Фундаментальная и прикладная математика (2002, №4)

1- . . . . . 517.968 ...

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1- . .

. . .

517.968

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Abstract

A. F. Voronin, Volterra convolution equation of rst kind on segment, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 4, pp. 955{966.

In the paper we obtain the decidability conditions for Volterra convolution equation of .rst kind on a segment and the solution of this equation (in quadratures).

1- 0 b], b > 0: Zx k(x ; t)u(t) dt = f(x) x 2 0 b] (0.1) 0

"# p = x + iy %& F k(p) :=

Zb

k 2 L1 (0 b) f 2 L2 (0 b):

(0.2)

eipt k(t) dt | % ( ) {+% , - k:

0

. /, / # 0 s > ;1, y0 > 0 0%#1# 12 3 #: lim t;s k(t) = C0 6= 0 (0.3) t!+0 1+s (0.4) jxlim j!1 j(x + iy0 ) F k(x + iy0 )j > C1 > 0: / ' 0 %% 1 $ /223 99-01-00540.

, 2002, " 8, 4 4, . 955{966. c 2002 !" #$ %&', ( $ ) *

956

. .

( # (0.1) % # (0.2){(0.4). 9 %21 0 / # / , -: 1, 2] % / 0 ( : , ) # 3 # (0.1) , # 3 #. / (% # (0.2){(0.4)), : 0 , / ;/3 3, . 415].

1.

0 (0 % 0 # :. 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

93 (2.28) & 3 % , / v(x) = 0, x 2 (0 b). ; (2.25) % / (0.1) , -1 u. ; 2 (% % 0). J , 3.

3.

9 1 (0.3) & /, / # k % (0 b) 12 : k(t) = (C0 + (t))ts t 2 (0 b) (3.1)

(t) 2 L1 (0 b) (t) = o(1) % t ! +0: (3.2) ipt ;, # (3.1) % t 0 b e , % / F k(p) = C0

=&

Zb 0

eipt ts dt +

Zb

eipt ts (t) dt:

(3.3)

0

(t) := ;C0 % t > b

(3.3) F k(p) = C0I1 (p) + I2 (p)

Imp > 0

(3.4)

1-

Z1

Z1

0

0

I1 (p) = eipt ts dt I2 (p) =

963

eipt ts (t) dt:

? (3.4) % / jF k(p)j > jC0j jI1(p)j ; jI2 (p)j Im p > 0: (3.5) G- 0 % : / (3.5). = 0: (/0: 5, c. 35], I1 (p) = (s + 1)(;ip);(s+1) Imp > 0: (3.6) "# % 12# / # - : jI2 (p)j 6

; (3.5){(3.7) % /

Z1

e;yt ts j (t)j dt y > 0:

(3.7)

0

1 x s+1 s+1Z ;yt s s +1 y e t j (t)j dt j(x+iy) F k(p)j > j(s+1)C0 j; + i y

y > 0: (3.8)

0

= L( # 5, c. 157] # y ! 1, / # (3.2), % / y

1+s

Z1

e;yt tsj (t)j dt (s + 1)j (+0)j = o(1):

(3.9)

0

? (3.8){(3.9) # jxj 6 y / j(x + iy)s+1 F k(x + iy)j > j(s + 1)C0j % y ! 1: (3.10) 9 (3.10) # % x. ; (3.10), # , - F k(p) ( / : / % % Imp > 0, % 0 # jxj 6 y. G # /# jxj > y > y0 . +1( 1 / p = x + iy ( jxj > y > y0 ( % # 12 : p = x + ijxj tg( ) + iy0 0 < 6 4 : ; (3.5){(3.6) # y > y0 % / jps+1 F k(p)j > j(s + 1)C0j ; s+1 Z1 y 0 s +1 jxj expf;tjxj tg( )ge;y0 t tsj (t)j dt: (3.11) ; sgn(x) + i tg( ) + i jxj 0

964

. .

= L( # 5, c. 157] # jxj ! 1, / # (3.2), % / 1+s

jxj

Z1

Z1 ; y t s 1+s 0 expf;tjxj tg( )ge t j (t)j dt 6 jxj ts j (t)j dt = o(1): (3.12)

0

0

? (3.11){(3.12) # 0 < 6 4 / j(x + iy)1+s F k(x + iy)j > j(s + 1)C0j % jxj ! 1 (3.13) %/ (3.13) 0%# # % 2 0 4 ] # 1( 0 2 (0 4 ). ? # (0.4) , / (3.13) 0%# # # = 0 (y = y0 ). = /, / (3.13) 0%# # % 2 0 4 ] # y > y0 . + .

4. ! 4.1. =

k(t) = 1 % t 2 0 b] f 2 L2 (0 b): ; (0.1) ( 12: : Zx

u(t) dt = f(x)

(4.1)

x 2 0 b]:

0

?

eipb ; 1 : ip 9 , # s = 0 (m = 0), y0 > 0 # (0.3){(0.4) 0%#1#, F k(p) =

u (x) = ; 2i

Z1

;1

e;ix( +i) 1F;f(eib+( +i) i) d

x 2 (0 b)

(4.2)

> y0 . =& , / % # / (4.2) ;if(x). , % / %& , -1 f (b 1) % b (f(x) = = f(x + b), x 2 (0 1)). ? 12# , # ( +% % / , -:: Z1 0

e;pt f(t) dt =

b

1 Z e;pt f(t) dt 1 ; e;bp

p = ;i( + i):

(4.3)

0

; (4.3) % % /, / 1 f(t) = 2

Z1

;1

e;it( +i) 1F;f(eib+( +i) i) d:

(4.4)

1-

965

? (4.4) (4.2)

u (x) = ;if(x) x 2 (0 b): (4.5) ? (4.5), 2, % /, / 3 # (0.1) % (4.1) 2 L2 (0 b) , f 0 2 L2 (0 b) xlim !+0 f(x) = 0

% F u(x) = f 0 (x), x 2 (0 b). 4.2. = %&, / # # (0.1), : (0.2), (1.3), 0% 12 : k0 2 L1 (0 b) k(0) = 1 k(b) = 0: (4.6) ; % , # % /# b

1 1 + Z eipt k0 (t) dt : F k(p) = ; ip

(4.7)

0

? (4.7) / 3 # Zb

eiptk0 (t) dt ! 0 % Imp ! 1 ( % Re p 2 R)

0

0 , / (1.5) 0%# # # m = 1 / (3 > 0. ; (1.1) (4.7) % / 1 @ u (x) = 2 @x

Z1

;1

F f( + i)d e;ix( +i) ( + i)(1 + F k0( + i)) 2 L2 (0 b):

? (4.8) , / / , /

f 0 2 L2 (0 b)

(4.8) (4.9)

1 + F k0(p) > C > 0 % Imp >

u 2 W21 (0 b): (4.10)

, 3 (4.10) 0 (4.8){(4.9) 12 : i (f(b)eib( +i) ; F f 0 ( + i)) 2 R F f( + i) = ; + i Z1 ib( +i) e;ix( +i) ( + i)2e(1 + F kd0 ( + i)) = 0 x 2 (0 b):

;1

966

. .

; 0 3 % /, / (0.1) % # (0.2), (1.3), (4.6), (4.9) 3 L2 (0 b). = F 3 F f 0 ( + i) d t ; 1 u(x) = e F (4.11) 1 + F k0( + i) (x) x 2 (0 b): 9 : 0, % 0% 0 # (0.2), (1.3) (4.6), (4.9). ;, ,, - # (0.1) % x, % / Zx

u(x) + k0 (x ; t)u(t) dt = f 0 (x) x 2 (0 b):

(4.12)

0

? , / 2- (4.12) 3 L2 (0 b), , 3 . J , % b ! 1 3 # (4.12) & , (4.11). L % L. +. . : ( & : (0.

" 1] . ., . . . | .: , 1978. 2] $ . . %&'( &) *), +,) + %) // ). . | 1958. | /. 13, 1 5 (83). | 4. 3{120. 3] /7)8 9. $. :* ; %& '. | .-=.: +*, 1948. 4] =' . ?., @A B. :. *( CD )&% )%. | .: , 1987. 5] : * E&' B., B)) F. GD 7& *% A+ =&. | .: =, 1952.

"+ 2000 #.

. .

. . .

519.95

: , , , , .

, ! "# $ , " $% " # , $ , " $% "

.

Abstract

V. Sh. Darsalia, Relative completeness for functional systems of polynomials, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 4, pp. 967{977.

For functional systems of polynomials with natural, integer and rational coe/cients we solve the problem of completeness of sets, containing all monomials, and sets, containing all polynomials of one variable.

N, Z Q | ( 0), X 2 fN Z Qg. PX

!, " # , $ # # X . %

X (" . ., . . . . )& , # , " " $ X , (" X 2 fN Z Qg, ) . ., . . . . . U = fu1 u2 : : : um : : :g, (" um | # X (m = 1 2 3 : : :), | . . " # U " x, y, z , t, " . . ") # , , /, r " PX : (f )(x1 : : : xn) = f (x2 x3 : : : xn x1) (f )(x1 : : : xn) = f (x2 x1 x3 : : : xn) (/f )(x1 : : : xn;1) = f (x1 x1 x2 : : : xn;1) n > 1 (f ) = (f ) = (/f ) = f n = 1 (rf )(x1 x2 : : : xn xn+1) = f (x2 x3 : : : xn xn+1): , 2002, 8, 0 4, . 967{977. c 2002 , !" #$ %

968

. .

4, f (x1 x2 : : : xn) g(xn+1 : : : xn+m ) PX , # ( (f g)(x2 : : : xn xn+1 : : : xn+m ) = f (g(xn+1 : : : xn+m ) x2 : : : xn): % # . 5( FX = (PX 6), (" 6 = f / r g, ) ( . .)

X . . . FX I , ! " # " M (M PX ) # " I (M ), 7 !, # M # 7 ( # #! 6. 8 ) M , I (M ) = M , , I (M ) = PX . 9 (" # , "7 " # " , ( & , . . FX M T , T M M # FX . 8:! "! " . . FN, FZ FQ ! # , "7 " , , "7 " ! # !. ; . . FN. 8. #. f (x1 : : : xn) , " i j (1 6 i < j 6 n) !" ak E2 = f0 1g (k = = 1 : : : i ; 1 i + 1 : : : j ; 1 j + 1 : : : n), f (a1 : : : ai;1 xi ai+1 : : : aj ;1 xj aj +1 : : : an) = xi + xj : 8. #. f (x1 : : : xn) , " i j (1 6 i < j 6 n) !" ak E2 = f0 1g (k = = 1 : : : i ; 1 i + 1 : : : j ; 1 j + 1 : : : n), f (a1 : : : ai;1 xi ai+1 : : : aj ;1 xj aj +1 : : : an) = xixj : . ") "7 : V0 PNn f0g& V1 | . #. !, " ! " & V+ | "" . #. !& V | # . #. !. =# " "7 # >1]: . . FN M (M PN) , M * V0 M * V1 M * V+ M * V : 1. M . . , . . , . . FN , M .

. M # FN& (" # M * V+ . =" , M " "" .

969

@# " . M , "7 . #. " , " "" & (" M * V+ . @ 0 2 M , 1 2 M , xy 2 M , M * V0, M * V1 M * V . =" , # M # ! FN. @ " .

