Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://dfgm.math.msu.su/spec/Rashevskii/tsvta_XXVII.pdf Äàòà èçìåíåíèÿ: Tue Mar 6 18:23:33 2012 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 22:46:10 2016 Êîäèðîâêà: IBM-866 Ïîèñêîâûå ñëîâà: m 97

. .



,
XXVII

-


2011


514; 515.1 22.15 78

, . . XXVII
. . . . , - , . 2011. : . . ( ), . . , . . , . . , . . , . . (), . . , . . (. ), . . , . . , . . , . . , . . , . . , . . , . . .
ë ¨ 1933 . . . . , -- . , . - . , , . , , , , , , , , , , , , , . , , .

ISBN 978-5-211-06253-5

? , 2011 .


514; 515.1 22.15 78

, . . XXVII
. . . . , - , . 2011. : . . ( ), . . , . . , . . , . . , . . (), . . , . . (. ), . . , . . , . . , . . , . . , . . , . . , . . .
ë ¨ 1933 . . . . , -- . , . - . , , . , , , , , , , , , , , , , . , , .

ISBN 978-5-211-06253-5

? , 2011 .



. . . . . . . . . . . . . . . . . . . . . , ( 2009 í 2011) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. B. , . . . , . . . . . . . . . . . . . . . . , . . , . . ,, . . , . . , . . , . . , . . , . . . . . . . , . . . . . . . . . . . . . . , . . G - - G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X (t) = K (t)X (t) . . . 4 9 41 42 51 83 106 150 173



. . . . . . . . . . . . . . . . . . . . . , ( 2009 í 2011) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. B. , . . . , . . . . . . . . . . . . . . . . , . . , . . ,, . . , . . , . . , . . , . . , . . . . . . . , . . . . . . . . . . . . . . , . . G - - G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X (t) = K (t)X (t) . . . 4 9 41 42 51 83 106 150 173



(3 1936 í 23 2011)

ë ¨ . 2011 , . , , 1950- . , -- . 3 1936 . -- -- ; . - , -- , , -- , . 1941 , , , . 1943 . ( -- ). 1943 1953 . 407, . , ( . . . . ), -- ( . . ). , ( í 1951, í 1952). 1953 . . . . . , . . , . . , . . . , 1958 . ë ¨ . . . , . . . . .




5

ë ¨. . 1958 . . . . . 1962 . ë ¨. . , . . 1964í1981 . ( 1966 1981 . -- ). 1981 2009 . (íí) , 1982í1983 . . 1999/2000 . (, ). 1982 . . íí , 13 . 1980- . , , , ë ¨ - - : , , . , , . , , . , ëMathematica¨ , .


6



. . -- , . . , -- . 1968 . . . í. ëTensor and Vector Analysis. Geometry, Mechanics, and Physics¨, . . , . . , . . Gordon and Breach 1998 ., . . ( ). 1960- . ( K - .). . . . -- . , K - , . ë ¨ (1973). . 1981 . , ë ¨ ( ). . . 1950- . , 1983 . -- ( . . , . . , . . , . . ). ,




7

. . 1. . , , , . . 2. . , . , , . 3. . . XX . , , . . , , , . 2000 . . . . . ë ¨, . . ë ¨ (.: , 1965; . . , . . , . . ; 100 000 .), ë , ¨ ( 2- .; .: , 1979, 1981; . . , . . , . . ; 280 000 .), ë ¨ / (.: , 1986; . . ; 70 000 .), ë ¨ / (.: , 1990; 70 000 .), ë ¨ (.: , 1992; 19 000 .). (, , , ), . 1996 . ë ¨.


8



. . , , , , . , , , 2010 . , , . . , . 2009 ., , . , , , , , . . . . . . . . .


, 2009 . 2011 .


10



. .
29 2009 .

F -- G L . W L , F (G) . ( ), -- L ( W (x1 , . . . , xn ) , x = x1 = . . . = xn ) . L , L . , L W (x1 , . . . , xn ) = Tr(x1 § . . . § xn ) , x1 , . . . , xn -- L , ë § ¨ , ë Tr ¨ . . , , . G/H -- , g, h -- G, H . Ad(H ) g , G -- () . () Ad(H ) g = h + b1 + . . . + bk , Ad(H ), F1 , . . . , Fk . W (x1 , . . . , xn ) = Tr(x1 § . . . § xn ) bj , j = 1, . . . , k , Fj . .




11

. . ,
5 2009 .

, .. 4- , . , , . 4- . , .. . , 4- ë¨ , .. . . , . , , , 4- , -- , . , . . .


12



. . : ,
12 2009 .

-- , . . (1885). d -, E d (.. ) . -- . -- . (, , .), ( .). , .




13

. . , ,
19 2009 .

, -: G , G . -- , , , -, . , . : , , . , , , . , , -, , 4T- . ( , , ) , -- . . 1. , () . , , , ? 2. , sl(n) -? , 1 2 : ë¨ .


14



, , - so(3) . , -- 1 -- . M n ½ n . {1, . . . , n} M I J , MI MJ . , 0 , .. , d -, .




15

. . ,
26 2009 .

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


16



. . , SO3 -
3 2009 .

G -- . G s1 , s2 , s3 , s1 s2 s3 = e (.. ). , ( . . ) , G . ( ) SO3 . , H G G . , , SO3 ( SO3 -), G . SO3 - . SO3 - , G 5 (.. 1, 3 5). SO3 - 3- , . . . SO3 - 2005 . , SO3 -, , . . . 1960- . . . , . G/H , H -- , . , , . , , R C H . , s1 , s2 , s3 G X = G/K , K = ZG (s1 ) . s1 --




17

o = eK , s2 -- p . x, y X , (x, y ) G - (o, p) . , - , . , K/L , L = ZK (s2 ) . . ë ¨ . X . d , , , , d . 1980í1990- . . . , . . . g G , , J , To (X ) J J . X J . g J . , J -- , 1964 ., . . (, , ) . , . , , ( ). . . 1960í1970- . ( ). . . .


18



. .
10 2009 .

, ë¨ . Rn , , , . , R2 , R3 ; , . , , . 1. ? 2. , ? . , , R2 . ( ) , . , , -- , .




19

. .
17 2009 .

M , f1 , . . . , fs , M . . K , , . , . -- , K . , , . , , , , .


20



. . ,
18 2010 .

. , : sign X = 22k L(X ), [X ], L(X ) , L(X ) =
j

tj /2 , th(tj /2)

tj -- , k (t2 , . . . , t2 ) = pk (X ), 1 n pk (X ) . , , , . , (, , , , ), , , , , . , , . , , , . .




21

. .
4 2010 .

. , / (, ..) , , ë¨ . : 1) ; 2) ; 3) . -- , . ( ) , , , . 1. , ( ) , . 1, , -- . 2. , , . ( ). . . . , ( ) , ( , , .). , ,


22



. . , .




23

. . H
18 2010 .

( G ; a(g , z ) ), G - a - A(G, a) . , . , ( ) . , (G, a) , A(G, a) . , , , . , C2 , í.


24



. .
1 2010 .

T QS(S1 ) S1 (- ). , S1 , . T S Diff + (S1 ) S1 . QS(S1 ) Diff + (S1 ) 1/2 H := H0 (S1 ; R) . T S , T S . , , (), . S Vect(S1 ) Diff + (S1 ) . F (H ) , 1/2 H = H0 (S1 ; R) . Diff + (S1 ) H () Vect(S1 ) F (H ) , S . T , QS(S1 ) T . , QS(S1 ) . Derq (QS) , , F (H ) . dq h H , h QS(S1 ) .




25

. .
15 2010 .

, . , ( í, ) , . , . , , , , . , ( ) .


26



. . í
29 2010 .

, , -- , . , . . . . , , ( ) . (, , .), , í , . , , , . -- í. , . , K - .




27

. .
13 2010 .

. (1867). , ë ¨ , , . ë¨ . 1980- . , , . , .. . .


28



. . , . . .
23 2010 .

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




29

. .
7 2010 .

Lien n - ( K ; K = R, C ). Lien , . , . , . Lien . . , . , (.. ) .


30



. .
21 2010 .

( ) p V i p2 2 =- + - mc + V - + t m x x p 2m (( ) ) + p+i + kT m , p x p (x, p, t) . (x, p, t) = |(x, p, t)|2 , , . , 1/ ( , = /m -- ) . . . , ( ), (x, p, t) , .




31

. .
11 2010 .

, , . , í. , , , , , . , í , í. -. , í 4 - R4 , í. í , 1984 . , . , , G - í R4 G , G -- . .


32



. . -
25 2010 .

- , . , . .




33

. .
( Skype)

,
9 2010 .

. , . sl2 , , . , , S3 ½ S1 , .


34



. .
23 2010 .

, . . , , -- , . , . í . , , .




35

. .
21 2011 .

. M n -- n - C 1 - R , Sm -- m - . k - Lk Sm , 0 k - V -k L

{V | } -- Lk Sm , |V | -- m - V , |Lk | k - Lk . M n , Sm Lk . k = 0 m = n (V ) = f (x1 , . . . , xn ) dx1 . . . dxn
- V

V = (x , . . . , x0 ) M n , n 1 f (x1 , . . . , xn ) = f (x0 , . . . , x0 ). lim 1 n 0 |V | V L
- V

L0

0 1

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


36



. .
14 2011 .

, XX ., -- , . , . , , , . . 2007 . . , . . -- . , . 2 , . , , -- , . . -, . . . . . - , , . - .




37

. .
28 2011 .

, , , , . ë¨ . , , . . , . . . 2, .. . , -- . .
Parshin A. N. Representations of higher adelic groups and arithmetic, Proc. ICM Hyderabad, 2010 (arXiv:mathNT/1012.0486).


38



. .
11 2011 .

() , . , . , 1) ; 2) , (, , ); 3) . , . . . , , .
1. . . , . . 2010. 201, 5. 65--110. 2. Manturov V. O. A functorial map from virtual knots to classical knots and generalisations of parity, arXiv:math.GT/1011.4640.




39

. .
25 2011 .

, , , . -- A = {a1 < a2 < . . .} , (ai+1 - ai ) . í , |A + A| |A|3/2 . (Z/pZ) ½ (Z/pZ) .


40



. . . .
16 2011 .

. . Rn ( , 1-ULC, ë¨). , , . , n - 2 . , , , Rn . 1 . . AncelíCannon , , , , AncelíCannon . , , , .





. .


, . ( ), .

, . , . , , XIX ., . ( , ) , , . -- -- ( , .. ). 1907 . . . ( . . ) , S D/ , D -- C , C = S2 H -- ( -- C ), Hol(D) -- Hol(D) D . , Hol(D) D , .. D ( Hol(D) ), D = Hol(D)/K , K --




43

Hol(D) ( , D = C ). \G/K , G -- , K -- G , -- ( D ). S (, D = G/K ). , -- 22- . , , , . M -- ( ). M , ( ) M = \G/H , G -- (, , ), H -- G , -- , G/H . ( G/H ) () M . G/H = M -- , M , -- . M G/H . Ui M Ui Vi G/H , Ui Uj G ( G G/H ). [1]. [12] G/H - M : , U f : U V G/H , - f 1 f = f : (U U ) (U U ) G . G/H , M . , G/H0 ( H0 -- H ) G/H -. G/H , G/H -. , , , . . M , F \G/H , F, H -- G , -


44

. .

F G/H , . , , H F \G (.. -- ). , , . M , M = \G/H , -- () Diff (G/H ) G/H , G/H . , . , . , . 1. (.. , ) . , , . 2. M , ( ) , . , , ( ). , CP 2 # CP 2 -- -- (. [8], 6 ). 3. M 2 3, ( n = 3 ) . -- . í è (.. ) . 4. . , T n -- ( ). -- \R , -- ( Zn ) ( , ) R (. [2]). 5. -- -




45

. M 4 , , M 4 = \X , X = G/K ( . [10]). M . dim(M ) = 1 , M R1 S1 . , . M -- ( ). ( I II -- -- ). , , , M . , G M = \G/H ( , M ), . 3. , , . . . : ? . M = (S2 ½ S1 ) # (S2 ½ S1 ) -- S2 ½ S1 . , ( ) ( ). , . . 1 , 1 , i (i = 1, 2) , i -- i , /1 1 i /2 . , 2 , ( ). , . -- . -- , a U = Z[] -- . - U . (FL) (.. ), - Z -


46

. .

, U -. cd() n . n = cd() , H q (, A) = {0} q > n - A , H n (, A ) = {0} - A . , cd() = . ( ), . (VFL) (.. (FL) ), , , / (FL) . , , (FL) . -- (VFL) vcd() = cd( /) ( , ). , , .. . vcd() . , vcd(1 (M )) dim(M ),

vcd (., , [3]). 1. M 3 -- . , vcd(1 (M )) = 0, 1 3, : 1) vcd(1 (M )) = 0 , M S3 M = S3 /D , D -- , S3 ( -- ., , [4]); 2) vcd(1 (M )) = 1 , 1 (M ) Z (, ) M S2 ½ S1 ; 3) vcd(1 (M )) = 3 , M R3 ( , M ) 1 (M ) .




47

. M = \G/H , -- Diff (G/H ) -- G/H . G . G/H M . G/H0 ( H0 -- H ) -- G/H ( G , H0 ). G/H0 -- M . G/H0 -- . M = G/H0 S3 , S2 ½ R R3 [5]. . M = S3 , M = S3 /D , D -- , S3 . vcd(1 (M )) = 0 . M = S2 ½R , M S2 ½ S1 [11]. , 1 (M ) , Z , , vcd(1 (M )) = 1 . M = R3 , M vcd(1 (M )) = 3 . . , 1 ë ¨: S3 , S2 ½ S1 R3 , . , . (3) 1 , [7]. , M = R3 1, . M n , M =Rn , , , ( Rn ). Diff (Rn ) ( n = 3 ). , ( ., , [3]). 1. (S2 ½ S1 ) # (S2 ½ S1 ) . . (S2 ½ S1 ) # (S2 ½ S1 ) - F (2) = Z Z . -


48

. .

1. M = (S2 ½ S1 ) # (S2 ½ S1 ) , 1 , Z . F (2) . . (S2 ½ S1 ) # (S2 ½ S1 ) : , , ë¨ . 1. [11]. ( ). M = S2 ½ S1 , e(M ) 2 . 1 (M ) , Z ( . [9]). . , , : N (S2 ½ S1 ) # (S2 ½ S1 ) ½ N . , , , , F (2) ½ . , -- ( ) . , Fg ½ S 3 Fg g 2 ( ) (, , SU(2) ), ( . [6]). M ( M = \G/H ), -- G . M , vcd() 3. : D , vcd(D) 3 ? ? : ) D -- 3 ( , Z § Z2 ); ) SL2 (C) ( , SL2 (R) ) A ( SL2 (R) );




49

) g g ½ Z , g -- Fg g 2 . . (. [6]). , ( ) M . , M , ( ). . 1 . 2 , , 2 . 2. M -- n - , . : 1) vcd(1 (M )) n;
n

2) vcd(1 (M )) = n , M = R ( M ); 3) vcd(1 (M )) = n - 1 ;

4) vcd(1 (M )) = 1 , 1 (M ) Z. M 3 . M 3 , M R3 S2 ½ R ( , 1, M = S3 M ). M = R3 , M , cd() 2 . , (, M = F ½ R , F -- ( ) Z Z2 Z2 ( K 2 ). M = S2 ½ R . M , . , 2 (, RP 2 ½ R) .


50

. .

1. . . . , . 1991. 2. A u s l e n d e r L. An exposition of the structure of solvmanifolds. Bull. Amer. Math. Soc. 1973. 79, 2. 227í285. 3. . ., . . . . . . . . . 1988. 29. 147í259. 4. . . , . 1982. 5. . . . . . . 1977. 18, 2. 280í293. 6. . ., . . . . . . . . . 1988. 20. 103í240. 7. D a v i s M. W. Groups generated by reflections and aspherical manifolds not covered by Euclidean space. Ann. of Math. 1983. 117. 293í325. 8. K a g a T., W a t a b e T. Simply connected 6-manifolds of large degree of symmetry. Sci. Rep. Niigata Univ., Ser. A. 1975. 12. 15í32. 9. ., . . , . 1977. 10. . . , . 1986. 11. T o l l e f s o n J. The compact 3-manifolds covered by S 2 ½ R . Proc. Amer. Math. Soc. 1974. 45. 461í462. 12. T o m a s s i n i A. G -Geometrie omogenee e uniformizzazione. Boll. Unione Mat. Ital., Sez. A, Mat. Soc. Cult. (8). 1998. 1, Suppl. 71í74.


. . , . .



, . .
. . : ( .) , , ? , ( ..) , . .


ç 1.

( [1, 2, 14]) , ( ) X X X , , X . X X . , X -- G , X G . ë¨ ( . . ) --
E-mail addresses: graev_36@mtu-net.ru, glitvinov@gmail.com. 10-01-00041-a. CNRS ().


52

. . , . .

G . , , -- . . -- [3] R2 , f (x, y ) R2 . R2 y = ax + b,


f (x, y ) (Rf )(a, b) =
-

f (x, ax + b) dx.

, : f (x, y ) (Rf )(a, b) =
R
2

f (x, y ) (y - ax - b) dx dy .

, f = Rf : f (x, y ) = c
R
2

(a, b)|y - ax - b| da db,

c (R2 ) , . R2 R2 -- . . . , , .


. .

53

, R2 c R2 : f (x, y ) (F f )( , ) = f (x, y )ei(x+y) dx dy .
R2

R2 , : = F f , f (x, y ) = ( , )e-i(x+y) d d .
R2

, . , : (Rf )(a) f . , , .. ë¨ . , , . : 2 |f (x)| dx = (F f )( ) d .
R
2

(R2 )

, Rn ( ); , Rn , ( . . ) ( Rn ). , , .. ë¨ , f (x) f () = f (x)e(x, ) d²(x), (1)
X

, , , , .




54

. . , . .

f () f (x) =
X

f ()e(x, ) d²(),

(2)

X X -- d² d² , x X , X , e(x, ) X ½ X . ( ) X X . [4], , . ç 2. R2 ë¨ , (§) a(§) . , , , . a(§), , . ( ), . , (1) (2) . . .
ç 2.

( . . ) ( ) . . . . ( ) X X . , X , y X


. .

