
6. Boiko V.M., Papyrin A.N., Wolinski M., Wolanski P. Dynamics of dispersion and ignition of dust
layers by a shock wave. Progress in Astronautics and Aeronautics. 1984. Vol. 94. pp. 293–301.
7. Borisov A.A., Gel’fand B.E., Timofeev E.I., Tsyganov S.A., Khomic S.V. Ignition of dust sus-
pensions behind shock waves. Progress in Astronautics and Aeronautics. 1984. Vol. 94. pp. 332–339.
8. Baranyshyn Y.A., Belaziorava L.I., Kasparov K.N., Penyazkov O.G. Photoemission measure-
ments of soot particles temperature at pyrolysis of ethylene. Nonequilibrium Phenomena: Plasma, Com-
bustion, Atmosphere. Moscow: Torus Press Ltd, 2009. pp. 87–93.
9. Ryabinin Yu.N., Sobolev N.N., Markevich A.M., Tamm I.I. Optical properties of gases at ultra
high pressures. Journal of Experimental and Theoretical Physics. 1952. Vol. 23, no. 5. pp. 564–575.
10. Al'tman I.S. Fizika goreniya i vzryva. 2004. Vol. 40, no. 1. pp. 75–77.
11. Fedorov A.V., Shul'gin A.V. Fizika goreniya i vzryva. 2011. Vol. 47, no. 6. pp. 98–100.
12. Leshchevich V.V., Penyaz'kov O.G., Fedorov A.V., Shul'gin A.V., Rosten Zh.-K. Inzhenerno-
fizicheskiy zhurnal. 2012. Vol. 85, no. 1. pp. 139–144. (in Russ.).
13. Bolobov V.I., Podlevskikh N.A. Fizika goreniya i vzryva. 2001. Vol. 37, no. 6. pp. 46–55. (in Russ.).
14. Kholla Dzh. (ed.), Uatta Dzh. (ed.) Sovremennye chislennye metody resheniya obyknovennykh
differentsial'nykh uravneniy (Contemporary numerical methods of solution of ordinary differential eqau-
tion). Moscow: Mir, 1979. 312 p. (in Russ.). [Hall G., Watt J.M. Modern Numerical Methods for Ordi-
nary Differential Equations. Oxford: Clarendon press. 312 p.]
15. Fedorov A.V., Kharlamova Yu.V. Fizika goreniya i vzryva. 2003. Vol. 39, no. 5. pp. 65–68. (in
Russ.).
8 2013
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1, [10]. -
,
, 10- . -
,
5- .
1.1. . , ,a b - , ,A B
. A ijA , T
A – A , det A
– A . a b ×a b T
a b
. a [ ]×a -
, [ ]× = ×a b a b b . I , 0
– , ⋅ – .
2.
2.1 . -
jix , jiy , jiz iQ j - , 1, 2j = ,
1, , 5i = ( . 1).
1 1 2 0j j jx y x= = = 1, 2j = .
.
3 5× :
1 5
1 5
1 5
j j
j j j
j j
x x
y y
z z
=A (1)
2 2 1j j j j j j′′ ′= =A H A H H A , (2)
1jH 2jH – , 1jx , 1jy 2jx . -
:
1
1 1
2 2 2
1 1 1 1 1sign( )
j
j j
j j j j j
x
y
z z x y z
=
+ + +
h ,
2
2 2
2 2 2 2 2sign( )
0
j
j j j j j
x
y y x y
′
′ ′ ′ ′= + +h .
, (2), , -
. ,
20- (12) 10- (13).
2.2. . -
[11–13]. - – ( , , )O π P ,
π – , P – 3-
π , O – ( P ). –
O π , O π . -
, (
) .
- ( , , )j j jO π P 1, 2j = . -
[ ]1 =P I 0 , [ ]2 =P R t , SO(3)∈R –
[ ]T
1 2 3t t t=t – , 1=t .
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
33. . .
,
2013, 5, 2 33
1 iO Q 2 iO Q ( . 1) 3
, -
R t . [12]:
[ ]
1
2 2 2 1
1
0
i
i i i i
i
x
x y z y
z
=E , (3)
1, , 5i = , [ ]×
=E t R .
2.3. .
.
1 ([10]). SO(3)∈R
2 kπ π+ , k ∈ , R :
1
u u
v v
w w
−
× ×
= − +R I I , (4)
, ,u v w∈ .
R (4) [ ]( , , , )u v w ×
=E t t R – .
1.
1 3 2
2 1 3
3 2 1
,
,
,
t vt wt
u
t wt ut
v
t ut vt
w
δ
δ
δ
− − +
′ =
− − +
′ =
− − +
′ =
(5)
1 2 3ut vt wtδ = + + , ( , , , ) ( , , , )u v w u v w′ ′ ′ = −E t E t .
. SO(3)′ = − ∈tR H R ,
T
2= −tH I tt . [ ]×
′ ′= = −E t R E . ′R (4) , ,u v w′ ′ ′ ,
, ,u v w′ ′ ′ :
' '
' '
' '
u u u u
v v v v
w w w w× × × ×
− + = − − +tI I H I I ,
, , (5).
(3) t , -
:
=S t 0 , (6)
i - 5 3× S [ ]
2
T
1 1 1 2
2
i
i i i i
i
x
x y z y
z ×
R .
R (4) 3 3×
S . :
4 3 2 2 3 4 3 2
2 3 2 2
[0] [0] [0] [0] [0] [1] [1]
[1] [1] [2] [2] [2] [3] [3] [4] 0,
if u u v u v uv v u u v
uv v u uv v u v
= + + + + + +
+ + + + + + + + =
(7)
1, ,10i = , [ ]n n w , [0] – . -
, if , 4.
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
34. . « . . »34
1. 3 3× S 3
/iF Δ ,
2 2 2
1 u v wΔ = + + + iF – 6- . , iF
i iF f= Δ , if -
iF .
2.4. . (7) :
=B m 0 , (8)
B – 10 35×
T4 3 3
1u u v u w v w=m –
.
(8) iuf , ivf 1, , 5i = , iwf
1, ,10i = . , :
′
′ =
m
B 0
m
, (9)
′B – 30 50×
T4 3 3 2 2 3 4 5
u w u vw u w v w vw w′ =m
– . , (9) (8).
′B –
( ).
( ):
3 2
u w 3
u w 3
u 3 2
v w 3
v w 3
v uv u v 1
1g 1 [3] [4] [4] [5]
2g 1 [3] [4] [4] [5]
3g 1 [3] [4] [4] [5]
4g 1 [3] [4] [4] [5]
5g 1 [3] [4] [4] [5]
6g 1 [3] [4] [4] [5]
28 .
1 6, ,g g :
1 21
2 32
4 53
5 64
( )
1
g gh uv
g gh u
w w
g gh v
g gh
≡ − = =C 0 , (10)
( )wC :
[4] [5] [5] [6]
[4] [5] [5] [6]
( )
[4] [5] [5] [6]
[4] [5] [5] [6]
w =C . (11)
2. ′B ,
« » – ′B .
24 ′B . -
.
(10) , (8) ,
det ( ) 0w =C . det ( )W w= C . 20- -
w .
2. :
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
35. . .
,
2013, 5, 2 35
10
10 10
0
( )k k
k
k
W p w w+ −
=
= + − , (12)
kp ∈ .
. 1 1 0j jx y= = , 33 0E = . ,
23
2 1 1
13
R vw u
t t t
R uw v
+
= =
−
.
(5), 1
w w−
′ = − . , w
1
( )w−
− ( ) . ,
10 10
1 10 10
10
01
( )( ) ( )k k
i i k
ki
W p w w w w p w w− + −
==
= − + = + −∏ .
1
w w w−
= − 10- :
10
0
k
k
W p w
=
= , (13)
kp 2
0
( 1)
k
k i i k
i
k
w w
i
−
=
= − . :
10 1
2
2
k i
i k
i k
i
p p
i kk
'
=
+
−
=
−
, (14)
i k 10, mod 2 0i k− = . ,
0k = (14)
10
0
2 i
i
p'
=
.
2.5. . -
, (4), – , -
. .
, , [14]
[15] . -
.
0w – W . 2
0 0 0 0/ 2 sign( ) ( / 2) 1w w w w= + +
W , 0| | 1w ≥ . , u - v -
-
0( )wC (11).
, (4) R . R , t -
0 0 0( , , )u v wS (6). 1=t .
T
2= −tH I tt ′ = − tR H R . [12, 6],
: [ ]A =P R t , [ ]B = −P R t , [ ]C ′=P R t [ ]D ′= −P R t . -
(3). , -
.
, 2P , -
, , -
, . , 1Q .
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
36. . « . . »36
1 2
1 2 1 33 3
13 23
,
t t
c c c R t
R R
= − = − = + . (15)
,
• 1 0c > 2 0c > , 2 A=P P ;
• 1 0c < 2 0c < , 2 B=P P ;
• 1 0c′ > 2 0c′ > , 2 C=P P ;
• 2 D=P P .