2. , , M . . ! . . F .

N

. A #" "" ! ", :( , " . #. "" !. =" , :( , " " "" . . 1, M " "" ,

. #. " # , " "" ! ,

. #. "

# ! . @ " . 3. M . . , . . , . . FN , M

.

. M # FN& (" # M * V+ M * V . =" , M " ""

# . @# " . M , "7 . #. " ! # !, " "" # & (" , M * V0 , M * V1 , M * V+ M * V . =" , # M # ! FN. @ " .

4. , , M . . ! . . F .

N

. A #" "" ! ", :( , " . #. "" !. A #" # ! ", :( , " . #. # !. =" , :( , " " M "" # . . 3, M " "" # ,

. #. ! " ! # ! # , " "" !

# ! ,

. #. ! " ! # ! # ! . @ " .

970

. .

; . . FZ .

1. " . . f (t1 : : : tn) M , ! . . . . >f 2 (t1 : : : tn )+1] (x ; y) . . F , f (t1 : : : tn) Z.

Z

. M | # . 4 # , f (t1 : : : tn) Z& (" f 2 (t1 : : : tn)+1 > 2 " ! # t1 : : : tn. =" , >f 2 (t1 : : : tn ) + 1] (x ; y) # ! Z( , 1). 4, # # " # "

H1 H2 H3 : : : ! PZ . C ". H1 = M . A " ! # ". # H1 : : : Hl & (" Hl+1 #" ## ! " g(h1 : : : hm ), (" g | M , h1 : : : hm | # , 1Hl . D , S Hl = I (M ). l=1

= # 7 ! " # l # , S Hl l=1 " ! ! . #. " x y + c, (" c | # Z. C ". " , H1 " " x y + c. A " ! # ". Hl " " x y + c& (" # , Hl+1 " " x y + c. ; # ## h = g(h1 : : : hm ) Hl+1 , (" g | M , h1 : : : hm | # , Hl . . "7 . 1. g = >f 2 (t1 : : : tn) + 1] (x ; y)& (" , m = n + 2 h = = g(h1 : : : hm ) = >f 2 (h1 : : : hn) + 1] (hn+1 ; hn+2 ). % ## # 1& " , " x y + c. 2. g = dz1k1 : : :zsks , (" d | , k1 : : : ks | & (" , m = s h = g(h1 : : : hm ) = dhk11 : : :hks s . E( , $ ## ! " x y + c, ) d = 1& ) 7 i (1 6 i 6 m), g(z1 : : : zm ) = zi + a (a | ) hi ! " x y + c. 8 hi | # , Hl , " x y + c, . . h " x y + c. =" , Hl+1 " " x y + c. 1

971

A, " x y + c " S Hl = I (M )& l=1 # $ I (M ) 6= PZ . # . F , M | # , f (t1 : : : tn) Z. @# " . f (t1 : : : tn) Z, . . 7 ! c1 : : : cn Z, f (c1 : : : cn) = 0. @ c1 : : : cn 2 M , >f 2 (c1 : : : cn) + 1] (x ; y) = x ; y 2 I (M ). @ , I (M ) " # " fx ; y xy 1g, # ! FZ>1]. =" , M | # . E

" . 1

5. # , , . . ! . . F .

Z

. 4 # # & # 7 ( 5, # 7!, # . #. !

. #. " # . . , 5 # ), " $ f>f 2(t1 : : : tn)+1] (x ; y)g, (" f (t1 : : : tn) | # PZ ,

. #. " # . 8 ("

1 7 ( , # 7!, # . #. f (t1 : : : tn) Z, . . 7 ( " : # ( " . % # >2]. @ " . 2. " . . f (t1 : : : tn) M ,

! . . . .

>f 2 (t1 : : : tn) + 1] (x ; y), >f 2 (t1 : : : tn) + 1] (xy), . . FZ

, f (t1 : : : tn) Z. . M | # . 4 # , f (t1 : : : tn) Z& (" f 2 (t1 : : : tn) + 1 > 2 " ! # t1 : : : tn. =" , >f 2 (t1 : : : tn) + 1] (x ; y) >f 2 (t1 : : : tn) + 1] (xy) # ! Z ( , 1). 4, # # " # "

H1 H2 H3 : : : ! PZ . C ". H1 = M . A " ! # ". # H1 : : : Hl & (" Hl+1 #" ## ! " g(h1 : : : hm ), (" g | M , h1 : : : hm | # , Hl . 1 D , S Hl = I (M ). l=1

972

. .

= # 7 ! " # l # , S Hl l=1 " ! ! . #. " x y + c, (" c | # Z. C ". " , H1 " " x y + c. A " ! # ". Hl " " x y + c& (" # , Hl+1 " " x y + c. ; # ## h = g(h1 : : : hm ) Hl+1 , (" g | M , h1 : : : hm | # , Hl . . "7 . 1. g = >f 2 (t1 : : : tn) + 1] (x ; y)& (" , m = n + 2 h = = g(h1 : : : hm ) = >f 2 (h1 : : : hn) + 1] (hn+1 ; hn+2 ). % ## # 1& " , " x y + c. 2. g = >f 2 (t1 : : : tn) + 1] (xy)& (" , m = n + 2 h = = g(h1 : : : hm ) = >f 2 (h1 : : : hn) + 1] (hn+1 hn+2). % ## # 1& " , " x y + c. 3. g | " M , . . " h = g(z ) = ck z k + ck;1z k;1 + : : : + c1 z + c0& (" , m = 1 h = g(h1 : : : hm) = g(h1 ) = ck hk1 + ck;1hk1 ;1 + : : : + c1 h1 + c0 : E( , h ! " x y + c, ) g(z ) = z + c0 & ) h1 ! ! ! " x y + c. 8 h1 | # , Hl , " x y + c, . . g(h1 : : : hm ) " x y + c. =" , Hl+1 " " x y + c. S1 A, " x y + c " Hl = I (M )& l=1 # $ I (M ) 6= PZ . # . F , M | # , f (t1 : : : tn) Z. @# " . f (t1 : : : tn) Z, . . 7 ! c1 : : : cn Z, f (c1 : : : cn) = 0. @ c1 : : : cn 2 M , . #. >f 2 (c1 : : : cn) + 1] (x ; y) = x ; y 2 I (M ) . #. >f 2 (c1 : : : cn) + 1] (xy) = xy 2 I (M ). @ , I (M ) " # " fx ; y xy 1g, # ! FZ . =" , M | # . E

" . 1

6. # , , . . ! . . F .

Z

973

. 4 # # & # 7 ( 5, # 7!, # . #. !

. #. ! " ! # ! # . @ (", , 5 # ), " $ f>f 2 (t1 : : : tn) + 1] (x ; y) >f 2 (t1 : : : tn) + 1] (xy)g, (" f (t1 : : : tn) | # PZ ,

. #. ! " ! # ! # . 8 ("

2 7 ( , # 7!, # . #. f (t1 : : : tn) Z, . . 7 ( " : # ( " . % # >2]. @ " . !") . c. FQ. 3. " . . f (t1 : : : tn) M , ! . . f 2 (t1 : : : tn) (x ; y)2 + (x ; y), . . FQ , f (t1 : : : tn)

Q. . M | # . 4 # , f (t1 : : : tn) Q. # " # "

H1 H2 H3 : : : ! PQ. C ". H1 = M . A " ! # ". # H1 : : : Hl & (" Hl+1 #" ## ! " g(h1 : : : hm ), (" g | M , h1 : : : hm | # , 1Hl . S D , Hl = I (M ). 1 l=1 = # 7 ! " # l # , S Hl l=1 " ! ! . #. " x y + c, (" c | # Q. C ". " , H1 " " x y + c. A " ! # ". Hl " " x y + c& (" # , Hl+1 " " x y + c. ; # ## h = g(h1 : : : hm ) Hl+1 , (" g | M , h1 : : : hm | # , Hl . . "7 . 1. g = f 2 (t1 : : : tn) (x ; y)2 + (x ; y)& (" , m = n + 2 h = g(h1 : : : hm ) = f 2 (h1 : : : hn) (hn+1 ; hn+2 )2 + (hn+1 ; hn+2 ) = = (hn+1 ; hn+2 )>f 2 (h1 : : : hn) (hn+1 ; hn+2 ) + 1]. E( , h

974

. .

! " x y + c, (" hn+1 ; hn+2 1 f 2 (h1 : : : hn) (hn+1 ; hn+2) + 1 x y + c hn+1 ; hn+2 x y + c f 2 (h1 : : : hn) (hn+1 ; hn+2 ) + 1 1: . # f 2 (h1 : : : hn) x y + c ; 1. 8 $ & " , ## g(h1 : : : hm ) " x y + c. . f 2 (h1 : : : hn)(x y + c) 0 ( $ , f (t1 : : : tn) 6= 0 " ! # t1 : : : tn) f 2 (h1 : : : hn)(x y + c) 2 ( )& " , ## g(h1 : : : hm ) " x y + c. 2. g = dz1k1 : : :zsks , (" d | , k1 : : : ks | & (" , m = s h = g(h1 : : : hm ) = dhk11 : : :hks s . E( , $ ## ! " x y + c, ) d = 1& ) 7 i (1 6 i 6 m), g(z1 : : : zm ) = zi + a (a | ) hi ! " x y + c. 8 hi | # , Hl , " x y + c, . . h " x y + c. =" , Hl+1 " " x y + c.1 A, " x y + c " S Hl = I (M )& l=1 # $ I (M ) 6= PQ. # . F , M | # , f (t1 : : : tn) Q. @# " . f (t1 : : : tn) Q, . . 7 ! c1 : : : cn Q, f (c1 : : : cn) = 0. @ c1 : : : cn 2 M , f 2 (c1 : : : cn) (x ; y)2 + (x ; y) = x ; y 2 I (M ). @ , I (M ) " # " fx ; y xy 1=2 1=3 : : : 1=p : : :g ((" p | # # ), # ! FQ>2]. =" , M | # . E

" .

7. # , , . . ! . . F .

Q

. 4 # # & # 7 ( 5,

# 7!, # . #. !

. #. " # . . , 5 # ), " $ f>f 2 (t1 : : : tn)(x ; y)2 +(x ; y)g, (" f (t1 : : : tn) | # PQ,

. #. " # . 8 ("

3 7 ( , # 7!, # . #. f (t1 : : : tn) Q, . . 7 ( " : # ( " . % # >2]. @ " .

975

4. " . . f (t1 : : : tn) M , ! . . f 2 (t1 : : : tn) (x ; y)2 + (x ; y), f 2 (t1 : : : tn) (xy)2 + xy, . . F , f (t1 : : : tn) Q.

