1- . .
. . .
517.968
: , .
!" 1-$ % & ' " % ! ( % &).
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). = & / , 12 1 , %& 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 12 1 .
. (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
9 3 (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. :. *( C D )&% )%. | .: , 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 1 0. | -.: ! 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 ! 2 1 & 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:
[email protected]
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 + g z ] = ]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), g z ](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) B a (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<
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 ) + (g z ] ')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 + g z ] = ] 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 + g z ] ; 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 ) (g z ];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 ) (g z ] ; (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)! + It dV (kv k2& ; ( )& )] + k=1 + It (g z ] ; (1 ; ` )Dt u(a ) ; ` Dt u(b ) v )& t 2 Tj Tj +1): (3.7) t
k
k
k b
j
5 A8 , ( ) jIt(g z ] ; (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 kL k ]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
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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 g z ] + + It(j ) ( =V )& ; (u v =V )& ; (g z ] ` )& ] (x t) 2 Qk , 0 6 j 6 k < n, It(j ) = I (uj ; u ) + (u ` )& ; (uj `j )& + It(j ) I g z ] + + Bjk + It(j ) ( =V )& ; (u v =V )& ; (g z ] ` )& ] (3.13) k X Bjk = ((ui `~i ; `i )& ;1 ; (ui `i)! ; (ui 1)! ):
t
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1003
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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), G N ;1] 6 G ] 6 G N ] () 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 ) g z ] (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 g z ] 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 g z ]:
1004
. .
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+ 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),
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(3.2). 5
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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
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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 ) g z ] ! g z ] 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 ) + (g z ] '")Q : & ' ! 0, a " ! 0, (1.8). J 1.1 ) . a
'
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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.
...
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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 .
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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 9 F (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
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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
. .
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* ( . 2 ( ) . 3 ) , , )* ,
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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 , (, - -'( ) ( Q S ( ) . ( f (x) ,
( .
(
1] Berman A., Plemmons R. J. Nonnegative matrices in the mathematical sciences. | New York: Academic Press, 1979. 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 >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]. 6 3 5& H(~n) 5 & # . 1. 7 # 8&& 9 , 8&3 9 && && +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 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' 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 . 6 3 # 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 @
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: [email protected]
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 -. I A - # (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
I A - # (12), #! # #! /%! j p^(). < ( # #! # x^() (8) | x() | (9) # ## u(),
, x^() A # (1). I A
#
-. : 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 ( - # #
. # . I A # # # (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 ( - -. #. I A
# (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
I A # # (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 )) -. % # % #
. I A # (# (
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 $ #88 9 . | ., 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 ( ) = xn n : 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 j jp (f ) ln j j 0 ln ln kf ( )kp 0 ln kf ( )k0p
0p (f ) = lim ( f ) = lim : p !1 ln j j !1 j jp (f ) E
ln kf ( )kp {p (f ) = lim j jp(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 ) + ")j jq (f ) ; ({p(f ) ; ")j jp(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 jj jj 6> 11 j'( )j = kBkf(f( ( )k))kp 6 Cpj jq 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: [email protected]
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 = 2 i 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 = (2 i)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) = (2 i)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 > 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 + ' () W v + 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
: , , , .