55

dy x . X ~(y ) = J : f (x) f f (x) dy x,
y X

: ~ J : f (x) f (y ) = f (x)a(x, y ) dx,
X

(3)

a(x, y ) (, ) X ½ X , a(x, y ) = y (x) , supp a(x, y ) = = {(x, y ) X ½ X | x y } dx -- X . ~ ~ J -1 : f (y ) f (x) = f (y )b(x, y ) dy ,
X

b(x, y ) -- X ½ X dy -- X . . J , F : f (x) f (y ) = f (x)e(x, y ) dx, (4) e(x, y ) , L2 (X, dx) -- X ½ X . L2 (X , dy ) , : f (x)g (x) dx = f (y )g (y ) dy . (5)

, - g (x) = x (§) ; (4) , x = e(x, y ) , : f (x) = f (y )e(x, y ) dy . (6)


56

. . , . .

, (5) (4) (6). , L2 (X, dx) S (), S ( ) S = F (S ) L2 (X , dy ) . , S S (4) -, M M , () S S . . . [5, 6]. ( (4) (6) , , (5)) F ( ), e(x, y ) e(x, y ) -- . 1. X : S , . [4]; , , . y R f (x) = f () § e(y , ) § e(x, ) d, X (x) = e(x, ) X . x() = e(x, ) X . , . X , X ; , X X F -1 (y ) = e(x, y ) . . , X = X = Rn , dx dy -- , F -- , e(x, y ) -- eix,y , S . , . M () . ç 6 S X ,


. .

57

. X . , L , F = LJ , X X . {J } (3) a (x, y ) , a0 = a(x, y ) , J0 = J , J1 = J -1 0 1 ( , ) , = ² , J² , X X .
ç 3.

x, y X X . (4) (6), f g f (x) g (x) X f g =F
-1

(f § g ),



f = Ff



g = F g.

, X . . M M . , , . [12]. , ( ) , . [5, 6, 12]. X f (x) f (x, y ) = = Rx f (y ), x, y X , Rx f (y ) -


58

. . , . .

x y f . , y R f (x) = f () § e(y , ) § e(x, ) d, X (x) = e(x, ) X . x() = e(x, ) X . ., , [12]. Rx , Rx Ly = Ly Rx x y X , Ly Ly f (x) = Rx f (y ) . . 1. x0 X ( ), R
x0

= I,

(7)

I -- . , x0 ; , . (7) , e(x0 , y ) = 1 y X . . , . . () ( ). , , [4í12]. , ( ) G ; Rx Rx : f (t) f (tx), tx -- , .. G . x y ² G . -, , ² . G . , ; , . , , ., , [6, 12].


. .

59

ç 4. , R3 4.1.






R

3

R3 f R3 Rf R3

(. [3]) R3 , .. R3 .

x3 = a1 x1 + a2 x2 + a3 , ( , x3 ) dx1 dx2 . a = (a1 , a2 , a3 ) R3 . = Rf (a) = (a1 , a2 , a3 ), (a1 , a2 , b) = f (x1 , x2 , a1 x1 + a2 x2 + b) dx1 dx2 .
R2

-, R3 : (a1 , a2 , a3 ) = f (x1 , x2 , x3 ) (x3 - a1 x1 - a2 x2 - a3 ) dx1 dx2 dx3 .
R3

. R3 , . (., , [14]), : (a) = Rf (a), 2 (a1 , a2 , a3 ) f (x) = da1 da2 , a2 3 a3 =x3 -a1 x1 -a2 x2
R
2

, -, f (x) = (a1 , a2 , a3 ) (x3 - a1 x1 - a2 x2 - a3 ) da1 da2 da3 .
R
3

, , .. f x , .


60

. . , . .

4.2.







R3

1. u(t) R Ju f R3 (Ru f )(a) = f (x1 , x2 , x3 )u(x3 - a1 x1 - a2 x2 - a3 ) d²(x),
R3

d²(x) = dx1 dx2 dx3 . Ru . , - (t) u(t) . f (x) , u(t) = Ru f . f ( ) u(s) f (x) u(t) (a1 , a2 , c) a3 (a1 , a2 , a3 ) , .. f ( ) = f (x)ei,x dx, u(s) = u(t)eist dt,
R3

(a1 , a2 , a3 )e
R

R ica
3

(a1 , a2 , c) =

da3 .

1. = Ru f , f ( ) , u f , u (a1 , a2 , c) = f (-a1 c, -a2 c, c)u(-c). . (a) = f (x)u(x3 - a1 x1 - a2 x2 - a3 ) dx
R3

(8)

f u f u , (a) = f (a1 c, a2 c, -c)u(c)eica3 dc.
R

(8) a3 .


. .

61

1. u -- , , f = Ru f . , (8) , f , f , . 2. u 1, f = Ru f : f (x) = (a)U (x3 - a1 x1 - a2 x2 - a3 ) da1 da2 , (9)
R
2

U ( t) =


R

[u(c)]-1 c2 e

ict

dc.

. f (x) = f ( )e-,x d =
R
3

=
R
3

f (-a1 c, -a2 c, c)e

ic(a1 x1 +a2 x2 -x3 ) 2

c dc da1 da2 .

, (8), f (x) = (a1 , a2 , c)[u(-c)]-1 c2 e
R
3

ic(a1 x1 +a2 x2 -x3 )

dc da1 da2 .

, (9).
4.3.


,









R3

3. u(t) R , c R , Ru , |u(c)| = |c| c R. (10)


62

. . , . .

. 2 , Ru , U (t) = u(t), U (t) =
R

[u(c)]-1 c2 e

ict

dc.

(11)

(11) -1 2 [u(c)] c = u(t)e(-ict)dt = u(c),
R

.. (10).
4.4.


3 u(c) = = |c|1+i sgn c , R = 0, 1 . [13], (, ) = (0, 1) |t|-2-i sgn (t) , (, ) = (0, 1) -- (t) . , , R3 , f (x) f (x) (p(a, x)) dx
R
3

f (x)


R3

|p(a, x)|-

2-i

sgn(p(a, x))dx

(, ) = (0, 1),

p(a, x) = x3 - a1 x1 - a2 x2 - a3 .
ç 5. , Rn C
n

R3 Rn Cn n . . Rn Cn f . xn = a1 x1 + . . . + an
-1 xn-1

+ an


. .

63

a = (a1 , . . . , an ) , Rn Cn (Rf )(a) = f (x) (xn - a1 x1 - . . . - an-1 xn-1 - an ) d²(x),
L
n

L = R L = C , d²(x) -- Ln . Rn Cn , u(t) R C , (Ru f )(a) = f (x)u(xn - a1 x1 - . . . - an-1 xn-1 - an ) d²(x).
L
n

= Ru f , f u f u an : (a1 , . . . , a
n-1

, c) = f (-a1 c, . . . , -a

n-1

c, c)u(-c).

, u -- , , , f = Ru f . R3 f (x) = (a)U (xn - a1 x1 - . . . an-1 xn-1 - an ) da,
L
n

U (t) =


L

[u(c)]-1 |c|k e

iRe ct

d²(c),

k = n - 1 L = R,

k = 2(n - 1) L = C.

R3 4. Ru n-1 , |u(c)| = |c| 2 L = R |u(c)| = |c|n-1 L = C .


64

. . , . .

. 4 Ln , L -- . . L = C 4 u(c) = c c² , , ² C , Re( + ²) = n - 1 , - ² = n Z . (, ²) = (k , n - 1 - k ) , k = 0, 1, . . . , n - 1 n-1 Ru u(t) = k n-1-k (t) , t t (t) -- - C . -²-1 u(t) = t--1 t . L = R 4 n-1 u(c) = |c| 2 +i sgn (c) , R , = 0, 1 . , n -- , n = 2k + 1 u(c) = ck , u(t) = k (t) . n+1 u(t) = |t|- 2 -i sgn (t) .
ç 6. , Sn Rn+1 6.1.


Sn Rn

+1

Sn , ( S2 ). , = 0 , F ( ) (J F )( ) =
S
n

F ( ) ( , ) d ,

(12)

d -- . , J F 0 . (12) . (J F )( ) =
S
n

F ( )sgn( , ) ( , ) d .


. .

65

Ju , , (Ju F )( ) = F ( )u( , ) d ,
S
n-1

u(§) -- R , .
6.2.


,

:

, . , , , Sn , Rn . (. . 6.3) Rn+1 . . Rn J Sn . , Rn Sn , Rn Rn+1 -- xn+1 = 1 : i = xi , |x| i = 1, . . . , n,
n+1

=

1 , |x|

( n 2 )1/2 . |x| = 1 + xi
i=1

Rn Sn . , R Rn Sn (12), Ju -- Ju u . , Rn 5. Sn (J F )( ) =
S
n

F ( )| , |

-

n+1 2

+i

d ,

R,

.


66

. . , . .

, Sn n+1 (J, F )( ) = F ( )| , |- 2 +i sgn( , ) d .
S
n

5 . F Sn Rn+1 - n+1 - i n - 2 d Sn -- n - dx Rn+1 n + 1 . F n - Rn+1 : (x; ) = F (x)| , x|-
n+1 2

+i

dx.

, -- 0 Rn+1 , - n+1 + i 2 Rn+1 . -- Rn+1 , , . , n+1 (13) ( ) = F (x)| , x|- 2 +i dx


. , ( ) , (Sn ) (Rn+1 ) (13), (Rn+1 ) - n+1 + i 2 n - d (Sn ) -- n - dx Rn+1 n + 1 . n+1 F (x) = ( )| , x|- 2 -i dx , (14)



-- (Rn+1 ) , , , d -- , (Rn+1 ) , . , -- xn+1 = 1 -- an = 1, (13) , Rn , (14) --


. .

67

. F = F . , F = F . , (13) , -- , .
6.3.


,

:

, Rn+1 . C Sn : ( ) - 2 (J f )( ) = ( +1 ) f ( )|, | d . 2
Sn

Re > -1 C . , J-1 = J . Sn : ( ) - 2 (J, f )( ) = ( +1 ) f ( )|, | sgn(, ) d . 2
S
n

6. J : = J f , f =J
-+n-1

.

J, . . Rn+1 :


( ) =
S
n

f (r )e
0

ir, n

r dr d , (15)

f (r ) =
S
n

( )e
0

-ir, n

d d.


68

. . , . .

, f -- Rn+1 , f r : f ( ) = f (r )r dr, ( ) = ( ) d, C.
0 0

Re > -1, Re < -1 . (15) . || (. [13]): ) ( +1 2 F (|| ) = ( ) |s|--1 . -2 ( ) +1 2 ( ) = ( ) f-+n-1 ( ), |--1 d . -2
S
n

, (15) r r , ( +) -2 n f-+n-1 ( ) = ( -n+1 ) ( ), |-n d. 2
S
n

-+n-1

f ( ) = f+n ( ) ( ) = --1 ( ) , - + 1 ( ) - 2 ( ) = ( +1 ) f ( ), | d , 2 Sn ( +n+1 ) f ( ) = ( -2 n ) ( ), |--n-1 d. - 2
Sn

, = J f f = J--n-1 , . 2. J , - - n - 1 = , .. = - n+1 + i , R . 2 , , .


. .

69

ç 7. , C3 , 7.1.
C3 J

-- C3 K -- ( ) C3 , . , , -- , (x0 , x1 , x2 , x3 ) x0 = 0 , x1 = u1 (t)x3 , x2 = u2 (t)x3 , t C.

K (x1 , x2 , x3 ) x1 = u1 (t)x3 + 1 , x2 = u2 (t)x3 + 2 , t, 1 2 C . t, 1 , 2 K . K J : f (x) (J f )(, t) = (, t), (a, t) = f (u1 (t)x3 + 1 , u2 (t)x3 + 2 , x3 ) d²(x3 ),
C

(16)

d²(x3 ) -- . , J f C3 K . (16) , : J f (, t) = f (x) (x1 -u1 (t)x3 -1 ) (x2 -u2 (t)x3 -2 ) d²(x), (17)
C
3

(§) -- - C , d²(x) -- C3 . , K [2], .. 1) f = J f ; 2) f x , x ( ).


70

. . , . .

7.2.



J

a

La = (1 , 2 ) (, t) = (1 , 2 , t) a(, t) = a(1 , 2 , t) : La : (a, t)
C2

( + s, t)a(s, t) d²(s).

Ja f C3 , a(s, t) , () Ja = La J J , K , La . (Ja f )(, t) = f (x) a(x1 -u1 (t)x3 -1 , x2 -u2 (t)x3 -2 , t) d²(x), (18)
C3

.. Ja (17) J (s1 ) (s2 ) a(s1 , s2 , t) . a(s, t) .
7 . 3 .

J

a

, f S , S -- . C3 . = Ja f . f ( ) = f (1 , 2 , 3 ) f , (1 , 2 , t) a(1 , 2 ; t) -- (1 , 2 , t) a(1 , 2 , t) . (18) 7. f , a (1 , 2 , t) = f (1 , 2 , -u1 (t)1 - u2 (t)2 ) § a(-1 , -2 , t). (19)

3. a -- , f , , f = Ja f , a(1 , 2 , t) = 0 C3 .


. .

71

4. = Ja f : (1 , 2 ; t) a(-1 , -2 , t ) = (1 , 2 ; t ) a(-1 , -2 , t) (1 , 2 , t) (1 , 2 , t ) , u1 (t)1 + u2 (t)2 = = u1 (t )1 + u2 (t )2 . 2 (., , [14]). , , Cr ( ) = Cr (1 , 2 , 3 ) , t u1 (t)1 + u2 (t)2 + 3 = 0 , .. 1 x1 + 2 x2 + 3 x3 = 0 . Cr ( ) , Cr . , C3 Cr = 0 . . L3 , L -- . , L = C , , , . 8. a(1 , 2 , t) -- , , Ja : = Ja f , f (x) = (, t) A(x1 -u1 (t)x3 -1 , x2 -u2 (t)x3 -2 ; t) d²() d²(t), (20) 1 A(s1 , s2 , t) = Cr
C2

|u (t)1 + u (t)2 |2 1 2 e a(1 , 2 ; t)

iRe(s1 1 +s2 2 )

d²( ).

. C3 f (x) = = 1 Cr
C
3

f ( )e

-iRex,

d²3 ( ) =

f (1 , 2 , -u1 (t)1 - u2 (t)2 )|u (t)1 + u (t)2 |2 d²( )d²(t). 1 2


72

. . , . .

(19) , f (x) = 1 = Cr (1 , 2 ; t)

|u (t)1 +u (t)2 |2 1 2 e a(-1 , -2 ; t)

-iRe[x1 (t)1 +x2 (t)2 ]

d²( )d²(t),

xi (t) := xi - ui (t)x3 , i = 1, 2 . , (20).
7.4. ,


K

9. a(1 , 2 ; t) 8, Ja , (18), , K , , |a1 , 1 ; t| = Cr
-1/2

|u (t)1 + u (t)2 |. 1 2

(21)

. 8 , Ja , , a(s, t) A(s, t) (20) A(s1 , s2 ; t) = a(s1 , s2 ; t), .. 1 Cr
C3

|u (t)1 + u (t)2 |2 1 2 e a(1 , 2 ; t)

iRe(s1 1 +s2 2 )

d²( ) = a(s1 , s2 ; t).

(22)

(22) |u (t)1 + u (t)2 |2 1 2 = Cr a(1 , 2 ; t) a(s1 , s2 ; t)e
-iRe(s1 1 +s2 2 )

d²(s).

a(1 , 2 ; t) , (21), .
7.5.
-1/2

1. a( ; t) = Cr (u (t)1 + u (t)2 ) (u (t)1 + u (t)2 ))² , 1 2 1 2 Re( + ²) = 1 , - ² -- .


. .

73

, (, ²) = (1, 0), (0, 1) , ) ( )--1( s1 s1 -1/2 a(s; t) = Cr u (t) u (t) 1 1 (, ²) = (1, 0) a(s; t) = Cr (, ²) = (0, 1) a(s; t) = Cr
-1/2 -1/2 -1/2

-²-1

(u (t)s2 - u (t)s1 ); 1 2

( ) u1 (t) + u2 (t) (s1 , s2 ); s1 s2 ) ( + u (t) (s1 , s2 ). u (t) 2 1 s1 s2

2. a( ; t) = Cr (u (t)1 + u (t)2 )(1 / 1 )1 (2 / 2 )2 , 1 , 2 = 0 . 1 2 ( -1/2 a(s; t) = Cr u (t)s-1 -2 s1 -1 s-2 -1 s1 2 -1 + 1 1 2 1 ) - - +u (t)s1 1 -1 s1 1 -1 s2 2 -2 s1 2 -1 . 2 z z ² C (. [2]).
ç 8. , R3 ,

, C3 , R3 , . K x1 = u1 (t)x3 + 1 , x2 = u2 (t)x3 + 2 , t R,

t, 1 , 2 R , K f R3 -- (Ja f )(, t) = f (x)a(x1 - u1 (t)x3 - 1 , x2 - u2 (t)x3 - 2 , t) d²(x).
R
3

, f (1 , 2 , 3 ) f = Ja f a , ..


74

. . , . .

(1 , 2 , t) a(1 , 2 , t) , (19), Ja . , Cr ( ) , .. t u1 (t)1 + u2 (t)2 + 3 = 0, . R3 -- Ja . : 10. a -- , R3 , f = Ja f , Cr ( ) . , f (19) . 11. 10, f Ja f f (x) = (, t)A(x1 - u1 (t)x3 - 1 , x2 - u2 (t)x3 - 2 ; t) d²() d²(t),
R3

A(s1 , s2 , t) = |u (t)1 + u (t)2 | 1 2 e = Cr (1 , 2 , -u1 (t)1 - u2 (t)2 ) a(1 , 2 ; t)
R2

i(1 s1 +2 2 )

d²( ).

5. Ja , |a(1 , 2 , t)|2 = |u (t)1 + u (t)2 | 1 2 . Cr (1 , 2 , -u1 (t)1 - u2 (t)2 ) (23)

, u1 (t) u2 (t) -- , Cr ( ) 1 , (23) |a(1 , 2 , t)| = |u (t)1 + u (t)2 |1/2 . 1 2 . Cr ( ) 1; |a(1 , 2 , t)| = |u (t)1 + u (t)2 |1/ 1 2 a(s; t) = s1 (t) u1
-3/2-i 2+i

.

(u (t)s2 - u (t)s1 ). 1 2


. .

75

ç 9. , k - Cn 9.1.


K

ç 7 C3 k - Cn n k < n. Cn Cn = Ck Cl , k + l = n , x = (x1 , . . . , xk ) y = (y1 , . . . , yl ) k - Ck Cl x y = u(t)x, u(t) = uij (t) -- (l, k ) -, -- t = (t1 , . . . , tk ), ti C . : yi = ui1 (t)x1 + . . . + uik (t)xk , i = 1, . . . , l .