Q

. M | # . 4 # , f (t1 : : : tn) Q. # " # "

H1 H2 H3 : : : ! PQ. C ". H1 = M . A " ! # ". # H1 : : : Hl & (" Hl+1 #" ## ! " g(h1 : : : hm ), (" g | M , h1 : : : hm | # , 1Hl . D , S Hl = I (M ). 1 l=1 = # 7 ! " # l # , S Hl l=1 " ! . #. " x y + c, (" c | # Q. C ". " , H1 " " x y + c. A " ! # ". Hl " " x y + c& (" # , Hl+1 " " x y + c. ; # ## h = g(h1 : : : hm ) Hl+1 , (" g | M , h1 : : : hm | # , Hl . . "7 . 1. g = f 2 (t1 : : : tn) (x ; y)2 + (x ; y)& (" , m = n + 2 h = g(h1 : : : hm ) = f 2 (h1 : : : hn) (hn+1 ; hn+2 )2 + (hn+1 ; hn+2 ) = = (hn+1 ; hn+2 )>f 2 (h1 : : : hn) (hn+1 ; hn+2 ) + 1]. E( , h ! " x y + c, (" hn+1 ; hn+2 1 f 2 (h1 : : : hn) (hn+1 ; hn+2) + 1 x y + c hn+1 ; hn+2 x y + c f 2 (h1 : : : hn) (hn+1 ; hn+2 ) + 1 1: . # f 2 (h1 : : : hn) x y + c ; 1. 8 $ & " , ## g(h1 : : : hm ) " x y + c. . f 2 (h1 : : : hn)(x y + c) 0 ( $ , f (t1 : : : tn) 6= 0 " ! # t1 : : : tn) f 2 (h1 : : : hn)(x y + c) 2 ( )& " , ## g(h1 : : : hm ) " x y + c.

976

. .

2. g = f 2 (t1 : : : tn) (xy)2 + xy& (" , m = n + 2 h = = h(g1 : : : gm ) = f 2 (h1 : : : hn) (hn+1 hn+2)2 + hn+1 hn+2 = (hn+1 hn+2 ) >f 2 (h1 : : : hn) (hn+1 hn+2) + 1]. E( , h ! " x y + c, (" hn+1 hn+2 1 f 2 (h1 : : : hn) (hn+1 hn+2 ) + 1 x y + c hn+1 hn+2 x y + c f 2 (h1 : : : hn) (hn+1 hn+2) + 1 1: . # f 2 (h1 : : : hn) x y + c ; 1. 8 $ & " , ## g(h1 : : : hm ) " x y + c. . f 2 (h1 : : : hn)(x y + c) 0 ( $ , f (t1 : : : tn) 6= 0 " ! # t1 : : : tn) f 2 (h1 : : : hn)(x y + c) 2 ( )& " , ## g(h1 : : : hm ) " x y + c. 3. g | " M , . . " h = g(z ) = ck z k + ck;1z k;1 + : : : + c1 z + c0& (" , m = 1 h = g(h1 : : : hm) = g(h1 ) = ck hk1 + ck;1hk1 ;1 + : : : + c1 h1 + c0 : E( , h ! " x y + c, ) g(z ) = z + c0 & ) h1 ! ! ! " x y + c. 8 h1 | # , Hl , " x y + c, . . g(h1 : : : hm ) " x y + c. =" , Hl+1 " " x y + c. 1 A, " x y + c " S Hl = I (M )& l=1 # $ I (M ) 6= PQ. # . F , M | # , f (t1 : : : tn) Q. @# " . f (t1 : : : tn) Q, . . 7 ! c1 : : : cn Q, f (c1 : : : cn) = 0. @ c1 : : : cn 2 M , . #. f 2 (c1 : : : cn) (x ; y)2 +(x ; y) = x ; y # " I (M ) . #. f 2 (c1 : : : cn)(xy)2 +(xy) = xy # " I (M ). @ , I (M ) " # " fx ; y xy 1=2 1=3 : : : 1=p : : :g ((" p | # # ), # ! FQ. =" , M | # . E

" .

8. # , , . . ! . . . . F .

Q

977

. 4 # # & # 7 ( 5, # 7!, # . #. !

. #. " # . . , 5 # ), " $ f>f 2 (t1 : : : tn)(x ; y)2 + (x ; y), f 2 (t1 : : : tn) (xy)2 + xyg, (" f (t1 : : : tn) | # PQ,

. #. " # . 8 ("

4 7 ( , # 7!, # . #. f (t1 : : : tn) Q, . . 7 ( " : # ( " . % # >2]. @ " . . ( ( " " " 5@8 ;I, # .. C. 9" # " # # "".

1] . . , // !. . . | 1996. | %. 2, . 2. | '. 365{374. 2] - . /. . 0 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. 0, 0 T ). ua (t) = ua (T ) t > T . %' o t+1 1n R =k T Ma = t 2 0 T ): ua (t0) dt0 > 0 . ;, . k=1 t , (0 T ) +, a (t) > 0, ua =a 2 L1 (0 T ). 6 0 T ] )) +,) Rt a(t) = ; ua (t0 )=a(t0 ) dt0 . 0 < &

Dt = Du Q, Dt u = D + gz ] = ]Du ; p] = 1 Q, Dt xe = u Q, Zb(t)

a(t)

ZX

(x t) dx = 0 (x) dx (0 T )

(1.1) (1.2) (1.3) (1.4)

0

+, z (x t) = ((x t) u(x t) xe(x t)), ' Q = QT = f(x t): a(t) < x < b(t) 0 < t < T g, +, b(t), ' 0 T ]. 5

) &: D = @=@x, Dt = @=@t. , ](x t) = ((x t) x), p](x t) = p((x t) x), gz ](x t) = g((x t) u(x t) xe(x t) x t).

...

995

@ , && (1.1){(1.4): x | a, t | . | 1' ( | ), u | , xe | ( . | , p | . g | & . | (++, . a(t) b(t) | , . 4 (1.1){(1.4) ( u xe)jt=0 = (0 (x) u0(x) x0e (x)) (0 X ), b(0) = X > 0 (1.5) Rx

xe0(x) = 0 (x0) dx0, 0

ujx=a(t) = ua(t) ujx=b(t) = ub (t) (0 T )

(1.6)

jx=a(t) = a (t) xejx=a(t) = 0 Ma . (1.7) 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 1. ( +,

X T . % & N . +, , p g ) ) 6,8]. A1 . @, ( x) p( x) F . . ( &) x 2 (A X ) . , > 0 p > 0. A2 . k kL1( ) + k1= kL1( ) + kpkL1 ( ) 6 C ( ), kD pkL1 ( ) 6 C ( ) & > 1. A3 . G() 6 G( x) 6 G(), G() ! +1 ! +1 G() ! ;1 ! +0. A4 . ( x) 6 c0 (E+ ( x) + L2 ( x) + maxf 1g) F. A5 . p( x) 6 c0(E+ ( x) + L2 ( x) + maxf 1g) F. A6 . E; ( x) 6 "L2 ( x) + C (";1 ) (1 +1) (A X ) & " 2 (0 1]. A7 . @, g( u xe x t) a a R+ R R Q . . (x t) 2 Q ( u xe). A8 . jg( u xe x t)j 6 g0(t)juj + g1(x t) R+ R R Q, kg0kL1 (0T ) + + kg1 kL2 1(Q) 6 c0 . ) ) . B1 . 0 2 L1 (0 X ), u0 2 L2 (0 X ), ' N ;1 6 0 (x) 6 N ) (0 X ) ku0 k(0X ) 6 N . B2 . ua ub 2 V 0 T ], ' ua ub , 0 T ). , kuakV 0T ] + kuX kV 0T ] 6 N . B3 . a 2 L1 (Ma ), ' N ;1 6 a(t) 6 N ) Ma . %' (A X ) +, xe , (x) = a (ta (x)), xe (x) = 0 x 2 (A 0] (x) = 0 (x), xe (x) = x0e (x) x 2 (0 X ). I +, (A X ). 4 , +, ta - (A 0] Ma , ( Ma) +, a. J +, a ) , (. 11]) . ( , a (ta (x)) . K ' +, z = ( u xe) 2 N (Q) V2 (Q) S211 W (Q) +,) b 2 W11 (0 T ) ' (1.1){(1.7), 1) (1.1) (1.3) ) L2 (Q). 2) (1.4) & t 2 (0 T ). 3) o ;(u Dt')Q + ( D')Q = (u0 'jt=0)(0X ) + (gz ] ')Q (1.8)

...

997

+, ' 2 W 12(Q), 'jt=T = 0, 'jx=a(t) = 'jx=b(t) = 0. 4) (1.6) +, a V2 (Q). 5) jt=0 = 0 , xe jt=0 = x0e (1.7) ) ) : +, xe . . x 2 (A X ) ) t , (x ta(x)) = (x) xe(x ta (x)) = xe (x). 7+ . 1.1. A1{A8 B1 {B3 . (1.1){(1.7), jj jjN (Q) + kuk h1 1 4i + kxe kS21 1W (Q) + kbkW11 (0T ) 6 K (N ): (1.9) V2 (Q)

=

4 1.1 + 2{4.

x

2. "

< ' 0 T ] n = T=n Tj = j , 0 6 j 6 n. n , 6 (3N );1X . %' A = xn 6 6 xn;1 6 : : : 6 x0 = 0, xj = a(Tj ), 0 6 j 6 n. 6 0 T ) -) +,) a , )) a (t) = a(Tj ) t 2 Tj Tj +1), 0 6 j < n. !aj = (xj xj ;1] ( xj = xj ;1, !aj = ?). J +, a(t) ) , Rx a (ta (x)) dx

x = a(t) 12], x +1 + j

j

Zxj

x +1

a(ta (x)) dx =

j

TZ +1 j

T

ua (t) dt 0 6 j < n:

(2.1)

j

6 (A X ) -) +,) ta , )) ta (x) = Tj x 2 !aj , 1 6 j 6 n ta (x) = 0 x 2 0 X ). K , a ! a L1 (0 T ), ta ! ta L1 (A X ) ! 0 , , lim sup kE ta kL1(AX ) = 0: !+0

(It w)(x t) =

Zt

t (x) a

Zt

w(x t0) dt0 (It(j ) w)(x t) = w(x t0) dt0 T

j

998

. .

(I w)(x t) =

Zx

a (t)

w(x0 t) dx0:

tR+

%' u(c ) = 1 uc (t0 ) dt0 ( c = a b), o uc (t) = uc (T ) t t > T . B , u(c ) 2 W11(0 T ) , ku(c ) kC 0T ] 6 sup juc j 6 N kDt u(c ) kL1 (0T ) 6 var uc 6 N: 0T ]

0T ] Lj = (xj yj ), Qj = Lj (Tj Tj +1), y0 = X ( L0 = (0 X ). j = 0 1 : : : n ; 1 Qj - Dt = Du (2.2) Dt u = D + gz ] = ] Du ; p ] = 1= (2.3) Dt xe = u (2.4) y T Z Z +1 (x Tj +1) dx = u(b ) (t) dt (2.5) y +1 T j ( u xe )jt=T = ( (x) uj (x) xj (2.6) e (x)) Lj ( ) u jx=x = u(a ) (t) u jx=y = ub (t) (Tj Tj +1) (2.7) 0 0 0 +, z = ( u xe ). B = , u = u0 , xe 0 = x0e . % B1 , B2 j = 0 (2.8) kj ( t)kL1 (& ) ; It(j ) u(a ) + It(j ) u(b ) > N ;1X ; N > (2N );1X: j

j

j

j

j

j

j

j

( 8] ' z = ( u xe ) 2 N (Qj ) V2 (Qj ) S211 W (Qj )

(2.2){(2.7), ) , j jjN (Q ) + ku kV2 (Q ) + kxe kS 1 1 W (Q ) 6 Kj (N ): 2 M , u 2 C (Tj Tj +1]. L2(Lj )), xe 2 C (Qj ). ( Lj j +1 = jt=T +1 , uj +1 = u jt=T +1 , xj e +1 = xe jt=T +1 , ' Kj (N );1 6 j +1 6 Kj (N ) kuj +1k& 6 Kj (N ): B , j = 0 yj +1 , xj < yj +1 6 yj (2.5). 4 , (2.2) (xj yj ) (Tj Tj +1), kj +1kL1 (& ) = j

j

j

j

j

j

j

j

= kj kL

TZ +1 j

1 (&j )

+

T

j

(ub (t) ; ua (t)) dt > (2N );1 X + ( )

( )

TZ +1 j

T

j

u(b ) (t) dt: (2.9)

...

