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# Фундаментальная и прикладная математика (2002 №4) (2002).pdf

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Abstract
A. F. Voronin, Volterra convolution equation of rst kind on segment, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 4, pp. 955{966.
In the paper we obtain the decidability conditions for Volterra convolution equation of .rst kind on a segment and the solution of this equation (in quadratures).
1- 0 b],
b > 0:
Zx
k(x ; t)u(t) dt = f(x) x 2 0 b]
(0.1)
0
"# p = x + iy %&
F k(p) :=
Zb
k 2 L1 (0 b) f 2 L2 (0 b):
(0.2)
eipt k(t) dt | %(
){+% ,- k:
0
. /, / # 0 s > ;1, y0 > 0 0%#1# 12
3#:
lim t;s k(t) = C0 6= 0
(0.3)
t!+0
1+s
(0.4)
jxlim
j!1 j(x + iy0 ) F k(x + iy0 )j > C1 > 0:
/ '0 %%1 \$ /223 99-01-00540.
, 2002, " 8, 4 4, . 955{966.
c 2002 !" #\$ %&',
(
\$ )
*
956
. . ( # (0.1) % # (0.2){(0.4). 9 %21
0 / # / ,-: 1, 2] %/0
(
: ,) # 3 # (0.1) , # 3#. / (% # (0.2){(0.4)), : 0 , / ;/3 3, . 415].
1. 0 (0 %
0 # :. </ c
0 .
1. k f 2 L1 (0 b) (0.3){(0.4). (0.1) L1 (0 b).
=& / ,
121 , %&
F f(p):=
Zb
0
eiptf(t) dt
@
u (t):= 2i @t
Z1
;1
F f(+i) d
e;it( +i) (+i)
m+1 F k(+i) (1.1)
> y0 m | / m 2 1 + s 2 + s):
> W2m (0 b) ( (/ %
9(
,
f 2 W2m (0 b) \$ f (l) 2 L2 (0 b) l = 0 1 : : : m
f (l) | ((2# %
# l- %# (0 b).
? 2. (0.2){(0.4). , > y0 :
u 2 W2m (0 b)
(1.2)
lim f(x) = 0
(1.3)
x!+0
(0.1) L2 (0 b).
. (0.1)
L2 (0 b), (1.3) > y0 ,
(1.2). !
" (0.1) #
\$
"
u(x) = im u(m) (x) # x 2 (0 b):
(1.4)
;0 1{2 %1 121 .
. (0.3){(0.4). > y0 ,
" "
j(x + iy)m F k(x + iy)j > C > 0 # x 2 R, y > :
(1.5)
1- 957
; 2 0%## ( (2 /, 3. (0.2). # " m " > 0 % &{( k (1.5), %"
(0.1) L2 (0 b)
#
"
) (1.2){(1.3). !
" (0.1) #
\$
" (1.4).
2. 1{3
=%&, / (
0 %
&). " , ( /, / % 0 m 2
> 0 %
(1.5).
= 3 # (0.1) 2
Lq (0 b), q = 1 ( q = 2.
=&
Zb
v(x) := k(x ; t)u(t) dt x > b
v(x) := 0 x < b
(2.1)
0
u(x) = f(x) = k(x) = 0 x > b:
(2.2)
; (0.1) %## 1 %%#1 (0 1)
12 (:
Zx
k(x ; t)u(t) dt = f(x) + v(x)
x 2 (0
1):
(2.3)
0
A
(2.3) & 0 ,- u v. ? B
Z1
k(x ; t)u(t) dt
q
0
6 kkk1kukq kukq =
Z1
1=q
ju(t)jq dt
0
, /
v(x) 2 Lq (b 2b) q = 1 2 (v(x) = 0 x > 2b):
=
1 (2.3) %(
){+% (0
/
F k(p)F u(p) = F f(p) + F v(p) Imp > F u(p) =
Zb
0
eipt u(t) dt
F v(p)
=
Z2b
b
eipt v(t) dt:
(2.4)
1), %(2.5)
958
. . (2.5) % p = +i, 2 R. A&
&
e;ib F k(1+ i)
(/
+ i)
F ;() := e;ib F u( + i) G() := e;ib ( + Fi)f(
m F k( + i) 2 R (2.6)
( + i)m
F + () := e;ib F v( + i) ( + i)
(2.7)
m F k( + i) 2 R
%/
F ; () = F + () + ( + i)m G() 2 R:
(2.8)
=: % %
0 1. D&
/, / f = 0. ; (2.8) G = 0 %# ,- G
(2.6). ? / 1 %
1 / # (2.8) %%# E+ E; , E := fx + iy : x 2 R y > 0g.
?
0
Z
F ; () = e;ib F u( + i) = eit e;(t+b) u(t + b) dt:
;b
(2.9)
? (2.9) , / # / # (2.8) #
## /: /: ,-: %% E; %0
: % -0.
"# %
: / # (2.8) , -%
0,
Zb
e;ib F v( + i) = eit e;(t+b) v(t + b) dt
(2.10)
0
-
0, ,-# ( + i)m F k( + i) /# #
(
# (1.5)) %% E+ . 9
, %
# /
# (2.8) #
## /: ,-: %% E+ , %0
: % -0 2: F: %% (0
/ M j jm % j j ! 1, M = const. ; % %-% %0
%&# 4, c. 87] 2
-# ,-# F (), # F () % 2 E .
: %: % -# ,-# F () (0 /
M j jm % j j ! 1. ; % +
# 4]
F () = Qm () Qm | %
0: % % m:
9
, F () = Qm (), 2 R. ; (2.9) %/
Z0
;b
eit e;(t+b) u(t + b) dt = Qm ()
2 R:
(2.11)
= {+( # / (2.11) # 1
% ! 1. 9
, %
# / F & &
1- 959
## 1 (/, / & 3 % Qm 0. ;
u 0. ; 1 .
# 0 2. G(/ / P %-0
%0 1, c. 154], P : L2 (R) ! L2 (R),
P G(p) = P fG(t)g(p) := 1
2i
Z1
;1
G(t) t ; (pdt i0)
p 2 R:
G 12 :
:
P + + P ; = I (/0: %) (P )2 = P P ;P + = P + P ; = 0:
; # (2.8) / ,0
G(t) = P + G(t) + P ; G(t) P G 2 L2 (R) \ C(R)
(2.12)
%/
F ;() ; ( + i)m P ; G() = F + () + ( + i)m P + G() 2 R: (2.13)
H, / %0
,-: P G(t) (2.12) / ,- G(z) % j Imz ; j < % > 0.
= 0 1 (0 %, / F () | /
,- %%# E %0
0 % -0. J , ,-# F ;() / E; , ,-# F + () %% E+ (0 / M j jm % j j ! 1, M = const. ;,
/ 3: (2.12), %/, / # %
# / # (2.13) #
#1# / ,-# %%# E; E+
%0
0 % -0. J , F %%# (0 / M j jm % j j ! 1. ; % %-%
%0
%&# 4, c. 87] +
# %/
F () ; ( + i)m P G() = Qm () 2 R:
(2.14)
? (2.14) (2.6) F u( + i) = eib (Qm () + ( + i)m P ; G()) 2 R:
(2.15)
+
# / (2.15) c# 0 % ! 1 % {+(. 9
, ;( + i)m P ; G() Qm () % ! 1:
(2.16)
? (2.16) , / %# % Qm m ; 1 , /
P ; G(t) ! 0 % t ! 1:
H/, Qm = Qm;1 .
=&
U() := eib ( + i)m P ;G():
(2.17)
960
. . ; (2.15) (2.17) F u( + i) = U() + eib Qm;1() 2 R:
(2.18)
=&, / (2.18) 1 2
# (1.2) , (1.4).
=
# / (2.18) %& L2 (R) , /
# / F L2 (R) % %1. =
(2.18)
( %(
), # %/ x 2 R %/
u(x)e;x = F ;1fU() + eib Qm;1 ()g(x):
(2.19)
? (2.19), / 0 68 3, c. 122], # n = 1 : : : m n @ n Z1
i
u(x) = 2 @xn e;ix( +i) (U() + eib Qm;1 ()) ( +di)n 2 L2 (R): (2.20)
;1
=& %, /
u (x) = ex F ;1feib P ;G()g(x) x 2 (0 b):
(2.21)
, %# ,- G (2.6) # (1.5) ,
/ eib G() 2 H 2(E+ ). ; % =F{ 2
,-# g 2 L2(0 1), # /
Z1
G() = eit g(t + b) dt 2 R g(t) = F ;1 G(t ; b):
;b
=
% %
# P ; %/ , Z0
;
P ; G() = eitg(t + b) dt 2 R:
;b
eib P ;G() =
Zb
0
eit g(t)dt
2 R:
=
03#2 ( %(
), F ;1 feib P ; G()g(t) = F ;1 G(t ; b) t 2 (0 b):
? %/ , / (1.1), 0 %
(2.21).
=&
Z1
1
Jm (x) := 2 e;ix( +i) eib Qm;1 () ( +di)m :
;1
1- 961
; (2.20){(2.21) %/
@ m (u (x) + J (x)) 2 L (0 b):
u(x) = im @x
(2.22)
m
2
m =&, /
Jm (x) = 0 % x < b:
(2.23)
, Qm;1 ()(+i);m 2 H 2(E+ ) H 2 (E+ ) | K %% E+
(2.23) 1, c. 23], 3, c. 170]. ; (2.22) (2.23)
%/
@ m u (x) 2 L (0 b):
u(x) = im @x
(2.24)
2
m ? (2.24) %
(1.2) (1.4). ? 2
# 3#
# (0.1) L2 (0 b) (1 0 (1.3).
2 .
"# % 0 2 % /
: (1.2){(1.3) # 2
# 3# # (0.1). = %
0 # (1.2){(1.3). =&, / ,-# u(x), %# ,: (1.4), | 3 # (0.1), / 3 0 2.
=&
Zx
v(x) := k(x ; t)u(t) dt ; f(x)
x>0
(2.25)
0
u(x) = f(x) := 0 x > b:
=
1 (2.25) %(
){+%, %/
# (2.5):
F k(p)F u(p) = F f1 (p) + F v(p) Imp > 0
F u(p) =
Zb
0
eipt u(t) dt
F v(p)
=
Z2b
b
eipt v(t) dt
F f1 (p) =
Zb
0
eipt (f(t) + v(t)) dt:
" &# &, % 3 # (2.5), %/ # (2.18):
(2.26)
F u( + i) = eib ( + i)m P + G1() ; Q1m;1 ()]
G1() := e;ib ( +Fi)f1m(F+k(i)+ i) 2 L2(R)
Q1m | %
0: % % m.
962
. . )-# u (1.3) # 1 (2.18) % %1. ;
(2.26) 0/ (2.18), %/
0 = ( + i)m (P ; G1() ; P ; G()) + Q1m;1() ; Qm;1()
P ; G() ; P ; G1() = ;( + i);m (Q1m;1 () ; Qm;1 ()) 2 R: (2.27)
+
# / (2.27) H 2 (E; ) , %
# | H 2 (E+ ). 9
, # %
# / 0 1 (
/ %& +
#. H/,
Q1m;1() ; Qm;1 () = 0 2 R G1 ; G 2 H 2 (E+ ):
; %: ,-: G, G1 Z b
G1() ; G() = e;ib ( + i)m F1 k( + i) ei( +i)x v(x) dx 2 H 2 (E+ ):
0
H/,
Zb
e;ib eix v(x) dx 2 H 2 (E+ ):
(2.28)
0
93 (2.28) & 3 % , / v(x) = 0, x 2 (0 b). ;
(2.25) %/ (0.1) ,-1 u. ; 2 (% %
0). J , 3.
3. 9 1 (0.3) & /, / # k %
(0 b) 12 :
k(t) = (C0 + (t))ts t 2 (0 b)
(3.1)
(t) 2 L1 (0 b) (t) = o(1) % t ! +0:
(3.2)
ipt
;, # (3.1) % t 0 b e , %/
F k(p) = C0
=&
Zb
0
eipt ts dt +
Zb
eipt ts (t) dt:
(3.3)
0
(t) := ;C0 % t > b
(3.3) F k(p) = C0I1 (p) + I2 (p)
Imp > 0
(3.4)
1- Z1
Z1
0
0
I1 (p) = eipt ts dt I2 (p) =
963
eipt ts (t) dt:
? (3.4) %/
jF k(p)j > jC0j jI1(p)j ; jI2 (p)j Im p > 0:
(3.5)
G- 0 %
: / (3.5). =
0: (/0: 5, c. 35], I1 (p) = (s + 1)(;ip);(s+1) Imp > 0:
(3.6)
"# %
12# /
# -:
jI2 (p)j 6
; (3.5){(3.7) %/
Z1
e;yt ts j
(t)j dt y > 0:
(3.7)
0
1
x s+1 s+1Z ;yt s
s
+1
y
e t j
(t)j dt
j(x+iy) F k(p)j > j(s+1)C0 j; + i
y
y > 0: (3.8)
0
= L(# 5, c. 157] # y ! 1, / # (3.2), %/
y
1+s
Z1
e;yt tsj
(t)j dt (s + 1)j
(+0)j = o(1):
(3.9)
0
? (3.8){(3.9) # jxj 6 y / j(x + iy)s+1 F k(x + iy)j > j(s + 1)C0j % y ! 1:
(3.10)
9 (3.10) # % x. ; (3.10), #
,- F k(p) (/ : / %%
Imp > 0, %
0 # jxj 6 y.
G # /# jxj > y > y0 . +1(1 / p = x + iy
( jxj > y > y0 ( %
# 12 :
p = x + ijxj tg() + iy0 0 < 6 4 :
; (3.5){(3.6) # y > y0 %/
jps+1 F k(p)j > j(s + 1)C0j ;
s+1
Z1
y
0
s
+1
jxj
expf;tjxj tg()ge;y0 t tsj
(t)j dt: (3.11)
; sgn(x) + i tg() + i jxj
0
964
. . = L(# 5, c. 157] # jxj ! 1, / # (3.2), %/
1+s
jxj
Z1
Z1
;
y
t
s
1+s
0
expf;tjxj tg()ge t j
(t)j dt 6 jxj
ts j
(t)j dt = o(1): (3.12)
0
0
? (3.11){(3.12) # 0 < 6 4 / j(x + iy)1+s F k(x + iy)j > j(s + 1)C0j % jxj ! 1
(3.13)
%/ (3.13) 0%## % 2 0 4 ] # 1(
0 2 (0 4 ). ? # (0.4) , / (3.13) 0%## #
= 0 (y = y0 ). =/, / (3.13) 0%## %
2 0 4 ] # y > y0 . + .
4. !
4.1. =
k(t) = 1 % t 2 0 b] f 2 L2 (0 b):
; (0.1) ( 12: :
Zx
u(t) dt = f(x)
(4.1)
x 2 0 b]:
0
?
eipb ; 1 :
ip
9
, # s = 0 (m = 0), y0 > 0 # (0.3){(0.4) 0%#1#, F k(p) =
u (x) = ; 2i
Z1
;1
e;ix( +i) 1F;f(eib+( +i)
i) d
x 2 (0 b)
(4.2)
> y0 .
=&, / %
# / (4.2) ;if(x). ,
%/ %& ,-1 f (b 1) % b (f(x) =
= f(x + b), x 2 (0 1)). ?
12# , # ( +%
%/ ,-::
Z1
0
e;pt f(t) dt =
b
1 Z e;pt f(t) dt
1 ; e;bp
p = ;i( + i):
(4.3)
0
; (4.3) %
%/, /
1
f(t) = 2
Z1
;1
e;it( +i) 1F;f(eib+( +i)
i) d:
(4.4)
1- 965
? (4.4) (4.2) u (x) = ;if(x) x 2 (0 b):
(4.5)
? (4.5), 2, %/, / 3 # (0.1) % (4.1) 2
L2 (0 b) , f 0 2 L2 (0 b) xlim
!+0 f(x) = 0
% F u(x) = f 0 (x), x 2 (0 b).
4.2. =%&, / # # (0.1), : (0.2), (1.3), 0% 12 :
k0 2 L1 (0 b) k(0) = 1 k(b) = 0:
(4.6)
; % , # % /# b
1 1 + Z eipt k0 (t) dt :
F k(p) = ;
ip
(4.7)
0
? (4.7) /
3#
Zb
eiptk0 (t) dt ! 0 % Imp ! 1 (
% Re p 2 R)
0
0, / (1.5) 0%## # m = 1 /
(3 > 0. ; (1.1) (4.7) %/
1 @
u (x) = 2
@x
Z1
;1
F f( + i)d
e;ix( +i) ( + i)(1
+ F k0( + i)) 2 L2 (0 b):
? (4.8) , / / , /
f 0 2 L2 (0 b)
(4.8)
(4.9)
1 + F k0(p) > C > 0 % Imp > u 2 W21 (0 b):
(4.10)
, 3 (4.10) 0 (4.8){(4.9) 12 :
i (f(b)eib( +i) ; F f 0 ( + i)) 2 R
F f( + i) = ;
+ i
Z1
ib( +i)
e;ix( +i) ( + i)2e(1 + F kd0 ( + i)) = 0 x 2 (0 b):
;1
966
. . ; 0 3 %/, / (0.1) % # (0.2), (1.3),
(4.6), (4.9) 3 L2 (0 b). = F 3 F f 0 ( + i) d
t
;
1
u(x) = e F
(4.11)
1 + F k0( + i) (x) x 2 (0 b):
9 : 0, % 0%0 # (0.2), (1.3) (4.6), (4.9). ;, ,,-# (0.1) % x, %/
Zx
u(x) + k0 (x ; t)u(t) dt = f 0 (x) x 2 (0 b):
(4.12)
0
?
, / 2- (4.12) 3 L2 (0 b),
, 3 . J , % b ! 1
3 # (4.12) & , (4.11).
L
% L. +. .: (& : (0.
"
1] . ., . . . | .: , 1978.
2] \$ . . %&'( &) *), +,) + %) // ). . | 1958. | /. 13, 1 5 (83). | 4. 3{120.
3] /7)8 9. \$. :* ; %& '. | .-=.: +*,
1948.
4] =' . ?., @A B. :. *( CD )&% )%. | .: , 1987.
5] : * E&' B., B)) F. GD 7& *%
A+ =&. | .: =, 1952.
"+ 2000 #.
. . . . . 519.95
: , , , , .
, ! "# \$, "\$%
"
#
, \$, "\$% "
.
Abstract
V. Sh. Darsalia, Relative completeness for functional systems of polynomials,
Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 4, pp. 967{977.
For functional systems of polynomials with natural, integer and rational coe/cients we solve the problem of completeness of sets, containing all monomials, and
sets, containing all polynomials of one variable.
N, Z Q | (
0),
X 2 fN Z Qg. PX !, "
#, \$ # # X . % X (" . ., . . . . )& , # , "
" \$ X , (" X 2 fN Z Qg, ) . ., . . . . .
U = fu1 u2 : : : um : : :g, (" um | # X
(m = 1 2 3 : : :), | . . "
# U " x, y, z , t, ".
.
") # , , /, r " PX :
(f )(x1 : : : xn) = f (x2 x3 : : : xn x1)
(f )(x1 : : : xn) = f (x2 x1 x3 : : : xn)
(/f )(x1 : : : xn;1) = f (x1 x1 x2 : : : xn;1) n > 1
(f ) = (f ) = (/f ) = f n = 1
(rf )(x1 x2 : : : xn xn+1) = f (x2 x3 : : : xn xn+1):
, 2002, 8, 0 4, . 967{977.
c 2002 ,
!"
#\$ %
968
. . 4, f (x1 x2 : : : xn) g(xn+1 : : : xn+m ) PX , #(
(f g)(x2 : : : xn xn+1 : : : xn+m ) = f (g(xn+1 : : : xn+m ) x2 : : : xn):
% # .
5( FX = (PX 6), (" 6 = f / r g, ) (
. .) X .
. . FX I , !
" #"
M (M PX ) #"
I (M ), 7 !, # M #7
( # #! 6. 8
) M , I (M ) = M , , I (M ) = PX .
9(" # , "7 " #"
, (
& , . . FX M T , T M M # FX .
8:! "! "
. . FN, FZ FQ!
# , "7 ", , "7 "! #!.
; . . FN.
8. #. f (x1 : : : xn) , " i j (1 6 i < j 6 n) !" ak E2 = f0 1g (k =
= 1 : : : i ; 1 i + 1 : : : j ; 1 j + 1 : : : n), f (a1 : : : ai;1 xi ai+1 : : : aj ;1 xj aj +1 : : : an) = xi + xj :
8. #. f (x1 : : : xn) , " i j (1 6 i < j 6 n) !" ak E2 = f0 1g (k =
= 1 : : : i ; 1 i + 1 : : : j ; 1 j + 1 : : : n), f (a1 : : : ai;1 xi ai+1 : : : aj ;1 xj aj +1 : : : an) = xixj :
.
") "7 :
V0 PNn f0g&
V1 | . #. !, "! "&
V+ | ""
. #. !&
V | #
. #. !.
=#
"
"7 # >1]: . . FN
M
(M PN) , M * V0 M * V1 M * V+ M * V :
1. M . . , . . , . . FN
, M .
. M # FN& (" # M * V+ . ="
, M " ""
.
969
@# " . M , "7 . #. ", " ""
& (" M * V+ . @ 0 2 M ,
1 2 M , xy 2 M , M * V0, M * V1 M * V . ="
, # M #! FN. @ ".
2. , , M . . ! . .
F .
N
. A #" ""
! ", :(
, " . #. ""
!. ="
, :(
, " " ""
.
. 1, M " ""
, . #. "
# , " ""
! , . #. "
#! . @ ".
3. M . . , . . , . . FN
, M .
. M # FN& (" # M * V+ M * V . ="
, M " ""
#
.
@# " . M , "7 . #. "! #!, " ""
#
&
(" , M * V0 , M * V1 , M * V+ M * V . ="
, # M #! FN. @ ".
4. , , M . . ! . . F .
N
. A #" ""
! ", :(
, " . #. ""
!. A #" #
! ", :(
, " . #. #
!. ="
, :(
,
" " M ""
#
.
. 3, M " ""
#
, . #. ! "! #! # , " ""
! #
! , . #. !
"! #! #! . @ ".
970
. . ; . . FZ
.
1. " . . f (t1 : : : tn) M , ! . . . . >f 2 (t1 : : : tn )+1] (x ; y) . . F , f (t1 : : : tn) Z.
Z
. M | # . 4#, f (t1 : : : tn) Z& (" f 2 (t1 : : : tn)+1 > 2 " !
# t1 : : : tn. ="
, >f 2 (t1 : : : tn ) + 1] (x ; y) # ! Z(
, 1).
4, # # " #"
H1 H2 H3 : : :
! PZ
.
C ". H1 = M .
A"
! #". # H1 : : : Hl & (" Hl+1 #" ##! "
g(h1 : : : hm ), (" g | M , h1 : : : hm | #, 1Hl .
D, S Hl = I (M ).
l=1
= #7 ! " # l #, S Hl
l=1
" !! . #. " x y + c, (" c | #
Z.
C ". ", H1 " "
x y + c.
A"
! #". Hl " "
x y + c& (" #, Hl+1 " "
x y + c.
; #
## h = g(h1 : : : hm ) Hl+1 , ("
g | M , h1 : : : hm | #, Hl .
. "7 .
1. g = >f 2 (t1 : : : tn) + 1] (x ; y)& (" , m = n + 2 h =
= g(h1 : : : hm ) = >f 2 (h1 : : : hn) + 1] (hn+1 ; hn+2 ). % ## # 1& "
, " x y + c.
2. g = dz1k1 : : :zsks , (" d | , k1 : : : ks | & (" , m = s h = g(h1 : : : hm ) = dhk11 : : :hks s . E(
, \$ ## ! " x y + c, ) d = 1&
) 7
i (1 6 i 6 m), g(z1 : : : zm ) = zi + a (a | ) hi ! " x y + c.
8 hi | #, Hl , " x y + c, . . h " x y + c.
="
, Hl+1 " " x y + c.
1
971
A, " x y + c " S Hl = I (M )&
l=1
#\$ I (M ) 6= PZ
. #
. F, M | #
, f (t1 : : : tn) Z.
@# " . f (t1 : : : tn) Z, . . 7
! c1 : : : cn Z, f (c1 : : : cn) = 0. @ c1 : : : cn 2 M , >f 2 (c1 : : : cn) + 1] (x ; y) = x ; y 2 I (M ). @ ,
I (M ) " #" fx ; y xy 1g, #! FZ>1].
="
, M | # . E ".
1
5. # , , . . ! . .
F .
Z
. 4# #
& # 7
( 5,
#7!, #
. #. !
. #. "
# . . ,
5 #), "\$ f>f 2(t1 : : : tn)+1] (x ; y)g,
(" f (t1 : : : tn) | #
PZ
, . #.
"
# . 8 (" 1 7
(,
#7!, #
. #. f (t1 : : : tn) Z,
. . 7
( " : #
( " .
% #
>2]. @ ".
2. " . . f (t1 : : : tn) M ,
! . . . . >f 2 (t1 : : : tn) + 1] (x ; y), >f 2 (t1 : : : tn) + 1] (xy), . . FZ
, f (t1 : : : tn) Z.
. M | # . 4#, f (t1 : : : tn) Z& (" f 2 (t1 : : : tn) + 1 > 2 " ! # t1 : : : tn. ="
, >f 2 (t1 : : : tn) + 1] (x ; y)
>f 2 (t1 : : : tn) + 1] (xy) # ! Z (
,
1).
4, # # " #"
H1 H2 H3 : : :
! PZ
.
C ". H1 = M .
A"
! #". # H1 : : : Hl & (" Hl+1 #" ##! "
g(h1 : : : hm ), (" g | M , h1 : : : hm | #, Hl .
1
D, S Hl = I (M ).
l=1
972
. . = #7 ! " # l #, S Hl
l=1
" !! . #. " x y + c, (" c | #
Z.
C ". ", H1 " "
x y + c.
A"
! #". Hl " "
x y + c& (" #, Hl+1 " "
x y + c.
; #
## h = g(h1 : : : hm ) Hl+1 , ("
g | M , h1 : : : hm | #, Hl .
. "7 .
1. g = >f 2 (t1 : : : tn) + 1] (x ; y)& (" , m = n + 2 h =
= g(h1 : : : hm ) = >f 2 (h1 : : : hn) + 1] (hn+1 ; hn+2 ). % ## # 1& "
, " x y + c.
2. g = >f 2 (t1 : : : tn) + 1] (xy)& (" , m = n + 2 h =
= g(h1 : : : hm ) = >f 2 (h1 : : : hn) + 1] (hn+1 hn+2). % ## # 1& "
, " x y + c.
3. g | " M , . . "
h = g(z ) = ck z k + ck;1z k;1 + : : : + c1 z + c0&
(" , m = 1 h = g(h1 : : : hm) = g(h1 ) = ck hk1 + ck;1hk1 ;1 + : : : + c1 h1 + c0 :
E( , h ! " x y + c, ) g(z ) = z + c0 &
) h1 !! ! " x y + c.
8 h1 | #, Hl , " x y + c, . . g(h1 : : : hm ) " x y + c.
="
, Hl+1 " " x y + c.
S1
A, " x y + c " Hl = I (M )&
l=1
#\$ I (M ) 6= PZ
. #
. F, M | #
, f (t1 : : : tn) Z.
@# " . f (t1 : : : tn) Z, . . 7
! c1 : : : cn Z, f (c1 : : : cn) = 0. @ c1 : : : cn 2 M , . #. >f 2 (c1 : : : cn) + 1] (x ; y) = x ; y 2 I (M ) . #. >f 2 (c1 : : : cn) + 1] (xy) = xy 2 I (M ). @ , I (M )
" #" fx ; y xy 1g, #! FZ
. ="
, M | # . E ".
1
6. # , , . . ! . .
F .
Z
973
. 4# #
& # 7
( 5,
#7!, #
. #. !
. #. ! "! #! # . @(", , 5 #), "
\$ f>f 2 (t1 : : : tn) + 1] (x ; y) >f 2 (t1 : : : tn) + 1] (xy)g, (" f (t1 : : : tn) | #
PZ
, . #. ! "!
#! # . 8 (" 2 7
(,
#7!, #
. #. f (t1 : : : tn) Z,
. . 7
( " : #
( " .
% #
>2]. @ ".
!") . c. FQ.
3. " . . f (t1 : : : tn) M , ! . . f 2 (t1 : : : tn) (x ; y)2 + (x ; y),
. . FQ
, f (t1 : : : tn) Q.
. M | # . 4#, f (t1 : : : tn) Q.
# " #"
H1 H2 H3 : : :
! PQ.
C ". H1 = M .
A"
! #". # H1 : : : Hl & (" Hl+1 #" ##! "
g(h1 : : : hm ), (" g | M , h1 : : : hm | #, 1Hl .
S
D, Hl = I (M ).
1
l=1
= #7 ! " # l #, S Hl
l=1
" !! . #. " x y + c, (" c | #
Q.
C ". ", H1 " "
x y + c.
A"
! #". Hl " "
x y + c& (" #, Hl+1 " "
x y + c.
; #
## h = g(h1 : : : hm ) Hl+1 , ("
g | M , h1 : : : hm | #, Hl .
. "7 .
1. g = f 2 (t1 : : : tn) (x ; y)2 + (x ; y)& (" , m = n + 2
h = g(h1 : : : hm ) = f 2 (h1 : : : hn) (hn+1 ; hn+2 )2 + (hn+1 ; hn+2 ) =
= (hn+1 ; hn+2 )>f 2 (h1 : : : hn) (hn+1 ; hn+2 ) + 1]. E( , h
974
. . ! " x y + c, ("
hn+1 ; hn+2 1 f 2 (h1 : : : hn) (hn+1 ; hn+2) + 1 x y + c
hn+1 ; hn+2 x y + c f 2 (h1 : : : hn) (hn+1 ; hn+2 ) + 1 1:
. #
f 2 (h1 : : : hn) x y + c ; 1. 8 \$ &
"
, ## g(h1 : : : hm ) " x y + c.
. f 2 (h1 : : : hn)(x y + c) 0 ( \$ , f (t1 : : : tn) 6= 0 " ! # t1 : : : tn) f 2 (h1 : : : hn)(x y + c) 2 ( )& "
, ## g(h1 : : : hm ) " x y + c.
2. g = dz1k1 : : :zsks , (" d | , k1 : : : ks | & (" , m = s h = g(h1 : : : hm ) = dhk11 : : :hks s .
E( , \$ ## ! "
x y + c, ) d = 1&
) 7
i (1 6 i 6 m), g(z1 : : : zm ) = zi + a (a | ) hi ! " x y + c. 8 hi | #, Hl , " x y + c,
. . h " x y + c.
="
, Hl+1 " " x y + c.1
A, " x y + c " S Hl = I (M )&
l=1
#\$ I (M ) 6= PQ. #
. F, M | #
, f (t1 : : : tn) Q.
@# " . f (t1 : : : tn) Q, . . 7
! c1 : : : cn Q, f (c1 : : : cn) = 0. @ c1 : : : cn 2 M , f 2 (c1 : : : cn) (x ; y)2 + (x ; y) = x ; y 2 I (M ). @
, I (M ) " #" fx ; y xy 1=2 1=3 : : : 1=p : : :g ((" p |
#
# ), #! FQ>2]. ="
, M | # . E ".
7. # , , . . ! . .
F .
Q
. 4# #
& # 7
( 5,
#7!, #
. #. ! . #. "
# . . , 5 #), "\$ f>f 2 (t1 : : : tn)(x ; y)2 +(x ; y)g,
(" f (t1 : : : tn) | #
PQ, . #.
"
# . 8 (" 3 7
(,
#7!, #
. #. f (t1 : : : tn) Q,
. . 7
( " : #
( " .
% #
>2]. @ ".
975
4. " . . f (t1 : : : tn) M , ! . . f 2 (t1 : : : tn) (x ; y)2 + (x ; y), f 2 (t1 : : : tn) (xy)2 + xy, . . F , f (t1 : : : tn) Q.
Q
. M | # . 4#, f (t1 : : : tn) Q.
# " #"
H1 H2 H3 : : :
! PQ.
C ". H1 = M .
A"
! #". # H1 : : : Hl & (" Hl+1 #" ##! "
g(h1 : : : hm ), (" g | M , h1 : : : hm | #, 1Hl .
D, S Hl = I (M ).
1
l=1
= #7 ! " # l #, S Hl
l=1
" ! . #. " x y + c, (" c | #
Q.
C ". ", H1 " "
x y + c.
A"
! #". Hl " "
x y + c& (" #, Hl+1 " "
x y + c.
; #
## h = g(h1 : : : hm ) Hl+1 , ("
g | M , h1 : : : hm | #, Hl .
. "7 .
1. g = f 2 (t1 : : : tn) (x ; y)2 + (x ; y)& (" , m = n + 2
h = g(h1 : : : hm ) = f 2 (h1 : : : hn) (hn+1 ; hn+2 )2 + (hn+1 ; hn+2 ) =
= (hn+1 ; hn+2 )>f 2 (h1 : : : hn) (hn+1 ; hn+2 ) + 1]. E( , h
! " x y + c, ("
hn+1 ; hn+2 1 f 2 (h1 : : : hn) (hn+1 ; hn+2) + 1 x y + c
hn+1 ; hn+2 x y + c f 2 (h1 : : : hn) (hn+1 ; hn+2 ) + 1 1:
. #
f 2 (h1 : : : hn) x y + c ; 1. 8 \$ &
"
, ## g(h1 : : : hm ) " x y + c.
. f 2 (h1 : : : hn)(x y + c) 0 ( \$ , f (t1 : : : tn) 6= 0 " ! # t1 : : : tn) f 2 (h1 : : : hn)(x y + c) 2 ( )& "
, ## g(h1 : : : hm ) " x y + c.
976
. . 2. g = f 2 (t1 : : : tn) (xy)2 + xy& (" , m = n + 2 h =
= h(g1 : : : gm ) = f 2 (h1 : : : hn) (hn+1 hn+2)2 + hn+1 hn+2 = (hn+1 hn+2 ) >f 2 (h1 : : : hn) (hn+1 hn+2) + 1]. E( , h ! " x y + c, ("
hn+1 hn+2 1 f 2 (h1 : : : hn) (hn+1 hn+2 ) + 1 x y + c
hn+1 hn+2 x y + c f 2 (h1 : : : hn) (hn+1 hn+2) + 1 1:
. #
f 2 (h1 : : : hn) x y + c ; 1. 8 \$ &
"
, ## g(h1 : : : hm ) " x y + c.
. f 2 (h1 : : : hn)(x y + c) 0 ( \$ , f (t1 : : : tn) 6= 0 " ! # t1 : : : tn) f 2 (h1 : : : hn)(x y + c) 2 ( )& "
, ## g(h1 : : : hm ) " x y + c.
3. g | " M , . . "
h = g(z ) = ck z k + ck;1z k;1 + : : : + c1 z + c0&
(" , m = 1 h = g(h1 : : : hm) = g(h1 ) = ck hk1 + ck;1hk1 ;1 + : : : + c1 h1 + c0 :
E( , h ! " x y + c, ) g(z ) = z + c0 &
) h1 !! ! " x y + c.
8 h1 | #, Hl , " x y + c, . . g(h1 : : : hm ) " x y + c.
="
, Hl+1 " " x y + c.
1
A, " x y + c " S Hl = I (M )&
l=1
#\$ I (M ) 6= PQ. #
. F, M | #
, f (t1 : : : tn) Q.
@# " . f (t1 : : : tn) Q, . . 7
! c1 : : : cn Q, f (c1 : : : cn) = 0. @ c1 : : : cn 2 M , . #. f 2 (c1 : : : cn) (x ; y)2 +(x ; y) = x ; y #" I (M ) . #. f 2 (c1 : : : cn)(xy)2 +(xy) = xy #" I (M ).
@ , I (M ) " #" fx ; y xy 1=2 1=3 : : : 1=p : : :g
((" p | #
# ), #! FQ. ="
, M | # . E ".
8. # , , . . ! . .
. . F .
Q
977
. 4# #
& # 7
( 5,
#7!, #
. #. ! . #. "
# . . , 5 #), "
\$ f>f 2 (t1 : : : tn)(x ; y)2 + (x ; y),
f 2 (t1 : : : tn) (xy)2 + xyg, (" f (t1 : : : tn) | #
PQ,
. #. "
# . 8 (" 4 7
(, #7!, #
. #.
f (t1 : : : tn) Q, . . 7
( " : #
( " . % #
>2]. @
".
. ( (" " " 5@8 ;I, # .. C. 9"
#
" # #"".
1] . . , // !. . . | 1996. | %. 2, . 2. |
'. 365{374.
2] -
. /. . 0 10. | -.: !3, 1993.
& ' 1997 .
. . 512.644
: , .
, ! "#
. % & ' & "#
.
Abstract
V. P. Elizarov, Systems of linear equations over modules, Fundamentalnaya i
prikladnaya matematika, vol. 8 (2002), no. 4, pp. 979{991.
Some necessary conditions for solvability of linear equation systems over modules
are studied. In some situations these conditions are also su/cient.
,
| . R M
: Rmn | " # (m n)-
'
R ( # # | Rk = R1k, R(k) = Rk1)) M(k) (Mk ) |
" # - () k '
M.
M * AX # = # (0.1)
- A 2 Rmn # 2 M(m) , .
| M(n). /
M = R
R, . AX # = B # (0.2)
- B # 2 R(m) . 1
(0.1) * 1 , " L(A # )
# * .
.
23] , #
# .
(0.2), .
" , # .
.
1 0 1!
2
3, 4"& 1
&5 1!
21& 3 611,13].
, 2002, 8, 9 4, . 979{991.
c 2002 ,
! "#
!!
\$
980
. . 7 " , .
*# "
(0.1). 8 | "
. 9 , .
9
, (0.1),
.
, , 27], -- , KX # = # R- L, - K | (m n)-
'
L # 2 L(m) , .
| R(n) 25,12].
. 0 R M. < - .
x 1. AX # = # (0.1)
- A 2 Rmn # 2 M(m) , * , - .
I. 1
(0.1) .
.
!
II.1. =
C 2 Rm J | R, !
!
(CA 2 J n ) =) (C # 2 J M):
!
II.2. =
C 2 Rm d 2 R, !
!
(CA 2 dRn) =) (C # 2 dM):
!
II.3. =
C 2 Rm , !
!
!
(CA = 0 ) =) (C # = ):
IV.1. =
J1 : : : Jm | R, 0 0 J n 11
0 0 J M 11
1
1
@A 2 @ : : :AA =) @# 2 @ : : : AA :
Jmn
Jm M
IV.2. =
d1 : : : dm | ' R, 0 0 d Rn 11
0 0 d M 11
1
1
@A 2 @ : : : AA =) @# 2 @ : : : AA :
dm M
dm Rn
981
IV.3. =
J | R, (A 2 Jmn ) =) ( # 2 J M(m) ):
IV.4. =
d 2 R, (A 2 dRmn) =) ( # 2 dM(m)):
1 II.2 II.3 -
-- 1 - 217, x 105, 106] #, - M = R = Z| # M = R | , #
#
# .
(0.2). 1 II.1 23] M = R.
1 II.1 -
@
.
1.1. (0.1) II.1 !
.
II.1.1. C 2 Rn J1 : : : Jn | R, !
!
(CA 2 J1 : : : Jn ) =) (C # 2 J1M + : : : + Jn M):
!
II.1.2. C 2 Rm d1 : : : dn | R, !
!
(CA 2 d1R : : : dnR) =) (C # 2 d1M + : : : + dnM):
!
II.1.3. C 2 Rm , !
!
C # 2 CAM(n) :
B, II.1.3 =) !II.1.1 =) II.1.
II.1 =) II.1.2. / CA = (d1 r1 : : : dn!rn), ri 2 R, J | !
,
"* '
d1 : : : dn. 7- CA 2 J n . / II.1 C # =
= j11 + : : : + jk k , - ji 2 J i 2 M. 7 ji = d1u1i + : : : + dm uni ,
usi 2 R, !
C # =
n
X
s=1
ds(u1s1 + : : : + uksk ) 2 d1M + : : : + dnM:
!
II.1.2 =) II.1.3. <
. A = (A#1 : : : A#n ) "
CA#i = di . 7-
!
!
#
CA
=
(d
1 : : : dn), II.1.2 C = d11 +: : :+dn n, i 2 M. <
,
!
!
C # 2 CAM(n).
1 II.1.3 " C. C. D.
1.2. (0.1) I =) II.1 =) II.2 =) II.3
+
+
+
+
IV.1 =) IV.2
IV.3 =) IV.4
(1.1)
982
. . ! " (1.1) =) # , .
B, I =) II.1 =) II.2 =) II.3, IV.1 =) IV.2, IV.1 =) IV.3,
IV.2 =) IV.4 IV.3 =) IV.4.
II.1 =) IV.1. /
0J n 1
1
A 2 @ : : :A
Jmn
!
!
!
C i | i- E
.
7-
C
A
=
Ai 2 Jin, i = 1 m.
m
i
!
/ II.1 C i # = i 2 Ji M.
C-
, II.2 =) IV.2.
/
, , II.3 =) II.2, II.2 =) II.1 II.1 =) I - , 23].
) 2 1
x x1 = 1
Z2x] IV.4, IV.3 J = (2 x),
IV.2 | d1 = 2, d2 = x.
) x 2x 1
1
2 x x2 = 1
Z2x] IV.2, IV.3 J = (2 x).
) 2 2
4 x1 = 2
Z
IV.3, IV.2 d1 = 2, d2 = 4.
-) 1 2
2 x1 = 6
Z
IV.1, II.3 c1 = 2, c2 = ;1.
) 2x1 = 1 Z
II.3, IV.4 d = 2.
) 2x1 + xx2 = 1 Z2x] II.2, IV.3
J = (2 x).
E
I =) II.1 =) II.2 =) II.3 M = R 23].
=
A (0.1), R R M
"
-
, - (1.1) - . F R , " -
983
"* -, (QF-), - - - I -
- J Annl Annr I = I Annr Annl J = J:
1.3.
) (0.1) m=1, IV.1 () IV.3 IV.2 () IV.4.
) (0.1) n = 1, IV.1 () IV.2.
) R | \$
, II.1 () II.2 IV.1 () IV.2.
-) M = R R | QF-, I () II.3.
J"
) ) .
) II.2 =)! II.1. / CA 2 J n. / , "*
'
CA#i , i = 1 n, -. =
" ' d 2 R,
!
n d 2 J. / II.2 !
CA
2
dR
C # = d, 2 M, ,
!
C # 2 J M.
C-
, IV.2 =) IV.1.
-) /
M = R, - R | QF-, II.3 =) I 22].
1
UAV Y # = U # (1.2)
- U 2 Rmm , V 2 Rnn | , * (0.1) 26, x 20]. 1
(0.1) (1.2) .
.
, .
L(A # ) = V L(UAV U # ):
, II.1{II.3, IV.1{IV.4 # (0.1) * -' .
1.4.
) (0.1) -# II.1{II.3, IV.1
IV.2, # % # "
(1.2).
) (0.1) -# IV.1 IV.2,
# % # "
AV Y # = # :
(1.3)
1
- # .
!
) / (0.1) II.1. =
CUAV
2 J n, !
!
CU A 2 !
J n V ;1 J n . / II.1 (0.1) - CU # 2 J M.
<
, C U # 2 J M, (1.2) II.1. C-
II.2 II.3.
984
. . ) / (0.1) IV.1. =
0J n 1
1
AV 2 @ : : : A Jmn
0J n 1
0J n 1
1
1
A 2 @ : : :A V ;1 @ : : : A :
Jmn
Jmn
/ IV.1 (0.1) -
0J M1
1
# 2 @ : : : A :
Jm M
<
, (1.3) IV.1. C-
IV.2.
1 IV.3 IV.4 - # # -' , IV.1 IV.2 | # -'
(1.2).
1.5. 2 1
1 x1 = 2
Z
IV.3 IV.4. 2x1 = 1 # .
1.6. a 0 11
R = a21 a22 aij 2 GF(2)
e1 = ( 00 01 ), e2 = ( 10 00 ) a = ( 01 00 ). e e 2
2
(1.4)
a x1 = 0
IV.1, , IV.2, IV.3
IV.4. 1
(1.4) U = ( ae ae ) V = (e) 0 a
e2 x1 = e2 :
* IV.4 d = e2 , IV.1{IV.3.
985
x 2. M
, # II.1{II.3, IV.1{IV.4
.
(0.1).
2.1.
) ax1 = (2.1)
& II.1, II.2, IV.1{IV.4.
) a1x1 + : : : + an xn = (2.2)
& II.1, IV.1 IV.3.
) R | \$
, (2.2) &
II.1, II.2, IV.1{IV.4.
1.2 ) ) , IV.4 =) I, ) | IV.3 =) I.
) (2.1) IV.4 d = a ea = ae
= e = a, 2 M.
) / (2.2) IV.3. =
J | ,
"* '
a1 : : : an, A 2 J1n. 7- = j11 + : : : + jk k ,
- ji 2 J i 2 M. /'
=
k
X
n
X
s=1
l=1
(a1ds1 + : : : + an dsn)s =
al (d1l 1 + : : : + dkl l )
- dsl 2 R.
) 1 ) "
1.3 ).
N 2.1 (M = R IV.1{IV.4)
23, 3]. /
) 1.2 , O 2.1 ) .
M IV.1 (IV.2) (0.1) 2.2. ' (0.1) (" RM | \$
) & " ", " %
IV.1 (IV.2).
=
(0.1) IV.1 (IV.2), "
1.4 ) ' "- * . / 2.1
'
.
.
9 , " (0.1) .
0J n 1
1
A 2 @ : : :A :
Jmn
986
. . !
!
7 Ai 2 Jin Ai #i = i , - #i 2 M(n) , i 2 Ji M, i = 1 m, 0J M1
1
# 2 @ : : : A :
Jm M
<
, (0.1) IV.1.
C-
O IV.2.
/" II.1.
2.3. (0.1) II.1 !
II.1.4. C 2 Rm , & !
!
CAX # = C # :
, II.1.4 II.1.3, "
1.1.
7
, II.1 , .
"
(0.1), .
" !
F X # = (2.3)
!
- (F ) (A # ) '@@
R. /
-) 1.2 , .
!"-! .
(2.3)
!
(F ) = (A1 1) + (A2 2 ).
P
* , " *
' - - ' . O -
, A G 2 Rmn , G = UAV , - U 2 Rmm V 2 Rnn | .
2.4. A (0.1) (" RM | \$
) ( , & " ", " % II.1 (II.2).
1.2 " "
.
/ U, V | G = UAV | . 7- GY # = U #
(2.4)
-' (0.1). / "
1.4 ) * II.1 (II.2). / "
2.2 " * .
. 7 G | , .
(2.4), .
(0.1).
987
C-
2.5. A (0.1) " , & " ", " %
II.2.
J"
2.4 2.5 M = R 23, 7 8]. 9
, , '
-,
O 210].
x 3.
M
R ,
R | "
R | . D R jD1# : : : Dk#;1 # j
(3.1)
- Di# 2 R(k) # 2 M(k) , -, '
M, "
(3.1) 27].
A 2 Rmn M(k) (A) " # * k, I(k)(A) | R, "* "
M(k)(A), M(k) (A # ), - # 2 M(m) , | " # k (A # ), (3.1). /"
"
M~ (k)(A # ) = M(k)(A # ) n M(k)(A). A (
(A # )) .
* # 0, #
( | rang A rang(A # )).
* , - (0.1):
~ (k)(A # ) I(k)(A)M, 1 6 k 6 minfm ng.
III.1 M
III.2. rang A = rang(A # ).
=
M = R, III.1 AX # = B #
(0.2)
I(k)(A) = I(k)(A B # )
* XIX R = Z214, 15], - @
D9 " M(k)(A) M(k)(A B # ). # 216, . 21] #
- # .
3.1. (0.1) III.1 III.1.1. I | R, (M(k)(A) I) =) (M~ (k)(A # ) I M) 1 6 k 6 minfm ng:
988
. . III.1.1 =) III.1. "
I = I(k) (A).
III.1 =) III.1.1. =
M(k)(A) I, I(k) (A) I. 7- III.1
M~ (k) (A # ) I M:
3.2. )
(0.1) II.1.
*"
) III.1+
) m > n, M~ (n+1)(A # ) = fg.
) T(j ) =
#
= jD1 : : : Dk#;1 Uj#j, - j = k n Uj# = (a1j : : : akj )T , Di# = (a1i : : : aki)T
Un#+1 = (1 : : : k )T . =
M(k)(A) I, T(j ) 2 I, j = k n. /
!
C = (T1 : : : Tk 0 : : : 0), - Ti | - '!
aij T(j ) , CA = (0 : : : 0 T(k) : : : T(n)) 2 I n . / II.1
!
C # = T(n+1) 2 I M. <
, III.1.
!
!
(j ) = 0, j = 1 n. /' CA = 0 )
/
m
>
n
k
=
n
+
1
T
!
C # = .
3.3. (0.1) RM, " R |
, I =)
II.1
=) II.2 =) II.3 =) III.2
+
+
+
+
IV.1 =) IV.2
(3.2)
+
+
+
III.1 =) IV.3 =) IV.4
! " (3.2) =) # , .
1.2 "
3.2 " , II.3 =) III.2
III.1 =) IV.3.
II.3 =) III.2. / rang A = t. =
t < minfm ng, , "
3.2 ), T(j ) k = t + 1, j !
= t + 1 n
!
!
J! = 0. /
C = (T1 : : : Tt+1 0 : : : 0) CA = 0.
/ II.3 C # = , . . T(n+1) = . <
, M(t+1) (A # ) = f0 g rang(A # ) = t. =
t = m 6 n, rang(A # ) = t. =
t = n!< m, , "
3.2 ), k = n + 1 C # = , . .
M(n+1)(A # ) = fg rang(A # ) = t.
III.1 =) IV.3. / A 2 Jmn . 7- I(1)(A) J, III.1
(1)
M~ (A # ) J M. <
, # 2 (J M)(m) .
989
- (1.1) " , 1.2 .
/
, , III.2 II.3, * 23].
) e 0
e x1 = e
III.1 IV.1,
III.2.
) 1 3
2 x1 = 3
Z
III.1, d1 = 1, d2 = 2 IV.2.
) 2 1x 1
1
1 2 x2 = 1
Z
IV.1, III.1:
I(2)(A) 6= I(2) (A # ).
7 3.3 , III.1 III.2, (0.2) 21,4,7{9], .
# # - (3.2) " -
.
3.4.
) (0.1) m 6 n, III.1 =) III.2.
) R | \$
, II.2 =) III.1.
) / rang A = t. =
t = m, rang(A # ) = t. =
t < m, I(t+1)(A) = f0g. / III.1 - M~ (t+1) (A # ) = fg, rang(A # ) = t.
) / "
1.3 ) II.1 () II.2.
E 2.1 3.3 3.5. RM | , ) ax1 = & II.1, II.2, III.1, IV.1{IV.4+
) a1x1 + : : : + an xn = (3.3)
& II.1, III.1, IV.1 IV.3+
) R | \$
, (3.3) & II.1, II.2, III.1, IV.1{IV.4.
" "
1.4.
990
. . 3.6. (0.1) IV.3 IV.4, "
UAV Y # = U # .
E UAV = dH UAV 2 Jmn A = U ;1dHV ;1 = dH1 A 2 U ;1Jmn V ;1 2 Jmn . 7- IV.4 IV.3 # 2 dM(m)
# 2 (J M)(m) . <
, U # 2 UdM dM(m) U # 2 (J M)(m) .
1 III.1 III.2 - # # .
3.7. 2 0x 2
1
0 0 x2 = 2
Z4 III.1 III.2, - * # .
", III.1 III.2 # #
-' .
3.8. U 2 Rmm V 2 Rnn | # , rang UAV = rang A I(k) (UAV ) = I(k)A, 1 6 k 6 minfm ng.
3.9. U 2 Rmm V 2 Rnn, M~ (k)(AV # ) M~ (k)(UA U # ), 1 6 k 6 minfm ng # M~ (k) (A # ), m > n, k = n + 1.
. =
C | , # k A, ! #
!
A1V1 : : : A1Vk#;1 1 !: : : : : : ! : : : : : : = jCV1# : : : CVk#;1 U#j =
Ak V1# : : : AkVk#;1 k n
X
=
ij =1
j =1k
vi1 1 : : :vi ;1 k;1jCi#1 : : : Ci# ;1 U# j
k
k
, M~ (k)(AV # ). C-
"
M~ (k) (UA U # ) k = n + 1, m > n.
3.10. (0.1) III.1
III.2, "
UAV Y # = U # .
3.8 3.9.
#
1] . . //
. . | 1993. | #. 48, ' 2. | . 181{182.
991
2] . . *
+
// -. . . | 1995. | #. 1, ' 2. | . 535{539.