C u(t) n - K , k - Cn , yi = ui1 (t)x1 + . . . + uik (t)xk + i , : y = u(t)x + . , K -- n - k - (.. K ) t u(t) Ck (l, k ) -. = (1 , . . . , k ) t = (t1 , . . . , tk ) K . . y = u(t)x, t Ck Cn k - (k - 1) - x0 = 0. K k - Cn , (k - 1) - , , k = 1 K Cn , . n = 3 ç 7. i = 1, . . . , l . (24)


76

. . , . .

9.2.

, K

K J , f (x, y ) Cn : J f (, t) = f (x, u(t)x + ) d²(x),
C
k

d²(x) -- Ck . J J f (, t) = f (x, y ) (y - u(t)x - ) d²(x) d²(y ), (25)
C
n

(§) -- - Ck . : f = J f . (, t) : ( , t) = (, t)eiRe, d²(), , =
l i=1 C
l

i i . (, t) =
(Cl )



( , t)e

-iRe,

d²( ),

(26)

(Cl ) -- , Cl , d²( ) , . (25) 12. f f ( , t) = f (- u(t), ), u(t) -- k - j =
l i=1

(27)

i uij (t),

j = 1, . . . , k .


. .

77

6. k - Cn , (24), f = J f . , (27), f f ( , f ) , ( , t) Cl ½ Ck ( , t) (- u(t), ) Ck Cl , .. ( , ) t = - u(t) . u(t) . J . 3. , K , CrK ( , ) Ck Cl , t = - u(t) . , CrK . k = 1 , n = 3 ç 7. , , uij (t) , . , Cr < . , , u(t) . Cr . f = J f (27) : f (x, y ) = f ( , )e-iRe[x,y,] d²( ) d²( ) = 1 = f (- u(t), ) | ( , t)|2 e-iRey-u(t)x, d²( ) d²(t), CrK
( -- , .. ( , t) = ,t) . , ( , t) -- t k 1 , . . . , l . , (26), 1 ( , t)| ( , t)|2 e-iRey-u(t)x, d²( ) d²(t) = f (x, y ) = CrK 1 = (, t)| ( , t)|2 e-iRey-u(t)x-, d²( ) d²(t) d²(). CrK


78

. . , . .

13. J : 1 f (x, y ) = CrK
Ck

, : , CrK < , ( )( ) i , t i , t (, t)

d²(t).
=y -u(t)x

9.3.

, K

K , (25) J - (s) Cl a(s, t) : (Ja f )(, t) = f (x, y ) a(y - u(t)x - , t) d²(x) d²(y ). (28) , Ja f = J f a(s, t) : (Ja f )(, t) = ( + s, t) a(s, t) d²(s).
C
l

14. = J f = Ja f ( , t) = ( , t) a(- , t), a(- , t) (29) J f (29)

-- a(, t) . , f Ja f . f = Ja f . (27) (29) ( , t) = f (- u(t), ) a(- , t). (30)

a -- , , (30) f . , , :


. .

79

15. CrK < , a( , t) -- Cn , , f = Ja f : f (x, y ) = (, t) A(y - u(t)x - , t) d²() d²(t), 1 A(s, t) = CrK
9.4. ,



| ( , t)|2 e a( , t)

iRes,

d²( ).

(31)



K

16. 15 Ja , a( , t) s a(s, t) |a( , t)| = Cr
-1/2 K

| ( , t)|,

( , t) =

( u(t)) . t

(32)

. 15, Ja , A(s, t) (31). , Ja , A(s, t) = a(s, t). (33)

A(s, t) a(s, t) s , | ( , t)|2 = a(s, t)e-iRes, ds = a( , t), Cr a( , t) (32).
9.5.




Ja a(s, t), s -- a( , t) l p -p -1/2 a( , t) = CrK ( , t) p p , p C.
p=1


80

. . , . .

a(s, t) . ( , t) -- 1 , . . . , l k , a( , t) l [ mp +p -p ] a( , t) = um1 ,...,ml (t) p ) . (p
m1 +...+ml =k p=1

, a(s, t) =
m1 +...+ml =k

[ u

m1 ,...,m

l

(t)

l

F (

mp + p

p



- p

p

] ),

p=1

F -- . [2], m + - p p = 0, p = 1, . . . , l , F (p p p p ) , , sp p p sp p . 1 = . . . = l = 0 a(s, t) ( ) -1/2 a(s, t) = CrK ; t (s). s
-m - -1 -1

ç 10. , k - Rn

Rn . K -- n - k - Rn ( ), K , Cn , (24). = (1 , . . . , l ) Rl t = (t1 , . . . , tk ) Rk K . , K f = Ja f (28), a(s, t) -- Rn , Rn . : 1) Ja 2) Ja , . f a f a s . (30),


. .

81

, , , a s , , . K . , , , , K . K 15. 17. Cr ( , ) K , a( , t) -- , , f = Ja f : f (x, y ) = A(s, t) = (, t)A(y - u(t)x - , t) d²() d²(t),

| ( , t)| e a( , t)

is,

d²( ).

7. Ja , a( , t) s a(s, t) |a( , t)|2 = Cr
-1

( , u(t))| ( , t)|.

, u(t) -- , , |a( , t)| = | ( , t)|1/2 . . , k = 1 , Cr ( , ) 1 a( , t) = n- 1 1/2+i , ui (t)i =
i=1

a(s; t) =

s1 (t) u1

-3/2-i

n- 1 i=2

(u (t)si - u (t)s1 ). 1 i


82

. . , . .

1. . . . . . 1960. 15, 2. 155í164. 2. . ., . ., . . , . 5. . , . 1962. è è 3. R a d o n J. Uber die Bestimmung von Funktionen durch ihre Integralwarte lngs è è gewisser Mannigfaltigkeiten. Ber. Verh. Sachs. Akad. 1917. B, 69. 262í281. ‡ 4. D e l s a r t e J. Hypergroups et operateur de permutation et de transformation. Colloque International du CNRS. 1956. 71. 274í290. 5. . . . , . 1973. 6. L e v i t a n B. M., L i t v i n o v G. L. Generalized displacement operators. In: Encyclopaedia of Mathematics, 4. Kluwer Acad. Publ., Dordrecht. 1989. 224í228. 7. J e w e t t R. I. Spaces with an abstract convolution of measures. Adv. Math. 1975. 18. 1í101. 8. . ., . . . , . 1992. 9. R o s s K. A. Signed hypergroups -- a survey. Contemporary Mathematics. 1995. 183. 319í329. 10. B l o o m W. B., H e y e r H. Harmonic analysis of probability measures on hypergroups. De Gruyter, Berlin e.a. 1995. 11. S c h w a r t z A. L. Three lectures on hypergroups. In: International conference è on harmonic analysis. Birkhauser, Basel. 1997. 93í129. 12. L i t v i n o v G. L. Hypergroups and hypergroup algebras. J. Soviet Math. 1987. 38, 4. 1734í1761. 13. . ., . . . , . 1959. 14. . ., . ., . . . , . 2000.


. . , . . , . . , . . , . . , . . , . . , . . , . .



-- . , -- . ë ¨, - .

ç 1.

: . . -- . , , , . (, , ;
( 10í01í00748-), ( -3224.2010.1), ë ¨ ( 2.1.1.3704) ë - ¨ ( 02.740.11.5213 14.740.11.0794).


84

. . , . . , .

. [1]), , , , -. , , , , . 1934 . [2] , . . , : , . , ë ?¨ [3], , . [4] . M -- , . W , M , .. , M . W d , M . W = (W, d) M = (M , ) . , W , , . , , . [5], . . [6]. , , , , , , . -- , . , , . , , , .. M . -




85

-- , ( ), . M -- G = (V , E ) -- . , G M , M V . , M -- G . M = (M , ) -- ( , ), G = (V , E ) -- , M , : E R+ -- , G = (G, ) . G (G ) , . V d , , G , . p q M (p, q ) d (p, q ) , G M , G -- . mf (M) , inf (G ) G M , , G , (G ) = mf (M) , -- . , , mpf (M, G) . -- mf (M) , mpf (M, G) () .
ç 2.

G = (V , E ) -- X = (X, d) -- . : V X X , G = (V , E ) G . G . : v w X d((v ), (w)) , d() -- . , V G = (V , E ) , , G . , , , . G . M X --


86

. . , . . , .

M (V ) , , M . , , . smt(M ) = inf {d() | -- , M } . , M , d() = smt(M ) , ; . [8] [9]. , , M , M . , . -- (). , , . , . , , . G = (V , E ) -- G : G X -- , X = (X, d) -- . [G, ] : V X G , = . mpn(G, ) =
[G,]

inf

d()

. [G, ] , d() = mpn(G, ) , G . 3. X = (X, d) -- , M -- X . smt(M ) = inf {mpn(G, ) | ( G) = M }. , .
smt Steiner Minimal Tree -- , , ( ) , , .




87

M = (M , ) -- G = (V , E ) -- , M . , G M . (M, G) : E R+ , (G, ) -- M . mpf (M, G) =
(M,G)

inf

(G)

G M . (M, G) , (G) = mpf (M, G) , (G, ) G M . 4. M = (M , ) -- . mf (M) = inf mpf (M, G).
G

, M -- -, G , -, G , ( mpf (M, G) ) .
ç 3. :

M = {p1 , p2 , p3 } , (pi , pj ) = ij . (G, ) p1 , p2 , p3 x , x , + - (xpi ) = j i ik j k . , 2 pi pj ij . d (pi , pj ) (pi , pj ) . 23 12 +2 +31 . , , .. . -, -- ë¨ ,


88

. . , . . , .

ë¨ ( , ). , p1 , p2 , p3 x . 1 , 2 , 3 , 1 + 2 + 3 . 1 + 2 12 , 2 + 3 23 , 3 + 1 31 ,

, , 21 + 22 + 23 12 + 23 + 31 . , 23 , , , mf (M , ) = 12 +2 +31 .
ç 4.

M = (M , ) -- , () G = (V , E ) . , (M, G) , : E R+ , G = (G, ) M , m (M, G) -- , , G -- M . 5. m (M, G) (M, G) -- RE E . : 1. M G G M . . . , M , : ( e) (p, q ) , -- G , p q , M , (e) 0 , e E , . S ( ) = (e) (M, G) ,
e E e




89

, . (M, G) , . , , (M, G) -- (e) , (, e (e) max (p, q ) ). (M, G) 0 . (M, G) {w | S ( ) S (0 )} RE , .. . S . , S (M, G) {w | S ( ) S (0 )} , (M, G) . 1. .
ç 5.
p,q M

, . , , , , , G . M = (M , ) -- . 1. G = (V , E ) , M (.. ), V \M 3 . 1. |E | = 2|M | - 3 . 2. M . , M , . : 1. M -- , G = (G, ) -- . G = (G , ) -- M , G (G ) (G) .


90

. . , . . , .

[10]. . . G , , , . , M . , , . , - (.. M ), . , - , . , G1 = (G1 , 1 ) , M , G1 M . , 3 , 2 , . G = (G , ) , M . , , (G ) (G) . , , , mf (M) = inf mpf (M, G) = inf mpf (M, G),
G G

. G1 , . . . , Gn -- , M . 5 Gi i , (Gi , i ) M . , mf (M) = inf mpf (M, G) = min wi (Gi ).
G i

i , , , (Gi , wi ) M . , 2. M -- . , , , .




91





1 2 , . , , .
ç 6. ,

G = (V , E ) -- M . G e , G1 G2 -- . Mi = M Gi . 3. , Mi . {M1 , M2 } M P G ( e) . S -- . S : S S. M G G , e E Mi PG (e) p Mi , (p) Mi . 6. G = (G, ) -- M -- M . d (p, (p)) 2 (G).
pM

, , -- G . . p G , p (p) . d (p, (p)) = (p ) . , e G p q M , p q e . PG (e) = {M1 , M2 } . , Mi pi , (pi ) Mj , j = i . p1 p2 p q . , , e p . . M = (M , ) -- , -- M .


92

. . , . . , .

M P (M, ) = (p, (p)),
pM

min P (M, ) , M , M P (M) . , P (M) -- , [11]. , p(M, ) M p(M) M : p(M, ) = P (M, )/2, p(M) = P (M)/2.

, . G = (G, ) -- M = (M , ) , M , G , . . G , O(G) O(G ) . , () M . 6 . 2. G = (G, ) -- M = (M , ) O(G) . (G) p(M, ) , , (G) , , mf (M) min max p(M, )
G O(G) O(G)

max p(M, )

p(M),

M , M . , (G) = max p(M, ),
O(G)

G -- mpf (M, G) = max p(M, ).
O(G)




93

ç 7. :

. . , M = (M , ) , M G = (G, ) , d M . G M . . , : pi , pj , pk , pl (pi , pj ) + (pk , pl ) , (pi , pk ) + (pj , pl ) , (pi , pl ) + (pj , pk ) , . 7 [12, 13]. , . , , , . 8 [14, 15]. , . . 3. . . G = (G, ) -- M = (M , ) . , G -- M . G = (G , ) -- M , -- M G . , 6, d (p, (p)) = 2 (G ).
pM

, p q M d (p, q ) (p, q ) = d (p, q ) , , , G -- M . d (p, (p)) d (p, (p)) = 2 (G ).
pM pM


94

. . , . . , .

, 6, d (p, (p)), 2 (G)
pM

(G) (G ) , , G -- M , . . M = (M , ) -- G = (G, ) -- M . , G -- M . G = (G, ) -- M . , G M , (G) = (G) . -- G . , 6, 2 (G) d (p, (p)) d (p, (p)) = 2 (G),
pM pM

, (p, (p)) = d (p, (p)) d (p, (p)) (G) = (G) , (p, (p)) = d (p, (p)) p M . , p q M G , -- , (p, q ) = d (p, q ) p q M , . . G M . . G = (G, ) -- M . , , G (M , d ) , , 3 -- . , . 9. G = (G, ) -- M = (M , ) . G (M , d ) . 3. . , , ( , ). , -




95

. . . . 2. M = (M , ) , x, y , z M (x, y ) max((x, z ), (y , z )).

(. [18]), . : 10. . , G M , (G, ) M . 4. M = (M , ) M . - . . [19]. 11. , .
ç 8.

, . M = {p1 , . . . , pn } M = (M , ) . ij = (pi , pj ) . n n - (v 1 , . . . , v n ) = max{|v 1 |, . . . , |v n |}, -- § n , .. (v , w) = w - v . M : M n M (pi ) = pi = (i1 , . . . , in ). ï 12. M .


96

. . , . . , .

M . G = (G, ) -- M = (M , ) , G = (V , E ) . G , , M : ({x, y }) = (x, y ), x, y M .

d V , ï . 13. w, v V d (v , w) d (v , w) . v , w M , ï d (v , w) = d (v , w) = (v , w) . ï G : V n G , G (v ) = (d (v , p1 ), . . . , d (v , pn )). ï ï G . 13 . 14. G = M (M ) . , . 13 . 15. G (V , d ) n , . 2. 15 [6]. (X, d) , G , , .. (v w) = d((v ), (w)) . 5. G = (G, ) -- (M , ) , = G -- . = . . (M , ) n , (G, ) (M , ) . 15, e E (e) (e) . , (G, ) , . . -- X , G , H = (H, ) -- . , H , H G = (G, ) .




97

5 , , [7] . 4. M = (M , ) -- , n , M : M n -- . G M , , , G M . , G M (M ) G M . 5. M = (M , ) -- , n , M : M n -- . M G , G n , M (M ) . , M (M ) M . , M = M (M ) n -- mf (M ) = smt(M ) . , : 6. n -- , K n -- . mf (K ) = smt(K ) .
ç 9.

, . , , . . , ( . [16]). , , .. .


98

. . , . . , .

-- . , ç 10. . (G, ) = = (V , E , ) , : E R -- . d : V ½ V R : d (u, v ) ( ) u v . d . M = (M , ) G , M , u, v M (u, v ) d (u, v ) . mpf - (M, G) G M inf (G ) M G . G , , G . mf - (M) inf mpf - (M, G) , inf G , M , G , .. M . , inf , . 7. M mf - (M) = mf (M) . 3. . M G , mpf - (M, G) < mpf (M, G) .
ç 10.

, , . [17].




99

10.1.



S -- n . k S : Znk S , 1) j Znk (j ) = (j + 1) , 2) s S k . G -- M . , e E , ç 6. e E -- G . G , G1 G2 . Mi = M Gi . PG (e) = {M1 , M2 } M . M G G , l , e E Mi PG (e) l p Znk , (p) Mi , (p + 1) Mi . l / , l l -. G T (G) . 4. G ç 6 1 - G . , M 1 -, . 4. , . i, j M ij i j . 5. M l , e E (j )(j +1) 2l e . 6. l - G , v V -- 3 , e1 , e2 , e3 -- . (j )(j +1) , 0 j < nl l (e1 , e2 ) , (e2 , e3 ) , (e1 , e3 ) .


100

. . , . . , .

, M = (M , ) -- , -- k . M
nk-1 1 ( (j ), (j + 1)). p(M, ) = 2k j =0

10.2.



2. 1. G = (G, w) -- M G , -- G . w(G) p(M, ).

. 8. M = (M , ) -- , G -- M . mpf - (M, G) = max p(M, ).
T (G)

. 9. M = (M , ) -- . mf (M) = min max p(M, ).
G T (G)

. , mf - (M) = min mpf - (M, G) = min max p(M, ).
G G T (G)

M , 7 mf - (M) = mf (M) .




101

10.3.



. , , , . M = (M , ) G M . 3. G , p(M, ) = max p(M, ) .
T (G)

4. G = (G, w) -- M . G , , , G M . m - G , (k)(k+1) , 0 k < mn , .. w((k)(k+1) ) = ( (k ), (k + 1)) . [10]. 7. -- G . : 1 . . 2 . M G . 3 . - M G . 6. G -- , G ( , ) . 7 , , G , G . , . 8. ab G . G , .


102

. . , . . , .

U {uij } ij . M = (M , ) n u U M u = (i, j )uij .
i, j

n U , ij + j k - ik 0. , . , , , n n - . [10] , , , (.. ) . [16] 7, , (, ). , , . 10. , , .
ç 11.