999

+, j +1, uj +1, xj e +1 c Lj Lj+1, j +1(x) = ( ) = a(ta (x)), uj +1(x) = ua (Tj +1 ) +1 xj e (x) =

TZ +1 j

( )

(ua (t) ; ua (t)) dt +

Zx

x +1

0

j +1 (x0) dx0

(2.10)

j

x 2 !aj +1 . B , + (2.10) Lj+1. 4 ' x 2 Lj (2.2) (xj x) (Tj Tj +1), (2.4) (2.6). % +1 j xj e (x) = xe (x) ;

Zx

x

(j +1 (x) ; j (x)) dx +

j

TZ +1 j

T

u(a ) (t) dt:

j

(2.1), + (2.10) & j . j = 0 + . j > 0 ,. 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 = 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). 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). n, (m n) = 1, m n 9 , " >1 = 2mnF (11) 2 n2 d = m2 ; n2 k = 1 c = m + 2 2 " m n 2 N, m > n, (m n) = 1, m n 9 , " >2 = mnF (13) k = 1, = m, = n2 , " m n 2 N, m > n, (m n) = 1, m n 9 , " >3 = 2nF (15) k = 1, = m2 , = n, " m n 2 N, m > n, (m n) = 1, m n 9 , " >4 = 2m. H (21) (9) 2 2 k = 1 a = (2m m) b = (22nm) " m n 2 N, m2 > 2n2, (m n) = 1, " >1 = (22mn m) F (11) 2 2 2 2 k = 1 c = m(2+ m2)n d = m(2; m2)n " m n 2 N, m2 > 2n2, (m n) = 1, " 4mn >2 = (2 m) F (13) k = 1 = m = 2n2 " m n 2 N, m2 > 2n2, (m n) = 1, " >3 = (22 nm) F

1016 (15)

. .

2 k = 1 = (22mm)2 = (22 nm)

" m n 2 N, m2 > 2n2, (m n) = 1, " 4m : >4 = (2 m)2 H (23) (9) 2 2 k = 1 a = (22nm) b = (2m m) " m n 2 N, 2n2 > m2 , (m n) = 1F (11) 2 2 2 2 k = 1 c = 2n(2+mm) d = 2n(2;mm) " m n 2 N, 2n2 > m2 , (m n) = 1F (13) 2 k = 1 = (22 nm) = (22mm)2 " m n 2 N, 2n2 > m2 , (m n) = 1F (15) k = 1 = 2n2 = m " m n 2 N, 2n2 > m2 , (m n) = 1. D , >1 = 2mn >2 = 2mn >3 = 4m 2 >4 = 4n (2 m) (2 m) (2 m) (2 m) 2 2 " m n 2 N, 2n > m , (m n) = 1. J " , 4 4 4 4 x = kmn y = k2 m 2; n z = k2 m 2+ n " k m n 2 N, m > n, (m n) = 1, m n 9 , 4 4 4 4 x = k (22mn y = k2 m ; 4n2 z = k2 m + 4n2 m) (2 m) (2 m) 2 2 " k m n 2 N, m > 2n , (m n) = 1, 4 4 4 4 x = kmn y = k2 m 2; n z = k2 m 2+ n " k m n 2 N, m > n, (m n) = 1, m n 9 , 4 4 4 4 2 4n ; m z = k2 4n + m x = k (22mn y = k 2 2 m) (2 m) (2 m) 2 2 " k m n 2 N, 2n > m , (m n) = 1, 9 (1) (2).

1017

1] . . . | ., 1978. 2] #$ % &. '( )( %. | ., 1959.

3] , -( #. .. ) // -0. % ., $ 12. 70-)5 1 - 1 $ . 6. 7. 8 . #8$) % $( (% ))) 8. .%8, 1993 (.: 60 % . | #8$) %, 1993. | #. 17{19. 4] 0

(2)

h 1 6 j 6 n, uh = 16min s6n u 2) A ! p, h = pu, h , (Ah)k < 0, " k, uh = 1max 6s6n u

h , (Ah)l > 0 " l, hu = 16min s6n u 3) # A , (1) (2) . 5( ) 1 2= . 9 ) Rn;1 (( .. . - ) = (n ; 1)- ( >. ? (n ; 2)- >i , i = 1 : : : n. - ) V i | 2 ( >, >i , i = 1 : : : n. @ ( w, ( >, ) w 2 Int >. -, * ( w ) ( , * 2 V i V k , ( >k >i . 7 ( vki vik . A ( i , 2 vij , j = 1 : : : n, j 6= i, >i ( >i . - ( (. n = 3 ( . 1. 5 ( h 2 Rn;1 (h x) ( ( x. 0 (h x)

. ( Rn;1. B ) ) ) ReInt. 1. h 2 Rn;1. , $ (h x) % V i , i = 1 : : : K , % V j , j = L : : : n, 1 6 K L 6 n. j

j

s s

k k

s s

l l

s s

1022

. .

:. 1

1) i , i = 1 : : : K , (h w0 ; w) 6 0 w0 2 i ; (3) 0 0 (h w ; w) < 0 w 2 ReInt i ; (4)

j , j = K + 1 : : : n, (h w0 ; w) > 0 w0 2 j ; (5) 0 0 (6) (h w ; w) > 0 w 2 ReInt j ; 2) wi , i = 1 : : : K , wi 2 i , i = 1 : : : K , , ! 1 6 k 6 K , (h wk ; w) < 0; (7) wj , j = L : : : n, wj 2 j , j = L : : : n,

, ! L 6 l 6 n, (h wl ; w) > 0: (8) F (7) (8) ). 1. ' $ (h x)

% , . . K = 1 L = n, (h w0 ; w) < 0 w0 2 1; (9) 0 0 (h w ; w) > 0 w 2 n: (10)

F (h V i ) = 1max (h V s) i = 1 : : : K; 6s6n (h V j ) = 16min (h V s) j = L : : : n: s6n

1023

1.

(11) (12)

-()( ( > ) n ; 1 h 6= 0, 2 V 1 : : : V n ( , ), (11), (12) : (h V 1) > (h V n ): (13) @ , ( vis ;w V s ;V i ( vis ( . -' ( ) is

vis ; w = is(V s ; V i) i s = 1 : : : n i 6= s: (14) @( (* ( h ( '* * ( (. A = ' (11), 2 v1s ( 1 (h v1s ; w) = (h 1s(V s ; V 1 )) = 1s (h V s ; V 1 ) = = 1s (h V s ) ; 1s (h V 1) 6 0 s = 2 : : : n: (15) - ) w0 2 1. H w0 ( ( 2 ( 1:

w0 =

X 16s6n s6=1

F (16) (h w0 ; w) =

h

s v1s s > 0 s = 2 : : : n

X 16s6n s6=1

sv1s ;

X 16s6n s6=1

s

! !

w =

X 16s6n s6=1

X 16s6n s6=1

s = 1:

(16)

s (h v1s ; w): (17)

B (15), (17) (3) i = 1. J ( (3) i = 2 : : : K; ( ( (5). ? w0 2 ReInt i, ('.. s (16) ), , ()( (h v1n ; w) = 1n (h V n ; V 1) < 0, (17) (h w0 ; w) < 0. K (4) (. J ( (6). - ( 1) 1 (. 5( ( 2). - ) ) ( wi, i = 1 : : : K, wi 2 i , i = 1 : : : K. H ( ( wi, i = 1 : : : K, (

( 2 ( i :

1024

wi =

. .

X 16s6n s6=i

isvis

is > 0 s = 1 : : : n s 6= i

X 16s6n s6=i

is = 1 i = 1 : : : K: (18)

A ) (14) ! X X wi ; w = isvis ; is w = =

X

16s6n s6=i

16s6n s6=i

is(vis ; w) =

(h wi ; w) =

X 16s6n s6=i

16s6n s6=i

X

16s6n s6=i

isis (V s ; V i ) i = 1 : : : K

isis (h V s ; V i ) i = 1 : : : K:

(19) (20)

4 , (15) = (11) (h vis ; w) = is (h V s ; V i ) 6 0 s = 1 : : : n s 6= i: (21) - , (7) , . . (h wi ; w) = 0 i = 1 : : : K: H (20), (21), = ) ('.. is , ( i = 1 : : : K is(h V s ; V i ) = 0 s = 1 : : : n s 6= i: (22) -()( 2 V i . ( (h x) ( , (22) , is ) ) 2) s = 1 : : : K, s 6= i. H (19) , K ( (wi ; w), i = 1 : : : K, K ; 1 ( V s ; V i , s = 1 : : : K, s 6= i. -' ( wi ; w, i = 1 : : : K, . K ' . -' 1 6 k 6 K, ( (h wk ; w) < 0, ( (7) (. K (8) ( . L 1 ) (. 7 ai (, i- ( A: i a = (ai1 : : : ain). 2. A 2 Mn u 0 Au = 0. X (23) ai = u12 aik uk(vik ; u) i k6=i

1025

2 +u2 u i k vik = u1 : : : uk;1 uk uk+1 : : : ui;1 0 ui+1 : : : un 1 6 i k 6 n i 6= k: . K , 2 u i vik ; u = 0 : : : 0 uk 0 : : : 0 ;ui 0 : : : 0 1 6 i k 6 n i 6= k: 2 @ ) uu k- , ;ui | i- . 5 1 X ai u (v ; u) = 1 X ai u 0 : : : 0 u2i 0 : : : 0 ;u 0 : : : 0 = i u2i k6=i k k ik u2i k6=i k k uk 2 X = aik uk 0 : : : 0 uui 0 : : : 0 ;ui 0 : : : 0 = k k6=i i X = 0 : : : 0 aik 0 : : : 0 ; akuuk 0 : : : 0 = i k6=i X i 1 i i i i a uk ai+1 : : : an : (24) = a1 : : : ai;1 ; u i k6=i k P P - Au = 0, . . aik uk = 0, aii ui = ; aik uk , i = 1 : : : n. k k6=i A = ' 2 , ) (24) ( ai. L 2 (. i k

2. A )* , * ( ) ( 2) 1, ( * * , 2. K ( '* * 2.5 2] 8.3.1 3], ( ( *. - , ' - {0 , . , )* . 2. A 2 Mn . ! , a) b) ! , ! c) > Re . ' , , A , b) c)

:

1026

. .

b0 ) , c0) > Re , 6= . 3. A 2 . A ( (. 5 ) A ) (A). 1. 1 (n ; 1)- ( >0 = fx 2 Rn+ : (u x ; u) = 0g: n 2 ' ( V i , i = 1 : : : n, )* (* * Rn (: P u2 ! k k (25) V i = 0 : : : 0 u 0 : : : 0 : i