1c′ 2c′ 1c 2c (15) R ′R .
,
12 11ini T
2 22 21 2( )
1
=
H H 0
P H H P
0
,
1jH 2jH 2.1.
3.
5- [6]. -
C/C++. -
. [6], . .
1
0,5
0,1
352 288×
45°
6
10 35 -
/ , – 24 / Intel Core i5 2,3 Ghz.
2 2ε = −P P , 2P – -
.
. 2. -
6
10 . , - , –
0,5 , , - , , -
, – -
. , -
, ,
.
. 2. . : , , −
× 13
1 56 10
, −
× 10
2 94 10 . : -
, , −
× 2
1 52 10 , −
× 3
7 17 10
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
37. . .
,
2013, 5, 2 37
. 3 ( )
.
, 0 1 . ,
.
4.
5- . -
. -
, 5-
. ,
.
,
/ . , , ( , , )ϕ θ ψ ,
R , ,
2ctg( )w ϕ ψ= − + .
10- W .
1. Kruppa, E. Zur Ermittlung eines Objektes aus zwei Perspektiven mit Innerer Orientierung /
E. Kruppa // Sitz.-Ber. Akad. Wiss., Wien, Math. Naturw. Kl. Abt. IIa. – 1913. – Vol. 122. – P. 1939–
1948.
2. Demazure, M. Sur Deux Problemes de Reconstruction / M. Demazure // INRIA. – 1988. –
RR-0882. – P. 1–29.
3. Faugeras, O. Motion from Point Matches: Multiplicity of Solutions / O. Faugeras, S. Maybank //
International Journal of Computer Vision. – 1990. – Vol. 4. – P. 225–246.
4. Heyden, A. Reconstruction from Calibrated Camera – A New Proof of the Kruppa-Demazure
Theorem / A. Heyden, G. Sparr // Journal of Mathematical Imaging and Vision. – 1999. – Vol. 10. –
P. 1–20.
5. Philip, J. A Non-Iterative Algorithm for Determining all Essential Matrices Corresponding to
Five Point Pairs / J. Philip // Photogrammetric Record. – 1996. – Vol. 15. – P. 589–599.
6. Nistér, D. An Efficient Solution to the Five-Point Relative Pose Problem / D. Nistér // IEEE
Transactions on Pattern Analysis and Machine Intelligence. – 2004. – Vol. 26. – P. 756–777.
7. Li, H. Five-Point Motion Estimation Made Easy / H. Li, R. Hartley // IEEE 18th Int. Conf. of Pat-
tern Recognition. – 2006. – P. 630–633.
8. Kukelova, Z. Polynomial Eigenvalue Solutions to the 5-pt and 6-pt Relative Pose Problems /
Z. Kukelova, M. Bujnak, T. Pajdla // Proceedings of the British Machine Conference. – 2008. – P. 56.1–
56.10.
9. Stewénius, H. Recent Developments on Direct Relative Orientation / H. Stewénius, C. Engels,
D. Nistér // ISPRS Journal of Photogrammetry and Remote Sensing. – 2006. – Vol. 60. – P. 284–294.
10. Cayley, A. Sur Quelques Propriétés des Déterminants Gauches / A. Cayley // J. Reine Angew.
Math. – 1846. – Vol. 32. – P. 119–123.
. 3. ( ) ( ) -
,
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
38. . « . . »38
11. Faugeras, O. Three-Dimensional Computer Vision: A Geometric Viewpoint / O. Faugeras //
MIT Press, 1993. – 663 p.
12. Hartley, R. Multiple View Geometry in Computer Vision. / R. Hartley, A. Zisserman. –
Cambridge University Press, 2004. – 655 p.
13. Maybank, S. Theory of Reconstruction from Image Motion / S. Maybank // Springer-Verlag,
1993. – 261 p.
14. Hook, D.G. Using Sturm Sequences to Bracket Real Roots of Polynomial Equations /
D.G. Hook, P.R. McAree // Graphic Gems I : . . . – Academic Press, 1990. – P. 416–422.
15. Numerical Recipes in C. Second Edition / W. Press, S. Teukolsky, W. Vetterling, B. Flannery. –
Cambridge University Press, 1993. – 1020 p.
ALGORITHMIC SOLUTION OF THE FIVE-POINT POSE PROBLEM
BASED ON THE CAYLEY REPRESENTATION OF ROTATION MATRICES
E.V. Martyushev
1
A new algorithmic solution to the five-point relative pose problem is introduced. Our approach is
not connected with or based on the famous cubic constraint on an essential matrix. Instead, we use the
Cayley representation of rotation matrices in order to obtain a polynomial system of equations from epi-
polar constraints. Solving that system, we directly obtain positional relationships and orientations of the
cameras through the roots for a 10th degree polynomial.
Keywords: five-point pose problem, calibrated camera, epipolar constraints, Cayley representation.
References
1. Kruppa E. Zur Ermittlung eines Objektes aus zwei Perspektiven mit Innerer Orientierung. Sitz.-
Ber. Akad. Wiss., Wien, Math. Naturw. Kl. Abt. IIa. 1913. Vol. 122. pp. 1939–1948.
2. Demazure M. Sur Deux Problemes de Reconstruction. INRIA. 1988. no. RR-0882. pp. 1–29.
3. Faugeras O., Maybank S. Motion from Point Matches: Multiplicity of Solutions. International
Journal of Computer Vision. 1990. Vol. 4. pp. 225–246.
4. Heyden A., Sparr G. Reconstruction from Calibrated Camera – A New Proof of the Kruppa-
Demazure Theorem. Journal of Mathematical Imaging and Vision. 1999. Vol. 10. pp. 1–20.
5. Philip J. A Non-Iterative Algorithm for Determining all Essential Matrices Corresponding to Five
Point Pairs. Photogrammetric Record. 1996. Vol. 15. pp. 589–599.
6. Nistér D. An Efficient Solution to the Five-Point Relative Pose Problem. IEEE Transactions on
Pattern Analysis and Machine Intelligence. 2004. Vol. 26. pp. 756–777.
7. Li H., Hartley R. Five-Point Motion Estimation Made Easy. IEEE 18th Int. Conf. of Pattern Rec-
ognition. 2006. pp. 630–633.
8. Kukelova Z., Bujnak M., Pajdla T. Polynomial Eigenvalue Solutions to the 5-pt and 6-pt Relative
Pose Problems. Proceedings of the British Machine Conference. 2008. pp. 56.1–56.10.
9. Stewénius H., Engels C., Nistér D. Recent Developments on Direct Relative Orientation. ISPRS
Journal of Photogrammetry and Remote Sensing. 2006. Vol. 60. pp. 284–294.
10. Cayley A. Sur Quelques Propriétés des Déterminants Gauches. J. Reine Angew. Math. 1846.
Vol. 32. pp. 119–123.
11. Faugeras O. Three-Dimensional Computer Vision: A Geometric Viewpoint. MIT Press, 1993.
663 p.
12. Hartley R., Zisserman A. Multiple View Geometry in Computer Vision (Second Edition). Cam-
bridge University Press, 2004. 655 p.
13. Maybank S. Theory of Reconstruction from Image Motion. Springer-Verlag, 1993. 261 p.
14. Hook D.G., McAree P.R. Using Sturm Sequences To Bracket Real Roots of Polynomial Equa-
tions. Graphic Gems I (A. Glassner ed.). Academic Press, 1990. pp. 416–422.
15. Press W., Teukolsky S., Vetterling W., Flannery B. Numerical Recipes in C (Second Edition).
Cambridge University Press, 1993. 1020 p.
18 2013 .
1
Martyushev Evgeniy Vladimirovich is Cand.Sc. (Physics and Mathematics), Mathematical analysis department, South Ural State University.
E-mail: zhmart@mail.ru
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
39. 2013, 5, 2 39
515.126
. . 1
– ,
. -
g:A →→→→ B -
f : X →→→→ X. , -
, , g
f : X →→→→ X .
: , ,
, .
. -
[1]. ,
.
. X ≈ Y , X Y . w(X) – -
X. ω; {0,1,2, }ω = .
, -
. { : }na n ω∈
a X∈ , ( ) { } { : }nS a a a n ω= ∪ ∈ .
X , a b X -
:f X X→ , ( )f a b= . -
, 1- - .
, . -
, -
, , . , -
.
[2].
1. X – , -
. X
{ : }na n ω∈ { : }nb n ω∈ , limn na a→∞ = , limn nb b→∞ = ( ) ( )S a S b = ∅ .
- U a -
V a :f X X→ , , : V U⊂ , ( )f a b= ,
( )f V V = ∅ , Xf f id= , ( )f x x= x V∉ ( ( ))i ii a V b f V∀ ∈ ⇔ ∈ .
. X ,
:g X X→ , a b . - W
a , W U⊂ ( )g W W = ∅ .
: { : , ( )}a i iJ i a W b g Wω= ∈ ∈ ∉ { : , ( )}b i iJ i a W b g Wω= ∈ ∉ ∈ .
aJ bJ – , .
ai J∈ - iQ ia ,
iQ W⊂ ( ) { }i iQ S a a= . , { : }i aQ i J∈
, { : }na n ω∈ .