. , smt(M ) = inf {d() | -- , M }, mst(M ) = inf {d() | -- M }. M X M




103

, .. sr(M ) = smt(M )/ mst(M ). sr(M ) M X X sr(X ) (. [20]). -- (. [8] [1]). , , , , . 5. M , . X = (X, ) -- , M X -- . í M sgr(M ) = mf (M , )/ mst(M , ). inf sgr(M ) , X n , sgrn (X ) í n X ; , inf sgrn (X ) , n > 1 , í X sgr(X ) . M ssr(M ) = mf (M , )/ smt(M , ). inf ssr(M ) , X n > 1 , ssrn (X ) n X ; , inf ssrn (X ) , n > 1 , X ssr(X ) . í , . 16 [10]. X = (X, ) -- , . sgr3 (X ) = 3/4 . 17 [10]. í 1/2 . , í 1/2 .


104

. . , . . , .

18 [10]. ssr3 (Rn , ) = ssr3 (Rn ), n 2, -- , 3/2 . , . , . : ; ; . -- , . - . . . 19. X -- . sgr(X ) = = 1/2 , ssr3 (X ) = 3/4 , , ssr4 (X ) = 2/3 . 20. n X ssrn (X ) 2(n-1) . X -- n - 1 , . F -- X . ssr(F ) F inf M F ssr(M ) . - . . [21]. 21. F -- . ssr(F ) = 3/2 .
1. . ., . . . , , . 2003. 2. J a r n i k V., K o s s l e r M. O minimalnich grafeth obeahujicich n danijch bodu. Cas. Pest. Mat. a Fys. 1934. 63. 223í235. 3. ., . ? . , . 2001. 4. G r o m o v M. Filling Riemannian manifolds. J. Diff. Geom. 1983. 18. 1í147. 5. . ., . ., . . . , , . 2004.




105

6. . . . . ..-.., -. 2009. 7. I v a n o v A. O., T u z h i l i n A. A. Minimal networks: The Steiner problem and its generalizations. CRC Press N.W., Boca Raton, Florida. 1994. 8. . ., . . . . . 2002. 11. 27í48. 9. . . . . . 2010. 87, 4. 514í518. 10. . ., . . . . ( ). 11. ., . : . , . 1989. 12. . . . . 1965. 20. 90í92. 13. S i m o e s - P e r e i r a J. M. S. A note on the tree realizability of a distance matrix. J. Combinatorial Th. 1969. 6. 303í310. 14. . . . . . . . . 1962. 2, 2. 371í372. 15. H a k i m i S. L., Y a u S. S. Distane matrix of a graph and its realizability. Quart. Appl. Math. 1975. 12. 305í317. 16. . ., . ., . ., . . . . -, . . . ( ). 17. . . . . ( ). 18. D e z a M. M., D e z a E. Encyclopedia of Distances. Springer-Verlag, Berlin, Heidelberg. 2009. 19. . . . . -, . . . ( ). 20. G i l b e r t E. N., P o l l a k H. O. Steiner minimal trees. SIAM J. Appl. Math. 1968. 16. 1í29. 21. . . 4 . . -, . . . ( ).


. . , . .




(C2 , , H ) F = Im f , = = Re(dz dw) , H = Re f (z , w) , f (z , w) = z 2 + Pn (w) -- , n N . n 3 f -1 (a) . f -1 (a) n f . . , Pn (w) , .

ç 1.

(M 2n , , H ) f1 , . . . , fn (. 1). = (f1 , . . . , fn ) : M 2n Rn , -1 (1 , . . . , n ) , (1 , . . . , n ) Rn . , sgrad fi , [1], -1 (1 , . . . , n ) ,
( 10í01í00748-), ( -3224.2010.1), ë ¨ ( 2.1.1.3704) ë - ¨ ( 02.740.11.5213 14.740.11.0794).




107

- . sgrad fi . . . . . sgrad fi , -- (R4 = C2 (z , w), Re(dz dw), Re(f (z , w))) F = Im(f (z , w)) , f : C2 C -- . . [2]. . [3], . . [4] [5í7]. , [5í12]. f (z , w) = z 2 + Pn (w) , n N . . [13] (. 1) , , ( 1). , , ( 2). ( ; . 1 2), ( ; . 2 4), C2 ë ¨ ë ¨ (. 5). , (. 3, 4, 5). ( 3, 4). , f -1 (0) ë -¨ . 3 [6]. . . . . . . . .


108

. . , . .

ç 2.

1. (M 2n , , H ) , M 2n -- , -- M 2n , H : M 2n R -- , ( ). , n f1 , . . . , fn : M 2n R , , : 1) f1 , . . . , fn M 2n , .. df1 , . . . , dfn M 2n , f1 = H ; 2) i, j = 1, . . . , n fi fj , .. fi f {fi , fj } = kl xk xj = 0 x1 , . . . , x2n , l kl -- , kl . 2. f : M 2n R sgrad f , g : M 2n R {f , g } = sgrad g (f ) . x1 , . . . , x2n sgrad f f (sgrad f )i = ij xj . . 3. (M 2n , , H ) x(t) = sgrad H |x(t) , t I -- I R . , .. t R , . [1]. 4. x M 2n ( ), df1 (x), . . . , dfn (x) . : M 2n Rn , : x (f1 (x), . . . , fn (x)) , . (x) x M 2n . Rn . 1. n = 1 x M 2 .




109

5. , , M 2n f1 , . . . , fn . ( ) (M 2n , , H ) f1 , . . . , fn T1 ,...,n = {x M 2n |f1 (x) = 1 , . . . , fn (x) = n } . ( ; T1 ,...,n , (f1 , . . . , fn ) n , ë ¨ n/2 , n [14, ç 2, ]; , , f (z , w) , .) T1 ,...,n , (.. df1 , . . . , dfn ), . (Mi2n , i , Hi ) i = (fi,1 , . . . , fi,n ) , -1 (1 , . . . , n ) , i = 1, 2 , i [1, 1.29], h1 : M1 M2 h2 : Rn Rn , 1 = h2 2 h1 . 6. 1 2 C - (M 2n , C , f ) sgradC f , M 2n -- dimC M = n , C -- 2- M 2n , f : M 2n C -- M 2n .
2.1.


M 4 = C2 (z , w) . R4 (x1 , y1 , x2 , y2 ) R4 (x1 , y1 , x2 , y2 ) C2 (z , w) , (x1 , y1 , x2 , y2 ) (x1 + iy1 , x2 + iy2 ) = (z , w) . R4 = dx1 dx2 - dy1 dy2 (, = Re(dz dw) ), H = Re(f (z , w)) : R4 R , f (z , w) -- . , (R4 , , H ) = (C2 (z , w), Re(dz dw), Re(f (z , w))) F = Im(f (z , w)) . (1)


110

. . , . .

1 [5, 2.1]. f (z , w) C2 , (1) F = Im(f (z , w)) , sgrad F = -i sgrad H . 2 [5, 2.2]. sgrad H sgradC f f (z , w) C = dz dw C2 (z , w) , .. sgrad H = sgradC f (. 6). 2 (C2 , C , f ) 4 , , H ) = (C2 , Re( ), Re f ) . (R C C - . 1 2 . 7. C - f1 ,f2 , fi : Mi C , i = 1, 2 , ( ), ( ) h : M1 M2 , f2 h = f1 . ç 3 ç 4 f = f (z , w) ( ), .. ( ) . ç 5. , (. 5, 7). 8. () ds2 T = f -1 ( ) f 1 ds2 := = ( + ) 2 T , 1- T (sgradC f |T ) = 1 . , ds2 , sgradC f |T i sgradC f |T . () T :T ½T R , x, y T , (x, y ) -- , T x, y , ds2 .




111

2.2.

,

r > 0 , 0 C
2 D0 ,r := { C | | - 0 | < r},

D

2 0 ,r

:= { C | | - 0 |

r }.

f : C2 C -- , , f = {i }s=1 , i C, i = 1, . . . , s ( i f (z , w) = z 2 + Pn (w) n , , s n ). C \ f s = |f | , 1 (C \ f ) Fs s . 2. () C \ f -- f , i -- C , i f , 1 i s , -1 . i := ,i § si , § ,i , si , = Di , -- > 0 , i , j j = i , j = 1, . . . , s , ,i -- i , si , ,i . i := [i ] 1 (C \ f , ) i , i = 1, . . . , s , 1 (C \ f , ) Fs . = () , f (z , w) = z 2 + Pn (w) -- . i, -- 2 si , = Di , ,i (. ()). , i Ti, -- ë ¨ (, ) f |f -1 (D2 ) , . 4, 5, 13 5. ,i i T . , (. [15, . I, ç 3.2, 2]), i , ,i . 9. E:= T , f |E : E C\f , , (E, C \ f , f |E ) ( , 5 ). (f |E ) : H1 (T ) C \ f ,
C\
f i ,

2

C\f


112

. . , . .

H1 (T ) := H1 (T ; Z) . C \ f , u 1 (C \ f , ) [Mu ] 0 (Homeo(T )) , , u , Mu : H1 (T ) H1 (T ) , , u . ² : 1 (C \ f , ) Aut(H1 (T )) , u Mu , . ²(1 (C \ f , )) Aut(H1 (T )) M . 10. p C2 f (z 1 , z 2 ) , 2 f (z 1 ,z 2 ) , zi zj
p

. , . 3 ( í; . [15, . I, ç 2, 4; ç 3, 2 3, 4]). f -1 (i ) , . í, Mi Aut(H1 (T )) , i 1 (C \ f , ) (. 2 () 9), Mi (h) = h + i , hi , h H1 (T ) , i H1 (T ) -- (.. ) ,i , . 2 (), i , h -- i h H1 (T ) . 11. p C2 f (z , w) ( ) ( f ) f z (z , w ), w (z , w ) 3 S3 , (z , w ) : S ( f ) , S3 -- f z (z , w ), w (z , w ) C2 p [15, . I, ç 3, 1].
2.3.
: ,

12. f (z , w) = = z 2 + Pn (w) , Pn (w) -- n N w , n ( w ), n (C2 , dz dw, f (z , w)) .




113

3 [6, 3]. C -- f (z , w) = z 2 + Pn (w) . T = f -1 ( ) C2 [ n-1 ] 2 n 3+(-1) . 2 4 ( : Ak-1 ; . [6, 4]). f (z , w) = z 2 + Pn (w) p C2 k - 1 4 > 0 U C2 p , 4 f |U 4 g : V ,k C , g (z , w) = z 2 + wk + f (p) , V
4 ,k

= {(z , w) C2 | |z 2 + wk |

, |w|

(2)1/k }.

(2)

2.4.



1 (. [13, 1]). kj (pi )i=1 , 1 j s , j kj i (pj + 1)
i=1

j s kj i n pj =
j =1 i=1

{j }s=1 j [1, s] n - 1,

P = Pn (w) n , {j }s=1 j (pi ) j
kj i=1

P

-1

(j ) j [1, s] .

f (z , w) = z 2 + Pn (w) s(f ) , Tj = f -1 (j ) -- kj , 1 j s(f ) . kj Tj (pi )i=1 , j kj i ²j := pj , 1 j s(f ) . (. 1)
i=1

. 1. f (z , w) = z 2 + Pn (w) n N . : 1) ²1 + § § § + ²s(f ) = n - 1 ; 2) kj ²j , kj + ²j n j = 1, . . . , s(f ) .


114

. . , . .

: s N {0} , k1 , . . . , ks , ²1 , . . . , ²s N , 1) 2) , kj (pi )i=1 ²j kj j , 1 j s , 1 , . . . , s C n = ²1 + § § § + ²s + 1 , s , 1 , . . . , s , Tj kj , p1 , . . . , pj j , j j = 1, . . . , s . (²1 , . . . , ²m ) Nm , ²1 ... ²m . ²1 , {i - 1 x i, j - 1 y j } i = 1, . . . , ²j , j = 1, . . . , m R2 x, y -- . 2. n N (²1 , . . . , ²m ) , n - 1 [ n ] , 2 .. : m 1) ²i = n - 1 ; 2) ²i [ n ] , i = 1, . . . , m . 2 f (z , w) = = z 2 + Pn (w) n , , s(f ) = m , i - ²i , 1 i s(f ) . : f (z , w) = z 2 + Pn (w) n , , , , n - 1 [ n ] . 2
ç 3. ( )
i=1 k

(2), n N , > 0 V V
4 ,n 4 ,n

:= {(z , w) C2 | |z 2 + wn | := {(z , w) C | |z + w |
2 2 n

, |w|

(2)

1/n 1/n

}, }.

, |w| < (2)

(3)




115

1 ( [6, 1]). n N > 0 0 C 4 g = gn : V ,n C , g (z , w) = z 2 + wn + 0 , 4 q = qn : M,n C , 4 = ([0, ] ½ S1 ½ S1 ½ ([-1, 0 ] [0 , 1]))/ , M,n - + n + 1 : (r, ,
t++2 k n

, 0- ) 1,k (r, , t-- n (0, , , h) 2 (0, 0, , h),

2 k

, 0+ ) ,

0

k < n,

(4)

2 , r [0, ] , = mod 2 , = mod 2 , t [- , ] , 4 h [-1, 0- ] [0+ , 1] . q (r, , , h)=rei +0 g (V ,n ) =
4 = q (M,n ) = D0 , . 0+ := 0 [0+ , 1] , 0- := 0 [-1, 0- ] . 4. () ( 4 ). M,n :=([0, ]½S1 ½S1 ½([-1, 0- ] [0+ , 1]))/ 4 M,n := [0, ]½ 1 ½ S1 ½ ([-1, 0 ] [0 , 1]) , ½S - + (4) ë¨ . 4 M,n {(r, mod 2 )} ½ S1 ½ ([-1, 0- ] [0+ , 1]) , 1 , 1,k (4), 0 k < n , n- 2 2 , ë¨ , n ( a1 , . . . , an ), ë¨ , n ( an , . . . , a1 ), /n , -- , . ({(r, mod 2 )} ½ S1 ½ ([-1, 0- ] [0+ , 1]))/1 ({(r, mod 2 )} ½ S1 ½ {0- , 0+ })/1 n a1 , . . . , an , , K2,n gn . n = 2 . 2 (4) 2


116

. . , . .

({(0, mod 2 )} ½ S1 ½ ([-1, 0- ] [0+ , 1]))/1 , mod 2 S1 ( ). (4) , (ë¨ ). , ( n = 2 ) (ë¨) . () () 1 , 2 - T = gn 1 ( ) , D0 , \ {0 } , ( n1 ) - = (prw |T )-1 0,wi, , prw : (z , w) w , wi, -- wn = - 0 , 0 i < n , a,b C a, b C . 2 - () ( ). T := gn 1 ( ) , D0 , . 1, gn |V
4 ,n

i=0

\T

:V
0

4 ,n

\T

0

D

2 0 ,

\ {0 } C

(5)

. {(0, 0)} (. () ()) ) ( 3 Z,n := {(0, 0)} . D
2 0 ,

\{0 }

. gn |V
4 ,n

\Z

3 ,n

:V

4 ,n

\Z

3 ,n

D

2 0 ,

C.

(6)

1 (6) D
2 0,

½ S1 ½ ([-1, 0) (0, 1]) D
2

2 0,

.

D0, ([0, ]½S1 )/ , (0, mod 2 ) (0, 0 mod 2 ) mod 2 S1 . (6) , 4 [M0 ] 0 (Homeo(T0 + , T0 + \ V,n )) (5) , 2 0 1 (D0 , \ {0 }, 0 +) (. 4 9). Homeo(T0 + , T0 + \ V,n ) Homeo(T0 + )




117

T0 + , 4 T0 + \ V,n , 4 M0 Homeo(T0 + , T0 + \ V,n ) . 1 , M0 ( ) M0 : (, 0, , h) , 0, + 2n (1 - |h|), h , ( , h) S1 ½ ([-1, 0- ] [0+ , 1]) , 2 , C (R) -- , |(-,0] = 0 |[1,+) = 1 . , [M0 ] ë ¨ 0 + T0 + . , n = 2 0 + T0 + . 2 ( [6, 2]). n N > 0 0 C 4 g = gn : V ,n C , g (z , w) = z 2 + wn + 0 , 4 q = qn : M,n C , 4 = ([0, ] ½ S1 ½ S1 ½ [-1, 0])/ , M,n n + 1 : ( ( ) ) r, , +t+2k , 0 1,k r, , -t+2k + 2 , 0 , 0 k < n, n n (7) (0, , , h) 2 (0, 0, , h), 2 4 , r [0, ] , = mod 2 R/2 Z , = mod 4 R/4 Z , t [- , ] , h [-1, 0] . 4 2 4 q (r, , , h) = rei + 0 g (V ,n ) = q (M,n ) = D0 , . 4 4 [6, 1]. h1 : V ,n M,n n ) {( ( ) | )|- 1/n r, , 2 arg (w) , - (2()w/n -rr1/n , (z , w) = (0, 0), 1 h1 (z , w) := (0, 0, 0, 0), (z , w) = (0, 0),
4 h1 (z , w) M,n 2 4 , z 2 + wn = rei , r [0, ] ,


118

. . , . .

0 < 2 , := 0 r = 0 , l = l(r, , w) Z [0, n-1] (w) = r,,l (w) , [6, 1] (. 2 3 ), (w) , |w| = (2)1/n , z 2 + wn = rei r [0, ] Im ( z )n < 0 .
(w)

5. () ( ). 4 4, M,n := ([0, ] ½ S1 ½ S1 ½ [-1, 0])/ 4 M,n := [0, ] ½ S1 ½ S1 ½ [-1, 0] , (7) ë4 ¨ . M,n 1 ½ [-1, 0] , {(r, mod 2 )} ½ S 1 , 1,k (7), 0 k < n , n-1 , ë¨ 2 , 2n ( a1 , . . . , an , a-1 , . . . , a-1 n 1 , 2n mod 2 ), . ({(r, mod 2 )} ½ S1 ½ [-1, 0])/1 ({(r, mod 2 )} ½ S1 ½ {0})/1 n a1 , . . . , an , , K2,n gn . ( n = 1 .) 2 (7) ({(0, mod 2 )} ½ S1 ½ [-1, 0])/1 , mod 2 S1 ( ). (7) , (ë¨ ). , (ë¨) . () () 2 , 2 - T = gn 1 ( ) , D0 , \{0 } , 4 (). () ( ). ) 4, ( -1 ( ) , D 2 , Z 3 := T := gn {(0, 0)} ,n 0 ,
D

{(0, 0)} (. () ()), . 2, (5) , -

2 0 ,

\{0 }




119

(6) D
2 0,

½ S1 ½ [-1, 0) D

2 0,

. 4 (), 4 [M0 ] 0 (Homeo(T0 + , T0 + \ V,n )) (5) , 2 0 1 (D0 , \ {0 }, 0 + ) . 2 , M0 ( M0 : (, 0, , h) , 0, +
2 n

) (1 - |h|), h ,

( , h) S1 ½ [-1, 0] , 4 , 4 (). , [M0 ] ë ¨ 0 + T0 + . 3 ( ). > 0 0 C 4 g : V ,2 C , g (z , w) = z 2 + w2 + 0 , 4 4 q : M C , M = [0, ]½S1 ½S1 ½([-1, 0- ] [0+ , 1])/ , 4 M : (r, , , 0+ ) 1 (r, , - + , 0- ), (0, , , v ) 2 (0, 0, , v ), 2 , 0+ := 0 [0+ , 1] , R/2 Z , = mod 2 q (r, , , h) = rei + 0 , 0- := 0 [-1, 0- ] , 0 r , = mod 2 R/2 Z , h [-1, 0- ] [0+ , 1] . 4 2 4 g (V ,2 ) = q (M ) = D0 , .