, ( i- , ( ( V i . H( u ) ( >0. - ( u , ) ( >0, ( ( , 1, ( > = ( >0 , ( w ( u. 0( ( h, ( p, ( h = pu. - ) (h0 x) | . (, ( () (u x ; u) = 0 . (, ( h. - , . ( (h0 x) ( 2* V i , i = 1 : : : K, ( >0 2* V j , j = L : : : n, ( >0, 1 6 K < L 6 n. - ) 1 6 i 6 K ( ai i- ( A. - , aii = 0. H, ()( A ) Au = 0 u 0, aij = 0, j = 1 : : : n. -' (Ah)i = (h ai ) = 0, (1), (2) . ? aii 6= 0, Au = 0 u 0 ) , aii < 0. - i = ; ua , i = 1 : : : n. ' ): i > 0, i = 1 : : : n. F) (23), 2 i i i

1027

X i ai + u = u12 i aik uk (vik ; u) + u = i

k6=i

X i 1 1 i = u2 i ak uk vik ; u2 i ak uk u + u = i k6=i i k6=i X i X i 1 1 = u2 i ak uk vik + 1 ; u2 i ak uk u: i k6=i i k6=i -()( Au = 0, X i aik uk = ; uaii (;aii ui) = u2i : X

(26)

i

k6=i

-' ( (* ( (26) , ) X iai + u = u2i aik uk vik (27)

i k6=i

i ai u v : (28) u2i k k ik -()( aik > 0, k 6= i, 1 6 i k 6 n, ('.. , ( (* (28), ). / , ( iai + u ( ( ( v ik , k = 1 : : : n, k 6= i. 5 , ( iai + u ( i: i ai + u 2 i . H )) (3) 1, = w0 = iai + u. - (h (iai + u) ; u) = (h iai ) 6 0 , ()( i > 0, (Ah)i = (h ai ) 6 0: (29) , ' ( ( h ( ( i, ( 2 (11). - 2 (11)

= , 2 ( >0 (25). H (11) P u2 P u2 k k hi ku > hk ku i i hi > hk k = 1 : : : n: ui uk A2 (1) (. J ( 2 (2). N ) 2) 1, ) ( i ai + u, 1 6 i 6 K. - ) , , ()( X i ai + u = k6=i

1028

. .

( u 0, ( Au = 0, A , (A) = 0. 5 ), ' (, . . (A) 6= 0, 2, b0), ( v 0, ( vA = (A)v . K : 0 = vAu = (A)(v u) 6= 0. 7 AK A, K ( . - ) B = A + sI , ) s )( )2, B > 0. 7 B K B , K ( . - ) B 0K | , B * ' , ** BK , , ' ' , * B K , . 7 C = B ; BK . -()( B , C > 0. F 15 8.4 3] (B ) = (B 0K + C ) > (B 0K ): -()( (B 0K ) = (B K ), (B ) > (B K ). H (A) = (B ) ; s > > (B K ) ; s = (AK ). -()( (A) = 0, (AK ) < 0, , ( ai , 1 6 i 6 K, K < n, . ' ( iai +u, 1 6 i 6 K, i > 0, 1 6 i 6 K, . - ) 2) 1 (. ) . 5( ( 3) 1. @ , , ( ' A ), ('.. , ( (* (28), ). -' ( iai + u ) ( i . ( )) 1, ( ) 3). H 1 ) (.

3. "

M- $

A . 1] M-. ' (, ) ) M-. H , (, ( 2) 1, ( = ) A, ( 5) 4.16 1]. 4. A A = sI ; B s > 0 B > 0 (30) ( s > (B ), (B ) | () B , M- . ? s > (B ), A M- ; s = (B ), A M- . F 4 , A M- , ;A , ) . O( M-

1029

(. 1]. K ( '* | 2) , ( ) . 3. A | , ;A . 1) A M- ,

A 2) A M- , A . H 2 3 *( ) ) M- , )* . 1.

1) * A M- , ! ;A . 2) * A M- , ! ;A . . - ) A | M-. H ;A | , ) . - 1 = . 7 , ; (;A) ) A. K 3, 1) ) A ). , ; (;A) > 0, ( (;A) < 0. ( 1) 1 (. 5( . - ) ;A | , ) . H = (;A). 4( 2, c), (;A) > ) ;A. H ) , ) ;A )( , ( ; ( )) A. -' ), ; (;A) 6 ) A. - , (;A) < 0. 7 0 < ; (;A) 6 . / , ) A ). K , 3, 1), A

M- . - ( 1) 1 ) (. 5( ( 2). - ) A | M-. H, M-, () B (30) s: (B ) = s. -()( B | ) , () (B ) ( B 3]. ? v > 0 | (( - ( B (B ), Bv = (B )v. 7 (30) , v |

1030

. .

( ) ;A : ;Av = 0. B , ) A. 5 ), ' (, , 2, c), (A) ) )2 ) ;A; , ) (;A) > 0. K ( , ;Au = (;A)u, u > 0 | (( - (, (;A). F (30) (;sI ; B )u = (;A)u, Bu = ;( (;A) + s)u. -()( (;A) ), , B , (;A) + s, * () B : (;A) + s = (;A) + (B ) > (B ) . -' ) ;A. 1, 2) (. 5( . - ) ) | ;A. -()( )2 = ) , 6 0. 4( 2 , , ) ;A )( , ( ; ) A. K ; > 0 , ) A ). H 3, 2) , A M- . - 1 ) (. H ) ( , M- ) 1. 5 M- ' . 1] . ! . A | M- n. Ax > 0 Ax = 0. F 1, 2 1 ' . 2. B 2 Mn | M-

u 0 | ( ) , ! ( ) ! ;B . h, ! p, h = pu, h , (Bh)k > 0, " k, hu = 1max 6s6n u

h , (Bh)l < 0. " l, hu = 16min s6n u . -()( B , A = ;B . H( (( A , 2, b0) =

( u 0. H( , A k k

s s

l l

s s

1031

( 2) 1. - , A ;B , , ( *.

4. & " - -' $ ' ( * - -'( ) (-. (, -( * (* . ( , ) . 5. 0 ( f(x), = Rn , ( , G+ (f r) = fx 2 Rn : f(x) > rg ( * r, ( ( , G;(f r) = fx 2 Rn : f(x) 6 rg ( * r. - ) f (x) = (f 1 (x) : : : f n (x)) | (-. (, f i (x) Rn , i = 1 : : : n. ) 2 ), . ( : 1) f i (x) .. Rn, i = 1 : : : n; 2) f i (x) xk , k i = 1 : : : n, k 6= i, xi , i = 1 : : : n, ), * *, fii (x) 6 0 (31) fki (x) > 0 k 6= i i = 1 : : : n: 6. ( x 2 Rn ) () - , y 2 Rn, ( f (y) > f (x). ( x 2 Rn () - , y 2 Rn, ( f (y ) f (x). ) ()* - ( x P+ , ()* - ( x P0+ . F : P+ P0+ . J ) (, ) ) - . 7 P; P0; ) ) - . A P; P0; . = f 0 (x) n n, ( i- ( j- fji (x) | . ( f i (x) xj , i j = 1 : : : n. 7 , . ( f 1 (x) : : : f n (x) (31), f 0 (x) ) ; ' (f 0(x)) f 0 (x), 1.

1032 4.

. .

$

f 1 (x) : : : f n (x) (31).

1) x 2 P+

x 2 P; , (f 0 (x)) = 0 10 ) x 2 P0+

x 2 P0; f 0 (x) , (f 0 (x)) = 0 2a) $

f i (x), i = 1 : : : n, ! z 0, Az = 0, (f 0 (x)) = 0 x 2 P0+ 2a0) $

f i (x), i = 1 : : : n, f 0 (x) , (f 0 (x)) = 0 x 2 P+ 2b) $

f i (x), i = 1 : : : n, ! z 0, Az = 0, (f 0 (x)) = 0 x 2 P0; 2b0 ) $

f i (x), i = 1 : : : n, f 0(x) , (f 0(x)) = 0 x 2 P; . . 5( 1). - ) x 2 P+ x 2 P; . - , (f 0(x)) 6= 0. - ) v > 0 | (( - ( f 0 (x), (f 0 (x)): f 0 (x)v = (f 0 (x))v . 7 = sign (f 0 (x)) ( f 0 (x). F .. . ( 2 t!0 f (x + tv) ; f (x) = f 0(x)tv + o(t) , ()( f 0 (x)v = (f 0 (x))v > 0, f (x + tv) > f (x) (32) f (x ; tv) < f (x) * * )* t, * , (( ) (), ( )

) - ( x. - ( 1) (. 5 ( ) ( 10 ) ), f 0 (x) ( v, 1, b0 , : v 0. -' (32) ( . 5( 2a). - ) ( x (f 0(x)) = 0, x 2= P0+ . H y0 , ( f (y0 ) f (x). F . ( f i (x), i = 1 : : : n, ( V ( y0 , ( f (y) f (x) * y 2 V . 7 v = y ; x. F v 2 V ; x. F , f (x + tv) f (x) * 0 < t 6 1 i = 1 : : : n: (33) F . ( f i (x), i = 1 : : : n, X f i (x + tv ) ; f i (x) 6 fji (x)tvj * 0 < t 6 1 i = 1 : : : n: (34) j

1033

-()( (34) * v, * ( V ; x, * = ( v 0, ( p, ( v0 = pz . H 1, 1) ( 1 6 k 6 n, ( X k 0 fj (x)vj 6 0: j

7 (34) , * * t > 0 f k (x + tv0 ) ; f k (x) ). K ' (33). - ( 2a) 4 (. - ( 2b) ( . 5( 2a0). - ) ( x (f 0 (x)) = 0, x 2= P+ . H y0 , ( f (y0 ) > f (x). F . ( f i (x), i = 1 : : : n, ( V ( y0 , ( f (y) > f (x) * y 2 V . 7 v = y ; x. F v 2 V ; x. F ( , ( x +tv = x +t(y ; x) = (1 ; t)x +ty G+ (f i f i (x)). A ), f i (x + tv ) > f i (x), f (x + tv) > f (x) * 0 < t 6 1 i = 1 : : : n: (35) A , ) ) .. ) . ( f i (x), i = = 1 : : : n, ): X X f i (x + tv ) ; f i (x) = fji (x)tvj + j (t)tvj j

j

j (t) | . (, ( t ! 0, t > 0. 1 t > 0, ' t: f i (x + tv) ; f i (x) = X f i (x)v + X (t)v i = 1 : : : n: j j j j t j j

(36)

-()( (36) * v, * ( V ; x, * = v0 , ( p, ( v0 = pz . H 1, 2) ( 1 6 k 6 n, ( X k 0 fj (x)vj < 0: j

7 (36) , * * t > 0 f (x+tv );f (x) ). K ' (35). t - ( 2a0 ) 4 (. - ( 2b0) ( . 3. $

f i (x) S .

(f 0 (x)) = 0, x 2 P+ . . 5 ( ) ), G+ (f r) ( * . ( f(x). k

0

k

1034

. .

5 * 4 5 ( ( 0 , (, - -'( ) ( QS ( ) . ( f (x) ,

( .