1
– - , , -
, - .
E-mail: medv@math.susu.ac.ru
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, ( )ig a ( )S b .
1 2a a aJ J J= ∪ , 1 { : ( ) ( )}a a iJ i J g a S b= ∈ ∉ 2 { : ( ) ( )}a a iJ i J g a S b= ∈ ∈ .
: 1ai J∈ . - iD
ia , i iD Q⊂ ( ) ( )ig D S b = ∅ . ig g= .
: 2ai J∈ , . . ( )i jg a b= j . –
, ( ) ( )ig Q S b – , i ic Q∈ ,
( ) ( )ig c S b∉ . :i X Xψ → , ia ic . -
- i iD Q⊂ ia , ( )i i i iD D Qψ ⊂ ,
( )i i iD Dψ = ∅ ( ( )) ( )i ig D S bψ = ∅ . :ig X X→ :
1
( ), ,
( ) ( ), ( ),
( ), ( ( )).
i i
i i i i
i i i
g x x D
g x g x x D
g x x X D D
ψ
ψ ψ
ψ
−
∈
= ∈
∈
, ig ( ) ( )i ig D S b = ∅ .
*:g X X→ :
( ), ,
*( )
( ), { : }.
i i a
i a
g x x D i J
g x
g x x X D i J
∈ ∈
=
∈ ∈
, *g – . , * { : }i aW W D i J= ∈ – -
- a . *( )g a b= ( ) ( ) ( ) *( *)S b g W S b g W= .
, aJ = ∅ , *W W= *g g= .
, bi J∈ - iE ib
, *( *)iE g W⊂ , ( ) { }i iE S b b= , 1
( ) ( )i iS a h E−
= ∅ , { : }i bE i J∈
. :ih X X→ *g
, ig g .
1
* { ( ): }i i bV W h E i J−
= ∈ – - a .
:h X X→ , :
1
1
( ), ( ) ,
( )
*( ), { ( ): }.i
i i i b
i b
h x x h E i J
h x
g x x X h E i J
−
−
∈ ∈
=
∈ ∈
:f X X→ :
1
( ), ,
( ) ( ), ( ),
, ( ( )).
h x x V
f x h x x h V
x x X V h V
−
∈
= ∈
∈
, f – . 1 .
1. X – , -
. X
{ : }na n ω∈ { : }nb n ω∈ , limn na a→∞ = , limn nb b→∞ = ( ) ( )S a S b = ∅ .
:f X X→ , Xf f id= , ( )f a b= ( )n nf a b=
n ω∈ .
. *
{ : }nU n ω∈ a , -
. 1 - 0U a
0 :g X X→ , , : *
0 0U U⊂ , 0( )g a b= , 0 0 0( )g U U = ∅ , 0 0 Xg g id= ,
0( )g x x= 0x U∉ 0 0 0( ( ))i ii a U b g U∀ ∈ ⇔ ∈ . -
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
41. . .
2013, 5, 2 41
, 1n ≥ - nU a -
:ng X X→ , , : *
1n n nU U U −⊂ , ( )ng a b= , ( )n n ng U U = ∅ , n n Xg g id= ,
( )ng x x= nx U∉ ( ( ))i n i n ni a U b g U∀ ∈ ⇔ ∈ . , -
{ : }nU n ω∈ a , { ( ): }n ng U n ω∈
b . n ω∈ { : }n i nJ i a Uω= ∈ ∉ .
j ω∈ :j X Xχ → , ( )j j ja bχ = .
- jO ja . ( )j jOχ – - -
jb . , { : }jO j ω∈
{ ( ): }j jO jχ ω∈ , lim { }j jO a→∞ = , lim ( ) { }j j jO bχ→∞ = , 1j k kO U U +⊂
1 1( ) ( ) ( )j j k k k kO g U g Uχ + +⊂ , 1j k ka U U +⊂ k ω∈ .
n ω∈ nJ , { : }n n j nV U O j J= ∪ ∈
( ) { ( ): }n n n j j nW g U O j Jχ= ∪ ∈ - , ( ) nS a V⊂ , ( ) nS b W⊂ , 1n nV V+ ⊂ ,
1n nW W+ ⊂ n nV W = ∅ . , ( )n n nf V W= :nf X X→ , -
:
1
1
( ), ,
( ), ( ),
( ) ( ), ,
( ), ( ) ,
, .
n
j
n n
n n
n j j n
j j n
g x x U
g x x g U
f x x x O j J
x x O j J
x
χ
χ χ
−
−
∈
∈
= ∈ ∈
∈ ∈
, n n Xf f id= .
:f X X→ ( ) lim ( )n nf x f x→∞= . , ( )f a b=
( )f b a= . , . , { : } { }nU n aω∈ =
{ ( ): } { }n ng U n bω∈ = , { , }x X a b∈ n , ( )n n nx U g U∉ .
1( ) ( )n nf x f x+ = ; lim ( )n nf x→∞ . ,
f – . 1 .
, { }a A a∈ ,
. d
A
. n ω∈ -
(n)
n : (0)
A A= ( 1) ( )
( )n n d
A A+
= . ,
n, ( 1)n
A −
≠ ∅ , ( )n
A = ∅ . ( )n
A = ∅ n ω∈ ,
, .
1.
2. X – .
X
. :g A B→ -
:f X X→ , Xf f id= .
. – , –
. f .
, . –
. n .
: 1n = . (1) (1)
A B= = ∅ , – .
, 1{ , , }kA a a= 1{ , , }kB b b= , ( )i ib g a= 1 i k≤ ≤ . ,
i :if X X→ , ( )i i if a b= . -
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
42. . « . . »42
- iU jV ia jb . -
, ( )i i if U V= . :f X X→ :
1
( ), ,
( ) ( ), ,
, { :1 }.
i
i i
i
i i
f x x U i
f x f x x V i
x x X U V i k
−
∈
= ∈
∈ ∪ ≤ ≤
, , n.
1n + . (1) (1)
n. , (1) (1)
( )g A B= .
g
:h X X→ , Xh h id= , (1) (1)
( )h B A= (1) (1)
( )h A B= .
(1)
A A { : }ma m ω∈ . (1)
{ ( ): }mB B g a m ω= ∈ . -
1, , ( ) ( )m kg a h a≠ m k.
(1)
( )h A A B = ∅ (1)
( )h B B A = ∅ . m ω∈
:m X Xχ → , ma ( )mg a . ( ) ( )A h A B h B
, - mU ( )m m mV Uχ= -
ma ( )mg a , mU , mV , 1
( ) ( )m mh U h U−
= 1
( ) ( )m mh V h V−
= -
, (1) (1)
A B . , (1)
(1)
– , , { : }jmU j ω∈
(1)
*a A∈ ⇔ { : }jmV j ω∈ (1)
( *) ( *)g a h a B= ∈ ⇔ -
{ ( ): }jmh V j ω∈ (1)
* ( *)a h h a A= ∈ . -
f :
1
1 1
1 1
( ), ,
( ), ,
( ) ( ), ( ) ,
( ), ( ) ,
( ), .
m
m m
m
m m
m m
x x U m
x x V m
f x h h x x h V m
h h x x h U m
h x
χ
χ
χ
χ
−
− −
− −
∈
∈
= ∈
∈
. 2 .
:g A B→
: f X X A→ .
2. X
{ : }nU n ω∈ { : }nV n ω∈ - , :
1) 0 1 nU U U⊃ ⊃ ⊃ ⊃ ,
2) 0U V = ∅ , { : }nV V n ω= ∈ ,
3) { : }nV n ω∈ X,
4) n ω∈ :n n ng U V→ .
{ : }nA U n ω= ∈ : f X X A→ ,
0 0( ) ( )f A g A V= ⊂ .
. : f X X A→ :
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
43. . .
2013, 5, 2 43
0 0
1
1 1
1
1
0
( ), ,
( ), ( ) ,
( )
( ), ( ) ,
, .
n n n n
n n n n
g x x U
g g x x g U n
f x
g x x V g U n
x x U V
ω
ω
−
+ +
−
+
∈
∈ ∈
=
∈ ∈
∉
, 0 1 { : }n nU A U U n ω+= ⊕ ∈ 1 1( ) ( )n n n nf V U U V+ += n ω∈ , -
, f X X A . -
1( )n ng U + 1 ( )n n nV g U + - nV ( , X),
0U V - X, ng , -
f . , 0 0( ) ( )f A g A V= ⊂ . 2 .
3. X
a b.
: { }f X X a→ , ( )f a b= .
. X , X -
{ : }nW n ω∈ , - . ,
{ : }nW W n ω= ∈ - X.
, a W∉ 0b W∈ . a *
{ : }nU n ω∈ , -
- *
0W U = ∅ .
X , 0 :g X X→ ,
a b . - 0U a , *
0 0U U⊂
0 0 0 0( )V g U W= ⊂ . , , 1n ≥ :ng X X→
- nU a : *
1n n nU U U−⊂
( )n n nV g U= - nW . , { : } { }nU n aω∈ = . -
3 2.