ç 4. ( )

l :=

k j =1

lj ,

l1 - 1, . . . , lk - 1 -- p1 , . . . , pk T0 = f -1 (0 ) . k = 0 l := 0 .


120

. . , . .

2 Mg,b g 0 , b 0 . g , b . 2 Mg,h,b , 2 Mg,b h .

5 [6, 5]. T0 = f -1 (0 ) -- ( ) f (z , w) = z 2 + Pn (w) n 2 , k 0 p1 , . . . , pk T0 , l1 - 1, . . . , lk - 1 , l1 , . . . , lk 2 . l n , l < n + k 0 > 0 , 4 (0, 0 ] Uj, pj f
-1

(D

2 0 ,

), 1

j

k , :
4 ,l
j

() f |U

4 j,

glj : V

C ,

glj (z , w) = z 2 + wlj + 0 , . (2) (3), 4 qlj : M,lj C (. 1 2), j = 1, . . . , k ; k () f (f -1 (D2 ))\ U 4 0 , j =1 j,

f0 : D Ln,k
,,l1 ,...,l
k

:= T0 \

2 0 , k

½ Ln,k U
4 j,

,,l1 ,...,l

k



C , ( , x)



,

-- -

j =1

2 , dimC Ln,k,,l1 ,...,lk = 1 , Mg,h,b n > l lj , 2 2 M0,1,k M0,1,k n = l ( k > 0 ) lj , k k [ ] [ lj ] 3+(-1)lj 3+(-1)n g = n- 1 - , b= . 2 2 , h= 2 2 j =1

, , l := +V
4 j,

k

j =1

lj , > 0 , V , |w| = (2)

4 j,

:= V

4 ,l

j =1

j

,
4 ,l
j

:= {(z , w) C2 | |z 2 + wlj |

1/lj

}=V

4 ,l

j

\V

,

2 2 2 1 j k , . (3), Ln,k,,l1 ,...,lk Mn,k,l1 ,...,lk , Mn,k,l1 ,...,lk := Mg,h,b 2 ( n > l lj ), Mn,k,l1 ,...,lk := 2 2 := M0,1,k M0,1,k ( n = l lj ) 5. 2 Mn,k,l1 ,...,lk Ln,k,,l1 ,...,lk




121


n,k,l1 ,...,l
k

:

k j =1

(R/(3 - (-1)lj ) Z) ½ {j } ½ {(-1)lj , -1} L

n,k,,l1 ,...,l

k

, Ln,k,,l1 ,...,lk j, ( mod (3 - (-1)lj ) ) := := n,k,l1 ,...,lk ( mod (3-(-1)lj ) , j, ) Ln,,l1 ,...,lk , {(-1)lj , -1} , 1 j k. 2 ( ; [6, 2]). T0 = f -1 (0 ) -- ( ) f (z , w) = z 2 + Pn (w) n 2 , k 0 p1 , . . . , pk T0 , l1 - 1, . . . , lk - 1 , l1 , . . . , lk 2 . l n , l < n + k 0 > 0 , (0, 0 ] f f -1 (D2 ) fn,k
4 Mn,k ,,l1 ,...,l ,l1 ,...,l
k

:M V )

4 n,k,,l1 ,...,l

0 , k

C . (D
2 0 ,

k

=

( k
j =1

4 j,

)
n,k


,,l1 ,...,lk

½L
k

n,k,,l1 ,...,l

k

) := (x)) k,

:=

(( k
j =1

V

4 j,

(D

2 0 ,

½L

n,k,,l1 ,...,l

)/ ) (x

n,k,,l1 ,...,l

k

V D x V
+ 4 j, 2 0 ,

4 j,

,1

j

½L

n,k,,l1 ,...,l

k




n,k,,l1 ,...,l
k

:

k j =1

+V

4 j,

D

2 0 ,

½ L

n,k,,l1 ,...,l

k

,


n,k,,l1 ,...,l

(z , w) := (k := z 2 + wlj + 0 , n,k

,l1 ,...,lk 4 j,

( (arg w) mod 2 , j, sgn Im

z )) w
lj /2

lj (z , w) + V
n,k,,l1 ,...,lk

,
n,k,l1 ,...,l 4 j,

(z , w) := (z 2 + wlj + 0 ,

lj (z , w) + V

(2 arg( w) mod 4 , j, -1)) , w k


122

. . , . .

l 1 f j

j

(z , w) + V

4 j,

Im
k

z ( w)lj

< 0,

k , fn,k
k

,l1 ,...,l

(z , w ) V ( , x) D
4 j,

n,k,l1 ,...,l

|V

4 j,

(z , w) = z 2 + wlj + 0 , ( , x) = ,
n,k,,l1 ,...,lk

,

1

j

k,
k

fn,k

,l1 ,...,lk |D

2 0 ,

½L

2 0 ,

½L

n,k,,l1 ,...,l

.

4 V j, M,lj 1 2 ë¨ n,k,,l1 ,...,lk | + V 4
j,

4

(([0, ] ½ S1 )/) ½ S1 ½ {(-1)lj , -1} D (r, , , ) (re
i

2 0 ,

½ L

n,k,,l1 ,...,lk

,

+ 0 ,

n,k,l1 ,...,l

k

(- , j, )),

2 (3-(-1)lj ) , {(-1)lj , -1} , (0, mod 2 ) (0, 0 mod 2 ) , mod 2 S1 , fn,k,l1 ,...,lk |V 4 j, i (r, , , h) re + 0 , 1 j k . fn,k
,l1 ,...,l
k

4 (Mn,k

,,l1 ,...,l

k

)=D

2 0 ,

.

ç 5.

0 C -- f = f (z , w) = z 2 + Pn (w) . T0 = f -1 (0 ) k 0 p1 , . . . , pk T0 , l1 - 1, . . . , lk - 1 4 , l1 , . . . , lk 2 . 0 > 0 U,j C -- 2, 1 j k . (0; 0 ] 2 D0 , . 13. D f |U 4 , 1
,j

2 0 ,

, j, U ,j T -- j k (. 2,

4

5 (), 4 (,) 5 (,). T , k 4 j, -- U ,j . :=
j =1




123

T , k j, f |f -1 (D2 ) . T j, , 1 j k (. 4 () 5 ()). 4 ( í). 2 2. D0 , [M0 ] 0 (Homeo(T )) (. 2 () 9), 0 , , , T0 = f -1 (0 ) (. 4 () 5 ()), .. T (. 13). i (t) , t [0; 1] , C , = 0 f i , 0 i s - 1 , 0 arg 0 (0) < § § § < arg s-1 (0) < 2 , .. 0 , . . . , s-1 ë ¨ (. [15, . I, ç 3, 3]). i s-1 1 (C \ {i }i=0 , ) , -1 ² : 1 (C \ {i }s=0 , ) Aut(H1 (T )) , i Mu = ²(u) M (. 2 () 9). T n 2 Mg,h,0 -- g = [ n-1 ] h = 3+(-1) 2 2 , 3. H1 (T ) Zn-1 . 2 = . 5. f n ( (C2 , dz dw) ) ë ¨ 0 , . . . , s-1 2 T0 Mg,h,0 i , 0 i s - 1 , i -- i (. 2 () 13), 0 , . . . , s-1 T0 T0 , -- T0 .
0 ,


124

. . , . .

m := e2i/m , m N . a,b C -- a, b C . 3. f (z , w) = z 2 - wn + nw , n 2 . 1) f n - 1 (0, k -1 ) C2 , 0 k n - 2 , ; n k = f (0, k -1 ) = (n - 1)k -1 C , n n 0 k n - 2; 2) k = (prw |T0 )-1 (0,wk,0 ) T0 T0 = f -1 (0) k (t) = tk , 1 0 t 1 (. 2 ()), wk,0 = n n-1 k -1 , n 0 k n - 2; 3) [k ] H1 (T0 ) k , 0 k n - 2 , Zn-1 ; H1 (T0 ) = 4) [k ] H1 (T0 ) , , [k ], [l ] = 1 0 l < k n - 2 , [k ], [k ] = 0 0 k n - 2 ; 5) [l ] , 0 l n - 2 , Mk Mk [l ] = [l ] + sgn(k - l)[k ] , 0 k, l n - 2 . . 1) f (z , w) ( (0, wk ) , wk -- fz ,w) = -nwn-1 + n , w f (z , w) (0, k -1 ) , n k = 0, . . . , n - 2 . k = f (0, k -1 ) = n = (n - 1)k -1 , k = 0, . . . , n - 2 . n 2) prw |T0 : T0 C , (z , w) w , w , -wn + nw = 0 , wn-1,0 = 0 wk,0 = n n-1 k -1 , k = 0, . . . , n - 2 . n k (t) = tk = t(n-1)k -1 n -wn + nw = k (t) t [0, 1] . k = --1 w , F () = (n - 1)t , n F () := -n + n . F () = -n(n-1 - 1) , F = F () [0, 1] [1, +) . ( 1) 1 F (0) = F n n-1 = 0 , a := n-1 F |[0,1] : [0, 1] [0, 1] [ ] 1 b := 1 F [ [0, 1] -- , 1 ] : 1, n n-1
n- 1 1,n
n-1 1

t [0, 1] prw |Ttk : T

tk

C




125

wn-1,tk = k -1 a-1 (t) wk,tk = k -1 b-1 (t) , n n wn-1,tk ,wk,tk . k = (prw |T0 )-1 (0,wk,0 ) tk , t [0, 1] . 3), 4) ; 5) 2), 4) í (. 3). . 4. ( n = 2 ) f (z , w) = z 2 + wn - 1 , n 2 . 1) f 1 (0, 0) ; n - 1 ; 0 = -1 ; 2) n k = (prw |T0 )-1 (0,k ) T0 , n k = 0, . . . , n - 1 , T0 = f -1 (0) , (t) = -t , 0 t 1 (. 2 ()); 3) [k ] H1 (T0 ) k := := k k+1 , 0 k n - 2 , H1 (T0 ) Zn-1 T0 = f -1 (0) ; = 4) [k ] H1 (T0 ) , , [k ], [k+1 ] = 1 0 k n - 3 , [k ], [l ] = 0 k = l ‘ 1 , 0 k , l n - 2, n-1 := n-1 0 ; 5) [k ] , 0 k n - 2 , M0 M0 [k ] = -[k+1 ] , 0 k n - 3, M0 [n-2 ] = [0 ] + . . . + [n-2 ] ; 6) M n , M Z/nZ . = . . 1) df (z , w) = = 2z dz + nwn-1 dw 0 = f (0, 0) = -1 ; . 2) (. 4 () 5 ()); . 3), 4) , prw |T0 k n - 1 ; . 5) 2), 4), k , 0 n 4 (), 5 () 4; . 6) 3) 5). . 5. f (z , w) = z 2 + Tn (w) , n 3 , Tn (w) -- n , .. Tn (w) = cos(n arccos w) w [-1, 1] . ) ( 1) f n - 1 k k n - 1 ; ; 0, cos n , 1


126

. . , . .

, 0 = 1 1 =[ -1] , T1 = f -1 (1) n-1 2 ( ) 0, cos 2k , 1 2k [ ] n - 1 , n T-1 = f -1 (-1) n 2 ) ( 0, cos 2k+ , 1 2k + 1 n - 1 ; n 2) [k ] H1 (T0 ) k := := (prw |T0 )-1 (ak ,ak+1 ) T0 , 1 k n - 1 , k- H1 (T0 ) T0 = f -1 (0) , ak := cos 22n , 1 k n; [ n- 1 ] 3) 2l , 1 l , T0 2 0 , [n] 0 (t) = t , 0 t 1 ; 2l-1 , 1 l 2, T0 1 , 1 (t) = -t , 0 t 1 (. 2 ()); 4) [k ] H1 (T0 ) , 1 k n - 1 , , [k ], [k+1 ] = 1 1 k n - 2 , [k ], [l ] = 0 k = l ‘ 1 , 1 k, l n - 1 ; 5) [k ] , 1 k n - 1 , M0 , M1 M0 [2l ] = [2l ] M1 [2l ] = [2l ] + [2l-1 ] - [2l+1 ] 1 l [ n-1 ] , 2 M0 [2l-1 ] = [2l-1 ] + [2l-2 ] - [2l ] M1 [2l-1 ] = [2l-1 ] 1 l [ n ] , [0 ] = [n ] := 0 . 2 . 1) w (-1, 1) n arccos df (z , w) = 2z dz + n sin(1-w2 w) dw , ; n - 1 ; , ) ( . f 0, cos k = (-1)k , 1 k n - 1. n 2), 4) , T0 , k- prw |T0 : T0 C , ak = cos 22n , 1 k n . 3) , 1 k < n Pn |[ak+1 ,ak ] , , , (-1)k , 0. 5) 3), 4) 4, í. , 4 j [Mj ] , j (t) , t [0, 1]




127

(. 4 () n = 2 ). 3), 4) M0 , M1 . .
ç 6.

D

2 0,

:= { C | | | < } .

6 ( 2- dz dw ë ¨). f (z , w) = z 2 + P2n+1 (w) , P2n+1 (w) -- 2n + 1 , n N . > 0 2 2 h : D0, ½ (D0, \ {0}) C2 , f h( , u) = , h (dz dw) = u
2 0, 2n-2

d du
u0

2 ( , u) D0, ½ (D 2 D0,

\ {0}) , lim |h( , u)| =

4 2 , h M := f -1 (D0, ) C2 . , 2- 4 M 4 , M C2 2 ½ D2 2 h . D0, 0, C 4 4 2 M \ M D0, ½ {0} (ë ¨ p , 2 D0, ).

. [7], , . . 1 > 0 Rmax 2 P2n+1 (w) = D1 . 2 := min{1 , 1/(2 Rmax )} . 2 2 h1 : D0,2 ½ (D0,2 \ {0}) C2 h1 ( , u1 ) := ) - P2n+1 (u-2 ), u-2 = 1 1 ( = u-2n-1 u2(2n+1) ( - P2n+1 (u-2 )), u 1 1 (

-2 1

) ,


128

. . , . .

. h1 , u2(2n+1) ( - P2n+1 (u-2 )) 1 . h1 , h1 (1 , u1 ) = h1 (2 , u2 ) , 1 = 2 u1 = ‘u2 , u1 = -u2 1 - P2n+1 (u-2 ) = - 1 - P2n+1 (u-2 ) , 1 2 h1 (1 , u1 ) = h1 (2 , u2 ) . -u2n-2 d du1 1 . f h1 ( , u1 ) = , h (dz dw) = 1 2(2n+1) u1 ( - P2n+1 (u-2 )) 1
2 2 2 g : D0,2 ½ D0,2 D0,2 ½ C g ( , u1 ) := ( , v ( , u1 )) , v = v ( , u1 )

v ( , 0) = 0 v

2n-2 v u1

=-



u2 1
2(2n+1) u1

n-2

( -P

2n+1

(u-2 )) 1

. g -

, , . 0 < < 2 2 2 , D0, ½ D0, Im g . g := g -1 |D0, ½(D0, \(C½{0})) . ~ 2 2 h := h1 g . ~ 14. () ds2 T = f -1 ( ) , p , : (0; 1) T ds2 T (. 8), lim (t) = lim (t) = p (. 6). () : (0; 1) T : D
2 0, t0
+

t0-

½ (0; 1) C2 ,

( , t) (t),

2 ( , t) D0, ½ (0; 1) (. 6). 3 ( ). f (z , w) = = z 2 + P2n+1 (w) -- , .. P2n+1 (w) = (w - a1 ) . . . (w - a2n+1 ) , ai R , i = 1, . . . , 2n + 1 , a1 < a2 < . . . < a2n+1 , n N . C - (C2 , dz dw, f )




129

(. 5) > 0 , : 2 1) D0, T = f -1 ( ) n ; 2) - 2 U := f -1 (D0, ) T0 , .. T0 2 D0, ; 3) U 2n I1 , . . . , In , J1 , . . . , Jn : U C, 2n e
,1

,...,e

,1

,d

,1

...,d

,1

: (0; 1) T ,

2 D0, ,


: U := U \


D
2 0,

n i=1

(e,i (0; 1) d,i (0; 1)) C

, ) Ik , Jk : U C Ik = Ik (f ) Jk = Jk (f ) f , D,k := Ik (U ), D,k := Jk (U ) C C , U Ik = Ik (f (I )), Ik = Ik (f (J )), Jk = Jk (f (I )), Jk = Jk (f (J )),

f (Ik ) f (Jk ) -- , Ik (f ) Jk (f ) , k = 1, . . . , n ; ) e,i , d,j : (0; 1) T , ds2 (. 8), p 2 (. 14), D0, , i, j = 1, . . . , n ; 2 ) D0, ) ( n (e,i (0; 1) d,i (0; 1)) C W := (T U ) = T \
i=1


130

. . , . .