3] . . / // -. . . | 2000. | #. 6, ' 3. | . 777{788.
4] 34
5. 6. . |
7.. . . . *.-. . | 8 9
, 1986.
5] :; <. <. 8
; *
+
, =/ // -. . . | 1995. | #. 1, ' 1. |
. 229{254.
6] >
8. <. <9
. | 3.: :, 1988.
7] Camion P., Levy L. S., Mann H. B. Linear equations over a commutative ring //
J. Algebra. | 1971. | Vol. 17, no. 3. | P. 432{441.
8] Ching W.-S. Linear equations over commutative rings // Linear Algebra and Appl. |
1977. | Vol. 18, no. 3. | P. 257{266.
9] Hermida J. A., Sanchez-Giralda T. Linear equations over commutative rings and
determinantal ideals // J. Algebra. | 1986. | Vol. 99, no. 1. | P. 72{79.
10] Kaplansky I. Elementary divisors and modules // Trans. Amer. Math. Soc. |
1949. | Vol. 66. | P. 464{491.
11] Kertesz A. The general theory of linear equation systems over semisimple rings //
Publ. Math. Debrecen. | 1955. | Vol. 4, no. 1{2. | P. 79{86.
12] Kertesz A. Systems of equations over modules // Acta scient. math. Szeged. |
1957. | Vol. 18, no. 3{4. | P. 207{234.
13] Kuhn H. W. Solvability and consistency for linear equations and inequalities // Amer.
Math. Monthly. | 1956. | Vol. 63, no. 4. | P. 217{232.
14] Smith H. J. S. On systems of linear indeterminate equations and congruences //
Phil. Trans. Royal Soc. London. | 1861. | A 151. | P. 293{326.
15] Smith H. J. S. On the arithmetical invariants of a rectangular matrix of which the
constituents are integral numbers // Proc. London Math. Soc. | 1873. | Vol. 4. |
P. 236{249.
16] Steinitz E. Rechteckige Systeme und Moduln in algebraischen Zahlkorpern. I //
Math. Ann. | 1912. | B. 71, N. 3. | S. 328{354.
17] Van der Waerden B. L. Moderne Algebra. V. 2. | Berlin: Springer, 1931.
% ! & 2001 .
. . (
)
e-mail: mm@mpei.ru
517.958+533.7
: , !, "# \$ , % &'"&(( !.
) &(!" &&*( + *,-( , &(+"
(! * - & ( * .
&(* % &'"&(( ! **, "# \$ ! *!
&*'! * . ..
Abstract
K. O. Kazyonkin, Existence of the global generalized solution of one-dimensional
problem of a viscous barotropic gas ow, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 4, pp. 993{1007.
The article is devoted to research of the boundary value problem of a viscous
barotropic gas 5ow through a channel of 6xed length. The existence theorem of
a global generalized solution for the case of nonsmooth data is established.
( ) 1{5]. % & & '& . ( &
) : ) +, C 1+ , C 2+ W21 . , . , , ).
% 6,7] ' 1' &
) (* - &( 7 )&& & - -'%*,.
&&*( ( 00-01-00207).
, 2002, % 8, 8 4, &. 993{1007.
c 2002 !" #\$,
%&
'( )
994
. . &. % 1{5] , ua .
% + . 4 ( ' & & & , 6]. 5 & & &. 6 & ) ,
' ) ,, ub . 7 ua ub
,
& & 0 T ], ) ,
.
4 ' 6].
, & 8{10].
x
1. 0 T ] +, , ua (t) > 0,
0 T ). ua (t) = ua (T ) t > T . %'
o
t+1
1n
R =k
T
Ma =
t 2 0 T ):
ua (t0) dt0 > 0 . ;, .
k=1
t
, (0 T ) +, a (t) > 0, ua =a 2 L1 (0 T ). 6 0 T ] )) +,)
Rt
a(t) = ; ua (t0 )=a(t0 ) dt0 .
0
< & Dt = Du Q,
Dt u = D + gz ] = ]Du ; p] = 1 Q,
Dt xe = u Q,
Zb(t)
a(t)
ZX
(x t) dx = 0 (x) dx (0 T )
(1.1)
(1.2)
(1.3)
(1.4)
0
+, z (x t) = ((x t) u(x t) xe(x t)), '
Q = QT = f(x t): a(t) < x < b(t) 0 < t < T g, +, b(t),
' 0 T ]. 5
) &: D = @=@x, Dt = @=@t. , ](x t) = ((x t) x),
p](x t) = p((x t) x), gz ](x t) = g((x t) u(x t) xe(x t) x t).
...
995
@ , && (1.1){(1.4): x |
a, t | . | 1' ( | ), u | , xe | ( . | , p | . g | & . | (++, . a(t) b(t) |
, .
4 (1.1){(1.4) ( u xe)jt=0 = (0 (x) u0(x) x0e (x)) (0 X ), b(0) = X > 0
(1.5)
Rx
xe0(x) = 0 (x0) dx0, 0
ujx=a(t) = ua(t) ujx=b(t) = ub (t) (0 T )
(1.6)
jx=a(t) = a (t) xejx=a(t) = 0 Ma .
(1.7)
Ba (1.1){(1.7) (
) , + ,, ' ) , co ) ua , ) & ) ub . (
;a(t) X ; b(t) ) , t, b(t) ; a(t) ' ,
& t.
A = a(T ) ' (A X ] +,) ta (x) =
= maxft 2 (0 T ): a(t) = xg x 2 (A 0) ta (x) = 0 x 2 0 X ].
C D Lq (Q) D Lqr (Q) kvkL (Q) = kvkRL (a(t)b(t))L (0T ) q r 2 1 1]. kvkG = kvkL2 (G) (v w)G = vw dG. V2 (Q)
G
S211 W (Q) | & kvkV2 (Q) =
= kvkL2 1 (Q) + kDvkL2 (Q) kvkS21 1 W (Q) = (kvk2W 12 (Q) + kDDt vk2Q )1=2. %'
N (Q) & & Q +, j vjjN (Q) = kvkL1 (Q) +k1=vkL1 (Q) +kDt vkQ . V 0 T ] | +, , 0 T ] kvkV 0T ] = sup jvj + 0var
v.
T ]
0T ]
+, v(x t) G R2, > 0, > 0.
4 (x t) 2 G ' E v(x t) = v(x + t) ; v(x t) (x + t) 2 G
( E v(x t) = 0) E( ) v(x t) = v(x t + ) ; v(x t) (x t + ) 2 G ( E( ) v(x t) = 0). 6 kvkh2021=4i = sup ;1=4 kE( ) vkQ 0<<T
h
01=4i
kvkV2h1 1 4i (Q) = kvkV2 (Q) + kvk22 . %' Rt
(It w)(t) = w(t0 ) dt0.
0
R+ = (0 +1). F = R+ (A X ), F = (;1 ) (A X )
> 0. %' F +,
qr
=
q
r
996
. . Z
Z p
Z
1
1
1
G( x) = ;1 ( x) d L( x) =
;1 ( x) d E ( x) = ; p( x) d
E+ ( x) = maxfE ( x) 0g E;( x) = maxf;E ( x) 0g:
H K (N ) ( ) ) +, N > 1. ( +, X T . % & N .
+, , p g ) ) 6,8].
A1 . @, ( x) p( x) F . .
( &) x 2 (A X ) . , > 0 p > 0.
A2 . k kL1( ) + k1= kL1( ) + kpkL1 ( ) 6 C (), kD pkL1 ( ) 6 C () & > 1.
A3 . G() 6 G( x) 6 G(), G() ! +1 ! +1 G() ! ;1 ! +0.
A4 . ( x) 6 c0 (E+ ( x) + L2 ( x) + maxf 1g) F.
A5 . p( x) 6 c0(E+ ( x) + L2 ( x) + maxf 1g) F.
A6 . E; ( x) 6 "L2 ( x) + C (";1 ) (1 +1) (A X ) & " 2 (0 1].
A7 . @, g( u xe x t) a a R+ R R Q . . (x t) 2 Q ( u xe).
A8 . jg( u xe x t)j 6 g0(t)juj + g1(x t) R+ R R Q, kg0kL1 (0T ) +
+ kg1 kL2 1(Q) 6 c0 .
) ) .
B1 . 0 2 L1 (0 X ), u0 2 L2 (0 X ), ' N ;1 6 0 (x) 6 N )
(0 X ) ku0 k(0X ) 6 N .
B2 . ua ub 2 V 0 T ], ' ua ub ,
0 T ). , kuakV 0T ] + kuX kV 0T ] 6 N .
B3 . a 2 L1 (Ma ), ' N ;1 6 a(t) 6 N ) Ma .
%' (A X ) +, xe , (x) = a (ta (x)),
xe (x) = 0 x 2 (A 0] (x) = 0 (x), xe (x) = x0e (x) x 2 (0 X ).
I +, (A X ). 4
, +, ta - (A 0] Ma ,
( Ma) +, a. J +, a ) , (. 11]) . ( , a (ta (x)) .
K' +, z = ( u xe) 2 N (Q) V2 (Q) S211 W (Q) +,) b 2 W11 (0 T ) ' (1.1){(1.7), 1) (1.1) (1.3) ) L2 (Q).
2) (1.4) & t 2 (0 T ).
3) o
;(u Dt')Q + ( D')Q = (u0 'jt=0)(0X ) + (gz ] ')Q
(1.8)
...
997
+, ' 2 W 12(Q), 'jt=T = 0,
'jx=a(t) = 'jx=b(t) = 0.
4) (1.6) +, a V2 (Q).
5) jt=0 = 0 , xe jt=0 = x0e (1.7) )
) : +, xe . . x 2 (A X ) ) t , (x ta(x)) = (x) xe(x ta (x)) = xe (x).
7+ .
1.1. A1{A8 B1 {B3 . (1.1){(1.7), jj jjN (Q) + kuk h1 1 4i
+ kxe kS21 1W (Q) + kbkW11 (0T ) 6 K (N ): (1.9)
V2
(Q)
=
4
1.1 + 2{4.
x
2. "
<
' 0 T ] n = T=n Tj = j ,
0 6 j 6 n. n , 6 (3N );1X . %' A = xn 6
6 xn;1 6 : : : 6 x0 = 0, xj = a(Tj ), 0 6 j 6 n. 6 0 T )
-) +,) a , )) a (t) = a(Tj ) t 2 Tj Tj +1), 0 6 j < n. !aj = (xj xj ;1] ( xj = xj ;1, !aj = ?).
J +, a(t) ) , Rx
a (ta (x)) dx x = a(t) 12],
x +1
+
j
j
Zxj
x +1
a(ta (x)) dx =
j
TZ +1
j
T
ua (t) dt 0 6 j < n:
(2.1)
j
6 (A X ) -) +,) ta , )) ta (x) = Tj x 2 !aj , 1 6 j 6 n ta (x) = 0 x 2 0 X ).
K , a ! a L1 (0 T ), ta ! ta L1 (A X ) ! 0 , ,
lim sup kE ta kL1(AX ) = 0:
!+0
(It w)(x t) =
Zt
t (x)
a
Zt
w(x t0) dt0 (It(j ) w)(x t) = w(x t0) dt0
T
j
998
. . (I w)(x t) =
Zx
a (t)
w(x0 t) dx0:
tR+
%' u(c ) = 1 uc (t0 ) dt0 ( c = a b), o uc (t) = uc (T )
t
t > T . B, u(c ) 2 W11(0 T ) ,
ku(c ) kC 0T ] 6 sup juc j 6 N kDt u(c ) kL1 (0T ) 6 var uc 6 N:
0T ]
0T ]
Lj = (xj yj ), Qj = Lj (Tj Tj +1), y0 = X ( L0 = (0 X ).
j = 0 1 : : : n ; 1 Qj - Dt = Du (2.2)
Dt u = D + gz ] = ] Du ; p ] = 1= (2.3)
Dt xe = u (2.4)
y
T
Z
Z +1
(x Tj +1) dx = u(b ) (t) dt
(2.5)
y +1
T
j
( u xe )jt=T = ( (x) uj (x) xj
(2.6)
e (x)) Lj (
)
u jx=x = u(a ) (t) u jx=y = ub (t) (Tj Tj +1)
(2.7)
0
0
0
+, z = ( u xe ). B
= , u = u0 ,
xe 0 = x0e . % B1 , B2 j = 0 (2.8)
kj ( t)kL1 (& ) ; It(j ) u(a ) + It(j ) u(b ) > N ;1X ; N > (2N );1X:
j
j
j
j
j
j
j
j
( 8] ' z = ( u xe ) 2 N (Qj ) V2 (Qj ) S211 W (Qj )
(2.2){(2.7), ) ,
j jjN (Q ) + ku kV2 (Q ) + kxe kS 1 1 W (Q ) 6 Kj (N ):
2
M , u 2 C (Tj Tj +1]. L2(Lj )), xe 2 C (Qj ). ( Lj j +1 = jt=T +1 , uj +1 = u jt=T +1 , xj
e +1 = xe jt=T +1 , '
Kj (N );1 6 j +1 6 Kj (N ) kuj +1k& 6 Kj (N ):
B, j = 0 yj +1 , xj < yj +1 6 yj (2.5). 4
, (2.2) (xj yj ) (Tj Tj +1), kj +1kL1 (& ) =
j
j
j
j
j
j
j
j
= kj kL
TZ +1
j
1 (&j )
+
T
j
(ub (t) ; ua (t)) dt > (2N );1 X +
( )
( )
TZ +1
j
T
j
u(b ) (t) dt: (2.9)
...
999
+, j +1, uj +1, xj
e +1 c Lj Lj+1, j +1(x) =
(
)
= a(ta (x)), uj +1(x) = ua (Tj +1 ) +1
xj
e (x) =
TZ +1
j
( )
(ua (t) ; ua (t)) dt +
Zx
x +1
0
j +1 (x0) dx0
(2.10)
j
x 2 !aj +1 . B, + (2.10) Lj+1. 4 ' x 2 Lj (2.2) (xj x) (Tj Tj +1),
(2.4) (2.6). % +1
j
xj
e (x) = xe (x) ;
Zx
x
(j +1 (x) ; j (x)) dx +
j
TZ +1
j
T
u(a ) (t) dt:
j
(2.1), + (2.10) & j . j = 0 + . j > 0 ,.
< (2.2){(2.7) j = 1 : : : n ; 1 ' , Q = QT = f(x t): a (t) < x < b (t) 0 < t < T g
+, z , b (t) | - +,, )
b (t) = yj t 2 Tj Tj +1), 0 6 j < n. ( ) 3.1 (2.8) & j , a 3.1 ' yj +1 , xj < yj +1 6 yj (2.5). ;, z 2 N (Q ) V2 (Q ) S211 W (Q ).
%' +,) ^b 2 C 0 T ], ) & Tj Tj +1],
0 6 j < n, )) ^b (Tj ) = yj , 0 6 j 6 n. !bj = yj yj ;1) ( yj = yj ;1 , !bj = ?). 6 L (t) = Lj RTj
t 2 Tj Tj +1). %' (It w)(x t) = w(x t0) dt0 x 2 !bj .
t
x
3. \$
%
"
O '& z , ^b ,.
3.1. ! 1.1. " z , ^b !
^ 1 (0T ) 6 K (N ) (3.1)
jj j N (Q ) + ku k h1 1 4i
V2
(Q ) + kxe kS21 1 W (Q ) + kb kW1
kE kL21(Q )
= K (N ).
=
6 K (N )(kE kL2(AX ) + kE ta k1L=12(AX ) +
+ kE GkL2 (AX 'C ;1 ]) + kE pkL2 (AX 'C ;1 ]) + 1=2 ) (3.2)
1000
. . 4
' (. 7 ,
+,) V (t) = k ( t)kL1(& (t)). 6 V 0 = k0 kL1(0X ) .
3.1. # (2N );1X 6 V (t) 6 NX + (2N );1 X t 2 0 T ):
(3.3)
. 5 (2.2) & (2.7) t 2
2 Tj Tj +1 ), 0 6 j < n, Dt V = (Dt 1)& = (Du 1)& = u(b ) ; u(a ) :
5 ( t, +
j
j
V (Tj ) ; V (Tj ; 0) =
ZTj
T ;1
ua(t) dt ;
j
ZTj
T ;1
u(b ) (t) dt
j
( (2.1) (2.5)), V = V 0 + It (ua ; u(a ) ) + It(j ) u(b ) ; It(j ) u(a )
(3.4)
Tj Tj +1). B, jIt(ua ; u(a ) )j 6 N=2, 0 6 It(j ) u(a ) 6 N 0 6 It(j ) u(b ) 6 N , (3.4) V 0 ; 3N=2 6 V 6
6 V 0 + 3N=2. J N ;1X 6 V 0 6 NX 3N 6 N ;1X , , (3.3).
3.1. \$
a (t) < b (t) % t 2 0 T ],
! & b 0 T ].
. 4 , , (3.3) (2.9) & j , yj +1, xj < yj +1 6 yj (2.5).
7) ( ,.
%' +, u; = (1 ; ` )u(a ) + ` u(b ) , ` = I =V v = u ; u; .
`j = ` jt=T , `~j = ` jt=T ;0, vj = v jt=T , v~j = v jt=T ;0. B, 0 6 ` 6 1 +
Dt u; = (1 ; ` )Dt u(a ) + ` Dt u(b ) + dV v Du; = dV (3.5)
dV = (u(b ) ; u(a ) )=V .
3.1. \$
' kE+ ]kL1 1(Q ) + ku k2L2 1(Q ) + kL ]k2L2 1(Q ) +
j
j
j
j
+ kDt L ]k2Q +
nX
;1
j =1
kuj k2!j
b
6 K0 (N ): (3.6)
. ...
1001
E (t) = (E ] 1)& (t) + 05kv k2& (t) . J +, v ' V 10(Qj ) Dt v = D + gz ] ; Dt u; Qj ,
v jt=T = vj (x) Lj ,
v jx=x = u(a ) (t) v jx=y = u(b ) (t) Tj Tj +1],
13]
1 kv k2 +I (j ) ( Dv ) = 1 kvj k2 +I (j ) (gz ];D u v ) T 6 t < T :
&
t ;
&
j
j +1
&
t
2 & t
2
7 ' Dt = Du + (3.5) j
j
j
j
j
j
j
(j )
= E (Tj ) + It(j ) dV (kv k2& ; ( )& )] +
+ It(j ) (gz ] ; (1 ; ` )Dt u(a ) ; ` Dt u(b ) v )&
0 6 j < n, E (t) + It kDtL ]k2&j
j
X
2
E (t) + kDt L ]kQ +
(kv~k k2& ;1 ; kvk k2& ) =
k=1
j
X
= E (0) + (E ] 1)(x 0) ; (E k ] 1)! + ItdV (kv k2& ; ( )& )] +
k=1
+ It (gz ] ; (1 ; ` )Dt u(a ) ; ` Dt u(b ) v )& t 2 Tj Tj +1):
(3.7)
t
k
k
k
b
j
5 A8 ,
( )
jIt(gz ] ; (1 ; ` )Dt u(a ) ; ` Dt ub v )& j 6 K1 It (g0 + 1)kv k& ]
(3.8)
1
=
2
= ( ] ) Dt L ] ; p ] A4, A5 ,
jIt dV ( )& ]j 6 K1 ("1 kDt L ]k2Q +
+ (";1 1 + 1)c0It kL ]k2& + kE+ ]kL1(& ) + V + 1]) (3.9)
"1 > 0.
7 +
k 2
kv~k k2& ;1 ; kvk k2& = kv~k k2! + (~
vk + vk uk
~k
; ;u
; )& \& ;1 ; kv k! :
J `k ; `~k = (V (Tk ; 0));1 ku(a ) kL1(T ;1 T ) +(1 ; V (Tk )=V (Tk ; 0))`k ,
B2 k 2
j(~
vk + vk uk
~k
; ;u
; )& \& ;1 j + kv k! 6
6 K2 kv kL2 1 (Q ) + ((u(a ) (Tk ))2 + (u(b ) (Tk ))2 )(xk;1 ; xk ): (3.10)
t
k
k
k
k
b
k
k
k
t
k
a
k
k
k
a
1002
. . 5 A6 (2.5) ,
;
j
X
(E k ] 1)! 6
k
b
k=1
j
X
k=1
(E;k ] 1)! 6 "2
k
b
j
X
k=1
1
kLk ]k2!k + C (";
2 )It ub
( )
b
6
6 "2 (kL]k2(AX ) + 2kL ]DtL ]kL1(Q ) ) + C (";2 1 )NT 6
6 "2 (kL ]k2Q + kDt L ]k2Q ) + K4 :
(3.11)
;
1
(3.7) , (3.8), (3.9) "1 = (2K1 ) , p
kL ]k2& (t) 6 kT (kL]k(AX ) + kDtL ]k2Q ) kT = (1 + T )2
A6 c " = (4kT );1 , , (3.10) (3.11) c "2 = 1=4, &
t
t
t
t
Y (t) kE+ ]kL1 (& (t)) + 21 kv k2& (t) + (4kT );1 kL ]k2& (t) +
X j 2
+ 14 kDt L ]k2Q +
kv k! 6 K5 (It (g0 + 1)Y ] + kv kL2 1 (Q ) + 1):
T <t
O a kvj k2! 6 kuj k2! + 2N 2(yj ;1 ; yj ), ku kL2 1 (Q ) 6
6 kv kL2 1(Q ) + 2N Q, , (3.6).
7) 3.1 ) , +, .
3.2. \$
kIt kL1 (Q ) 6 K (N ):
(3.12)
. R 6] (x t) 2 Qj , 0 6 j < n, +
It(j ) = I (uj ; u ) + (u ` )& ; (uj `j )& + It(j ) I gz ] +
+ It(j ) ( =V )& ; (u v =V )& ; (gz ] ` )& ]
(x t) 2 Qk , 0 6 j 6 k < n, It(j ) = I (uj ; u ) + (u ` )& ; (uj `j )& + It(j ) I gz ] +
+ Bjk + It(j ) ( =V )& ; (u v =V )& ; (gz ] ` )& ] (3.13)
k
X
Bjk =
((ui `~i ; `i )& ;1 ; (ui `i)! ; (ui 1)! ):
t
j
b
j
j
b
t
j
b
j
j
j
j
j
j
B, jBjk j 6 K1
i
i=j +1
X
k
i=j +1
j
i
a
kuik2!i
b
i
b
+ ku kL2 1(Q ) + 1 () , (3.6), (3.13) , (3.12).
...
1003
3.2. \$
K0 (N );1 6 6 K0 (N ):
(3.14)
. J Dt = Du , = Dt G ] ; p ]. 5 t, . . x 2 (xn;1 X ) G ] = G ] + It p ] + It :
(3.15)
, (3.12), GN ;1] 6 G] 6 GN ] () A3, B1 , B3 ) p ] > 0, )) ,
;K1 6 G ] 6 It p ] + K1 , (3.14) (. 8, 3.2]).
3.3. \$
ku kV2 (Q ) 6 K (N ) k kQ 6 K (N ):
(3.16)
. % (3.14) A2 , K1;1 6 ]= 6 K1 0 6 p ] 6 K1 . ( , (3.6)
) ku kL2 1 (Q ) 6 K
K1;1 kDu k2Q 6 ( ]= Du Du )Q = kDt L ]k2Q 6 K
k kQ 6 K2 (kDu kQ + kp ]kL1(Q ) ) 6 K:
O )
, +, , u ^b .
3.3. \$
ku kh2021=4i 6 K (N ).
. B 9, 2.1] +, It(j ) )
DIt(j ) = u ; uj ; It(j ) gz ]
(3.17)
L21 (Qk ), 0 6 j 6 k < n.
<
' Q Pi = f(x t): x 2 !bi ta (x) < t < Ti g,
1 6 i 6 n ; 1, Pn = f(x t): xn;1 < x < yn;1 ta (x) < t < Tn g. Pi 6= ?.
J +, It )
DIt = ui ; u ; It gz ]
L21 (Pi ) (( (3.17)) , 10, 1.2], Pi . M , (3.12) , kIt kL1 (Q ) 6 K (N ).
( L21(Q ) E( ) u = ;DE( ) It ; E( ) It gz ]:
1004
. . O L2 (Q ) E( )u , & kE( ) u k2Q = (E( ) It DE( )u )Q ; (E( ) It gz ] E( )u )Q +
+
nX
;1
TZ +1
i
i=0 T
nX
;1 TZ +1
E( )It (xi t)E( ) u(a ) (t) dt ;
i
i
;
E( )It (yi t)E( )u(b ) (t) dt +
i=0 T
nX
;1 TZ +1
i
+
i
i=0 T +1 ;~
E( ) It (xi t)(u (xi t + ) ; u(a ) (t + )) dt
i
~ = minf g. ,
kE( )It kQ 6 k kQ kE( )It gz ]kQ 6 1=2 kgz ]kL2 1(Q ) kE( )u k2Q 6 2 k kQ kDu kQ + 1=2 kgz ]kL2 1(Q ) kE( ) u kQ +
+ 2kE( )It kL1 (Q ) (kE( ) u(a ) kL1 (0T ) + kE( ) u(b ) kL1 (0T ) + kDu kL1 (S ) )
S j
S =
!a (Tj + ; ~ Tj + ). % , (3.16), A7 16j<n
kE( ) u(c ) kL1 (0T ) 6 var uc c = a b
0T ]
kDu kL1 (S )
6 (mes S )1=2kDu kQ 6 K1 1=2
kE( )u k2Q 6 K2 1=2 (kE( )u kQ +1), ,.
3.4. \$
(3.2).
. 5 + (3.15) . . x 2 (xj yj ; ) &
t 2 ta (x) Tj +1], 0 6 j < n, G +E ] ; G ] = G +E ] ; G] ; (E G) +E ]+(E G) +E ]+
+ It (p + E ] ; p ]) + It (E p) + E ] + It E + 1 + 2 (3.18)
(E G)w](x t) = (E G)(w(x t) x), (E p)w](x t) = (E p)(w(x t) x),
1 (x t) =
tZ(x)
a
t (x+)
tZ(x)
a
(x + t) dt 2 (x t) =
p ](x + t) dt:
t (x+)
5 , (3.14) A2 , jp ]j 6 K1 , K1;1 jE j
6 jG + E ] ; G ]j 6 K1 jE j, jp + E ] ; p ]j 6 K1 jE j.
a
a
6
...
1005
B, (x t) 2 !ai (Ti Tj ), 0 6 i < j < n, (It E )(x t) = (E It(i) )(x t). ( + (3.17) , ku kL2 1 (Q ) 6 K0 jIt E j 6 K2 1=2. M , , (3.12) (3.16) k1 k& + k2 k& 6
6 K2 kE ta kL1=12(A0). % (3.18) kE k& 6 K3 (It kE k& + d ( ))
d ( ) ) ) (3.2). 5
Q, ) ,.
3.5. \$
k^b kW 11 (0T ) 6 K (N ).
. ;, k^b kC 0T ] 6 maxf;A X g (. 3.1)
Dt^b = (yj +1 ; yj )= t 2 (Tj Tj +1 ). % (2.5) , (3.14)
jyj +1 ; yj j 6 K0 N . ( kDt^b kL1 (0T ) 6 K0 N .
J 3.1 ) .
x
4. % & = n = T=n ! 0.
+, z & QT Q~ T = f(x t):
a(t) < x < X 0 < t < T g, ZTj
( ) 0
0
(x t) = a(ta (x)) u (x t) = u(a ) (t) xe (x t) = xj
e (x) ; ua (t ) dt
t
x 2 !aj , t < Tj (x t) = j (x)
u (x t) = u(b ) (t)
xe (x t) = xj
e (x) +
Zt
u(b ) (t0 ) dt0
T
j
x 2 !b , t > Tj , 1 6 j < n. +, ) (2.2), (2.4) Q~ T . M , & , (3.1) (3.2),
& Q Q~ T . B, , (3.2) , sup kE kL2 (Q~) ! 0 ! 0. % < L2(Q~ T ), + j
- , ' & ( z ,
^b ( & ), ! Lp (Q~ T ), p 2 1 +1), - L1 (Q~ T ), Dt ! Dt L2 (Q~ T ). u ! u L2 (Q~ T ) - L21 (Q~ T ), Du ! Du L2 (Q~ T ).
xe ! xe S211W (Q~ T ). ^b ! b C 0 T ], Dt^b ! Dt b - L1 (0 T ).
1006
. . M , ! = ]Du ; p] L2 (Q~ T ) gz ] ! gz ] L21(Q~ T ).
R 6,8].
;, +, z = ( u xe) +, b ) , (1.9). O (1.1) (1.3), , )
L2 (Q). M , ) jt=t (x) = , xe jt=t (x) = xe (1.6). & (3.4) , a ! a,
b ! b L1 (0 T ), (1.4) & t 2 (0 T ).
6
(1.8).
' 2 W21 (Q), '
'jx=a(t) = 'jx=b(t) = 0, 'jt=T = 0. " > 0 a(t) + 2" < b(t) ; 2"
& t 2 0 T ]. '" = 'e" , e" (x t) = 0 x < a(t) + " x > b(t) ; ", e" (x t) = 1 x 2 (a(t) + 2" b(t) ; 2"), e" (x t) = (x ; a(t))=" ; 1
x 2 a(t) + " a(t)+2"] e" (x t) = (b(t) ; x)=" ; 1 x 2 b(t) ; 2" b(t) ; "].
K , '" 2 W21 (Q) '" ! ' W21 (Q) " ! 0, '
supp '" QT \ Q < 0(").
5 (2.3) ;(u Dt '")Q + ( D'" )Q = (u0 '"jt=0)(0X ) + (gz ] '")Q :
& ' ! 0, a " ! 0, (1.8).
J 1.1 ) .
a
'
a
1] . . !" \$ \$ // & '. | 1981. | +'. 50. |
C. 3{14.
2] . . 2 ' " \$ " " " // & '. | 1982. |
+'. 56. | C. 22{43.
3] . . 4 ' " //
& '. | 1983. | +'. 59. | C. 23{38.
4] + " +. 5. 6' "' " " " // & '. | 1990. | +'. 97. | C. 3{21.
5] Belov S. Ya. On the initial-boundary value problems for barotropic motions of a viscous gas in a region with permeable boundaries // J. Math. Kyoto Univ. | 1994. |
Vol. 34, no. 2. | P. 369{389.
6] 5 5. 5., 9: 9. 2. ;<: ;' " " ' // +
=>?. | 1995. | @ 6. | C. 5{21.
7] 9: 9. 2. A ;;D:" ;<: ;' " " ' // +
=>?. | 1996. | @ 6. | C. 71{78.
8] 5 5. 5., 4 5. 5. " 'E " " ' // &FF.
. | 1994. | G. 30, @ 4. | C. 596{608.
...
1007
9] 5 5. 5., 4 5. 5. A ;;D:'E " 'E " // =.
. | 1994. | G. 55, @ 6. | . 13{31.
10] 5 5. 5., 4 5. 5. 2 :'E E " \$ ; ' ;' !D ' // H+=
=I. | 1996. | G. 36, @ 2. | . 87{110.
11] 6 ?. L. G F D . | L;.: Q, 1999.
12] I., :F-6 . Q F . | =.: =,
1979.
13] Q'\$ 2. 5., +. 5., R 6. 6. Q ' ' ;" . | =.: =, 1979.
* "+ 2000 .
. . ,
. 511.5
: , ,
, .
! " , #\$" , & \$#" ' ( \$" ", ) ' \$" # ' !
&( .
Abstract
S. Sh. Kozhegel'dinov, On some diophantine equation, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 4, pp. 1009{1017.
Using the arithmetic functions introduced by the author we study the equivalence of four parametrizations of the solutions of a diophantine equation in natural
numbers such that every one of them gives all the solutions of this equation.
x4 + y2 = z 2 x, y, z .
, x2 , y, z ! " "" ".
# x4 + y2 = z 2 (1)
"
x y z 2 N
(2)
, ! ,
(x2 )2 + y2 = z 2 (1)
"
x y z 2 N
(2)
!
" ", | ) " .
*
(1) (2) +
,1{8].
, 2002, 8, 3 4, . 1009{1017.
c 2002 ! ",
#\$
%
1010
. . 12+ , x4 + y2 = z 2 ! ) ! , x = 2, y = 3,
z = 5. 5 6" ) ! x,
y, z +7 ,1].
12" 8 " ! " ", 9 !" . 2" 8 " ! " ", " 7 ,2].
, " ) , 8 . :
,].
!, )" a > 1 a a = p1 1 p2 2 : : :pk k " p1 p2 : : : pk | , 1 2 : : : k | .
; n | . <", , 1
2
k
p1 n ] p2 n ] : : : pk n ]
a n adeg n. < , +
k
1 2
adeg n = p1 n ] p2 n ] : : : pk n ] :
5 !, 1) ) ! a1 a2 : : : am + ) ! deg n , deg n j a1 deg n j a2 : : : deg n j am :
2) ddeg n ) ! a1 a2 : : : am ! 8 n a1 a2 : : : am , + " 8 n
6 .
5 8 n a1 a2 : : : am (a1 a2 : : : am )deg n .
< , (a1 a2 : : : am )deg n = ddeg n , 1) ddeg n > 0 ),
2) ddeg n j a1 , ddeg n j a2,. . . , ddeg n j am ,
3) ddeg n j a1 , ddeg n j a2,. . . , ddeg n j am , ddeg n j ddeg n .
,3], t2 x2 + y2 = z 2
(3)
"
t x y z 2 N
(4)
1011
+
2 2
2
2c2 + b2 )
t = k 2>ab x = k 2>bc y = k2 2b(a >c 2; b ) z = k2 2b(a >
(5)
2
"
k a b c 2 N ac > b (ac b)deg 2 = 1 > = (2b(2b(ac b) a2c2 ; b2 ))deg 2 : (6)
(ac b)deg 2 = 1, ! (3) (4) 6 .
; c = a (5) (6) 4
2
4
2
t = k 2>ab x = k 2>ab y = k2 2b(a>;2 b ) z = k2 2b(a>+2 b ) (7)
"
k a b 2 N a2 > b (a2 b)deg 2 = 1 > = (2b(2b(a2 b) a4 ; b2))deg 2 : (8)
A (7) (8) t = x. 5 t = x (3) (4) (1) (2). < , (1) (2) +
(5) (6)
c = a, . . (7) (8). (a2 c)deg 2 = 1, ! (1) (2) 6
.
B ", + +8 .
1. (1) (2) 2
2
2
2
(9)
x = k 2>ab y = k2 2ab(a>2; b ) z = k2 2ab(a>2+ b ) 1
1
1
k a b 2 N a > b (a b) = 1 >1 = (2ab(2 a ; b))deg 2 :
(10)
(1) (2) .
2. (1) (2) 2
2
2
2
4
4
x = k c >; d y = k2 2cd(c>2; d ) z = k2 c >;2d (11)
2
2
2
k c d 2 N c > d (c d) = 1 >2((c2 ; d2)(2 c ; d))deg 2 :
(12)
(1) (2) .
3. (1) (2) 4
2 ) z = k2 2 (4 + 2 ) (13)
x = k 2> y = k2 2 (>;
2
>23
3
3
1012
. . k 2 N 2 > (2 )deg 2 = 1 >3 = (2 (2 (2 )4 ; 2 ))deg 2 : (14)
(1) (2) .
4. (1) (2) 2
4
2
4
x = k 2> y = k2 2 (>2; ) z = k2 2 (>2+ ) (15)
4
4
4
k 2 N > 2 ( 2 )deg 2 = 1 >4 = (2 (2 ( 2 ) 2 ; 4))deg 2: (16)
(1) (2) .
D (9) ( ) .
. E ! x = 60, y = 3250, z = 4850 (1) (2) (9) k = 5, a = 9, b = 4,
" >1 = 6F (11) k = 5, c = 13, d = 5, " >2 = 12F
(13) k = 5, = 3, = 4, " >3 = 2F (15)
k = 5, = 9, = 2, " >4 = 3.
* 1{4 , , -, 3, 8, ! (. (7) (8)), -, 5. (10), (12), (14),
(16), (9), (11), (13), (15) ! " (1) (2) # .
. ;! 6 (9) (11).
; (9)
k = k a = (2c c+;dd) b = (2c c;;dd) " k c d 2 N, c > d, (c d) = 1. <" (9) (11).
A , 2(c2 ; d2) 2
k = k >1 = (2 2>
2
ab
=
c ; d)2
(2 c ; d)2
2
2
4
4
2ab(a2 ; b2) = 8cd(c ; d4 ) 2ab(a2 + b2) = 4(c ; d 4) (2 c ; d)
(2 c ; d)
(9) (11).
; (11)
k = k c = (2a a+;b b) d = (2a a;;b b) 1013
" k a b 2 N, a > b, (a b) = 1. <" (11) (9).
*, 4ab
1
2 ; d2 =
k = k >2 = (2 2>
c
2
a ; b)
(2 a ; b)2
8ab(a2 ; b2) c4 ; d4 = 8ab(a2 + b2 ) 2cd(c2 ; d2 ) =
(2 a ; b)4
(2 a ; b)4
(11) (9).
< , (10) (12), (9) (11) 6.
;! 6 (9) (13). ; (9)
2
k = k a = (2 ) b = (2 ) " k 2 N, 2 > , (2 )deg 2 = 1. <" (9) (13). A , 22 3
k = k >1 = (2>
2
ab
=
)2
(2 )2
2
4
2
2
4
2
2ab(a2 ; b2) = 2 (2 ;4 ) 2ab(a2 + b2 ) = 2 (2 +4 ) ( )
( )
(9) (13).
; (13)
k = k = a0a1 = a0b
" k a0 a1 b 2 N, a21 | , 8 a, a0 | , , a = a0 a21, a > b, (a b) = 1. <" (13)
(9). *, k = k > = a0 >1 2 = 2a2 a b
0 1
a1
2 (4 ; 2 ) = 2a30b(a2 ; b2) 2 (4 + 2 ) = 2a30b(a2 + b2 )
3
(13) (9).
, (10) (14), (9)
(13) 6.
5), ! 6 (9) (15). ; (9)
2
k = k a = ( 2 ) b = ( 2 ) " k 2 N, > 2 , ( 2 )deg 2 = 1. <" (9) (15). A , 1014
. . 2
k = k >1 = ( > 24)2 2ab = (2 2 )2 2 2 ( 2 ; 4 )
2 2 ( 2 + 4)
2
2
2ab(a2 ; b2) =
2
ab
(
a
+
b
)
=
( 2 )4
( 2 )4
(9) (15).
; (15)
k = k = ab0 = b0b1 2
" k a b0 b1 2 N, b1 | , 8 b, b0 | , , b = b0b21 , a > b, (a b) = 1. <" (15)
(9). *, k = k >4 = b0 >1 2 = 2ab20b1 b1
2 ( ; ) = 2ab (a ; b ) 2 ( 2 + 4 ) = 2ab30(a2 + b2)
2
4
3
0
2
2
(15) (9).
< , (10) (16), (9) (15) 6.
< (10), (12), (14), (16) !
(11), (13), (15) 6 (9), 6 ! , , (9), (11), (13), (15) 6. < 5
.
; + +8 :
x y z 2 N (x y) = 1 x 9F
(17)
(18)
x y z 2 N (x y) = 1 x 9:
<" + (1), +8 +
(17) (18), .
+ +8 .
6. (1) (17) 4
4
4
4
x = mn y = m 2; n z = m 2+ n (19)
m n 2 N m > n (m n) = 1 m n \$ :
(20)
7. (1) (18) 4
4
4
4
x = (22mn
y
= m ; 4n2 z = m + 4n2 (21)
m)
(2 m)
(2 m)
m n 2 N m2 > 2n2 (m n) = 1:
(22)
1015
8. (1) (18) 4n4 ; m4
4n4 + m4
x = (22mn
y
=
z
=
(23)
m)
(2 m)2
(2 m)2 m n 2 N 2n2 > m2 (m n) = 1:
(24)
9. (22) (24), (21) (23) # .
H (19), (21), (23) +
! (9), (11), (13), (15). I
6.
H (19) (9) k = 1, a = m2 , b = n2 , "
m n 2 N, m > n, (m n) = 1, m n 9, " >1 = 2mnF (11) 2
n2 d = m2 ; n2 k = 1 c = m +
2
2
" m n 2 N, m > n, (m n) = 1, m n 9, " >2 = mnF (13) k = 1, = m, = n2 , " m n 2 N, m > n, (m n) = 1, m n
9, " >3 = 2nF (15) k = 1, = m2 , = n,
" m n 2 N, m > n, (m n) = 1, m n 9, " >4 = 2m.
H (21) (9) 2
2
k = 1 a = (2m m) b = (22nm) " m n 2 N, m2 > 2n2, (m n) = 1, "
>1 = (22mn
m) F
(11) 2
2
2
2
k = 1 c = m(2+ m2)n d = m(2; m2)n " m n 2 N, m2 > 2n2, (m n) = 1, "
4mn
>2 = (2 m) F
(13) k = 1 = m = 2n2 " m n 2 N, m2 > 2n2, (m n) = 1, "
>3 = (22 nm) F
1016
(15) . . 2
k = 1 = (22mm)2 = (22 nm) " m n 2 N, m2 > 2n2, (m n) = 1, "
4m :
>4 =
(2 m)2
H (23) (9) 2
2
k = 1 a = (22nm) b = (2m m) " m n 2 N, 2n2 > m2 , (m n) = 1F (11) 2
2
2
2
k = 1 c = 2n(2+mm) d = 2n(2;mm) " m n 2 N, 2n2 > m2 , (m n) = 1F (13) 2
k = 1 = (22 nm) = (22mm)2 " m n 2 N, 2n2 > m2 , (m n) = 1F (15) k = 1 = 2n2 = m
" m n 2 N, 2n2 > m2 , (m n) = 1.
D, >1 = 2mn >2 = 2mn >3 = 4m 2 >4 = 4n (2 m)
(2 m)
(2 m)
(2 m)
2
2
" m n 2 N, 2n > m , (m n) = 1.
J " , 4
4
4
4
x = kmn y = k2 m 2; n z = k2 m 2+ n " k m n 2 N, m > n, (m n) = 1, m n 9,
4
4
4
4
x = k (22mn
y
= k2 m ; 4n2 z = k2 m + 4n2 m)
(2 m)
(2 m)
2
2
" k m n 2 N, m > 2n , (m n) = 1, 4
4
4
4
x = kmn y = k2 m 2; n z = k2 m 2+ n " k m n 2 N, m > n, (m n) = 1, m n 9,
4
4
4
4
2 4n ; m z = k2 4n + m x = k (22mn
y
=
k
2
2
m)
(2 m)
(2 m)
2
2
" k m n 2 N, 2n > m , (m n) = 1, 9 (1) (2).
1017
1] . . . | ., 1978.
2] #\$
% &. '( )(
%. | ., 1959.
3] ,-(
#. .. )
// -0.
%
., \$12. 70-)5 1 -
1 \$. 6. 7. 8
. #8\$)
%
\$((% ))) 8. .%8, 1993 (.: 60 %. | #8\$)
%, 1993. | #. 17{19.
4] <8 '. 7
1 \$ ) ) 0. | .:
=%,
1992.
5] #\$
% &. . | ., 1961.
6] ,-(
#. .. =%) ?8
) ) ) \$-
1 0. | ., 1993.
7] ,-(
#. .. @2 8 (
)(
%
)-
. II. | %. ))) ?%)
)
%. | ., 1990. | A\$.
&7=767 1990, B 8-&90.
8] ,-(
#. .. 0, 10
\$(8 )%8 // -0. %
., \$12. 150-)5 1 -
1 @1. @% )
1. #8\$)
% \$((% ))) 8. .%8, 1991 (.: 60 %. |
#8\$)
%, 1991. | #. 131{132.
& 1999 .
. . -
512.643+512.742
: -, , , !"## \$%## &##, '#-()#.
# %*% * #&+*
-,. - "#%!"# % # ( %*% &#, #*%
* , !\$* %#) #& %# \$ #%* %#), \$ %#), ) . \$, )# +# .
\$, #. - "#* # % )&,%# +#
# )#, #&!"# %#). - )#%# +# # %# '#-()# %#)-/), )*
,. &% ,. ## % %. 0 %# %# )#)* ##, 1# %#%# ! &% *
% ()#* )# /). ' )&#%# &! #,,
)!"# ###). %*% n-# #) #!"# #,* .)#.
Abstract
L. I. Krechetov, On extremal properties of the dominant eigenvalue, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 4, pp. 1019{1034.
The property of almost monotonicity for the non-singular irreducible M-matrix
is speci8ed. In its existing form the property means that the result of application
of the above matrix to a vector is either the zero vector or a vector with at least
one component positive and one component negative. In this paper the positive
and the negative components are explicitly indicated. As an application, a criterion
of Pareto-extremality for a vector function with essentially non-negative matrix of
partial derivatives is derived. The criterion is a counterpart of the classical Fermat
theorem on vanishing of the derivative in an extremal point of a function. The
proofs are based on geometric properties of n-dimensional simplex described in two
lemmas of independent nature.
-. 1, 4.16, . (5)]
' , ( ) (, (, ( *
( ) * ). ) ( ) ) (
) (.
, 2002, 8, 9 4, . 1019{1034.
c 2002 !
"#\$,
%&
'( )
1020
. . ( --'() (-.(, (
* * ) . / ((
0 '()
( .(.
1) ( 1) . 1
( , )* , ( (( ' .( , () ) (,
2 (, , 2) ( -{0. 4 , ( .( . 5 , ) (), ( (* n- ( )
*(.
2 () .
3 ) ,
, )* , .(
( M-.
4 . ( (
--'().
- , ) *( '() .
/ ) --)* *
(
.
1. 7 Mmn )* m n,
Mn | )* (* n n.
9 )) :
Rn | ( n;
x = (x1 : : : xn) | ( Rn;
Rn+ = fx 2 Rn : xi > 0 i = 1 : : : ng;
(h x) | ( ( h x 2 Rn ;
x > 0 )( , ( xi > 0, i = 1 : : : n;
x > 0 )( , ( xi > 0, i = 1 : : : n, = x 6= 0;
x 0 )( , ( xi > 0, i = 1 : : : n.
1. A 2 Mn )
, = ' ).
2. A 2 M , ( S 2 Mn 1 6 r 6 n, (*
C D O F
STAS =
1021
C 2 Mr , F 2 Mn;r , D 2 Mrn;r O 2 Mnr ;r | .
1. A 2 Mn u 2 Rn ,
u 0, , Au = 0. 1) h 2 Rn (Ah)i 6 0
(1)
h ,
1 6 i 6 n, hu = 1max
6s6n u
i
i
s
s
(Ah)j > 0
(2)
h 1 6 j 6 n, uh = 16min
s6n u
2) A ! p, h = pu, h , (Ah)k < 0,
" k, uh = 1max
6s6n u
h , (Ah)l > 0
" l, hu = 16min
s6n u
3) # A , (1) (2) .
5() 1 2= .
9 ) Rn;1 (( .. . -) = (n ; 1)-
( >. ? (n ; 2)- >i ,
i = 1 : : : n. -) V i | 2 ( >, >i ,
i = 1 : : : n.
@ ( w, ( >, )
w 2 Int >. -, * ( w ) (, * 2 V i V k , ( >k >i .
7 ( vki vik . A( i , 2 vij , j = 1 : : : n, j 6= i, >i ( >i . - ( (. n = 3 ( . 1.
5 ( h 2 Rn;1 (h x) (
( x. 0 (h x) .( Rn;1.
B ) ) ) ReInt.
1. h 2 Rn;1. , \$
(h x) %
V i , i = 1 : : : K , %
V j , j = L : : : n, 1 6 K L 6 n. j
j
s
s
k
k
s
s
l
l
s
s
1022
. . :. 1
1) i , i = 1 : : : K , (h w0 ; w) 6 0 w0 2 i ;
(3)
0
0
(h w ; w) < 0 w 2 ReInt i ;
(4)
j , j = K + 1 : : : n, (h w0 ; w) > 0 w0 2 j ;
(5)
0
0
(6)
(h w ; w) > 0 w 2 ReInt j ;
2) wi , i = 1 : : : K , wi 2 i ,
i = 1 : : : K , , ! 1 6 k 6 K , (h wk ; w) < 0;
(7)
wj , j = L : : : n, wj 2 j , j = L : : : n,
, ! L 6 l 6 n, (h wl ; w) > 0:
(8)
F (7) (8) ).
1. '
\$
(h x) %
, . . K = 1 L = n, (h w0 ; w) < 0 w0 2 1;
(9)
0
0
(h w ; w) > 0 w 2 n:
(10)
F
(h V i ) = 1max
(h V s) i = 1 : : : K;
6s6n
(h V j ) = 16min
(h V s) j = L : : : n:
s6n
1023
1.
(11)
(12)
-()( ( > ) n ; 1 h 6= 0, 2
V 1 : : : V n ( , ), (11), (12) :
(h V 1) > (h V n ):
(13)
@, ( vis ;w V s ;V i ( vis (
. -' ( ) is vis ; w = is(V s ; V i) i s = 1 : : : n i 6= s:
(14)
@( (* ( h ( '* * ( (. A = ' (11), 2 v1s ( 1 (h v1s ; w) = (h 1s(V s ; V 1 )) = 1s (h V s ; V 1 ) =
= 1s (h V s ) ; 1s (h V 1) 6 0 s = 2 : : : n: (15)
-) w0 2 1. H w0 (
( 2
( 1:
w0 =
X
16s6n
s6=1
F (16) (h w0 ; w) =
h
s v1s s > 0 s = 2 : : : n
X
16s6n
s6=1
sv1s ;
X
16s6n
s6=1
s
! !
w =
X
16s6n
s6=1
X
16s6n
s6=1
s = 1:
(16)
s (h v1s ; w): (17)
B (15), (17) (3) i = 1. J ( (3) i = 2 : : : K; ( ( (5).
? w0 2 ReInt i, ('.. s (16) ), , ()( (h v1n ; w) = 1n (h V n ; V 1) < 0, (17) (h w0 ; w) < 0.
K (4) (. J ( (6). -( 1) 1
(. 5( ( 2).
-) ) ( wi, i = 1 : : : K, wi 2 i ,
i = 1 : : : K. H (
( wi, i = 1 : : : K, (
(
2 ( i :
1024
wi =
. . X
16s6n
s6=i
isvis
is > 0 s = 1 : : : n s 6= i
X
16s6n
s6=i
is = 1 i = 1 : : : K: (18)
A ) (14) !
X
X
wi ; w =
isvis ;
is w =
=
X
16s6n
s6=i
16s6n
s6=i
is(vis ; w) =
(h wi ; w) =
X
16s6n
s6=i
16s6n
s6=i
X
16s6n
s6=i
isis (V s ; V i ) i = 1 : : : K
isis (h V s ; V i ) i = 1 : : : K:
(19)
(20)
4 , (15) = (11) (h vis ; w) = is (h V s ; V i ) 6 0 s = 1 : : : n s 6= i:
(21)
-, (7) , . . (h wi ; w) = 0 i = 1 : : : K:
H (20), (21), = ) ('.. is , ( i = 1 : : : K
is(h V s ; V i ) = 0 s = 1 : : : n s 6= i:
(22)
-()( 2 V i .( (h x) (, (22)
, is ) ) 2) s = 1 : : : K, s 6= i.
H (19) , K ( (wi ; w), i = 1 : : : K,
K ; 1 ( V s ; V i , s = 1 : : : K, s 6= i. -'
( wi ; w, i = 1 : : : K, . K ' . -' 1 6 k 6 K, (
(h wk ; w) < 0, ( (7) (. K (8) ( . L 1 ) (.
7 ai (, i- ( A:
i
a = (ai1 : : : ain).
2. A 2 Mn u 0 Au = 0. X
(23)
ai = u12 aik uk(vik ; u)
i k6=i
1025
2 +u2
u
i
k
vik = u1 : : : uk;1 uk uk+1 : : : ui;1 0 ui+1 : : : un 1 6 i k 6 n i 6= k:
. K , 2
u
i
vik ; u = 0 : : : 0 uk 0 : : : 0 ;ui 0 : : : 0 1 6 i k 6 n i 6= k:
2
@) uu k- , ;ui | i- .
5 1 X ai u (v ; u) = 1 X ai u 0 : : : 0 u2i 0 : : : 0 ;u 0 : : : 0 =
i
u2i k6=i k k ik
u2i k6=i k k
uk
2
X
= aik uk 0 : : : 0 uui 0 : : : 0 ;ui 0 : : : 0 =
k
k6=i
i
X
=
0 : : : 0 aik 0 : : : 0 ; akuuk 0 : : : 0 =
i
k6=i
X i
1
i
i
i
i
a uk ai+1 : : : an :
(24)
= a1 : : : ai;1 ; u
i k6=i k
P
P
- Au = 0, . . aik uk = 0, aii ui = ; aik uk , i = 1 : : : n.
k
k6=i
A = ' 2 , ) (24) ( ai. L 2 (.
i
k
2. A
)* , * () ( 2) 1, ( * * ,
2. K( '* * 2.5 2] 8.3.1 3], ( ( *. -
, ' -{0, . , )* .