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'( d % d > 1 %) " $ . *
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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
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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: A A] ]+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: [email protected]
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 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 ; ? . B C - . @ 4, . 1. | 2B ; 2, 1997. | . 111{119. 9] -./0 1. 2. 34 0 4/ E 4; 4; // @4 VI B4 ; ? . B C - . @ 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) = 2 iq1 (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) = ; 2 iqz (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) : = 2 iq1 (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) = ; 2 iq (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 2 iq1(z) t;z t q1(z) ;
1 2 iq2(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 = ; 4 i t q2(0) 4 i 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) = 4 iq (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 4 iq2(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 2 i 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
2 m (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 + 2 i 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 + 2 i 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 + 2 i 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 + 2 i 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 4 qk (t0 ) m t ; t0
1139
;
exp(;i N ) Z fk (t) exp(i N2 )Rk (t) = 4 R (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 = 4 R (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 ) = 4 R1 (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) = 4 R (t ) sin(( (31) 0 ; 0 )=2) 2 2 0 G ,
L
8>1 > q1+ (t) = tN=2 R1 (t) >:;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
4 q1+ (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 4 q2+ (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 = ;2 C: (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 ) + 2 c0 = 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) = 2 Q^ 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 ; 2 C +
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 = 4 Rj (t0 ) sin ;20 d j = 1 2: (46) 2 qj (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 = 4 qj (t0 ) =
1
L
Z
4 qj+ (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 4 itj2 4 t 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 4 i 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 = 4 R (t ) cos( ; 0 ) + 2 sin 2 d = sin ;20 1 0 LZ 1(t) d = K~ (t ) = 4 R1 (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 ) ; 4 R1 (t ) (Q1(t) + Q;2 (t))R 0 sin 2 1 0 a b
Z 2(t) d + 4 R1 (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 ) ; 4 R (t ) d ; sin ;20 1 0
(49)
b
; 4 R1 (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 + + 4 R1 (t ) (Q2 (t) ; Q;1(t))R 4 R2(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 ; + 4 R1 (t ) (Q2 (t) ; Q;1(t))R 4 R2(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 = ;2 C 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 4 R (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 4 R (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 = 2 qj+ (t0 ) L
,
L
L
Zb
= Ij (t)ifj (t)qj+ (t) d j = 1 2 (58) a
dt0 1 Z Ij (t) = 2 i j = 1 2: qj+ (t0 )(t ; t0) L
D , Ij (t)
6 + , , . $ , 6 +
Vj (z), ,
: 1 Z dt0 Vj (z) = 2 i 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) = + + 2 i + 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 = + + 2 i qj (t0 )(t0 ; z) 2 i 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 2 i qj ()( ; z) = 2 i 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: [email protected]
511.3+512.62+530.145.61
: , (") " "$ "%, % &$'"( )" *%+,, +", - + *.{ ", 0 , 1( 2 3 , $+% . 4" 44 ( , +" 2. . 2 5+ .
6" $, +"$, " , %,$' . ( " "$ , "
a 6;1547 % "'+ *,5 ' $ .5 (a > 4) , 5 $' , ) $. (' +"$"'+*8 10 ). " ) ", $+" +", 0 , 1( 2 3 , 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. 1 9 ) xk = R3 ch(i'k ) = R3 cos 'k p R3 = 2 p3 r = 2 c + 2 | , , 'k = ( + 2 k)=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 = ( + 2 k)=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. 1 9 ! 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. 1 9 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 = ( + 2 k)=3 (k = 1 2 3). 1 9 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) , '
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) ! '
!! 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 % / '
) '
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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
. .
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Xk X1 X2 X3 X4 X5 X6 X7
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!k 117811 247460 254581 263756 275581 291733 40803
p
R8 = 7 c + 7 | , '! Tk . \
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* xk !k ' = 2 = 14, . . * (n l) = (9 0) (n l) = (15 6) ) 8. O P90(r) a . 8.
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1] # $. %. & ' . | ).: , 1977. 2] , -., . /. # 0 1 2 1 1 10 3 1 . | ).: 5 1 ' , 1960. 3] , 1 -., / $. &6 7 8 7 . 9. 1. | ).: , 1973.
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xk x1 x2 x3 x4 x5 x6 x7 x8
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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.