1. X
. X – .
3. - -
*U { : }iW i ω∈ , -
- . -
*U U⊂ { : }iV W i ω⊂ ∈ , A U⊂ .
. , { : }iA a i ω= ∈ . i
i ib W∈ :ig X X→ , ia ib . -
iU ia , : *iU U⊂ ( )i i ig U W⊂ .
{ : }iU i ω∈ { : }iU i k≤ .
iU { : }i jU U j i< , , { : }iU i k≤
- . { : }iU U i k= ≤
{ ( ): }i iV g U i k= ≤ – . 3 .
4. X – , -
. X -
.
: f X X A→ , ( )f A B= .
. ,
, - , A B .
{ : 1, }niW n i ω≥ ∈ . { : 1, }niW W n i ω= ≥ ∈ .
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
44. . « . . »44
2 :g X X→ , ( )g A B= . – -
, - 0U ,
: 0( )B g U⊂ , 0U W = ∅ , 0( )g U W = ∅ 0 0( )U g U = ∅ .
3 1n ≥ -
nU , nV :n n ng U V→ , , nA U⊂ , 1n nU U+ ⊂ { : }n niV W i ω⊂ ∈ .
, { : }nA U n ω= ∈ ,
. 4 2.
. 2
, 4 .
5. X – -
, . -
Z * , -
*X X Z, * .
. Z, , -
, -
Z. 4 : f X X A→ , ( )f A B= .
X A . 1
* ( )X f X A−
= , -
.
1. van Douwen, E.K. A compact space with a measure that knows which sets are homeomorphic /
E.K. van Douwen // Adv. in Math. – 1984. – Vol. 52. – Issue 1. – P. 1–33.
2. Engelking, R. General topology / R. Engelking. – Berlin: Heldermann Verlag, 1989. – 540 p.
ABOUT EXTENSION OF HOMEOMORPHISMS
OVER ZERO-DIMENSIONAL HOMOGENEOUS SPACES
S.V. Medvedev
1
Let X be a zero-dimensional homogeneous space satisfying the first axiom of countability. We
prove the theorem about an extension of a homeomorphism g: A → B to a homeomorphism f: X → X,
where A and B are countable disjoint compact subsets of the space X. If, additionally, X is a non-
pseudocompact space, then the homeomorphism g is extendable to a homeomorphism f: X → X A.
Keywords: homogeneous space, homeomorphism, first axiom of countability, pseudocompact space.
References
1. van Douwen E.K. A compact space with a measure that knows which sets are homeomorphic.
Adv. in Math. 1984. Vol. 52. Issue 1. pp. 1–33.
2. Engelking R. General topology. Berlin: Heldermann Verlag, 1989. 540 p.
30 2013 .
1
Medvedev Sergey Vasiljevich is Cand.Sc. (Physics and Mathematics), Associate Professor, Mathematical and Functional Analysis Depart-
ment, South Ural State University.
E-mail: medv@math.susu.ac.ru
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
45. 2013, 5, 2 45
519.633
. . 1
, . . 2
, . . 3
,
,
.
.
: ,
, .
1.
-
[1]
[ ]11 12( , ) ( , ) ( , ) ( , ) ( , ), ( , ) [0, ] 0,t t
tv x t v x t w x t v x w x x t Tπ= Δ − Δ + Φ ⋅ + Φ ⋅ ∈ × , (1)
[ ]21 220 ( , ) ( ) ( , ) ( , ) ( , ), ( , ) [0, ] 0,t t
v x t w x t v x w x x t Tβ π= + + Δ + Φ ⋅ + Φ ⋅ ∈ × , (2)
[ ](0, ) ( , ) (0, ) ( , ) 0, 0,v t v t w t w t t Tπ π= = = = ∈ , (3)
[ ]( , ) ( , ), ( , ) ( , ), ( , ) [0, ] ,0v x t x t w x t x t x t rϕ ψ π= = ∈ × − , (4)
( , ) ( , ), ( , ) ( , )t t
v x s v x t s w x s w x t s= + = + [ ],0 , 0.s r r∈ − >
1 1 2 2: ( , ) ( , ), : ( , ) ( , )t t
i i i iv x z x t w x z x tΦ ⋅ → Φ ⋅ → [ ]0,x π∈ , [ ]0,t T∈ -
[ ]( ,0 ; )C r R− R.
[ ]( ) ( ), ,0 ,u t h t t r= ∈ − (5)
-
[ ]( ) ( ) ( ), 0,t
Lu t Mu t u f t t T= + Φ + ∈ , (6)
. . . . [2, 3]. U, F –
, ( ) ( )t
u s u t s= + [ ],0s r∈ − , :L U F→ , [ ]: ( ,0 ; )C r U FΦ − →
, { }ker 0L ≠ , M: domM F→ , U,
[ ]: 0,f T F→ . (6) ,
,
. -
(5), (6), -
.
,
- - ,
(6), . . [4–6]. ,
, -
1
– , - ,
« ».
E-mail: omelchenko_ea@mail.ru
2
– , , - -
.
E-mail: mariner79@mail.ru
3
– , , .
E-mail: davydov@csu.ru
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
46. . « . . »46
, ,
.
[2, 3, 5, 6]
(1)–(4). , -
,
( 0, , 1, 2).ij i jΦ = =
, .
, -
, [5, 6]
. , , -
.
2.
-
[ ]( , ) ( , ) ( , ), ( , ) [0, ] 0,tv x t v x t w x t x t Tπ= Δ − Δ ∈ × , (7)
[ ]0 ( , ) ( ) ( , ), ( , ) [0, ] 0,v x t w x t x t Tβ π= + + Δ ∈ × , (8)
[ ](0, ) ( , ) (0, ) ( , ) 0, 0,v t v t w t w t t Tπ π= = = = ∈ , (9)
( ,0) ( ), [0, ],v x x xϕ π= ∈ (10)
0β < , v, w – . , ( ),0w x -
w . , [7], (7)–(10) .
( ),0v x , ( ),0w x -
(7)–(10) -
.
[ ]0,π / ,h Nπ=
, 0, , .nx nh n N= = [ ]0,T
0,τ > ,mt mτ= 0, , .m M=
v, w ( , )n mx t m
nv , m
nw . -
1
1 1 1 1
2 2
2 2
, 0, , 1,
m m m m m m m m
n n n n n n n nv v v v v w w w
m M
h hτ
+
+ − + −− − + − +
= − = − (11)
1 1
2
2
0 , 0, , ,
m m m
m m n n n
n n
w w w
v w m M
h
β + −− +
= + + = (12)
1, , 1n N= − ,
0
( ), 0, , ,n nv x n Nϕ= =
0 0 0, 0, , .m m m m
N Nv v w w m M= = = = =
(11), (12) ( ),m m m
n n nξ ηΨ = ,
1 1 1
2
1 1
2
( , ) ( , ) ( , ) 2 ( , ) ( , )
( , ) 2 ( , ) ( , )
,
m n m n m n m n m n m
n
n m n m n m
v x t v x t v x t v x t v x t
h
w x t w x t w x t
h
ξ
τ
+ + −
+ −
− − +
= − +
− +
+
1 1
2
( , ) 2 ( , ) ( , )
( , ) ( , ) ,m n m n m n m
n n m n m
w x t w x t w x t
v x t w x t
h
η β + −− +
= + +
( , )n mv x t , ( , )n mw x t v, w (7)–(10) -
. , 1 2p p
hτ + ,
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
47. . ., . .,
. .
2013, 5, 2 47
C, τ h, 1 2
2
( )p pm
n
R
C hτΨ ≤ + ,
1, , 1,n N= − 0, , 1m M= − .
1. v, w (7)–(10) , v -
t, v, w x. -
(11), (12) 2
.hτ +
. ( ),v x t , ( ),w x t
2 21 1 1
( ), .
2 12 12
m m
n tt xxxx xxxx n xxxxv h v w h wξ τ η= − + + = −
.
1. 0β < . (11), (12) , 2
.hτ ≤
. m
qρ , m
qσ q- m- ,
( , ) , ( , ) , 0, 1, 2,n niqx iqxm m
n m q n m qv x t e w x t e qρ σ= = = ± ±
(11), (12)
1
2
(2 (cos 1) 2 (cos 1))m m m m
q q q qqh qh
h
τ
ρ ρ ρ σ+
− = − − − , (13)
2
2
0 (cos 1)
m
qm m
q q qh
h
σ
ρ βσ= + + − .
2
2
(1 cos )
m
qm
q
qh
h
ρ
σ
β
=
− −
(14)
(13),
2
1
2 2
2
1 (1 cos ) 1 .
2(1 cos )
m m
q q
h
qh
h qh h
τ
ρ ρ
β
+
= − − −
− −
(15)
, 0β < ,
2
2 2
2
1 (1 cos ) 1 1
2(1 cos )
q
h
r qh
h qh h
τ
β
≡ − − − ≤
− −
q N∈ . , 1qr ≥ − q N∈ . ,
2
2 2
(1 cos ) 1 1.