(

1] Berman A., Plemmons R. J. Nonnegative matrices in the mathematical sciences. | 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 >:

(1)

ni > 0 i = 1 w ni | ! ! i' qik | ! k- &, 3 &$ i !3 & & & ' m | ! & , ' w | ! ' Lk | !# & k (Lk > 0). 4! (1) ! $ 5 $ && # , & ! # $ ( , & % & ) && $ +1,3]. 63 5& H(~n) 5 & # . 1. 7 # 8&& 9 , 8&3 9 && && +4]. 2. 7 , $ & ! && ! && , 3 5& ! && +1] 8 ! 9 ! && +4]. ; #$ & & +3]. 3. 7 % # ! 5 , $ && $ , & 3$ #$ # 5 +1]. 4. &% & &% !& $ $ . 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 k = 0 m:

H # 0 0, # D k , k = 1 m, # & , ! ! & % & $ D . 6 & 0 = 1. ; & & ! (1) %

8> @H(~n) P m >> 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 0, j 2 J), , &%

!.

3. ! . ' () H(~n) (n = n(L~ )) @H > 0 k = 1 m: @Lk . L 5, 3 $ -

1, 3 Rm+ = fL~ = (L1 : : : Lm ) j Lk > 0 k = 1 mg 2m ; 1 % S J , J f1 : : : mg. S J ! (1) ! (4). M S J | & # S J , & j > 0, j 2 J +2]. E # $ ! ! (4) S J @H @H j @Lj = j 2 J @Lk = 0 k 2= J j | &%$ # D . 63 # 5 55 & 5&$ +9]. M & &$

8> ;P

>< @(;@nH(~n)) + P @ =1 q@nn ;L j = 0 i = 1 w' >> Pw qj n = Ljj2J+ lj j 2 J: : i i w

i

i

i=1

j i

i

i

j

(11)

6&# ~n | > ! (4), ~ | # D , &%$ & >%. ; > &$ (11) # (~n ~ ~l) ! (~n ~ 0) (~n = (n1 : : : nw ), ~ ~l # j lj , j 2 J, ), #& &! (11) & & ! (4). 6 $ 5& (11) ( $ $ & $ +9])

1041

3 &%$ &#: && > 0 5& ni (~l) i (~l), i = 1 w (ni (~l) 2 C 1, i (~l) 2 C 1 S(0 )), ! Pw j ~ j @ @(;H+~n(~l)]) + X j (~l) i=1 qi ni (l) ; L = 0 i = 1 w (12) @n @n

i

i

j 2J

w X i=1

qij ni(~l) ; Lj = lj j 2 J:

6 ~n(0) = ~n , ~(0) = ~ . ) # & (12), # @

Pw

qij ni ; Lj i=1 = 0 i = 1 w k 2 J @ni

@ni (~l) @(;H(~n)) + X @ni (~l) j (~l) k @lk @ni j 2J @l Pw j j @ q n ; L i i @(;H+~n(~l)]) = ; X @ni (~l) j (~l) i=1 i = 1 w k 2 J: k @lk @ni j 2J @l L & $ , 55& (13), &!

8> ; P ~

; P q n ;L

@ q n ( l ) ;L @ < 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 . < 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. | ".: , -

.: & > &% // ? @. | 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. 0 x+ = 0 x < 0. nX ;1

% , " 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, 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]). 4 A (14) #

: " A - : , A n = 2 3 - k- " A % # t = 1. +

+

+

+

n) (t) = sgnsin mt, ;"#! ! xmn () n '# ! #, x(mn t 2 R, * * ! ( ,-. ; ! ! n ' ( ! ! ! n + 1. 1

1058

$

. .

1] - . ., . . . | .: "#$ %&'', 2000. 2] Borsuk K. Drei S*atze u*ber die n-dimensionale euklidische Sph*are // Fund. Math. | 1933. | Bd. 20. | S. 177{191. 3] 01 2. 3. 4 56 7 $ #889 . | ., 1979. 4] Shoenberg I. J., Cavaretta A. S. Solution of Landau's problem concerning higher derivatives on the half line // Proc. of the Intern. Conf. on Construction Function Theory, Golden Sands (Varna), May 19{25, 1970. | So?a: Publ. House Bulgarian Acad. Sci., 1972. | P. 297{308. & ' 2002 .

, . H.

517.55

: , ,

- , .

!" # , $$"% $ & . !

$ ' ! ! # , $$"% $ , ' $ %(

& , #"%$ #.

Abstract S. N. Mishin, Operators, commuting with operators of nite order, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 4, pp. 1059{1067.

Linear continuous operators commuting with operators of 2nite order are studied in this work. The proved theorems contain as a special case some well-known results about the linear operators commuting with di3erential and also with generalised di3erential operators, acting in the spaces of analytical functions.

, ,

, , XX . (., , #5{7,15,16]). + ,. -. . #8] . + , A , H . 0 1 2- , . , 2002, $ 8, 4 4, . 1059{1067. c 2002 , !" # $

1060

. H.

+ , 2 +. 0. 4 #2, 3] 7 7 #11{14]. +

B : H1 ! H , A (H1 H | 1 k P A H ), B = ck A , k=0 ck | 1

B < 1 B . + .

x

1.

0 H | A : H ! H | . = H 2 kxk1 6 kxk2 6 : : : 6 kxkp 6 : : : (1.1) 0 0 0 kxk1 6 kxk2 6 : : : 6 kxkp 6 : : :: (1.2) > , (1.1) (1.2) ( ). (H kkp) H , (H kk0p) | H1. > , An : H1 ! H , n = 1 2 : : :, . ? 8n 8p 9Cp(n) 9q(p n): kAn(x)kp 6 Cpkxk0q 8x 2 H1: @ , A : H1 ! H , 2 fcng,

fcn Ang , . . 8p 9Cp 9q : kcnAn(x)kp 6 Cpkxk0q 8x 2 H1 8n: @ , , , cn = nan, a 2 R ( ). 0 kAn (x)k p (p q n) = sup 0 k x k kxkq 6=0 q ( (p q n) = +1 ). p- p , p- p A (. #3,14]): ln (p q n) fpg p (q) = nlim !1 n ln n p = qlim !1 p (q) = sup (p) 8 fpg p = 8p ;p ((p q n))1=n p = lim p (q) = <sup (p) n p(q) = nlim !1 q!1 :0 p , (2.1) (1.2) ( , f H1) f () , g(), (2.2), g() = '()f (), '() | . ? - fxng p H , xn = 6 0 8n,

A(xn) = xn;1, n = 1 2 : : :, n A(x0) = 0, nlim k x k = 0 8p. D 1 A n p !1 ,

7 (. #13]). C p (f ), p (f )

f (1.1),

0p (f ), p0 (f ) |

f (1.2): ln ln kf ()kp (f ) = lim ln kf ()kp p (f ) = lim p !1 !1 jjp (f ) ln jj 0 ln ln kf ()kp 0 ln kf ()k0p

0p (f ) = lim ( f ) = lim : p !1 ln jj !1 jjp (f ) E

ln kf ()kp {p (f ) = lim jjp(f ) 0

!1

1062

. H.

2 p- - f p- p (f ),

p (f ) p0 (f ) | p- - f p- p (f ) 0p (f ). 0 1 (f ) = supfp (f )g, (p) { (f ) = supf{p(f )g | f (p) (f ). 0 B : H1 ! H | , A: 8p 9Cp 9q : kB(x)kp 6 Cpkxk0q 8x 2 H: B f AB = BA B (f ()) = '()f () 8 2 C : (2.3) '() | . > 2

B . B (2.3) B '() 8p 8" > 0 9Cp(") 8q > q0(p) 8 2 C : j'()j < Cp exp (q0 (f ) + ")jjq (f ) ; ({p(f ) ; ")jjp(f ): (2.4) 0 Lfxng | 1 - f (). @ :

B D

', x 2 Lfxng B (x) = D(x) , fxng H1,

,

, . E , B (f ()) = D(f ()) = '()f () 8 2 C . F Q = B ; D. @ , Q(f ()) 0< 1 P , Q(f ()) = Q(xn)n . @ , Q(xn ) = 0 8n. n=0 C ( Q) . . B , A, 1 X ck Ak (2.5) 0

k=0

ck | 1

. D 1 S , x 2 Lfxng S (x) = B (x). D fxn g H1 (2.5) , | B . G, B

| , 1 (2.5). D

B ,

, 1063

A : H1 ! Hf1 p- p p- p , 1 n P

- ( ) f () = xn n=0 . "

B A

' ( ) # 1) (') < 1p 8p 2) (') = 1p (') < e1p= 8p, (2.5) x 2 H1 1 P x 2 Lfxng ck Ak (x) = B (x)1 . $ , fxng H1 , k=0 (2.5)

B

H1.

1.

.

0 B A. H ( - A ) B

', (2.3). 0 ' . 0 jckj < Kk; ('k)+" 8k 8" > 0: 0 p-

kAk(x)kp < Lpk(p+")k kxk0q 8x 2 H1 8k 8" > 0 8p q = q(p): 0 , fSn g (2.5) . B N N kSN (x)kp 6 X kckAk(x)kp 6 Mp X k(p; (1') +")k kxk0q 6 Cpkxk0q 8x 2 H1 8N 8p: k=0

k=0

0 (. #1]) fSN g S : H1 ! H ,

S ', 1 x 2 Lfxng S (x) = B (x), fxng H1 , x 2 H1 S (x) = B (x). @ 2 . H . 2. 1 fxng H1

0p (f ) = p (f ) = (f ) = 1 0 (f ) ; { (f ) < 1= 8p: ep 1

5 ('), (') | ' ( ) ($. 79, 10]).

1064

. H.

B

A, (2.5) E , (2.4) 1. % ,

, .

x

3.

1. 0 H1 = H = s | , kxkp = max fjxkjg. C A(x0 x1 : : :) = (x1 x2 : : :) P1 k6p , - f () = en n n=0 (en | ). 0 ,

B , A, | . E , B , , 8p 9Cp 9q : kB(f ())kp 6 Cpkf ()kq 8 2 C : I , ( kf ()kp = 1j jp jjjj 6> 11 j'()j = kBkf(f(()k))kp 6 Cpjjq 8 2 C : p

+ '() | . B

, B A, , . 2. 0 H1 = H = H (C ) |

: kF kp = jmax jF (z)j. C zj6p d A = dz , 1 P - f () = ez = znn! n . 0 ,

n=0 , dzd , | 1 . E , - f () = ez p- p (f ) = 1 p- p(f ) = {p (f ) = p < 1. + (2.4) '() 2 #1 1). C dzd p- p = 1 p- p = 0. H

, 2

: B

dzd H (C ), ,

, 1065

B=

1 dk X ck dz k : k=0

(3.1)

3. 0 H = # ], > 1, | ,

, . + 2 H1 # ], 6 . H # ] # ] ;( +"p)r 8F 2 # ] lim "p = 0 kF kp = sup max j F ( z ) j e (3.2) p!1 r>0 jzj6r ;( +"p )r 8F 2 # ] lim "p = 0: (3.3) kF k0p = supjmax j F ( z ) j e p!1 zj6r r>0

0 # ] # ] # ] . C dzd # ] - f () = ez , # ]. J

1 (3.2) (3.3)

p (f ) = 0p (f ) = ; 1 p (f ) = {p (f ) = ; 1 ( ( + "p )) 1 1 p0 (f ) = ; 1 ( ( + "p )) 1 1 0(f ) = ; 1 ( ) 1 1 { (f ) = ; 1 ( ) 1 1 : ;

;

;

;

# ] ! # ] p- p-

1 1 1 1 1

; 1 p = p = e ( ) 1 ; + " : p B (2.4) ,

B : # ] ! # ], dzd : # ] ! # ],

1 1 '() 2 ; 1 ; 1 ; 1 2. @ , B : # ] ! # ] dzd : # ] ! # ], , (3.1). 4. 0 H = H (G) | , G. + 2 H1 H (D) ,

D, D | d- 7 G (. .)1 . H H (G) H (D)

C

d dz :

;

;

;

;

1d

$ ' ".