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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, hk vi = hke vi () k = ke: K (= %H =) % %& ?& %+ v > 2m + 1: d;2 (kd;1 ; ekd;1)) k ) + : : : + v ((k1 ; e 1 e jkd ; kd j = 6 vd;1 d ; 2 2m(1 + v + : : : + v (v ; 1)(1 + v + : : : + vd;2 ) ) =1 6 < vd;1 vd;1 =) (kd ; ekd ) = 0 ( ? % , " ? ,, )H
d;3 d;1 e e e jkd;1 ; ekd;1 j = ((k1 ; k1 ) + : : : + v (kvd;d;22; kd;2)) + v (kd ; kd ) 6 d;3 d;3 6 2m(1 + vv+d;:2: : + v ) < (v ; 1)(1 +vvd;+2 : : : + v ) = 1 =) (kd;1 ; ekd;1 ) = 0H ::: (k1 ; ek1 ) = (v(k2 ; ek2) + : : : + vd;1 (kd ; ekd )) = 0: . , T hk1 vi : : : T hkn vi ? T hke 1 vi : : : T hke ne vi , (k1 : : : kn), ;(k1 : : : kn ) , & (ke1 : : : kene ), % . . > , % Gmd fT (1:::vd;1 ) j v > 2m + 1 T 2 G 1 g. K% , % h 2 Gmd , U ;1hU 2 Gmd ) U 2 K, % d;1 Gmd f(UT U ;1)(1:::v ) j v > 2m + 1 T 2 G 1 U 2 Kg: (2.1) T d d 2. > G = Gm, % m2N Gmd ) ) m. -), , % Gmd G .
Z
d-
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> Rmnn~ | (k1 : : : knH ke1 : : : kene ), ) Gmd + hk1 : : : hkn hke 1 : : : hke ne ( $) d- ) ) , m (k1 : : : kn) ;(k1 : : : kn) , & (ke1 : : : kene )). . \ \ \ Gmd = fh 2 K(d) j hk1 : : : hkn ? hke 1 : : : hke ne g
N N
n~ 2 n2 k2Rmnn~ , % Gmd G , ,-) &: h ! (hk1 hk2 : : : hkn hke1 hke2 : : : hkene ) K(d) Kn+~n (-
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 =
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<& , & % H0 , $ ) % . ) fh 2 K(d) j ha ? h;a g ,-) a, % ) H0 . A a 2 Zd nf0g. > h ! ha K(d) K fh 2 K(d) j 8b 2 Zn f0g : hab ? h;ab g fT 2 K j 8k 2 Zn f0g : T k ? T ;k g. > , 3], Q, & (
, U 2 K U QU ;1 = Q). > H00 | & Q h ! ha . ., -), H00 fT 2 K j 8k 2 Zn f0g : T k ? T ;k g, -), H00 | G , & $ . E ?, H00 &d;1 ) ; 1 (1 v:::v f(UT U ) j T 2 Q U 2 K : 8k T k ? T ;k g ) d ; 1 v: h(1 v : : : v ) ai 6= 0 (% v > dhjaj 1i). F ) v & , , H00 , 1.
1188
. .
. , % & % (2.2) . 4 , H0 %+ % . . % -+ & && Zd. > C(T) | ) ,-) & ( )), ,-) T 2 K. 7 T . 4. & Wd def fh 2 K(d) j 8a 2 Zd n f0g : C(ha ) = b d = clfh j b 2 Z gg W. . F W1 W , % , U 2 K U W U ;1 = W 4]. < a 2 Zd n f0g Wa | W h ! ha . . , $ G - . ; , f(UTU ;1)v j U 2 K T 2 W hv ai 6= 0g, , 1. A%, Wa . < , , h C(ha ) fhb j b 2 Zdg, %, C(ha ) clfhb j b 2 Zdg, Wa fh 2 K(d) j C(ha ) = clfhb j b 2 Zdgg: % a, % \ \ Wd = fh 2 K(d) j C(ha ) = clfhb j b 2 Zdgg Wa :
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I I J $& % % % %+ . F % | $ & | % % , . P (M I BI I ) % N 8 (M i Bi i). ; % M I i2I % . < %& ? k : I ! Zd n f0g & h 2 K(d) hk , &,- (M I BI I ) hk (m) = (hk(1)m1 hk(2)m2 : : :). ; %, UT | & , ,-& T 2 K, T | & & $ .