2(1 cos )
h
qh
h qh h
τ
β
− − ≤
− −
2
hτ ≤ .
(14) (15)
2
1 2 2
1
2 2
2
2
2
1 (1 cos ) 1
2(1 cos )
2 2
(1 cos ) (1 cos )
2 1
1 (1 cos ) 1 .
2
(1 cos )
m
qm
qm
q
m
q
h
qh
h qh h
qh qh
h h
qh
h qh
h
τ
ρ
ρ β
σ
β β
τ
σ
β
+
+
− − −
− −
= = =
− − − −
= − − −
− −
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
48. . « . . »48
qr -
2
hτ ≤ .
1. 1 , 0β ≥ -
.
, -
[8], 1 1.
2. 0β < (11), (12) ,τ h , 2
hτ ≤ . -
(7) – (10) 2
hτ + .
3.
- (1)–(4).
[ ]0,π /h Nπ= , 0, ,nx nh n N= = .
[ ],r T− . , /T r – -
, / /T M r Lτ = = , ,L M N∈ ,
, , ,0, , .mt m m L Mτ= = − ,m m
n nv w v, w
nx kt :
( ){ } ( ){ }, , : , 0, , , 0, , .m m m m
n n n n
k
v w v w k L m k n N k M= − ≤ ≤ = =
,
( ){ } ( ) [ ] [ ]: , , ,0 ,0m m k k
n n n n
k
I v w g h Q r Q r→ ∈ − × − . 0, , ,n N= [ ],0Q r− – -
[ ],0r− -
. : [ ]
[ ]
,0
,0
sup ( )Q r
s r
g g s−
∈ −
= [ ],0 .g Q r∈ − ,
ijΦ [ ],0Q r− .
[4, 5], , -
p
τ v, w, C1, C2,
0, , ,n N= 0, ,k M= [ ],k kt t r t∈ −
1 2( ) ( , ) max ( , ) ,k m p
n n n n m
k L m k
g t v x t C v v x t C τ
− ≤ ≤
− ≤ − +
1 2( ) ( , ) max ( , ) .k m p
n n n n m
k L m k
h t w x t C w w x t C τ
− ≤ ≤
− ≤ − +
1
1 1 1 1
11 122 2
2 2
,
m m m m m m m m
m mn n n n n n n n
n n
v v v v v w w w
g h
h hτ
+
+ − + −− − + − +
= − + Φ + Φ (16)
1 1
21 222
2
0 ,
m m m
m m m mn n n
n n n n
w w w
v w g h
h
β + −− +
= + + + Φ + Φ (17)
1, , 1,n N= − 0, , 1m M= − (16) 0, ,m M= (17),
0 0
( ,0), ( ,0), 0, , ,n n n nv x w x n Nϕ ψ= = = (18)
[ ]0 0
( ) ( , ), ( ) ( , ), 0, , , ,0n n n ng t x t h t x t n N t rϕ ψ= = = ∈ − , (19)
0 0 0, 0, , .m m m m
N Nv v w w m M= = = = = (20)
[2, 3] (1) – (4)
ϕ ψ
21 220 ( ,0) ( ) ( ,0) ( , ) ( , ), (0, ).x x x x xϕ β ψ ϕ ψ π= + + Δ + Φ ⋅ + Φ ⋅ ∈
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
49. . ., . .,
. .
2013, 5, 2 49
, .
(16), (17) ( , )m m m
n n nψ ξ η= ,
1 1 1
2
1 1
11 122
( , ) ( , ) ( , ) 2 ( , ) ( , )
( , ) 2 ( , ) ( , )
( , ) ( , ),m m
m n m n m n m n m n m
n
t tn m n m n m
n n
v x t v x t v x t v x t v x t
h
w x t w x t w x t
v x w x
h
ξ
τ
+ + −
+ −
− − +
= − +
− +
+ − Φ ⋅ − Φ ⋅
1 1
21 222
( , ) 2 ( , ) ( , )
( , ) ( , ) ( , ) ( , ).m mt tm n m n m n m
n n m n m n n
w x t w x t w x t
v x t w x t v x w x
h
η β + −− +
= + + + Φ ⋅ + Φ ⋅
, 4 [5], 1
(11) , (12), 1 3 [5] .
3. 0β < , 2
hτ ≤ , [ ]2 2
: ,0L
I R Q r+
→ −
0p
τ , m
nψ 1 2p p
hτ + .
(1)–(4) { }0 1 2min ,p p p
hτ + .
, ijΦ
, .
4.
( 1)ij ijg a gΦ = − [ ]1,0 ,g Q∈ − i, j = 1, 2. , (1)–(4)
[ ]11 12( , ) ( , ) ( , ) ( , 1) ( , 1), ( , ) [0, ] 0,tv x t v x t w x t a v x t a w x t x t Tπ= Δ − Δ + − + − ∈ × , (21)
[ ]21 220 ( , ) ( ) ( , ) ( , 1) ( , 1), ( , ) [0, ] 0,v x t w x t a v x t a w x t x t Tβ π= + + Δ + − + − ∈ × , (22)
(3), (4). (16), (17)
, , - -
1
1 1 1 1
11 122 2
2 2
,
m m m m m m m m
m L m Ln n n n n n n n
n n
v v v v v w w w
a v a w
h hτ
+
− −+ − + −− − + − +
= − + + (23)
1 1
21 222
2
0 ,
m m m
m m m L m Ln n n
n n n n
w w w
v w a v a w
h
β − −+ −− +
= + + + + (24)
1, , 1, 0, , 1n N m M= − = − (23) 0, ,m M= (24), (18), (19)
(20). -
τ , - – 2
τ [9].
1, .
2. v, w (21), (22), (3), (4) , v
t, v, w
x. (23), (24) 2
hτ + .
3
1. 0β < , 2
hτ ≤ , - - .
(1)–(4) 2
hτ + .
. 1 (v,w) β = –0,75, T = 10, M = 1000, N = 16
c ( ,0) sin , [0, ]v x x x π= ∈ (7)–(10).
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. .
2013, 5, 2 51
NUMERICAL SOLUTION OF DELAYED LINEARIZED QUASISTATIONARY
PHASE-FIELD SYSTEM OF EQUATIONS
E.A. Omelchenko
1
, M.V. Plekhanova
2
, P.N. Davydov
3
For delayed linearized quasistationary phase-field system of equations the numerical method of so-
lution was proposed. The convergence of explicit difference scheme that takes account of delay in the
system under investigation was thoroughly studied. On the basis of the results obtained the implementa-
tion of the method was realized.
Keywords: Sobolev type equation, quasistationary phase-field system of equations, difference
scheme.
References
1. Plotnikov P.I., Klepacheva A.V. Siberian Mathematical Journal. 2001. Vol. 42, no. 3. pp. 551–
567.
2. Fedorov V.E., Omelchenko E.A. On solvability of some classes of Sobolev type equations with
delay. Functional Differential Equations. 2011. Vol. 18, no. 3–4. pp. 187–199.
3. Fedorov V.E., Omel’chenko E.A. Siberian Mathematical Journal. 2012. Vol. 53, no. 2. pp. 335–
344.
4. Lekomtsev A.V., Pimenov V.G. Russian Mathematics (Izvestiya VUZ. Matematika). 2009.
Vol. 53, no. 5. pp. 54–58.
5. Pimenov V.G. Trudy Inst. Mat. i Mekh. UrO RAN. 2010. Vol. 16, no. 5. pp. 151–158. (in Russ.).
6. Pimenov V.G., Lozhnikov A.B. Proceedings of the Steklov Institute of Mathematics (Supplemen-
tary issues). 2011. Vol. 275. Suppl. 1. pp. 137–148.
7. Fedorov V.E., Urazaeva A.V. Trudy Voronezhskoy zimney matematicheskoy shkoly (Proc. of the
Voronezh Winter Mathematical School). Voronezh: VGU. 2004. pp. 161–172.
8. Richtmyer R.D., Morton K.W. Difference Methods for Initial-Value Problems. Interscience, New
York, 1967. 405 p.
9. Kim A.V., Pimenov V.G. i-Gladkiy analiz i chislennye metody resheniya funktsional'no-
differentsial'nykh uravneniy (i-differential analysis and numerical methods of solutions of functional and
differential equations). Izhevsk: RKhD, 2004. 256 p. (in Russ.).
15 2013 .
1
Omelchenko Ekaterina Aleksandrovna is Senior Lecturer, Department of Humanities and socio-economic disciplines, Ural Branch of Russian
Academy of Justice.
E-mail: omelchenko_ea@mail.ru
2
Plekhanova Marina Vasilyevna is Associate Professor, Differential and Stochastic Equations Department, South Ural State University.
E-mail: mariner79@mail.ru
3
Davydov Pavel Nikolaevich is Post-graduate student, Mathematical Analysis Department, Chelyabinsk State University.
E-mail: davydov@csu.ru
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53. . .,
. .