2. A 2 Mn . !
, a) b) ! ,
!
c) > Re .
'
, , A , b) c) :
1026
. . b0 ) , c0) > Re , 6= .
3. A 2 . A
( (.
5 )
A ) (A).
1. 1 (n ; 1)-
(
>0 = fx 2 Rn+ : (u x ; u) = 0g:
n 2 ' ( V i , i = 1 : : : n, )* (* * Rn (:
P u2
!
k
k
(25)
V i = 0 : : : 0 u 0 : : : 0 :
i
, ( i-
, ( ( V i . H( u )
( >0.
- ( u , ) ( >0, ( (, 1, ( > = ( >0 , ( w ( u.
0( ( h, ( p, ( h = pu. -) (h0 x) | .(, (
() (u x ; u) = 0 .(, ( h. -, .( (h0 x) ( 2* V i , i = 1 : : : K, ( >0 2* V j ,
j = L : : : n, ( >0, 1 6 K < L 6 n.
-) 1 6 i 6 K ( ai i- ( A.
-, aii = 0. H, ()( A )
Au = 0 u 0, aij = 0,
j = 1 : : : n. -' (Ah)i = (h ai ) = 0, (1), (2) .
? aii 6= 0, Au = 0 u 0 )
, aii < 0. - i = ; ua , i = 1 : : : n. '
): i > 0, i = 1 : : : n. F) (23),
2 i
i
i
1027
X
i ai + u = u12 i aik uk (vik ; u) + u =
i
k6=i
X i 1
1
i
= u2 i ak uk vik ; u2 i
ak uk u + u =
i k6=i
i
k6=i
X i
X i 1
1
= u2 i ak uk vik + 1 ; u2 i ak uk u:
i k6=i
i k6=i
-()( Au = 0, X
i aik uk = ; uaii (;aii ui) = u2i :
X
(26)
i
k6=i
-' ((* ( (26) , ) X
iai + u = u2i aik uk vik (27)
i k6=i
i ai u v :
(28)
u2i k k ik
-()( aik > 0, k 6= i, 1 6 i k 6 n, ('.., ((*
(28), ). / , ( iai + u (
(
( v ik , k = 1 : : : n, k 6= i. 5 ,
( iai + u ( i: i ai + u 2 i . H )) (3) 1, = w0 = iai + u.
-
(h (iai + u) ; u) = (h iai ) 6 0
, ()( i > 0,
(Ah)i = (h ai ) 6 0:
(29)
, ' ( ( h ( ( i, ( 2 (11). -2 (11) = , 2 ( >0 (25). H (11)
P u2
P u2
k
k
hi ku > hk ku i
i
hi > hk k = 1 : : : n:
ui uk
A2 (1) (. J ( 2 (2).
N ) 2) 1, ) ( i ai + u, 1 6 i 6 K. -) , , ()(
X
i ai + u =
k6=i
1028
. . ( u 0, (
Au = 0, A , (A) = 0. 5
), ' (, . . (A) 6= 0, 2, b0), ( v 0, (
vA = (A)v . K : 0 = vAu = (A)(v u) 6= 0.
7 AK A, K ( . -) B = A + sI , ) s
)( )2, B > 0. 7 B K B , K ( . -) B 0K |
, B * ', ** BK , , ' ', * B K , . 7
C = B ; BK . -()( B , C > 0.
F 15 8.4 3] (B ) = (B 0K + C ) > (B 0K ):
-()( (B 0K ) = (B K ), (B ) > (B K ). H (A) = (B ) ; s >
> (B K ) ; s = (AK ). -()( (A) = 0, (AK ) < 0, , ( ai , 1 6 i 6 K, K < n, .
' ( iai +u, 1 6 i 6 K, i > 0, 1 6 i 6 K,
. - ) 2) 1 (. ) .
5( ( 3) 1. @, , ( '
A ), ('.., ((* (28), ). -' ( iai + u )
( i . ( )) 1, ( ) 3).
H 1 ) (.
3. " M-
\$
A
. 1] M-. '
(, ) ) M-. H, (, ( 2) 1, ( = ) A, ( 5) 4.16 1].
4. A A = sI ; B s > 0 B > 0
(30)
(
s > (B ), (B ) | ()
B , M-
.
? s > (B ), A M-
; s = (B ), A M-
.
F 4 , A M-
, ;A , )
. O(
M-
1029
(. 1]. K ( '* | 2) , ( ) .
3. A | , ;A . 1) A M-
, A 2) A M-
, A .
H 2 3 *() ) M- , )* .
1.
1) *
A M-
, ! ;A .
2) *
A M-
, ! ;A .
. -) A | M-. H ;A |
, ) . - 1 = . 7, ;(;A) ) A. K 3, 1) ) A ). , ;(;A) > 0, ( (;A) < 0. ( 1)
1 (. 5( .
-) ;A | , ) . H = (;A). 4( 2, c), (;A) > ) ;A. H) , ) ;A )( , ( ; (
)) A. -' ),
;(;A) 6 ) A. - , (;A) < 0. 7 0 < ;(;A) 6 .
/ , ) A ). K , 3, 1), A M-
. -( 1) 1 ) (. 5( ( 2).
-) A | M-. H, M-, ()
B (30)
s: (B ) = s. -()( B | ) , ()
(B ) ( B 3]. ? v > 0 |
((
- ( B (B ),
Bv = (B )v. 7 (30) , v |
1030
. . ( )
;A : ;Av = 0. B, ) A. 5
), ' (, , 2, c), (A) )
)2 ) ;A; , ) (;A) > 0. K ( , ;Au = (;A)u, u > 0 | ((
- (,
(;A). F (30)
(;sI ; B )u = (;A)u, Bu = ;((;A) + s)u. -()( (;A)
), , B , (;A) + s,
* ()
B :
(;A) + s = (;A) + (B ) > (B )
. -' ) ;A. 1, 2) (. 5( .
-) ) | ;A. -()( )2 = ) , 6 0. 4( 2, , ) ;A )( , ( ; ) A. K ; > 0
, ) A ). H 3, 2) , A M-
. - 1 ) (.
H) (, M- ) 1. 5 M- ' . 1] .
!
. A | M-
n. Ax > 0 Ax = 0.
F 1, 2 1 ' .
2. B 2 Mn | M-
u 0 | (
) , !
( ) ! ;B . h, ! p, h = pu,
h , (Bh)k > 0,
" k, hu = 1max
6s6n u
h , (Bh)l < 0.
" l, hu = 16min
s6n u
. -()( B , A = ;B . H( (( A ,
2, b0) = ( u 0. H( , A k
k
s
s
l
l
s
s
1031
( 2) 1. - , A ;B , , ( *.
4. & "
-
-'\$
' ( * --'()
(-.(, -(
* (*
.(
, ) .
5. 0( f(x), = Rn , (
, G+ (f r) = fx 2 Rn : f(x) > rg ( * r, ((
, G;(f r) = fx 2 Rn : f(x) 6 rg ( * r.
-) f (x) = (f 1 (x) : : : f n (x)) | (-.(, f i (x) Rn , i = 1 : : : n.
)
2 ), .( :
1) f i (x) .. Rn, i = 1 : : : n;
2) f i (x) xk , k i = 1 : : : n, k 6= i, xi ,
i = 1 : : : n, ), * *,
fii (x) 6 0
(31)
fki (x) > 0 k 6= i i = 1 : : : n:
6. ( x 2 Rn ) () -, y 2 Rn, (
f (y) > f (x).
( x 2 Rn () -, y 2 Rn, (
f (y ) f (x).
) ()* - ( x P+ , ()* - ( x P0+ . F: P+ P0+ .
J ) (, ) ) -. 7 P; P0; ) ) - . A P; P0; .
= f 0 (x) n n, (
i-
( j- fji (x) | .( f i (x) xj , i j = 1 : : : n. 7, .( f 1 (x) : : : f n (x) (31), f 0 (x) ) ; ' (f 0(x)) f 0 (x), 1.
1032
4.
. . \$
f 1 (x) : : : f n (x) (31).
1) x 2 P+ x 2 P; , (f 0 (x)) = 0
10 ) x 2 P0+ x 2 P0; f 0 (x) , (f 0 (x)) = 0
2a) \$
f i (x), i = 1 : : : n, ! z 0,
Az = 0, (f 0 (x)) = 0 x 2 P0+ 2a0) \$
f i (x), i = 1 : : : n, f 0 (x) , (f 0 (x)) = 0 x 2 P+ 2b) \$
f i (x), i = 1 : : : n, ! z 0,
Az = 0, (f 0 (x)) = 0 x 2 P0; 2b0 ) \$
f i (x), i = 1 : : : n, f 0(x) , (f 0(x)) = 0 x 2 P; .
. 5( 1). -) x 2 P+ x 2 P; . -
, (f 0(x)) 6= 0. -) v > 0 | ((
- ( f 0 (x), (f 0 (x)): f 0 (x)v = (f 0 (x))v .
7 = sign(f 0 (x)) ( f 0 (x). F .. .(
2 t!0
f (x + tv) ; f (x) = f 0(x)tv + o(t)
, ()( f 0 (x)v = (f 0 (x))v > 0, f (x + tv) > f (x)
(32)
f (x ; tv) < f (x)
* * )* t, * , (( )
(), ( )
) - ( x. -( 1) (.
5 () ( 10 ) ), f 0 (x) ( v, 1, b0 , : v 0.
-' (32) ( .
5( 2a). -) ( x (f 0(x)) = 0, x 2= P0+ . H y0 , ( f (y0 ) f (x). F .(
f i (x), i = 1 : : : n, ( V
( y0 , (
f (y) f (x) * y 2 V .
7 v = y ; x. F v 2 V ; x. F ,
f (x + tv) f (x) * 0 < t 6 1 i = 1 : : : n:
(33)
F .(
f i (x), i = 1 : : : n, X
f i (x + tv ) ; f i (x) 6 fji (x)tvj * 0 < t 6 1 i = 1 : : : n: (34)
j
1033
-()( (34) * v, * ( V ; x, * = ( v 0, ( p, ( v0 = pz . H 1, 1) ( 1 6 k 6 n, (
X k 0
fj (x)vj 6 0:
j
7 (34) , * * t > 0 f k (x + tv0 ) ; f k (x) ). K ' (33). -( 2a) 4 (. -( 2b) ( .
5( 2a0). -) ( x (f 0 (x)) = 0, x 2= P+ . H y0 , (
f (y0 ) > f (x). F .(
f i (x), i = 1 : : : n, ( V
( y0 , (
f (y) > f (x) * y 2 V .
7 v = y ; x. F v 2 V ; x. F (
, ( x +tv = x +t(y ; x) = (1 ; t)x +ty G+ (f i f i (x)). A), f i (x + tv ) > f i (x), f (x + tv) > f (x) * 0 < t 6 1 i = 1 : : : n:
(35)
A , )) ..) .(
f i (x), i =
= 1 : : : n, ):
X
X
f i (x + tv ) ; f i (x) = fji (x)tvj + j (t)tvj
j
j
j (t) | .(, ( t ! 0, t > 0.
1 t > 0, ' t:
f i (x + tv) ; f i (x) = X f i (x)v + X (t)v i = 1 : : : n:
j
j j
j
t
j
j
(36)
-()( (36) * v, * ( V ; x, * = v0 , ( p, ( v0 = pz . H 1, 2) (
1 6 k 6 n, (
X k 0
fj (x)vj < 0:
j
7 (36) , * * t > 0 f (x+tv );f (x) ). K ' (35).
t
-( 2a0 ) 4 (. -( 2b0) ( .
3. \$
f i (x) S .
(f 0 (x)) = 0, x 2 P+ .
. 5 () ), G+ (f r) ( * .(
f(x).
k
0
k
1034
. . 5 * 4 5 ((
0, (, --'()
( QS () .( f (x) , ( .
(
1] Berman A., Plemmons R. J. Nonnegative matrices in the mathematical sciences. |
2] Seneta E. Non-negative matrices. | New York: Wiley, 1973.
3] ., . . | .: , 1989.
* + , 2000 !.
. . , . . . . . 517.97
: , , , !, !" !#.
\$ %! & , # !
, # %
!"& & '
. ( )& '#
*
+
, | !) '
! %
! '
! %
! . .!! '#
& '
%!
/& %! /#( !. \$ !
, !"( ! 0 , #
'
#( &
!#. 1
& 0)! !
, !"
!
& . 2#
" &
0 &
'
!( *'(, , . . *, #
.
Abstract
A. P. Levich, P. V. Fursova, Problems and theorems of variational modeling in
ecology of communities, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002),
no. 4, pp. 1035{1045.
The formulations of variation problem of ecological community, the existence
and uniqueness theorems of the variation problem solutions and the strati;cation
theorem are given in this article. For ecological application *the Gibb's theorem, is
proved | it is an analogue of equivalence of maximum entropy with ;xed energy
level problem and minimum energy with ;xed entropy level problem. Monotone
increasing property of extended entropy functional is formulated and proved. In the
case, when the number of limiting resources is greater than the number of species,
balance equations are su=cient for ;nding the population size. The compatibility
condition for the corresponding system is found. The property of variation problem
for *close, species, i. e. species with *almost, proportional quotas are described.
>
)
>..? 02-04-48085 02-04-06044.
, 2002, 8, A 4, . 1035{1045.
c 2002 ,
!"
#\$ %
1036
. . , . . 1. . !, #\$ # #
! , % & &
# &#. \$ # , ' ( & &# . )
&! &#& , \$
! &, - !.
*&& ! &\$ & +1,2]:
8>
Pw Pw Pw
><H(n1 : : : nw) = i=1 ni ln i=1 ni ; i=1 ni ln ni ! max'
Pw
>i=1 qik ni 6 Lk k = 1 m'
>:
(1)
ni > 0 i = 1 w
ni | ! ! i' qik | !
k- &, 3 &\$ i !3 & & & ' m | ! &, ' w | ! ' Lk |
!# & k (Lk > 0).
4! (1) !\$ 5
\$ && # , & ! #\$ ( , &
% &) &&\$ +1,3].
63 5& H(~n) 5& # .
1. 7 # 8&&9 , 8&39 && && +4].
2. 7, \$ &! && ! && , 3 5& ! && +1] 8! 9 ! && +4]. ; #\$ & & +3].
3. 7% # ! 5,
\$ &&\$ , & 3\$ #\$
# 5 +1].
4. < & 2 \$ , \$\$ #\$ ! & &
1037
# % &% & & 5 ! #>&% &&% !&
\$\$ .
5. 7## 5& H(~n), % & , # # , ! & & +5].
6 5&\$ #\$ ! (1) & w > m &! &#, &% +2].
1. @& ! (1) !, ni > 0, i = 1 w.
2. ; & : % L~ 2 Rm+ = fL~ 2 Rm j Lk > 0 k = 1 mg
> ! (1) &&, 3 5&\$ \$ &&
ni = n exp(;~ q~i) i = 1 w
(2)
Pw
q~i = (qi1 : : : qim ), !# n =
ni i=1
~ = (1 : : : m ) # > ! &\$
8> Pw
><i=1 exp(;~ q~i) = 1'
w k
k nP
k
~
q
exp(
;
q
~
)
;
L
= 0 k = 1 m'
i
>> i=1 i
:k
(3)
> 0 k = 1 m:
3. ; 5: 3 Rm+ = fL~ = (L1 : : : Lm ) j Lk > 0 k = 1 mg
( % S J (),
-
! & J f1 2 : : : mg , ! \$ L~ = (L1 : : : Lm ) & Rm+ , &%& J 2 f1 2 : : : mg, ! (1) ! 8>H(~n) ! max'
< Pw j
j
(4)
>:i=1 qi ni = L j 2 J'
ni > 0 i = 1 w
~n = (n1 : : : nw ).
1038
. . , . . 4, ! ! (1) (&&) 2m ; 1 ! (4) S J > ! (2) # & ( \$ ), & J. 6!3 & ( # )
% #% (% &%), . . &% !\$ % .
\$ 5&&% % \$
! (1), 5& 3\$ H(~n) > ! (1)
&! w 6 m.
2. B
3 5& +6].
~ = (1 : : : m ), . m k k
P
(2) (1), c = L . Lk
k=1
S J , k > 0 k 2 J , k = 0 k 2= J . :
8 Pw j
>> qi ni ! min j 2 J'
<i=1
n) > c'
>>H(~
w k
P
:i=1 qi ni 6 Lk k = 1 m k 6= j:
(5)
"# (2) (1) (5).
\$ % : ~n = (n1 : : : nw ) | Pw
(5), ~n % (1) Lj = qij ni .
i=1
6#& ! & ! & 5 & !
& 5 & 3 C +7], 5& & 8\$ C9 +6].
. 4> 5& D &
& ! (1) (5).
E ! (1):
L = 0 (;H(~n)) +
w
m X
X
k
k=1
i=1
qik ni ; Lk (6)
1039
8> @H(~n) P
m
><;0 @n + k=1 qik k = 0 i = 1 w'
Pw
>>k i=1 qikni ; Lk = 0 k = 1 m'
:k
i
(7)
> 0 k = 0 m:
H # 0 0, # D k ,
k = 1 m, # &, ! ! &% & \$ D. 6& 0 = 1. ; & & ! (1) % 8> @H(~n) P
m
><; @n + k=1 qikk = 0 i = 1 w'
w k
k P
k = 0 k = 1 m'
q
n
;
L
i
>> i=1 i
:k
i
(8)
> 0 k = 1 m:
E ! (5) 5& D L~ = 0
w
X
i=1
qij ni + l
X
m
k=1
X
m X
w
k
k Lk ; H(~n) +
k6=j
i=1
qik ni ; Lk :
(9)
J! & # l = 1. &! & & ! (5) % \$ &\$
8> @H(~n) Pm k k 0 j
>; @n + qi + qi = 0 i = 1 w'
>> m
k6=j
< P k Lk ; H(~n) = 0'
>>k=1k Pw k
k
q
n
;
L
= 0 k = 1 m k 6= j'
>>: i=1 i i
k > 0 k = 1 m:
i
(10)
4, ! 5& \$ && (2) 5&m
P
& H(~n) &! H(~n) = k Lk , . . & (10)
k=1
, # & (10) (8) % !#% !\$ \$ D (%! (8) & k = j). ; , > ! (1) #% ! (5).
Pw
< , Lj = qij ni &!, ! j
Pw
qij ni ; Lj
i=1
&% !\$ j , # & & ! (1) % &% & ! (5) !#% !\$ \$ D ~ ~.
i=1
1040
. . , . . Pw
6#& \$ qik ni, k = 1 m, 5& H(~n) i=1
% & 5&, D &
5& ! (1) (5) ! & (# 0 6= 0, & & # j > 0, j 2 J), , <&{; +8],
\$ % > &% !.
3. !
. ' () H(~n) (n = n(L~ )) @H > 0 k = 1 m:
@Lk
. L 5, 3\$ -
1, 3 Rm+ = fL~ = (L1 : : : Lm ) j Lk > 0 k = 1 mg
2m ; 1 % S J , J f1 : : : mg.
S J ! (1) ! (4). M S J | &# S J , & j > 0, j 2 J +2]. E #\$ ! ! (4) S J @H
@H
j
@Lj = j 2 J @Lk = 0 k 2= J
j | &%\$ # D. 63 # 5 55 &
5&\$ +9].
M & &\$
8>
;P
>< @(;@nH(~n)) + P @ =1 q@nn ;L j = 0 i = 1 w'
>> Pw qj n = Ljj2J+ lj j 2 J:
: i i
w
i
i
i=1
j
i
i
i
j
(11)
6&# ~n | > ! (4), ~ | # D, &%\$ & >%. ; > &\$ (11) #
(~n ~ ~l) ! (~n ~ 0) (~n = (n1 : : : nw ), ~ ~l # j lj , j 2 J, ), #& &!
(11) & & ! (4).
6 \$ 5& (11) ( \$ \$ &\$ +9])
1041
3 &%\$ &#: && > 0 5& ni (~l) i (~l), i = 1 w
(ni (~l) 2 C 1, i (~l) 2 C 1 S(0 )), !
Pw j ~ j @
@(;H+~n(~l)]) + X j (~l) i=1 qi ni (l) ; L = 0 i = 1 w
(12)
@n
@n
i
i
j 2J
w
X
i=1
qij ni(~l) ; Lj = lj j 2 J:
6 ~n(0) = ~n , ~(0) = ~ .
)#
& (12), # @
Pw
qij ni ; Lj
i=1
= 0 i = 1 w k 2 J
@ni
@ni (~l) @(;H(~n)) + X @ni (~l) j (~l)
k
@lk
@ni
j 2J @l
Pw j
j
@
q
n
;
L
i
i
@(;H+~n(~l)]) = ; X @ni (~l) j (~l) i=1
i = 1 w k 2 J:
k
@lk
@ni
j 2J @l
L &\$ , 55& (13), &!
8> ; P ~ ; P q n ;L @
q
n
(
l
)
;L
@
<I = =1 @l
=1
= @n
j 2 J j = k'
@n
@l
;
P
>: @n @ =1 q n ;L
0 = @l
j 2 J j 6= k:
@n
w
i
j
i
w
i
k
i
k
w
j
i
i
k
j
i
i
j
(13)
i
j
i
i
i
(14)
j
(15)
i
( (14) (15), &!
@
k~
~
(16)
@lk (;H+~n(l)]) = ; (l):
M &% z k = Lk + lk . )% &% >:
@H = @H @z k ' @H = @H @z k ' @H = @H n @lk @z k = k :
@Lk @z k @Lk @lk @z k @lk @Lk @z k n @lk @Lk
6#& S J D > ! (4) #,
j
> 0, j 2 J, S J ! (1) (4) , @H=@Lk > 0,
S
k = 1 m S J ! ! (1).
J
, ! & Lk ! +10] H- O# !\$ &
.
1042
. . , . . 4. # w 6 m
6 ! 5&# &, 3 3 . D&% &, \$
#%, ! ! !
&% ! ! (1)
. ; 5 \$ & &
# , % &%.
. ) 8H(~n) ! max'
>< w
P qk n 6 Lk k = 1 m'
(17)
>:i=1 i i
ni > 0 i = 1 w:
"# * (. . *, * ) % %
, , * #, w %
*
.
. &, ! &% 5
#> !& , ! (17) , #& &% !
% \$ !\$, & 3 : 8 w
< P qij ni = Lj j 2 J J f1 2 : : : mg w 6 m jJ j > w'
(18)
:in=1i > 0 i = 1 w:
> & !\$, ! (18). )
% w &\$ % ! ni , i = 1 w: ni = Qi=Q, Q | # , \$
!\$ \$ w w &, Qi | # , \$ i- , \$ & Lk , k = i1 : : : iw ( w &), # &% \$ (, ! &
\$ ! \$ \$ ).
6 &! ! ni > jJ j ; w &\$ !\$, &! > & &:
q1l QQ1 + : : : + qwl QQw = Ll l = iw+1 : : : ijJ j;w :
; , &!, ! w #> !&
& m, S J jJ j > w % & &\$
w
X
i=1
1043
qij ni = Lj j 2 J jJ j > w:
)
\$ ! \$ . L S J jJ j < w &, & \$ 5 +2,11], ! 5& \$
&& (2).
5. % %
6&# & , % & L1 L2 . < , &# #, . . q11 =q21 = d, q12 =q22 = d + ", " d. L &!
\$ # !\$ \$ +2], 55 & , !% , !% 5&
1 xq11 + q1xq21
1 yq12 + q1yq22
q
q
1
1
2
0
0
= 2 q11 2 q21 ' = 2 0q12 22 0q22 (19)
q1 x0 + q2 x0
q1 y0 + q2 y0
x0 | # & xq11 +xq21 = 1, y0 | # & yq12 +yq22 = 1.
# ! \$ % +11]
8> n1 q11 n2 q21
LL21 < '
>< n = x02 n = x20
q1 n2
q2
n1
(20)
LL21 > '
>>: nn = y0 qn2 ;=q1y0 n
1
1
1
q
;
q
L
2
2
1 2
2
1
n = q22 ;q21 +q11 ;q12 n = q22 ;q21 +q11 ;q12 6 L2 6 :
4# = L1 =L2 .
> # ! , % &
> \$, #
& 5& (19) (20).
1
1 "xq11
q
q
2
2
k
= (qi ") = q2 ; 2 q11 q021
1 2 q2 (dx0 + x0 + "xq01 )
1
1 "yq12
q
q
2
2
k
= (qi ") = q2 ; 2 q12 q022
2 2 q2 (dy0 + y0 + "y0q1 )
(qik ") 6 6 (qik ")
n1 =
q22 ; q21
n2 = dq21 ; dq22 ; q22 " :
n (1 ; d)(q22 ; q21) ; q22 " n (1 ; d)(q22 ; q21 ) ; q22 "
1044
. . , . . B\$3 # ! :
1"xq11
1 "xq11
n1 =
q
q
q11 n1 q12 2
2
0
0
=
=
x
=
y
1
1
1
1
0
0
n = q21 "xq01 + "q21 xq02 q21 "(xq01 + xq02 )
n =
n2 = xq21 n2 = yq22 :
0
0
n
n
=
M " ! 0:
=
q21 lim
=
lim
=
"!0
"!0
q2
2
21
11 q22
12
q
q
q
, " ! 0 y0 = x0 , y0 = x0 , #& y0
&\$ xdq21 + xq21 = 1 ydq22 + yq22 = 1.
x0 & ; , &!, & 8!9
#, # & 5 &!,
# ! :
n1 = xq11 n2 = xq21 :
0
0
n
n
J % &&% ## . D. J& ! , #
\$ , &.
&#
1] . . , . | ".: \$%- ". &-, 1982.
2] . ., *. ., +& *. . " , % - . // " % . | 1994. | . 6, 2 5. | 4. 55{76.
3] Levich A. P. Variational theorems and algocoenosec functioning principles // Ecological Modelling. | 2000. | Vol. 131, no. 2{3. | P. 207{227.
4] . . 9 && // &%
:*, ;. *. 2. | ".: <
% :=;, 2001. | 4. 163{176.
5] . . 4&& - .. | ".: \$%- ". &-,
1980.
6] . ., *. . 9 - > , - .: & > &% // ?@. | 1997. | . 42, . 2. |
4. 534{541.
7] B C. *. D , . | ".: B%, 1946.
8] B 9. "., *. ". < & - > %. |
".: \$%- ". &-, 1989.
1045
9] ? C. = , % . | ".:
F% >, 1987.
10] . . * > : % &, >
// <&, : & G
@ . H> 1. "%, %. | ".: \$%-
". &-, 1996. | 4. 235{288.
11] I& . *. F % , % - . // " % . |
>.
& ' 2002 .
. . . . . e-mail: dmikhalin@mtu-net.ru
517.518.855
: , , , !, "#! !, \$#%&! !, '&! !,
! !, , ! , ( )*.
+ (, '(\$ '\$ '(! \$
! '(! *(! ", ' '(* (, !
", ', ', *%-.
Abstract
D. A. Mikhalin, Optimal recovery of values of smooth functions and their
derivatives using inexact information on a segment, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 4, pp. 1047{1058.
We consider the problem of recovery of certain function's derivative value at the
speci5ed point when the function is smooth, belongs to a speci5ed class and values
of this function on a segment are given with an error.
1. n 2 N, k 2 Z+, 0 6 k < n, > 0, T = R+ T = R. k- x(),
, , ! "
#
W1n (T ;) = fx() j x(n;1)() 2 C(T ) jx(n;1)(t0 ) ; x(n;1)(t00)j 6 jt0 ; t00j
;g
% , # & T y() 2 C(&), kx() ; y()kC () 6 . '
2 T n &, # %# (
( x(k)()).
, 2002, 8, 6 4, . 1047{1058.
c 2002 ,
!"
#\$ %
1048
. . ,% , -. x() - y() 2
2 C(&), "# FC () (( ). /-" ': FC () (W1n (T ;)) ! R ,
e(x(k) () W1n (T ;) FC () ') :=
sup
x2W1n (T;)
y2FC() (x)
jx(k)() ; '(y())j
# '. 1 e(x(k)() W1n (T ;) FC () ') ! min # '
"
E( k n T ; &) = E(x(k)() W1n (T ;) FC () )
#, # ( ! % %, | . (3".- #. "
41, c. 125].)
8# "#, " E( k n T ; &) #
# .
9 " (
, & | 4;1 1]
40 1], (T ;) | ( " (R;?) (. . #
W1n (R) % : ), " (R+ ;0) (% #- W1n (R+), x(s) (0) = 0, 0 6 s < n), > 1.
;
#
% # " % .
1. <
x() 2 C (n;1)(T ) n # 1 <
< 2 < : : : < m #, 4j j +1], j =
= 1 : : : m ; 1, # n-- - x(n) = (;1)j c, . . -.
;%# #, | m
X
j
n
j
n
x(t) = aj t + c0 t + 2 (;1) (t ; j )+ j =0
j =1
n
n
n
, x+ = x x > 0 x+ = 0 x < 0.
nX
;1
%, "
<#
# - #.
1 ( ).
1
m 2 Z+
(
(
n
)
(
n
)
n
xmn () 2 W1 (R)
jxmn()j = 1 xmn ()
m
xmn ()
n +m +1
&
kxmn()kC () kxm;1n()kC () (kxmn ()kC ()
m ! 1)
(
)
n) ()j = 1 x(n) ()
jx(mn
m
mn
m+n
&
.
, ,
)
, >0
, &
, ,
.
xmn () 2 W1n (R)
& xmn ()
,
1049
2
m 2 Z+
xmn; () 2 W1n (R ;0)
.
m
n
&
> 0
kxm;1n; ()kC () (kxmn; ()kC ()
(
)
m
n
m
, 0
,
kxmn; ()kC () m ! 1), 2 W1n (R ;0), 0
3
&
0
.
x
xmn; ())
.
"
0
xmn; ()
%
- ,
- , fj grj=1
\$
, n k
,
,
,
-
xmn; () 2
0
-
(xmn() xmn ()
,
,
| .
E(x(k)() W1n (T ;) FC () ) = jx(k)()j
m+1
0
| 0
x(k)() Pr y( )
j =1
j
j
, j
|
-
, .
. @# " A -
# # xmnpq; () xmnpq; ()) p = q = 1, -.:
A ". :! #
%A E(x(k)() Wpn(T ;) FLq () ).
2. !
2.1. B#
# A# - ( : #. 41, c. 126{128]).
:
x(k)() ! max kx()kC () 6 x() 2 W1n (T ;) 0 6 k 6 n ; 1: (1)
(1) | (#
(. #
% # %#), A x^() .
#! # /%! ( # #. 41, c. 83{87]).
<%
/%!, ## (, #!
0 = ;1, 2 R p(), /%!
L((x() u()) p() ;1) = ;x(k)() + kx()kC () +
Z
p(t)(x(n) (t) ; u(t)) dt
T
(
n
)
% % ### u() = x () # "#, ju(t)j 6 1 x^().
# L((x() u()) p() ;1) ! min, ju(t)j 6 1 #
B# : , ! "
# 1050
. . C(&) ! #, jx()j #! % % ### # : fi gri=1 ,
:# !
x(k)() = r
X
i=1
Pr
Z
i sgn x^(i )x(i ) + p(t)(x(n)(t) ; u(t)) dt = 0 8(x() u()) (2)
T
% i > 0, i = 1 jx^(i )j = 1 6 i 6 r (
(2) "# i=1
# !#).
(# u^() # 1, . . A # #.
9 & = 40 1], T = R+, ; = ;0 , : x^()
% : ### ###, , : u^() # , #% A . ,
: # n- #A % , # G" A#H, , : (
- | G
#H.
9 & = 4;1 1], T = R, ; = ;? , : x^()
% : ### ###, #! " "
A ,
: u^() # , " n + 1, " n. , : # n- n + 1 #A
% , # G" A#H #, | G
#H.
9 " A ( !. ;!#, # %#.
2.2. ;!# ".- # " : : .
B | #! L1 (4a b]). <# !# (
Rt n;
# x() 2 B - (Tn x)(t) = (t(;n;)1)! x() d. 8% (Tn x)(n)() = x()
a
(Tn x)(j )(a) = 0 : j = 0 : : : n ; 1. I## # J = fPn;1() + (Tn x)() j Pn;1 2 Pn;1 x 2 Bg:
;!#, (# # . # #. ;
(% # " -. .
m ( , . "2]).
f 2 C(Sm Rm) f(;x) = ;f(x)
x S
x^ 2 Sm
f(^x) = 0
<#
# ".- # " .
1
"
&
. % , .
2 (m).
1051
' ': S ! B
y^ 2 J
4a b] (m + n + 1)
%.
I## -"- Sm. !#
m+P
n;1
x := '(). cj ()tj | # A% "
! j =0
m + n ; 1 (Tn x )() # # 4a b]. <# ( cn() : : : cm+n;1 (). # "! Sr Rm, , # K, # ,
. . ^ 2 Sm, cn(^) = : : : = cm+n;1 (^) = 0. 8%
nP
;1
y^(t) = (Tn x^)(t) ; cj (^)tj # (m+n+ 1)-
#
j =0
L" A.
9## B , #-. 1 #-. # : fj gmj=1 . 8% (
## #! J
" #
n m # 4;1 1]. # # #! B - #, # xmn (),
#-. (m + n + 1)-
.
9 # % 40 1], #-.: t = 0 n-% . ;
(% ## ! #! B ( 40 1]), J ## # f(Tn x)() j x 2 Bg.
M # 2, # # A% "
! m ; 1 (Tn x )() : x 2 B, # xmn; () = (Tn x^)(), #-. (m + 1)-
40 1] n-% . 8# "#, . " A: # 1 - .
# . : . 9 % : - : " A: !# t t = 1, #!#
!# 4;1 1]. # xmn (), %
= kxmn ()kC (#;11]).
I## , % 40 1] % ;0 . (# -" m # # tm ; 2 (t ; )m + 2 (t ; )m ; : : : + (;1)m 2 (t ; )m 1 + m!
2+
m+
m! m!
m!
% fi gmi=1 | . L" xmn; (), ! " , " #
(m + 1)-
. '
! # ! xmn; (), #! #!
x(m;1)n; () xmn; (), 42 1] ! " m-
. 9
-, # 2 B ! #!, 1 , J #
f(Tn x)() j x 2 Bg. , # . : ,
: # 1.
, (
, -.
0
0
0
0
0
1052
. . 2.3. "
&'( ). I## & = 4;1 1], T = R,
; = ;? . A# - /%! Z1
L((x() u()) ;1 () p()) = ;x(k) () +
;1
x(t) d
(t) +
Z1
;1
p(t)(x(n)(t) ; u(t)) dt
% # () : x(). 9 # /%!, x^() | A (1) % %,
% x():
x(k) () =
Z1
;1
x(t) d^
(t) +
Z1
;1
p^(t)x(n)(t) dt 8x 2 W1n (R)
(3)
% # ^() : x^()O
! ### u():
;
#
Z1
;1
p^(t)u(t) dt ! min ku()kL1 (R) 6 1
(4)
u^() = sgn p^():
(5)
!#, A# ( - " A xmn () #! xmn(), .
: .# . I## . fj gmj =1+n | x^(). 8% # ^()
(: :.
<
# ! p^().
9- :, x^() " 41 ], (# "# ,
p^() (# . "# (3)
-.# "#:
x(k)() =
mX
+n
j x(j ) + p^()x(n;1)()j ; p^_()x(n;2)()j + : : : +
1
j =1
n
+ (;1) ;k;1p^(n;k;1)()x(k)()j1 + : : : +
Z
(
n
;
1)
n
n
;
1
+ (;1) p^ ()x()j1 + (;1) p^(n) (t)x(t) dt:
1
1
(6)
( ! " : x W1n (R), ##
(
p^ n)() 0 p^(j )() = p^(j ) (1 ) = 0 0 6 j 6 n ; 2, j 6= n ; k ; 1, !
(;1)n;k;1p(n;k;1)() = 1, p(n;k;1)(1 ) = p(n;1)() = 0. 8% (6) , p^(n;1)() ! # : j j
1053
j = 1 : : : m + n, , 41 ] p^() #
-. :
(t ; )n;k;1 mX
+n (t ; )n;1 j;
n
;
k
;
1
p^(t) = (;1)
(n ; k ; 1)! ; j =1 j (n ; 1)! :
(# (6) " : # : x().
;
% " " (5), ":#, " p^() "
: fj gmj=1 | : x^(). 8# "#,
: : # # m + n m + n
: j :
8
mP
+n
>
l = 0 : : : n ; 1
< (l;l!k)! (1 ; )l;k ; j=2 j (1 ; j )l = 0
(7)
m+n
>
: (n(;n;k;1)!1)! (l ; )n;k;1 ; P j (l ; j )n;;1 = 0 l = 1 : : : m
j =1
% %# : k : ( : l < k) # # -.
9 " A# . (## | 0 = ;1 ,
, A # : (7). (# !#. Q "
# #! A, ! 0 = 0. 8%
" A -.
IA - # (7), #! # #!
/%!
j p^(). < (# #!
# x^() (3) | x() | (4) ### u(), , x^()
A# (1).
#!
/%! - #
# ## (#. 41, c. 127]), #! A #
% :
mX
+n
x(k)() j y(j )
j =1
% j j - xmn ().
&' . I## & =
= 40 1], T = R+, ; = ;0 . R
% -, "# .#
, #, x^() A# (1) % %, % x():
x(k) () =
Z1
0
Z1
x(t) d^
(t) + p^(t)x(n) (t) dt 8x 2 W1n (R+ ;0)
0
(8)
% # ^() : x^()O ! ### u():
1054
. . Z1
; p^(t)u(t) dt ! min ku()kL1(R) 6 1
0
#
(9)
+
u^() = sgn p^():
(10)
!#, A# ( - " A xmn; () #! xmn; (), . : 2.2. I## . fj gmj=1 | x^(). 8% # ^()
(: :. M , fj gmj=1 | x^().
<
# ! p^().
9- :, x^() " 40 ], (# "# ,
p^() (# . "# (8)
-.# "#:
0
x(k)() =
m
X
j =1
0
j x(j ) + p^()x(n;1)()j0 ; p^_()x(n;2)()j0 + : : : +
+ (;1)n;k;1p^(n;k;1)()x(k)()j0 + : : : +
+ (;1)n;1 p^(n;1)()x()j0 + (;1)n
Z
0
p^(n) (t)x(t) dt:
(11)
( ! " : x W1n (R+ ;0),
## p^(n)() 0 p^(j )() = 0 0 6 j 6 n ; 1, j 6= n ; k ; 1, !
(;1)n;k;1p(n;k;1)() = 1 ( t = 0 x() ". ). 8% (11) , p^(n;1)() ! # :
j j j = 1 : : : m, , 40 ] p^()
# -. :
m
n;k;1 X
; j )n;;1 p^(t) = (;1)n;k;1 (t(n;;)k ; 1)! ; j (t(n
:
; 1)!
j =1
(# (11) " : # : x().
;
% " " (10), ":#, " p^() "
: fj gmj=1. 8# "#, # # m m : j :
m
(n ; 1)! ( ; )n;1 ; X
j (l ; j )n;;1 = 0 l = 1 : : : m:
(12)
l
(n ; k ; 1)!
j =1
9 " A# . (## | 0 , , A # : (12). (# !#. Q "
# #! A, ! 0 = 0. 8%
" A -.
1055
IA - # (12), #! # #!
/%! j p^(). < (# #!
# x^() (8) | x() | (9) ### u(), , x^()
A# (1).
IA # -. :
x(k)() m
X
j =1
j y(j )
% j j - xmn; ().
0
2.4. \$ S " A: : #! A
#
: n. 9 : : , -##, # #. #-. ## ( #
n- ) % ( % %- !- #%: : #), ! # " A: , ":# ! : ( j = j sgn x^(j ) #
!. Q ( - #
# .
# .
IA# # # (7) k = 0, n = 3. 9 (-. p^(m ) = 0) ( m+2 m+3 , m = m+1 . j = m + 1 m + 2 m + 3 " " : # # (7) = j j-# "#,
, M#, i(j ) = ij j = m + 1 m + 2 m + 3.
" #- A , j #-
! , , # | "
. T : m+1 , m+2 , m+3 ##
x(k ) mX
+n
j =1
j (k )y(j ) = y(j ) k = m + 1 m + 2 m + 3
. . #
# | #%
# y() : : : xm3 .
I## " A k = 0, n = 4, m = 5. 8 !,
n = 3, , #
# |
# , : : : x54
-. y(). T ( - -. #. IA # (5), # -. #!
/%!:
1056
. . 0 ;16531 10;6 + 60081 10;6 ; 71534 10;6 2 + 27984 10;6 3
BB 0000038178 ; 000013876 + 000016521 2 ; 0000064629 3
BB ;000084248 + 00030619 ; 00036456 2 + 00014262 3
BB
0018505 ; 0067256 + 0080077 2 ; 0031326 3
() = B
;040629 + 14766 ; 17581 2 + 068777 3
BB
89199 ; 32418 + 38598 2 ; 151 3
BB
;18982 + 85427 ; 11686 2 + 5041 3
B@
24306 ; 11826 + 18066 2 ; 86709 3
;12855 + 63839 ; 10073 2 + 50741 3
1
CC
CC
CC
CC :
CC
CC
A
9## # - y(t) = t4 ; t2 . 8% #
# ##
"
x() ;02624 + 15444 ; 42824 2 + 30004 3:
T . 1 % y() #
% # x() ( . 8# # : :
: x54.
7. 1. y(t) = t4 ; t2 #! ( '(\$ ' *\$! k = 0, n = 4, m = 5
IA# # (12) k = 0, n = 3, m = 4. # -. #!
/%!:
0 ;73044 10;12 + 15319 10;11 ; 81286 10;12 2 1
CC :
B
11745 ; 24975 + 1323 2
() = B
A
@
;26326 + 66016 ; 3969 2
15581 ; 41041 + 2646 2
9## # - y(t) = t5 ; t2 . 8% #
# ##
"
x() = 2572 ; 78448 + 52727 2:
T . 2 % y() #
% # x() ( .
1057
7. 2. y(t) = t5 ; t2 #! ( '(\$ *\$ ;0 k = 0, n = 3, m = 4
3. #
# , :
%
: %: #%": | #, # !.
R. . K
" 43] A
:! E(x(0)
_ W1n (R) FC (R
))
-.% #
% # . IA# (# (
1 .
R
% # ## A !, A # = m , m 2 N, % m | # m-% (
% n.
T # :! E(x(k)() W1n (R+) FC (R) )
#
% # . R # x(k)() ! max kx()kC (R) 6 x() 2 W1n (R+) 0 6 k 6 n ; 1: (13)
9 ( "
A = 0, " #, " #. U"% M
" 44] A
- -. :
jx(k)(0)j ! max kx()kC b(R) 6 1 kx(n)()kL1 (R) 6 2n;1n!: (14)
;, n > 4 A (14) # : " A-
: , A
n = 2 3 - k- " A%
# t = 1.
+
+
+
+
n) (t) = sgnsin mt,
;"#! ! xmn () n '# !#, x(mn
t 2 R, * * ! ( ,-. ; ! ! n
'(! ! ! n + 1.
1
1058
\$
. . 1] - . ., . . . |
.: "#\$ %&'', 2000.
2] Borsuk K. Drei S*atze u*ber die n-dimensionale euklidische Sph*are // Fund. Math. |
1933. | Bd. 20. | S. 177{191.
3] 01 2. 3. 4 56 7 \$ #889. | .,
1979.
4] Shoenberg I. J., Cavaretta A. S. Solution of Landau's problem concerning higher
derivatives on the half line // Proc. of the Intern. Conf. on Construction Function
Theory, Golden Sands (Varna), May 19{25, 1970. | So?a: Publ. House Bulgarian
Acad. Sci., 1972. | P. 297{308.
& ' 2002 .
, . H. 517.55
: , , -
, .
!" # , \$\$"%
\$ & . ! \$ '
! ! # , \$\$"% \$ , ' \$ %(& , #"%\$ #.
Abstract
S. N. Mishin, Operators, commuting with operators of nite order, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 4, pp. 1059{1067.
Linear continuous operators commuting with operators of 2nite order are studied
in this work. The proved theorems contain as a special case some well-known results
about the linear operators commuting with di3erential and also with generalised
di3erential operators, acting in the spaces of analytical functions.
, , , , XX . (., , #5{7,15,16]). + ,. -. .
#8] .
+ , A , H . 0 1 2-
, .
, 2002, \$ 8, 4 4, . 1059{1067.
c 2002 ,
!" # \$
1060
. H. + ,
2 +. 0. 4
#2, 3] 7 7 #11{14]. + B : H1 ! H , A (H1 H | 1 k
P
A H ), B =
ck A , k=0
ck | 1 B < 1 B .
+ .
x
1. 0 H | A : H ! H | .
=
H 2 kxk1 6 kxk2 6 : : : 6 kxkp 6 : : :
(1.1)
0
0
0
kxk1 6 kxk2 6 : : : 6 kxkp 6 : : ::
(1.2)
> , (1.1) (1.2) (
). (H kkp) H , (H kk0p) | H1. > , An : H1 ! H ,
n = 1 2 : : :, . ? 8n 8p 9Cp(n) 9q(p n): kAn(x)kp 6 Cpkxk0q 8x 2 H1:
@
, A : H1 ! H , 2 fcng, fcn Ang
, . .
8p 9Cp 9q : kcnAn(x)kp 6 Cpkxk0q 8x 2 H1 8n:
@ , , , cn = nan, a 2 R ( ).
0
kAn (x)k p
(p q n) = sup
0
k
x
k
kxkq 6=0
q
(
(p q n) = +1 ). p- p , p- p
A (. #3,14]):
ln (p q n)
fpg
p (q) = nlim
!1 n ln n p = qlim
!1 p (q) = sup
(p)
8
fpg p = 8p
;p ((p q n))1=n p = lim p (q) = <sup
(p)
n
p(q) = nlim
!1
q!1
:0
p < 8p:
0
, 1061
B p-
p-
8p 8" > 0 9Cp(") 9q(") 8x 2 H1 8n : kAn(x)kp < Cp(p + ")nnpnkxk0q (1.3)
8p 8" > 0 8C 8q 9nk(") ! 1 9fxk(")g H :
kAnk (xk)kp > (p ; ")nk nkpnk kxkk0q : (1.4)
x
2. , 0 H | (1.1). 0 A : H ! H | . C
H1 H A H (H1 H ). 0 H1 (1.2) H1 1 . 0
H1 (1.1) Hf1.
0
1
X
f () = xnn : C ! H |
(2.1)
n=0
- A, . . , A(f ()) = f () 8 2 C
(2.2)
(. #4,13]).
> , (2.1) (1.2)
(
, f H1) f () , g(), (2.2), g() = '()f (), '() | . ? - fxng p H , xn =
6 0 8n, A(xn) = xn;1, n = 1 2 : : :,
n
A(x0) = 0, nlim
k
x
k
=
0 8p. D 1 A n p
!1
, 7 (. #13]).
C
p (f ), p (f ) f (1.1),
0p (f ), p0 (f ) | f (1.2):
ln ln kf ()kp (f ) = lim ln kf ()kp p (f ) = lim
p
!1
!1 jjp (f )
ln jj
0
ln ln kf ()kp 0
ln kf ()k0p
0p (f ) = lim
(
f
)
=
lim
:
p
!1 ln jj
!1 jjp (f )
E ln kf ()kp
{p (f ) = lim
jjp(f )
0
!1
1062
. H. 2 p-
- f p- p (f ),
p (f ) p0 (f ) | p-
- f p- p (f ) 0p (f ). 0 1 (f ) = supfp (f )g,
(p)
{ (f ) = supf{p(f )g | f
(p)
(f ).
0 B : H1 ! H | , A:
8p 9Cp 9q : kB(x)kp 6 Cpkxk0q 8x 2 H:
B f AB = BA B (f ()) = '()f () 8 2 C :
(2.3)
'() | . > 2 B . B (2.3) B
'()
8p 8" > 0 9Cp(") 8q > q0(p) 8 2 C :
j'()j < Cp exp (q0 (f ) + ")jjq (f ) ; ({p(f ) ; ")jjp(f ): (2.4)
0 Lfxng | 1
- f (). @
: B D ', x 2 Lfxng B (x) = D(x) , fxng H1, , , .
E , B (f ()) = D(f ()) = '()f () 8 2 C . F
Q = B ; D. @ , Q(f ()) 0<
1
P
, Q(f ()) =
Q(xn)n . @
, Q(xn ) = 0 8n.
n=0
C
( Q) .
.
B , A,
1
X
ck Ak (2.5)
0
k=0
ck | 1 . D 1 S , x 2 Lfxng
S (x) = B (x). D fxn g H1 (2.5) , | B . G, B
| , 1 (2.5). D B , , 1063
A : H1 ! Hf1 p-
p p- p ,
1 n
P
- ( ) f () =
xn n=0
. " B A ' ( ) # 1) (') < 1p 8p
2) (') = 1p (') < e1p= 8p,
(2.5) x 2 H1 1
P
x 2 Lfxng ck Ak (x) = B (x)1 . \$ , fxng H1 ,
k=0
(2.5) B H1.
1. .
0 B A. H
( - A )
B ', (2.3). 0 ' . 0 jckj < Kk; ('k)+" 8k 8" > 0:
0 p-
kAk(x)kp < Lpk(p+")k kxk0q 8x 2 H1 8k 8" > 0 8p q = q(p):
0
, fSn g (2.5) . B
N
N
kSN (x)kp 6 X kckAk(x)kp 6 Mp X k(p; (1') +")k kxk0q 6 Cpkxk0q 8x 2 H1 8N 8p:
k=0
k=0
0 (. #1]) fSN g S : H1 ! H , S ', 1 x 2 Lfxng S (x) = B (x), fxng H1 , x 2 H1
S (x) = B (x).
@
2 .
H
.
2. 1 fxng H1 0p (f ) = p (f ) = (f ) = 1 0 (f ) ; { (f ) < 1= 8p:
ep
1
5 ('), (') | ' (
) (\$. 79, 10]).
1064
. H. B
A,
(2.5)
E
, (2.4) 1.
%, , .
x
3. 1. 0 H1 = H = s | , kxkp = max
fjxkjg. C
A(x0 x1 : : :) = (x1 x2 : : :) P1 k6p
, - f () = en n
n=0
(en | ). 0
, B , A, | . E , B , ,
8p 9Cp 9q : kB(f ())kp 6 Cpkf ()kq 8 2 C :
I
, (
kf ()kp = 1j jp jjjj 6> 11
j'()j = kBkf(f(()k))kp 6 Cpjjq 8 2 C :
p
+ '() | .
B
, B A, , .
2. 0 H1 = H = H (C ) | : kF kp = jmax
jF (z)j. C
zj6p
d
A = dz , 1
P
- f () = ez = znn! n . 0
, n=0
, dzd , | 1
. E , - f () = ez p- p (f ) = 1 p- p(f ) = {p (f ) = p < 1.
+ (2.4) '() 2 #1 1). C
dzd p- p = 1
p- p = 0. H
, 2 : B
dzd H (C ), , , 1065
B=
1 dk
X
ck dz k :
k=0
(3.1)
3. 0 H = # ], > 1, | , , . + 2
H1 # ], 6 . H # ] # ] ;(+"p)r 8F 2 # ] lim "p = 0
kF kp = sup
max
j
F
(
z
)
j
e
(3.2)
p!1
r>0 jzj6r
;(
+"p )r 8F 2 # ] lim "p = 0: (3.3)
kF k0p = supjmax
j
F
(
z
)
j
e
p!1
zj6r
r>0
0
# ] # ] # ]. C
dzd # ] - f () = ez , # ].
J
1 (3.2) (3.3) p (f ) = 0p (f ) = ; 1 p (f ) = {p (f ) = ; 1 (( + "p )) 1 1 p0 (f ) = ; 1 (( + "p )) 1 1 0(f ) = ; 1 () 1 1 { (f ) = ; 1 () 1 1 :
;
;
;
;
# ] ! # ] p- p-
1 1 1 1 1
;
1
p = p = e () 1 ; + "
:
p
B (2.4) , B : # ] ! # ], dzd : # ] ! # ],
1 1 '() 2 ; 1 ; 1 ; 1 2. @
, B : # ] ! # ] dzd : # ] ! # ], , (3.1).
4. 0 H = H (G) | , G. + 2 H1 H (D) ,
D, D | d-
7 G (. .)1 .
H H (G) H (D) C
d
dz :
;
;
;
;
1d
\$' ".
1066
. H. kF kp = zmax
jF (z)j 8F 2 H (G)
2G
G1 G2 : : :
kF k0p = zmax
jF (z)j 8F 2 H (D)
2D
D1 D2 : : :
p
p
1
p=1
Gp = G
(3.4)
Dp = D:
(3.5)
1
p=1
0
H (D) H (G) HD . C
dzd H (G) - f () = e(z;z0 ) , z0 2 G, H (D). C
p- p (f ) = 0p (f ) = 1 { (f ) 6 (f ) 6 0(f ). C
dzd : H (D) ! HD
p- p = 1 p- p = ed1p (dp | Gp D).