1066

. H.

kF kp = zmax jF (z)j 8F 2 H (G) 2G

G1 G2 : : :

kF k0p = zmax jF (z)j 8F 2 H (D) 2D

D1 D2 : : :

p

p

1 p=1

Gp = G

(3.4)

Dp = D:

(3.5)

1

p=1

0 H (D) H (G) HD . C dzd H (G) - f () = e(z;z0 ) , z0 2 G, H (D). C p- p (f ) = 0p (f ) = 1 { (f ) 6 (f ) 6 0(f ). C dzd : H (D) ! HD p- p = 1 p- p = ed1p (dp | Gp D). + (2.4)

B : H (D) ! H (G), dzd : H (G) ! H (G), '() 2 #1 0(f ) ; { (f )]. D '() 2 #1 d],

(3.1). F . 0 G | r, D | r + d. + 1 { (f ) = (f ) = r, 0 (f ) = r + d, , '() 2 #1 d] , dzd , ,

(3.1).

"

1] . . . | .: , 1967. 2] "#$ . %. %#& '( ) ( #'# #* #& ( +,+' $ &$ // /0 1112. | 1986. | 4. 228, 5 1. | 1. 27{31. 3] "#$ . %. %#& '( ( #'# '#8 // 98 (+. | :# : :"9, 1999. | 1. 6{23. 4] "#$ . %. 0) #* & 4 # // 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 ) 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-;& 2. | ?. 178{186. 8] 8! . 6. 3! ; CR-;& 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. ;; : : : | :'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< _ î | " 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] . . ! ! 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]. 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 ( )

> (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): 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, > ;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) >: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 // ; 4) , 5 $' , )$. (' +"$"'+*8 10 ). ")", $+" +", 0, 1( 23, 4., 4 9 %. 1 F1 (;47 c7 z) = 0, 8+ % '%"( "'"% ( . %'5 +""%) % &$'"( )" *%+, FEL-+4. . &" *9( "+$'"( ."'+ ( 4 a > 3) '4 . ( xk = zk ; (c + a ; 1) "' y = 0 %4%, %%".+'. ;"%,< , ' '" Ra = (a; 1)pc + a ; 1. 1+" 4")%" .+ )++* "+", "'"&"'+ '4 . =+5 ( 5 ;"&)"%< | +"$ Tk "3"'+. . ' $% a = 3 a = 4 4" $, +"$, ; "%,< '4+"+ +"$ Tk 4 2 6 c < 1. 6 %,$' ( ""$ %,.% , ;"'"&,< ' $ (ac) = (4 6) (6 4) (8 14): : :.

Abstract V. F. Tarasov, Zeroes of Schrodinger's radial function Rnl (r) and Kummer's function 1 F1 (;a7 c7 z) (n < 10, l < 4), Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 4, pp. 1159{1178.

Exact formulae for calculation of zeroes of Kummer's polynomials at a 6 4 are given7 in other cases (a > 4) their numerical values (to within 10;15) are given. It is shown that the methods of L. Ferrari, L. Euler and J.-L. Lagrange that are used for solving the equation 1 F1 (;47 c7 z) = 0 are based on one (common for all methods) equation of cubic resolvent of FEL-type. For greater geometrical clarity of (nonuniform for a > 3) distribution of zeroes xk = zk p; (c + a; 1) on the axis y = 0 the ;circular< diagrams with the radius Ra = (a ; 1) c + a ; 1 are introduced for the Crst time. It allows to notice some singularities of distribution of these zeroes and their ;images 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) , '

/ > ) ? ', ! (a c) = (6 4) (8 14) (10 30) :: :. [

) ! '

!! a = 5 8, ! )! / ! I

' /

* 13{21].

1173

-& 5

Xk X1 X2 X3 X4 X5

) ' = 5 c=2 !k c=4 ;5382969146721729 123325 ;6509445054813172 ;3887034041421476 246626 ;4418666187096746 ;1389166848982466 261849 ;1373003703176392 2399066971204843 284173 2944418003467669 8260103065920831 32537 9356696941618640

!k 125124 247010 263029 285085 34205

1 . X'* 1 F1(;5% c% z) = 0

z 5 ; 5(c + 4)z 4 + 10(c + 3)2z 3 ; 10(c + 2)3 z 2 + 5(c + 1)4z ; (c)5 = 0 5 Y (z ; zk ) = 0 k=1

!

) z = x + (c + 4) x5 ; 10(c + 4)x3 ; 20(c + 4)x2 + 15(c + 2)(c + 4)x + 4(c + 4)(5c + 14) = 0 5 Y (x ; xk ) = 0: k=1

p

) xk = R5 cos !k , R5 = 4 c + 4 | , '! Tk % / '

) '

) !k ) 5. E * (nl) = 60 71 : : : ( . . 6s 7p : : :)% / )

Pnl (r) ) . 5.

G'. 5

1174

. .

2 . X'* 1 F1(;6% c% z) = 0

z 6 ; 6(c + 5)z 5 + 15(c + 4)2 z 4 ; 20(c + 3)3z 3 + + 15(c + 2)4 z 2 ; 6(c + 1)5z + (c)6 = 0 6 Y (z ; zk ) = 0 k=1

!

) z = x + (c + 5) x6 ; 15(c + 5)x4 ; 40(c + 5)x3 + 45(c + 3)(c + 5)x2 + + 24(c + 5)(5c + 19)x ; 5(c + 5)(3c2 + 4c ; 31) = 0 6 Y (x ; xk ) = 0: k=1

(c = 2) (c = 4) ,p /'

( ) !

* / ' > '? c0 = (;2 + 97)=3

/ > ? = 26162859. 3!

; * '

p ' 4 . ) xk = R6 cos !k , R6 = 5 c + 5 | , '! Tk % / '

) '

) !k ) 6. E * (nl) = 70 81 : : : ( . . 7s 8p : : :)% / ) Pnl (r) ) . 6.

G'. 6

3 . X'* 1 F1(;7% c% z) = 0

z 7 ; 7(c + 6)z 6 + 21(c + 5)2 z 5 ; 35(c + 4)3z 4 + + 35(c + 3)4z 3 ; 21(c + 2)5 z 2 + 7(c + 1)6z ; (c)7 = 0 7 Y (z ; zk ) = 0: k=1

1175

-& 6

Xk X1 X2 X3 X4 X5 X6

) ' = 6 c=2 !k c=4 ;6472331878288871 119292 ;7703580796555735 ;5203700190356591 246836 ;5906001619244266 ;3123358479523088 256343 ;3338714779588498 ;0081183433295277 269648 0167097269514240 4234610429083116 288669 4941345374521041 1064596355238071 36412 1183985455135321

!k 120902 246812 257139 270638 289233 37877

U ) xk = zk ; (c + 6) = R7 cos !k p R7 = 6 c + 6 | , '! Tk % / '

) '

) !k ) 7. E * (nl) = 80 91 : : : ( . . 8s 9p : : :)% / ) Pnl (r) ) . 7.

G'. 7

4 . X'* 1 F1(;8% c% z) = 0

z 8 ; 8(c + 7)z 7 + 28(c + 6)2 z 6 ; 56(c + 5)3z 5 + 70(c + 4)4z 4 ; ; 56(c + 3)5z 3 + 28(c + 2)6z 2 ; 8(c + 1)7z + (c)8 = 0 8 Y (z ; zk ) = 0: k=1

U )

xk = zk ; (c + 7) = R8 cos !k (c = 2)

(c = 14)

1176

. .

-& 7

Xk X1 X2 X3 X4 X5 X6 X7

) ' = 7 c=2 !k c=4 ;7538975780195004 116374 ;8852299909125201 ;6436413810345738 247711 ;7272916485939195 ;4647949497463266 254104 ;5044531279324631 ;2083702750979461 262947 ;2063517391427633 1420699383021589 274802 1845514252547855 6194165548007463 291407 7025619215128048 1309217690795441 39514 1436213159814075

!k 117811 247460 254581 263756 275581 291733 40803

p

R8 = 7 c + 7 | , '! Tk . \

) '

* xk !k ' = 2 = 14, . . * (n l) = (9 0) (n l) = (15 6) ) 8. O P90(r) a . 8.

G'. 8

] ), @. 4. 4

!

) ,

, , X. D. ]

]. @. 4 9/ !: .

,--

1] # $. %. & ' . | ).: , 1977. 2] , -., . /. # 0 1 2 1 1 10 3 1. | ).: 5 1 ' , 1960. 3] , 1 -., / $. &6 7 8 7. 9. 1. | ).: , 1973.

1177

-& 8

xk x1 x2 x3 x4 x5 x6 x7 x8

) ' = 8 c=2 !k c = 14 ;8590616426796815 114146 ;1535518628423683 ;7615036815196860 248738 ;1224652088718598 ;6043745443831138 253273 ;8731064552665336 ;3818056898959927 259524 ;4658558657501770 ;0838290311854182 267712 0117906997030555 3070055126837155 278406 5816135545473467 8249735526148988 293131 1284039940328846 1558595524365278 42081 2221688843579742

!k 118599 247556 254205 261649 270210 280446 293596 46164

4] . $. $., 9 %. )., : &. ;. # 0 1 2 . | ).:

, 1979. 5] & & ,. 1 1 6 ). 9 0 ' ? 1 8 @ 1 1 1. | ).: ), 1966. 10] , . $' , 1 '@ 0, 0@ '. | ).: , 1965. 11] > 5., = 1 /. = @ ' 8. | ).: ), 1977. 12] # 5. < 7 3 6 0 . | ).:

, 1989. 13] ) 18 A. < 7 3 -8 702 / = . C. %. ) . | ).: ), 1988 (E1 1 >. F 6 ' @ 2 1GH 3 - . .. 362{370). 14] ,'

K . &., < 2 K K | 8 (M B ). 4 K + & ) U(T A ") = fS 2 K j (SA 4 TA) < "g, A 2 B, T 2 K, " > 0. ( $& K | . E K(d) && Zd 8 (M B ) &" , & s 2 Zd h ! hs K(d) K. 7 h fhs gs2Z d 2 K(d) %(1 0 ::: 0) d (h h(010:::0) : : : h(0:::01)). % d = 1 & h h1 . F ,-) %&: % Zd-&& , % & (r p q : : :)H | % (r p q : : :)H i- ,- % (, r, er,