2013, 5, 2 53
, [6, 7], Q
: ( ( , ))Q Q x tψ= , ( , ) constx tψ = – -
( , )Q x t . (3) (4)
''' 3 '' '
( 3 ) ( ) 0x x t x xx xt xxx xkQ Q k Q k qψ ψ ψ ψ ψ ψ ψ ψ+ − + + − + + = , (5)
''' '3 3 '' '
( ) ( 3 ) ( ) 0x x t x xx xt xxx xkQ Q Q k Q k qα ψ ψ ψ ψ ψ ψ ψ ψ− + − + + − + + = . (6)
( )' ψ . (5) (6), -
, ( , )x tψ , ,
0xψ ≠ [6, 7]
12
3
( ),t xx
x
k
f
ψ ψ
ψ
ψ
− +
= 23
( )xt xxx x
x
k q
f
ψ ψ ψ
ψ
ψ
− + +
= . (7)
1( ),f ψ 2 ( )f ψ – . , (7)
. 2
1( )/(3 ).xx t xf kψ ψ ψ= + -
x xxxψ -
. xtψ ,
2
1
,
3
t x
xx
f
k
ψ ψ
ψ
+
= 3
3 1
3 1
2 3
xt x x x tq f f
k
ψ ψ ψ ψ ψ= + + , ' 2
3 1 1 2
1 1 3
2 3 2
f f f f
k
= + − . (8)
(8) ,
( xxt xtxψ ψ= ). ,
2 2 4
1 3 4
3 1
3 ,
2 3
tt t t t x xq f f f
k
ψ ψ ψ ψ ψ ψ= + + + '
4 1 3 32 3f f f kf= + . (9)
xtψ ,
. , xtt ttxψ ψ= . -
,
2 2 ' 4
4 3 4 1 46 6 3 ( / ) 0t x t xf f k f f f kψ ψ ψ ψ+ + + = . (10)
, 3 0f = , (10) . -
:
1. (7) , ( , )x tψ -
(8), (9) ' 2
2 1 1/3 2 /(9 )f f f k= + , 1( )f ψ – .
(5), (6)
''' '' ' ' 2
1 1 1[ /3 2 /(9 )] 0kQ Q f Q f f k+ + + = ; ''' '3 '' ' ' 2
1 1 1( ) [ /3 2 /(9 )] 0kQ Q Q f Q f f kα− + + + = .
3 0f ≠ , , (10) -
(8), (9), , 1( )x gψ ψ= , 2 ( ).t gψ ψ= -
, 2 1( ) ( ), const.g Cg Cψ ψ= = -
2. 3 0f ≠ , ( ),ax btψ ψ= + consta = , constb = . ,
'
( ) ( )x xu Q Q ψ ψ ψ= = , ( )u u ax bt= + (1) (2)
2
0,z zzbu ka u qu− + + = 2 3
0,z zzbu ka u qu uα− + + − = .z ax bt= +
, 1 constf = 2
2 12 /(9 ) constf f k= = . (10) -
. (9) tψ , , x – . ,
0tψ ≠ , (9) 1(1/ ) 1,5 (1/ ) /(3 )t t tq f kψ ψ+ = . -
, 11,5 ( )exp(3 / 2)/[1 ( )exp(3 )/(3 )]t qC x qt f C x qt kψ = − . (8)
xψ . ,
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54. . « . . »54
1exp(3 / 2)/[1 ( )exp(3 / 2)/(3 )].x xC qt f C x qt kψ = −
(8), -
( )C x : /(2 )xxC qC k= . 1 2( ) exp[ /(2 )] exp[ /(2 )].C x A x q k A x q k= ± +
1 const,A = 2 constA = . , 1 1A = , 2 0A = -
tψ xψ
1
3 exp[(3 / 2) /(2 )]/ 2
,
1 exp[(3 / 2) /(2 )]/(3 )
t
q qt x q k
f qt x q k k
ψ
±
=
− ± 1
/(2 ) exp[(3 / 2) /(2 )]
1 exp[(3 / 2) /(2 )]/(3 )
x
q k qt x q k
f qt x q k k
ψ
± ±
− ±
.
xψ
. , ,
1 13 ln{1 exp[(3 / 2) /(2 )]/(3 )}/k f qt x q k k fψ = − − ± . (11)
2 0f = . 1( ) 3 /(2 )of kψ ψ ψ= + , constoψ = ( . 1). ,
( , )x tψ (8), (9).
(9) (8) , x – , ,
2 [ ( ) 3 2 / 2],t o oM x qψ ψ ψ ψ ψ= + + + ( ) 2 exp(3 / 2)x oN x qtψ ψ ψ= + .
(8),
2
2
2
2 3
exp ,
2 2 2 3 2
ok q q M
qt x
k k kq
ψ
ψ = ± ± − − const,M = constoψ = . (12)
, .
2 0f = , constt xxk q Aψ ψ ψ− − = = . ,
2 2
4 /(9 )o M qψ = , ( , )x tψ (2) (1) 0A = . ,
1. 2 2 2
( , ) (2 / ){ /(2 ) exp[3 / 2 /(2 )] /(3 )} / 2ou x t k q q k qt x q k M k ψ= ± ± − −
t xxu ku qu C= + + , constM = , constoψ = ,
( , , )oC C q M ψ= , (2).
2. 2 2 2
( , ) (2 / ){ /(2 ) exp[3 / 2 /(2 )] /(3 )} / 2ou x t k q q k qt x q k M k ψ= ± ± − −
(1) (2),
2 2
4 /(9 )o M qψ = .
, '2
2 ( )f Qα ψ= , 1 (7) -
,
' 2 '2
1 1 2/3 2 /(9 )f f k f Qα+ = = . (13)
, (6) ''' ''
1 0kQ Q f+ = . -
1f (13),
'''' '' '''2 ''2 '2
5 /3 3 / 0Q Q Q Q Q kα− + = .
, '
02 / (3 )Q k α ψ ψ= + , 0 constψ = . -
(13) , 1 06 /(3 )f k ψ ψ= − . , (8), (9).
,
3
01 2 3
exp ,
3 3 2 2 3
q
qt x c
q k
ψ
ψ = ± − − const.c = (14)
( )Q ψ ( 1 constf = )
''' '' ' ' 2
1 1 1[ /3 2 /(9 )] 0kQ Q f Q f f k+ + + = ; ''' '3 '' ' ' 2
1 1 1( ) [ /3 2 /(9 )] 0kQ Q Q f Q f f kα− + + + = .
( )Q ψ , ,
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55. . .,
. .
2013, 5, 2 55
'
1 1 2 1exp[ /(3 )] exp[ 2 /(3 )]Q C f k C f kψ ψ= − + − , 1 const,C = 2 constC = .
, (1) '
x xu Q Qψ= = ,
1
/(2 ) exp[3 / 2 /(2 )]
1 exp[3 / 2 /(2 )]/(3 )
q k qt x q k
u
f qt x q k k
± ±
=
− ±
1 1 2 1{ exp[ /(3 )] exp[ 2 /(3 )]}.C f k C f kψ ψ− + −
ψ (11),
1
1 2
3 3
exp 1 exp .
2 2 2 3 2 2
fq q q q q
u t x C C t x
k k k k
= ± ± + − ±
( )Q ψ . , '
( )Q p ψ= , ,
'
( )p r p= . 3 2
1 12 /(9 )pkrr f r p f p kα+ = − .
2
r ap bp= + , const,a = constb = . ,
/(2 )a kα= ± , 1 /(3 )b f k= . ,
' 1
1
/(3 )
,
/(2 ) exp[ /(3 )]
f k
Q
k C f kα ψ
= −
± −
const.C =
, (2) '
x xu Q Qψ= = , -
(2)
1
0 1
[ /(3 )] / exp[3 / 2 /(2 )]
,
1 [ /(3 )]exp[3 / 2 /(2 )]
f k q qt x q k
u
C f k qt x q k
α± ±
=
− − ±
0 .
2
C C
k
α
= ± (15)
, -
1
0 1
[ /(3 )] / exp[3 / 2 /(2 )]
,
1 [ /(3 )]exp[3 / 2 /(2 )]
f k q qt x q k
u
C f k qt x q k
α +
=
− − ±
1
0 1
[ /(3 )] / exp[3 / 2 /(2 )]
,
1 [ /(3 )]exp[3 / 2 /(2 )]
f k q qt x q k
u
C f k qt x q k
α− +
=
− − ±
1
0 1
[ /(3 )] / exp[3 / 2 /(2 )]
,
1 [ /(3 )]exp[3 / 2 /(2 )]
f k q qt x q k
u
C f k qt x q k
α −
=
− − ±
1
0 1
[ /(3 )] / exp[3 / 2 /(2 )]
.
1 [ /(3 )]exp[3 / 2 /(2 )]
f k q qt x q k
u
C f k qt x q k
α− −
=
− − ±
(16)
, 2 0f = (5) ''' ''
3 /(2 ) 0.oQ Q ψ ψ+ + = -
, , '
1 2 / 2 oQ C C ψ ψ= − + , 1 const,C = 2 constC = . -
, (12),
1 2
2 3 3 3
exp exp exp .