+ (2.4) B : H (D) ! H (G), dzd : H (G) ! H (G), '() 2 #1 0(f ) ; { (f )]. D '() 2 #1 d], (3.1).
F
. 0 G | r, D | r + d. + 1 { (f ) = (f ) = r, 0 (f ) = r + d, ,
'() 2 #1 d] , dzd , , (3.1).
"
1] . . . | .: , 1967.
2] "#\$ . %. %#& '( ) (#'# #* #& ( +,+'\$ &\$ // /0 1112. | 1986. | 4. 228, 5 1. | 1. 27{31.
3] "#\$ . %. %#& '( (#'# '#8 //
98 (+. | :#: :"9, 1999. | 1. 6{23.
4] "#\$ . %. 0) #*& 4# // <\$. (#. \$'. | 1999. |
4. 5, (. 3. | 1. 801{808.
, 1067
5] ?+ @. 1. :, (#'#, (#+'8 + #'\$ ##\$. | /++.. . . . .-\$. . | 9, 1981.
6] A#, B. <. :,C (#+'8 + (#'#\$ ##& (#'# (#+'#+' '8+ // <.
. ) (#*. | 1973. | 4. 7, 5 1. | 1. 74{76.
7] A#, B. <. : (#+' (#'#, (## +'DC (#+'#+' '8+ (#+'8 + (#'#\$ ##& // @. \$'\$. +'. ). 0H. | 1974. | 4. 5. |
1. 359{388.
8] A#, B. <. :(#'# +) 8+ +\$+'. | @- 2+'.
., 1983.
9] E . F. 2+(# # . | ., 1956.
10] E' 0. <. G . 2& I+('. | .: , 1983.
11] J 1. H. : (#& '( (#'# // /0. | 2001. | 4. 381, 5 3. |
1. 309{312.
12] J 1. H. :(#'# 8) (#& // 98 (+. (. 2. | :#:
:"9, 2001. | 1. 28{75.
13] J 1. H. : +('# (#'# 8) (#& // 98 (+.
(. 2. | :#: :"9, 2001. | 1. 96{115.
14] J 1. H. %#& '( (#'# (+'+' (#'#, +'DC ( (#+'#+' // 98 (+. (. 3. |
:#: :"9, 2002. | 1. 47{99.
15] ( . . :, (#'#, (#+'8 + ##\$, (#+'#+' ' + (#\$ // '. \$'. | 1978. |
4. 24, 5 6. | 1. 829{838.
16] Boas R. P. Functions of exponential type, III // Duke Math. J. | 1944. |
P. 507{511.
% & ' 2001 .
CR-
. . e-mail: simona@math.kgu.krasnoyarsk.su
517.55
: , CR-, !{#, \$%.
&'\$' (''( \$ ' CR-% !! ; \$'. *+
\$+ \$(! ; %, ,-! .
Abstract
S. G. Myslivets, The analytic representation of CR functions on the hypersurfaces with singularities, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002),
no. 4, pp. 1069{1090.
The validity problem of the analytic representation theorem for CR functions
on hypersurfaces ; with singularities is considered. Near singular points of ; the
boundary behavior of the functions that giving the analytic representation is investigated.
1. CR-
3, 22], , !" # CR-
(.,
, 21,23]). () , , * {, . -
.
.# / # # C n , n > 1, 0# 1
#, . . H1 (/ O) = 0 (
, # / #" ).
&\$ ( 6 &77*, 99-01-00790.
, 2002, ' 8, 8 4, . 1069{1090.
c 2002 ,
!"
#\$ %
1070
. . .!
, ; # ( C 1 ) # /, "6 / ! /+ /; .
7 # # ; = fz 2 /: (z) = 0g
# ( C 1 ) 6 / , d 6= 0 ;. /+ = fz 2 /: (z) > 0g, /; = fz 2 /: (z) < 0g. 9
; /+ , /+ ; # .
: , f 2 L1loc(;) CR-
;, Z
f @ = 0
(1)
;
# (n n ; 2) 1
C 1 (/) /.
- ; (1) , f;]01 @- (@(f;]01) = 0 /).
1 (
, , ). CR-
f 2 L1loc(;) h /, / n ;
@h = ;f;]01. , h / h ,
, f = h+ ; h; ;:
(2)
(2) ! ":
k(;),
1) ; 2 C k+1 , k | % , f 2 Cloc
k
0 < < 1, h 2 Cloc (; / ) (2) " ;&
2) ; 2 C 1 f 2 Lploc(;), p > 1, z 0 2 ; % U , lim
"!+0
Z
;\U
j(h+ (
+ "(
)) ; h;(
; "(
))) ; f(
)jp d=2n;1(
) = 0
=2n;1 | (2n;1)- ' ;, (
) | " %"
; , " /+ . , (2)
" ( ' f %"( "( h .
k(;) C k , . Cloc
k ( # ) " # "
*)# .
*
h #" # ,{>
f (.
9,23]).
CR-
1071
(
f 2 L1loc(;) CR-
; n F, F | !, ,
6 , (. 1 2 !). 9 # !
#, H1 (/ n F O) ! # #. (
; # D, 6 !
CR-
f ; n F # D. 0
A (. *. B 12], 19,25]).
D !. E, F | / # n #A , 8]. D 1 # , H1 (/ n F O) = 0 , #, . 7 # !, CR-
.
G
CR-
CR- ; # 2,7,10,14].
I#" " CR-
" f, f ) # ! F.
.
1 ; ! ! # ! F. : , h F .
E, ; " ) , 11].
-) .
1 (. 22]). .# / # , / = fz = (z1 z2) 2 C 2 : jz1j < 1 jz2 j < 1g:
*
# ; ; = fz 2 /: Imz2 = 0g, F = fz 2 /: z1 =
= Imz2 = 0g. .!
/+ = fz 2 /: Im z2 > 0g, /; = fz 2 /: Imz2 < 0g.
;
" f = z11 . - #, f 2 L1(;) f #
CR-
; n F.
(
f 1, f = h+ ; h; ; n F, h | / .
lim
t!+0
Z
h (z1 t) dz1 = 0
jz1 j=1=2
:A
(
h (z1 t) z1 1072
. . t 6= 0), Z
jz1 j=1=2
1
z1 dz1 = 2i 6= 0
!
" 1.
7 , ! ! F CR-
.
2 (. 8]). .# / # ( 1).
>! ; ; = fz 2 /: jz1 j = jz2 jg:
- O ;. 0
#, (z) = z1 z 1 ; z2 z 2, ; = fz: (z) = 0g d(z) = 0 # O.
;
! / = fz 2 /: (z) > 0g "
f = z11z2 . 7 CR-
; n O. : ,
f 2 L1 (;). 0
#, ; Re z1 = r cos '1 Imz1 = r sin '1 Re z2 = r cos '2 Imz2 = r sin '2 0 < r 6 1, 0 6 '1 '2 6 2. * G ; 02 0 0 1
G = @0 r2 0 A 0 0 r2
1
p
p
d=3 = det G dr d'1 d'2 = 2r2 dr d'1 d'2:
E#,
Z
Z 1
p
2
jf j d=3 =
d=
3 = 2(2) :
jz z j
;
;
1 2
.# f 1, f = h+ ; h; ; n O.
. :A
Z
jz1 j=1=2
jz2 j=1=2
Z
jz1j=1=2
jz2j=1=2
h(z) dz1 ^ dz2 = 0
1 dz ^ dz = (2i)2 6= 0
z1 z2 1 2
!
".
CR-
1073
.
, # ; , ! # # CR-
. D # ; 2 " CR-
, ! , ; n O (. 10, Prop. 1]). D6, , # " CR-
(. 7]). , ; 1.
2. D #A # "6" ". 9# / C n
(n > 1) , H1 (/ O) = 0. K ! ; / / ! /+ /; : ; = M F, F /
=2n;1(F ) = 0, M | ( C 2n;1) #
/ n F . , #, M = fz 2 / n F : (z) = 0g
| 6 C 2n;1 / n F , d 6= 0 M, / = fz 2 / n F : (z) > 0g.
9
L1loc(;) "6
: f 2 L1loc(;), f 2 L1loc (M) " K /
sup
Z
">0
M \K nfz : d(zF )<"g
jf j d=2n;1 < +1
d(z F ) | z ! F.
(
; , L1loc(;), # =2n;1(F) = 0.
(
; # " (2n ; 1)-" B (=2n;1(; \ K) < +1 " K /), f (. . f 2 L1 (;)) ! L1loc(;).
:! f 2 L1loc(;) f;] 1 / "6
:
Z
f f;]() = "!
lim+0
M nfz : d(zF )<"g
| # (2n ; 1) 1
/.
;
6" #" " '(z) C 1 (/), " F = fz 2 /: '(z) = 0g. 6 (.,
, 15, 1.4.13]). .#
M" = fz 2 M : '(z) > "g " > 0:
C 1 (/)
1074
. . . E (., , 15, 1.4.6]) M" # @M" " > 0 ( ! #
E !
' M).
.# K | # /, ! ' M \K , #, 1 !
!
' ! . .1 ! !
# ",
@M" \ K # , .
(
f 2 L1loc(M), " > 0 f 2 L1loc (@M" ). 0
#,
" 2 a b], 0 < a 6 b, ! M" # @M" , ! Mab = fz 2 M : a 6 '(z) 6 bg M
(., , 24, . 248])
Z
Mab\K
jf j d=2n;1 =
Zb
a
d"
Z
@M"\K
jf j d=2n;2
=2n;2 # (2n ; 2)- B @M" , = jd=2n;1=d"j. 6= 0 @M" \ K, Z
jf j d=2n;2 < +1
@M"\K
" 2 a b].
9
f 2 L1loc(M) K /
!
Z
Sf (K ") =
jf j d=2n;2:
@M" \K
2. )% f 2 L1loc (;) % CR- M -
!
Sf (K ") = o(1) " ! 0
(3)
!"( K /. * 1 " f .
+ , f = h+ ; h; M
h 2 O(/ ), h ;
(. . M) , 1.
. > ! , f 2 L1loc(;) f;] 1 /. .!, @f;]01 = 0 /:
(4)
1
1. , f 2 Lloc (M) CR- M , ( " > 0 Z
Z
f @ = f M"
@M"
1075
CR-
( -( %"( (n n;2) .
C 1 (/) " /.
. .# z 2 M. ;
A B(z r) z r, "6
F , CR-
f
! L1, . . 6 ## Pk , Z
jf ; Pk j d=2n;1 ! 0
M \B(zr)
k ! 1:
E6
#
# 6 ,
4] !
. 0 . ! 22], 9, 6.6].
.!
, z 2 Mab , supp B(z r) B(z r) \ M Mab . .
M
Z
Mab\B(zr)
jf ; Pk j d=2n;1 =
Zb
a
Z
d"
@M"\B(zr)
jf ; Pk j d=2n;2 ! 0
k ! 1. .1 6 ## ks, Z
@M"\B(zr)
jf ; Pks j d=2n;2 ! 0
" 2 a b].
E#,
Z
lim
s!1
. E
@M"\B(zr)
Z
M"
s ! 1
jf ; Pks j d=2n;2 = 0:
Pk @ =
Z
Pk :
s
s
@M"
. s ! 1, "
!
M" \ B(z r). . "" A B(z r), !
6 .
0# 1 # 8.2
9].
.!
# 2. . 1
@f;]01() = f;]01(@) = "!
lim+0
Z
M"
f @ = "!
lim+0
Z
@M"
f :
1076
.#
. . Z Z
f 6 c
@M"\K
@M"
jf j d=2n;2 = cSf (K ") ! 0
K = supp , (4) f;]01.
(
; | , (4) , f CR-
;, # f CR-
, "6
" .
D , F ! ;, (4) !
# CR-
.
.!, f, "6 (4), . 0# # 1 , 23], 9, x 6].
.#
n
X
M = Mk @z@ |
k=1
k
, 1
Mk " C n
X @Mk
= k=1 @z k
# -
0
0. . T 2 E 0pq (C n ) # XX
T=
TIJ dzI ^ dz J I
J
(j1 : : : jq )
I = (i1 : : : ip ) J =
| "6
#
p q N
dzI = dzi1 ^ : : : ^ dzi dzJ = dzj1 ^ : : : ^ dzj TIJ | C n .
D 6] ) " - XXX
M \T =
(M TIJ )IJ (j) dzI ^ dz J nj p
I
J j 2J
q
| , J n j # #
q ; 1,
"6
J j, IJ (j) dz j ^ dzI ^ dz J nj = IJ (j) dzI ^ dzJ :
(
q = 0, " M \ T = 0, M \ T 2 D0pq;1 (C n ).
E
@- (. 6])
(5)
T = @(M \ T) + M \ @T:
CR-
1077
D M #) ,{>
:
n
X
M = (n2;n1)! j
jk2n @ @
k
k=1
1
1 " # C n M \ (z ) d=2n = U(
z)
z # -
z (., , 9, x 6]). M U(
z) ,{>
:
n
(n
; 1)! X
U(
z) = (2i)n (;1)k;1 j
k;;zzj2kn d
k] ^ d
k=1
d
= d
1 ^ : : : ^ d
n d
k] d
d
k .
: , 1
M \ T T.
H01(/ O) = 0 @f;]01 = 0, 0# f;]01 = ;@h,
h | /. .# supp f;]01 ;, *
h / n ;.
.# a 2 ;. ;
U a !6
/, U 0 a,
!6
U. .# | C 1 (U) U, 1 U 0. T = f;]01
!
E 001(C n ) @T = 0 U 0. . (5) ,{>
M T = @(M \ T) + M \ @T:
@T = @(M \ @T) = 0 U 0 M \ @T 1
U 0, *
M \ @T = @'
' # C 1 U 0. T = @(M \ T + ') U 0:
M
' ;. @h = f;]01 = T U 0 , #
h ; (M \ T + ')
0
U *
. E#, (. . # h+ ; h; ) ; #" M \ T.
1078
. . : 9, x 6]
M \T =
Z
;\U
(
)f(
)U(
z) z 2= ;:
9" ,{>
(., , 9, . 1]) # .
.
1 2 ", (3) ! #, 6 ,
A. D 2 @;" = fz 2 ;: jz1j = jz2j = "g, 1 f = z11z2
Z
p
p
Sf (") = 2"2 "12 d'1 d'2 = 2(2)2 :
@ ;"
Q 1.
1. )% ; % ! (2n ; 1)-! ' =2n;2(F ) = 0. , f L1
loc(;) CR- M , f 2.
. 0 " K / j'(z)j = j'(z) ; '(
)j 6 C jz ; j 6 Cd(z F) z 2 K 2 F:
=2n;2(F ) = 0, " " > 0 6 F \ K
A
, 6
> 0, 6
1
A #A ". 9" , Sf (K C) ! 0 " ! 0.
0 ! F ) 2 ! # .
.# 6 / g, F # ! , . . F = fz 2 /: g(z) = 0g. #
! 2n ; 2 (., , 24, . 25]),
=2n;1(F) = 0.
2. )% f 2 L1loc(M) CR- M k > 0, f(z)dk (z F) 2 L1loc(;), k0 > 0, fgk0 2 L1loc (;) @(fgk0 ;]01) = 0 / (. .
fgk0 % CR- ;).
. .#
F" = fz 2 /: d(z F ) 6 "g:
" K /
jg(z)j = jg(z) ; g(
)j 6 C(K)d(z F ) 6 C(K)"
z 2 K \ F" , 2 F \ K. .1 # k0 > k + 1 fgk0 2 L1loc (;).
CR-
1079
.# # # (n n ; 2) 1
C 1 (/) , !6
K. @(fgk0 ;]01()) =
Z
fgk0 @ =
;
Z
fgk0 @(1 ; " )] +
;
Z
fgk0 @(" )
;
" | C 1 (/), 1 F"=3 \ K " F2"=3\K ( ! # " ) ! F"=2 \ K ). @ C
" 6 1 j = 1 : : : n
@zj
"
(., , 5, x 4.5]). .1
Z
Z
fgk @(") 6 C2"k ;k;1 jfdk(z F)j d=2n;1 ! 0
0
0
;\K
;
" ! +0.
S Z
;
Z
fgk0 @(1 ; " )] = f @gk0 (1 ; ")] = 0
;
# # gk0 (1 ; " ) !
/ n F.
3. )% F % / g.
) , f 2 L1loc(M) CR- M k > 0, f(z)dk (z F) 2 L1loc (;), f
2 .
. S 2 # 2 , fgk0 , . . h 2 O(/ ), ; (. . M) fgk0 = h+ ; h; . f +
;
f = ghk0 ; ghk0 M
! M, 1.
(
; F | , f 2 L1loc (;), 3 6 !
1 10] .
.
2 3. 0
#, O !
! F = fz 2 /: z1 = z2 g. - ! ; n F , 1
+
;
z1z2 = h ; h 1080
. . h+ = z2 ;1 z1 z11 , h; = z2 ;1 z1 z12 . , " ; n O
z11z2 = h+ ;h; , 1 ;nO
6) .
2. )% ( " 3 ; / C 1 , CR- f ; CR- f~, " F " CR- F .
. ;
H = gh 0 , / .
S# B
(., , 13, . 4]), , " K / C > 0, > 0, jg(z)j > Cd(z F) > Cd(z ;) z 2 K:
C1 > 0 > 0
C1 z 2 K \ / jH (z)j 6 d (z ;)
# ,{>
f ! ;.
.1 H " CR-
f~ ; (., , 20]). 9) !
# f~ = f~+ ; f~; .
G!
, ; | , 7].
k
3. !" D 1 h ! F. : ! #, 1 #" ,{>
f. .1 ! # 1 !. (
; (
- )
, ,{>
; A (., , 9] 1 ).
0 # ! # !. .1 ! F M, !. .
) ! ; #
16{18].
.# Y | p C 1
Rp+1, 0 6= Y (
, Y | p- ).
>! U Rp+2 C 1 Rp+2 nf0g
CR-
1081
U = fx 2 Rp+2 : x1 = '(t)y1 : : : xp+1 = '(t)yp+1 xp+2 = t
y = (y1 : : : yp+1 ) 2 Y 0 6 t 6 "0 g
'(t) # C 1 0 "0], '(0) = 0, '(t) > 0 '0 (t) > 0 t 2 (0 "0]. 0 U, Y t = 0 0.
(
'0 (0) 6= 0, 1 , '0(0) = 0, 1
# (
)) 16{18].
.# =p | p- B Y , =p+1 | (p + 1)- B U.
3. / U " c1'p (t) dt d=p 6 d=p+1 6 c2 'p (t) dt d=p
(6)
"( c1 c2 > 0.
. K
(#) Y "6
:
8
>
<y1 = 1(u)
p
>y : : := (u) u = (u1 : : : up) 2 U R :
p+1
p+1
U | ! Rp, !
= (1 : : : p+1) | !
C 1 (U), "6 # p U.
! U # ! # ! :
8x = '(t) (u)
>
> 1: : : 1
<
t 2 0 "0] u 2 U:
>
xp+1 = '(t)p+1 (u)
>
:xp+2 = t
.# G" | * U, A , p
p
d=p+1 = det G" dt du = det G" dt du1 : : : dup:
S
0'2(0 0 ) : : : '2(0 0 ) ''0(0 ) 1
1 p
1
BB ...1 1 . . .
CC
..
..
.
.
G" = B
@'2(p0 10 ) : : : '2(p0 p0 ) ; ''0(p0 ) CA ''0 ( 10 ) : : : ''0 ( p0 ) 1 + ('0 )2 ( )
j0 = ((1 )0u : : : (p )0u ), (j0 k0 ) | j0 k0 , j k = 1 : : : p.
j
j
1082
. . >
* GY = ((j0 k0 ))pjk=1. p
d=p+1 = pdet G" dt d=p = 'p (t)V(t u) dt d=p
det GY
p
) V(0 u) 6= 0 U (# d=p = det GY du), 1
0 < c1 6 V(t u) 6 c2 < 1 0 "0] U:
, #A #, ; = U X
X | ! Rq, )
p + q + 1 = 2n ; 1:
F = Op+2 X Op+2 = (0 : : : 0) 2 Rp+2
M M = (U X) n (Op+2 X):
E , . . q = 0, p = 2n ; 2, 11].
. C n ! #A R2n =
= Rp+2 Rq. (
x = (x1 : : : x2n) 2 R2n, x0 = (x1 : : : xp+1) 2 Rp+1,
x00 = (xp+3 : : : x2n) 2 Rq, x = (x0 xp+2 x00).
: !, / n ; = /+ /;, ) ! /+ xp+2
#A 0.
>! ;, , # " (2n ; 1)-" .
>! ! # 6" ", ; !
U X /. .
1 , ) !, ,
, . 0 !# , 1 #.
: , #, f 2 L1 (U), #A (
!) # /. M
1 #, f p
! 0 "0] U X d=2n;1 = det G" dt du dx, dx = dx1 : : :dxq . , ! #, =2n;1(;) < 1 (. . X | ! Rq).
D ! M" # !
M" = U" X
U" = f(x0 xp+2 2 Rp+2 : x1 = '(t)y1 : : : xp+1 = '(t)yp+1 xp+2 = t
y = (y1 : : : yp+1 ) 2 Y 0 < " 6 t 6 "0 g:
@M" = @U" X, @U" = f(x0 xp+2 ) 2 Rp+2 : x1 = '(")y1 : : : xp+1'(")yp+1 xp+2 = " y 2 Y g:
CR-
1083
.1 , 3, , B =2n;2 @M"
c1 'p (") d=p (y) dx00 6 d=2n;2 6 c2'p (") d=p (y) dx00
(7)
d=p (y) | B Y , dx00 = dxp+3 : : :dx2n = d=q (x00) # B X.
(
!
sf (") =
Z
Y X
jf j d=p(y) dx00 2 (7) !
3. )% f 2 L1(;) CR- M = ; n F . ,
" ! +0
sf (") = o 'p1(")
f 1 .
D , 3 CR-
f L1 (;).
D (q = 0, p = 2n;2) 3 6
2.1 11].
.# f 2 L1(;). ;
Z
2n;1 (
)
Pm (x) = f(
) d=
x 2 / n ;
j
; xjm
;
= (
0 p+2 00) = (
1 : : : 2n), m > 0.
- # 1 x, F ,
f ! F. ,
#, x 2 /+ . 0 x 2 /; .
.# x~ | # x R Rq,
# x~ = (00 xp+2 x00). 0 x 2 /+ xp+2 > 0 jx
~j 6 jxj 6 cjx~j
!
# c x. 0
#,
jxj2 = jx0j2 + x2p+2 + jx00j2 6 x2p+2 + '2 (xp+2 )jyj2 + jx00j2 6 c1 x2p+2 + jx00j2
# Y , ' 2 C 1 0 "0]. .1
2
c x2 + jx00j2
1 6 jjxx~jj2 6 1x2p+2+ jx00j2 6 c:
p+2
E#, jx~j, jxj. .1 #A ,
x = x~.
1084
. . D 3 Y ( ! , Y !
) Z
d=2n;1(
)
jPm(x)j = f(
)
((
p+2 ; xp+2 )2 + j
0j2 + j
00 ; x00j2)m=2 6
Z"
;
0
6 d1 'p (t) dt
0
Z"
0
Y X
= d1 'p (t) dt
0
6 d1 'p (t) dt
Z
Y X
0
Z"
Z
Y X
0
Z"
Z
jf(
)j
d=p(y)d
00
((t ; xp+2 )2 + j
0j2 )m=2 =
jf(
)j
d=p(y)d
00
((t ; xp+2 )2 + '2 (t)jyj2 )m=2 6
jf(
)j
d=p (y)d
00
((t ; xp+2 )2 + d2'2 (t))m=2 6
0
dt
6 d3 'p (t) ((t ; x s)f2(t)
2
m=2 :
+
d
p+2
2' (t))
0
,
jPm (x)j 6 d3
Z"
0
0
p
dt
'p (t) ((t ; x s)f2(t)
2
m=2 :
+
d
p+2
2' (t))
(8)
K d2'(t) '(t) xp+2 x, #A # Z"0
(t) dt
Imp (x) = 'p (t) ((t ; x)s2f +
'2 (t))m=2 x > 0:
0
0 # #.
.# t = t (x) | g(t) = (t ; x)2 + '2 (t) 0 "0].
4. 0 " "
lim x = 1 + ('0 (0))2
x!+0 t (x)
(t ; x)2 + ('(t ))2 =
1
lim
2
x!+0
('(x))
1 + ('0 (0))2 :
. D x ; t (x) = '(t )'0 (t )
1
x ; 1 = '(t ) '0 (t ):
t (x)
t
CR-
1085
t > x g(t) , 0 6 t (x) 6 x. .1
t (x) ! 0 x ! +0. E#,
'(t )
x
0
lim
= lim 1 + t ' (t ) = 1 + ('0 (0))2 :
x!+0 t (x) x!+0
0 # : '0(0) = 0
'0 (0) > 0.
.# '0(0) = 0. ), lim x = 1:
x!+0 t (x)
.!, lim '(x) = 1:
x!+0 '(t (x))
0
#,
'(x) = '(x) ; '(t (x)) + '(t (x)) = 1 + '(x) ; '(t (x)) =
'(t (x))
'(t (x))
'(t (x))
)'0((x))
= 1 + (x ; t'(t
= 1 + '0(t (x))'0 ((x)):
)
. !
1 x ! 0, # t (x) ! 0 (x) ! 0
x ! 0.
E#,
(t ; x)2 + ('(t ))2 = lim ('(t ))2 (1 + ('(t ))2 ) = 1:
lim
x!+0
x!+0
('(x))2
('(x))2
.# '0(0) > 0. ;
(x))
' 1+('x0 (0))2
'(t
= xlim
= 1 + ('10 (0))2
lim
!+0
x!+0 '(x)
'(x)
B
. .1
(t ; x)2 + ('(t ))2 = lim ('(t ))2 (1 + ('(t ))2 ) =
lim
x!+0
x!+0
('(x))2
('(x))2
'(t) 2
1
0 2
= xlim
!+0 '(x) (1 + (' (0)) ) = 1 + ('0 (0))2 :
9
Js (x) =
Z"
0
0
dt
((t ; x)2 + ('(t))2 )s=2 x > 0:
1086
. . 5. Js (x) " ! :
1) s > 2, 1
Js (x) = O ('(x))
x ! +0N
s;1
2) 1 6 s < 2, j ln '(x)j Js (x) = O ('(x))s;1
x ! +0N
3) s < 1, Js (x) = O(1) x ! +0:
.
1) .# s > 2. 1 Z
('(x))s;2 dt
Js(x) = ('(x))
s;2 ((t ; x)2 + ('(t))2 )s=2 :
0
S# 4, "0
Z
s;2 dt
c
Js (x) 6 ('(x))s;2 ((t ; x)2 + ('(t('(x))
))2 )(s;2)=2((t ; x)2 + ('(t))2 ) 6
"0
0
C
6 ('(x))
s;2
Z"
0
dt
(t ; x)2 + ('(t))2 (9)
0
!
#
c C, 6
x.
0!, !
# a A, 6
t x 0 "0], x)2 + ('(t))2 6 A:
(10)
a 6 (t(t ;; x)
2 + ('(x))2
.!
y = '(x). '0 (x) > 0 x > 0, 6 x = g(y), y > 0. .!
(
t ; x = v
t = u:
.
(t ; x)2 + ('(t))2 = v2 + '2 (u + v) :
(t ; x)2 + ('(x))2
v2 + '2 (u)
9
'(u) = w, u = g(w). K (
v = Wv ) vW2 + wW2 = 1:
w = w
W
CR-
1087
v2 + '2 (u + v) = v2 + '2 (g(w) + v) = vW2 + '(g(w)
W + Wv) 2 =
v2 + '2 (u)
v2 + w2
'(g(w)
2
W + Wv ) ; '(g(w))
W + '(g(w))
W
= vW2 +
=
2
0
W 2 < A:
= vW2 + ' ()Wv + wW = vW2 + (Wv '0 () + w)
;
" # ! !
, (t ; x)2 + ('(x))2 6 1 (t ; x)2 + ('(t))2 a
A # (10).
:
(9) (10), 1 Z
dt
Js (x) 6 Ca ('(x))
s;2 (t ; x)2 + ('(x))2 =
0
" C 1
C
1
t
; x 0
= a ('(x))s;1 arctg '(x) 6 a ('(x))s;1 :
0
2) .# 1 6 s < 2. , # (10), "0
1 Z
('(x))s;1 dt
Js(x) = ('(x))
s;1 ((t ; x)2 + ('(t))2 )s=2 6
"0
0
Z"
0
dt
1
6 b1 ('(x))
s;1 p(t ; x)2 + ('(x))2 =
0
"0
p
1
2
2
= b1 ('(x))s;1 ln j(t ; x) + (t ; x) + ('(x)) j 6
0
p2
1
1
6 b2 ('(x))s;1 j ln( x + ('(x))2 ; x)j 6 b3 ('(x))s;1 j ln '(x)j:
3) .# s < 1. , # (10), Js (x) 6 b4
Z"
0
0
Z dt
dt
6
b
((t ; x)2 + ('(x))2 )s=2 4 jt ; xjs 6 C:
"0
0
0# A.
4. )% f 2 L1(;) sf (") = O(1='N (")) " ! +0 N (N 6 p), 1088
. . 1) N > 2 + p ; m, 1
Pm (x) = O ('(j(x0 x )j))m+N ;p;1
j(x0 xp+2 )j ! 0 x 2 / N
p+2
2) 1 + p ; m 6 N < 2 + p ; m, 0
Pm (x) = O ('(j(xj ln0 x'j(x )jx))pm+2+)Njj;p;1
j(x0 xp+2 )j ! 0 x 2 / N
p+2
3) N 6 1 + p ; m, Pm (x) = O(1) j(x0 xp+2 )j ! 0 x 2 / :
. S# (8), jPm (x)j 6 d3
Z"
0
Z"
0
dt
'p (t) ((t ; x s)f2(t)
6
p+2 + d2'2 (t))m=2
0
6
6 d4 'p;N (t) ((t ; x )2 dt
p+2 + d2'2 (t))m=2
0
Z"
0
6 d5 ((t ; x )2 + ddt'2 (t))(m;p+N )=2 :
p+2
2
0
.
5, .
0 (q = 0, p = 2n ; 2) 4 ) 4.3
11].
9
M(x) ,{>
f 2 L1 (;):
Z
M(x) = f(
)U(
x) x 2= ;:
;
4. 1 ( " 4 :
1) N > 3 + p ; 2n, 1
j(x0 xp+2 )j ! 0 x 2 / N
M(x) = O ('(j(x0 x )j))2n+N ;p;2
p+2
2) 2 + p ; 2n 6 N < 3 + p ; 2n, j(x0 xp+2 )j)j
M(x) = O ('(jj(xln'(
j(x0 xp+2 )j ! 0 x 2 / N
0 xp+2 )j))2n+N ;p;2
3) N < 2 + p ; 2n, M(x) = O(1) j(x0 xp+2)j ! 0 x 2 / :
CR-
5. )% f
1089
% CR- M ! K / ! Sf (K ") = O('s (")) " ! +0 s > 0, f " 2,
h , !( f , " ! :
1) s 6 2n ; 3, 1
h (x) = O ('(j(x0 x )j))2n;s;2
j(x0 xp+2)j ! 0 x 2 / N
p+2
2) 2n ; 3 < s 6 2n ; 2, j(x0 xp+2)j)j
h (x) = O ('(j jln'(
j(x0 xp+2)j ! 0 x 2 / N
(x0 xp+2)j))2n;s;2
3) 2n ; 2 < s, h (x) = O(1) j(x0 xp+2)j ! 0 x 2 / :
0# 4 3.
0 4 5 6" 4.5 4.6 11] .
#
2 L1loc(;)
1] . ., . . !!
! !! . | \$
: \$&, 1979.
2] Anderson J. T., Cima J. A. Removable singularities for Lp CR functions // Michigan
Math. J. | 1994. | Vol. 41. | P. 111{119.
3] Andreotti A., Hill C. D. E. E. Levi convexity and the Hans Lewy problem. I // Ann.
Scuola Norm. Super. Pisa. | 1972. | Vol. 26, no. 2. | P. 325{363.
4] Baouendi M. S., Treves F. A property of the functions and distributions annihilaited
by a locally integrable system of complex vector 0elds // Ann. Math. | 1981. |
Vol. 113. | P. 387{421.
5] 2
!
! 3. 4
, ! ! 5&
. | 6.: 6
, 1968.
6] Harvey R., Lawson H. B. On boundaries of complex analytic varieties. I // Ann.
Math. | 1975. | Vol. 102. | P. 223{290.
7] 8! . 6. 9 &
&!: CR-;&< //
6. . | 1988. | =. 136, > 2. | ?. 178{186.
8] 8! . 6. 3!
; CR-;&< ! : // . \$ ???4. ?
. !!. | 1990. | =. 54, > 6. |
?. 1320{1330.
9] 8! . 6. 2:
{6
!. | \$
: \$&, 1992.
1090
. . 10] Kytmanov A. M., Rea C. Elimination of L1 singularities of H@older peak sets for CR
functions // Ann. Scuola Norm. Super. Pisa. | 1995. | Vol. 22, no. 2. | P. 211{226.
11] Kytmanov A. M., Myslivets S. G., Tarkhanov N. N. Analytic representation of CR
functions on hypersurfaces with singularities. | Preprint 99/29. | Institut f@ur Mathematik, Universit@at Potsdam, 1999.
12] Lupacciolu G. A theorem on holomorphic extension of CR-functions // Paci0c J.
Math. | 1987. | Vol. 124, no. 1. | P. 177{191.
13] 6
2. ;;
<
&!: ;&<. | 6.: 6
, 1968.
14] Merker J., Porten E. Enveloppe d'holomorphie locale des variWetWes CR et Welimination
des singularitiWes pour les fonctions CR intWegrables // C. R. Acad. Sci. Paris, Ser. 1. |
1999. | Vol. 328. | P. 853{858.
15] \$
!: 4. : !: !
:. |
6.: 6
, 1971.
16] Rabinovich V. S., Schulze B.-W., Tarkhanov N. N. A calculus of boundary value
problems in domains with non-Lipschitz singular points. | Preprint 9. | Univ.
Potsdam, 1997.
17] Schulze B.-W. Pseudo-diYerential boundary problems, conical singularities, and
asymptotics. | Berlin: Akademie Verlag, 1994.
18] Schulze B.-W. Boundary value problems and singular pseudo-diYerential operators. | Chichester: J. Wiley, 1998.
19] Stout E. L. Removable singularities for the boundary values of holomorphic functions // Proc. of the Mittag-LeZer Institute, 1987{1988. | Princeton, NJ: Princeton
Univ. Press, 1993. | P. 600{629.
20] Straube E. J. Harmonic and analytic functions admitting a distrubution boundary
value // Ann. Scuola Norm. Super. Pisa. | 1984. | Vol. 11, no. 4. | P. 559{591.
21] [ 3. 6. 6 : !! //
& :. ?
! ! !!. 5&!
. =. 7. | 6.: \\$=, 1985. | ?. 23{124.
22] ]
^. 6. CR-;&< // 6. . |
1975. | =. 98, > 4. | ?. 591{623.
23] ]
^. 6. : ! // 4. [
. 3!
;
< : <. | 6.: 6
, 1979. | ?. 122{155.
24] ]
^. 6. 8! !. | 6.: \$&, 1985.
25] Chirka E. M., Stout E. L. Removable singularities in the boundary // Aspects of
Mathematics. | 1994. | Vol. E26. | P. 43{104.
& ' 2000 .
. . . . . 519.214
: , ,
, .
! "#\$% # , " \$
'( d % d > 1 %) "\$ . *
# +,- .
,
#\$ #
, \$ .\$ .
Z
Abstract
A. N. Nazarova, Logarithmic velocity of convergence in CLT for stochastic linear
processes and elds in a Hilbert space, Fundamentalnaya i prikladnaya matematika,
vol. 8 (2002), no. 4, pp. 1091{1098.
Z
In the paper the sums of linear random 4elds de4ned on d for any d > 1
and taking values in a Hilbert space are studied. The convergence velocity in CLT
for such 4elds is discussed. We obtain easily veri4able su5cient conditions for
logarithmic velocity of convergence.
(., , 1{7] # \$%). ' % , #(
) . ' 8] + + ) (,-.). / 0 . -
, % \$+% + + +.
'( . - H | , (H ) | , % H H . 2 A
(H ) A = sup Ax : x H x = 1 ( +
#, ++ 0 H , ++ 0 ). - k k Zd | , (
(5 P) % kk L
2 L
k
k
fk
k
2
f
k k
2
g
g
F
, 2002, 8, 6 4, . 1091{1098.
c 2002 !,
"#
\$% &
1092
. . H , ak k Zd | (H ). 9
% , Pk , + , )
, 0 < E 0 2 < , aj 2 < .
:
) ( ) + % :
X
Xk = aj(k;j ) k Zd:
(1)
f
2
g
L
f
k
k
g
1
k
k
1
Z
; (. 3.2 9]), \$
=
2
j2
d
# (1) L2(5),
\$+), #% 5 H % + .
> X
Sn =
Xk n > 1 = (1 : : : 1) Nd
2
16k6n
+ k = (k1 : : : kd) n = (n1 : : : nd ) k 6 n , ki 6 ni i = 1 : : : d.
2 % + z = (z1 : : : zd ) Rd+ z z] = z1 z1] : : : zd zd ]
( z z) = ( z1 z1 ) : : : ( zd zd ):
' 8] +
1. k k Zd | . , X
aj < :
(2)
2
;
;
;
;
f
2
;
g
Z
k
j2
;
k
1
d
nj;1=2Sn ;D
! N (0 AC0 A )
n
(3)
P
{ , A = aj, A | j2 d
!
,
A, C0 | 0 , n = (n1 : : : nd)
n = n1 : : :nd
, n "
#.
' # +, (2) =. 2 )+ + + + ( 0 , +# # k .
2. 1, A = 0. " > 0 sup P( n ;1=2 Sn < r) P( < r) = O((log n ); ) n
(4)
j
Z
N
j
j
! 1
! 1
! 1
6
r>0
j
j
j
k
k
;
k k
j
j
j
! 1
1093
#\$:
1) \$ 0 < < 1, E 0 2+ < ,
2) \$ c0 = c0 ( ) > 0, X
aj 6 c0(log( n + 1));3
k
j2=(;nn)
k
k
k
1
j
& n Nd.
j
2
>
2.
= jnj;1=2S
n
n
X
= jnj;1=2A
n
16k6n
k :
; , n Nd r > 0 " > 0
P( n < r ") P( < r) P( n n > ") 6
6 P( n < r) P( < r) 6
6 P( n < r + ") P( < r) + P( n n > "):
- % , 8
k
k
2
;
k
k
k
k
8
;
8
k k
;
;
k
;
k
k k
;
k k
k
;
k
P(k
n k < r) ; P(kk < r) 6 P(knk < r + ") ; P(kk < r + ")] +
+ P(kk < r + ") ; P(kk < r)] + P(k
n ; nk > ")
+ 8n 2 Nd 8" > 0 sup P( n < r) P( < r) 6 sup P( n < r) P( < r) +
r>0
j
k
k
k k
;
j
r>0
j
k
k
;
k k
j
+ sup P(r < < r + ") + P( n n > ") = I1 + I2 + I3 : (5)
r>0
j
k k
j
k
;
k
.+ ++ k , Ak | #, 0 + , , )+ 3] 1]. >%, 1), n Nd
X
;
1=2
I1 = supP n Ak < r P( < r) 6 c1( n );=2 (6)
r>0
f
g
f
g
2
j
j
;
16k6n
k k
j
j
# c1 + A 0 .
2 )+ I2 % .
1 (10]). | 0 " " H. \$ C, a R+ " > 0
(x H : a 6 x 6 a + ") 6 C":
8
2
k
2
k
8
1094
. . .+ ++ (0 AC0 A ) 1, I2 = sup P(r < < r + ") 6 c2"
(7)
N
j
r>0
k k
j
+ c2 + A C0 .
> ) I3 . - C
=(
I3 = P( n n > ") 6 ";2 E n n 2:
(8)
2. ' 2, E n n 2 = O((log n );3 ) n
:
-# +
0 , +#, ++ ( % + = + 2. D (5){(8) % " > 0
+ c3 = c3 ( ) > 0 = n
sup P( n < r) P( < r) 6 c1 n ;=2 + c2 " + c3";2 (log n );3 :
k
k
r>0
j
k
;
k
;
k
k
k
;
j
k k
j
;
j
j
k
! 1
j
j
j
'
" = "(n) = (log n ); , n > 1. .
sup P( n < r) P( < r) 6
j
j
r>0
k
k
;
j
j
k k
j
j
6 c1 n ;=2 + c2 (log n ); + c3 (log n ); = O((log n ); ) n
j
j
j
j
j
j
j
j
(9)
! 1
+% )+ (4).
d
2. '( P baj,. j Z , # b0 =
= a0 A bi = ai i = 0. ;, A =
j
j2 d
/, X
n 1=2(
n n ) = Sn A
k =
16k6n
X X
X X
X X
bk;j j :
=
ak;j A j +
ak;j j =
;
j
j
Z
6
;
;
16j6n 16k6n
;
j2=1n]
.
d
X Ek
n ; n k2 6 jnj;1Ek0k2
6
j
nj;1Ek0 k2
2
j2
X ZX
16k6n
X
1;j6k6n;j
j2=(;2n2n) 1;j6k6n;j
2
= Ek0k (J1 + J2 ):
j2
2
bk 6
2
bk +
X
Z
d
16k6n
X
j2;2n2n;1] 1;j6k6n;j
(10)
2 bk =
(11)
1095
) J1.
2
2
X X
X X
;
1
;
1
J1 = n
bk 6 n
bk 6
j2=(;2n2n) 1;j6k6n;j
j2=(;2n2n) 1;j6k6n;j
X
X X
6 n ;1
bk
bk 6
j2=(;2n2n) 1;j6k6n;j
X
Xk2=(;nn)
X
6 n ;1
bk
bk =
k2=(;nn)
j2 d 1;j6k6n;j
X X
=
bj
bk :
j
j
j
j
j
j
j
k
Z
Z
k
.+ ++
X
j2
k
d
bj =
k
X
j6=0
k
k
k
k
k2=(;nn)
k
;
bk =
k
k
k
k
k
k
aj + a0 A 6
X
2), k
k2=(;nn)
d
Z
% , j2
k
k
j
k
X
Z
j2
X
k2=(;nn)
k
k
d
k
k
aj +
k
X
Z
k
j2
d
aj 6 2
k
X
j2
Za
k
d
jk
<
1
ak 6 c0 (log( n + 1));3
j
k
j
J1 = O((log n );3 ) n
:
2 )+ J2 ( \$+)%
X
2
hn (t) = bk;jnjt] t 2 2] 2 = (2 : : : 2) Nd
j
k
j
! 1
2 ;
16k6n
(12)
2
a] )% a ( + Rd ) +). /, X
2 X
2
0 6 hn (t) 6
bk;jnjt] 6
bk = B < :
(13)
16k6n
2,
J2
= jnj;1
=
X
k
X
Z
j2;2n2n;1] j=jnj(j+1)=jnj]
k
k2
j2;2n2n;1] 1;j6k6n;j
X
Z
k
2
bk =
;1
jnj
hn (t) dt =
k
1
d
X
j2;2n2n;1]
Z
;2n=jnj2n=jnj]
hn
j
nj
j
hn (t) dt 6
=
Z
;22]
hn (t) dt:
(14)
>)+ (11), (12) (14) % =% + 2.
1096
. . 3.
Z
hn (t) dt = O((log n );3 ) n
j
j
! 1
;22]
. E #
:
An () = t 2 2]: hn (t) > Bn () = t 2 2]: hn(t) 6 n Nd = (n) > 0.
- | : Rd. ., (13), Z
Z
Z
hn (t) dt =
hn (t) dt +
hn (t) dt 6
f
2 ;
g
f
2 ;
g
2
;22]
An ()
Bn ()
6 sup hn (t) (An ()) + sup hn(t) (Bn ()) 6
t2An ()
j
j
t2Bn ()
j
j
6 B (An ()) + ( 2 2]) 6 B (An ()) + 4d :
(15)
2
;
'( (1=2 1) = c0 (d log( n + 1)) 6 .
>, = O((log n );3 ) n
. F, (An ()) = O((log n );3 ) n
:
(16)
;
2
j
j
j
j
! 1
j
j
! 1
2 0 # + t +
2 2]. - = (n) > 0, #.
) - i0 , 1 6 i0 6 d, + ti0 6 . G 1 6
6 k 6 n, ki0 n ti0 ] > 1 + n ]. . d-
H)
I
+ k n t] 1 6 k 6 n +
# # k Zd :
k0 > 1 + n ] . -0
X
2 X
2 X
2
hn (t) 6
bk;jnjt] 6
bk =
ak : (17)
;
;
;
f
;
j
j
j
j
j
j
j
j
g
f
2
g
16k6n
k
k
k
k
k
k
k0 >1+jnj]
k0 >1+jnj]
> f (x) = c0(log(x + 1));3 , x > 0. . % 2) hn (t) 6 f 2 ( n ]d):
(18)
) - ( i0 , 1 6 i0 6 d, + ti0 > 1 + . G 1 6 k 6 n, ki0 n ti0 ] 6 ni0 n n ] 6 n ]. . % ) X
2
hn(t) 6
ak 6 f 2 ( n ]d):
(19)
j
; j
j
; j
j; j
j
; j
j
j
jki0 j6;jnj]
k
k
j
j
% d = 1 d > 2.
1 ) - d = 1 6 t 6 1 . . n nt] > 0. J ,
1 nt] 6 1 n ] 6 0 = n \$+) n (n) n. K , ;
;
;
;
1097
0
k nt] 1 6 k 6 n = n, 0
X
2 X
2
n
n
hn(t) = bk;nt] = ak;nt] A :
2 f
;
g
k=1
;
k=1
1
P
', A =
aj , j=;1
2 X
hn(t) = ak;nt] 6
k60 #\$# k>n+1
X
k60 #\$# k>n+1
k
ak;nt]
2
k
:
.+ ++ k nt] 6 n ] k 6 0 k nt] > n + 1 n + n ] = 1 + n ] k > n + 1, X
2
hn (t) 6
aj 6 f 2 (n ])
(20)
;
;
;
jj j>n]
k
;
k
= n.
2 ) - d > 2 % i, 1 6 i 6 d, 2 6 ti 6 1.
. % i ki n ti] 6 ni n 2 ]. '
i = i0 , + ni0 = min
n 6 n 1=d:
i i
; j
j
; j
j
j
j
- + ki0 n ti0 ] 6 n 1=d 2 n ] 6 n ] = n , \$+) n (n) ( , n 1=d. 2, ++ + ), hn (t) 6 f 2 ( n ]d):
(21)
;
1+
'
= (n) = 2 n
, 1=2 < < 1. . %
= (n), = n
f 2 ( n ]d) = c20 (log(2d n d + 1));6 6 = c20((d log( n + 1));6 :
(22)
. (18){(22) , An () ( 1 + ) 1 ] d = 1
An () ( M 1 + M) 2M 1] d > 2 M = ( : : : ):
.+ , (An ()) 6 (1 + 2 )d (1 2 )d 6 4d2d;1 = O((log n );3 ) n
: (23)
= (15) (23) =% + 3, +# 2, 2.
' +% # \$ N. '. O+ .
;
j
j
j
j
j
j
j
j
;
j
j
j
f ;
;
j
j
j
j
j
;
n
;
; j
j
n
f ;
j
j
j
j
j
j
g
g
j
j
! 1
1098
. . 1] . ! // #
. !. | 1984. | (. 24,
+ 4. | ,. 29{48.
2] ., . /., 0
., 1
. (
2
. ! // 42 . , ! . / . (. 81. | 5.:
474(4, 1991. | ,. 39{139.
3] 9
. . : . 2 ! // (. . . | 1985. | (. 30, + 4. | ,. 662{670.
4] 9
. ., , . ., ? . . 0 . . 2 ! //
5. !. | 1989. | (. 180, + 12. | ,. 1587{1613.
5] 9 . 5. !
@ . 2 . // (. . . | 1966. | (. 11, . 1.
6] A
. . : 2 ! B // (. . . | 1982. | (. 27, .2. | ,. 270{278.
7] Senatov V. V. Normal approximation: new results, methods and problems. | VSP
Science, 1999.
8] 7 . 7. 7 . .
2 ! // 5. . | 2000. | (. 68,
. 3. | ,. 421{428.
9] Araujo A., GinDe E. The central limit theorem for real and Banach valued random
variables. | New York: John Wiley and Sons, 1980.
10] Kuelbs J., Kurtz T. Berry{Essen estimates in Hilbert space and applications to the
law of the iterated logarithm // Ann. Probability. | 1974. | Vol. 2, no. 3. |
P. 387{407.
' ( ( 2000 .
. . 519.48
: , , , ! , !
" .
#\$% , & %' &, ! &, &, !
" & (
"\$.
Abstract
A. G. Pinus, On the diagrams of classes of conditionally termal functions, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 4, pp. 1099{1109.
We give some conditions under which the classes of conditionally termal, elementary conditionally termal, positive conditionally termal and existential positive
conditionally termal functions coincide.
1{4] | , , 9+ - , " #\$, |
%&% #\$ '. ' " #\$ & & )' #\$ '* |
#\$, &% ' , %& \$ ' ' #\$ '. ' #\$ &
' \$, #\$, &%
' &% ': '# ('# - '), # (# - '), "#, #. /% % & - " . (\$ %%/
.\$
(% (%&
%\$ ( 0 99-01-00571).
, 2002, 8, 0 4, . 1099{1109.
c 2002 !,
"# \$% &
1100
. . #\$ '. / 1
#\$ \$.
2 (
) ' && & - '(x) (&&
"& # '(x)) ' . 2 (
) - (" ) f'1 (x) : : : 'n(x)g, #
Wn 'i(x) %, & i j # 'i(x)&'j (x) i=1
. 2 (
) '
\$, #\$ ' , %& \$ &%' ( ): f'1 (x) : : : 'n(x)g | (" ), ft1 (x) : : : tn(x)g | (" ) , 8>
<'1(x) ! t1(x)
|
t(x) = > : : :
:'n(x) ! tn(x)
- (" ) .
2 (9+ - ) ' && & - ' (&& & 9-# "' ' ). 2 (9+ - ) A '
\$, ' 1 ( ) &% ( ): f'1(x) : : : 'n(x)g | (9+ -), ft1(x) : : : tn(x)g | (9+ -) _n
A j= 8 x
& i, j
i=1
'i (x) A j= 8x ('i (x) & 'j (x) ! ti (x) = tj (x))
8
><'1(x) ! t1(x)
t(x) = > : : :
|
:'n(x) ! tn(x)
- (9+ -) ' A.
5- 1 % t(x)
' A (
, 1101
9+ -
) #\$ t(x). 2 " t(x) ( ) ( ) & a b 2 A A j= t(a) = b \$
' i 6 n A j= 'i (a) A j= ti (a) = b.
6 . . #\$ ' A #\$, %& ' , ' #\$ \$ ' ( ). 7 #\$ -
, , 5].
9' \$ #\$ - &% , 2,4,6].
,
A.
A = hA< i | '(x1 : : : xn) | A, ' ( ) A , A ' ' ( ) A.
2. A = hA< i | '(x1 : : : xn) | A, ' 9+ - ( ) A , A ' ' ( ) A.
1.
= & ' A T (A) #\$ " ', PCT(A) | #\$, CT(A) | ,
9+ CT(A) | 9+ - , ECT(A) |
" #\$. 7 &% ' & - " #\$ & ' A:
+ CT(A) T(A) PCT(A) 9 CT(
()
A) ECT(A)
/ , ' A & - #\$, % " '.
>1 , T(A) = CT(A) ' A (. . , #\$
#\$ ' A). 4] , ' A CT(A) = ECT(A) &: &
# ' A - #.
? Iso A (Ihm A, End A, Aut A) # ( '#, "#,
1102
. . #) ' A. ? End1 A (Ihm1 A) "# ( '#) ' A,
".
1.
:
! "# # A "\$ -
1) 9+ CT(A) = ECT(A),
2) End A = Aut A End1 A.
. @\$ 2) ! 1) 1 ' \$ #\$
9+ CT(A) ECT(A ' A. @\$& 1) ! 2) - '. 2 2 End A n (Aut A End1 A). 2 fa1 : : : ang | - ' A a = ha1 : : : ani. 2 (a1) 6= (a2 ). B\$&
f(x1 : : : xn) 2 ECT(A) &% :
(
f(x1 : : : xn) = '(x1 : : : xn) ! x1
:'(x1 : : : xn) ! x2 ' '(x1 : : : xn) | "' - a,
&' - b = hb1 : : : bni " ' A, '
A j= '(b), - (ai ) = bi # ' A.