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V K(d), h 2 V ! # : X hk - - hk jX ) hv , X Bi i 2 I, k(i) = v. . % V + ) Zd-&&, -) G d H 2 3 . E , % V | K(d). > L02(BI ) & ) fLE gEI E "#$%&$# , N LE ? f i , f i 1 i 2= E f i 2 L02 (i ) i2I i 2 E . E , % LE UQ hk , %+ & & Uhk jLE
+ (hk(i) ). i2E 4 & , hk jX | hv , & & Uhk jL02 (X ) , hv . > 2 & hv Uhk LE , E 6= fig L i 2 I , % k(i) 2 fv ;vg. . , L02(X ) Lfig. i : k(i)2fv;vg P < A 2 X , 0 < (A) < 1, 1A ; (A) = fi , fi 2 Lfig . i2I < , % fi , , ,. E, , % fi1 , fi2 -, C1 C2 Bfi1 g D1 D2 Bfi2 g & , % fi1 (C1) \ fi1 (C2) = fi2 (D1 ) \ fi2 (D2 ) = ?, (1A ; (A))(m) = fi1 (mi1 ) + fi2 (mi2 ) + f (3.1) f | ? , - fmi gi2I nfi1 i2 g . A ? f, , % (3.1) mi1 mi2 . 8 % " ) %) %&H , ( mod 0) & %. > fi1 (x)+fi2 (y) 6= fi1 (z)+fi2 (y) x 2 C1, z 2 C2, y 2 Bfi1 g + + % , fi1 (x) + fi2 (y) 6= fi1 (x) + fi2 (t) x 2 Bfi1 g , y 2 D1 , t 2 D2 , ,% , % fi1 (z) + fi2 (y) = fi1 (x) + fi2 (t) (mod 0) x 2 C1, z 2 C2 , y 2 D1 , t 2 D2 . B %, fi1 (x)+fi2 (t) 6= fi1 (z)+fi2 (t) fi1 (x)+fi2 (t) 6= fi1 (x)+fi2 (y), % fi1 (z) + fi2 (t) = fi1 (x) + fi2 (y) (mod 0) x 2 C1, z 2 C2 , y 2 D1 , t 2 D2 . % , % fi2 (t) = fi2 (y) (mod 0) y 2 D1 , t 2 D2 , % % ,. 4 , A 2 Bi i. > , % i A. > , , % A1 2 X \ Bi1 , A2 2 X \ Bi2 i1 6= i2 0 < (A1 ) (A2 ) < 1. E $ % A1 A2 X Bi | %. A%, X Bi i, %+ k(i) 2 fv ;vg. . hv ? h;v ) h 2 V , k(i) = v.
1190
. .
> J I % %+. < / ? : J ! I ) ,- : (M I BI I ) ! (M J BJ J ) & ( (m))j = m (j ) : N < S : J ! K % % S N & S(j), &,- (M J BJ J ). j 2J E , % ? C(T ) T 2 K ) ,-) & ( )), ,-) T. 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 , %, - ;1 1;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 hg , 5 F & ) - , B1 . E B1 \ B2 , , F . . ; a 2 Zd+ n f0g Zd-& h + d Z -& g, gab = hb ) b 2 Zd. 2. < a 2 Zd+ n f0g - & h Zd, ,- %+ ) & a. > h 2 Wd \ V . A p | - ha 1i % . < & ? c: N ! f1 ha 1ig , c N, ? & & fpc(i)gi2N, %+ & ? % % %. Zd-& hc : hbc hcb1i (hb hb hb : : :). $ & , c ) c. &
1192
. .
hac = cha1i (ha ha : : :) $ , cha1i , % ? , p & ? % % %. . & g: gb hab c . 3. < a b 2 Zd+ nf0g, a b, - & h Zd, a b. > c | "& -& a b ( : ? ). . , % a b, %, % - i, % ai bi . > h 2 Wd \V | ? % bi $ . & g: gf hf ai (hfa hfa : : : hfa) M 1 : : : M bi . ; 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: [email protected] 519.172.2+519.173+519.177
: , , !" , #$ , $% .