2 2 2 2 2 3 2 2
k q q q M q
u C qt x qt x C qt x
q k k k k k
= ± ± ± − − ±
2 0f = , (6) ''' '3 ''
( / ) 3 /(2 ) 0.oQ Q k Qα ψ ψ− + + = -
: '
2 / [1/(2 )]oQ k α ψ ψ= ± + .
(6)
1
2 3 3
exp exp .
2 2 2 2 2 2
q k q q q M
u qt x qt x
k k k k kα
−
= ± ± ± − (17)
(14), (2)
1
/ exp[3 / 2 /(2 )]{exp[3 / 2 /(2 )] } ,u q qt x q k qt x q k cα −
= ± ± ± − const.c = (18)
u
( , , ).
, (1) (2) constt = .
, (2) ( , )u x t
( , )t u x [8].
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56. . « . . »56
2 2 2 3 3
( 2 ) ( ) 0u u xx x u xu x uu ut k t t t t t t t qu u tα+ − + − − = . (19)
(19) ( ) ,u xϕ ξ− = x η= :
2 2 2 2
[ ( 2 ) 2 ( )( ) ( ) ]x x xx x x xt k t t t t t t t t t t t t tξ ξ ηη ξη ξξ ξ ξ η ξ ξη ξξ ξξ η ξϕ ϕ ϕ ϕ ϕ ϕ+ − + − − − − + −
3 3
[ ( ) ( ) ] 0.q tξξ ϕ α ξ ϕ− + − + =
, constξ = – (19), -
tξξ . , 0tη = . ,
constξ = constt = . , (19) ,
2 3 3 3
[ ( ) ( ) ] 0.xxt kt q tξ ξ ξϕ ξ ϕ α ξ ϕ− − + − + =
, , 0α = ( (1)), uξ ϕ+ = , , -
constt =
0 0 1( , ) { ( ) ( )sin[ ( ( )) / ]}/ ,u t x w t c t x c t q k q= + ± +
0 ( ) const,c t = 0 ( ) const,w t = 1( ) const.c t = (20)
(2). , ,
( , )x u t –
2 3 3
( ) 0t u uu ux x kx qu u xα− + − = . (21)
0α ≠ , constt = , -
1
4 2
0 0/(4 ) /(4 )
du
x c
u k qu k w u cα
= ±
− + +
. (22)
, , 1 1 2 2 0 0( ), ( ), ( )c c t c c t w w t= = = ,
(21),
1 1 2 2 0 0( ), ( ), ( )c c t c c t w w t= = = . -
( , )u x t , (22):
4 2 2 2 2
0 0/(4 ) /(4 ) [ /(4 )][ 2 ( ) ( / 2 ( ) ( ))][ 2 ( )u k qu k w u c k u a t u q a t b t u a t uα α α− + + = + − − − −
2
( / 2 ( ) ( ))]q a t b tα− − +
(22) -
, [9]. -
a constt = ( , ) ( )u x t x=℘ ( ( )x℘ – -
). [10], , -
, , , (20)
(1).
(2) , , -
, , (
(15)–(18) ) .
. 1 (1:
t = 0; 2: t = 0,5; 3: t = 1; 4: t = 1,5; 5: t = 2; 6: t = 2,5). . 2
(1: t = 0; 2: t = 0,1; 3: t = 0,5; 4: t = 1; 5: t = 1,5).
(15), (17), (18) , ,
, –
( . . 3, ( , )u x t
c 1t = (1: c = 100; 2: c = 1000; 3: c = 2500;
4: c = 4000; 5: c = 7000).
{ 0, 0}x t= = . , (18) [ / (0,0)]/ (0,0).c q u uα= ± − (0,0) / ,u q α= 0c =
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57. . .,
. .
2013, 5, 2 57
( , ) const /u x t q α= = . ,
( . 4, ( , )u x t
c , 1: c = 1600; 2: c =
2800; 3: c = 4900). , (15)
, , (15)
( . (16)). (0,0)u
( . 1,
. 2).
0,00 2,00 4,00 6,00 8,00 x
0,00
0,04
0,08
0,12
u
6
5
4
3
2
1
. 1.
12345
30,0
u
20,0
10,0
0,0
–4,00 –2,00 0,00 2,00 x
. 2.
0,00 4,00 8,00 x
0,00
4,00
8,00
u
12,00
1
2 3
4
5
. 3.
0,0
100,0
200,0
u
300,0
1
2
3
7,0 8,0 9,0 x10,0
. 4.
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2013, 5, 2 59
TOWARDS THE DIFFERENCES IN BEHAVIOUR OF SOLUTIONS OF LINEAR AND
NON-LINEAR HEAT-CONDUCTION EQUATIONS
L.I. Rubina
1
, O.N. Ul’yanov
2
The linear and non-linear heat-conduction equations are analyzed by the previously initiated geo-
metrical method of analyzing linear and non-linear equations in partial derivatives. The reason of the
difference in behavior of solutions of equations under consideration was stated, as well as the reason of
aggravation of the non-linear equation. A class of solutions of linear equations that represents the sur-
faces of the levels of non-linear heat-conduction equations was excluded.
Keywords: non-linear equations in partial derivatives, heat-conduction equations, exact solutions,
surfaces of the level.
References
1. Kurdyumov S.P., Malinetskiy G.G., Potapov A.B. Nestatsionarnye struktury, dinamicheskiy
khaos, kletochnye avtomaty. Novoe v sinergetike. Zagadki mira neravnovesnykh struktur (Unsteady
structures, dynamic chaos, cellular automata. New in synergetics. Mysteries of the world of nonequilib-
rium structure). Moscow: Nauka, 1996. 111 p. (in Russ.).
2. Samarskiy A.A., Galaktionov V.A., Kurdyumov S.P., Mikhaylov A.P. Rezhimy s obostreniem v
zadachakh dlya kvazilineynykh parabolicheskikh uravneniy (Blow-up regimes in problems for quasilin-
ear parabolic equations). Moscow: Nauka, 1987. 477 p. (in Russ.).
3. Vazquez J.L., Galaktionov V. A Stability Technique for Evolution Partial Differential Equations.
A Dynamical System Approach. Birkhauser Verlag, 2004. 377 p. (ISBN: 0-8176-4146-7)
4. Berkovich L.M. Vestnik SamGU – Estestvennonauchnaya seriya. 2005. no. 2(36). pp. 32–64. (in
Russ.).
5. Kurkina E.S., Nikol'skiy I.M. O rezhimakh s obostreniem v uravneniyakh
div( grad )tu u u uσ β
= + (About blow-up regimes in equations div( grad )tu u u uσ β
= + ). Different-
sial'nye uravneniya. Funktsional'nye prostranstva. Teoriya priblizheniy: Tezisy dokladov mezhdunarod-
noy konferentsii, posvyashchennoy 100-letiyu so dnya rozhdeniya Sergeya L'vovicha Soboleva. (Ab-
stracts of the International Conference dedicated to the 100th anniversary of the birth of Sobolev “Dif-
ferential Equations. Functional Space. Theory of Approximation”) Novosibirsk, 2008. p. 512.
6. Rubina L.I., Ul’ianov O.N. Trudy Inst. Mat. i Mekh. UrO RAN. 2010. Vol. 16, no. 2. pp. 209–
225. (in Russ.).
7. Rubina L.I., Ul’yanov O.N. Proceedings of the Steklov Institute of Mathematics (Supplementary
issues). 2008, Vol. 261. Suppl. 1. pp. 183–200.
8. Rubina L.I. Journal of Applied Mathematics and Mechanics. 2005. Vol. 69. Issue 5. pp. 829–836.
(in Russ.).
9. Lomkatsi Ts.D. Tablitsy ellipticheskoy funktsii Veyershtrassa. Teoreticheskaya chast' (Weier-
strass elliptic function charts. Theoretical part). Moscow: VTs AN SSSR, 1967. – 88 p. (in Russ.).
10. Gradshteyn I.S., Ryzhik I.M. Tablitsy integralov, summ, ryadov i proizvedeniy (Table of inte-
grals, sums, series and compositions). Moscow : Fizmatlit, 1962. 1100 p.
17 2012 .
1
Rubina Liudmila Ilinichna is Cand. Sc. (Physics and Mathematics), Senior Staff Scientist, Institute of Mathematics and Mechanics of the
Russian Academy of Sciences (Ural branch).
E-mail: rli@imm.uran.ru
2
Ul’yanov Oleg Nikolaevich is Cand. Sc. (Physics and Mathematics), Senior Staff Scientist, University’s academic secretary, Institute of
Mathematics and Mechanics of the Russian Academy of Sciences (Ural branch).
E-mail: secretary@imm.uran.ru
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
60. . « . . »60
519.8
. . 1
. -
-
, ( ), . -
«
» . .
, ,
. -
.
: , , ,
, .