6' A j= '((a)) , , (f(a)) = (a1 ) 6= (a2 ) ; f((a)). 6 \$ 9+ CT(A)-#\$ f 2= 9+ CT(A), 9+ CT(A) 6= ECT(A) .
2. ! # # A "\$ :
1) PCT(A) = CT(A),
2) Ihm A = Iso A Ihm1 A.
= "' - 1, " 2 Ihm A n (Iso A Ihm1 A), dom = fa1 : : : ang '(x1 : : : xn) | ' ' A, A j= '(a1 : : : an) A j= :'((a1 ) : : : (an)).
\$ 9+ - PCT-#\$ - (' - 3 4]) ' A,
9+ CT(A) = PCT(A), ', &% &: & '# ' A - "#.
7, - '# ' A 9+ CT(A) = PCT(A). 7 , &% , .
E
D
1. 2 A = A S B< f 1 h1 , ' A = fa1 a2 a3g, B = fb1 b2g,
f(a1 ) = f(a3 ) = a2 , f(a2 ) = a1 , f(b1 ) = b2 , f(b2 ) = b1 , h(a1 ) = a2, h(a2) = a1 ,
1103
h(a3) = a3, h(b1 ) = b2 , h(b2 ) = b1 . 7- ': fa1 a2g ! fb1 b2g, '(ai ) = bi , # ' A, - "# ' A. F, , 9+ CT(A) = PCT(A).
=, \$ P(x) = (f(x) =
= h(x)) :P (x) = (h(x) = x), , 9
" # 1 ' =
= hf(x) h(x) P (x) :P (x) P1(x) P2(x)i, ' P1(x) = 9z(:P(z) & x = f(z)),
P2(x) = 9z(:P(z) & x = f 2 (z)). 7 -, - ,
(ai ) = ai , (bi ) = ai (ai) = ai , (b1) = a2 ,
(b2 ) = a1 , & "# ' A - ' A ('% A ' ). 6 , & 9+ -
'(x1 : : : xn) ' A - &% 0
0
0
0
0
0
0
r1
&=1
i
& x +1 = x & P1(x +1) &
& & x +1 = x & P2 (x +1 ) & & x +1 = x &
= +1
= +1
&P(x +1 ) & & x +1 = x & P (x +1 ) & f(x +1 ) = x +1
= +1
x1 = xi & :P(x1) &
r3
i r2
r2
i=r1 +1
i
n
r3
r2
i
r1
r4
r2
r4
i r4
r1
r3
i r3
i
r4
i
r4
r3
r1, r2, r3, r4, r1 6 r2 6 r3 6 r4 . / , &% 9+ CT(A)-#\$& g(x1 : : : xn), '(x1 : : : xn) - - 9+ - ' (x1 : : : xn), - hc1 : : : cni A, ci = a2 1 6 i 6 r1, ci = b1(b2 )
r1 +1 6 i 6 r2 , r3 +1 6 i 6 r4 ci = b2 (b1) r2 +1 6 i 6 r3, r4 +1 6 i 6 n.
6' ' (x1 : : : xn) 0
0
r1
&=1
i
&
x1 = xi & :P (x1) &
&
r3
i=r2 +1
& x +1 = x & P(x +1) &
r2
i=r1 +1
r1
i
r1
xr2 +1 = xi & P(xr2 +1 ) & f(xr2 +1 ) = xr1 +1 &
& P (xr3 +1 ) &
& x +1 = x &
r4
i=r3 +1
r3
& x +1 = x & P(x +1) & f(x +1) = x +1:
n
i=r4 +1
r4
i
r4
r4
i
r3
6 ' (x1 : : : xn) ' , "
A j= 8x1 : : : xn ('(x1 : : : xn) ! ' (x1 : : : xn)). 6 & 9+ CT- ' A 9+ CT- '(x1 : : : xn)
- PCT- ' (x1 : : : xn) - #\$, . . & 9+ CT(A)-#\$ PCT(A)-#\$.
0
0
0
1104
. . 1 1. / '& \$& ' A, +
9 CT(A) = PCT(A).
>1 7] ' A
( A): ' A , . . T (A) =
= CT(A), ' ', ' A ' M(A), - ' A, #. J ' , T(A) = CT(A) K&\$ T (A) = PCT(A)
PCT(A) = CT(A).
4] T(A) = PCT(A) PCT(A) = CT(A). 2 \$ PCT(A) = T(A) ( 2) #
' M(A), -, T (A) = PCT(A) # M(A). J&% , " .
2. 2 A = hA< i | & ' 1 '"\$ Con A , '# A - A. 2 A | '% A ' ' A PCT(A)-#\$. 6' \$ #\$ PCT(A) Ihm A = Ihm A , , PCT(A ) = PCT(A) = T(A ). - Con A = Con A, , ' A PCT(A ) = T(A ), ' M(A ) #.
L 1 '(x1 : : : xn) &i=j ij6n xi 6= xj , -:
End A 6= Aut A End1 A ! CT(A) 6= 9+ CT(A):
6 , & ' A \$
CT(A) = 9+ CT(A) ! CT(A) = ECT(A) = 9+ CT(A):
/, ' A ", T(A) = Enf(A), ' Enf A | #\$ - ' A, &% "#
' A. & 9+ CT(A) Enf(A) - "
' & 9+ CT(A) = T(A) ( , -). M ' =", , N, 21 . @ , , , ("). 5 , (., , 8])
" &: & " - ( ' '), & , &
', & 1, &% 1 , & ' " m, &
- ' Zm2 .
2-, , 1, 9+ CT(A) = T(A) " ' A.
0
0
0
0
0
6
0
0
0
0
0
1105
3. 2 ' A = hf0 1 a b cg< _ ^ fa fbi , hf0 1 a b cg< _ ^i | " 1 M3 , #\$ fa , fb & a b . 6 A " ', End A = Aut A. B\$& fc - f0 1 a b cg ' #\$ fa fb . 6' # , , "# ' A. 2
A | '% ' A ' 9+ CT(A)-#\$.
6 , 9+ CT(A ) = 9+ CT(A) = T (A ), Sub A = Sub A, fc 2= End A =
= End A fc 2= T (A ), ' f0 1 a bg ' A 0
0
0
0
0
0
0
fc .
% - #\$,
% ' ( ), , T(A) = PCT(A) ' - ' ( ). =, & ' A '%& A PCT(A)-#\$, Ihm A = Ihm A ( , Sub A = Sub A, Iso A = Iso A, End A = End A, Aut A = Aut A) ' A ' ( ), '& ' ( ) A T(A ) = PCT(A ).
2- , PCT(A) = 9+ CT(A) 9+ CT(A) = CT(A), CT(A) = ECT(A).
4. 2 A | & " '. " A & T(A) = PCT(A) = 9+ CT(A).
>1 , A , , 9+ CT(A) = T (A) 6=
6= CT(A). B\$& '(x) &% :
0
0
0
0
0
0
0
0
0
(
'(x) = At(x) ! x
:At(x) ! 0
' At(x) | " #, &% ' A.
6 ' 2 ECT(A), ' # ' A, ' 2= CT(A), . . CT(A) 6= ECT(A).
2-, \$
CT(A) = 9+ CT(A) = ECT(A) ! PCT(A) = 9+ CT(A)(PCT(A) = CT(A))
.
D
E
5. 2 ' A = A = A1 S A2< f g , ' f | ,
g | #\$, , hA1 < f i hA2 < f i | \$ 4 2, #\$ g &% :
(
d(x y) = x x y 2 A1 , x y 2 A2 , x 2 A1 y 2 A2 y y 2 A1 x 2 A2:
1106
. . / , End A = Aut A , ,
9+ CT(A) = ECT(A). 6 & # ' A - #, CT(A) = ECT(A). 2 a1 2 A1 , a2 2 A2 - h: A1 ! A2 f , h(a1 ) = a2. 7- h '# ' A. 7 A
#\$& '(x) &% :
(
'(x) = f(x) x 2 A1 x
x 2 A2 :
7, ' 2 ECT(A). 9 ', '(h(a1 )) = '(a2 ) = a2 6=
6= f(a2 ) = f(h(a1 )) = h('(a1 )), ' 2= PCT(A), . . PCT(A) 6= CT(A) =
= 9+ CT(A) = ECT(A).
- PCT(A) = CT(A) ", 2, Ihm A = Iso A Ihm1 A , , End A = Aut A End1 A, ", 1, 9+ CT(A) = ECT(A).
Q& 1 - #\$ ' ( ), # &% -.
3.
1. & T(A), PCT(A) ( ) '" # ,
#\$ ( ).
2. ( , #\$# ( ), "\$ :
9+ CT(A) PCT(A)
jj
ECT(A) !PCT(A)
CT(A) + CT(A) PCT(A) =9 CT(
A) ECT(A) 6!PCT(A)
9+ CT(A) =
jj
ECT(A)
CT(A) =
= 9+ CT(A) = ECT(A)
= CT(A) =
+ CT(A) PCT(A) =9 CT(
A) = ECT(A)
9+ CT(A) =
jj
PCT(A)
ECT(A) 6!PCT(A)
CT(A) =
+ CT(A) PCT(A) =9 CT(
A) ECT(A) !PCT(A)
= 9+ CT(A) = ECT(A)
= CT(A) =
9+ CT(A) = ECT(A)
= CT(A) 1107
J ' ( ) % . /, \$ " ' - #
&% : ' A1 = hA1 < 1i A2 = hA1 < 1i (A1 'r:e: A2 ) ' ', ' % \$ - A1 - A2, T (A1) = T (A2 ) , ' T(A2 ) |
#\$ - A1 , -- #\$ T (A1 ).
? A]r:e: ', \$ " ' A. 9' ' A1 A2 & (
,
9+ -
, ), % \$ - A1 A2 , CT(A1 ) = CT(A2 ) (PCT(A1 ) = PCT(A2 ) , 9+ CT(A1 ) = 9+ CT(A2 ) ,
ECT(A1 ) = ECT(A2 ) ). ? A]c:r:e: (A]p:c:r:e:, A] + c:r:e:, A]e:c:r:e:) ' ( , 9+ -, "
) \$ " ' A. 7, &
' A '
A]r:e: A]p:c:r:e: AA] ]+c:r:e: A]e:c:r:e:
( )
c:r:e:
L : ' ( & " ' A)
& ' ( ) . 9] , & " ' A & A]r:e: A]c:r:e: . 2 4] ', &% & CT(A) = ECT(A), 1, 2 ' &.
Q A]c:r:e: = A]e:c:r:e: '
A = hA< i. 7, " :
1) - A ECT(A) = CT(A)<
2) ' B = hA< i ECT(B) = ECT(A), A CT(B) = CT(A).
& ECT(A) CT(A) 1) 1) ECT(A) = CT(A). 6 , 1 A]c:r:e: = A]e:c:r:e: ' A = hA< i &% :
) & # ' A - #<
) & ' B = hA< i, Sub B = Sub A, Aut B = Aut A,
& # ' B - # ' B.
6 , A]c:r:e: = A]e:c:r:e: " ' A ( A]r:e: = A]c:r:e:), -, ,
9
9
0
0
1108
. . , & ' A " ', " ' - #.
9' , & ' A = hA< i A] +c:r:e: = A]e:c:r:e: ) End A = Aut A End1 A (: 9+ CT(A) = ECT(A))<
) & ' B = hA< i, Sub B = Sub A, Aut B = Aut A,
End A = Aut A End1 A (: ' B = hA< i ECT(B) = ECT(A), A 9+ CT(B) = 9+ CT(A)).
6 , A] +c:r:e: = A]e:c:r:e: " ' A, -, , , & " '.
6 - & ' A = hA< i A]p:c:r:e: = A]c:r:e: ) Ihm A = Iso A Ihm1 A<
) & ' B = hA< i, Iso B = Iso A, Ihm B = Ihm A.
6 A]p:c:r:e: = A]c:r:e: "
' A, -, , , & "
'.
Q A]p:c:r:e: = A]c:r:e:, - , '
A = hA< i ) PCT(A) = 9+ CT(A)<
) & ' B = hA< i, Sub B = Sub A, End B = End A,
Ihm B = Ihm A.
6 , A]p:c:r:e: = A] +c:r:e: " ' A, -, , , & " " '.
7 2. - A]r:e: = A]p:c:r:e: " '?
9
0
0
9
0
0
9
1] Pinus A. G. On the conditional terms and conditional identities on the universal
algebras // Siberian Advances in Math. | 1998. | Vol. 8, no. 2. | P. 96{109.
2] . . ! . | #\$.
3] . . N -! n-! \$&\$!'\$( )* \$!' //
+ . ,. #-. \$ .. /\$\$*\$. | 1999. | 0 1. | 1. 36{40.
4] . . 3 *&(4, *564 !\$ -\$ \$7 \$!- // 1-. \$. 8. | 2000. | 9. 41, 0 6.
1109
5] Pinus A. G. Conditional terms and their applications // Algebra. Proceedings of
Kurosh conference. | Berlin, New York: Publ. Walter de Gruyter. | 2000. |
P. 291{300.
6] . . <\$\$*\$&( ! \$!'4 *&7 // 1-. \$. 8. |
1997. | 9. 38, 0 1. | 1. 161{165.
7] Pixley A. F. The ternary discriminator function in universal algebra // Math. Ann. |
1971. | Vol. 191, no. 3. | P. 167{180.
8] Davey B. A., Pitkethly J. G. Endoprimal algebras // Alg. univ. | 1997. | Vol. 38,
no. 3. | P. 266{288.
9] . . 3 \$&\$!' ! \$&\$!' )* \$!4 \$!-\$4. |
#\$.
' ( 2000 .
-
. . -
e-mail: lnp@positsel.dnttm.ru
519.83
: , !, "#\$!, "! !
#%&%"!, &!&" & '#&, !\$#(.
)*+ !\$#a %, ! " %&- "&-, &&# %" &.&./!
\$&!"!+"&- &- ! " &&#( -"%! !\$#&&%. )\$#
!""&% " &+! *#! #*!+(, #!#!% &!&"!. 0%(
&! "! !! "-#%&%"! !\$#( " %&- "&-, "-#%&%"&&"&%&"! "! !- ! "-#%&%"&- &"&%&"! "#\$!-.
&*& "/"%&%! "! !- "-#%&%"! ! &#( "&%!, #! &&#(, !\$# ! "! !! #%&%"!. 1%&%"( ! "-#%&%"( "#\$!! -( % %& %!. '&*&, +& -( "-#%&%"( "#\$!!
%2" "-"!!(!. '&+( "&%!, #! &&#(, #%&%"( "#\$!! %2" "!!(!. '#!%( "&%!, #! &&#(, #%&%"(
#!! &!( & '#&.
Abstract
L. N. Positselskaya, Equilibrium and Pareto-optimality in noisy non-zero sum
discrete duel, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 4,
pp. 1111{1128.
We study a non-zero sum game which is a generalization of the antagonistic
noisy one-versus-one duel. Equilibrium and "-equilibrium points are presented in
explicit form. It is shown that the "-equilibrium strategies of both players coincide
with their "-maxmin strategies. We give the conditions under which the equilibrium
strategy is a maxmin strategy. Pareto optimal games are investigated.
1. . ! ! . "
t , t (% , , 2002, & 8, 9 4, ". 1111{1128.
c 2002 ,
! "# \$
1112
. . ) . ,
), ) , .
* ! % + !
% ,1]. , (
), % (
), , , ,5, 6]. ,2, 5, 7], !
+
, , +. 1% , 1982 , ! ,4]. 5 %% ,1,8].
1
!
! . % ! .
5 , %% , ,9]. 5 %
.
5 % a , %% ) ,2,3].
2. C ; = fX Y K1 (x y) K2(x y)g
X, Y | ! , Kj (x y), j = 1 2, | , X Y ) j- x 2 X, | y 2 Y . , , K1 (x y) + K2 (x y) = 0 % x 2 X, y 2 Y . 5 . < ! X Y . ; = (= > K(' )) ;, = >
! ! X Y ;, ! + K j (' ) !
+ Kj (x y) ' 2 =, 2 >. ;
, x 2 X - 1113
y 2 Y . 5 ' 2 =, 2 >
.
(x y) . ? (xe ye )
, % x 2 X, y 2 Y K1 (x ye ) 6 K1 (xe ye) K2 (xe y) 6 K2 (xe ye ):
(1)
5 (v1 v2 ), v1 = K1 (xe ye ), v2 = K2 (xe ye), , (xe ye ). ?, -% , .
? xm 2 X , + m1 (x) = yinf
K (x y) %)
2Y 1
. ? ym 2 Y , + m2 (y) = xinf
K (x y) 2X 2
. 5
w = (w1 w2) w1 = sup inf K1 (x y) w2 = sup inf K2 (x y)
x2X y2Y
y2Y x2X
. 5 wj ) ) j- . @ , . * . 5 .
1% K ! ! K=(K1 (x y)K2 (x y))
S | ! s = (x y). 1) ! ! K S % :
K 1 K 2 8j 2 f1 2g Kj1 > Kj2 & 9j 2 f1 2g Kj1 > Kj2 C
s1 s2 K 1 K 2 si =(xi yi ) K i =(K1 (xi yi ) K2(xi yi )) i=1 2:
(2)
? sp = (xp yp ) !, s = (x y), s sp .
5 ". " , , ", . . % (x1 y1 ) (x2 y2 ) 2 S K1 (x1 y1 ) < K1 (x2 y2 ) () K2 (x1 y1 ) > K1 (x2 y2 ))C
(3)
K1 (x1 y1 ) = K1 (x2 y2 ) () K2 (x1 y1 ) = K1 (x2 y2 )):
1114
. . 3. "-
,
"-" D % ! %
. ? (x" y") "- , %
x 2 X, y 2 Y K1 (x y" ) ; " 6 K1 (x" y") K2 (x" y) ; " 6 K2 (x" y" ):
(4)
? , -% "-, "- . @ , "-, ", %
.
" " | ! "n .
"
f(x"n y"n )g, n 2 N, "- , (x"n y"n ) | "n - (n 2 N) vej = nlim
K (x y ) (j = 1 2):
!1 j "n "n
5 (ve1 ve2 ) , f(x"n y"n )g.
? x"m 2 X "- ,
m1 (x"m ) > w1 ; ", m1 (x) = inf
K (x y), w1 | Y 1
. E "- .
4. \$
* , % %% ) .
%% +. 1
, ! ,0 1]. <++ j- + Pj (t) (j = 1 2), !
t .
5 ! t
Pj (t) (j = 1 2). F Pj (t) , , Pj (0) = 0,
Pj (1) = 1, 0 < Pj (t) < 1 t 2 (0 1). @ % ,
. @ , ,
% , t = 1, . "% j- Aj , % Bj ,
Aj > 0 Bj > 0 Aj + Bj 6= 0 j = 1 2:
(5)
- 1115
5) 0, % . < # \$ .
G j- tj , . H! !
,0 1]. " (t1 t2) . "! + j- Kj (t1 t2) !
), j- , tj (j = 1 2). F Kj +
8
><M1(t1) = A1P1(t1) ; B1(1 ; P1(t1))
t1 < t2
K1 (t1 t2 ) = >L1 (t1 ) = A1 P1(t1 )(1 ; P2(t1 )) ; B1 (1 ; P1 (t1))P2 (t1 ) t1 = t2
:N1(t2) = ;B1 P2(t2) + A1(1 ; P2(t2))
t1 > t2:
8
><M2(t2) = A2P2(t2) ; B2(1 ; P2(t2))
t2 < t1
K2 (t1 t2 ) = >L2 (t2 ) = A2 P2(t1 )(1 ; P1(t1 )) ; B2 (1 ; P2 (t1))P1 (t1 ) t1 = t2
:N2(t1) = ;B2 P1(t1) + A2(1 ; P1(t1))
t2 > t1:
(6)
?) ,0 1], % , .
"! A = (A1 A2), B = (B1 B2 ) A ), B | ) . 5
-+ ++ P (t) = (P1(t) P2(t)). 1 % ;11(P A B), ) ) | ;11(P A B). G ;11(P A B) c t %
% It .
" ;11(P A B) ) ) A1 = B2 , A2 = B1 , . . ) !
)
. I
) (6) , K1 (t1 t2) = ;K2 (t1 t2),
. e. . < % ,2, 3]. "! A = (A2 A1), KA = K1 (t1 t2) %
% ;11(P A A) ) , ) A.
5. "-
;11(P A B )
" t11 | ) P1(t) + P2(t) = 1:
(7)
1116
. . "!
min A1 +1 B1 C A2 +1 B2
=
:
2
5 " > 0 1 2 (t11 1) P2(1 ) < P2(t11 ) + ":
(8)
(9)
1
'" , ,t11 1]. E, " ,t11 2], 2 2 (t11 1) P1(2 ) < P1(t11 ) + ":
(10)
;11(P A B) f('"n "n )g
"-. ve = (ve1 ve2 ) ve1 = (A1 + B1 )P1 (t11) ; B1 = A1 ; (A1 + B1 )P2(t11)C
(11)
ve2 = (A2 + B2 )P2 (t11) ; B2 = A2 ; (A2 + B2 )P1(t11):
1.
. D %
):
K 1(' "n ) 6 K 1 ('"n "n ) + "n % ' 2 =C
K 2('"n ) 6 K 2 ('"n "n ) + "n % 2 >C
lim K ('"n "n ) = ve1 C
n!1 1
lim K ('"n "n ) = ve2 :
n!1 2
D!, % t 2 ,0 1] K 1(It "n ) < ve1 + "2n :
" t 2 ,0 t11]. " "n ,t11 2 ], K 1(It "n ) =
Z
Z
2
t11
M1(t) d"n () =
2
Z
(12)
(13)
(14)
(15)
(16)
2
t11
((A1 + B1 )P1 (t) ; B1 ) d"n () 6
6 ((A1 + B1 )P1 (t11) ; B1 ) d"n () = (A1 + B1 )P1(t11) ; B1 = ve1 :
t11
" t 2 (t11 2 ). I
- K 1 (It "n ) =
=
Zt
t11
Zt
N1 () d"n () +
Z
2
1117
M1 (t) d"n () =
t
Z2
"
n
(A1 ; (A1 + B1 )P2 ()) d () + ((A1 + B1 )P1(t) ; B1 ) d"n () 6
t11
t
6 (A1 ; (A1 +B1 )P2 (t11)) t ;;tt11 + ((A1 +B1 )(P1 (t11)+
"n ) ; B1 ) 2;;t t =
2 11
2 11
t
;
t
;
t
;
t
"
11
2
2
n
= ve1 ; t + ve1 ; t + "n (A1 + B1 ) ; t < ve1 + 2 :
2 11
2 11
2 11
" t 2 ,2 1]. I
K 1(It "n ) =
Z
2
t11
N1 () d"n () =
Z
2
Z
2
(A1 ; (A1 + B1 )P2()) d"n () <
t11
< (A1 ; (A1 + B1 )P2(t11)) d"n () = ve1 < ve1 + "2n :
t11
, (16) . D! K 1('"n It) > ve1 ; "2n % t 2 ,0 1]:
" t 2 ,0 t11]. " '"n ,t11 1], K 1('"n It) =
Z1
Z
1
t11
N1 (t) d"n () =
Z
(17)
1
t11
(A1 ; (A1 + B1 )P2(t)) d'"n () >
> (A1 ; (A1 + B1 )P2(t11 )) d'"n () = A1 ; (A1 + B1 )P2(t11) = ve1 :
t11
" t 2 (t11 1 ). I
K 1 ('"n It) =
=
Zt
t11
Zt
M1 () d'"n () +
Z
1
N1 (t) d'"n () =
t
Z1
"
n
((A1 + B1 )P1() ; B2 ) d' () + (A1 ; (A1 + B1 )P2(t)) d'"n () >
t11
t
> ((A1 +B1 )P1(t11 ) ; B1 ) t ;;tt11 + (A1 ; (A1 +B1 )(P2 (t11)+
"n )) 1;;t t =
1 11
1 11
t
;
t
;
t
;
t
"
11
1
1
n
1
1
1
1
= ve ; t + ve ; t ; "n (A1 + B1 ) ; t = ve > ve ; 2 :
1
11
1
11
1
11
1118
. . " t 2 ,1 1]. I
K 1('"n It) =
Z
1
t11
M1 () d"n () =
Z
1
Z
1
t11
((A1 + B1 )P1 () ; B1 ) d'"n () >
> ((A1 + B1 )P1 (t11) ; B1 ) d'"n () = ve1 > ve1 ; "2n :
t11
, (17) . (16) , K 1 (' "n ) < ve1 + "2n % ' 2 =:
(18)
5 , ' = '"n , K 1 ('"n "n ) < ve1 + "2n =) ve1 > K 1 ('"n "n ) ; "2n :
(19)
(17) , (20)
K 1 ('"n ) > ve1 ; "2n % 2 >:
5 , = "n , K 1 ('"n "n ) > ve1 ; "2n =) ve1 < K 1 ('"n "n ) + "2n :
(21)
" (18), (21), K 1 (' "n ) < ve1 + "2n < K 1 ('"n "n ) + "n % ' 2 =:
I % , (12) . (19), (21) (14).
5 ) (13), (15) ) (12), (14).
1 , ;11(P A A) '" , " "-. 5 A1 = B2 ,
A2 = B1 (8) , ++ , "- , + = 1=2(A1 + A2 )1 .
1. , (6), !
Mj (t11) = Nj (t11) " A B j = 1 2C
(22)
Lj (t11 ) > Nj (t11) () Aj 6 Bj Lj (t11) > Nj (t11 ) () Aj < Bj j = 1 2:
(23)
1 0 #.& :3] #! &#!! "-&!(, "#\$!- \$&!"!+"&- ! " "!#!+(! ! (A1 6= A2 ) &/ &+&": + *%!"!&" &
A1 , A2 .
- 1119
. * (22) (6) (7). D (23) L1 (t11 ) ; N1 (t11 ) = A1 P1(t11 )(1 ; P2 (t11)) ; B1 (1 ; P1(t11))P2 (t11) +
+ B1 P2(t11) ; A1 (1 ; P2(t11)) = (B1 ; A1 )P1(t11)P2 (t11): (24)
?) (23) (24). ? j = 2 .
2. ;11(P A B) # It11 " . \$ %
1) A1 > B1 , ('"n It11 ) "-,
(11)&
2) A2 > B2 , (It11 "n ) "-,
(11)&
3) A1 6 B1 , A2 6 B2 , (It11 It11 ) ,
v = (v1 v2) v1 = A1 P1(t11 )(1 ; P2(t11)) ; B1 (1 ; P1(t11))P2 (t11)C
v2 = A2 P2(t11 )(1 ; P1(t11)) ; B2 (1 ; P2(t11))P1 (t11):
.
1) D 1 %
):
K1 (t t11) 6 K 1('"n It11 ) + "n % t 2 ,0 1]C
(25)
"
"
K 2(' n t) 6 K 2(' n It11 ) + "n % t 2 ,0 1]C
(26)
lim K ('"n It11 ) = ve1 C
(27)
n!1 1
"
2
lim K (' n It11 ) = ve :
(28)
n!1 2
J, '"n ,t11 1 ], Z
1
K 1 ('"n It
11
) = N1 (t11) d'"n () = N1 (t11) = A1 ; (A1 +B1 )P2 (t11) = ve1 C (29)
t11
Z1
K 2 ('"n It11 ) = M2 (t11) d'"n () = M2 (t11) = (A2 +B2 )P2(t11) ; B2 = ve2 : (30)
t11
1
(27) (28).
D! , K1 (t t11) 6 ve1 % t 2 ,0 1]:
(31)
" t 2 ,0 t11), K1(t t11) = M1 (t) = (A1 + B1 )P1(t) ; B1 < (A1 + B1 )P1(t11) ; B1 = ve1 :
1120
. . " t 2 (t11 1]. I
K1 (t t11) = N1(t11 ) = A1 ; (A1 + B1 )P2(t11) = ve1 :
" t = t11. I
A1 > B1 , 1, :
K1 (t11 t11) = L1 (t11) 6 M1 (t11) = (A1 + B1 )P1 (t11) ; B1 = ve1 :
I % , (31) . 1
, (29), (25).
D (26) , (16) (32)
K 2('"n It) < ve2 + "2n % t 2 ,0 1]:
1
, (29), (26). , 1 ).
2) K!
2 1 .
3) D 3 %
K1 (t t11) 6 K1 (t11 t11) = L1 (t11 ) % t 2 ,0 1]C
(33)
K2 (t11 t) 6 K2 (t11 t11) = L2 (t11 ) % t 2 ,0 1]:
(34)
D! (33). " t 2 ,0 t11). I
, 1,
K1 (t t11) = (A1 + B1 )P1 (t) ; B1 < (A1 + B1 )P1(t11 ) ; B1 = M1 (t11) < L1 (t11):
" t 2 (t11 1]. I
, 1, K1 (t t11) = N1 (t11) < L1 (t11):
I % , (33) . L (34) (33)
.
6. *" "-" ;11(P A B )
3. ' ;11(P A B) (11), '" " "-.
.
I %. D!, K ('" ) > ve1 ; "C
sup inf K1 (' ) = ve1 C inf
2 1
'2 2
sup inf K2 (' ) = ve2 C 'inf
K 2(' " ) > ve2 ; ":
2
2 '2
(35)
(36)
- 1121
5 , 1.
(18) inf K (' ) < ve1 + " % ' 2 = =) sup inf K1 (' ) 6 ve1 + ":
2 1
'2 2
(20) inf K ('" ) > ve1 ; " =) sup inf K1 (' ) > ve1 ; ":
2 1
'2 2
?
, (35). ?) (36) (35) .
II %. 5 ;11 (P C C),
C1 = A1 , C2 = B1 . "! + K 1 (' ). 5 , "- . " ;11(P C C) ve1 , "- '" . ?
, ) (35),
. . ;11(P A B) ve1 , '"
"- .
K!
.
4. ( It11 j -
(j = 1 2) #, % !
! Aj 6 Bj .
. D! j = 1.
1) " A1 6 B1 . D!, It11 , . . (37)
inf K (I ) > ve1 :
2 1 t11
D !, K1 (t11 t) > ve1 % t 2 ,0 1]:
(38)
" t 2 ,0 t11). I
K1 (t11 t) = N1 (t) = A1 ; (A1 + B1 )P2 (t) > A1 ; (A1 + B1 )P2(t11) = ve1 :
" t 2 (t11 1]. I
K1 (t11 t) = M1 (t11) = ve1 :
" t = t11. I
1 A1 6 B1 , K1 (t11 t11) = L1(t11 ) > M1 (t11) = ve1 :
, (38) . (38) (37).
1122
. . 2) " A1 > B1 . I
It11 , 1 A1 > B1 , K1 (t11 t11) = L1(t11 ) < M1 (t11) = ve1 :
? j = 2 .
7. -" ;11(P A B )
" p = (t1 t2) , ,
. H! ) % P . P ! ! ;11 (P A B) ! : 1) (tp 1), tp 2 ,0 1)C 2) (1 tp ), tp 2 ,0 1)C 3) (tp tp ), tp 2 ,0 1].
" p1 2 P !, p2 2 P , p2 p1. " p1 p2 ,
p1 p2 p2 p1. " p1 p2 #, K(p2 ) = K(p1 ).
2. \$ t t 2 ,0 1). \$ " A B : 1) (t 1) (t 1)& 2) (1 t) (1 t)& 3) (t 1) (1 t )& #*
(t 1) (1 t) t = t = t11 %.
. J, t 2 ,0 1) K1 (t 1) = M1 (t)C K1 (1 t) = N1 (t)C K2 (1 t) = M2 (t)C K2 (t 1) = N2 (t):
" t < t . 5 Pj (t) K1 (t 1) = (A1 + B1 )P1 (t ) ; B1 < (A1 + B1 )P1 (t ) ; B1 = K1 (t 1)C
K2(t 1) = A2 ; (A2 + B2 )P2(t ) > A2 ; (A2 + B2 )P2(t ) = K2 (t 1): (39)
(39) , (t 1) (t 1) (t 1) (t 1), . . !
.
E, Pj (t) K1(1 t ) = A1 ; (A1 + B1 )P2(t ) > A1 ; (A1 + B1 )P2(t ) = K1 (1 t)C
K2 (1 t) = (A2 + B2 )P2 (t ) ; B2 < (A2 + B2 )P2 (t ) ; B2 = K2 (1 t): (40)
(40) , (1 t ) (1 t) (1 t 1) (1 t ), . .
!
.
D !
:
N1 = K1 (t 1) ; K1 (1 t) = (A1 + B1 )(P1 (t ) + P2(t ) ; 1)C
N2 = K2 (t 1) ; K2 (1 t) = (A2 + B2 )(1 ; P1(t ) ; P2(t ):
" sgn N1 = ; sgn N2, (t 1) (1 t) . "
, t = t = t11, (7) Kj (t11 1) = Kj (1 t11) (j = 1 2), . .
(t11 1) (1 t11) .
- 1123
3.
1) + 8j 2 f1 2g Aj > Bj 9j0 2 f1 2g Aj0 > Bj0 , (t11 1) (t11 t11),
(1 t11) (t11 t11) , , (t11 t11) \$.
2) + 8j 2 f1 2g Bj > Aj 9j0 2 f1 2g Bj0 > Aj0 , (t11 t11) (t11 1),
(t11 t11) (1 t11) , , (t11 1) (1 t11) \$.
.
1) " Aj > Bj Aj0
> Bj0 . I
1 Lj (t11) 6 Nj (t11) = Mj (t11 ) (j = 1 2) Lj0 (t11 ) < Nj0 (t11):
K, Kj (t11 t11) = Lj (t11) Kj (1 t11) = Nj (t11 ) = Kj (t11 1)
Kj (t11 t11) 6 Kj (1 t11) = Kj (t11 1) Kj0 (t11 t11) < Kj0 (1 t11) = Kj0 (t11 1):
< , (t11 1) (t11 t11), (1 t11) (t11 t11).
2) " Aj 6 Bj Aj0 < Bj0 . I
1 Lj (t11) > Nj (t11) = Mj (t11 ) (j = 1 2) Lj0 (t11 ) < Nj0 (t11):
K, Kj (t11 t11) = Lj (t11) Kj (1 t11) = Nj (t11 ) = Kj (t11 1)
Kj (t11 t11) > Kj (1 t11) = Kj (t11 1) Kj0 (t11 t11) > Kj0 (1 t11) = Kj0 (t11 1):
< , (t11 t11) (t11 1), (t11 t11) (1 t11).
4. + 8j 2 f1 2g Aj 6 Bj 9j0 2 f1 2g Aj0 < Bj0 , "
tp 2 ,0 1)
1) (tp 1) (t11 t11),
2) (1 tp) (t11 t11).
.
1) " tp 2 ,0 t11).
I
, M1 (t) 1, K1 (tp 1) = 1 (tp ) < 1 (t11) 6 L1(t11 ) = K1 (t11 t11):
" tp 2 (t11 1). I
, % N2 (t) 1,
K2 (tp 1) = N2 (tp ) < N2 (t11) 6 L2(t11 ) = K2 (t11 t11):
, 8tp 2 ,0 t11) (t11 1) (tp 1) (t11 t11). " 2 3 (t11 t11) (t11 1). ?
, 8tp 2 ,0 1) (tp 1) (t11 t11).
1124
. . 2) " tp 2 ,0 t11). I
, M2 (t) 1, K2 (1 tp ) = 2 (tp ) < 2 (t11) 6 L2(t11 ) = K2 (t11 t11):
" tp 2 (t11 1). I
, % N1 (t) 1,
K1 (1 tp ) = N1 (tp ) < N1 (t11) 6 L1(t11 ) = K1 (t11 t11):
, 8tp 2 ,0 t11) (t11 1) (1 tp ) (t11 t11). " 2 3 (t11 t11) (1 t11). ?
, 8tp 2 ,0 1) (1 tp ) (t11 t11).
5. + A1A2 > B1 B2 , 8tp 2 ,0 1] (tp tp ) (t11 1),
(tp tp ) (1 t11).
. * Nj (tp ) = Kj (tp tp) ; Kj (t11 1):
N1(tp ) = A1 P1(tp )(1 ; P2 (tp )) ; B1 (1 ; P1 (tp ))P2(tp ) ; (A1 + B1 )P1(t11 ) + B1 =
= A1 (P1(tp ) ; P1 (t11) ; P1 (tp )P2 (tp )) ; B1 (P2(tp ) ; P2(t11 ) ; P1(tp )P2 (tp ))C
N2(tp ) = A2 P2(tp )(1 ; P1 (tp )) ; B2 (1 ; P2 (tp ))P1(tp ) ; (A2 + B2 )P2(t11 ) + B2 =
= A2 (P2(tp ) ; P2 (t11) ; P1 (tp )P2 (tp )) ; B2 (P1(tp ) ; P1(t11 ) ; P1(tp )P2 (tp )):
"! j (tp ) = Pj (tp ) ; Pj (t11) ; P1(tp )P2(tp ) (j = 1 2). I
N1 (tp ) = A1 1 ; B1 2 C N2(tp ) = A2 2 ; B2 1 :
(41)
"!, j (tp ) < 0 % tp 2 ,0 1], j = 1 2. * .
1) tp 6 t11 . I
Pj (tp ) 6 Pj (t11) , , j < 0.
2) tp > t11 . "% ! j (tp ) % :
1 (tp ) = P1 (tp )(1 ; P2(tp ) ; P1(t11 )) ; P1(t11)(1 ; P1(tp )) =
= P1 (tp )(P2(t11 ) ; P2(tp )) ; P1 (t11)(1 ; P1(tp ))C
(42)
2 (tp ) = P2 (tp )(1 ; P1(tp ) ; P2(t11 )) ; P2(t11)(1 ; P2(tp )) =
= P2 (tp )(P1(t11 ) ; P1(tp )) ; P2 (t11)(1 ; P2(tp )):
5 + Pj (t) (42) , j (tp ) < 0.
" (t11 1) (1 t11) , , (tp tp ) (t11 1).
"
!, % ++ Bj 0. I
(41) j (tp ) , Nj (tp ) < 0, . . (tp tp) (t11 1).
@ % ++ Aj 0, A1 A2 > B1 B2 B1 B2 = 0, ! .
" Bj > 0, Aj > 0, j = 1 2. "
!, !
, . . tq 2 ,0 1), (tq tq ) (t11 1). I
Nj (tq ) > 0
j = 1 2, % . 1
, j (tq ) < 0, A1 6 2 6 B2 B1 1 A2
- 1125
% . ?
,
A1 < B2 () A A < B B 1 2
1 2
B1 A2
! .
5.
1) + % ! ! ;11(P A B)
A1 A2 > B1 B2 , (1 t11) (t11 1) \$.
2) + 8j 2 f1 2g Aj > Bj 9j0 2 f1 2g Aj0 > Bj0 , (1 t11) (t11 1) \$, (t11 t11) \$-.
3) + 8j 2 f1 2g Aj 6 Bj 9j0 2 f1 2g Aj0 < Bj0 , \$ (to to ), to 2 ,0 1]& % (t11 t11) (t11 1), (t11 t11) (1 t11) , , (1 t11) (t11 1) \$-.
.
1) " (1 t11) (t11 1) , %
(t11 1). " 2 (tp 1) (t11 1) (1 tp) (t11 1). "
5 % tp 2 ,0 1] (tp tp) (t11 1). I % , % , (t11 1), . . (t11 1) ".
2) Aj > Bj (j = 1 2) A1 A2 > B1 B2 . ?
, 1 , (1 t11) (t11 1) ". " 1 3 (t11 t11) "-.
3) @ Aj 6 Bj 9j0 2 f1 2g Aj0 < Bj0 , 4 (tp 1) (t11 t11) (1 tp) (t11 t11). 5 ! .
) " % tp 2 ,0 1] (tp tp ) (t11 t11). I
(t11 t11) ".
%) ? tq 2 ,0 1], (tq tq ) (t11 t11). "!
T = ft j t 2 ,0 1] Li (t) > Lj (t11 )gC L(t) = L1 (t) + L2 (t):
5 ! T + L(t) T %) Lm . " to 2 Arg max
L(t). " to 2 T
t2T
Lm > L(tq ) > L(t11 ), (to to ) (t11 t11). "!, (to to )
". " (tp tp ) to to ).
I
(tp tp ) (t11 t11) , , tp 2 T. ? , ) (tp tp) to to ) , L(tp ) > L(to ), to . "
!, tp 2 ,0 1] (tp 1) (to to ).
I
(tp 1) (t11 t11), 4. E, tp 2 ,0 1], (1 tp) (to to ), (1 tp ) (t11 t11), 4.
1126
. . ?)
(t11 t11) (t11 1) (t11 t11) (1 t11)
2 3.
6 ( ). +
% ;11(P A B) #, :
1) (A1 ; B1 )(A2 ; B2 ) < 0C 2) Aj = Bj j = 1 2:
(43)
. "
!, (43) . I
! .
1) 8j 2 f1 2g Aj > Bj 9j0 2 f1 2g Aj0 > Bj0 . I
2 5
(t11 t11) "- , , .
2) 8j 2 f1 2g Aj 6 Bj 9j0 2 f1 2g Aj0 < Bj0 . I
3 5
(1 t11) (t11 1) "- , ,
.
7 ( ). +
;11(P A B) % ! ! !
A1A2 = B1 B2 (44)
#.
. J, ) (44), > 0, t1 t2 2 ,0 1] K1 (t1 t2) = ;
K2 (t1 t2):
(45)
D, ! A1A2 > 0. I
(44) B1 B2 > 0
A1 =B2 = B1 =A2. 5 + (6) , (45)
= A1=B2 = B1 =A2.
" A1 A2 . @ A1 = 0, (5)
B1 6= 0, (44) B2 = 0, A2 6= 0. I
+ (6) , (45) = B1 =A2. E, A2 = 0, (5) B2 6= 0, ) (44)
B1 = 0, A1 6= 0. " = A1 =B2, (45).
, > 0, (45), . (45) , (3), . . .
- 1127
8. - ". " ;11(P A B )
", ("-), ("- ).
" A1A2 > B1 B2 . I
j 2 f1 2g, Aj > Bj . @
A1 > B1 , 1 2 ('" It11 ) "-.
? "- (1 t11) ".
? 1 5 (1 t11) ". @ A2 > B2 , 2 2 (It11 " )
"-. ? "- (t11 1) ! ". ? 1 5 (t11 1) ".
" A1 A2 < B1 B2 . I
8j 2 f1 2g Aj 6 Bj 9j0 2 f1 2g Aj0 <Bj0 ,
3 2 (It11 It11 ) .
@ (t11 t11). ? 2 3 ) (t11 t11) % ", ve , "- ('"n "n ), ) (t11 1) (1 t11) ( 1). I , (t11 t11) !
% ". < .
". " A1 = A2 = 0, B1 > 0, B2 > 0. I
v1 = K 1 (t11 t11) = ;B1 (1 ; P1(t11))P2 (t11) < 0C
v2 = K 2 (t11 t11) = ;B2 (1 ; P2(t11))P1 (t11) < 0:
" (t11 t11) ", K(0 0) = (0 0) (v1 v2):
< ! %%, % ! , , %
. " p = (0 0) %%, t = 0 ! . " , %%, .
/
1] . , . | .: , 1964.
2] Fox M., Kimeldorf G. S. Noisy duels // SIAM J. Appl. Math. | 1969. | Vol. 17. |
P. 353{361.
3] Fox M. Duels with possibly assymetric goals // Zastisovanie matematiky. | 1980. |
Vol. XVII, no. 1. | P. 15{25.
1128
. . 4] Kimeldorf G. Duels: an overview // Mathematics of con)ict. | North-Holland,
1983. | P. 55{71.
5] * +. ,., -./0 1. 2. 34 4 // .: 56 72 8, 1982. |
. 27{38.
6] -./0 1. 2. 9: ; < 0 4 // * = . | .: 52>>
>, 1983. | . 260{266.
7] Radzik T. General noisy duels // Math. Japonica. | 1991. | Vol. 36, no. 5. |
P. 827{857.
8] -./0 1. 2. *4/ / 4. ? // .
. +0. @4 IV B4; ?. BC-. @ 4, . 1. | 2B; 2, 1997. | . 111{119.
9] -./0 1. 2. 340 4/ E 4; 4; // @4
VI B4; ?. BC-. @ 6, . 1. | 2B;
2, 1999. | C. 77{85.
% & 2002 .
,
. . , . . . . . 517.956.225
: , ,
.
! " #
\$ , %&' % . ( "
% ', | (#.
* +{-." ' #,
# . / \$# '#
.
Abstract
A. I. Sgibnev, P. A. Krutitskii, On the mixed problem for Laplace equation outside cuts, placed along a circumference in a plane, Fundamentalnaya i prikladnaya
matematika, vol. 8 (2002), no. 4, pp. 1129{1158.
The boundary value problem for harmonic functions outside cuts lying on the
arcs of a circumference is considered. The Dirichlet condition is given on one side
of each cut and Neumann condition is speci5ed on the other side. The problem is
reduced to the Riemann{Hilbert problem for complex analytic function, which is
solved in a closed form. An explicit solution of the original problem is obtained.
1. . , !
, , ! . . "
, .
\$ %8{10] !+
, , 4-, . .
!, /. \$ %8] , 2002, 8, 6 4, . 1129{1158.
c 2002 !,
"#
\$% &
1130
. . , . . 1
. \$ %10] 1
{2, , 1
, , |
2. 4 2 | %10]. \$ %9]
, , , 1
, , | 2. %8{10] +
,
, , .
/ , 6+
, , %1{3]. 4
, {7
6+
, | , 6.
4 ,
6+
, 2, %6] %11] . 4 ,
/ , , 1
, , | , 2, %7]. 4
, ,
!
! ,
.
\$ %5] ,
,
6+
, , ,
1
, , | 2.
\$ ,
, . ; ,, . < 6+
, ,. ,
, .
2. (x1 x2) 2 R2 (r ): x1 = r cos , x2 = r sin . ,, z = x1 + ix2 . 4
N ,: N1 , +
,
(a1n b1n), N2 ,
+
(a2n b2n), N = N1 + N2 . B , ,
, +. \$ ,:
L1n = fr = 1 2 (a1n b1n)g n = 1 : : : N1
L2n = fr = 1 2 (a2n b2n)g n = 1 : : : N2
L1 =
N1
n=1
L1n L2 =
N2
n=1
1131
L2n :
"
, , ,:
(Lk ) =
Nk
fr = 1 0 2 (akn bkn)g k = 1 2 L = (L1 ) (L2 ) :
n=1
D
+ ; 6+
, .
B ,
, 6+
u(x) HL0 , :
1) u(x) 2 C 0 (R2 n L)
2) ru(x) 2 C 0(R2 n L n X), ,
X=
2 Nk
((cos akn sin akn ) (cos bkn sinbkn )) |
k=1 n=1
+ L,
3) x ! d 2 X jruj < cjx ; dj (1)
, c 0, ;1.
M. 2
6+
u(x) 2 HL0 , ,
L, ,
uj(L1 )+ = Q^ 1 (t) (L1 )+ (2)
@u 1;
(3)
@r (L1 ); = Q2 (t) (L ) @u 2+
(4)
@r (L2 )+ = Q2 (t) (L ) uj(L2); = Q^ 1 (t) (L2 ); (5)
, t = ei 2 L, 6+
Q^ 1 (t) 2 C 1(L), Q2 (t) 2 C 0(L) 7 2 (0 1]. ,, 6+
u(x) :
;1
;2
u(x) = C ln jxj + O(1) @u(x)
(6)
@ jxj = C jxj + O(jxj ) jxj ! 1
C | . (\$ C = 0, , (6) ,
6+
u(x) .)
1.
M .
1132
. . , . . . F
, M u1(x)
u2 (x). ;, 6+
u0 (x) = u1(x) ; u2 (x) M ,
(2){(5) u0 (x) = O(1) @@ujx0j = O(jxj;2) jxj ! 1:
(7)
"
Lkn, n = 1 : : : Nk , k = 1 2.
B ,
!
. "
lr , r + . G
,
ru0 + L, 6 7
/:
Z @u0
2
kru0kL2 (Dr nL) =
u0 @ dl
+
;
L L lr
, Dr | , r, | ,
+ Dr n L.
F (@u0 =@)jL = (@u0 =@r) , Z + @u0 + ; @u0 ; Z @u0
2
dl + u0 @ dl:
kru0kL2(Dr nL) =
u0 @r ; u0 @r
L
lr
F , ,
. F ,:
Z
lr
Z @u0
@u
0
u0 @ dl = r u0 @ jxj d jxj = r
2
0
, (7) O(r;1 ). F, r ! 1, kru0k2L2 ( 2nL) = 0. G
(2),
, u0(x) 0. ; .
R
3. !"# \$ ! M {7
. \$ 3, {7
, M.
\$ ,
h0L . B ,
, 6+
w(z) h0L , 1) -,6 L,
2) ,
,
3) L , , , + L, , ,
, . . jw(z)j < cjz ; dj , , c > 0, > ;1 | ,
d = exp (iakn ) d = exp (ibkn ), n = 1 : : : Nk , k = 1 2.
1133
1
66+
(2), (5) 6+
Q1 (t) =
= @ Q^ 1(t)=@ 2 C 0(L), , (2) !
@u i
1
(8)
@ (L1)+ = Q1 (t) t = e 2 L u(cos a1n sin a1n) = Q^ 1(exp (ia1n )) n = 1 : : : N1
(9)
(5) !
@u i
2
(10)
@ (L2 ); = Q1(t) t = e 2 L u(cos a2n sin a2n) = Q^ 1(exp (ia2n )) n = 1 : : : N2:
(11)
F u(x) | M. \$ 6+
H(z) = u(x) + iv(x)
(12)
, v(x) | (,) ,
6+
, u(x) {
. ;,
(13)
I(z) = zHz (z) = z(ux1 ; iux2 ) |
0
6+
, hL ( ,
, jI(z)j = jz j jHz (z)j = jxj jruj = O(1) z ! 1) ,
Re(iI+ (t)) = Q1(t) (L1 )+ Re(I+ (t)) = Q2(t) (L2 )+ (14)
;
1
;
;
2
;
Re(I (t)) = Q2(t) (L ) Re(iI (t)) = Q1(t) (L ) (15)
(3), (4), (8), (10). (G
(9), (11) .) D, 6+
I(z) 6
{7
.
R. 2
6+
I(z) 2 h0L , ,
(14), (15) , I(0) = 0.
4 {7
, , %1, 2]. 4
, 6+
Y (z) h0L , 6+
Y (z) Y (1=z) h0L Y (t) = Y (t):
(16)
F I(z) | R. \$ 6+
1(z) = I(z) 2 (z) = I (z)
(16), , 6+
1(z), 2(z) : 6+
1(z), 2 (z) h0L
,
:
+1 (t) = ;2 (t) ; 2iQ1(t) +2 (t) = ;;1 (t) + 2Q2(t) L1 +1 (t) = ;;2 (t) + 2Q2(t) +2 (t) = ;1 (t) + 2iQ1 (t) L2 :
1134
,
. . , . . "J
!
:
+1 (t) = g(t);2 (t) + Q~ 1 (t) +2 (t) = ;g(t);1 (t) + Q~ 2(t)
(17)
2Q2(t)
1(t)
~;
g(t) = ;11 Q~ 1 (t) = ;2iQ
2Q2(t) Q (t) = 2iQ1 (t) :
4 6
, , , 6+
L1 , , , 6+
L2 .
%1], R (17) 6 I(z) = (1 (z) + 2(z))=2.