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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.
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& $ ( 2]: 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 , N(A) A] \ B = ?, jC j 6 2r + 1 (1) maxfw(A) w(B)g 6 2w(G) 3 : 1 &( " " " " ( ( . 0 ( ( * " " "( . ) " | ( Kn , Kmn , ( 2 | " ( * " " 1
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1196 w(X) =
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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, $ $ @ $ e $ 1 Ce . GT "( , * 1 ,
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1197
1
. 0 1 (6) @ $ 2] w(2Ce ) . ; " 1
" ( "! $ 6]. @$, ( 1 g " . 0($ " $ , 1 h. K $, "! w(Ax ) > w(Bx ) h(x) = jF(Ax)j. 0 @$ 1 Cx , , @$ GS . F "! Ax Bx $ (5), . -* ""! " , w(G) > 3w(Bx ) + 3w(Cx) ; wV (s) + wE (x) +2 wE (y) + wE (z) + 2wF (Fx ) : (8) $ x 1 3- 1
GT . 3-, "! 1 Cx, $ y z, 1 ( * 3-
( . 1 a ). M Cx
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& ( s. -$ !( ( !, ""! (8) "
($ $ . 2.1. *! fy z g . . 9 " ""! , fy z g \ E(GS ) = ?. 1 Cy , $ ( $ $ y GS ,
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! 1 Cz . ) $
, 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 * !(
( 1
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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 ($ $ jF(Az )j > jF (Ax)j, ( " . 2.3. fy z g 2 E(GS ). . 0
! 2.2 " & ( u & v " GS !
$ y. 9 " ""! , z 2= E(GS ) ( . 1 ). 1 Cz , $ ( $ $ z GS , ! & Az Bz , "!(
! 1 Cz . 9 , w(Bx ) = w(Bz ) ; wV (u) ; wE (x) ; wE (y) ; wF (Fx ) (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)
-"! w(Az ) 6 w(Bz ) "
(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) ; (30) , g(z) = g(x). 0 " 1
h ($ $ jF(Az )j > jF (Ax)j, ( " . $, @$ y z (, fy z g E(GS ) ( . 1 ). 9 , w(Ax ) = wF (Fx),
w(G) = w(Bx ) + w(Cx) + wF (Fx ) 6 6 3w(Bx ) + 3w(Cx) ; wV (s) + wE (x) +2 wE (y) + wE (z) + 2wF (Fx ) (31) "
(8). = " & (.
1201
P , @$ ( ( ( 2. ?
, "" @$ wE0 (e) = (1 ; 2 )wE (e) (32)
X wF0 (f) = wF (f) + wE (e) (33) e2Ef
Ef E(GT ) | ! @$, 1 ( fH 6 1=2 | " , , $ " , 1 ( . 9 , 0 w(C) w0(A) + w (C) (34) 2 = w(A) + 2 0 w(C) w0(B) + w (C) = 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 " & "$ (32){(33). 0 , " = 1=2 ( . Q , & @$ ( ( 2. ?
, "" & c0V (v) = (1 ; 2 )cV (v) (38)
@$ X c0E (e) = cE (e) +
cV (v) (39) v2Ve
Ve V (GT ) | ! & , 1 ( $ eH 6 1=2 | " , , $ " , 1 @$. 9 , c0 (C) = c(C) $ 1 C GT . , " C " & "$
(38){(39). 0 , " = 1=2, & .
4. !
2 " ( " ( . ?
$( " " " G, $ !
" $
@$ ( , $ 1 GT . 9 ( $ " * " G !
1202
. .