( 4n -, . . 1 n = 3 , ei := s2i+1 ) C , 4n - A1 ( ) . . . A4n ( ) , U C2 W := { } ½ (W \ {A1 ( ), . . . , A4n ( )})
D
2

C2 : U U

( U ) , (f |U , ) = = IdU

(tA3 ( ) + (1 - t)A2 ( )) = (tA4n ( ) + (1 - t)A1 ( )) = e,1 (t), (tA2k+1 ( ) + (1 - t)A2k ( )) = = (tA4n-2k+3 ( ) + (1 - t)A4n-2k+4 ( )) = e,k (t), 1 < k n, (tA2k ( ) + (1 - t)A2k-1 ( )) = = (tA4n-2k ( ) + (1 - t)A4n-2k+1 ( )) = d,k (t), 1 k < n, (tA2n ( )+(1-t)A2n-1 ( )) = (tA2n+1 ( )+(1-t)A2n+2 ( )) = d,n (t) t (0; 1) , 4n - W A1 ( ) + A3 ( ) = 0, A3 ( ) - A2 ( ) = A4n ( ) - A1 ( ) = e,1 , A2k+1 ( ) - A2k ( ) = A4n-2k+3 ( ) - A4n-2k+4 ( ) = e,k , 1 < k n, A2k ( ) - A2k-1 ( ) = A4n-2k ( ) - A4n-2k+1 ( ) = d,k , 1 k < n, A2n ( ) - A2n-1 ( ) = A2n+1 ( ) - A2n+2 ( ) = d,n , n k e,k :=2 (-1)i-k Ii ( ), d,k :=2 (-1)i-k Ji ( ), 1 k n;
i=k


i=1

) (dz dw)|U = df d ; ) ë ¨ Ik = Ik (f ) Jk = Jk (f )
a2
k+1

1 Ik ( ) =
a



( )



-P

2n+1

(y ) dy ,

2k

( )

Jk ( ) =

1
a

a ( ) 2k



- P2

n+1

(y ) dy ,

2 D0, ,

2k-1

( )




131

. 1. 4n - W0 n = 3

i -P -P
2n+1 2n+1



, )/2) > 0 ,

((a ((a

2k

+ a2

k+1

2k-1

+ a2k )/2) < 0 ,

ai ( ) -- P2n+1 (w) = , ai ; ) (f , ) U f = 0, = 1; 4) C2 C2 , (z , w) (-z , w) , f , (f , ) 2 : D0, { } ½ (A2k-1 ( )A2k ( )A2k+1 ( )A4n-2k+2 ( )) W, 1 k n, { } ½ (A2k+1 ( )A4n-2k ( )A4n-2k+1 ( )A4n-2k+2 ( )) W, 1 k < n, , ( , ) ( , 2c( ) - ) , c( ) -- . , 3.


132

. . , . .

7. f (z , w) = z 2 + PN (w) , PN (w) -- N 1 , , T0 = f -1 (0) . : () prw |T0 : T0 C , (z , w) w , sgradC f |T0 ‘(prw ) (sgradC f |T0 ) , , .. (z1 , w), (z2 , w) T0 , z1 = z2 , (prw ) (sgradC f (z1 , w)) = -(prw ) (sgradC f (z2 , w)) ; () PN , ‘(prw ) (sgradC f |T0 ) Cw , .. Sym : Cw Cw , w w . . () (z , w) T0 sgradC f (z , w) = (-PN (w), 2z ) , (prw ) (sgradC f (z , w)) = 2z . w , (z1 , w), (z2 , w) T0 w Cw z1 = -z2 . prw |T0 (prw ) (sgradC f (z1 , w)) = 2z1 = w = -2z2 = -(prw ) (sgradC f (z2 , w)) . w () sym : T0 T0 : (z , w) (z , w) . sym , z 2 + PN (w) = 0 , z 2 + PN (w) = z 2 + PN (w) = 0 , , PN (w) . (prw ) (sgradC f (z , w)) = ((prw ) sgradC f )(w), Sym prw = prw sym,
sym (sgradC f (z , w)) = (-PN (w), 2z ) = sgradC f (sym(z , w))

, Sym = = = (((prw ) sgradC f )(w)) = Sym ((prw ) (sgradC f (z , w))) = (prw ) (sym (sgradC f (z , w))) = (prw ) (sgradC f (sym(z , w))) = (prw ) (sgradC f (z , w)) = ((prw ) sgradC f )(w) = ((prw ) sgradC f )(Sym(w)).

6. f (z , w) = z 2 + P2n+1 (w) , P2n+1 (w) = (w - a1 ) . . . (w - a2n+1 ) , ai R , i = 1, . . . , 2n + 1 , a1 < a2 < . . . < a2n+1 , n N . sgradC f |T0 , , . :




133

. 2. sgradC f |

T0

prw | n = 3

T0

: T0 C

w

() sgradC f |T0 ( prw |T0 : T0 C , (z , w) w , . 7 ()) . 2 n = 3 ; 2n - 1 s1 , . . . , s2n-1 T0 ( n S1 , . . . , Sn C , s1 = (prw |T0 )-1 (S1 ) , s2k-2 s2k-1 = (prw |T0 )-1 (Sk ) , k = 2, . . . , n ), T0 n c1 , . . . , cn , S1 ½ (0, 1) ( C n C1 , . . . , Cn C , [a2k , a2k+1 ] Ck , ck = (prw |T0 )-1 (Ck ) , k = 1, . . . , n ); ck
ak 2
+1

Tk =
a
2k



dw -P
2n+1

(w)

,

k = 1 , . . . , n;

p0 T0 , ds2 0 ( C - n (-1)i-k Ti k = 2, . . . , n ) |s2k-2 | = |s2k-1 | = n i=k (-1)i-1 Ti ; |s1 | =
i=1


134

. . , . .

() T0 d1 , . . . , dn i sgradC f , p0 , dk ck prw |T0 (dk ) = Dk prw |T0 : T0 Cw Ck , k = 1, . . . , n - 1 Dk [a2k , a2k+1 ] Cw , Dn [a2n+1 , +) Cw (. . 3 n = 3 );

. 3. i sgrad f | n = 3

T0

Cw

() s2k-2 s2k-1 = = (prw |T0 )-1 (Sk ) . () dk (prw |T0 )-1 (Dk ) . () : ) } {( T++ := -P2n+1 (w), w | Im w 0 T0 0 ( , -P2n+1 ((a2 + a3 )/2) > 0 ) [ n-1 ] [n] I (d1 ), I (s4k ), I (d2k+1 ), 1 k , s4k-2 , d2k , 1 k , 2 2 I : C2 C2 -- , I (z , w) = = (-z , w) ; s2k-1 ek , k = 1, . . . , n ; e1 . . . en d1 . . . dn T0 , , 4n - . 1. . , u := sgradC f |T0 , p0 , N Ip0 . T0 \ Ip0 = ci , ci -- T0 \ Ip0 , i = 1, . . . , N . u|ci .
i=1




135

1. , i ci 1 i N , i T > 0 . , ci T . , u u = sgradC f |T0 , v ds2 := Sym(0 0 ) v = i sgradC f |T0 . 0 [u, v ] = 0 , > 0 , (-, ) gv i -- u T , gv -- v R . T - u|ci T0 ( ci ), i . , i ci ci . g ci i . u|ci g ci , > 0 t : [0, T ] ½ [-, ] T0 , (t, ) gv gu (g ) . g , u T (.. i ). ( gv (g ) , [-, ] ) j 0 j (, j = 0 j ) , gvj (g ) i . [u, v ] = 0 , t T : (t, ) gu gv (g ) , (T , j ) = gu gvj (g ) = gvj (g ) = (0, j ) . , (T , 0) = (0, 0) = g , .. g g T ( T /k k N ). T (.. k = 1 ) (T /k ) - (. ). g i , i ci . i = , ci , i = ci . 2. prw : T0 Cw , (z , w) w . a1 , . . . , a2n+1 , Cw . a1 < a2 < . . . < a2n+1 , P2n+1 |(a1 ,a2 )(a3 ,a4 )§§§(a2n+1 ,+) > 0 P2n+1 |(-,a1 )(a2 ,a3 )§§§(a2n ,a2n+1 ) < 0 . ( ) ‘(prw ) (sgrad f |T0 ) = = ‘2 -P2n+1 (w) / w Cw R C (-, a1 ) (a2 , a3 ) . . . (a2n , a2n+1 ) R (a1 , a2 ) (a3 , a4 ) . . . (a2n+1 , +) R .


136

. . , . .

3. p0 T0 sgradC f |T0 2n - 2 - , p0 2(2n - 1) , p0 , ( ) ‘(prw ) (sgrad f |T0 ) Cw 2n - 1 . ‘(prw ) (sgrad f |T0 ) Cw . ( S1 ), (-, a1 ] (. 2). n , [a2k , a2k+1 ] , k = 1, . . . , n . , ‘(prw ) (sgrad f |T0 ) n , 1, , k = 2, . . . , n ak, [a2k-1 , a2k ] R Cw , - [a2k-1 , a2k ] (k - 1) - ( [a2k-2 , a2k-1 ] ), - , Sk . Ck k - , k = 1, . . . , n . , S2 , . . . , Sn R Cw , , R Cw , , Cw . , R Cw , Sk , ak, , k = 2, . . . , n , Cw . , ‘(prw ) (sgrad f |T0 ) Cw . , Cw S1 , . . . , Sn C1 , . . . , Cn , . 4. k - ( Ck ) a2k+1 dw Tk = , k = 1, . . . , n , |Sk | a2
k

-P

2n+1

Sk |Sk | =

n

(w)

(-1)i

-k

Ti k = 1, . . . , n .

sgradC f |T0 . 2. , 2n - 1 si , 1 i 2n - 1 , T0 n ck , k = 1, . . . , n , ck , . -

i=k




137

. 4. ‘(prw ) (sgrad f |

T0

) C

w

p0 T0 ; s2k-2 s2k-1 |Sk | prw |T0 : T0 Cw Sk ‘(prw ) (sgrad f |T0 ) , k = 2, . . . , n , s1 |S1 | S1 ‘(prw ) (sgrad f |T0 ) . ck Tk k - ‘(prw ) (sgrad f |T0 ) ( Ck ), [a2k , a2k+1 ] Ck , k = 1, . . . , n . i sgradC f |T0 . , sgradC f |T0 . 2. 7. f (z , w) = z 2 + P2n+1 (w) , n N . ds2 -- 0 T0 = f -1 (0) (. 8). : (0; 1) T0 -- , p0 (. 14 ()). 2 > 0 , D0, T , : (0; 1) T ds2 (. 8), , p (. 14), 0 = . . 6 > 0 , 2 1 > 0 , D0, 2 U T p T


138

. . , . .

2 2 u : U D1 , u (p ) = 0 (u ) (sgradC f |

2 U

2 u -- D1 C . ((u )-

1

) = u2-2n u , ( ) ) ds2 |U 2 =


= |u|4n-4 |du|2 -- 2 , , D1 C(u) , -1 -1 2 (u , u ) = (u-1 , u-1 ) D1 , ds2 , 2 U . , 2 2 u = 0 D1 , D1 , 2 ( D1 (u-1 , u-1 ) ) r1 = 41 -3 > 0 n 2 T0 T0 \ U0 , 1-. 2 , 3 > 0 - 0 0 := {z0 C |
2 4n-
3

. 0 |T0 \U0 -- 2 : 0 T0

Re z0

1 - 2 , | Im z0 |

3 } C,

0 |[2 ,1-2 ] = |[2 ,1-2 ] , 2 0 0 U0 2 ) dz = ( ) . (.. 0 (2 ‘ i3 ), 0 (1 - 2 ‘ i3 ) U0 0 0 0 , 4 > 0 - : 0 T , C , | | < 4 , 0 , 2 U , dz0 = ( ) , ( , z0 ) (z0 ) -. 5 > 0 , C , | | < 5 , 0 C(z0 ) ( , ds2 ), 2 ( )-1 (u )-1 ( D1 ) , 0 2 C . U D1 u-1 (0) = p , (. 5). 3. 1) 2) 5: T0 . 3). 1. Ii , Ji , i = 1, . . . , n , ). ), ..




139

. 5.

Ik ( ), Jk ( ) . ë ¨ Ik . , 1 Ik ( ) = - ( a2k+1
a 2k+1 j =2k
2k

( )

-P2n ( ) -P2

+1

(w) dw - )
a
2k+1

( ) j aj

(-1)

n+1

(aj ( ))

1 = 2
a



( )


2k

dw -P2
n+1

(w)

.

( )

, > 0 , | | < Ik ( ) , Ik = Ik ( ) . , Ik (0) = 0 , , Ik |D0, 2
2 D0, D,k := Ik (D a2 ( ) k dw Jk ( ) = 21 . ) . a
2k-1

2 0,

) .

( )

-P

2n+1

(w )

2. 6 T0 ei , di , i = 1, . . . , n , . 7 ei , di , i = 1, . . . , n e,i , d,i : (0; 1) U , e,i (0; 1), d,i (0; 1) T . 6 T0 U = T0 \(e1 . . .en d1 . . .dn ) , T U , 1- |T U . := |T U C d = |T U , (0, a2 ( )) = 0 .


140

. . , . .

) e,i , d,i . ) , df d = df (z , w) dw = dz dw . 2z ) ) ). 3. ). 6 , T U T e,i , d,j , i, j = 1, . . . , n , , (T U ) C . 6 , 0 : T0 U C , W0 4n -. W -- 4n -, , e,i , d,j , i, j = 1, . . . , n := -1 W . 6 (), 4n - W 0 , 0 d1 , e 1 , d 2 , e 2 , d 3 , e 3 , . . . , dk , e k , d d
-1 n

,

1 d--1 n

,e

-1 n

,d

-1 n- 2

,e

-1 n-1

,...,d

-1 k

,e

k+1 , ek+1 , . -1 -1 -1 k+1 , dk-1 , ek

. . , dn , e n , ,...,d
-1 1

, e-1 , e 2

-1 1

( sgradC f |T0 i sgradC f |T0 ). , 2 D0, 4n - W , e,i , d,j , i, j = 1, . . . , n . 4n - W . d = |T U , 4n - 1- , , .. := .


, T . ,k := (prw |T )-1 ([a2k ( ), a2k+1 ( )]) T ,
,k

:= (prw |T )-1 ([a

2k-1

( ), a2k ( )]) T



, 0,k 0,k sgradC f i sgradC f (. 6), k = 1, . . . , n .
,k = 2 Ik ( ),



,k

= 2 Jk ( )




141

Ik ( ) Jk ( ) (. 1). 4 1- T {p } M (. 6) ,

e d

,n

,1

= ,n = 2 In ( ), e,k + e,k+1 = ,k = 2 Ik ( ), = ,1 = 2 J1 ( ), d,k + d,k+1 = ,k+1 = 2 Jk+1 ( )

1 e
,k

k < n .
= 2 (Ik ( ) - I k n-k In ( )), k+1 ( ) + . . . + (-1) k -1 Jk-1 ( ) + . . . + (-1) J1 ( )),

1 1

k k

n, n.

d,k = 2 (J ( ) -

), 3), . 4). , I : (z , w) (-z , w) f , C - dz dw , n T \ (dk, I (dk, ) ek, I (ek, )) , k=1

(0, ak ( )) , 1 k 2n + 1 . , , I . , -- . 3 .
ç 7.
2 D0, := { C | | | < } . 8 ( 2- dz dw ë ¨). f (z , w) = z 2 + P2n+2 (w) , P2n+2 (w) -- 2n + 2 , n N . > 0 2 2 hj : D0, ½ (D0, \ {0}) C2 , j {+, -} , ,

f hj ( , u) = ,
2 ( , u) D0, ½ (D

h (dz dw) = un j
2 0,

-1

d du

\ {0}) , lim |hj ( , u)| =
u 0


142

. . , . .

2 D0, , 4 := f -1 (D 2 ) C2 . h+ , h- M 0, 4 , 2 - M , 4 M C2 2 ½ D2 2 h , h . D0, +- 0, C 2 4 4 2 M \ M (D0, ½ {0}) (D0, ½ {0}) (ë ¨ 2 p,+ , p,- , D0, ). . , 6, [7]. 6, 2 2 h1,j : D0,2 ½ (D0,2 \ {0}) C2 , j {+, -} , ) ( h1,j ( , u1 ) := j - P2n+2 (u-1 ), u-1 = 1 1 ( ) = j u-2n-2 u2(2n+2) ( - P2n+2 (u-1 )), u-1 , 1 1 1

. 15. ds2 T = f -1 ( p,+ , p,- , : (0; 1) T i, j
t0-

() ) , ds2 T (. 8), {+, -} lim (t) = p,i ,
t0+

lim (t) = p

,j

(. 8).

() : (0; 1) T , p,+ , p,- , , 14 (). 4 ( ). f (z , w) = = z 2 + P2n+2 (w) -- , .. P2n+2 (w) = (w - a1 ) . . . (w - a2n+2 ) , ai R , i = 1, . . . , 2n + 2 , a1 < a2 < . . . < a2n+2 , n N . C - (C2 , dz dw, f ) (. 5) > 0 , :




143

2 1) D0, T = f -1 ( ) n ; 2) - 2 U := f -1 (D0, ) T0 , .. T0 2 D0, ; 3) U 2n+1 I1 , . . . , In , J1 , . . . , Jn+1 : U C,

2n + 1 e
,1

,...,e

,n

,d

,1

...,d

,n+1

: (0; 1) T ,

2 D0, ,


: U := U \

(
D
2 0,

d

,n+1



n

) (e,i (0; 1) d,i (0; 1)) C

i=1

, ) Ik , J : U C Ik = Ik (f ) J = J (f ) f , D,k := Ik (U ), D, := J (U ) C C , U Ik = Ik (f (Ii )), Ik = Ik (f (Jj )), J = J (f (Ii )), J = J (f (Jj )),

f (Ii ) f (Jj ) -- , Ii (f ) Jj (f ) , 1 i, k n 1 j, n + 1 ; ) e,i , d,j : (0; 1) T , ds2 (. 8), p,+ , p,- 2 (. 14), D0, , i = 1, . . . , n , j = 1, . . . , n + 1 ;


144

. . , . .