B , %1]. F
i
G(t) = exp i 2 g(t) = ;i 1(z) = exp ;i 4 1 (z) 2 (z) = exp i 4 2(z)
, ,
L 6+
1 (t), 2(t) +1 (t) = G(t);2 (t) + exp i 4 Q~ 1 (t) +2 (t) = G(t);1 (t) + exp ;i 4 Q~ 2(t):
F
1(z) = 1 (z) + 2 (z), 2(z) = ;1 (z) + 2 (z),
6+
1(z) 2(z) 2 h0L ,
L
1+ (t) = G(t)1;(t) + F1 (t) 2+ (t) = ;G(t)2; (t) + F2(t)
(18)
, 6+
(2 exp(;i )(Q1(t)+Q2(t)))
4
(19)
F1(t)=exp i 4 Q~ 1(t)+exp ;i 4 Q~ 2(t)=
2 exp(i 4 )(Q1 (t)+Q2 (t))
(2 exp(;i )(Q2(t);Q1(t)))
~
4
~
: (20)
F2(t)=exp ;i 4 Q2 (t) ; exp i 4 Q1(t)=
2 exp(i 4 )(Q1 (t) ; Q2 (t))
D 6 , , F1(t) = ig(t)F1 (t) F2(t) = ig(t)F2 (t):
\$ (18) , %3] 1(z) = q1;1(z), 2(z) = q2;1(z), ,
a1n + 3b1n N Y
N1
exp ;i 8
(z ; exp(ia1n ))1=4(z ; exp(ib1n))3=4 q1(z) = exp ;i 2
n=1
N
2
2
2
Y
exp ;i 3an + bn (z ; exp(ia2n ))3=4(z ; exp(ib2n ))1=4
8
n=1
1135
N1
Y
3a1n + b1n (z ; exp(ia1 ))3=4(z ; exp(ib1 ))1=4 q2(z) = exp ;i N
exp
;
i
n
n
2 n=1
8
a2 + 3b2n N2
Y
exp ;i n
(z ; exp(ia2n ))1=4(z ; exp(ib2n ))3=4:
8
n=1
D qk(z) = z1N qk (z) k = 1 2:
4
, t = ei 2 L ( , , < akn > bkn, , 2 %akn bkn], k = 1 2):
akn + 3bkn k
1
=
4
k
3
=
4
(t ; exp (ian)) (t ; exp (ibn )) = 2i exp i 2 exp i 8
sign( ; akn ) k 3=4 exp (i 3 )
k 1=4 4 sin ; an sin ; bn 1
2
2
k k
(t ; exp (iakn))3=4(t ; exp (ibkn ))1=4 = 2i exp i 2 exp i 3an 8+ bn sign( ; akn ) ; akn 3=4 ; bkn 1=4 exp(i )
4 sin
1
2 sin 2 , k = 1, n = 1 : : : N1 k = 2, n = 1 : : : N2 . F! t = ei 2 L
(exp(i 3 )) N
( exp(i )) N
+
+
4
q1 (t) = exp(i ) t 2 R1(t) q2 (t) = exp(i 34 ) t 2 R2(t)
(21)
4
4
, 6+
:
N1 ; a1 1=4 ; b1 3=4
Y
N
sin 2 n sin 2 n sign( ; a1n) R1(t) = 2
n=1
N2 ; a2 3=4 ; b2 1=4
Y
n sin
n sign( ; a2 )
sin
(22)
n
2
2
n=1
N1 1 1=4
1 3=4 Y
R2(t) = 2N sin ;2 an sin ;2 bn sign( ; a1n) n=1
N
2
2 1=4 2 3=4
Y
sin ; an sin ; bn sign( ; a2n):
(23)
2
2
n=1
L
, qk (t) qk+ (t), k = 1 2. D (21) ,
+
+
+
+
q1+ (t) = ;ii q1tN(t) = ig(t) q1tN(t) q2+ (t) = ;ii q2tN(t) = ;ig(t) q2tN(t) :
1136
. . , . . L, %3] (18) Z
+
k (t) dt + Pk (z) k = 1 2
k (z) = 2iq1 (z) Fk (t)q
t;z
qk (z)
k
PN
L
, 6+
Fk (t) (19), (20), Pk (z) = Bnk z n (k = 1 2) | n=0
N.
G
6+
qk (z), k(z):
+ ;N
N Z
N
1 (t)t ) ;dt + z P1(z) =
1 (z) = ; 2iqz (z) ig(t)F1 (t)(ig(t)q
2
1=t ; 1=z
t
q1(z)
1
L
Z
+ N
1 (t) z N +1 dt + z P1 (z) :
= 2iq1 (z) F1(t)q
t;z
t
q1(z)
1
L
M,
,
Z F2(t)q2+(t) z N +1 zN P2(z)
1
2(z) = ; 2iq (z)
dt + q (z) :
t;z
t
2
2
L
F
k (z) k (z), 6+
I(z):
exp(;i 4 )
I(z) = 12 (1 (z) + 2(z)) =
2 (1 (z) + 2 (z)) =
exp(;i 4 )
=
1 (z) + 1(z) ; 2(z) + 2 (z)) =
4 (
exp(;i 4 )
1 Z F1(t)q1+ (t) 1 + z N +1 dt + P1(z) + z N P1(z) ;
=
4
2iq1(z)
t;z
t
q1(z)
;
1
2iq2(z)
Z
L
L
+
F2 (t)q2 (t) 1 + z N +1 dt + ;P2(z) + z N P2(z) :
t;z
t
q2 (z)
"
XN 1 1 n XN 1 n
P 1 (z) = 1 (P1(z) + z N P1(z)) = 1
2
2 n=0(Bn + BN ;n )z = n=0 Dn z XN 2 2 n XN 2 n
1
1
2
N
P (z) = (;P2 (z) + z P2 (z)) =
2
2 n=0(;Bn + BN ;n )z = n=0 Dn z :
"
, !66
+
Dn1 = (1=2)(Bn1 + BN1 ;n ), Dn2 =
= (1=2)(;Bn2 + BN2 ;n ) Dn1 = DN1 ;n Dn2 = ;DN2 ;n :
(24)
1137
F I(0) = 0, , !66
+
D01 :
(0) D2 D01 = I ; qq1(0)
(25)
0
2
1 Z F1 (t)q1+ (t) dt + q1(0) 1 Z F2(t)q2+ (t) dt:
I = ; 4i
t
q2(0) 4i
t
L
L
1 f1 (t) = Q1(t) + Q2 (t) f2 (t) = g(t)(Q2 (t) ; Q1 (t))
(26)
;i
h(t) = 1 :
;, exp(;i 4 )Fk (t) = 2h(t)fk (t), k = 1 2. D,
Z h(t)f1(t)q1+ (t) z N +1
;i P 1(z) ;
1
I(z) = 4iq (z)
1
+
dt
+
exp
t;z
t
4 2q1(z)
1
L
Z
h(t)f2 (t)q2+ (t) 1 + z N +1 dt + exp ;i P 2 (z) : (27)
; 1
4iq2(z)
t;z
t
4 2q2(z)
,
L
N+
I(z) ,
(14),
(15), ,
, , qk (z) = O(z N )
z ! 1, k = 1 2.
1. I(z) (27) Dn1
(n = 1 : : : N), Dn2 (n = 0 : : : N), (24), D01
(25) R.
4. % , R, M . F
6+
I(z) ,
. F , 6+
I(z)=z L (
I(z) L I(0) = 0) ( I(z) ,
), , :
I(z) = H (z) = 1 Z (t) dt :
(28)
z
z
2i
t t;z
L
D 6 L+, , (t) = I+ (t) ; I; (t). D,
(28) z 6
(12), (13), 1 Z (t) ln(z ; t) d + c
H(z) = u(x) + iv(x) = ; 2
L
1138
. . , . . , c | . "
Z
1
u(x) = Re H(z) = 2 %; Re (t) ln r(x ) + Im(t)!(x )] d + c0 (29)
L
i
, t = e , c0 | , !(x ) = arg(z ; t) | ,,
+
%2], r(x ) = jz ;tj = (x1 ; cos )2 + (x2 ; sin )2 . N+
!(x )
p
2m (m +) 6
x1 ; cos cos !(x ) = p
(x1 ; cos )2 + (x2 ; sin )2
x2 ; sin sin !(x ) = p
:
(x1 ; cos )2 + (x2 ; sin )2
P
x 2 R2 n L, !(x ) 6
! 6+
, L. F
!(x ) 6+
u(x) . 1 6+
u(x)
%1]:
Z
Lkn
Im(t) d = 0 n = 1 : : : Nk k = 1 2:
1 , 6+
(t) , I(z) L, 6
L+,{F:
1
t0 N +1
+
h(t
)f
(t
)q
(t
)
1+ t
+
I+ (t0 ) = +1
2q1 (t0 ) 2 0 1 0 1 0
0
Z h(t)f1(t)q1+ (t) t0 N +1 1
+ 2i
1+ t
dt ;
t ; t0
N +1
1
+ (t0 ) 1 + t0
h(t
)f
(t
)q
+
; +1
0
2
0
2
t0
2q2 (t0 ) 2
Z h(t)f2(t)q2+ (t) t0 N +1 1
+ 2i
1+ t
dt +
t ; t0
L P 1(t0) P 2(t0)
+ exp ;i 4 + + exp ;i 4 + 2q1 (t0 )
2q2 (t0 )
L
1 ; 1 h(t )f (t )q+ (t ) 1 + t0 N +1 +
t0
2q1;(t0 ) 2 0 1 0 1 0
Z
N
+1
1 h(t)f1 (t)q1+ (t) 1 + t0
+ 2i
dt ;
t ; t0
t
I; (t0 ) =
L
1 ; 1 h(t )f (t )q+ (t )1 + t0 N +1 +
t0
2q2;(t0 ) 2 0 2 0 2 0
Z
N
+1
+
1 h(t)f2 (t)q2 (t) 1 + t0
+ 2i
dt
+
t ; t0
t
L
1
2
+ exp ;i 4 P ;(t0 ) + exp ;i 4 P ;(t0 ) :
2q1 (t0 )
2q2 (t0 )
\$ (t0 = ei0 2 L)
Z fk (t)qk+ (t) t0 N +1
1+ t
dt =
Kkm (t0) +1
4qk (t0 ) m t ; t0
1139
;
exp(;i N ) Z fk (t) exp(i N2 )Rk (t)
= 4R (t2 )0 exp(i)
; exp(i0 ) (1 + exp(i(N + 1)(0 ; )))i exp(i) d =
k 0 m
Z L fk (t)Rk(t) N + 1
1
= 4R (t ) sin(( ; )=2) cos 2 ( ; 0 ) d k = 1 2 m = 1 2:
k 0 m
0
L
L
"
. 2 K1 (t0) = K11 (t0 ) + K12 (t0 ) K2 (t0 ) = K21 (t0 ) ; K22 (t0):
"
, Z f1(t)R1(t) N + 1
K1 (t0 ) = 4R1 (t ) sin((
cos 2 ( ; 0 ) d
(30)
; 0 )=2)
1 0
LZ
1
g(t)f2 (t)R2(t) cos N + 1 ( ; ) d:
K2(t) = 4R (t ) sin((
(31)
0
; 0 )=2)
2
2 0
G
, L
8>1
>
q1+ (t) = tN=2 R1 (t) <i
2
q1+ (t0) tN=
0 R1(t0 ) >
>:;1i
8>1
>
q2+ (t) = tN=2 R2 (t) <;i
2
q2+ (t0) tN=
0 R2(t0 ) >
>:i1
t 2 L1 t 2 L1 t 2 L2 t 2 L2 t0 2 L1
t0 2 L2
t0 2 L1
t0 2 L2
t 2 L1 t 2 L1 t 2 L2 t 2 L2 t0 2 L1
t0 2 L2
t0 2 L1
t0 2 L2
1140
. . , . . 1
4q1+ (t0)
Z
h(t)f1 (t)q1+ (t) 1 + t0 N +1 dt =
t ; t0
t
(;iKL 1 (t ) ; iK 2 (t ))
1 0
1 0 = h(t )K (t )
0 1 0
1
K1 (t0 ) + K12 (t0)
1
h(t)f2 (t)q2+ (t) 1 + t0 N +1 dt =
t ; t0
t
4q2+ (t0)
L
;iK21 (t0 ) + iK22 (t0)
=
= h(t0 )g(t0 )K2 (t0 ):
;K21 (t0 ) + K22 (t0)
D , , 1=q1; (t) = ;ig(t)=q1+ (t),
;
1=q2 (t) = ig(t)=q2+ (t), 1
I+ (t0 ) = h(t0 ) f1 (t0 ) ;2 f2 (t0) + exp ;i 4 P +(t0) +
2q1 (t0 )
2 (t0 )
P
+ exp ;i 4 + ; ih(t0 )K1 (t0 ) + ih(t0 )g(t0 )K2 (t0 )
2q2 (t0)
1
I; (t0) = ih(t0 )g(t0 ) f1 (t0 ) +2 f2 (t0) ; exp ;i 4 P +(t0 ) ig(t0 ) +
2q1 (t0 )
2
+ exp ;i 4 P +(t0 ) ig(t0 ) ; g(t0 )h(t0 )K1 (t0) ; h(t0 )K2 (t0 ):
2q2 (t0)
=
Z
(
)
2+, ,, ;1 exp(;i 4 )
1
exp(;i 4 )
1
1
= tN=2 R (t) ;i = itN=2 R (t) ;i q1+ (t)
q2+ (t)
1
2
1
1 (t)
exp ;i 4 P +(t) (1 + ig(t)) = (;1 ; i) 2tN=P2 R
2q1 (t)
1(t)
2
P 2 (t) :
exp ;i 4 P +(t) (1 ; ig(t)) = (1 ; i) 2itN=
2 R2(t)
2q2 (t)
;, , (t):
(t0 ) = I+ (t0 ) ; I; (t0 ) =
1(t0)
Q2(t0)
= h(t0 ) Q
;
ih(t
)g(t
)
0 0 Q1(t0 ) +
Q2 (t0)
+ h(t0)(;i + g(t0 ))K1 (t0 ) + h(t0 )(1 + ig(t0 ))K2 (t0 ) +
1
P 2 (t0) :
+ (;1 ; i) N=P2 (t0) + (1 ; i) N=
2t0 R1(t0 )
2it0 2 R2 (t0)
1141
Q 6 . ,
P 1 (t):
1 (D1 tn + D1 tN ;n ) = 1 tN=2 (D1 tn;(N=2) + D1 t(N=2);n ) =
n
N ;n
n
2 n
2
= tN=2 Re(Dn1 t(N=2);n ) = tN=2 1n cos N2 ; n + n1 sin N2 ; n , kn = Re Dnk , nk = Im Dnk , k = 1 2,
n = 0 : : : %N=2]. + . M,
,
1 (D2 tn + D2 tN ;n ) = tN=2 1 (D2 tn;(N=2) ; D2 t(N=2);n ) =
N ;n
n
2i n
2i n
= ;tN=2 Im(Dn2 t(N=2);n ) = tN=2 ;2n sin N2 ; n + n2 cos N2 ; n :
;,
N 1 N #N=
2]
X
1
1
=
2tN=2 R1(t) R1(t) n=0 n cos 2 ; n +n sin 2 ; n P 1(t)
(32)
#N=
2]
X
1
N
N
2
2
2itN=2 R2(t) = R2(t) n=0 ;n sin 2 ; n +n cos 2 ; n :
G
!
6, !66
+
K1 , K2 :
h(t0 )(;i + g(t0 )) = ;1 ; i, h(t0)(1 + ig(t0 )) = 1 ; i, t0 2 Lk , k = 1 2, 1(t0 ) ; ih(t0 )g(t0 ) Q2 (t0 ) +
(t0 ) = h(t0 ) Q
Q (t )
Q (t )
P 2 (t)
2 0
+ (;1 ; i)K1 (t0 ) + (1 ; i)K2 (t0 ) +
#N=2]
1
1 cos 0
+ (;1 ; i) R (t )
1 0 n=0 n
#N=2]
1
+ (1 ; i) R (t )
;2n sin 0
2 0 n=0
X
X
1 0
N 2 ;n
N + n1 sin 0 N2 ; n
+
+ n2 cos 0 N2 ; n
:
2 ;n
D, 6+
(t) Re (t0 ) = ;g(t0 )Q2(t0 ) ; K1 (t0) + K2 (t0 ) ;
#N=
X2] 1 cos 0 N ; n + 1 sin 0 N ; n +
; 1
n
R1(t0 ) n=0 n
2
2
#N=
X2] ;2 sin 0 N ; n
+ R 1(t )
n
2
2 0 n=0
+ n2 cos 0 N2 ; n
(33)
1142
. . , . . Im(t0) = ;g(t0 )Q1 (t0) ; K1 (t0 ) ; K2 (t0) ;
#N=
X2] 1 cos N ; n + 1 sin N ; n ;
; 1
0 2
0 2
n
R1(t0 ) n=0 n
X2] ;2 sin 0 N ; n + 2 cos 0 N ; n : (34)
1 #N=
n
n
R2(t0 ) n=0
2
2
L, (25) !66
+
10, 01 20, 02 :
(0) 2 ;Re q1 (0) 2 : (35)
10 = Re I ;Re qq1(0)
2 +Im qq1(0)
2 01 = ImI ;Im qq1(0)
0
q2 (0) 0
2(0) 0
2(0) 0
2
!66
+
1n, n1 , n = 1 : : : %N=2], 2n , n2 , n = 0 : : : %N=2], . P
N , %N=2] = (N ; 1)=2 !66
+
1 2N. P
N , %N=2] = N=2, !66
+
2N=2, N=
2
, , . F!
!66
+
, 2N. ; c0.
F
!66
+
6+
u(x) (29)
, ,
6+
,
(3), (4), (8), (10). "
!66
+
, , 6+
u(x) (29).
1) . u(x) (6), Z
Re (t) d = ;2C:
(36)
;
L
2) , u(x) , %1]
Z
Im(t) d = 0 m = 1 : : : Nk k = 1 2:
(37)
Lkm
3) ; (9), (11), u(cos akm sinakm ) = Q^ 1 (exp (iakm )) m = 1 : : : Nk k = 1 2:
(38)
G
(36) Re (t) (33) U 0 20 + V 0 02 +
X (B0(n)1 + R0(n)1 + S0(n)2 + T 0(n)2 ) = W 0:
#N=2]
n
n=1
n
n
n
(39)
G
(37) Im (t) (34) U(m k)20 + V (m k)02 +
X (B(n m k)1 + R(n m k)1 + S(n m k)2 +
#N=2]
n=1
n
n
n
+ T(n m k)n2 ) = W(m k) m = 1 : : : Nk k = 1 2: (40)
1143
G
(38) (29) ~ k)20 + V~ (m k)02 +
U(m
X (B(n
~ m k)1 + R(n
~ m k) 1 + S(n
~ m k)2 +
#N=2]
n
n=1
~ k)
+ T~(n m k)n2 ) + 2c0 = W(m
n
n
m = 1 : : : Nk k = 1 2: (41)
G
(39), (40), (41) (2N +1) 1
2
2
2
2
(2N +1) (11 : : : 1N=2, 11 : : : N=
2;1 , 0 : : : N=2;1, 0 : : : N=2 ,
1
1
1
1
2
2
c0, N , 1 : : : (N ;1)=2 , 1 : : : (N ;1)=2 , 0 : : : (N ;1)=2 ,
02 : : : (2N ;1)=2, c0, N ). 1 ( n = 1 : : : %N=2]):
(40) (m = 1 : : : Nk , k = 1 2):
B(m n k) = ;
S(m n k) =
Zbkm cos(( N2 ; n))
akm
bkm
Z
akm
bkm
Z
R1(t)
d R(m n k) = ;
Zbkm sin(( N2 ; n))
akm
bkm
R1(t)
Z cos(( N2 ; n))
sin(( N2 ; n))
d
R2 (t) d T (m n k) = ; k
R2(t)
am
W(m k) = (g(t0 )Q1(t0 ) + K1 (t0) + K2 (t0 )) d0 +
+
Z
bkm
akm
akm
N
Re I cos N
2 + ImI sin 2 d
R1(t)
U(m k) =
V (m k) =
sin N2 + R (t) d
2
Zbkm cos l cos N2 + sin l sin N2
R1(t)
akm
bkm
Z ; sin l cos N2 + cos l sin N2
R1(t)
akm
;
cos N
2 d
R2(t)
(41) (m = 1 : : : Nk , k = 1 2):
Z
N
~ n k) = cos(( 2 ; n)) B(m
R1(t)
L
d
(ln j exp (iakm ) ; exp (i)j ; arg(exp (iakm ) ; exp (i))) d
1144
~ n k) =
R(m
Z
L
. . , . . sin(( N2 ; n))
R1(t)
(ln j exp (iakm ) ; exp (i)j ; arg(exp (iakm ) ; exp (i))) d
N
~ n k) = sin(( 2 ; n)) S(m
Z
L
R2(t)
(ln j exp (iakm ) ; exp (i)j + arg(exp (iakm ) ; exp (i))) d
cos(( N2 ; n))
T~(m n k) =
R (t)
Z
L
2
(; ln j exp (iakm ) ; exp (i)j ; arg(exp (iakm ) ; exp (i))) d
W~ (m k) = 2Q^ 1(exp (iakm )) +
Z
+ (;g(t)Q2 (t) ; K1 (t) + K2 (t)) ln j exp (iakm ) ; exp (i)j d +
ZL
+ (g(t)Q1 (t) + K1 (t) + K2 (t)) arg(exp (iakm ) ; exp (i)) d +
+
ZL
L
N
Re I cos N
2 + Im I sin 2 R1(t)
(; ln j exp (iakm ) ; exp (i)j + arg(exp (iakm ) ; exp (i))) d
~ k) =
U(m
Z
L
N
cos l cos N
2 + sin l sin 2 R1(t)
(; ln j exp (iakm ) ; exp (i)j + arg(exp (iakm ) ; exp (i))) d +
+
Z
L
sin N
k
k
2
R2(t) (ln j exp (iam ) ; exp (i) j + arg(exp (iam ) ; exp (i))) d
V~ (m k) =
Z
L
sin l cos N2 ; cos l sin N
2 R1(t)
(ln j exp (iakm ) ; exp (i)j ; arg(exp (iakm ) ; exp (i))) d +
+
Z
L
cos N2
k
k
R2(t) (; ln j exp (iam ) ; exp (i)j ; arg(exp (iam ) ; exp (i))) d
(39):
B 0 (n) =
X2 XN B(m n k)
k=1 m=1
R0 (n) =
X2 XN R(m n k)
k=1 m=1
S 0 (n) = ;
W0 =
+
Z
X2 XN S(m n k)
k=1 m=1
T 0 (n) = ;
R1(t)
d
Z cos l cos N2 + sin l sin N2
L
V0 =
k=1 m=1
(g(t0 )Q2 (t0 ) + K1 (t0) ; K2 (t0 )) d0 ; 2C +
Z L Re I cos N2 + Im I sin N2
U0 =
X2 XN T(m n k)
1145
R1(t)
;
Z ; sin l cos N2 + cos l sin N2
L
L
F arg(exp (iakm ) ; exp (i))
R1(t)
sin N
2 d
R2(t)
N
2
+ cos
R (t) d:
2
6
! 6+
,
L. N+
f1(t), f2 (t) 1
(26), 6+
R1(t), R2(t) | (22), (23), g(t) = ;1 , 6+
K1 (t), K2 (t)
(30), (31). ,,
1 Z f (t)R (t) exp i N + d +
I = ; 2
1
1
2 2
L
N Z
1
+ 2 g(t)f2 (t)R2(t) exp i 2 + l d
L
N
1 X
N2
1
X
q1(0) = eil l =
1
1
2
2
q2(0)
4 n=1(bn ; an) + n=1(an ; bn ) :
2. (2N + 1) (39){(41) (2N + 1) (11 : : : 1N=2, 11 : : : (1N=2);1 ,
2 , c0 , N !, 11 : : : 1
20 : : : 2(N=2);1, 02 : : : N=
(N ;1)=2 ,
2
1
1
2
2
2
2
1 : : : (N ;1)=2, 0 : : : (N ;1)=2 , 0 : : : (N ;1)=2 , c0, N ! ) t = ei .
1 ,. F
, (39){(41) ~ kn, ~nk , c~0 . F
!
6 (33), (34), (29), , M 1 Z (; Re ~(t) ln r(x ) + Im ~(t)!(x )) d + c~ u~(x) = 2
(42)
0
L
1146
. . , . . ,
N 1 N #N=
2] X
1
1
Re ~(t0 ) = ; R (t )
~ cos 0 2 ; n + ~n sin 0 2 ; n +
1 0 n=0 n
#N=
X2] ;~2 sin 0 N ; n
+ R 1(t )
n
2
2 0 n=0
+ ~n2 cos 0 N2 ; n
#N=
X2] ~1 cos N ; n
Im ~(t0) = ; R 1(t )
0 2
1 0 n=0 n
+ ~n1 sin 0 N2 ; n
;
X2] ;~2 sin 0 N ; n + ~2 cos 0 N ; n :
1 #N=
n
n
R2(t0 ) n=0
2
2
\$
6 ,
6
,
+
,, +
%4], @~u + @~u ;
@~u + @~u ;
;
=
;
Re
~
(t
)
0
@r
@r L
@ ; @ L = Im ~(t0):
C , , M u~(x) 0:
(43)
" Re ~(t0 ) = 0, Im ~(t0 ) = 0, t0 2 L ;
X2] ~1 cos 0 N ;n + ~1 sin 0 N ;n 0 t0 = ei0 2 L
1 #N=
n
R1(t0 ) n=0 n
2
2
X2] ;~2 sin 0 N ;n + ~2 cos 0 N ;n 0 t0 = ei0 2 L:
1 #N=
n
n
R2(t0 ) n=0
2
2
G
6 (32), P 1(t0 ) =
XN D~ 1 tn 0
n=0
n0
P 2 (t0 ) =
XN D~ 2 tn 0
n=0
n0
t0 2 L
, D~ nk = ~ kn + i~nk , k = 1 2, n = 0 : : : % N2 ]. Q , !66
+
~ kn, ~nk , k = 1 2, . ;, (42), (43)
, c~0 = 0. D, , (39){(41)
. 4
, N, (39){(41) .
\$ c0 !66
+
kn, nk (11 : : : 1N=2 ,
1
1
1
2
2
2
2
1
1 : : :N=
2;1 , 0 : : :N=2;1, 0 : : :N=2 , c0 , N , 1 : : : (N ;1)=2 ,
1147
11 : : : (1N ;1)=2, 20 : : : 2(N ;1)=2 , 02 : : : (2N ;1)=2 , c0, N ) (39){(41), , 2. !66
+
10 , 01 6 (35). F
!
(29), (33), (34). ; 6+
u(x) (29) . 2 , u(x) HL0 M.
2. # M \$ ! % (29),
(33), (34), &%% kn , nk , c0 ' (39){(41), 2, 10, 01 (' % (35).
4
, M (1) + L
c = ;3=4. Q 6 %3].
. M (N1 = 1, N2 = 0 N1 = 0, N2 = 1) %5].
5. ' (I) F N1 = 1, N2 = 0, C = 0. ;, L = L11 = fr = 1 2 (a b)g q1(z) = ;i exp(;i'1 )(z ; exp(ia))1=4(z ; exp(ib))3=4 '1 = a +8 3b q2(z) = ;i exp(;i'2 )(z ; exp(ia))3=4(z ; exp(ib))1=4 '2 = 3a 8+ b ;a 1=4 ;b 3=4
;a 3=4 ;b 1=4 (44)
R1(t) = 2 sin 2 sin 2 R2(t) = 2 sin 2 sin 2 l = b ;4 a :
L
(39){(40) :
U 020 + V 0 02 = W 0 U(1 1)20 + V (1 1)02 = W (1 1)
(45)
,
Z
Z Re I cos 2 + ImI sin 2
W 0 = fQ2(t) + K1 (t) ; K2 (t)g d +
d
R1 (t)
L
Z
L
W(1 1) = fQ1(t) + K1 (t) + K2 (t)g d +
Z
Re I cos 2 + Im I sin 2
d
R1(t)
Z L sin 2
L
Z
cos cos l + sin sin l
U 0 = ; R (t) + 2 R (t) 2
2
1
L
cos ; cos 2 sin l + sin 2 cos l
V 0 = R (t)2 +
R1(t)
2
L
d
d
1148
U(1 1) =
V (1 1) =
Z sin 2
. . , . . cos 2 cos l + sin 2 sin l d
+
R2(t)
R1(t)
Z ; cos 2
L
R2(t) +
; cos 2 sin l + sin 2 cos l
d
R1(t)
L
Z
Z
1
1
Re I = 2 f1 (t)R1 (t) sin 2 d + 2 f2 (t)R2(t) cos 2 + l d
LZ
LZ
1
1
ImI = ; 2 f1 (t)R1 (t) cos 2 d + 2 f2 (t)R2 (t) sin 2 + l d
L
K1 (t),
L
K2 (t) (30), (31), 6+
f1 (t), f2 (t)
(26). ,, t = exp(i), t0 = exp(i0 ).
D ,6 7 (I), Z exp(i 2 ) p
Rk (t) d = 2 exp(i'k ) k = 1 2:
L
B , , ,, U 0 , V 0 , U(1 1), V (1 1). F , , ,
, '1 ; l = '2, p
U 0 = 2(; sin '2 + cos '1 cos l + sin '1 sin l) = ;2 sin '2 ; 4 p
V 0 = 2(cos '2 ; cos '1 sin l + sin '1 cos l) = 2 cos '2 ; 4 p
U(1 1) = 2(sin '2 + cos '1 cos l + sin '1 sin l) = 2 cos '2 ; 4 p
V (1 1) = 2(; cos '2 ; cos '1 sin l + sin '1 cos l) = 2 sin '2 ; 4 :
" . 1 !, ,
Z fj (t)qj+ (t) dt 1 Z fj (t)Rj (t)
K~ j (t0) = t+0
t ; t0 t = 4Rj (t0 ) sin ;20 d j = 1 2: (46)
2qj (t0 )
L
L
L, ,6 7 (I)
Z ~
jZ
(
;
1)
Kj (t0 ) d0 = 2
fj (t) d j = 1 2:
L
L
(f1 (t)
(47)
G
Q2;j (t) =
+ (;1)j f2 (t))=2, (26), , , W 0 (j = 1) W(1 1) (j = 2) Z
L
(Q2;j (t) + K1 (t) + (;1)j +1K2 (t)) d =
+
(;1)j +1 ; f2 (t)
2 + K2 (t)
Z f1(t)
1149
2 + K1 (t) +
ZL
d = (TK1(t0 ) + (;1)j +1 TK2(t0 )) d0 L
, (47) 6+
Z fj (t)qj+ (t) t20 t0 TKj (t0) Kj (t0) ; K~ j (t0 ) = +1
t ; t0 1 + t2 ; 2 t dt =
4qj (t0 )
=
1
L
Z
4qj+ (t0 )
L
fj (t)qj+ (t)
t2 (t ; t0 ) dt j = 1 2:
;, W 0, W(1 1) Z
Z f (t)q+ (t) Z t ; t0 Z fj (t)qj+ (t)
dt dt =
TKj (t0 ) d0 = j 4itj2
4t J(t) d: (48)
t0 qj+ (t0) 0
L
L
L
L
4 Z dt0 Z dt0
2
J(t) = t
; +
=
(exp(i) exp(;i'k ) ; exp(i'k )) =
t0qk+ (t0 )
qk (t0 ) 1 + i(;1)k+1
L
L
4i
1
=
2
= 1 + i(;1)k+1 t sin 2 ; 'k k = 1 2:
(F
J(t) 6 ,, . 7 (III, II).) D +, J(t) 6 (48), Z
Z f (t)q+ (t) TKj (t0 ) d0 = i j t j t1=2 sin 2 ; 'j 1 + i(;1 1)j +1 d =
L
L
Z
1
= ; p fj (t)Rj (t) sin 2 ; 'j d j = 1 2:
2L
D , Z
W 0 = f2 (t)R2(t) ; cos 2 sin '2 ; 4 + sin 2 cos '2 ; 4 d
ZL
W(1 1) = f2 (t)R2(t) cos 2 cos '2 ; 4 + sin 2 sin '2 ; 4 d:
L
; , , 1 Z f (t)R (t) cos d 2 = 1 Z f (t)R (t) sin d
20 = 2
2
2
2
2
0 2
2
2
L
L
1150
. . , . . (45). D 6 (35) 1 Z f (t)R (t) sin d 1 = ; 1 Z f (t)R (t) cos d:
10 = 2
1
1
0
2
2 1 1
2
L
L
F ,
, Z
1
0
0
1
1
0 cos 2 + 0 sin 2 = 2 f1 (t)R1 (t) sin ;2 0 d:
L
F +
(33), (34):
1 0 1 0 1
K1 (t0 ) + R (t ) 0 cos 2 + 0 sin 2 =
1 0Z
f
(t)R
(t)
1
;
1
1
0
2
= 4R (t )
cos( ; 0 ) + 2 sin 2 d =
sin ;20
1 0
LZ
1(t) d = K~ (t )
= 4R1 (t ) f1 (t)R
1 0
sin ;20
1 0
L
1
0
0
2
2
K2 (t0 ) + R (t ) ;0 sin 2 + 0 cos 2 = K~ 2 (t0):
2 0
D, N1 = 1, N2 = 0, C = 0 M 6 (29), , 6+
Re (t0 ), Im (t0 ) Re (t0 ) = ;Q2 (t0 ) ; K~ 1 (t0) + K~ 2 (t0 ) =
Z
1(t) d +
= ;Q2 (t0 ) ; 4R1 (t ) (Q1(t) + Q;2 (t))R
0
sin 2
1 0
a
b
Z
2(t) d
+ 4R1 (t ) (Q2(t) ; Q;1 (t))R
0
sin
2 0
2
a
Im(t0) = ;Q1 (t0) ; K~ 1 (t0 ) ; K~ 2 (t0) =
b
Z (Q1(t) + Q2(t))R1(t)
1
= ;Q1 (t0 ) ; 4R (t )
d ;
sin ;20
1 0
(49)
b
; 4R1 (t )
2 0
Z
b
a
(Q2(t) ; Q1 (t))R2(t) d:
sin ;20
a
N+
R1 (t), R2(t) (44), c0 (41):
1 Z (Re (t) ln j exp (ia) ; exp (i)j ;
c0 = Q^ 1 (exp(ia)) + 2
L
; Im (t) arg(exp (ia) ; exp (i))) d: (50)
1151
Q %5].
(II) F N1 = 0, N2 = 1, C = 0, L = L21 = fr = 1 2 (a b)g. U , ,
(I), , . F u(x) | . 5 (I). 6+
1 Z (Im(t) ln r(z t) + Re (t)!(z t)) d + c v(x) = ImH(z) = ; 2
(51)
1
L
, Re (t), Im (t) (49). N+
v(x) u(x) {
@v @u = ;r @v :
r @u
=
@r @ @
@r
/, , v(x) , ,
,
. G
{
, , 6+
v(x) ,
:
@v = ;Q (t) @v = Q (t):
1
2
@r L+
@ L;
\$ 6+
v(x) : ;Q1 (t)
Q2 (t), Q2 (t) Q1 (t). F 6+
V (x). N+
V (x) 6 (51),
Re (t0 ) = ;Q1(t0 ) +
Z
1 Z (Q1(t) + Q2(t))R2 (t) d
1 (t) d +
+ 4R1 (t ) (Q2 (t) ; Q;1(t))R
4R2(t0)
sin 20
sin ;20
1 0
a
a
Im(t0 ) = Q2 (t0) +
b
b
Z
1 Z (Q1(t) + Q2(t))R2 (t) d:
1 (t) d ;
+ 4R1 (t ) (Q2 (t) ; Q;1(t))R
4R2(t0)
sin 20
sin ;20
1 0
b
a
b
a
N+
V (x) ,
, v:
@V = Q (t) @V = Q (t):
2
1
@r L+
@ L;
c1 , V jL; = Q^ 1 (t). ;,
c1 = Q^ 1 (exp(ia)) +
1 Z (Im(t) ln j exp (ia) ; exp (i)j + Re (t) arg(exp(ia) ; exp(i))) d:
+ 2
L
1152
. . , . . F 6+
V (x) M .
(III) F N1 = 1, N2 = 0, Q^ 1 (t) = Q2(t) = 0, t 2 L, C 6= 0. \$ !
qj (z), Rj (t), 'j l , . (I). L
(39){(40)
U 020 + V 002 = ;2C U(1 1)20 + V (1 1)02 = 0
(52)
, U 0, V 0 , U(1 1), V (1 1) (I). " (35) 20 = C sin '2 ; 4 02 = ;C cos '2 ; 4 10 = ;C sin '1 ; 4 01 = C cos '1 ; 4 :
D, M N1 = 1, N2 = 0, Q^ 1 (t) = Q2(t) = 0,
t 2 L, C 6= 0 6 (29), , 6+
Re (t0), Im(t0 ) Re (t0 ) = ;C R 1(t ) sin 20 ; '1 + 4 + R 1(t ) cos 20 ; '2 + 4 2 0
11 0 0
Im(t0 ) = C ; R (t ) sin 2 ; '1 + 4 + R 1(t ) cos 20 ; '2 + 4 1 0
2 0
6+
R1(t) R2(t) (44), c0 | (50).
6. !"#
)* L
(39){(41) . 1
!, 6
(9), (11) I(z).
F (an bn), , n = 1 : : : N, | + ,, ,
Ln = fr = 1 2 (an bn)g n = 1 : : : N L =
N
n=1
Ln :
"
, L , L~ .
F t = ei 2 L~ . F
, , m, 1 6 m 6 N ; 1,
u(cos am sin am ) = Q^ 1 (exp(iam )):
(53)
;, (8) (10) u(cos bm sinbm ) = Q^ 1 (exp(ibm )). F! u(cos am+1 sin am+1 ) = Q^ 1(exp(iam+1 )) (53)
1153
!
Z
am+1
bm
@u d = Q^ (exp(iam+1 )) ; Q^ (exp(ibm )):
1
1
@
G
, (@u=@)jL~ = Re%iI]jL~ = ; ImIjL~ , (9), (11)
u(cos a1 sin a1) = Q^ 1(exp(ia1 ))
(54)
Z
am+1
bm
ImI(t) d = Q^ 1(exp(ibm )) ; Q^ 1(exp(iam+1 )) m = 1 : : : N ; 1: (55)
L ! 6
+
, (9), (11)
, (36), (37), , kn, nk .
F qk (t) = tN=2 Rk (t), k = 1 2, t 2 L~ , (27) Z f1(t0)R1(t0)tN=0 2 t N +1 1
I(t) = ;i 4R (t)tN=2 exp i 4
1+ t
dt0 +
t0 ; t
1
0
Z
+ exp i 4
t N +1 L1
N=
2
f1 (t0 )R1(t0 )t0 1 +
t0 ; t
t0
dt0 +
Z f2(t0)R2(t0)tN=0 2 t N +1 1
+ i 4R (t)tN=2 exp(;i 4 )
1+ t
dt0 ;
t0 ; t
0
2
L2
Z
; exp ;i 4
2 t N +1 f2 (t0 )R2(t0 )tN=
0
1+ t
dt0 +
t0 ; t
0
L1
L2
#
N=
;i 4 ) 2] 1
+ exp(
R1(t) n=0 n cos
exp(i ) #N=2]
+ R (t)4
;2n sin
2
n=0
= exp ;i 4 K11 (t) + exp
exp(;i ) #N=2] 1
+ R (t)4
n cos
1
n=0
) #N=2]
2
4
+ exp(i
R2(t) n=0 ;n sin
X
X
X
X
N 2 ;n
+ n1 sin N2 ; n
+
2 ;n
+ n sin 2 ; n
+
N 2 N 2 ; n + n cos 2 ; n =
2
;i K1 (t) + exp i K21 (t) ; exp i K22 (t) +
4N 1 4 N 4
N 2 ;n
+ n2 cos N2 ; n :
1154
. . , . . G
(55) I(t), t 2 L~ 2 + V~ (m) 2 +
~
U(m)
0
0
X (B(n
~ m)1 + R(n
~ m) 1 +
#N=2]
n
n=1
n
~ m)2n + T(n
~ m)n2 ) = W~ (m) m = 1 : : : N ; 1: (56)
+ S(n
\$ (56) ( n = 1 : : : %N=2]):
~ n) = ;
B(m
Z
am+1
bm
am+1
Z
cos(( N2 ; n))
~ n) = ;
d R(m
R1(t)
Z
am+1
bm
m
+1
a
sin(( N2 ; n))
R1 (t) d
Z cos(( N2 ; n))
sin(( N2 ; n))
~
d
R2(t) d T(m n) = m
R2(t)
m
b
p b
~
W(m)
= 2(Q^ 1 (exp(ibm )) ; Q^ 1 (exp(iam+1 ))) +
~ n) = ;
S(m
+
Z
Z
am+1
am+1
bm
bm
~ =
U(m)
V~ (m) =
(K1 (t) ; K2 (t)) d +
N
Re I cos N
2 + ImI sin 2 d
R1(t)
Z cos l cos N2 + sin l sin N2
am+1
bm
am+1
R1(t)
sin N
2
; R (t) d
2
Z ; sin l cos N2 + cos l sin N2
bm
R1(t)
cos N + R (t)2 d m = 1 : : : N ; 1:
2
G
(39), (40), (56) 2N 1
2
2
2N kn, nk (11 : : : 1N=2, 11 : : : N=
2;1 , 0 : : : N=2;1,
2 , N , 11: : :1
1
1
2
2
02 : : : N=
2
(N ;1)=2 , 1 : : :(N ;1)=2 , 0 : : :(N ;1)=2 ,
2
2
0 : : : (N ;1)=2, N ). c0 ! , (54):
Z
1
1
^
c0 = Q1(exp(ia )) + 2 (Re (t) ln j exp (ia1 ) ; exp (i)j ;
L
; Im(t) arg(exp (ia1 ) ; exp (i))) d: (57)
L
.
3. (39), (40), (56)
& (39){(41) .
/ 3 2 M.
1155
3. # M ! % (29), (33), (34), (57), &%% kn, nk ' (39), (40), (56), 3, 10, 01 (' % (35).
7. )
\$ ! ,, . 5 (I).
F, N1 = 1, N2 = 0, L = L11 = fr = 1 2 (a b)g, t = ei , t0 = ei0 .
(I) ,
Z
Hj = K~ j (t0 ) d0 j = 1 2
L
, 6+
Kj (t0) (46).
F Z ~
Z t0 Z fj (t)qj+ (t) dt dt0
Kj (t0 ) d0 =
t ; t0 t it0 =
2qj+ (t0 )
L
,
L
L
Zb
= Ij (t)ifj (t)qj+ (t) d j = 1 2 (58)
a
dt0
1 Z
Ij (t) = 2i
j = 1 2:
qj+ (t0 )(t ; t0)
L
D, Ij (t) 6+
, ,. \$ , 6+
Vj (z), , :
1 Z
dt0
Vj (z) = 2i
z 2= L j = 1 2:
qj+ (t0)(t0 ; z)
L
4
, 6 L+,{F 6+
Vj L 6
Z dt0
1
1
t 2 L:
Vj (t) = + + 2i +
2qj (t)
qj (t0)(t0 ; t)
L
F! , Ij (t) , 6+
Vj (t) 6
Ij (t) = ; 12 %V+j (t) + V;j (t)] t 2 L:
F
6+
Vj (z). 1 !, L
W, ,
L, lR , R + , 1156
. . , . . , ,
W lR , 1 Z
dt0
d
1 1 Z
=
+
+
2i qj (t0 )(t0 ; z) 2i qj ()( ; z) qj (z)
&
lR
(59)
, , 6+
z (,
, qj (z) L). F
! dt
Z 1
1 Z
d
1
1
0 = (1 + (;1)j +1 i)V (z)
;
j
2i qj ()( ; z) = 2i
qj+ (t0 ) qj; (t0 ) t0 ; z
&
L
, lR (59) R ! 1, Vj (z) = 1 + (;11)j +1 i q 1(z) :
j
"
1 1 ! 1
1
1 + (;1)j i = (;1)j +1 :
Ij (t) = ; 2 + + ;
=
; +1
j
+1
qj (t) qj (t) 1 + (;1) i
2qj (t) 1 + (;1)j +1i 2qj+ (t)
F ! (58), (;1)j Z f (t) d j = 1 2:
Hj = 2
j
b
(II) ,
Jk =
a
Z
exp(i 2 ) d k = 1 2:
Rk (t)
L
L , Jk , :
Z dt exp((;1)k+1i 4 ) Z d
k
+1
Jk = exp (;1) i
= 1 + i(;1)k+1
4
qk () qk+ (t)
&
L
, W | , L L. N+
1=qk (z)
W lR ,
+ ! . F Z d Z d
qk () + qk () = 0:
"
Z
L
&
lR
dt = ;
1
1 + (;1)k+1 i
qk+ (t)
Jk =
Z
d
2 exp(i'k )
qk () = 1 + i(;1)k+1 lR
2 exp(i'k )
p
k = 1 2:
(III) ,
Ik =
Z
L
1157
exp(;i 2 )
Rk (t) d k = 1 2:
F Ik Jk , p
Ik = 2 exp(;i'k ) k = 1 2:
4
, Z dt Ik = exp (;1)k+1 i 4
tqk+ (t)
L
Ik , . D 6
Z dt 2 exp(;i'k )
=
k = 1 2:
tqk+ (t) 1 + i(;1)k+1
L
Q . 5.
1] . . { ! // #. . | 1990. |
'. 2, * 9. | +. 114{123.
2] . . . . ./0 0 // #. . | 1990. | '. 2,
* 4. | +. 143{154.
3] #02 3. 4. +5./ 5/ .. | #.: 3
, 1968.
4] . . +. . 9
// ;<# #=. | 1991. | '. 31, * 1. | +. 109{121.
5] +5 . 4., . . +2. . . >
5 ! // <
#
. -. +. 3. =
. .. |
2000. | * 6. | +. 27{30.
6] . 3. @., . ., #/2 . A. B
/ 5 B
, !C 5 ! // =. . . | 1996. | '. 2, /. 1. | +. 147{160.
7] . +., . 3. @., . . +2. 5 . // =. . . | 2000. | '. 6, /. 4. | +. 1061{1073.
8] . . D 2 E0 . . 5 5 . // E3 +++. | 1992. | '. 325, * 3. | +. 428{433.
9] Krutitskii P. A. An initial-boundary value problem for the pseudo-hyperbolic equation of gravity-gyroscopic waves // Journal of Mathematics of Kyoto University. |
1997. | Vol. 37, no. 2. | P. 343{365.
1158
. . , . . 10] Krutitskii P. A. An explicit solution of the pseudo-hyperbolic initial problem in a multiply connected region // Mathematical Methods in the Applied Sciences. | 1995. |
Vol. 18, no. 11. | P. 897{925.
11] . . +
B
5 ./0 B
5 // ;<# #=. | 1990. | '. 30,
* 11. | +. 1689{1702.
' ( 2001 .
R (r)
1F1(;a cz) (n < 10, l < 4)
nl
. . e-mail: mumfordd@mail.ru
511.3+512.62+530.145.61
: , (") ""\$ "%, % &\$'"( )" *%+,, +", -+ *.{", 0, 1( 23, \$+%. 4"44 (, +" 2. . 25+.
6" \$, +"\$, " , %,\$' . ( ""\$ , "
a 6;1547 % "'+ *,5 ' \$.5 (a > 4) , 5 \$' , )\$. (' +"\$"'+*8
10 ). ")", \$+" +", 0, 1( 23, 4.,
4 9 %. 1 F1 (;47 c7 z) = 0, 8+ % '%"( "'"% (
. %'5 +""%) % &\$'"( )" *%+, FEL-+4. . &" *9(
"+\$'"( ."'+ (
4 a > 3) '4 . ( xk = zk ; (c + a ; 1) "' y = 0 %4%, %%".+'. ;"%,< ,
' '" Ra = (a; 1)pc + a ; 1. 1+" 4")%" .+ )++* "+", "'"&"'+ '4 . =+5 ( 5 ;"&)"%< | +"\$ Tk "3"'+. .
' \$% a = 3 a = 4 4" \$, +"\$, ; "%,< '4+"+ +"\$ Tk 4
2 6 c < 1. 6 %,\$' ( ""\$ %,.% , ;"'"&,<
' \$ (ac) = (4 6) (6 4) (8 14): : :.
Abstract
V. F. Tarasov, Zeroes of Schrodinger's radial function Rnl (r) and Kummer's
function 1 F1 (;a7 c7 z) (n < 10, l < 4), Fundamentalnaya i prikladnaya matematika,
vol. 8 (2002), no. 4, pp. 1159{1178.
Exact formulae for calculation of zeroes of Kummer's polynomials at a 6 4 are
given7 in other cases (a > 4) their numerical values (to within 10;15) are given.
It is shown that the methods of L. Ferrari, L. Euler and J.-L. Lagrange that are
used for solving the equation 1 F1 (;47 c7 z) = 0 are based on one (common for all
methods) equation of cubic resolvent of FEL-type. For greater geometrical clarity of
(nonuniform for a > 3) distribution of zeroes xk = zk p; (c + a; 1) on the axis y = 0
the ;circular< diagrams with the radius Ra = (a ; 1) c + a ; 1 are introduced for
the Crst time. It allows to notice some singularities of distribution of these zeroes
and their ;images<, i. e. the points Tk on the circle. Exact ;angle< asymptotics
of the points Tk for 2 6 c < 1 for the cases a = 3 and a = 4 are obtained.
While calculating zeroes xk of the Rnl (r) and 1 F1 functions, the ;singular< cases
(ac) = (4 6) (6 4) (8 14):: : are found.
, 2002, +" 8, D 4, '. 1159{1178.
c 2002 !
"#\$,
%& '( )
1160
. . x 1. .
. . 1]
2], H- !
!
Rnl(r) = Anl e;z=2 z l 1 F1(;a% c% z)
(1)
Anl | !' , n > 1, l = 0 n ; 1, a = n ; l ; 1 > 0,
c = 2l + 2 > 2, z = 2r, = Z=n > 0 | )* !
, Z > 1 |
,
a
X
(;a)k z k |
F
(
;
a%
c%
z)
=
(20 )
1 1
(c) (1)
k=1
k
k
!
'
!!
3] (
),
(c)k =
= ;(c + k)=;(c), (c)0 = 1, ! ./!!
). 1'
, '
0 6 Rnl (0) < 1 Rnl (+1) = 0. 2 * ) 2. 3. 4
n < 8 l < 4 4], . . a < 7 = 2 4 6 8.
8!, (20) (;1)a (c)a , '
! )* !'
)! ('
9:!) ;
!
a
X
Kac (z) = (;1)k Cak (c + a ; k)k z a;k (200)
k=1
= a!=(k!(a ; k)!) | !)* ;
. , (1) !'
(2) 9, !
, ) !,
, x = 0 x = 1.
= !
* (
)/) ! )/ '
, * !'
!!
.
.
! -!
/'
/ '
/ >? 2,4], >!
*? / !, )9
'
) * ,
.
= * 9 !) )'
* !'
!!
, / ' a 6 4, ! )/ !
(@. A{3. , C. D
, C. E*
F.-C. C,) 1,5,6]%
)/ '/ (a > 4) 9 '
;/ * ( '9 10;15). 2 I
* !
'
* (
a > 3) * zk = xk + (c + a ; 1) !'
!!
( py = 0) )
>
? !!) !
Ra = (a ; 1) c + a ; 1. E !
)
* xk / > ? | '
Tk , ,:/ ,, ,
' '
a = 3 a = 4 / >)
?
! c ! 1. 2 ,
, ' !
) D
, E*
C, I
1 F1(;4% c% z) = 0 !
9 * ( / !
) '
* ) FEL-.
k
a
1161
3
1,5,7{9], ' xk 4- ) ! k '
* ) * B4 A4 (A4 | '
9:
, jB4j = 4, jA4 j = 12), e | ' ,
1
2
1
2
1
2
1 = (14)(23)
2 = (13)(24)
3 = (12)(34)
:
4
3
4
3
4
3
1'
, ' k2 = e , 1 2 = 2 1 = 3 . .
E
!
) tk (k = 1 2 3) !
C, (!. . 3.3) ) ! ! lk mk / ! 10]:
1
2
1
2
l4 = (41)(42)(43)
m4 = (12)(13)(23)(4)
4
3
4
3
1
2
1
2
m1 = (23)(24)(34)(1)
l1 = (12)(13)(14)
4
3
4
3
1
2
1
2
l2 = (21)(23)(24)
m2 = (13)(14)(34)(2)
4
3
4
3
1
2
1
2
l3 = (31)(32)(34)
m3 = (12)(14)(24)(3)
:
4
3
4
3
. )'
* !'
!!
)
) > )
? '
(a c) = (4 6) (6 4) (814) (10 30) :: :. O
3, 6] *
!'
!!
!
) / A. Kienast (1921), A. Erdelyi (1938), W. C. Taylor (1939), S. U. V
(1941), F. Tricomi (1947) .
=
)
'
), !)/ '/.
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
x 2. 1F1(;a c z) a = 1 2 3
1 . . K1c (z) = z ; c = 0
e z1 = c. X
! !
z = x+c, !
! x = 0, . .
x1 = 0. S
!
'
; >,? ! R1 = 0. E!
'9 * (nl) = 20 31 42 53 :: :
(. . 2s 3p 4d 5f :: :)% / )
Pnl (r) = rRnl (r) ) . 1.
2 . . 2
Y
K2c(z) = z 2 ; 2(c + 1)z + (c)2 = 0
(z ; zk ) = 0:
p
k=1
U z12 = (c + 1) c + 1.
1162
. . G'. 1
X
! !
z = x + (c + 1), !
! 2
Y
x2 ; (c + 1) = 0
(x ; xk ) = 0%
p
k=1
p
xk = c + 1 = R2 cos 'k , R2 = c + 1 | ,, '1 = '2 = 0. S
!
'
; '
, ' xk !
9
> )? | ' Tk , ,:
( !