. -* $ ( @$ , ! F "@ ! 3- , $ (
" 1
F. ? & "1
* " ! $( (. 9" "* ! !. . 1 F " G , $ @ 3- 1 $( @$
G. 0 1
GT " G ( . 0 $, , 1
" "$ 20], "( ! , 1
. " 3. G F , #%! 3. 1) + % ! ! ! F . 2) , # ! F 3- (F1 F2) G, 3- G. 3) -# ! F , . 3- F1 F2 ( !! ! !( ( . . -!, ! "
1 " F, " (& , 3. ? * ($ " !( $ b1 b2 , 1 ( F ( . 2 ). 0 , & ( v1 v2 !( G. - @ F "
$ g1 ! & v1 v2 . 9 , $ g1 3- f1 1 @$ b1 b2 G. 0 , & ( v1 v2 ! !( G ( . 2 ), !( @$ b1 b3 & ( v0 v3 . - F " (& 3, G " , & ( v0 v3 !( G. 0 * " @ F "
$ g1 ! & v0 v3 . S & " " * . -* $ $, $
, "
$ g1 " F ! & v1 v2 . F F $( 4- , 1 &. 0 " @$ b3 b4 , !( $ g1 ( . 2 ). - F " (& 3, G " , $ & ( v2 v3 , $ & ( v1 v4 !( G. K $, $
, !( & ( v2 v3 . - @ F ! "
$ g2. 9 , $ g2 3- f2
1 $ b3 G. "1 " "
( @$ ( " ", " 1 $ &. ; ( = , $( m, $ " m ; 3 "
( @$, $ , m ; 2 3- . 9 , m ; 4 1 ( $ G, 3- f1 1 -
1203
2#. 2. 2 ' .%" F
@$. -* " 3- fm;2 1 @$ G. ; , ( " ( ! (. " ! . ? !( ( 3 $ fG 2 F(G). 0-" (, ! FfG 3- , $ ( "
1
fG , ! $( & , $. - 1
fG , " (& , 3,
3 $ 3- , ! ( 1 @$ fG . - ! * 3- , ( " F, ( 3- . - ! 3- G @ 3- GT , @ 3- GT . 9 , wF F(GT )] = wF F (G)]: (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.
? " " !
" " G ( " 2 1
GT . " $, ~ 9 , ( ,
G " G. ! C, , @$, ( & @$
1204
. .
~ " G~ " C~ 1
G~ T G, ~ ~ ( . 3). F $ E " C $ "1 1
~ " C & v 2 V (G), F~ F (G), ~ F. -* ! $ $ ! $( & (, . , " C~ @$, " @( "1 1
! , $ " (, $(
$ " " @ , " .
2#. 3. >$ C~ .%" G~T 0 # G~. @# 0 G 0% $.# ' ' $. ! ' )*'. 2* #$ C~ .%A
( " " ( @$ & . ? ( " ( @$, , !, & . - , ( " ( $ . 0 , ( " " ~ = , G " @$ G. 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 @ " ( "(, " $. ~
" ( G~ S G~ T . - $ 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~ ~ T ) w(C) ~ x=EF~ w( G ~ ~ minfw(A) w(B)g > 3 ; 2 ; (44) 6 wm | ( , 1 C~ " !, "! $& ( , F~e F (G~ T ) | , 1 $ e~, E~e | ! @$, 1 ( F~e. G F~e "~ B~ gH w(A) ~ = w(B) ~ ! $ !@ "! fA "! "! $& . ~ 1 ( ! F~C G, ( @$ C~ ( !, ). 0 (
G~ !
"! F~A F~B , " !, A~ B~ . 0
3 "1 & G~ T
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 " ( G ! , , ~ @$, ( @$ " , ( & 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 $ (. S $ & "( @$ " & ( " . 9$ , , . K , & "( "
,
&, @$ " . -* 4 ( 2 ( , .
6. $
> ( ( 2, ( 3 ( 4 (
" " " G, , n * . - , : 1. 0! " G " . 0: O(n) "
"
, "! $ 21]. 2. 1 " G. 0: O(n). 3. - GS & " .