) D

2 0,


,n+1

( ( W := (T U ) = T \ d



n i=1

(e,i (0; 1) d,i (0; 1))

))

C

( 4n +2 -, . . 6 n = 3 , ei := s2i+1 ) C , 4n+2 - A1 ( ) . . . A4n+2 ( ) , U C2 W := { } ½ (W \ {A1 ( ), . . . , A4n+2 ( )}) C2
2 D

: U U ( U ) , (f |U , ) = = IdU (tA = (tA (tA = (tA = ( ) + (1 - t)A2k ( )) = (tA4n-2k+3 ( ) + (1 - t)A4n-2k+2 ( )) = e,k (t), 1 k n, 2 ( ) + (1 - t)A3 ( )) = (tA1 ( ) + (1 - t)A4n+2 ( )) = d ,1 (t), 2k ( ) + (1 - t)A2k+1 ( )) = (tA4n-2k+6 ( ) + (1 - t)A4n-2k+5 ( )) = d,k (t), 1 < k n, 2n+1 ( ) + (1 - t)A2n+2 ( )) = (tA2n+4 ( ) + (1 - t)A2n+3 ( )) = d,n+1 (t)
2k-1

t (0; 1) , (4n + 2) - W A1 ( ) + A3 ( ) = 0, A2k-1 ( ) - A2k ( ) = A4n-2k+3 ( ) - A4n-2k+2 ( ) = e,k , 1 k n, A2 ( ) - A3 ( ) = A1 ( ) - A4n+2 ( ) = d,1 , A2k ( ) - A2k+1 ( ) = A4n-2k+6 ( ) - A4n-2k+5 ( ) = d,k , 1 < k n, A2n+1 ( ) - A2n+2 ( ) = A2n+4 ( ) - A2n+3 ( ) = d,n+1 , n+1 k (-1)i-k-1 Ii ( ), 1 k n, e,k := 2 (-1)k-i Ii ( ) = 2
i=1 i=k+1 k-i-1 Ji

d

,k

:= 2

k -1 i=0

(-1)

( ) = 2

n+1 i=k

(-1)i

-k Ji

( ),

1

k

n + 1;




145

. 6. 4n + 2 - W0 n = 3
) (dz dw)|U = df d ; ) ë ¨ Ik = Ik (f ) J = J (f )

a ( ) 2k

Ik ( ) =
a 0
2k-1

-P2n dy

a +2

2+1

(y )



( )

dy ,

J ( ) =
a2 ( ) n+1 +

-P2n

+2

(y )

dy ,

( )

a1 ( ) -

J ( ) =

2



- P2n

+2

(y )

,

J

( ) =
a
2n+2

2
( )



dy - P2
n+2

(y )

,

1 k n + 1 , 1 n , J0 = J0 ( ) Jn+1 = Jn+1 ( ) -- , , -P2n+2 (y ) i -P2n+2 (y )
y (a
2k-1

;a

2k

) )

> 0 Ik ( ), < 0 J ( ),

1 0

k

n+1, n+1,

y (a2 ;a

2+1

ai ( ) -- P2n+2 (w) = , ai , 1 i 2n+2 , a0 := - , a2n+3 := + ; ) (f , ) U f = 0, = 1;


146

. . , . .

4) C2 C2 , (z , w) (-z , w) , f , (f , ) 2 : D0, { } ½ (A2k-1 ( )A2k ( )A2k+1 ( )A4n-2k+4 ( )) W, 1 k n, { }½(A2k+1 ( )A4n-2k+2 ( )A4n-2k+3 ( )A4n-2k+4 ( )) W, 1 k n, , ( , ) ( , 2c( ) - ) , c( ) -- . 3 , 6 7. 8. f (z , w) = z 2 + P2n+2 (w) , P2n+2 (w) = (w - a1 ) . . . (w - a2n+2 ) , ai R , i = 1, . . . , 2n + 2 , a1 < a2 < . . . < a2n+2 , n N . sgradC f |T0 , , . : () sgradC f |T0 ( prw |T0 : T0 C , (z , w) w , . 7 ()), . 7 n = 3 ; 2n s1 , . . . , s2n T0 ( n S1 , . . . , Sn C , s2k-1 s2k = (prw |T0 )-1 (Sk ) , k = 1, . . . , n ), T0 n + 1 c1 , . . . , cn+1 , S1 ½ (0, 1) ( C n + 1 C1 , . . . , Cn+1 C , [a2k-1 , a2k ] Ck , ck = (prw |T0 )-1 (Ck ) , k = 1, . . . , n + 1 ); ck a 2k dw , k = 1, . . . , n + 1; Tk = -P2n+2 (w)
a
2k-1

s2k+(j -1)/2 p0,(-1)k j T0 , 1 k n , j = ‘1 , ds2 ( 0 C - ) |s2k-1 | = |s2k | = k n+1 (-1)i-k-1 Ti k = 1, . . . , n; = (-1)k-i Ti =
i=1 i=k+1




147

. 7. sgradC f |

T0

prw | n = 3

T0

: T0 C

w

() T0 d1 , . . . , dn+1 i sgradC f , dk p0,(-1)k p0,(-1)k-1 T0 , dk ck prw |T0 (dk ) = Dk prw |T0 : T0 Cw Ck , 1 k n + 1 , k = 2, . . . , n Dk [a2k-1 , a2k ] Cw , D1 (-, a1 ] Cw Dn+1 [a2n+2 , +) Cw (. . 8 n = 3 );

. 8. i sgrad f | n = 3

T0

Cw


148

. . , . .

() s
2k-1

s

2k

= (prw |T0 )-1 (Sk )

. () dk (prw |T0 )-1 (Dk ) . () ( ) : {( ) } T++ := -P2n+2 (w), w | Im w 0 T0 0 ( , -P2n+2 ((a1 + a2 )/2) > 0 ) [n] [ n+1 ] e1 , I (e2k ), I (d2k ), 1 k , e2k-1 , d2k-1 , 1 k , 2 2 I : C2 C2 -- , I (z , w) = = (-z , w) , ek s2k-1 , 1 k n; e1 . . . en d1 . . . dn+1 T0 , , (4n + 2) - . 6.

. 9.

9. f (z , w) = z 2 + P2n+2 (w) , n N . ds2 -- 0 T0 = f -1 (0) (. 8). : (0; 1) T0 -- , p0,i p0,j , i, j {+, -} , (. 15 ()). 2 > 0 , D0, T , : (0; 1) T ds2 (. 8),




149

, p,i p,j (. 15), 0 = . . 5 9 7.
1. . ., . . . , , . ë ¨, , 1999. 2. F l a s c h k a H. A remark on integrable Hamiltonian systems. Physics Letters A. 1988. 131, 9. 505í508. 3. G a v r i l o v L. Abelian integrals related to Morse polynomials and perturbations of plane Hamiltonian vector fields. Annales de l'Institut Fourier. 1999. 49. 611í652. 4. B a t e s L., C u s h m a n R. Complete integrability beyond LiouvilleíArnol'd. Rep. Math. Phys. 2005. 56, 1. 77í91. 5. . . . . . 2010. 201, 10. 109í136. 6. . ., . . . . . 2011. 202, 3. 69í106. 7. . ., . . . .: . . . ë ¨ (, , 2010). - . -, , 2011. (. arXiv:1107.1911v1 [math.DG] 11 Jul 2011.) 8. . C., . . . . . 1978. 12, 2. 49í59. 9. . ., . . . . 1984. 39, 2. 3í56. 10. . . . . 1986. 287, 5. 1071í1075. 11. . . . . 1989. 44, 1. 145í173. 12. F o m e n k o A. T. Symplectic geometry. Gordon and Breach, 1995. ‡ ‡ 13. T h o m R. L'equivalence d'une fonction differentiable et d'un polynome. Topology. 1965. 3, 2. 297í307. 14. . . . . . 1978. 12, 1. 51í61. 15. . . . , , 2000.


. . , .



G - - G
G , - , .. . .

ç 1.

í [2] G -. í : ) (, , ) , , G , ; ) G . , G . , H < G M H , H . H K < G , M K = , H = g H g -1 , .. M H = M H , 08-01-00034-, 10-01-92601-- 2.1.1/5031.





G -

151

M

H

M

H



= . -1 M = M gH g
g G

, G - , G H . M M = g (M H ),
[[g ]G/N (H )]

N (H ) -- H , M H - N (H )/H . , í G ( N (H ) ) -- - . (., , [3]) G . [4, 7.2], , K -, . , - . [6í11].
ç 2.

G - M :


M H G -- , M . G


152

. . , .

M - G0 = G/H : G½M II
II II II I$ M u: uu uu uu uu

G0 ½ M



, G0 ½ M M , H . : G½ G0 ½ M
/ /M

1. [5, p. 210] , G - H . H , : / H ½ M


M



k : H U(Vk ) () H . H - : (k Vk ), (1)
k

H k , Vk Vk H k . 1. (1) G , .. k Vk G .




G -

153

. G . x M . g G , x gx : (x, g ) : x gx . g1 , g2 G (x, g1 g2 ) = (g2 x, g1 ) (x, g2 ), (x, g1 g2 ) : x - - g --
(x,h) (x,g2 )
2

x

- - - g ---
(x,g )

(g2 x,g1 )

1 g2

x

.

, g2 = h H G , g2 x = hx = x . , -- -- (x, g h) : x - - x - - gx . , g1 = h H G , g1 g x = hg x = g x . , (x, hg ) : x - - gx - - - gx . -- - - (x, h) x M , (x, h) = (h) : (k,x Vk ) (k,x Vk ),
k k (x,g ) (g x,h)

, H Vk k , (h) = (Id k (h)).
k

, : (x, g h) = (x, g ) (h) = (x, g hg
-1

g ) = (g hg

-1

) (x, g ). (2)

(x, g ) x x = (k,x Vk ) : (x, g ) =
k

(x, g )1,1 § § § . .. . . . (x, g )1,k § § § . . .

(x, g )k . . . (x, g )k . . .

,1

§§§



,k

. §§§ .. .

k = l (x, g )k,l = 0 , .. (x, g ) , (x, g ) = (x, g )k,k : (k,x Vk ) (k,gx Vk ),
k k k,x k ,g x

(x, g )k,k : ( .

Vk ) (k

Vk ),


154

. . , .

ç 3. V

G - = 0 V M 0 V M - G H G . F 0 . H , V V H . 2. G - = 0 V F V , G0 , F V , .. W = G0 ½ (F V ) W = G0 ½ (F V ) G0 G G½W G ½ G0 .. G ½ (G0 ½ (F V )) G½G





/W /G



²

0

/ G0 ½ (F V ) /G

² 0

0

² G - G0 , (g1 , [g ]) : [g ] ½ (F V ) [g1 g ] ½ (F V ), (g1 , [g ]) = Id (u(g1 g )u-1 (g )), (3) g1 G, [g ] G0 ,




G -

155

u : G H -- H - , u(g h) = u(g )h, u(1) = 1, g G, h H.

2. (3) G . . , ) (3) , .. (g2 g1 , [g ]) = (g2 , [g1 g ]) (g1 , [g ]) g G , g1 G , g2 G ) (3) g h [g ] : (g1 , [g ]) = (g1 , [g h]), .. Id (u(g1 g )u
-1

(g )) = Id (u(g1 g h)u

-1

(g h))

g G , g1 G h H . , (g2 g1 , [g ]) = Id (u(g2 g1 g )u
-1

(g )) =
-1

= Id (u(g2 g1 g )u(g1 g )u

(g1 g )u

-1

(g )) = (g1 g )u
-1

= Id (u(g2 g1 g )u(g1 g )) Id (u = (g2 , [g1 g ]) (g1 , [g ]),

-1

(g )) =

). u(g h) = u(g )h g G h H , u(g1 g h)u
-1

(g h) = u(g1 g )hh-1 u-1 (g ) = u(g1 g )u-1 (g ),

). , , u : G H , , . 3. u : G H u : G H W W , .. W G



/W



0

G

0


156

. . , .

. G0 ½ ( F V )



/ G0 ½ (F V )

G0

G

0

([g ]) : [g ] ½ (F V ) [g ] ½ (F V ), ([g ]) = Id (u (g )u-1 (g )). (4)

(4) , g [g ] , .. [g ] ½ (F V )
([g ]) (g1 ,[g ])

/ [g1 g ] ½ (F V ) / [g1 g ] ½ (F V )
([g1 g ])

[g ] ½ (F V )



(g1 ,[g ])

. M {O } , M = O , [g ]O = O , [g ] G0 .


4. {O } , O : O = [g ]U ,
[g ]G
0

U , .. [g ]U [g ]U = [g ] = [g ],

= U [g ]U = [g ] G0 .




G -

157

, [g ] = [g ] , U [g ]U = . , O O U ½ G0 , O O O O (U [g ]U ) ½ G0 . (., , [1]). 1. = 0 V U . , G - : O ½ (F V ) |O , (5) |O
O



O ½ (F V ) U ½ (G0 ½ (F V ))

g


/ |O O



Id½(g )

U ½ (G0 ½ (F V ))

/ U ½ (G0 ½ (F V ))

. . {O } , 4. (5), H - ,1 : U ½ (F V ) |U , |U
O
,
1

g


/ |[

g ]U

O

, Id½(g ,[1])

[g ]

(6)

U ½ (F V )

/ [g ]U ½ (F V )


158

. . , .

g [g ] , , , . ,[g] , .. [g ] :
,[g ]

=

,[hg ]

,

h H.

, ,1 H -, : |U
O
,
1

h


/ |[

g ]U

O

, Id½(h,[1])

1

(7)

U ½ (F V )

/ U ½ (F V )

(6) (7) : |U
O

,1

h


/ |[g O

,1

g ]U


/ |[g O

]U



,[g ]

U ½ (F V )

Id½(h,[1])

/ U ½ (F V )

Id½(g ,[1])

/ [g ]U ½ (F V )

: |U
O
,
1

gh


/ |[

g ]U

O

, Id½(g h,[1])

[g ]

U ½ (F V )

/ [g ]U ½ (F V )

(g , [1]) (h, [1]) = (g h, [1]),

,[g] . AutG (G0 ½ (F V )) G0 ½ (F V ) G - G0 F V G . 1. (U ½ G0 ) (U ½ G0 ) = (U [g


]U ) ½ G0 ,




G -

159

.. = (U [g ]U ) ½ (G0 ½(F V )) (U [g ]U ) ½ G


-1


/ (U [g ]U ) ½ (G0 ½(F V )) / (U g U ) ½ G

Id 0

0

G , .. (x) (g1 , [g ]) = (g1 , [g ]) (x), x U [g ]U , g1 G , [g ] G0 .


(x) AutG (G0 ½ (F V )).

AutG (G0 ½ (F V )) . AutG (G0 ½ (F V )) -- Aa , (Aa , a) (G0 ½ (F V ))

A
a

/ G0 ½ ( F V ) / G0

G0

a

, .. a AutG (G0 ) G0 , .. a[g ] = [g a], Aa = {Aa [g ]}[g
]G

[g ] G0 ,
0

a G0 .

,

Aa [g ] : [g ] ½ (F V ) [g a] ½ (F V ), G : [g ] ½ (F V )
(g1 ,[g ]) Aa [g ]

/ [g a] ½ (F V ) / [g1 g a] ½ (F V )
(g1 ,[g a])

[g1 g ] ½ (F V )



Aa

[g1 g ]

(g1 , [g a]) Aa [g ] = Aa [g1 g ] (g1 , [g ]),


160

. . , .

.. (Id (u(g1 g a)u
-1

(g a)))Aa [g ] = Aa [g1 g ](Id (u(g1 g )u

-1

(g ))), (8)

[g ] G0 , g1 G . 5. : 1
/ GL(F )


/ AutG (G0 ½ (F V ))

pr

/G

0

/1.

. pr : AutG (G0 ½ (F V )) G Aa : G0 ½ (F V ) G0 ½ (F V ) a : G0 G0 , .. a AutG (G0 ) G0 . , pr , GL(F ) . , .. A1 . (8) (Id (u(g1 g )u
-1 0

(g )))A1 [g ] = A1 [g1 g ](Id (u(g1 g )u

-1

(g ))).

(9)

g1 = h H , (Id (u(hg )u-1 (g )))A1 [g ] = A1 [g ](Id (u(hg )u-1 (g ))). , A1 [g ] = B 1 [g ] Id. , (9) g = 1 , (Id (u(g )))A1 [1] = A1 [g ](Id (u(g ))), .. (Id (u(g )))(B 1 [1] Id) = (B 1 [g ] Id)(Id (u(g ))), (B 1 [g ] Id) = (B 1 [1] Id).

, ker pr GL(F ) .




G -

161

, [a] = 1 , Aa [g ] (8): g = 1 , ( g1 g ) (Id (u(g a)u ..
-1

(a)))Aa [1] = Aa [g ](Id (u(g ))),
-1

Aa [g ] = (Id (u(g a)u

(a)))Aa [1](Id (u

-1

(g ))).

(10)

, Aa [1] : [1] ½ (F V ) [a] ½ (F V ) g = 1 . , Aa [1] . H : [1] ½ (F V )
(h,[1]) Aa [1]

/ [a] ½ (F V ) / [a] ½ (F V )
(h,[a])

[1] ½ (F V ) , .. ..



Aa [1]

Aa [1] (h, [1]) = (h, [a]) Aa [1], Aa [1] (Id (h)) = (Id (g
-1

(a)hg (a))) Aa [1], (h)) Aa [1].

Aa [1] (Id (h)) = (Id

g (a)

, Aa [1] , , g (a) . (2) , g (a) . , g , () C (g ) , g (h) = (g -1 hg ) = C (g )(h)C -1 (g ). C (g ) ²g S1 C1 .


162

. . , .

, Aa [1] (Id (h)) = (Id C (g (a)) (h) C (Id C
-1 -1

(g (a))) Aa [1],

(g (a))) Aa [1] (Id (h)) =
-1

= (Id (h)) (Id C , (Id C ..
-1

(g (a))) Aa [1].

(g (a))) Aa [1] = B a [1] Id,

Aa [1] = B a [1] C (g (a)). (10),

Aa [g ] = (Id (u(g a)u ..

-1

(a)))(B a [1] C (g (a)))(Id (u-1 (g ))),

Aa [g ] = B a [1] ((u(g a)u-1 (a)) C (g (a)) (u-1 (g ))).

(11)

, B a [1] , Aa [g ] , (10). (8), .. (Id (u(g1 g a)u
-1

(g a)))Aa [g ] = Aa [g1 g ](Id (u(g1 g )u

-1

(g )))

(11): (Id (u(g1 g a)u
a -1

(g a)))
-1

(B a [1] ((u(g a)u = (B [1] ((u(g1 g a)u
-1

(a)) C (g (a)) (u
-1

-1

(g )))) =
-1

(a)) C (g (a)) (u



(g1 g )))) (g ))),

(Id (u(g1 g )u .. B a [1] ((u(g1 g a)u
a -1

(g a)) (u(g a)u-1 (a)) C (g (a)) (u (a))C (g (a))(u
-1

-1

(g ))) = (g ))).

= B [1]((u(g1 g a)u

-1

(g1 g ))(u(g1 g )u

-1




G -

163

B a [1] , a , g g1 : (u(g1 g a)u-1 (g a)) (u(g a)u-1 (a)) C (g (a)) (u-1 (g )) = = (u(g1 g a)u-1 (a)) C (g (a)) (u-1 (g1 g )) (u(g1 g )u (u(g1 g a)u
-1 -1

(g )),

(a)) C (g (a)) (u
-1

-1

(g )) =

= (u(g1 g a)u

(a)) C (g (a)) (u-1 (g )).