!
) !
;* ,. E! '9 * (nl) = 3s 4p 5d 6f : ::% / )
Pnl (r) ) . 2.
G'. 2
3 . . K3c(z) = z 3 ; 3(c + 2)z 2 + 3(c + 1)2 z ; (c)3 = 0
3
Y
(z ; zk ) = 0:
k=1
X
! !
z = x + (c + 2), !
! x3 + px + q = 0
3
Y
1163
(x ; xk ) = 0
k=1
p=3 = q=2 = ;(c+2), ! D = (q=2)2 +(p=3)3 = ;(c+1)2(c+2) < 0.
. !
=
!
! 1,5]
0 = x1 + x2 + x3 p = x1 x2 + x1x3 + x2 x3 < 0 ;q = x1x2x3 > 0:
3 !
A{ 1,5{8], I
I
! p
p
x = 3 C1 + 3 C2 (3)
p
pC12 = ;q=2 i ;D = r(cos i sin ) =p r exp(i)% p r =
=2 , cos = ;q=(2r) = 1= c + 2, sin =
= p (q=2)2 ; D = (c + 2)3p
;D=r =
p
= (c + 1)=(c + 2), tg = c + 1, arctg 3 6 arctg < =2 2 6 c < 1.
19 )
xk = R3 ch(i'k ) = R3 cos 'k p
R3 = 2 p3 r = 2 c + 2 | ,, 'k = ( + 2k)=3 (k = 1 2 3).
S
!
'
; '
, ' xk ( y = 0) !
9 > )? |
' TP
, ! R3 '
120 , )/
k , ,:
P
P
xk = cos 'k = sin 'k = 0. .'
! ' Tk >
? ;*
, !
!
c ( R3). @*
! / >9?
!, !
, ' 3 c ! 1: 1) c = 2, R3 = 4 '3 = 20 ,... % 2) c = 34, R3 = 12 '3 26801977,.. . , , ),
' 3 !
>
/99? , '^3 ! 30 ; 0. E! '9 * (nl) = 4s 5p 6d 7f : : :% / )
Pnl (r) ) . 3, xk ) 'k ) 1.
G'. 3
1164
. . -& 1
2
4
6
8
) ' = 3
x1 '1
x2 '2
x3 '3
;3064177772475911
;
0694592710667721 3758770483143634
140
260
20
;3858783723282275
;0684482873823212
4543266597105489
141968
261968
21968
;4523604490519944 ;0679753773246365 5203358263766308
143098
263098
23098
;5107249542250522
;0677010086744995
5784259628995517
143855
263855
23855
x 3. 1F1(;4 c z)
@ , !, 7]: > (5) ),9 '
;
) !: .
4) ! )) 9:/ ! / '
* , ! ,
'*,
; !
*
)
;
)?.
3.0. FEL-
. K4c (z) = z 4 ; 4(c + 3)z 3 + 6(c + 2)2z 2 ; 4(c + 1)3z + (c)4 = 0
4
Y
(z ; zk ) = 0:
(4)
k=1
X
! !
z = x + (c + 3), !
! 4
Y
x4 + px2 + qx + r = 0
(x ; xk ) = 0
k=1
(5)
p = ;6(c + 3), q = ;8(c + 3), r = 3(c + 1)(c + 3), ! 5] D4 =
= 16p4r ; 4p3q2 ; 128p2r2 +144pq2r ; 27q4 +256r3 = 21033(c+1)3 (c+2)2 (c+3) > 0.
. !
=
!
! 1,5]
0 = x1 + x2 + x3 + x4 p = x1 x2 + x1x3 + x1 x4 + x2x3 + x2x4 < 0
;q = x1x2 x3 + x1x2x4 + x1 x3x4 + x2 x3x4 > 0 r = x1x2x3 x4 > 0: (6)
Z
! I (4) (5) ! !, !
)
D
(1522{1565), E*
(1707{1783) C, (1736{1813), 1165
-& 2a
2
4
6
8
) ) FEL- (7)
w1 '1
w2 '2
w3 '3
;17119558864499 ;7570643011497 24690201875996
132587081302345 252587081302345 12587081302345
;25189503825942 ;10472902340380 35662406166322
133400988628857 253400988628857 13400988628857
;33224811663315 ;13388992759957 46613804423272
133803207369757 253803207369757 13803207369757
;41246163835753 ;16310175136243 57556338971996
134043471587122 254043471587122 14043471587122
)/, )
, , >
I9:
? | '
* ) FEL-.
1. 3
Y
w3 + p^w + q^ = 0
(w ; wk ) = 0
(7)
k=1
p^=3 = ;8(c+2)2 , q^=2 = ;16(c+2)2 (c+3), D3 = ;D4 =108 < 0.
wk = Rw cos 'k (8)
p
, 'k = ( + 2k)=3, tg =
pRw = 4 2(c + 2)2 | p
= (c + 1)=(c + 3) < 1, arctg 06 6 < =4 2 6 c < 1 (. . 4).
. )I
,
p3 p3 C
I
+
C2, (7) I
!
w
=
1
p
C12 = ;q^=2 i ;D3 = rpexp(i)%
r
=
(8(c
+
2)2 )3=2.
p
3
A
p !
!: Rw = 2 rp= 4 2(c + 2)2 , cos =
= (c + 3)=(2c + 4), sin = (c + 1)=(2c + 4).
S
!
'
; '
, ' wk ( y = 0) !
9 > )? | ' Tk , ,:
,
Rw '
120, )/
P w = P cos '!
P
G'. 4a
sin 'k = 0. .'
! ' Tk
k
k =
>
? ;* , !
!
c ( Rw ). @*
! / >9?p !, !
,
' 3 c ! 1: 1)
p c = 2, Rw = 8 10 '3 12587081,.. . %
2) c = 34, Rw = 24 74 '3 14734707 ,. . ., , ), ' 3
!
>
/99? , '^3 ! 15 ; 0. wk ) 'k ( ,*
' Tk ) 9 2.
1166
. . 3.1. 1
2 B4
1 . X' !! 4- z 4 + a1z 3 + a2z 2 + a3z + a4 = 0
4
Y
(z ; zk ) = 0
k=1
(9)
a1 a3 < 0, a2 a4 > 0. .
! 2 z 2 + 21 a1z = 14 a21 ; a2 z 2 ; a3 z ; a4:
= !
t 6= 0, ! ! '! ; ),
(z 2 + (1=2)a1z)t + (1=4)t2, !
!
2 1
1
1
1
1
2
2
2
2
z + 2 a1z + 2 t = 4 a1 ; a2 + t z + 2 a1t ; a3 z + 4 t ; a4 : (10)
, | )* /'
, ,! ! )! 9, . .
p p
Az 2 + Bz + C = (z A + C)2 (mz + n)2
p
B = 2 AC > 0. 19 !
t '
! 1
2 1
1
2
2
2 a1 t ; a3 ; 4 4 a1 ; a2 + t 4 t ; a4 = 0
!, 3
Y
t3 ; a2t2 + (a1 a3 ; 4a4)t ; (a23 + a4 (a21 ; 4a2 )) = 0
(t ; tk ) = 0: (11)
k=1
E | '
* ) !
D
/
(9). . t0 | * tk , (10) 2
z 2 + 21 a1 z + 12 t0 = (mz + n)2 p
p
m = A = ((1=4)a21 ; a2 + t0 )1=2, n = C = ((1=4)t20 ; a4)1=2. X ),
'
' !
z 2 + 12 a1 z + 12 t0 = (mz + n)
(12)
I
* /! ) *, )
*
1 2 B4 .
2 . 3/ (4), '! (11), . .
3
Y
t3 ; 6(c+2)2 t2 +12(c+1)4 t ; 8(c+1)3(c+3)(c2 +6c+4) = 0
(t ; tk ) = 0: (110)
k=1
1167
U !
t = w + 2(c + 2)2 > 0, '
! '
* ) FEL- (7). =) !
t0 t1 = w1 + 2(c + 2)2 ,
'
! !
1
2
z 2 ; 2(c + 3)z ; 21 t0 = (mz + n)
(120)
4
3
m = (w1 + 4(c + 3))1=2 , n = ((1=4)w12 + (c + 2)2 w1 + 2(2c + 3)(c + 2)2)1=2 .
1. .!
! !
D
!'
K42(z) = z 4 ; 20z 3 + 120z 2 ; 240z + 120 = 0:
[!
z = x + 5 4
Y
x4 ; 30x2 ; 40x + 45 = 0
(x ; xk ) = 0
r
r
r
r
k=1
x3 + x4 = ;(x1 + x2), D4 = 2143453 = 165888000. A (110)
!
3
Y
t3 ; 120t2 + 4320t ; 4800 = 0
(t ; tk ) = 0:
k=1
[!
t = w + 40 FEL-
3
Y
w3 ; 480w ; 3200 = 0
(w ; wk ) = 0
k=1
D3 = ;1536000. U ) wk = Rw cos 'k , w1 ;17119558%
t0 t1 22880442, m 16971864, n 32982809. 2
, I !
z 2 ; 10z + 21 t0 = (mz + n)
/! :
z41 5848593 51053012 > 0 z32 41514069 15797699 > 0:
. t2 t3 9 )
!).
A! !, ' (4) !
! * (nl) = 50 61 72 : :: (. . 5s 6p 7d : : :)% / )
Pnl (r)
) . 4, x = zk ; ( + 3) = R4 cos !k pc + 3, ) !k )
4.
R4 = 3
k
3.2. k
2 B4
1 . X' (9) ! !
z = x ; a1=4, !
!
x4 + px2 + qx + r = 0
(90 )
1168
. . G'. 4
p = a2 ; (3=8)a21, q = a3 + (1=8)a31 ; a1 a2, r = a4 ; (3=256)a41 + (1=16)a21a2 ;
; (1=4)a1a3 . =
! (
C. E*
) 9 !
9 t > 0:
p
p
x4 + px2 + qx + r = (x2 ; x t + )(x2 + x t + )
(900)
, | )
!
). .
!, ;
) )/ / xi, '
! !
p
fp = + ; t q = ( ; ) t r = g:
3 )/ / /! !
):
2 = p + t + pq 2 = p + t ; pq :
t
t
. / 9 , !
!
2
4r = (p + t)2 ; qt > 0
' !, -& 4
) ' = 4
x1 '1
x2 '2
x3 '3
x4 '4
;2428364992353722
0731178751689099
5953894312683190
2 ;4256708072018568
129386
248776
276257
27432
;2734394134343176
1057940683138001
6920931804019684
4 ;5244478352814508
131356
249848
277659
29313
;3
1334110346786258
7755103283846041
6 ;6089213630632299
132576
250528
278524
30494
1169
3
;t 0
q Y
t3 + 2pt2 + (p2 ; 4r)t ; q2 = 0 1 p + t = 0
(t ; tk ) = 0: (13)
q p + t 4r k=1
E | '
* ) !
E*
/ (90 ). U !
t = w ; 2p=3 > 0, !
!
1
3
Y
2
8
3
2
2
3
w ; 3 p + 4r w ; q + 27 p ; 3 pr = 0
(w ; wk ) = 0:
(14)
k=1
E (,
) I
!
A{. A, wk , tk = wk ; 2p=3 > 0, /! !
)
0 0 , t0 tk . X ), '
' !
p
p
fx2 ; x t0 + 0 = 0 x2 + x t0 + 0 = 0g
I
* /! ) *, )
!
k 2 B4 :
p 1 01=2
1
:
(15)
xk = 2 t0 4 t0 ; 0
2 . 3/ (5), '
! (13), . .
3
Y
t3 ; 12(c + 3)t2 + 24(c + 3)2 t ; 64(c + 3)2 = 0
(t ; tk ) = 0:
(130)
k=1
U !
t = w + 4(c + 3) > 0, (14) ! '
* ) FEL-
P (7). tk Q= wk +4(c+3) > 0 !
) 0, 0 ) 3a, tk = 12(c+3)
tk = 64(c + 3)2 , 00 = r = 3(c + 1)(c + 3).
2. 3/ !
1, !
! !
E*
!'
p
p
x4 ; 30x2 ; 40x + 45 = (x2 ; x t + )(x2 + x t + )
p
p
2 = (t ; 30) ; 40= t, 2 = (t ; 30) + 40= t, 180 = (t ; 30)2 ; 1600=t > 0,
. . 00 = 45 9 tk . A (15) '
! :
1
2
x 1 = 08485932 51053009
15797723
4
3
1
2
x2 = 17627645 41911296
24939434
4
3
1
2
x3 = 33425365 26113578
09141719
4
3
P
1'
, ' xk = 0 8k.
r
r
r
r
r
r
r
r
r
r
r
r
1170
. . -& 3a
2
4
6
8
.!
) !
E*
t1 1 1
t2 2 2
t3 3 3
28804411355013 124293569885028 446902018759959
;253439899807440 ;144582285170938 43533610112365
;17755688837547 ;31124144944035 103368408647594
28104961740578 175270976596204 636624061663219
;362966770274867 ;189245553291006 73219353206961
;28928267984556 ;55483470112791 143404708456258
27751883366846 226110072400434 826138044232720
;472224806229567 ;232653098515381 103461635313751
;40023310403587 ;81236829084185 182676408918969
27538361642471 276898248637568 1015563389719960
;581375932274353 ;275167488709014 134120143956616
;51085706083175 ;107934262653417 221443245763345
3.3. ! " lk mk
k
2 B4
1 . = ; !
, ;
!
5]:
t1 = ;(x1 + x4)(x2 + x3) = (x1 + x4)2 (1)
t2 = ;(x1 + x3)(x2 + x4) = (x2 + x4)2 (2)
t3 = ;(x1 + x2)(x3 + x4) = (x3 + x4)2 (3)
P
xk = 0. E ;
!
) ! '
3
Y
t3 + b1t2 + b2 t ; b3 = 0
(t ; tk ) = 0
k=1
(16)
;
) bi | ;
!
)
!!
'
tk ,
!
X
X
Y
b1 = tk = 2p b2 = ti tj = p2 ; 4r b3 = tk = q2:
A (13) , ), (14). 1'
) *:
1) t3 ; t1 = (x3 ; x1 )(x4 ; x2) > 0, t3 ; t2 = (x3 ; x2)(x4 ; x1) > 0,
t2 ; t1 = (x2 ; xp1 )(x4 ; x3) > 0%p
2) x1 +x4p= t1 , x2 +x3 = ; t1 , x2 +x4 = pt2, x1 +x3 = ;pt2, x3 +x4 = pt3 ,
x1 + x2 = ; t3.
19 9 I
1171
-& 3b
2
4
6
8
.!
) !
C,
L1
L2
L3
03374306346167 07009375679670 13291107319411
02768800610416 06914410923248 13177763448781
02404504599308 06863400402382 13119149840919
02154434939364 06831631597718 13083316644413
p
p
p
p
p
p
2x1 = t1 ; t2 ; t3 < 0 2x2 = t2 ; t1 ; t3 < 0
(17)
p p p
p p p
2x3 = t3 ; t1 ; t2 > 0 2x4 = t1 + t2 + t3 > 0:
A! !, !) (17) I
/ (5) (90).
(16), ,)* ;
!
tk * k 2 B4 , ) ! lk mk ( / !).
2 . X !! (5), (7) (130). 1 '! r
p
p
Lk = cos 'k + cos (c) = 2RRw = 13 4 2cc ++34 4
p4
p
4
(2=3) 01 6 (c) < (1=3) 2 2 6 < 1 ( 03748942 6 < 03964023). A !) (17) ! xk = R4 cos !k (170)
cos !1 = (L1 ; L2 ; L3 ) cos !2 = (L2 ; L1 ; L3 )
cos !3 = (L3 ; L1 ; L2 ) cos !4 = (L1 + L2 + L3 ):
X ), !
C, p)' ) !k ,* ' Tk , ! R4 = 3 c + 3 (!. . 4). \
)
'
Lk ) 3b% ; )
, )' >)
?
'
Tk 2 6 c < 1: !^k ; !k 1172 % 655 % 807 % 1148.
3.4. \$
1 F1 (;4& 6& z )
) 4, :
'* (a c) = (4 6),
!'
!!
!
(z2 = 6 x2 = ;3), . .
!
!
z 4 ; 36z 3 + 432z 2 ; 2016z + 3024 = (z ; 6)(z 3 ; 30z 2 + 252z ; 504) = 0
x4 ; 54x2 ; 72x + 189 = (x + 3)(x3 ; 3x2 ; 45x + 63) = 0:
(18)
1172
. . -& 2b
) (19)
t1 '1
t2 '2
t3 '3
;
7089213630632
0334110346786
6755103283846
6 152393585260486 272393585260486 32393585260486
O!! ; '* . X
! * !
x = t + 1, !
!
0 = x3 ; 3x2 ; 45x + 63 7! t3 ; 48t + 16 = 0:
(19)
.
'
!
! p=3 = ;16, q=2 = 8, D3 = ;4032. A I
!, (,
) (3), . . !
!
p
p
t = 3 C1 + 3 C2
p
3 = 43, cos =
C12 = r exp(i), r = ;(p=3)
p
p
= ; q=(2r)
p = 0 ;1=8,p3 sin = ;D3 =r
= 63=8,
tg = ; 63, R3 = 2 r = 8, . . = 97180756
G'. 4b
'k = ( + 2k)=3 (k = 1 2 3). 19 tk = R03 cos 'k !
9 > )? | ' Tk P,
!
R03 = 8
P
P
'
120 (!. 2b . 4b). [
tk = cos 'k = sin 'k = 0.
@ ;* , >!
!? ' T20 (. . ' x2 = ;3
t02 = ;4), R03 cos '02 = ;4, '02 = 240.
=: /! 9 (18), !) ,) !
,, . . R03 7! R4 = 9, xk )' !
(170) !
): '1 7! !1 , '02 7! !2, '2 7! !3 ,
'3 7! !4 .
A! !, *
, ' Tk , !
:9 , ), xk ,
( !* y = 0).
x 4. 1F1(;a c z) a = 5 8
%
&
'() &* (a c) = (6 4) (8 14)
5, c. 223], > :
n-* n > 4 I! /?. .;! )
' a > 5 )'9 '
( '9 10;15). 1)
, ' / > )
? ', !
(a c) = (6 4) (8 14) (10 30) :: :. [
) !'
!!
a = 5 8, ! )! / !
I
'
/
* 13{21].
1173
-& 5
Xk
X1
X2
X3
X4
X5
) ' = 5
c=2
!k
c=4
;5382969146721729 123325 ;6509445054813172
;3887034041421476 246626 ;4418666187096746
;1389166848982466 261849 ;1373003703176392
2399066971204843 284173 2944418003467669
8260103065920831 32537 9356696941618640
!k
125124
247010
263029
285085
34205
1 . X'* 1 F1(;5% c% z) = 0 z 5 ; 5(c + 4)z 4 + 10(c + 3)2z 3 ; 10(c + 2)3 z 2 + 5(c + 1)4z ; (c)5 = 0
5
Y
(z ; zk ) = 0
k=1
!
) z = x + (c + 4) x5 ; 10(c + 4)x3 ; 20(c + 4)x2 + 15(c + 2)(c + 4)x + 4(c + 4)(5c + 14) = 0
5
Y
(x ; xk ) = 0:
k=1
p
) xk = R5 cos !k , R5 = 4 c + 4 | , '! Tk % / '
)
'
) !k ) 5. E * (nl) = 60 71 : : : (. . 6s 7p : : :)% / )
Pnl (r) ) . 5.
G'. 5
1174
. . 2 . X'* 1 F1(;6% c% z) = 0 z 6 ; 6(c + 5)z 5 + 15(c + 4)2 z 4 ; 20(c + 3)3z 3 +
+ 15(c + 2)4 z 2 ; 6(c + 1)5z + (c)6 = 0
6
Y
(z ; zk ) = 0
k=1
!
) z = x + (c + 5) x6 ; 15(c + 5)x4 ; 40(c + 5)x3 + 45(c + 3)(c + 5)x2 +
+ 24(c + 5)(5c + 19)x ; 5(c + 5)(3c2 + 4c ; 31) = 0
6
Y
(x ; xk ) = 0:
k=1
(c = 2)
(c = 4)
,p
/'
() !
* /
'
>'?
c0 = (;2 + 97)=3 / >?
= 26162859. 3!
;* '
p
' 4 . ) xk = R6 cos !k , R6 = 5 c + 5 | , '! Tk % / '
)
'
) !k ) 6. E
* (nl) = 70 81 : : : (. . 7s 8p : : :)%
/ )
Pnl (r) ) . 6.
G'. 6
3 . X'* 1 F1(;7% c% z) = 0 z 7 ; 7(c + 6)z 6 + 21(c + 5)2 z 5 ; 35(c + 4)3z 4 +
+ 35(c + 3)4z 3 ; 21(c + 2)5 z 2 + 7(c + 1)6z ; (c)7 = 0
7
Y
(z ; zk ) = 0:
k=1
1175
-& 6
Xk
X1
X2
X3
X4
X5
X6
) ' = 6
c=2
!k
c=4
;6472331878288871 119292 ;7703580796555735
;5203700190356591 246836 ;5906001619244266
;3123358479523088 256343 ;3338714779588498
;0081183433295277 269648 0167097269514240
4234610429083116 288669 4941345374521041
1064596355238071 36412 1183985455135321
!k
120902
246812
257139
270638
289233
37877
U )
xk = zk ; (c + 6) = R7 cos !k p
R7 = 6 c + 6 | , '! Tk % / '
)
'
) !k ) 7. E *
(nl) = 80 91 : : : (. . 8s 9p : : :)% / )
Pnl (r) ) . 7.
G'. 7
4 . X'* 1 F1(;8% c% z) = 0 z 8 ; 8(c + 7)z 7 + 28(c + 6)2 z 6 ; 56(c + 5)3z 5 + 70(c + 4)4z 4 ;
; 56(c + 3)5z 3 + 28(c + 2)6z 2 ; 8(c + 1)7z + (c)8 = 0
8
Y
(z ; zk ) = 0:
k=1
U )
xk = zk ; (c + 7) = R8 cos !k (c = 2)
(c = 14)
1176
. . -& 7
Xk
X1
X2
X3
X4
X5
X6
X7
) ' = 7
c=2
!k
c=4
;7538975780195004 116374 ;8852299909125201
;6436413810345738 247711 ;7272916485939195
;4647949497463266 254104 ;5044531279324631
;2083702750979461 262947 ;2063517391427633
1420699383021589 274802 1845514252547855
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Abstract
S. V. Tikhonov, On relation of measure-theoretic and special properties of
Zpp.-actions
, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 4,
1179{1192.
d
It is shown how using the -mixing property one can construct 7nite measure-preserving d-actions possessing di8erent and even unusual properties. In the case of
a -classical time. this approach was applied by Lemanczik and del Junco as an alternative to the so-called Rudolf's -counterexamples machine., based on the notion
of joining.
Z
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1979 6] !" # \$% & . ( %
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h fhs gs2Z
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d- 1181
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(
d
)
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2. & h Zd + (E q), E 2 B, q 2 Nd, hi E \ hj E = ?, i j , 0 6 i j < q. F E ",
%
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; n v =
=
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% \$ & , \$ % , % ) . >
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m1 , m2, k1, k2 ,
1 (v2 :::vd vd nd ) ,-) 0 6 mj < n, 0 6 kj < n
v
j = 1 2 (m1 k1) 6= (m2 k2). .
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>
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N-+ ) p = kn + m (v
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,
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1,
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d d
d
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d- 1183
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v
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2
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4
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Q(s j) 1184
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1185
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t
P
4
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i
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2
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> K | ) 8 (M B ),
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& UT (- , &
1] 5]). > K(d) | + & & && Zd. A, % K(d) , .
, G - .
3. <
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, (k1 k2 : : : kn) ;(k1 k2 : : : kn)
& (ke1 ke2 : : : kene ).
T
d
d
> G = Gm .
m>0
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1. % m 2 N Gmd .
. > , % T 2 G 1, , v > 2m + 1
T v 2 Gmd , v = (1 v : : :vd;1 ).
1186
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T hk1 vi : : : T hkn vi ? T hke 1 vi : : : T hke ne vi (hk1 vi : : : hkn vi) ;(hk1 vi : : : hkn vi) & (hke1 vi : : : hkene vi). < , % \$ , % (k1 k2 : : : kn ), ;(k1 k2 : : : kn ) , & (ke1 ek2 : : : ekne ). <% , % ,)
k ke, ) ) , m,
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)
+
:
:
:
+
v
((k1 ; e
1
e
jkd ; kd j = 6
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d
;
2
2m(1 + v + : : : + v
(v ; 1)(1 + v + : : : + vd;2 ) )
=1
6
<
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vd;1
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( ?
%
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d;1
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e
e jkd;1 ; ekd;1 j = ((k1 ; k1 ) + : : : + v (kvd;d;22; kd;2)) + v (kd ; kd ) 6
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d
d
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Z
d- 1187
> Rmnn~ | (k1 : : : knH ke1 : : : kene ), ) Gmd + hk1 : : :
hkn
hke 1 : : : hke ne ( \$) d-) ) , m (k1 : : : kn) ;(k1 : : : kn) ,
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. \ \
\
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N N
n~ 2 n2 k2Rmnn~
, % Gmd G , ,-) &:
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n + n~ \$ K) H
fT 1 T 2 : : : T n S 1 S 2 : : : S n~ j T 1 : : : T n ? S 1 : : : S n~ g
G - Kn+~n 4, 1.4].
-), & % ,% (2.1) ,
K(d) , 1, +& T 2 G 1 V =
= f(1 v v2 : : : vd;1 )gv>2m+1. 4
, Gmd K(d).
4. < T S 2 K , /, (% T ? S), ) -
, ,- .
3. &
H0 fh 2 K(d) j 8a 2 Zdnf0g : ha ? h;a g H.
. \
A, %
H0 =
Z
a2 dnf0g
fh 2 K(d) j 8b 2 Zn f0g : hab ? h;ab g:
(2.2)
<&
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% ) H0 .
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1188
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> C(T) | ),-) & ( )), ,-) T 2 K. 7 T .
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b
d
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. F W1 W ,
% , U 2 K U W U ;1 = W 4].
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Wd =
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Z
a2 dnf0g
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& \$ .
Z
d- 1189
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# :
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i 2 E . E , % LE UQ
hk , %+ & & Uhk jLE + (hk(i) ).
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k(i) 2 fv ;vg. . , L02(X ) Lfig.
i : k(i)2fv;vg
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1190
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4
4 Wd = fh 2 K(d) j 8a 2 Zd n f0g : C(ha ) =
= clfhb j b 2 Zdgg .
6. , h 2 Wd \ V.
(1) k : I ! Zd n f0g v: J ! ZdNn f0g. *
) Q,
# hk hv , ( S) S(j) 2 clfhb j b 2 Zdg +, : J ! I, hv(j ) = hk(
(j )) .
(2) k: I ! Zd n f0g v: J ! Zd n f0g. 1 : I ! I 2 : J ! J | ,. 1 2 hk hv ,
*
) Q, # 1 hk N
v
2h , ( S) S(j) 2 clfhb j b 2 Zdg +,
: J ! I, 1 = 2 hv(j ) = hk(
(j )).
(3) k: I ! Zd n f0g 1 : I ! I | , . 1
hk , ,
1hk N
( S) , S(i) 2 clfhb j bN2 Zdg, : I ! I | +, , 1 hk , ( S) | 1 . - , I , ,
hk .
. A, % 2 "
) Q, 1 3 , ) % .
(1) Q;1(Bj ). 4
5 - i(j), %
;
1
Q (Bj ) Bi k(i(j)) = v(j). <
h 2 Wd ?
C(hk(i)) & ) &. \$ %, 4], hk(i) , Bi . 4
, Q;1 (Bj ) = Bi . >
(j) = i. > ) j -
Q;1 (Bj ) ,, | / ?. (, % 1 .
(2) > Q 1 hk 2 hv . . Q (1 hk )f
(2 hv )f ) ) f. > 1 2 | % , f , % f1 f2 . N
.
Q hfk hfv . 4
1, \$ % Q ( S)
& / ? : J ! I , S(j) 2 clfhb j b 2 Zdg. E ?, , 1 hk 2hv . <
Z
d- 1191
i = (j) O
;1
2 hv ( S)(I I : : : Bi I : : :) = (I I : : : B
2 (j ) I : : :)H
& ,
O
;1
( S)1 hk (I I : : : Bi I : : :) = (I I : : : B
;1 1 (j ) I : : :)
N
1 = 2. A, %, -
;11;1 (j), ( S)1 hk (I I : : :Bi I;1 : : :) (I I : : : B
2 (j ) I : : :). > 2 .
(3) E , % \$ &
?
hk . ;
N 1 - .
E , %
N
N
( S) N 1. O 1 hk ( S) = ( S)1hk . . hk ( N
S), 1, N hk ,, \$
N
N
1 ( S)hk = ( S) 1 hk , 1 ( S) = ( S) 1.
x
4. Zd- &
) % -& % .
. < Zd-& + /, , /, ) \$
, ,- \$
Zd.
1. < - && Zd, /, , - . > h 2 Wd \ V . & h h B1 B2 : h B1 h2 B2 , -, , %) B1 B2 . 4 & , & h h2 /, , -
B1 B2 . 4 & , - -
F B2 g 2 Zd n f0g, % (h2 )g jF = hg hg jF
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; a & ge : egf hf 1i (hf hf : : : hf ). >, % b -. <
\$ % , % ai (hai hai : : : hai ) N
bi . 4
6, & ( S) , S(j) 2 clfhf j f 2 Zdg, bi = ai . > ai bi , ) bi \$
H %, ) \$
. E bi | %. A%,
& g b.
'
1] Stepin A. M. Spectral properties of generic dinamical systems // Math. USSR
Izvestiya. | 1987. | Vol. 29. | P. 159{192.
2] Furstenberg H. Disjointness in ergodic theory, minimal sets and a problem in diophantine approximation // Math. Syst. Th. 1. | 1967. | P. 1{49.
3] Del Junco A. Disjointness of measure-preserving transformations, minimal self-joinings and cathegory // Ergodic Theory and Dynamical systems I. Progress in Math.
10. | Boston: Birkhauser, 1981. | P. 81{89.
4] Lemanczyk M., Del Junco A. Generic spectral properties of measure-preserving maps,
and applications // Proc. Amer. Math. Soc. | 1992. | Vol. 115, no. 3. | P. 725{736.
5] , , !". #\$%&'( )(. | *.: +,', 1980.
6] Rudolph D. J. An example of a measure-preserving map with minimal self-joinings
and applications // J. Anal. Math. | 1979. | Vol. 35. | P. 97{122.
7] Sinai Ja. G. On weak isomorphism of transformations with invariant measure // Mat.
Sbornik. | 1963. | Vol. 63. | P. 23{42.
8] -\$. /. +. / &0(1&) \$,00.\$ &).( &.", 2)", // *).
3")'. | 1989. | 4. 45, 5 3. | . 3{11.
9] Katznelson Y., Weiss B. Commuting measure preserving transformations // Israel
J. Math. | 1972. | Vol. 12. | P. 16{173.
10] Conze J. P. Entropie d`un groupe abelien des transformations // Z. Wahrscheinlichkeitstheorie Verw. Geb. | 1973. | B. 26. | S. 11{30.
11] Glasner E., King J. L. A zero-one law for dynamical properties // Contemporary
Mathematics. | 1998. | Vol. 215. | P. 231{242.
' ( ) 2002 .
. . , . e-mail: tishchen@kapella.gpi.ru
519.172.2+519.173+519.177
: , , !" , #\$ , \$% .
& ##'' \$% # "%( ' ', )*' '. & !+ \$%', , # )*
'- "%(.- #'#(. / #%., # 0% .%(
0! 1!%#( \$0'# \$'%(#( \$%., .%(. 2##' "%( \$
0 \$%#! .!%0! \$% '( #\$ 0# . 3 #!%(! !%##,#! \$' | Kn , Kmn ,
0' 2 | \$0'# #! 1!#( #\$%( #\$ 0% !" \$% .
Abstract
S. A. Tishchenko, Separators in planar graphs as a new characterization tool,
Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 4, pp. 1193{1214.
We consider planar graphs with non-negatively weighted vertices, edges, and
faces. We let vertices and edges have nonnegative costs. In the case of triangular
graphs with equal weights, the obtained results are proved to be equivalent and
optimal. The analysis of planar graphs with non-negativelyweighted faces for a given
plane embedding enables the separator search in dual graphs. We demonstrate
e;cient planar graph characterization by the separator method on several classical
examples: graphs Kn and Kmn , graphs of diameter 2.
1. . 1]. !
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1 (R. J. Lipton, R. E. Tarjan 2]). G
rP , w : V (G) ! R+,
w(X) =
w(v), X V . V (G) A, B
v2X
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maxfw(A) w(B)g 6 2w(G)
3 :
1 &( "
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1196
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X
v2V (X )
. . wV (v) +
X
e2E (X )
wE (e) +
X
f 2F (X )
wF (f) X GT :
(2)
C( "( " ". C( " !, &
( @\$ 1
GT 1 : cV : V (GT ) ! R+, cE : E(GT ) ! R+,
X
X
c(C) =
cV (v) +
cE (e) C GT :
(3)
v2V (C )
e2E (C )
0 \$, 1
c w (.
2 (R. J. Lipton, R. E. Tarjan, . . ). GT ,
!. " GT # # ! GS cmax . \$ % - , # e !!#! GS , !#% GT A, B , c(C) 6 cmax + cE (e)
(4)
;wmin + P wE (x) + 2wF (Fe )
w(C)
x=Ee
(5)
minfw(A) w(B)g > w(G)
3 ; 2 ;
6
wmin | C ' Fe | , ! e,
Ee | , ( Fe. ) Fe (! ! fA B g' w(A) = w(B) .
. -"!
, "(& 3, . \$ 2], "
( GS 1
GT . -
@\$ E(GS ) E(GT ) n E(GS ) ( @(
1 . - GS | jV (GS )j = jV (GT )j > 3, \$
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Ce . GT
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GT ) ( "! Ae Be , "!(
!
Ce . * 1
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cmax + cE (e). 9"
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(6)
g(e) def
= maxfw(Ae) w(Be )g + w(C
T
S
2
8>
w(Ae ) > w(Be )
<jF (Ae)j
def
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e 2 E(GT n GS ) (7)
:jF (Be)j
w(Ae ) < w(Be )
1197
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h. K \$,
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Cx ,
, @\$
GS . F
"!
Ax Bx \$
(5), . -* ""!
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, w(G) > 3w(Bx ) + 3w(Cx) ; wV (s) + wE (x) +2 wE (y) + wE (z) + 2wF (Fx ) : (8)
\$ x 1
3- 1
GT . 3-,
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Cx, \$ y z, 1
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. 1 a ). M
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&
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v, (
@\$
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2.1. *! fy z g .
. 9 "
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1
Cy , \$( \$
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Ay By , "!( !
1
Cy (
. 1 a ). ! 1
Cz , \$( \$
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Az Bz , "!( !
1
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, w(By ) = w(Bx ) + w(Az ) + wE (x) + wE (z) + w(Pv ) ; wV (s) + wF (Fx) (9)
w(Bz ) = w(Bx ) + w(Ay ) + wE (x) + wE (y) + w(Pu ) ; wV (s) + wF (Fx ) (10)
w(Ax ) = w(Ay ) + w(Az ) + wE (y) + wE (z) + w(Pt) ; wV (s) + wF (Fx ) (11)
w(Cx ) = w(Pu) + w(Pv ) + wE (x) ; wV (s)
(12)
w(Cy ) = w(Pu) + w(Pt ) + wE (y) ; wV (s)
(13)
w(Cz ) = w(Pt) + w(Pv ) + wE (z) ; wV (s):
(14)
;" (8) w(G) = w(Ax ) + w(Bx ) + w(Cx )
(15)
"
1198
. . 2#. 1. =!% Cx 3-( Fx . 2g(x) = 2w(Ax ) + w(Cx ) > w(Ax ) + 2w(Bx ) +
+ 3w(Cx) ; wV (s) + wE (x) +2 wE (y) + wE (z) + 2wF (Fx ) : (16)
9\$O
(9){(11), w(By ) + w(Bz ) =
= 2w(Bx ) + w(Ax ) + 2wE (x) + w(Pu ) + w(Pv ) ; w(Pt) ; wV (s) + wF (Fx ):
(17)
-
(12){(14) (17), w(Cz )
y)
w(By ) + w(C
2 + w(Bz ) + 2 =
E (y)+wE (z)+2wF (Fx ) : (18)
= w(Ax ) + 2w(Bx ) + 3w(Cx ) ; wV (s)+wE (x)+w
2
1199
9\$O
(16) (18), "
w(Cz )
y)
w(By ) + w(C
(19)
2 + w(Bz ) + 2 < 2g(x) 6 g(y) + g(z):
, \$ \$,
! "!
, w(Ay ) >
> w(By ). ;" (11){(14), * w(Cx )
y)
w(Ay ) + w(C
2 = g(y) > g(x) = w(Ax ) + 2 =
= w(Az ) + w(Cz ) ; wV (s) + wE (x) +2wE (y) + wE (z) + 2wF (Fx) +
w(Cy )
y)
+ w(Ay ) + w(C
(20)
2 > w(Ay ) + 2 :
; (20) , g(y) = g(x). 0 "
1
h (\$
\$ jF(Ay )j > jF (Ax)j, ( "
.
K \$,
"!
y 2 E(GS ). 0 * !(
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, (21)
w(Bx ) = w(Bz ) ; wE (x) ; wF (Fx)
w(Ax ) = w(Az ) + wV (t) + wE (y) + wE (z) + wF (Fx )
(22)
w(Cx ) = w(Cz ) ; wV (t) + wE (x) ; wE (y) ; wE (z):
(23)
;" (8), (15) (21){(23), " (&
s " &
u)
2g(z) > 2g(x) = 2w(Ax ) + w(Cx) > w(Ax ) + 2w(Bx ) +
+ 3w(Cx) ; wV (u) + wE (x) +2 wE (y) + wE (z) + 2wF (Fx) =
= 2w(Bz ) + w(Cz ) + w(Az ) + w(Cx) ; wV (u) ; wE2(x) + wE (y) + wE (z) >
> 2w(Bz ) + w(Cz ):
(24)
-"!
w(Az ) 6 w(Bz ) "
(24). , w(Az ) > w(Bz ). ;" (22) (23), w(Cx )
w(Cz )
z)
w(Az ) + w(C
2 = g(z) > g(x) = w(Ax ) + 2 = w(Az ) + 2 +
+ wE (z) + 2wF (Fx ) > w(A ) + w(Cz ) : (25)
+ wV (t) + wE (x) + wE (y)
z
2
2
1200
. . ; (25) , g(z) = g(x). 0 "
1
h (\$
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2.3. fy z g 2 E(GS ).
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(26)
w(Ax ) = w(Az ) + wE (z) + wF (Fx )
(27)
w(Cx) = w(Cz ) + wV (u) + wE (x) + wE (y) ; wE (z):
(28)
;" (8), (15), (26) (28), " (&
s " &
t)
x)
g(z) > g(x) = w(Ax ) + w(C
2 >
> 2w(Bx ) + w(Cx ) + ;wV (t) + wE (x) + wE2(y) + wE (z) + 2wF (Fx) =
z ) + w(B ) + w(Cz ) ; wV (t) ; wE (z) + wE (x) + wE (y) >
= w(Bz ) + w(C
x
2
2
w(C
)
z
> w(Bz ) + 2 :
(29)
-"!
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(29). , w(Az ) > w(Bz ). ;" (27) (28), z ) = g(z) > g(x) = w(A ) + w(Cx ) =
w(Az ) + w(C
x
2
2
w(C
)
w
(u)
+
w
(x)
+
w
(y) + wE (z) + 2wF (Fx ) >
z
V
E
E
= w(Az ) + 2 +
2
w(C
)
z
> w(Az ) + 2 :
(30)
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1
h (\$
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.
\$, @\$ y z (, fy z g E(GS ) (
. 1 ). 9
,
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6 3w(Bx ) + 3w(Cx) ; wV (s) + wE (x) +2 wE (y) + wE (z) + 2wF (Fx ) (31)
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=
w(B)
+
(35)
2
2
w0(GT ) = w(GT ):
(36)
-* (5) ;wmin + P wE0 (x) + 2wF0 (Fe )
0
0
w (C)
x=Ee
: (37)
minfw0(A) w0(B)g > w (G)
3 ; 2 ;
6
, " C " &
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(40)
0-(, \$ 3- f 2 FfG 1
\$( \$ ! E(G), @ wF (fT ) = 21 (jEf \ E(G)j ; 1)wF (f)
(41)
Ef | ! @\$, 1
( 3-
f.
C( ( \$(
@\$. 0(\$ " .
5. ? " "
!
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2 1
GT . "\$, ~ 9
,
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@\$
1204
. . ~ " G~
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G~ T G,
~
~
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. 3). F
\$ E " C \$ "1 1
~ " C &
v 2 V (G), F~ F (G),
~
F. -* ! \$ \$ ! \$( &
(, . ,
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! , \$
"
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"
"@ , "
.
2#. 3. >\$ C~ .%" G~T 0# G~. @# 0 G
0% \$.#' ' \$.!' )*'. 2* #\$ C~
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. - , ( "( \$
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3.
4. G , !. " G #
1205
# ! G~ S
! G~ T G G~ cmax .
\$ % C G, C~ G~ T , # e~ !!#! G~ S ,
!#% G A, B , c(C) 6 cmax + cE (e)
(42)
e | , e~, w(C) ;
minfw(A) w(B)g > w(G)
3 ; 2 P
~ j ; 1)wV (v)
;wmin + 3w(C2) +
wE (x) + (jE~e \ E(G)
x=EF~
;
(43)
6
wmin | C , C2 | G,
( !, ! ( ( G~ S , F~e F(G~ T ) | , ! e~, E~e | ,
~ , !!
( F~e , v | , ! F (G)
~
# Fe .
~ ( G.
. " G,
~
-
! * G , ( * G. 0 ( 3 ,
1
G~ T . 0
@ "
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\$ E(G)
~
~
E(GT ) n E(G) @( ( 1 . 0 ~ 6 cmax +cE (~e), \$
,
! 2 , 1
C~ G~ T , c(C)
~
~
~
F(GT ) "! A B, "!( !
C~
, P w (x) + 2w (F~ )
;
w
~
+
min
E
F e~
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G
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6
wm | ( , 1
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~ = w(B)
~ ! \$ !@ "! fA
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.
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( @\$ C~ (!, ). 0 ( G~ ! "! F~A F~B , " !,
A~ B~ . 0 3 "1
&
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1206
. . ~ ; wF (F~1) ; wF (F~2 )
wF (F~A ) > wF F(A)]
(45)
2
~ ; wF (F~1) ; wF (F~2)
(46)
wF (F~B ) > wF F(B)]
2
F~1 F~C | ! , 1
( ~ F~2 F~C n F~1. 9\$O
(45) (46), "
\$ C,
~ wF (F~B ) ; wF F(B)]
~ g > ; wF (F~1) ; wF (F~2): (47)
minfwF (F~A) ; wF F (A)]
2
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, ( &
V (C),
~
~
E(C)\E(G), &
, ( ! F~C (
. 3). D
* G ! A B. ! C1 C2 &
, ( ! F~1 F~2 G~ . 9
,
~ + wV (C1) + wV (C2):
w(C) = w(C)
(48)
0 (47) "1 &
G~ ~ w(B) ; w(B)
~ g > ; wV (C1 ) ; wV (C2 ):
(49)
minfw(A) ; w(A)
2
9\$O
(44), (48) (49), " (43). - &
C1 C2 "(& ( @\$ C~ ~ \ E(G)]
~
cE E(C)] = cE E(C)
(50)
" C \$
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S \$ &
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.
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GS & "
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1207
4. !
1
g @\$ ! E(GT ) n E(GS ) \$
". 0: O(n).
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~
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GS & "
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1
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G~ S &
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22] \$, " "\$. "( , "
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O(n lg n) 23].
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! GT . V (G) A,
B , jC j 6 d + 1, N(A) A] \ B = ? 6 minfjAj jB jg > 2jV (G)j ; 3jC j + 1:
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" . GT = G, C !!! G, GS G, , .
. ) (51) " (5) ! &
(.
?
!( ( ( 2 (
@\$.
1208
. . 2.2. ! G GS d
! GT . E(G n C) A, B C , jC j 6 d + 1, N(A) A] \ B = ? 6 minfjAj jB jg > 2jE(G)j ; 3jC j ; 3:
(52)
" . GT = G, C !!! G, GS G, , .
. ) (52) " (5) ! \$.
( ( 2 ( .
2.3. GT GS d. +% C , jC j 6 d + 1, !#% F(GT ) A B , 3 minfjAj jB jg > jF(GT )j ; 1:
(53)
. ) (53) " (5) ! .
5. , (51){(54) ..
. ?
, "
( "
( GT . ;" =, 2jE(GT )j = 3jF (GT )j = 6jV (G)j ; 12:
(54)
0 I \$ 1
C GT "
(
GT ) , "!( A B 1
. 9"
! &
VA = V (A), VB = V (B), @\$ EA = E(A), EB = E(B) FA = F (A), FB = F (B). "
( G0 ,
"( G A &
,
@ @\$
&
1
C. 9
,
jE(G0)j = jEB j + 2jC j
(55)
0
jF(G )j = jFB j + jC j
(56)
jV (G0)j = jVB j + jC j + 1:
(57)
-
(54) " G0, "
(58)
2(jEB j + 2jC j) = 3(jFB j + jC j) = 6(jVB j + jC j + 1) ; 12:
Q
, 2(jEA j + 2jC j) = 3(jFAj + jC j) = 6(jVA j + jC j + 1) ; 12:
(59)
9\$O
(54), (58) (59), "
1209
6 minfjVAj jVB jg ; 2jV (GT )j + 3jC j ; 1 =
= 2 minfjEAj jEB jg ; 32 jE(GT )j + jC j + 1 = 3 minfjFAj jFBjg ; jF(GT )j + 1:
(60)
=
(51){(53) " (60).
( 2.1 ( 1. 0-"(, " " " ( , 1
, -(, ""
(3jC j + 1)=2, ( ", 1 (5) "
.
0 ( &
4 !( .
4.1. G G~ S d ! G~ T G G~ . +%
- C G, !% d + 1 , !#% V (G n C) A B , N(A) A] \ B = ? minfjAj jB jg > jV j3; 1 ; jC21j ; jC2j
(61)
C1 C2 | G, ( ! G~ , !!
( # G~ S .
" . G~ T = G~ , C !!! , G~ S G~ , C # .
. ) (61) " (43) ! &
( V (G).
0 , @
" "
, @\$ ".
4.2. G G~ S d ! G~ T G G~ . +%
C G, jC j 6 d + 1, !#% V (G n C) A B , N(A) A] \ B = ? minfjAj jB jg > min jV j ; j2C j ; 1 jV j3; 1 ; jC21j ; jC2j (62)
C1 C2 | G, ( ! G~ , !!
( # G~ S .
. 0 4.1 , &
-@\$( " C 0 G, ,
\$ d+1 &
@\$, ,
V (G n C 0 ) ( ! A0 B 0 , ,
1210
. . (61). &
! , , &
( V (C 0 ) &
"(
@\$ ! E(C 0 ), \$, \$( ! A = A0 ; C B = B 0 ; C (62). ? * " , !
! fA0 B 0g \$ 1 &
( \$ \$
E(C 0). -* &
"(
\$ E(C 0) !
(\$ \$ V (C 0 ), \$ "! \$&
.
-
* maxfjAj jB jg ; minfjAj jB jg > 1, minfjAj jB jg = minfjA0j jB 0jg > jV j3; 1 ; jC21j ; jC2j
(63)
" &
"(
E(C 0 ) "
! "!
( . F
! maxfjAj jB jg ; minfjAj jB jg 6 1, minfjAj jB jg > jV j ; 2jC j ; 1 :
(64)
0 \$ &
( " (.
? 1
"
( (5) "
"
( "( ( d jV (G)j = 3k + (3d)=2 + 1, ( , " " 1
" ( C 6 d + 1, ,
minfw(A) w(B)g > k.
"
( "
( . 4 k > 0, d = 2 k = 0, d = 3.
? ( ,
"\$( "
@ (
"\$.
8. ( % 1. G( Kn , n > 5, "(.
. 9 "
, "( Kn , n > 5.
9
, jV (Kn )j = n > 3, K1n;1 2 . 0 2.1 , " C, jC j = 3,
\$
,
Kn ( ! A B, "
@
6 minfjAj jB jg > 2jV (G)j ; 3jC j + 1 = 2n ; 8 > 0:
(65)
, ! A B "(. P
, Kn @ \$(
" ( &
, .
% 2. G( Kmn , m > 3, n > 3, "(.
. 9 "
, "( (
Kmn , m > 3, n > 3. 9
, jV (Kmn )j = m + n > 3, ,
Kmn 3. 0 2.1 , \$
", \$ 1
C, 3 6 jC j 6 4, \$
,
Kmn (
! A B, "
@
6 minfjAj jB jg > 2jV (G)j ; 3jC j + 1 > 2m + 2n ; 11 > 0:
(66)
1211
2#. 4. '#" \$'%(# '.% (5). B' \$% # ''%(' 0'' !!# d = 2 3 minfw(A)w(B )g 6 k \$
#\$ 0% C 6 d + 1: ) k = 0, d = 2, jV j = 4D *) k = 1, d = 2, jV j = 7D
) k = 2, d = 2, jV j = 10D ) k > 0, d = 2, jV j = 3k + 4D 0) k = 0, d = 3, jV j = 5
, ! A B "(. 9
, &
( \$O
A B Kmn . (,
" C | \$ ", \$ 1
, "(& ,
4, "* ! ! \$ &
Kmn . , \$ m 6 2, \$ n 6 2, .
% 3. V
&
3- " 2 "(& 6.
. 9 "
, 3-( "(
G, jV (G)j > 8. - D(G) = 2, 2.1 ,
\$ ", \$ 1
C, jC j 6 5, \$
,
G (
! A B, "
@
(67)
minfjAj jB jg > 2jV (G)j ;6 3jC j + 1 > 0:
"* \$ \$,
&
( a1 2 A b1 b2 2 B.
, | ", , 2-"
&
( a1 &
b1 b2. K , " " 2, A = fa1 g,
1212
. . B = fb1 b2g, jC j = 5. G G 1
, \$!@ . 5. 9
, c2c4 2= E(G), " - " a1c3 b1c3 . Q
, c2 b2 2= E(G), " "
b1 c3. -* " c2 c5 "
&
a1 ,
a1c2 2 E(G). Q
" a1c4 2 E(G), deg(a1 ) > 4, .
2#. 5. E G \$' 3
9. *
+&
" Lipton, Tarjan 2] \$, G 1 &(
&
, @\$
. C( ! ",
&
( @\$ 1 . \$( 2] "
\$
: " " ".
C( "
" ( . = " @\$( &
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&
(), \$(( &
( "(. 0 (
( *
"
"
"( .
-
@( "
( \$
*
( 2 @ 2.1 1
"( . -
* " =, -
{>. ! , (
(5) " ( ". ), "( ( \$( \$\$,( "(
24,25]. -* " ! , , .
,!
1213
1] Aho A. V., Hopcroft J. E., Ulmann J. D. The Design and Analysis of Computer
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6] Djidjev H. N. On the problem of partitioning planar graphs // SIAM J. Alg. Discrete
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element meshes // SIAM J. Sci. Comput. | 1998. | Vol. 10, no. 2. | P. 364{386.
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and nearest neighbor graphs // J. ACM. | 1997. | Vol. 44, no. 1. | P. 1{29.
13] Spielman D. A., Teng S.-H. Disk packings and planar separators // 12th Annual
ACM Symposium on Computational Geometry. | 1996. | P. 349{358.
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16] Kuratowski C. Sur le probl.eme des courbes gauches en topologie // Fund. Math. |
1930. | Vol. 15. | P. 271{283.
17] /\$0!'% 1. 2. #%\$ 3'(4 ") !" 5'"', ,"- (6 = 3, D = 3) //
89'& . \$ 5"\$%. +. | 2001. | /. 7, (5. 1. | 1. 159{171.
18] /\$0!'% 1. 2. #%\$ 3'(4 ") !" ,"- &\$ !+" 2 -\$%\$"''4 :4!"4 ;"%+!"\$+\$%4 // 89'& . \$ 5"\$%. +. | 2001. | /. 7, (5. 4. |
1. 1203{1225.
19] Mitrinovic D. S., Pecaric J. E., Volenec V. Recent advances in geometric inequalities. | Kluwer, 1989.
20] Thomassen C. Triangulating a surface with a prescribed graph // JCT B. | 1993. |
Vol. 57. | P. 196{206.
21] Hopcroft J. E., Tarjan R. E. E<cient planarity testing // J. Assoc. Comput. Math. |
1974. | Vol. 21. | P. 549{568.