. , -
[1, 2]:
{1,2} 1 2 1 2 {1,2}{ } ,{ ( , ), ( , ), ( ), ( )}i i
i i i i V i S i iX f x x x x R x R xϕ∈ ∈ . (1)
(1) , i- « » in
i ix X comp∈ ⊆ R ; -
1 2 1 2( , )x x x X X= ∈ × ; 1 2( , )if x x ( {1,2}i∈ ) -
1 2X X× .
i-
1 2 1 2( ) max ( , ) ( , )
i i
i i i
x X
x f x x f x xϕ
∈
= − ( {1,2}i∈ ).
i-
1 2 1 2( ) max min ( , ) min ( , )
k k k ki i
i
V i i i
x X x Xx X
R x f x x f x x
∈ ∈∈
= − ( , {1,2}i k ∈ ,i k≠ ).
1 2 1 2( ) max ( , ) min max ( , )
i ik k k k
i
S i i i
x Xx X x X
R x x x x xϕ ϕ
∈∈ ∈
= − ( , {1,2}i k ∈ ,i k≠ ).
, ( )i
V iR x ( )i
S iR x
( {1,2}i∈ ) ,
i ix X∈ . 1 2( , )if x x
Xi , i(x1, x2),
( )i
V iR x , ( )i
S iR x ( {1,2}i∈ ) .
(1) i (x1, x2)
( )1 2 1 2( , ), ( , ), ( ), ( )i i
i i V i S if x x x x R x R xϕ , (2)
fi(x1, x2) ( xi ∈ Xi) -
, .
1. . :
(2) ?
-
( ). , -
:
{1,2} 1 2 {1,2}{ } ,{ ( , )}i i i iX f x x∈ ∈ , (3)
i- 1 2( , )x x « »
1 2( , )if x x .
1
– , ,
. E-mail: solnata@pochta.ru
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
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2013, 5, 2 61
1. (3) 1 2( , )e e
x x -
( ),
1 1 2 1 1 2 1 1
2 1 2 2 1 2 2 2
( , ) ( , ) ,
( , ) ( , ) ,
e e e
e e e
f x x f x x x X
f x x f x x x X
≥ ∀ ∈
≥ ∀ ∈
1 2( , )e e
x x .
, * *
1 2( , ) 0i x xϕ = ( {1,2}i∈ ), * *
1 2( , )x x .
-
.
2. (3) *
1x
,
2 2 2 21 1
1 1 2 1 1 2max min ( , ) min ( , )
x X x Xx X
f x x f x x∗
∈ ∈∈
= .
1
1( ) 0VR x∗
= .
, *
2x ,
1 1 1 12 2
*
2 1 2 2 1 2max min ( , ) min ( , )
x X x Xx X
f x x f x x
∈ ∈∈
= ,
2 *
2( ) 0VR x = .
,
.
3. , {1,2} 1 2{ } , ( , )i iX f x x∈
1 1 2 1 2( , ) ( , )f x x f x x= ,
2 1 2 1 2( , ) ( , )f x x f x x= − 1 2( , )e e
x x ( ).
1) ,
1 2( , ) 0e e
i x xϕ = ( {1,2}i∈ );
2) ,
( ) 0i e
V iR x = ( {1,2}i∈ );
3) 11 21( , )e e
x x 12 22( , )e e
x x – ( ), -
« » ,
11 21 12 22( , ) ( , )e e e e
f x x f x x= .
, ( ) 1 2( , )e e
x x -
( )1 2 1 2( , ), ( , ), ( ), ( )e e e e i e i e
i i V i S if x x x x R x R xϕ
( {1,2}i∈ ) 1 2( , )e e
x x (
2 1 2 1 1 2( , ) ( , )e e e e
f x x f x x= − ).
« » -
.
( )
{ } { } ( )1 1
1 1 1 2 2 2,..., , ,..., ,m n
ijX x x X x x A a= = = (4)
m n× -
11 12 1
21 22 2
1 2
n
n
m m mn
a a a
a a a
A
a a a
= ,
1 1
i
x X∈ i- A, 22
j
x X∈
j- ,
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
62. . « . . »62
1 2( , )ji
x x ija ( )ijA a= , ,
1 1 1 2 1 12 2 2 2( , ) ( , ) , ( , ) ( , )j j j ji i i i
ij ijf x x f x x a f x x f x x a= = = − = − . (5)
{ } { } ( ) ( ){ } { }
{ }1 2 1 2
1 12 21,2 1,2
, , , ( ) ( ), ( ), ( )j jk i i
A k ij k ij V V S Sk k
X A a R x R x R x R xϕ∈ ∈
Γ = = Φ = ,
{ }1
1 1 1,..., m
X x x= – , { }1
2 2 2,..., n
X x x= – -
, 1 2( , )ji
x x -
.
-
. ( )1
1 ijϕΦ = -
:
{ }1 1
1
1 1 1 1 1 12 2 2
1,...,
( , ) max ( , ) ( , ) max
i
j j ji i i
ij ij ij
i mx X
x x f x x f x x a aϕ ϕ
∈∈
= = − = − ,
( )2
2 ijϕΦ =
{ }22
2
2 1 2 1 2 12 2 2
1,...,
( , ) max ( , ) ( , ) min
j
j j ji i i
ij ij ij
j nx X
x x f x x f x x a aϕ ϕ
∈∈
= = − = − .
1
i
x
{ } { } { }1 1 2 22 2
1
1 1 1 1 12 2
1,..., 1,...,1,...,
( ) max min ( , ) min ( , ) max min min
j ji
j ji i i
V ij ij
j n j ni mx X x X x X
R x f x x f x x a a
∈ ∈∈∈ ∈ ∈
= − = − ,
2
j
x
{ } { } { }1 1 1 122
2
2 1 2 12 2 2
1,...,1,..., 1,...,
( ) max min ( , ) min ( , ) max min max
i ij
j j ji i
V ij ij
j ni m i mx X x Xx X
R x f x x f x x a a
∈∈ ∈∈ ∈∈
= − = − .
1
i
x
{ } { } { }1 12 22 2
1 1 1
1 1 1 1 12 2
1,...,1,..., 1,...,
( ) max ( , ) min max ( , ) max min max
ij j
j ji i i
S ij ij
i mj n j nx Xx X x X
R x x x x xϕ ϕ ϕ ϕ
∈∈ ∈∈∈ ∈
= − = − ,
2
j
x
{ } { } { }1 1 1 122
2 2 2
2 1 2 12 2 2
1,...,1,..., 1,...,
( ) max ( , ) min max ( , ) max min max
ji i
j j ji i
S ij ij
j ni m i mx X x Xx X
R x x x x xϕ ϕ ϕ ϕ
∈∈ ∈∈ ∈∈
= − = − .
(4). (4),
1 2( , )ji
x x (5).
( )1 1 22, ji
x x X X
∗∗
∈ × ( ),
1 1 1 22 2( , ) ( , ) ( , )j jk i i t
f x x f x x f x x
∗ ∗∗ ∗
≤ ≤
1 1
k
x X∈ 2 2
t
x X∈ .
kj i j i t
a a a∗ ∗ ∗ ∗≤ ≤
{ }1,..., ,k m∈ { }1,...,t n∈ .
. ( ) ,
. :
7 1 4 1
4 2 3 2
2 2 5 2
4 3 7 2
A
− −
=
− −
.
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2 2 2 4 3 2 3 4
1 2 1 2 1 2 1 2( , ),( , ),( , ),( , )x x x x x x x x , -
( ) .
:
1 2
1( ),SR x 1 3
1( ),SR x 2 2
2( ),SR x 2 4
2( )SR x .
( )1 1
S ijϕΦ = ( )
:
1
maxij ij ij
i
a aϕ = − .
1
0 3 11 1
3 0 4 0
5 0 2 0
3 5 0 4
SΦ = .
( )2 2
S ijϕΦ = :
2
max( ) ( ) minij ij ij ij ij
jj
a a a aϕ = − − − = − ,
, 2 1 2( , )ji
ijf x x a= − .
2
11 3 0 5
2 0 1 0
0 0 3 0
7 0 10 1
SΦ = .
:
1
min max 4,ij
i j
ϕ = 2
min max 3ij
j i
ϕ = .
1 1 1
1( ) max min maxi
S ij ij
ij j
R x ϕ ϕ= − 2 2 2
2( ) max min maxj
S ij ij
ji i
R x ϕ ϕ= − ,
:
1 2
1( ) 0,SR x = 1 3
1( ) 1,SR x = 2 2
2( ) 0,SR x = 2 4
2( ) 2SR x = .
, 2 2
1 2( , )x x ,
.
2. S-
(1) :
1 1
1 1 1 1,{ ( ), ( )}e
V SX R x R xΓ = − − ,
1
e
X – ,
2 2
2 2 2 2,{ ( ), ( )}e
V SX R x R xΓ = − − ,
2
e
X – .
1. 1 1
e e
x X∈ 2 2
e e
x X∈ (1)
s- , -
1Γ 2Γ .
1. (1) :
1) 1X 2X ;
2) 1 1 2( , )f x x 1 2X X× 1x
2 2x X∈ ;
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