1207
4. ! 1
g @$ ! E(GT ) n E(GS ) $ ". 0: O(n). - " "
( "1( " ( . -(
, & : 1. 0! " G " . 0: O(n) "
"
, "! $ 21]. 2. - G.~ 0: ~O(n). 3. 1 " G. 0: O(n). ~ 4. - GS & " . 5. ! 1
g @$ ! E(G~ T ) n E(G~S ) $ ". 0: O(n). 6. - " . 0: o(n), ""1 " . 0 * "(, & 3 " & 4 , , $( ("( O(n). - G~ S & " "$ " 22] $, " "$ . "( , " $ !@ & * "$( O(n) " O(n lg n) 23].
7. & ( 2 4 1 " ! 1 " "! . 0 ( " @ " (, "( , , * 1 ( . ( 2 4 ( ( " ! . 0!( ( ( 2 ( & . 2.1. ! G GS d ! GT . V (G) A, B , jC j 6 d + 1, N(A) A] \ B = ? 6 minfjAj jB jg > 2jV (G)j ; 3jC j + 1: (51) " . 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( " " ( . = "
@$( & -@$( " (, , & (), $(( & ( " (. 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 Algorithms. | Reading, MA: Addison-Wesley, 1974. 2] Lipton R. J., Tarjan R. E. A separator theorem for planar graphs // SIAM J. Appl. Math. | 1979. | Vol. 36, no. 2. | P. 177{189. 3] Smith W. D., Wormald N. C. Geometric separator theorems and applications // 39th Annual Symposium on Foundations of Computer Science | FOCS '98. | Palo Alto, CA, 1998. | P. 232{243. 4] Alon N., Seymour P., Thomas R. Planar separators // SIAM J. Discrete Math. | 1994. | Vol. 7, no. 2. | P. 184{193. 5] Chung F. R. K., Graham R. L. A survey of separator theorems // Paths, Flows, and VLSI Layouts / B. Korte et al. (eds.). | Springer, 1990. | P. 17{34 6] Djidjev H. N. On the problem of partitioning planar graphs // SIAM J. Alg. Discrete Math. | 1982. | Vol. 3, no. 2. | P. 229{240. 7] Miller G. L. Finding small simple cycle separators for 2-connected planar graphs // Journal of Computer and System Sciences. | 1986. | Vol. 32. | P. 265{279. 8] Kirkpatrick D. Optimal search in planar subdivisions // SIAM J. Comput. | 1983. | Vol. 12. | P. 28{35. 9] Chiba N., Nishizeki T., Saito N. Applications of the Lipton and Tarjan planar separator theorem // J. Info. Proc. | 1981. | Vol. 4, no. 4. | P. 203{207. 10] Ravi S. S., Hunt H. B., III. An application of the planar separator theorem to counting problems // IPL. | 1987. | Vol. 25, no. 5. | P. 317{322. 11] Miller G. L., Teng S.-H., Thurston W., Vavasis S. A. Geometric separators for nite element meshes // SIAM J. Sci. Comput. | 1998. | Vol. 10, no. 2. | P. 364{386. 12] Miller G. L., Teng S.-H., Thurston W., Vavasis S. A. Separators for sphere-packings 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. 14] Fellows M., Hell P., Seyarth K. Large planar graphs with given diameter and maximum degree // Discrete Appl. Math. | 1995. | Vol. 61. | P. 133{153. 15] ., !" #. "$% &'(! ) & *$ +!"$$ ," -. | #.: #$", 1998. 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
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). G 5 /. "/ T=G = (A=G)q | / 8 q6=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).
. G 5 /. " / 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 . . G 5 /. @ $ 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 ). . G 5 /. @ 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. ,. / . 9 5.
. | 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
,
,
. :
| ,
.
,
(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 j E 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; (r C)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 _ >: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
. .