1214
. . 22] Dreyfus S. E., Wagner R. A. The Steiner problem in graphs // Networks. | 1972. |
Vol. 1. | P. 195{207.
23] Berman P., Ramaiyer V. Improved approximations for the Steiner tree problem //
J. Algorithms. | 1994. | Vol. 17. | P. 381{408.
24] Gilbert J. R., Lipton R. J., Tarjan R. E. A separator theorem for graphs of bounded
genus // J. Algorithms. | 1984. | Vol. 5. | P. 391{407.
25] Alon N., Seymour P., Thomas R. A separator theorem for nonplanar graphs //
J. Amer. Math. Soc. | 1990. | Vol. 3, no. 4. | P. 801{808.
& ' '
2001 .
. . 512.541
: , , p-
, Z -
.
!
, "
Z -
# p-
# \$ p %!# # # .
Abstract
A. R. Chekhlov, On quasi-closed mixed groups, Fundamentalnaya i prikladnaya
matematika, vol. 8 (2002), no. 4, pp. 1215{1224.
We obtain a description of mixed Abelian groups in which the closure in the
Z -adic and p-adic topology for every prime p of any pure subgroup is a direct
summand of the initial group.
A qc- (cs-), Z - p- p ( ) A. , cs- !
.
" qc- | \$ , % (. &1, x 5]* &2, x 74] .). . qc-. . x 2 %
qc-. cs-
/ &3{6]. . &7, x 2] cs-. 3 &2, 74.9] ,
cs- / , . . cs- &2, x 39]. . x 3 %
5 cs-.
. / . 6 A | , 1 tA /* Ap | p-
* p! A = T pnA* A1 = T p! A*
n=1
+! \$ +,,- . 00-01-00876.
, 2002, 8, . 4, . 1215{1224.
c 2002 ,
!"
#\$ %
p
1216
. . E (A) | /! \$
8* rp (A) | p-
, . . 8- A=pA* p (A) | p-
, . . &2, x 16]
p-
* 9(A) | 5 5 p c pA 6= A* A&n] = fa 2 A j na = 0g* o(a) | \$
a* S (A)
(Sp (A)) | 5 5 (
p-
5) ; ) | Z - (
p-)
* H ^ (HpA
A H , A . ; A
p-!
, 5 p-5 . 6 p! A = 0, A p-!
* A | qc-, 5 5 p.
x
1. " / 5 .
1.1. A = L Ai (A = Q Ai), 9(Ai) \ 9(Aj ) = ? i 6= j,
i2I
i2I
L
H | Z - A, H = (H \ Ai )
i2I
(H = Q (H \Ai)). A qc-
(cs-
) i2I
, Ai qc-
(cs-
).
. 6 A = B G, 9(B) \ 9(G) = ?, x = b + g 2 H ,
b 2 B g 2 G, b + H = ;g + H 2 (A=H )1 = 0.
1.2. A | , G | . 1) G;p 2 Sp (A) ( G;p 2 S (A)), G^ 2 Sp (A)
2) p G 2 Sp (A) G;p 2 Sp (A), G 2 S (A) G^ 2 S (A). ! , p! A 2 Sp (A) (
p! A 2 S (A)) p, A1 2 S (A), . . A1 " # A
3) Ap , p! A 2 Sp (A), . . p- .
.
1) 6 pm x = Tg 2 G^ , g = pm z ;
z 2 Gp . "/ G^ = G;q , q
/, z 2 G;q q 6= p. 3 g = pm z =
= bn + qn yn , bn 2 G, yn 2 A. 6 / sn , tn | ! ,
pm sn + qntn = 1, z = sn bn + qn (sn yn ; tn z ). "/ \$ /
n, z 2 G;q . , (A=G;p )q = 0 q 6= p, , G;p 2 Sq (A). =
1). 2) 1).
3) > Ap , p! (Ap ) / D, D B = Ap . 3 A = D C , Ap = D (Ap \ C ), p! A = D p! C ,
1217
p! C = (p! A) \ C . ; Ap \ C , Ap , . "\$ ?
/, p! (Ap ) = 0. ", p! A 2 Sp (A). "/ pm x = a 2 p! A. @ /
n a = pn+m an , an 2 A, bn = x ; pn an 2 A&pm ]. "/
bn+1 ; bn = pn (an ; pan+1 ), Ap n
/
/ fbng1
n=1 5 b 2 Ap . " b ; bn 2 p Ap ,
!
m
!
m
p (Ap ) = 0, p b = 0. A x ; b 2 p A p (x ; b) = a.
1.3. A | G 2 Sp (A). 1) G + B 2 Sp (A) "\$ # B 2 Sp (A), Gp B ,
%
-
B=Bp p-
2) p- (G;p )p # G;p F , F = (Gp )^Ap # p-# Gp # G Z - # Ap
3) A = B D Bp = 0, G \ D 2 Sp (A)
4) Gp = 0 p! A | \$ , G;p \$ 5) %
-
A=Ap p-, G=Gp p-
p! A = 0 Gp Gp = G.
. 1) "/ pnx = g + b, g 2 G, b 2 B, x 2 A. > Gp B , /, g 2= Gp . . p- 8-
B=Bp \$
b \$
z 2 Bp , a 2 B , b = z + pn a.
6 o(z ) = pm , pn+m (x ; a) = pm g = pn+m y y 2 G,
c = g ; pn y 2 Gp B pn (x ; y) = c + b = pnd, d 2 B . 3
y + d 2 G + B pn (y + d) = pnx. . , G 2 S (A), G + B 2 S (A)
B 2 S (A), tG B B=tB .
2) "/ x 2 G;p . > /
n x = gn + pn an ,
gn 2 G, an 2 A. 6 pm x = 0, pm gn = ;pn+m yn 5 yn 2 G.
A gn ; pnyn = zn 2 G&pm ]. " x = zn + pn (yn + an ) 2 (Gp );pAp . >/ , p- q- Ap q 6= p Z -
.
3) @
/, G \ D 2 Sp (G). 6 pn x = g 2 G \ D, x 2 G,
x = b + y, b 2 B , y 2 D. > pnb = 0, b = 0. " x = y 2 G \ D.
> , tB = 0, G 2 S (A), G \ D 2 S (A).
4) "/ x 2 G;p , x = gn + pn an, gn 2 G, an 2 A. > p! A , Aq = 0 q 6= p. "\$
o(x) < 1, o(x) = pm /
m. 3
pm gn = ;pn+m an = pn+m bn, bn 2 G. "/ Gp = 0, gn = pnbn , x 2 (p! A)p = 0. B/
, G 2 S (A) G, A1 | , G^ .
5) . 1) G + Ap 2 Sp (A). "\$ (G + Ap )=Ap = G=Gp | p- . B/
, p! A = 0 Gp / G.
1218
x
. . 2. qc-
" / 5 , ?5 \$ .
1) &'
A qc-
,
"\$ G 2 S (A) DG Z - E ( p- ) "
DA=G | '
E ( p-
'
).
F, G^ =G = (A=G)1 , G^ 2 S (A) / , (A=G)1 2 S (A=G). \$ / , (A=G)1 | . .
2) A | qc-
, A1 | .
= , 0^ = A1 .
3) ("\$ \$ qc-
. qc-
, .
4) )
A qc-
, p G 2 Sp (A) G;p 2 Sp (A).
@
/ \$ L 1.2, . 1).
G5/. "/ T=G = (A=G)q | / 8q6=p
- A=G, p-
. > q-
(A=T )q
8- A=T 5 5 q 6= p. "\$
T 2 Sq (A) q 6= p. > T=G 2 Sp (A=G), G 2 Sp (A), T 2 Sp (A). B/
, T 2 S (A). @, T=G / p- , \$ Tp; = G;p .
F
, G;p 2 S (A), T .
5) ! A DG 2 Sp (A) G;p 2 S (A)E , "* #
A=p! A, p! A 2 S (A).
F
, 0;p = p! A G;p G
A.
6) &'
A qc-
,
tA | , A1 = 0 p DH 2 Sp (A) | \$ (" H = 0)E Hp; 2 Sp (A).
H
tA 1.3, . 2). @
/. 3
p! A = 0;p 2 S (A). "\$, 5), /, p! A = 0. . , Aq = 0 5 5 q 6= p. "/ G 2 Sp (A) Gp 6= 0. G
/, G;p 2 Sp (A). 6 D = (Gp )^Ap , D=Gp | p- (/ tA | ), \$ A=Gp = D=Gp R=Gp D 2 Sp (A). > G=Gp \ D=Gp = 0, R=Gp / , G=Gp R=Gp . @, G;p = D + N , N = G;pR N=Gp G=Gp R=Gp. "\$ /, N 2 Sp (R),
\$ N 2 Sp (A). > 1.3, . 1) /, D + N 2 Sp (A). 3 N=G = p! (R=G) Ap =Gp = D=Gp Rp =Gp, 1219
G=Gp \ Rp =Gp = 0. "/ G 2 Sp (A), (R=G)p = Rp=Gp . 6 Gp |
, 8- Ap =Gp &2, 74.5]. "\$ 1.2, . 3) N=G = p! (R=G) 2 Sp (R=G),
, N 2 Sp (R). 6 Gp , A = Gp B , G = Gp (G \ B ),
G \ B 2 Sp (A) | G;p = Gp (G \ B );p , (G \ B );p 2 Sp (A) ( ). F
, / G;p 2 Sp (A).
B 3) qc- %
. B
2.1. A | '
+ qc-
\$# p (A) p! A 2 S (A). Ap .
.
A, p (A) = p (A=p! A) (A=p! A)p = Ap
!
( p A 2 S (A) (p! A) \ Ap = p! (Ap ) = 0). "\$, 5),
/, p! A = 0. 6 p (A) , 1
1
L
L
hbii 2 Sp (A), o(bi ) = 1. " gi = bi ; pbi+1. >
hgii 2 Sp (A).
i=1
i=1
"/ fai g1
i=1 | /
/ H% Ap p- pm ai = 0 (i = 1 2 : : :). 6 ai+1 ; ai = pi xi ,
1
L hgi ; xi;1i 2 Sp (A), x0 = a1. " 1.3, . 4) F
i=1
p- \$ | S (A).
"\$ F Ap 2 S (A) 1.3, . 1). 3 gi = (gi ; xi;1) + xi;1 ,
gi ; xi;1 2 F , xi;1 2 Ap . > b1 = (g1 + pg2 + : : : + pi;1gi;1) + pibi
pm g1 + pm+1 g2 + : : : + pm+i;1 gi;1 2 F , pm b1 2 F . A b1 2 F Ap ,
, /
/ fai = a1 + px1 + : : : + pi;1xi;1g1
i=1 5 Ap ,
. . Ap .
2.2.
1) ,
qc-
, \$
.
2) Ap | A p, A
qc-
, tA | ,
\$ p (A) " # Ap .
3) A | + , tA . A qc-
.
4) &'
qc-
A % Ai , p!i Ai = 0, i \$ 9 #- , (Ai )pi = Api G;pi Ai 2 Spi (Ai ) "\$ G 2 Spi (Ai ).
. @ qc- A A1 , &2, 54.2] 1).
2) 6 A = Ap C , p! A = (p! A \ Ap ) p! C . F/ p! A \ Ap = p! (Ap ) |
( Ap ), p! C | p- ,
Cp = 0. 3, p p! A p-. "\$, 5), /, 1220
. . p! A = 0. "/ / H 2 Sp (A) . > (A=H )p = Ap . 6 Ap , 1.2, . 3)
p! (A=H ) = Hp; =H 2 Sp (A=H ). A , Hp; 2 Sp (A). "/
/ Ap . > p (H ) = n, /
F = hg1 i : : : hgn i | p-
H . 6 gi = ai + ci , ai 2 Ap , ci 2 C , \$
a1 : : : an K Ap . 3 Ap = K N F K = K R, R = (F K ) \ C .
B
1.3, . 1) F K 2 Sp (A). "\$ R 2 Sp (C ). F,
(F K );p = K R;p , R;p 2 Sp (C ), p-
C . "\$ (F K );p 2 Sp (A). "/
K | , ps K = 0 /
s. 3 ps (Fp; ) = (ps F );pps A = ps (R;p ) = (psR);pps A 2 Sp (ps A). > Fp; |
1.3, . 4), Fp; 2 Sp (A).
3) B
1.2, . 3) p! A 2 Sp (A). "\$ ? / . 2).
6 A | qc-, Ai / A=p!i A. "/
T p4)
! A = A1 = 0, \$8 i : A ! Ai !
8
p
Q
A ! Ai . A A \$ 8, \$ |
i2I
!A
, hAp (a) = hA=p
(a + p! A) a 2 A,
p
4).
x
3. cs-
J /, , /! E (A) A \$
, A = A. A, A | qc-, A cs- / , G^ | A G 2 S (A). G
, A , 9(A) = 9(G) G 2 S (A).
B
cs- ? .
3.1 (4, 2.1, L
2.3]). ! Q
'
cs-
\$ A Ap A Ap = S , A | p2
p2
S , E (A) # ' E (S ) - \$# #- ' 9 | #- 9(Ap1 ) \ 9(Ap2 ) = ? p1 6= p2, p1 p2 2 9 Ap | # p-
'
# cs-
#,
# A.
3.2 (4, 2.2, 2.4]). ! p-
'
cs-
\$ A , \$ p-
# A q- q 6= p.
1221
F
3.1 , , , cs- . H
cs- / &4{6]. " %
cs-.
3.3. A | + , p! A = 0, T = tA p-
%
-
A=T .
A cs-
, T A | p- Ab,
E (A) E (Ab), E (A) - \$# #- ' E (Ab).
. G5/. " / cs- A,
5 , cs-. A T . > A=T T p-, qA = A q 6= p. "\$ A / Ab, E (A), K
Ab, /! E (Ab),
E (Ab) / E (T ). "/ Ab = B G. >
T = (T \ B ) (T \ G), T \ B = tB B , K = (tB )^A | A = K N . 3 Ab = Kb Nb , B = Kb = tcB . "\$ | ! A K , Ab = B (, 2 E (Ab)).
@
/. "/ G 2 S (A). > Gb | Ab. 6
| ! Ab Gb , 2 E (A), A = A (1 ; )A, A = A \ Gb = G^A.
3.4. B | p-
'
cs-
\$ ,
F | cs-
3.3 (
'
p-
), A = B F cs-
, 1) "\$#- b 2 B , g 2 F hp (b) 6 hp (g) o(g) = 1 * % f : B ! F , fb = g
2) \$ p-
B \$
" ( ) F .
. G5/. @ \$
b, g 1) ?
, b 2 B n pB . > hb + gi 2 Sp (A), A = K N , K = hb + gi;p . > tA = (tA \ K ) (tA \ N ), 1.3, . 4) tA \ K = 0,
tA = tN , F N . . f / f = ()jB , , | ! A K , F . 3/ 8, A cs-, 1.3, . 4), , p-
A .
" / / /
\$ . "\$ rp (B ) = 1, , 2.1,
/, /
/ H% \$
Fp
5 F . "/ Fp F , F p-
, /
, p- 1222
. . . A/ / /, !
&2, 40.3].
@
/. > qF = F q 6= p, B |
cs-, , , / / , G;p | A, G 2 Sp (A), G^ , G 2 S (A). "/ G 2 Sp (A).
> 1.3, . 3) G \ F 2 Sp (A). 3 F = (G \ F );p M , V = (G \ F );p + G 2 Sp (A), 8- V=G = (G \ F );p =(G \ F )
;
p-. @ V = (G \ F )p R, R = V \ (M B ). >
G;p = (G \ F );p R;p , G;p A, R;p |
M B . " 1.3, . 2) (G;p )p = ((G \ F )p)^Fp . > Gp = (G \ F )p (Gp );p (G \ F );p , , , R | .
. 1.3, . 3), 5) R \ M = 0. "/ | ! M B B . >
R 2 Sp (B ). @/
, x = a + pb 2 R, a 2 M , b 2 B , a = y + pz
y 2 Mp z 2 M . A pm x = pm+1 (z + b), pm = o(y).
"/ R 2 Sp (A) , x = pc c 2 R c = b.
"/ rp (B ) < 1, / g = x + y 2 R n pR, x 2 M , y 2 B . " % hp (y) = 0. 3 B = hyi;p B1 . 6 o(x) = 1, , ? 8 f : hyi;p ! hxi;p fy = x,
, M hyi;p = M hgi;p . , o(x) < 1. " R;p = hgi;p (M B1 ) \ R;p , B = hyi;p B1
rp (B1 ) = rp (B ) ; 1. " ! R;p | . F, ^
;
hyi;
p = hyi . @/
, hyip ? , / ,
/
p-, p-!
B . B/
, hyi;p = hgi;p = hgi^ . "\$
;
^
G 2 S (A), hgip R . A/ , R^ , , G^ |
.
6 rp (B ) = 1, 3.2 B , F , q- q 6= p. . \$ Z - p-,
p-
/ /. "\$ / / / p-
5 p- . 6
F | p- , 5, 5 %, M , , p- , p- . "\$ (R)^
(R^ ). B/
, B = (R^ ) B1 , M (R^ ) = M R^ M B = R^ (M B1 ).
6 pn F = 0, F &2, 27.5]. "\$ F = (G \ F ) M G = (G \ F ) R, R = (M B ) \ G. "/ pn A = pn B = B , pn(R^ ) n
n
p R p B , pnB = pn (R^ )K .
", X Y = B , X = hpn(R^ )iB , Y = hK iB . "/ px = g + b,
g 2 X , b 2 Y . > pnx = pn;1g + pn;1b, pn;1g 2 pn(R^ ). B-
1223
/
, pn;1g = pn g1, g1 2 R^ . > X | , g = pg1. " pn;1b = pn(x ; g1 ). "\$ a = x ; g1 2 Y , ,
x = g1 + a 2 X Y . = ?
/ X Y B .
"/ pn (R^ ) = pnX , M X = M R^ , M B = M X Y =
= M R^ Y .
3.5. &'
+ L
A csQ-
, A S = Ai A Ai = S ,
i2I
i2I
9(Ai ) \ 9(Aj ) = ? i 6= j , Ai | cs-
, A, p!i Ai = 0 pi , "* \$ pi -
, \$ \$ , \$ + 3.3 3.4, A | S , E (A) E (S ) E (A) - \$# #- ' E (S ).
. G5/. @ p 2 9(A) A =
= R(p) D(p) , D(p) = 0;p = p! A, p! R(p) = 0 (
p , / p-
). "/ 9(A) = fp1 < p2 < : : :g . 3 D(p1 ) = D(p2 ) RL
(p2 ) , A = D(pn ) R(pn ) : : : R(p1 ) , R(pn+1 ) = D(pn ) =p!n+1D(pn ) . "\$ R(p) A, 9 9(A).
p2
H p-
Ap R(p). B/
, R(p) = (Ap )^ G(p) , (Ap )^ | cs- 3.3, G(p) | cs- . 3 q(Ap )^ = (Ap )^ 5 5 q 6= p, G(pn ) pm - pm < pn, \$ 3.2 9(G(pn ) ) \ 9(G(pm ) ) = ? .n 6= m. B-
n
L
/
, 9(R(p) ) \ 9(R(q) ) = ? p 6= q. L- A
R(pi )
. L i=1
pm - 5 m < n. F
, A
R(p) | p2
. " Ai = R(pi ) . 6 / f 2 E (A), f jS 2 E (S ) \$
\$
8 ( S S ) \$
8 f 2 E (S ), f jA = f . "\$ /
E (A) /! E (S ) = E (S ). 6 S = B G, , S 5
S , S = (B \ S ) (G \ S ). 3 B \ S B \ A
A = (B \ A) N /
N .
6 | ! A B \ A, S = S (1 ; )S , S = B
B \ A B .
@
/ 1.1 , 3.3.
1] . . // . . . . 17. . | .: "#, 1979. | &. 3{63.
1224
. . 2] +, -. .,/ . . 1. | .: , 19741 . 2. | .: ,
1977.
3] 2 . 3. 4 ,, // 5. | ,, 1984. | &. 137{152.
4] 2 . 3. CS - 9 / // 5. |
,, 1988. | &. 131{147.
5] 2 . 3. 4 CS - 9 / // 9. ,. /. 95.
. | 1990. | ; 3. | &. 84{87.
6] 2 . 3. 9 / /
p- c 5
9 , 5 // 5. | ,,
1991. | &. 157{178.
7] . . <., = . ., 2 . 3. 9 /, 9
/, // 5. | ,, 1994. |
&. 3{52.
& ' ( 2000 .
. . 517.53
: , , .
! " # \$
%
% % % % %.
Abstract
R. F. Shamoian, On some properties of partial sums of the Taylor series for
the analytical functions in the circle, Fundamentalnaya i prikladnaya matematika,
vol. 8 (2002), no. 4, pp. 1225{1233.
We give a complete characterization of the functions belonging to some classes
of holomorphic functions in the circle, in terms of modules of Cesaro mean values.
D = fz: jz j < 1g | C ,
T = fz: jz j = 1g | , dm2 (z) dm2 () | D T , I = (0 1], H(D) | "
#" D #.
%& '
(
% H p (. )1]) , + D #+ f ( H p (D) , k(n f)(z)kH = O(1), 1 < p < 1, n | '
1 n
Pk a z m , f(z) = P
# f, (nf)(z) = S0 (z)+:::n+S ;1 (z) , Sk (z) =
an z .
m
m=0
n=0
, )2] , f 2 .(D), 2 )0 1], , jmax
j( f 0 )(z)j 6 cn1;, n 2 N, c | + zj<1 n
(.(D) | "& '
& D, ., , )2]).
, '
/ "
#
' Hp1 (D), H1p(D) " '" "
j(nf)(z)j ( " +" p, ). 0+ # 1
' 2
#" D #
p
n
, 2002, 8, . 4, . 1225{1233.
c 2002 ,
! "# \$
1226
. . Z1
Hpq(D) = f 2 H(D): kf kqH = Mpq (D f r)(1 ; r) dr < 1
0
2 R q p 2 (0 1) > ;1 n
Hp1(D) = f 2 H(D): kf kH 1 = sup Mp (D f r)(1 ; r) < 1
r2(01)
o
2 R > 0 pq
p
D | ##
D : H(D) ! H(D),
X
+ + 1) ak z k > 0 f(z) = X a z k (D; D f) = f
(D f)(z) = ;(k;(k
k
+ 1);( + 1)
k>0
k>0
1 Z jf(r)jp dm() p 2 (0 1):
Mpp (f r) = 2
T
H p (D),
H p10(D)
, &
0
=
' C, C( p), C( p) ' '
(
, '+6
, , p.
1
1.
f 2 H(D) p 2 ( 21 1] > 0 2 R
. ,
,
,
, Z X
1
n=0
T
nrn;1 n( f)(r)
j D
j
p
. 1
dm()
p
f 2 Hp1(D)
p) r 2 (0 1): (1)
6 (1C(
; r)+2
2
f 2 H(D) > 0 2 R
) f 2 H11(D)
)
(1)
p=1
)
M1(n (D f) jz j)(1 ; jz j); +1 6 C1 ( )n ;1 1< < +1
3
0<p<1 2R
f 2 H1p(D)
. ,
,
. :
| ,
.
,
(2)
.
. ,
Z1
0
(1 ; r)
2p+
X
1
n=1
D
nrn;1M1 (n( f r))
p
< 1 > 2p ; 1:
, )4] , X
1
n
ak z k ; f = 0 f(z) = X ak z k f 2 X
lim
n!1
X
k=0
k=0
(3)
(4)
1227
X = H p X = H0pp , 1 < p < 1, ;1 < < 1. , (
+, ' )4], " H011(D)
H 1(D)6
/ # f,
P
1
P k
f(z) = ak z , , nlim
ak z k X 6
!1
k>0
k=n+1
/.
:
'
1" (
.
2.
s > 0 > ;1
1
X
;
s
k
lim D
ak z = 0
f f 2 H 1 (5)
n!1
1
H
k=n+1
1
X
;
s
k
lim D
ak z 1 1 = 0
f f 2 H011 : (6)
n!1
,
. H0 k=n+1
!"
!"
, '/
2
62
" " + <
H p (D) H0pq (D) = H pq(D) | =
{%.
?, (. )6]) '
+ =
{% (
, 6 cp kf k
1 < p < 1
sup jSn f()j
N
n2
Lp (dm( ))
Lp (dm( ))
f 2 Lp (dm(z)) (Sn f)() =
n
X
^ k n = 0 1 2 : : : 2 T:
f(k)
k=;n
, /6
2
"
" # g, g 2 H(D),
n
P
^ k '
" (Hg f)(z) = sup g^(k)f(k)z
n
2
N
k
=0
H pq H p H s 0 < p q 6 s 6 1.
3.
g 2 H(D) 0 < p q 6 s 6 1 > ;1
1)
Hg
H pq(D) H s(D)
,
,
. #
, sup Ms (Hg (Dm f) jz j)(1 ; jz j)m+1; 1 ;
jzj2I
m
| ,
2)
Hg
p
+1
p
< 1
(7)
1
m 2 N m > +1
q ;1+ p
H p (D) H s (D) 0 < p < s 6 1
,
#
,
,
, sup Ms (Hg (Dm f) jz j)(1 ; jz j)m+1; 1 < 1
p
jzj2I
(8)
m 2 N m > p1 ; 1
0'
1{3 /6" "
(
+".
m
| ,
,
.
1228
. . 1.
1
. f 2 H(D)
. 1
X
f(rz) = (1 ; r)2 nrn;1(nf)(z) z 2 D r 2 (0 1)
n=1
Z f(zr)
n(nf)(z) = (1 ; r)2 r;n ;n dm() z 2 D r 2 (0 1):
(9)
(10)
T
> p ; 2 f g 2 H(D) r 2 I G 2 H pp(+2)p;2 0 < p 6 1 > ;1
f(r')g(r')
C dm(') 6
2
1
. ,
,
,
,
,
.
1 Z
2 T
+ 1
Z1 Z
;
2( +1)
jD+1g(R)jjf(RC)j(1 ; R2) dm()R dR (11)
6 r
0 T
Z1 Z2
p
Z
jG(w)j(1 ; jwj) dm2(w) 6 C( p) jG(w)jp(1 ; jwj)p+2p;2 dm2(w):
D
0 0
2.
n
P
1
p(z) =
ak z k 0 6 m < n < 1 m n
1 k
k=m
P
f = ak z 0 < q 6 1 1 6 p < 1 ;1 < < 1
.
k=0
,
,
,
,
,
p
.
,
, k>0
3.
)
n
X
(13)
k>0
.
D~ : H(D) ! H(D)
,
. ak jz j2k wk =
k=0
!
1 Z
n;1 wn;1
z
C
z
w
C
= 2 (n f)(z) 1 +
1 ; n1 + : : : + 1 ; n;n 1 dm()E z = jz jE
T
,
, -
,
f 2 H(D)
,
p
| !!"
,
2 N f 2 H pq(D)
. f n kf kH 6 Mp (p r) 6 rm kf kH :
2 D f 2 H p 0 < p 6 1 > 0
~ f 2 HPp
~
D
D
P
(D~ f)(z) = (k + 1) ak z k f(z) = ak z k
(12)
) (nf)(jz j w) =
1 Z
= 2
fw (z) 1+ 1 ; n1 zC+: : :+ 1 ; n ;n 1 zCn;1 dm()E z = jz j:
2k
T
1229
F
" 1 3 . , (6"+ 1, . )5, . 20], )7, 4], )8], )3]. 2
' ' )9] )10].
0(
, + (1){(3) ". ++ (10) (11) Z f)(zr)(rC)n dm() 6
nj(n(D f))(z)j = r;2n (D (1
; r)2
T
Z Cj
l n
l;1 dm2(w) 6
6 c(l r) j(Dj1 ;fzw)(jw)
2 jD w j(1 ; jwj)
T
Z j(1 ; jwj)l;1 dm2(w)
6 c1(nl ) j(D fz )(w)
j1 ; wj2
D
z = R, c1 = c1(l), > l ; 1 > 0, r > 21 .
0
, + (12), X
1
p
n
;
1
I = Mp
nR jn(D (f(R)))j 6
n=0
Z Z jD f (w)jp (1 ; jwj)pl+p;2
c
z
2 (l p)
dm2(w) dm():
6 (l ; R)(l+1)p
j1 ; wj2p
TD
R
G/, '+ H j1;d'r'j 6 (1;cr(t))t;1 , t > 1, T
t
Z (1 ; jwj)(l;1)p
c
p ) 1 (l p )
I 6 (1 ; R)(l+1)p (1 ; jwjR)p djwj 6 (1c1;(lR)
(+2)p
0
p 2 21 1 > l ; 1 > 0 f 2 Hp1:
1
I, , " + (2).
F(+ &
, Z ;1
jn(D f)(z)j 6 C1()n;1 jD fz (w)j1 j;(1w;j2jwj) dm2(w):
D
0
, '+ (14), #, f 2 H11, H,
(z = jz j')
1230
Z
T
. . Z1 (1 ; jwj)( ;1);1
(1 ; jwjjz j) djwj 6
0
;1
C
(
6 (12 ; jz j))n
+1; > 1 > ; 1
jn(D f)(z)jdm(') 6 C2( )(n;1)
, (2) .
I " + (3).
J'+ 3 ), (13) kGR kH = Mp (G R), R 2 I,
G 2 H(D), 1 Z
1
2
;
2 n(D f)(jz j w) = 2 (D D f(z)) D2 1 + 1 ; n zC(w) + : : : +
T n ; 1
+ 1 ; n (Cz (w))n;1 dm() z = R w = r'E
Z
1
X
n
;
1
nR
jn(D f(R2r))j dm() 6
p
n=1
6c
1
X
n=1
T
nRn;1
Z
jD;2D f(Rt)j dm(t)
T
Z
T
jD
2
~
Gn(R')j dm(') 1
P
G~ n(R') = (1 ; nk )(R')k .
k=0
0
, , ' 2 Z
jD2G~n(R')j dm(') 6 (1 ;C1R)2 + n(1 C;2R)3 :
T
1 Z
1
X
n
;
1
nR
jn(D f(R2r))j dm() 6
n=1
6C
1
X
n=1
T
nRn;1
Z
6 (1 ;C3R)4
T
K
,
Z
T
jD;2D f(Rt)j dm(t)
jD;2D f(Rt)j dm(t):
C
C
2
1
(1 ; R)2 + n(1 ; R)3 6
1231
Z1 X
1
0
n=1
n(Rn;1)
Z
T
Z1
jn(D f(R))j dm()
6 C (1 ;
R);2p
0
Z
T
p
(1 ; R)2p+ dR 6
p
jD;2D f(Rt)j dm(t) dR 6 C1kf kH1 :
p
, ' '
%{ (., , )11]).
?
" + (3) .
0'
(1){(3) / ' 1 ( 1).
0(
, , + (2). J
+ T (9), Z
Z
1
X
2
n
;
1
j(D f)(rz)j dm() 6 (1 ; r) nr jn(D f)(z)j dm():
n=1
T
T
0
, + (2), Z
1
j(D f)(r2 )j dm() 6 C(1 ; r)2 X nrn;1(1 ; r);(;);1n;1 6 (1 ;Cr) :
n=1
T
L '.
2.
M
, X
1
D;
k=n+1
Z
1
X
ak z k r2k = fz (r)D;
(rC)k dm() z = R2':
k=n+1
T
0
, + 1 ( 2), H ++ r ! 1,
Z 1
D; X ak z k r2k dm(') 6
k=n+1
T
n+1 ZZ
w~
;
;1
6 C(r) jfz (w)
C j D D
~ dm(')
l ; w~ (1 ; jw~ j) dm2(w)
TD
Z n+1
Z 1
X
k
D;
ak z dm(') 6 C kfjzjkH 1 jj1w;j wj (1 ; jwj);1dm2 (w) (140)
k=n+1
T
fjzj (w) = f(wjz j), z w 2 D.
D
1232
. . I+ 1
&
Z1
1
n
+1
lim jwj
log 1 ; jwj (1 ; jwj);1djwj = 0 > 0
n!1
0
Z 1
X
;
k
lim D
ak z dm(') = 0 > 0:
n!1
(1400)
k=n+1
T
F H011 . J' (140) Z1 Z
Z1 Z 1
X
;
k
D
ak z dm(')(1 ; jz j)djzj 6 C
jf(z)j(1 ; jzj) djzj dm(') k=n+1
0 T
0 T
Z jwjn+1
j1 ; wj (1 ; jwj);1dm2(w) > 0 > ;1:
D
G2
+ '+ &
(140). <
2 '.
3.
0+ '
" (7) (8) '+
+
' #
1 +1
1
;(m+1; ; ) (1 ; rz)m+1 H 6 C(1 ; r)
m 2 N m > 1p + +q 1 ; 1 r 2 I
1
;(m+1; 1 ) m 2 N m > 1 ; 1 r 2 I:
(1 ; rz)m+1 H 6 C1 (1 ; r)
p
(
, (7) ++
+ . ? +
/6
, #
)8] ZDn :
jf(w)jS (1 ; jwj)S( +1 + 1 );2dm2(w) 6 C kf kH 0 < p q 6 S < 1: (15)
p
q
pq
p
p
q
p
pq
D
,'& 1, ja0 + C1a1(r2w) + : : : + Cnan(r2w)nj =
1
Z X
k
k
n
=
ak (r) w (1 + (r)C1 + : : : + (r) Cn ) dm() 6
T
6 C(r)
k=0
Z
D
n
jfw(z)j X zk lk Ck(1 ; jzj)m;1dm2(z)
k=0
1233
k+m+1)
+1
1
w = jwj', r 2 I, w 2 D, lk = ;(;(
m+1);(k+1) , m 2 N, m > q ; 1 + p .
0
, "+ r ! 1, + 1 ( 2), H + (7), X
S
X
S
Z
n
n
sup Ck ak wk 6 C jfw (z)jS sup z k lk Ck (1 ; jz j)S (m;1)+2S ;2dm2 (z)
n k=0
n k=0
D
S
Z X
Z
n
+1
sup Ck ak wk dm(') 6 C jfw (w)
~ jS (1 ; jw~ j)S ( + 1 );2dm2 (w)
~ 6
n
T
k=0
6 C 1 k f kH
q
p
D
0 < p q 6 S 6 1:
, ' (15).
0 + (8) + (
+, (15) '
" ' (. )11])
Z
jf(w)jS (1 ; jwj) ;2dm2(w) 6 C(p S)kf kH 0 < p < S 6 1:
pq
S
p
p
D
<
3 '.
1] . . | .: , 1963.
2] Bennett G., Stegenga D., Timoney R. Coe\$cients of Bloch and Lipschitz functions //
Ilin. Math. J. | 1981. | Vol. 25, no. 3.
3] Djrbashian A., Shamoian F. A. Topics in the theory of Ap spaces // Teubner Texte
zur Math. | 1988. | Vol. 105.
4] Kehe Zhu. Duality of Bloch spaces and norm convergence of Taylor series // Mich.
Math. J. | 1991. | Vol. 38. | P. 89{101.
5] Hedenmalm H., Korenblum B., Kehe Zhu. Theory of the Bergman spaces. |
Springer, 2000.
6] + ,. -. + . / 01. | 23 . 4. 15. | .: -25242, 1987.
7] 7+ 8. 0. 9 : : 3 3 // ., /., /, 3+. | 1999. | < 3/4. | ,. 361{371.
8] 7+ 8. 0. 9 1 / 3 =: 3 // >. . ?. | 2000. | < 10. | ,. 1405{1415.
9] Buckley S. M., Koskela P. and Vicoti^c D. Fractional integration and weighted
Bergman spaces // Proc. Camb. Society. | 1999. | P. 145{160.
10] Yevti^c M., PavloviEc M. Coe\$cient multipliers on spaces pf analytic functions // Acta
Sci. Math. | 1998. | Vol. 64. | P. 531{545.
11] MateljeviEc M., PavloviEc M. Multipliers of H p and BMOA // Pacif. J. Math. |
1990. | Vol. 146. | P. 71{89.
% & & 2001 .
. . 517.926
: , .
! # # \$ % #
%& \$ ' .
Abstract
V. I. Bulatov, On solvability of the initial problem for linear regular non-homogeneous dierential systems, Fundamentalnaya i prikladnaya matematika,
vol. 8 (2002), no. 4, pp. 1235{1238.
The solvabilitycriterion of the initial problem for linear regular non-homogeneous
di/erential systems is proved. Analytic representations for the solutions of this
problem are obtained.
A0x_ (t) = Ax(t) + f (t)
(1)
x(t) 2 Rn, f (t) 2 Rn, t 2 T = 0 +1, A0 A | (n n)-.
(1) !" #, \$ #%" &" ' ((
" (
)" x(t) (*
), \$"" (1).
+\$" # (1) ,\$-
" !,
x(0) = x0
(2)
x0 | !
# n-), ," !*#, \$ *
x(t) /# , \$" (2).
0\$- # ' "\$"" 1\$,
\$# !* ,\$-
# !, (2) \$" \$"
& (1) '
\$,)& 1\$
# & *
# /# !,, 2
& ,! 1\$-
,
(
), \$-, \$,
(
\$-
# ')
& (1), . . )& det A0 6= 0.
, 2002, 8, 0 4, . 1235{1238.
c 2002 !,
"#
\$% &
1236
. . 5 -, , (1) "\$"" \$"
#, \$
det(A0 ; A) 6 0
(3)
| )1\$)
" 1
".
6 \$-
#* 1
'" \$"
(1, c. 33]). (3) (n n)- 7(t), t, 8 _
><A07(t) = A7(t) + tnn! En
_ t)A0 = 7(t)A + tnn! En
(4)
7(
>:7(0) = 0
En | n.
. ! " x(t) (1) # f (t) "
Zt (t ; )n
Zt
0
0
n! x( ) d = 7(t)A0 x(0) + 7(t ; )f ( ) d
(5)
7(t) (4).
+#\$-
, \$ (1), (4) 1\$ " 1 ," 1\$,
Zt (t ; )n
Zt
0
0
Zt
_
n! x( ) d = 7(t ; )A0 x( ) d ; 7(t ; )Ax( ) d =
= ;7(t ; )A0 x( )]
=t
=0
+
Zt
0
7(t ; )A0 x_ ( ) d ;
0
= ;7(0)A0 x(t) + 7(t)A0 x(0) +
= 7(t)A0 x(0) +
Zt
Zt
Zt
7(t ; )Ax( ) d =
0
7(t ; )(A0 x_ ( ) ; Ax( )) d =
0
7(t ; )f ( ) d:
0
< . \$ (1) n # f (t) # (2) " # , 7(n) (0)Ax0 +
;
X
n 1
k =0
7(n;k)(0)f (k) (0) = 0
(6)
1237
" x(t) x(t) = 7
(n+1)
(t)A0 x0 +
Zt
7
(n+1)
(t ; )f ( ) d +
0
;
X
n 1
k =0
7(n;k)(0)f (k) (t)
(7)
7(t) (4).
. +\$" # \$"
# (1) n-)-(
)
Zt
y(t) = 7(t ; )f ( ) d
(8)
0
7(t) \$" (4). ?\$ f (t) "\$"" n ! ((
# )-(
)# t, \$ (8), !"# 1 t,
' (n + 1) ! ((
# (
)# t, \$" )# \$ (4)
1\$\$-
((
1 1\$ @#'
y
(n+1)
(t) = 7(0)f (t) +
_ f (t) +
= 7(0)
=
;
X
n 1
k =0
Zt
Zt
_ t ; )f ( ) d
7(
0
7% (t ; )f ( ) d
0
(n;k)
7
(0)f (t) +
(k)
Zt
(n) Zt
=
(n;1)
_ t ; )f ( ) d
7(
(n)
=
0
= ::: =
7(n+1)(t ; )f ( ) d:
0
A \$" *
" x(t) ,\$-
# !, (2) (5) 1\$,
x(t) =
Zt (t ; )n
0
n! x( ) d
=7
(n+1)
(n+1)
(t)A0 x(0) +
;
X
n 1
k =0
= (7(t)A0 x(0) + y(t))(n+1) =
(n;k)
7
(0)f (t) +
(k)
B\$" !- t = 0 ," (2), 1\$,
(7(n+1)(0)A0 ; En)x0 +
;
X
n 1
k =0
Zt
7(n+1)(t ; )f ( ) d:
0
7(n;k)(0)f (k) (0) = 0
, (6) , , ! (4) \$
7(n+1)(0)A0 ; En = 7(n) (0)A:
(9)
1238
. . B1\$2 1-, , ,\$-
\$ (2) \$"
# (1)
\$" *
(6). D, -1&, 1 t = 0 \$" )-(
) (7) \$ (9) x(0) = 7(n+1) (0)A0 x0 +
;
X
n 1
k =0
7(n;k)(0)f (k) (0) =
= (En + 7(n) (0)A)x0 +
;
X
n 1
k =0
7(n;k)(0)f (k) (0) = x0
-&, (4) \$" (7) 1\$,
A0 x_ (t) ; Ax(t) = (A0 7(n+2)(t) ; A7(n+1) (t))A0 x0 + A0 7(n+1) (0)f (t) +
Zt
+ (A07(n+1) (t ; ) ; A7(n+1)(t ; ))f ( ) d +
0
+
;
X
n 1
k =0
(A0 7(n;k)(0)f (k+1) (t) ; A7(n;k)(0)f (k) (t)) =
= A07(n+1) (0)f (t) + (A0 7(n)(0)f_(t) ; A7(n) (0)f (t)) +
= (A0 7(n;1)(0)f%(t) ; A7(n;1) (0)f_(t)) + : : : +
_ f (n) (t) ; A7(0)
_ f (n;1) (t)) =
+ (A0 7(0)
= (A0 7(n+1)(0) ; A7(n) (0))f (t) + (A07(n) (0) ; A7(n;1)(0)f_(t)) + : : : +
_ f (n;1) (t) + A0 7(0)
_ f (n) (t) =
+ (A0 7% (0) ; A7(0))
= (A0 7(n+1)(0) ; A7(n) (0))f (t) = f (t):
E
,, 1 1\$
(6) \$" \$"
# (1) n ! ((
# - f (t) (\$ (7) % ' *
x(t)
,\$-
# !, (2).
1] . . // ! " #\$# . | 2001. | !. 10. | (. 33{35.
' ( 2002 .
A-
. . . 517.51
: A-
, .
, g | , g 2 Lp (R), p > 1, #
~ A-
g~ % , f (x) 2 L(R), fg
& R Z
Z
~ dx = ;(L) f g~ dx:
(A) fg
R
Abstract
R
Anter Ali Alsayad, Hilbert's transformation and A-integral, Fundamentalnaya i
prikladnaya matematika, vol. 8 (2002), no. 4, pp. 1239{1243.
We prove that if g is a bounded function, g 2 Lp (R), p > 1, its Hilbert's trans~ is an A-integrable
formation g~ is also a bounded function, and f (x) 2 L(R), then fg
function on R and
Z
Z
~ dx = ;(L) f g~ dx:
(A) fg
R
R
, 2 -
f (x) L0 2 ],
Z f (x + t) ; f (x ; t)
1
lim+0
f (x) = ; "!
dt
2 tg 2t
"
\$ %&' \$\$\$ (%\$ (\$. 1, . 557]).
,. -. .' 2,3] (0( \$.
(. . ). 2 -
' ' , 2 -
f L0 2 ], f (x)'(x) A- Z2
Z2
0
0
(A) f' dx = ;(L) f ' dx:
1 ' %%0 ' 2. 3
\$4 4], \$. 1, . 587].
, 2002, & 8, 0 4, . 1239{1243.
c 2002 !",
#\$
%& '
1240
. 7
f~(x) = ; 1 "!
lim+0
Z1 f (x + t) ; f (x ; t)
t
"
dt
& f R.
9& \$ \$, :
( \$ ,. -. .', % ;'% R.
. g | , g 2 Lp (R), p > 1, ~ A-
g~ , f (x) 2 L(R), fg
R Z
Z
~ dx = ;(L) f g~ dx:
(A) fg
R
R
. <\$\$ f (x) > 0 \$, Z
Z
~
; f g~ dx = nlim
(1)
!1 fg dx
R
E
0
1
: En = R n Ln , fEngn=1 | '' \$, Ln | &
\$, L0n | & \$, 0 `0i | > (0\$ \$ `i >, j`0i j = 5j`ij (\$.
n
1, . 576{583]), '
f (x) = 'n (x) + An(x) + rn(x) ; An(x)]
f~(x) = 'n (x) + A~ n (x) + ~rn(x) ; A~ n(x)]
:
(
0 6 f (x) 6 n
'n (x) = f0(x) f (x) > n
(n
An(x) = 2 x 2 Ln 0 x 2 R n Ln R
rn(x) = f (x) ; 'n (x) (rn ; An ) dx = 0 (\$. 1, . 575{577]).
`
3 '~n (x) | ;'% :
'n (x) 2 L,
'~n (x) 2 Lq (R), q > 1, B A~ n (x). C r~n(x) ; A~ n(x), \$\$\$ En (\$. 1, . 577{579, 583]). E :
A :
Z
(~rn (x) ; A~ n (x))'(x) dx
i
En
1241
A-
\$ \$&. F :
Z
Z
Z
Z
~ dx = '~n g dx + A~ ng dx + (~rn ; A~ n)' dx = I1 + I2 + I3 :
fg
En
\$\$
En
I1 =
F
Z
En
En
En
Z
(2)
Z
'~n ' dx = '~n ' dx ; '~n ' dx:
Z
Ln
R
Ln
0
'~n ' dx = o(1)
0
(\$. 1, . 587]).
R
R
G : &, '~n g Lp , '~ngdx = An g~ dx |
\$ B : g 2 L2 , g~ 2 L2 , f 2 L2 \$\$
Z
Z
~ dx:
f g~ dx = ; fg
R
R
R
R
2 g 2 L2 \ Lp g~ 2 L2 \ Lp , f
Z
2 L2 \ Lq \$\$
~ dx 1 + 1 = 1:
f g~ dx = ; fg
p q
Z
R
R
,' 'n L1 . 3: j'nj2 6 C j'nj 2 L1 , 'n 2 L1 , , :
,
j'njq 6 C j'nj 2 L0n , q > 1. G', 'n 2 Lq , q > 1. 3: '~n 2 Lq
(\$. 5, . 160] 6, . 209]).
L ! g, : g~ ;;;;
L
3' ' gk 2 Lp \ L2, gk ;;;;
k k!1! g~ k!1
;' 1, . 32]
Z
Z
'~n gk dx = ; 'ng~k dx
p
p
R ??
R ??
R
R
? gk ;! g ?
Z y k!1 Z y
'~n g dx = ; 'n g~ dx
,
F
Z
En
Z
'~n g dx = ; 'ng~ dx + o(1):
R
Z
Z
Z
f g~ dx ; 'ng~ dx 6 K jf ; 'n j dx = o(1)
R
R
R
1242
'n (x) :
j'~(x)j, B\$
I1 =
Z
Z
En
'~n g dx = ; f g~ dx + o(1):
R
< I2 = o(1) I3 = o(1) &( , 1, . 587{588].
I%J & \$&, >\$
Z
Z
~ dx = ; f g~ dx + o(1)
fg
R
En
. . (1) .
K\$ ', Z
Z
~
~
nlim
!1 fg dx ; fg]n dx = 0:
R
En
, : \$\$
(3)
Z
fg
~ ]n dx 6 nL0n = o(1)
Ln
0
B\$ ', Z
~ dx ; fg
~ ]n g dx = 0:
lim ffg
(4)
n!1
En
~ j 6 n. I%
\$ ,&:' & (, jfg
~
Gn \$ > > x, : jfgj > n, \$ G0n = Gn L0 . E-&>,
\$\$
G0n 6 Gn = o(1=n)
-&>,
Z
Z
fg dx ; fg]n dx = ffg dx ; fg]ng dx 6
G
G
Z
Z
6 jfgj dx + jfg]n j dx = J1 + J2 :
0
n
n
Gn
F jJ2j 6 nG0n = o(1), jJ1j 6
Z
Gn
0
Gn
0
0
Z
Z
j'ngj dx + jAngj dx + jr~n ; A~ n j dx:
Gn
0
Gn
0
L& B> : ' o(1). 1 & , B I1 , I2 , I3 . , (4) . G (3) (1),
A-
>\$
Z
1243
Z
~ ]n dx = ; f g~ dx + o(1)
fg
R
R
R
R ~ dx 0 , (A) fg
Z
Z
~ dx = ; f g~ dx:
(A) fg
R
R
3\$ .
E (
& :%( %:' 3. ,. -4 \$ \$0' %.
1] . . . | .: , 1961.
2] #\$%& '. . A-\$ )* +,- // # /) .
,-. . | 1956. | . 181, 2 8. | 3. 139{157.
3] #\$%& '. . &) A-& // 78 3339. | 1955. |
. 102, 2 6. | 3. 1077{1080.
4] Titchmarsh E. C. On conjugate functions // Proc. London Math. Soc. | 1929. |
Vol. 29. | P. 49{80.
5] , '. <& & = )& H p . | .: , 1984.
6] 3> ., <> '. <& & > \$/ &\$&? )&?. | .: , 1974.
( ) ) 1997 .
. . . . . 512.552.51+517.982
: , , , !".
# \$ %\$" %!\$ ! !& \$\$ &\$
.
Abstract
A. A. Seredinskiy, An algebraic characterization for rings of continuous quaternion-valued functions, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002),
no. 4, pp. 1245{1249.
Rings of continuous quaternion-valued functions on compact spaces are characterized in purely ring-theoretic terms.
. A 1A , :
) I J K 2 A, !
J K
1
J K
I I ;1 K ;J
J J ;K ;1 I
K K J ;I ;1
) A0 A, 1) I J K 2
= A0,
2) A A = A0 + I A0 + J A0 +
+ K A0 "
1
1
I
I
, 2002, 8, , 4, \$. 1245{1249.
c 2002 !,
"#
\$% &
1246
. . A0 :
1) a b 2 A0 c 2 A0 , a2 + b2 = c2"
2) # a 2 A0 b 2 A0 c 2 A0 , a = b2 ; c2
bc = 0"
3) # a (1A + a2);1 "
4) a (bn 2 A0 j n 2 N), n2 (a2 + bn2 ) = 1A , a = 0"
5) # a b 2 A0 n 2 N, a2 + b2 = n2 1A "
6) (an 2 A0 j n 2 N) | , (mk 2 N j k 2 N), k2((am ; an)2 + b2 ) = 1A m n > mk b b(k m n), a 2 A0, # (nk 2 N j k 2 N), k2((a ; an )2 + c2 ) = 1A
n > nk c c(k n).
. A0 , 1){6) "
" \$%& \$ 0 : A0 ! C (Max A0 R), C (Max A0 R) | )
)*%%+
)+, ))\$)-)+, &") ) *) *%%)
, )+, A0 (. /1]). 1) -) *%
C (Max A0 Q), Q = (Q kk) | )% %
)) %
p %) )
. 2% *% )%)+ %\$, kxk a2 + b2 + c2 + d2 *% \$
) x a + Ib + Jc + Kd 2 Q, a b c d 2 R.
3*% )
%4) , * (a + Ib + Jc + Kd) 0 a +
+ i0 b + j0 c + k0 d +, a b c d 2 A0 . 3-
), %\$" %4) &") \$)-) %) ),.
54" 6)" a 2 A0 *
&") ^a : Max A0 ! R, "
- a^ 0 a. 8
, &") a^ \$)- -%\$ A^0 . 9 /1] + *\$), - )% %
)) % (A0 k kA0 ) *). 24)
*\$, - )4
A^0 \$)"+ )4
, %) -))+, &") ) *%%)
Max A0 ) ) %
)%) )%+.
) *) "
(2, . I, x 2.10.I]). & R #
#
Fb (Max A). '# R (!
#
(!
#
Fb.
:, )4
A^0 %;<-) % G , S
\$ )4
(G0 2 G 0 j 2 =) , 6)
\$ G 0 fcoz f j f 2 A^0g, ) %
))+, *% \$
)+ )4
=, %+ * ) )4
Max A0. 2*) , -
coz f fM 2 Max A0 j f (M ) 6= 0g.
1. & y (a + Ib + Jc + Kd) a b c d 2 A0 .
'# y 2 C (Max A0 Q).
. >% *% \$
)" &") y = (a + Ib + Jc +
+ Kd) 0 a + i0 b + j0 c + k0 d = ^
a + i^b + j c^ + kd^. ?"
M 2 Max A0 .
@
y(M ) = a^(M ) + i^b(M ) + j c^(M ) + kd^(M ) q 2 Q.
"-%) - q 2 Q.
1247
>% *% \$
)"
?4 3-
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M. M. Khapaev, On one ill-posed singularly perturbed problem, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 4, pp. 1251{1254.
The boundary value problem for a second order ordinary singularly perturbed
equation is under consideration. The solution of the problem with interfaces on the
boundary and within the segment is constructed.
d2 z = F (z x) 0 < x < 1 0 6 " < 1
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