Z

d-

1181

r1, r1 $ ri, rei , r1i, ri1 ). , -

, , d-. . % 0 1 (0 : : : 0) (1 : : : 1). ? , , , , , . . ) . 4 v s % hv si. < T 2 K v T v & h 2 K(d) , & hs T hvsi . + $ . I , % V fv j v 2 Nd v1 = 1g & , , q 2 Nd - v 2 V : (v2 : : : vd vd qd + 1) > vq. > & f(1 v v2 : : : vd;1 )gv2N. 1. , hm1 vi + hk1 n vi = hm2 vi + hk2 n vi ) m1 , m2, k1, k2 , 1 (v2 :::vd vd nd ) ,-) 0 6 mj < n, 0 6 kj < n v j = 1 2 (m1 k1) 6= (m2 k2). . hm vi + hnk1 vi = hnk2 vi ; n < m = m1 ; m2 < n: > k = k1 ; k2 , ,, % ,- : hnk + m vi = 0 ; (v2 : : : vd vd nd ) + n 6 kn = k1 n ; k2 n 6 (v2 : : : vd vd nd ) ; n:

v

v

N-+ ) p = kn + m (v : : : v v n ) (v : : : v v n ) 2 d d d 2 d d d hp vi = 0 ; +1 6 p = kn + m 6 ; 1:

v

v

" $ ? % % %& ?&. d;1 0)i < 1 jp j 6 vi+1 ; 1 jpd j = hp(1v2:::v i vd vi i = 1 : : : d ; 1. . pd | ? , , " 1, pd = 0. d;2 0vd )i = hp(1v2:::vd;2 00)i , p v = 0 v = 0. jpd;1j = hp(1v2:::v d d d vd;1 vd;1 O jpi j 6 viv+1i ; 1 i = 1 : : : d ; 2 jpd;1 j < 1 , , pd;1 = 0. > $ , % p = kn + m = 0. . jmj < n, jkj = jjmnjj < 1. A%, k = k1 ; k2 = 0 m = m1 ; m2 = 0. > % , % (m1 k1) 6= (m2 k2 ). O , !$# (E n) %) /& ,-) & . 4 , (E n) & ". N+ ? ,-) &: F " (E ndvd ) " 1 ; 2 . F ) $& " ", " % $& &. < " (E ndvd ) " 22n;d vd .

Z

d-

1183

F E % $&

" (E nd vd ), ) $ : v 1 v 1 v 2 3 d (E ) = n n v : : : n (E) > 1 2 2 d;1 vd;1 (1 + n2)) : : :(vd ; vd;1 (1 + nd;1)) 2 ; : > (v2 ; (1 + n1))(v3 v; v: :2:v 2nd vd 2 d;1 n1 : : : nd;1 F " (E n) % + , % $&, (E )n1 : : :nd >

> 1 ; 2 (v2 ; (1 + n1 ))(v3 ; v2(1v+:n: 2:v)) : : :(vd ; vd;1 (1 + nd;1 )) : 2

d

> V & , , s - v 2 V , % (v2 : : : vd ) > (sv1 (1 + n1 1 ) : : : svd;1(1 + nd;1 )). < v " (E n) " (1; 2 )(1; s )d;1 . % " s, % $ " 1 ; . O , - & h 2 HT (V ), ,- ", (E n) 1 ; . E , % h - " (E n) &, 1 ; . ( $ : % E ; ( E " $ % ,

, E , ,- n11:::n d , E -). > . 1. > & , K(d) (. 11]), % % % & g f(Ei " mi)gi=1k . E, h 2 2 f(Ei " mi))gi=1k %, ) i 2 f1 2 : : : kg , (hmi Ei 4 gmi Ei) < ". A % t, )- 8";1 P i2max jmij j. 12:::k j =1d

4 , 1, & g - " (S (t : : : t)) (S t), ,- " 1 ; 4" . F S % % fS g, ) % , Ei, i = 1 2 : : : k, $ " (S t) , Ei , , + . 1 , fS g % S\

\

\

06s fRg | ,-) R, ) fS g. ),- U, - hs S gs R: 8s 0 6 s < t 8 : Uhs S = gs R (mod 0): . U -. A i 2 f1 2 : : : kg. . (U ;1hmi UEi 4 gmi Ei ) 6 ( (A 4 (B t C) 6 (A 4 B) + (C))

6

U ;1hmi U

+

6

i U ;1 hm U

+ 4" + 4" 6 X

Ei \

Ei n

X

06m , % 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-

1187

> 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. , 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 . 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 (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- &

) % -& % .

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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 dùn 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

: , , !" , #$ , $% .

& ##'' $% # "%( ' ', )*' '. & !+ $%', , # )* '- "%(.- #'#(. / #%., # 0% .%( 0! 1!%#( $0'# $'%(#( $%., .%(. 2##' "%( $ 0 $%#! .!%0! $% '( #$ 0# . 3 #!%(! !%##,#! $' | Kn , Kmn , 0' 2 | $0'# #! 1!#( #$%( #$ 0% !" $% .

Abstract S. A. Tishchenko, Separators in planar graphs as a new characterization tool, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 4, pp. 1193{1214. We consider planar graphs with non-negatively weighted vertices, edges, and faces. We let vertices and edges have nonnegative costs. In the case of triangular graphs with equal weights, the obtained results are proved to be equivalent and optimal. The analysis of planar graphs with non-negativelyweighted faces for a given plane embedding enables the separator search in dual graphs. We demonstrate e;cient planar graph characterization by the separator method on several classical examples: graphs Kn and Kmn , graphs of diameter 2.

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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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1195

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1) & "! A ! & "! BH 2) * "! A 1 * "! B. " " 1 GT 1 &( & , @$ : wV : V (GT ) ! R+, wE : E(GT ) ! R+, wF : F(GT ) ! R+,

1196 w(X) =

X v2V (X )

. .

wV (v) +

X

e2E (X )

wE (e) +

X

f 2F (X )

wF (f) X GT :

(2)

C( "( " " . C( " !, & ( @$ 1

GT 1 : cV : V (GT ) ! R+, cE : E(GT ) ! R+, X X c(C) = cV (v) + cE (e) C GT : (3) v2V (C )

e2E (C )

0 $, 1

c w (.

2 (R. J. Lipton, R. E. Tarjan, . . ). GT , ! . " GT # # ! GS cmax . $ % - , # e !!#! GS , !#% GT A, B , c(C) 6 cmax + cE (e) (4) ;wmin + P wE (x) + 2wF (Fe ) w(C) x=Ee (5) minfw(A) w(B)g > w(G) 3 ; 2 ; 6 wmin | C ' Fe | , ! e, Ee | , ( Fe. ) Fe ( ! ! fA B g' w(A) = w(B) . . -"! , "(& 3,

. $ 2], " ( GS 1

GT . - @$ E(GS ) E(GT ) n E(GS ) ( @( 1 . - GS | jV (GS )j = jV (GT )j > 3, $ $ @ $ e $ 1 Ce . GT "( , * 1 ,

I, $ " ( GT ) ( "! Ae Be , "!( ! Ce . * 1 "(& cmax + cE (e). 9" e) e 2 E(G n G ) (6) g(e) def = maxfw(Ae) w(Be )g + w(C T S 2

8> w(Ae ) > w(Be ) <jF (Ae)j def h(e) = >maxfjF (Ae)j jF (Be)jg w(Ae ) = w(Be ) e 2 E(GT n GS ) (7) :jF (Be)j w(Ae ) < w(Be )

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

$ x GS " & ( u & v, ( @$ . & s * " , & ( t " GS . 9$ Pu, Pv Pt " " GS s u, v t . , & s ! " $ & ! fu v tg. 0 * ( ", , "

& ( s. -$ !( ( !, ""! (8) "

($ $ . 2.1. *! fy z g . . 9 " ""! , fy z g \ E(GS ) = ?. 1 Cy , $( $ $ y GS ,

! & Ay By , "!( ! 1 Cy ( . 1 a ). ! 1 Cz , $( $ $ z GS , ! & Az Bz , "!(

! 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

( . 1 , , ). 2.2. y 2 E(Cx). . 9 " ""! , " & ( u & v " GS !

$ y ( . 1 ). - GS 1 , t 2 Ax , z 2= E(GS ). 1 Cz , $( $ $ z GS , ! & Az Bz , "!(

! 1 Cz . ) $

, (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. !

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1202

. .

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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.

? " " !

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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 , & "( "

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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

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 | / 8q6=p - A=G, p- . > q- (A=T )q 8 - A=T 5 5 q 6= p. " $ T 2 Sq (A) q 6= p. > T=G 2 Sp (A=G), G 2 Sp (A), T 2 Sp (A). B /

, T 2 S (A). @, T=G / p- , $ Tp; = G;p . F , G;p 2 S (A),

T . 5) ! A DG 2 Sp (A) G;p 2 S (A)E

, "* # A=p! A, p! A 2 S (A).

F , 0;p = p! A G;p G A. 6) &' A qc- , tA | , A1 = 0 p DH 2 Sp (A) | $ (" H = 0)E Hp; 2 Sp (A). H

tA 1.3, . 2). @

/. 3 p! A = 0;p 2 S (A). " $ , 5), /, p! A = 0. .

, Aq = 0 5 5 q 6= p. "/ G 2 Sp (A) Gp 6= 0. G

/, G;p 2 Sp (A). 6 D = (Gp )Âp , 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 )Â | 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 = GÂ. 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] . . 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 $- , ( ) , $-, $, ( $- # ')

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1236

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. ! " 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

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0

= ;7(0)A0 x(t) + 7(t)A0 x(0) + = 7(t)A0 x(0) +

Zt

Zt

Zt

7(t ; )Ax( ) d =

0

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0

7(t ; )f ( ) d:

0

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-

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$$

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

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R

Z Z Z f g~ dx ; 'ng~ dx 6 K jf ; 'n j dx = o(1)

R

R

R

1242

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I1 =

Z

Z

En

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R

$ 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] , '. ., '. $/ &$ &? ) &?. | .: , 1974. ( ) ) 1997 .

. .

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: , , , !".

# $ %$" % !$ ! !& $$ &$ .

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 * &") â : Max A0 ! R, " - a^ 0 a. 8 , &") a^ $)- -%$ A^0 . 9 /1] + *$), - )% % )) % (A0 k kA0 ) *). 24) *$, - )4 A^0 $)"+ )4 , %) -))+, &") ) *%%) Max A0 ) ) % )%) )%+. ) *) "

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Новый Мир ( № 4 2002)

Новый Мир ( № 4 2002)

Advanced Functional Programming 4 conf., AFP 2002

Advanced Functional Programming 4 conf., AFP 2002

Газета День Литературы # 68 (2002 4)

Газета День Литературы # 68 (2002 4)

2002)

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Журнал "Вокруг Света" №4 за 2002 год

Журнал "Вокруг Света" №4 за 2002 год

EvoPLAN Kinsale, Ireland, April 3-4, 2002. Proceedings

EvoPLAN Kinsale, Ireland, April 3-4, 2002. Proceedings

Algorithm Engineering and Experimentation, 4 conf., ALENEX 2002

Algorithm Engineering and Experimentation, 4 conf., ALENEX 2002

Aid Activities in Europe and Oceania 2002 Volume 2004 4

Aid Activities in Europe and Oceania 2002 Volume 2004 4

Journal of Biblical Literature, Vol. 121, No. 4 (Winter 2002)

Journal of Biblical Literature, Vol. 121, No. 4 (Winter 2002)

Информационный листок украинской фантастики N 4-2002 (114)

Информационный листок украинской фантастики N 4-2002 (114)

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