B 1$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] + *$), - )% % )) % (A0 k kA0 ) *) . 2 4) *$ , - )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-
f ((^a ; (^a(M ) ; "=2)) _ 0) ^ (((^a(M ) + "=2) ; a^) _ 0) ^ ^ ((^b ; (^b(M ) ; "=2)) _ 0) ^ (((^b(M ) + "=2) ; ^b) _ 0) ^ ^ ((^c ; (^c(M ) ; "=2)) _ 0) ^ (((^c(M ) + "=2) ; ^c) _ 0) ^ ^ ((d^ ; (d^(M ) ; "=2)) _ 0) ^ (((d^(M ) + "=2) ; d^) _ 0): ), &") f *% )4 A^0 . B)- , G coz f 2 G . @ " a(M ) < a^(M ) + " ^a(M ) ; < ^ 2 2 ^b(M ) ; " < ^b(M ) < ^b(M ) + " 2
2
c^(M ) ; "2 < c^(M ) < c^(M ) + 2"
d^(M ) ; 2" < d^(M ) < d^(M ) + "2 M 2 G. ?" N 2 G. @ N 2 coz ^a ; a^(M ) ; 2" _ 0 \ coz a^(M ) + 2" ; a^ _ 0 \ " _ 0 \ coz ^b(M ) + " ; ^b _ 0 \ \ coz ^b ; ^b(M ) ; 2
2
2
2
" _ 0 \ coz c^(M ) + " ; c^ _ 0 \ \ coz c^ ; c^(M ) ; 2 2 " \ coz d^ ; d^(M ) ; _ 0 \ coz d^(M ) + " ; d^ _ 0
-<
a^(M ) ; 2" < a^(N ) < ^a(M ) + 2" ^b(M ) ; " < ^b(N ) < ^b(M ) + " 2
2
2
2
c^(M ) ; " < c^(N ) < c^(M ) + "
d^(M ) ; 2" < d^(N ) < d^(M ) + 2" : @
kf (M ) ; f (N )k = = ka^(M ) ; a^(N ) + i(^b(M ) ; ^b(N )) + j (^ c(M ) ; c^(N )) + k(d^(M ) ; d^(N ))k =
1248 =
. .
q
(^ a(M ) ; ^a(N ))2 + (^b(M ) ; ^b(N ))2 + (^c(M ) ; c^(N ))2 + (d^(M ) ; d^(N ))2
p
6 " =4 + " =4 + " =4 + " =4 6 ": : ,
2
2
2
6
2
y 2 C (Max A0 Q).
2. ) * : A ! C (Max A0 Q) . . ?" a b c d a0 b0 c0 d0 2 A0 . @ ((a + Ib + Jc + Kd) + (a0 + Ib0 + Jc0 + Kd0)) = = ((a + a0) + I (b + b0) + J (c + c0) + K (d + d0)) 0 (a + a0 ) + i0 (b + b0) + j0 (c + c0) + k0 (d + d0) = = (a + Ib + Jc + Kd) + (a + Ib + Jc0 + Kd0):
> %
((a + Ib + Jc + Kd)) (a0 + Ib0 + Jc0 + Kd0)) = = (aa0 ; bb0 ; cc0 ; dd0 + (ab0 + ba0 + cd0 ; dc0)I + + (ac0 ; bd0 + ca0 + db0)J + (ad0 + da0 ; cb0 + bc0)K ) 0 (aa0 ; bb0 ; cc0 ; dd0) + i0 (ab0 + ba0 + cd0 ; dc0) + + j0 (ac0 ; bd0 + ca0 + db0) + k0 (ad0 + da0 ; cb0 + bc0 ): 9 " - 0 | $ %& $ " )4) 6 )
i j k 2 Q * $ " % +, *%+" % )
*% )
((a+Ib+Jc+Kd))(a0 +Ib0 +Jc0 +Kd0)) = (a+Ib+Jc+Kd)(a0 +Ib0 +Jc0 +Kd0): @4 ), - (1A ) = 1, (I ) = i, (J ) = j (K ) = k. ?" (a + Ib + Jc + Kd) = (a0 + Ib0 + Jc0 + Kd0). @ 0 (a) + i0 (b) + + j0 (c) + k0 (d) = 0 (a0 ) + i0 (b0 ) + j0 (c0) + k0 (d0 ). 9 *$ ; ,
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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.
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