, , [a] G0 (Aa , a) AutG (G0 ½ (F V )) . , pr AutG (G0 ½ (F V )) - G0 . G - G0 ½ (F V ) () X , G , X B AutG (G0 ½ (F V )) . VectG (M , ) G - = 0 V M - G H < G , H 0 ; V V H . , g (h) = (g -1 hg ) g G . , (2), . VectG (M , ) M , V . G GL(F ) AutG (G0 ½ (F V )) 5 . 2. VectG (M , ) [M , U (F )].


164

. . , .

Bundle(X, L) L - X . 2. VectG (M , ) Bundle(M /G0 , AutG (G0 ½ (F V ))). (12)

. 1, VectG (M , ) : (U [g ]U ) AutG (G0 ½ (F V )), , [g ] = [g ] , U [g ]U = , [g ] = 1 , U [g ]U = U [g ]U = . , U U M M /G0 . - M /G0 , , G - G0 ½ (F V ) -. , G - h : O ½ (F V ) O ½ (F V ) h : U ½ (G0 ½ (F V )) U ½ (G0 ½ (F V )) O U ½ G0 , U M /G0 . , h : U AutG (G0 ½ (F V )), (14) (13)

U M /G0 . , (12) . , (14), U M /G0 , , , , U - M M /G0 U ½ G0 , , , (13). , (12) . (12), , , VectG (M , ) M M /G0 , , [M /G0 , B G0 ] .




G -

165

3. X , VectG (M , ). Bundle(X, AutG (G0 ½ (F V )))
M Bundle(X,G0 )

M Bundle(X,G0 )

. 2 VectG (M , ) Bundle(X, AutG (G0 ½ (F V ))). (15) (15). : (U U ) AutG (G0 ½ (F V ))

Bundle(X, AutG (G0 ½ (F V ))). 5 pr : AutG (G0 ½ (F V )) G0 . pr G0 , , G0 M M /G0 = X . , ( ) [g ]U , M (U ½ G0 )
[g ]G
0

[1]U [g


]U U U ,

[g ] = pr . , (U ½ (G0 ½ (F V ))),


U ½ (G0 ½ (F V )) U ½ (G0 ½ (F V )) (x, g , f v ) = (x, ([g ], f v )) = (x, [g g ], A [g ](f v )) , x U U . , 5 . U ½ G 0 [g ]U ,
[g ]G
0


166

. . , .

([g ]x, f v ) = ([g g , (U ½ G0 ) ½ (F V ) U ½ G
0

]x, A



[g ](f v )).

M . , G -, G G0 . , VectG (M , ) . , , G0 -, M . , (15) . 3. X , [X, B (AutG (G0 ½ (F V )))] [M , U (F )].
M [X,B G0 ]

. . , VectG (M , ) = M Bundle(X, B G0 ) . , M ½ F V VectG (M , ).
ç 4. , H < G

G - M
p

M H < G -- . , M , -1 M= M gH g (16)
[g ]G/N (H )



H . M H




G -

167

H M , N (H ) -- H G . -1 g M H = M gH g , lH l-1 = g H g -1 , g -1 l N (H ) . F G , .. F = {K < G |
K

= }.

, G . G ½ F F , (g , K ) g K g
-1

,

. 3. , H < G G - , g H g -1 F . , H < G -- . 6. H = g H g
-1

, M

H

M

gH g

-1

= .
H
-1

. x M M gH g , x , H g H g -1 , lH -1 l , l G . 7. (16), G - - . = [g] ,
[g ]G/N (H )

[g] = p-1 (M gH g ) -- - N (g H g -1 ) g G g : [1] g [1] = [g]


-1



K

= ,

g K g -1

= g

K

= .


168

. . , .

, .. / [1] N (H ) ½ [1]
sg ½g

N (g H g , sg : N (H ) N (g H g
-1

-1



g

) ½ [g

]

/ [g



(17)

]

) = g N (H )g

-1

,

(g , n) g ng

-1

.

. 6 , -1 M= M gH g
[g ]G/N (H )

, , =



[g ]G/N (H )

[g] .

G , g § [1] = [g] g G . [g] M , [g]
p



M

gH g

-1

[g] N (g H g N (g H g N (g H N (g H g -1 ) - ,
-1

-1

):

) ½ [g] [g] ,

g -1 ) = g N (H )g -1 , .. [g] g G . sg : N (H ) N (g H g -1 ) , / [1] / [g

N (H ) ½


[1]

N (g H g

-1

) ½ [g

]

]




G -

169

.. g ng -1 § g x = g § nx . , [1] [g] . (17), , n N (H ) , g G . N (H ) M H N (H )/H : N (H ) ½ [1]
/ [1] /M

N (H )/H ½ M

H

H

, H , N (H )/H ½M M , H , .. N (H ) - H . 4. , G - ( ) H . X () : H GL(F ) N (H ) . X (g ) g : g H g
-1 s
g -1

/H



/ GL(F ) ,

sg (n) = g ng -1 . N (g H g -1 ) X (g ) g H g -1 - ug : g H g ([g ], g1 ) = GX () :=
k -1 sg
-1

/H

u

/ N (H )

s

g

/ N (g H g

-1

)

(Id k (u(g1 g )u

-1

(g ))) = (u(g1 g )u-1 (g )).



[g ]G/N (H )

X (g ),

.. lH l-1 = g H g -1 , X (g ) X (l ) . .


170

. . , .

8. GX () G - ( ) H , GX () X () . , N (H )(X ()) = X () (GX ())g
Hg
-1

= N (g H g

-1

)/g H g

-1

.

. G ½ GX () GX () : g G g : X () X (g ) sg ½ Id : N (H )0 ½ F N (g H g
-1 -1

)0 ½ F,

N (H )0 = N (H )/H , , lH l = g H g -1 , l : X () X (l ) , X (g )
s-1 ½Id g

/ X ()
l
-1

g

X (l ) , ..

l

-1

/ X ()

l = (sg ½ Id) (g g
-1

-1

l),

l : X () X () = X (

g

-1

l

)
-1

g 4. g : AutN
(H )

l N (H ) . (18)

(X ()) - AutN



(g H g

-1

)

(X (g )),

[g ] G/N (H ) . . (17) = = GX () . AutN
(H )

(X ()) - Aut A g Ag



N (g H g

-1

)

(X (g ))

-1

,




G -

171

, l [g ] G/N (H ) , l-1 g N (H ) A AutN (H ) (X ()) . , g Ag
-1

= g (g

-1

l)(l

-1

g )Ag

-1

= g (g

-1

l)A(l

-1

g )g

-1

= l Al

-1

.

5. GX () , H < G . 9. AutG (GX ()) AutN
(H )

(X ()).

. , AutG (X ) -- Aa , (Aa , a) X

A
a

/X / G/H

G/H

a

a AutG (G/H ) a AutG (G/H ) N (H )/H, a[g ] = [g a], [g ] N (H )/H.

Aa = (Aa [g ])[g]N (H )/H AutN (H ) (X ()) . (Aa [g ])[g]G/H Aa [1] , 5 ( (10)). VectG (M , ) - G . 4. VectG (M , ) VectN (H ) (M H , ) . . 7 , N (H ) N (g H g -1 ) -1 M gH g /N (g H g -1 )0 B AutN (gH g-1 ) (X (g )) M H /N (H )0 B Aut
N (H )

(X ()),


172

. . , .

M H /N (H )0
g ï

/ B Aut

N (H ) g ï

(X ())

M

gH g

-1



/N (g H g
-1

-1

)

0

/ B Aut N


(g H g
-1

)

(X (g ))

g : H gH g -- , (18) g [g ] G/N (H ) .
1. L u k e G., M i s h c h e n k o A. S. Vector bundles and their applications. Kluwer Academic Publishers, Boston, 1998. 2. C o n n e r P., F l o y d E. Differentiable periodic maps. Springer-Verlag, Berlin, 1964. 3. A t i y a h M. F. K-theory. Benjamin, New York, 1967. 4. A t i y a h M. F., S e g a l G. B. Equivariant K-theory. Lecture notes, Oxford, 1965. ‡ 5. L e v i n e M., S e r p e C. On a spectral sequence for equivariant K-theory. K-Theory. 2008. 38, 2. 177í222. (arXiv:math/0511394v3 [math.KT] 19 Nov 2005). 6. M i s h c h e n k o A. S., M o r a l e s M e l e n d e z Q. Description of the vector G -bundles over G -spaces with quasi-free proper action of discrete group G . arXiv:0901.3308v1 [math.AT] 21 Jan 2009. 7. M o r a l e s M e l e n d e z Q. Description of G -bundles over G -spaces with quasi-free proper action of discrete group. II. arXiv:0912.5047v1 [math.KT] 27 Dec 2009. 8. . ., . G - G - - G . . 716-2009 24.11.2009. 9. . G - G - - . II. . 717-2009 24.11.2009. 10. . . . 718-2009 24.11.2009. 11. . . . . -, . ., . 2010. 2. 57í59.


. .



X (t) = K (t)X (t)
, . . . , í X = X K , K (t) so(3), su(2), e(2), sl(2, R) sl(2, R)+ . .

1. X =K X
ç 1.

, . . . í [1] [2]. , , SO(3) SU(2) . . . 2001 . . . , . , . . . . . , ( ) X = K X , K .
( 10-01-00748-), ( -3224.2010.1), ë ¨ ( 2.1.1.3704) ë - ¨ ( 14.740.11.0794).


174

. .

X = K X , K , , . , . . . , X (t) = K (t)X (t) (1)

, K (t) : § so(3) SO(3) , § su(2) SU(2) , § e(2) E(2) ( 2-), § sl(2, R) SL(2, R) ,
§ sl(2, R)+ SL(2, R)
+

( -

2-).

, , . -- ( , ) (1) , K (t) G . , , íí. , . , G A(X, Y ) G , eX eY = e
A(X,Y )

,

X Y G , e exp : G g G g . , eX = exp X . X Y , A(X, Y ) = X + Y , .. : eX eY = eX +Y . , X Y , A(X, Y ) . (1905) í (1906) í (1908) A(X, Y ) (0, 0) . . , B (X, Y ) ,




X (t) = K (t)X (t)

175

( ) G b(X, Y ) G , eX eY = e
b(X,Y )

.

b(X, Y ) = B (X, Y ) , X Y , G . B (X, Y ) b(X, Y ) , X Y . b: G ½ G G exp(X ) exp(Y ) . , 1 B (X, Y ) = X + Y + [X, Y ] + . . . , 2 . , X Y . , b(X, Y ) (0, 0) . , X Y , . so(3) . , , B (X, Y ) X, Y so(3) . so(3) , X Y = B (X, Y ) . so(3) (.. ) , (. ) so(3) . , so(3) , : SO(3) . SO(3) ( , ). , SO(3) RP 3 , 3- S 3 , . ( ) SO(3) , . : eX eY = eb(X,Y ) X1 , . . . , XN ? , X1 , . . . , XN


176

. .

G , . bN (K1 , . . . , KN ) G , e
K1 K2

e

...e

KN

=e

bN (K1 ,...,KN )

.

N (s, K1 , . . . , KN ) = bN (sK1 , . . . , sKN ), s -- , : 1 dn bN (sK1 , . . . , sKN ) . n! dsn s=0 K1 , . . . , KN nn bN (sK1 , . . . , sKN ) = s cN . cn (K1 , . . . , KN ) = N
n=1 n=1 n=1

s = 1 bN (K1 , . . . , KN ) = cn . N

cn (K1 , . . . , KN ), N

K1 , . . . , KN -- ( ) G . cn N () n bN . , N ( N = 2 ) G , K1 , . . . , KN , .
ç 2. X (t) = K (t)X (t) .

X (t) = K (t)X (t), a t b (), b > 0 . , - K (t) X (t) G .




X (t) = K (t)X (t)

177

G -- (m ½ m) -. -- GL(m, R) . (1) X (t) = e
H (t)

X (a),

H (t) G - [a, b] . , X (b) = eH (b) X (a) . , K (t) -- G , H (t) , X (t) . (1). K (t) ( ), .. K1 K (t) t . (1) X (t) = e
(t-a)K1

X (a),

, , H (t) = (t - a)K1 G . , K (t) ( G ), .. [K (t1 ), K (t2 )] = 0 t1 t2 , [§,§] . (1)
t

K (u)du

X (t) = e

a

X (a), t
a

, , H (t) =

K (u)du . , H (t) -

G , ( ). (K (u); 0 u t) H (t).

, H = H (K ) . , (1) íí, (product integral). , K (t) - ().


178

. .

K (t) - , Ki (ti-1 , ti ) , i = 0, . . . , N ; = ti - ti-1 = (b - a)/(N - 1) . , X (t) (1) . [t0 , t1 ] , K () K1 . dX (t) = K (t)X (t) = K1 X (t) dt X (t) = e , , X (t1 ) = e
K1 (t-a)K1

X (a),

a = t0

t

t1 ,

X (a),

= t1 - a.

, [t1 , t2 ] , K K2 . X (t) = e
K2

X (t1 ),
K K1

t

1

t

t2 .

, X (t2 ) = e
2

e

X (a),

. , X (b) = X (tN ) = e
KN KN

e

-1

...e

K2 K1

e

X (a).

, g = exp G , b K (t) e
t= a

K (t) . (1) (t ) K (u) X (t) = e X (a).
u=a




X (t) = K (t)X (t)

179

, X (b) =

(

b t= a

e

K (t)

) X (a).

, , g . ., , : J. D. Dollard, C. N. Friedman, ëProduct integration¨ (AddisoníWesley, 1979). (1) X (t) = e
H (t)

X (a),

H (t) -- . , . , Feynman [4] (1), , , . Magnus , , H (t) . H (t) Bialynicki-Birula, Mielnik, Plebanski [5]. 1976 . [6] H (t) , . Chen [7] H (t) . íí. X (t) = e X (t) = ,
t u=a H (t)

X (a) ) X (a),

(

t

e

K (u)

u=a

e

K (u)

=e

H ( t)

,
-1

, , e

H (b)

= lim e
N

KN eKN

...e

K2 eK1

.


180

. .

eKN eKN -1 . . . eK2 eK1 = ebN (K1 ,...,KN ) , , H (b) = lim bN (K1 , . . . , KN ) , N . , () íí. , H (b) Hn (b) , Hn H (b) = n adK , adK (X ) = [K, X ] -- g . ( ) [§, §] . , , Hn n , : Hn (n - 1) . , Hn n . : Hn (b) , H (b) ? , K (t) { K1 = const, a t 1 (a + b), 2 K (t) = 1 K2 = const, 2 (a + b) < t b. íí. Hn H1 , . . . , Hn-1 . [1] K1 , K2 , . . . , KN , , .. , ( ).
ç 3. . ()
n=1

, . k (z ) = z 1-e
-z

-

z 2




X (t) = K (t)X (t)

181

z = 0 : k2p z 2p . k (z ) = 1 +
p=1

2p = k2p (2p)! . , G | § | . 1 ( ; . . , . . [1]). G -- K (t) , a t b , -- - -, g . , b |K (u)| du < ,
a

( . [1]). Hn (b) , , H (b) = H1 (b) + H2 (b) + . . . : b H1 (b) = K (u) du, n (n + 1)H
n+1 n ( 1 [Hr , Tn-r ] + = Tn + 2 r=1 [Hm1 , [H + k2 p p1 2p r mi >0 m1 +...+m2p =r

1

a

m2

, . . . , [H

m2p

,T

n- r

) ] . . . ]] ,

T0 = H1 k b [ Tk (b) =
u1 =a

1

u 1 [ K (u1 ), K (u2 ), . . . u2 =a uk-2 [ uk
-1

] K (uk
+1

...,
uk
-1

K (uk ),
=a uk =a

)duk

+1

] duk . . . du2 du

]

1


182

. .

, Tk (b) = ...

k+1

[K (u1 ), [K (u2 ), . . . . . . , [K (uk
-1

), [K (uk ), K (u

k+1

)]] . . . ]]du1 . . . du

k+1

,

k+1 -- (k + 1) - Rk+1 , k
+1

(b) = {u = (u1 , . . . , u

k+1

):b

u

1

u

2

§§§

uk

+1

a}.

ç 4. . íí

( - N ): BN (K1 , . . . , KN ) =
n=1

cn (K1 , . . . , KN ), N

cn (K1 , . . . , KN ) -- n . N . , 1, K . ë¨ 2. TN (s, K1 , . . . , KN ) = K T
N N +1
2

N i=1

e

-sK

N

...e

-sKi

+1

Ki e

sK

i+1

...e

sKN

,

= 0 .
sK
2

=e

-sKN

...e

-sK

K1 e

...e

sK

N

+ ... + e

-sKN

KN

-1

e

sK

N

+ KN .

, s K1 , . . . , KN .
q DN = TN (0, K1 , . . . , KN ) = (q )

dq TN dsq

s=0




X (t) = K (t)X (t)

183

g (, ). , q = 0 ,
0 DN = TN (0, K1 , . . . , KN ) = K1 + § § § + KN . q DN ( K1 , . . . , KN ) . 2 ( ; . . , . . [1]). G K1 , . . . , KN , . n=1

bN (K1 , . . . , KN ) =

cn (K1 , . . . , KN ), N

bN (K1 , . . . , KN ) = log(eK1 eK2 . . . eKN ) cn -- N n . cn N :
0 c1 = DN = K1 + § § § + KN , N

n > 0 (n + 1) cn+1 N ( n ] 1 1[ r 1n n cN , DN-r + = DN + n! (n - r)! 2 r=1 [ [m + k2p cm1 , . . . , cN 2p , D N
p1 2p r mi >0 m1 +§§§+m2p =r

n- r N

]

) ... . ]

2, K (t) , K1 , . . . , KN . BN (K1 , . . . , KN ) =
n=0

cn (K1 , . . . , KN ), N

K1 , . . . , KN -- (.. ) G .


184

. .

N = 2 . 2 , c1 (X, Y ) = X + Y , 2 c2 (X, Y ) = 2 c3 (X, Y ) = 2 c4 2
1 12 1 12

[X, Y ], [X - Y , [X, Y ]],
1 48

1 (X, Y ) = - 48 [Y , [X, [X, Y ]]] -

[X, [Y , [X, Y ]]].

N 1 c2 (K1 , . . . , KN ) = [Ki , Kj ], N 2 1 i i=2 1 =0 2 =0

(



N -i

=0

p=0

½
q N

N -i p=0

) + ad
q -KN

-p ad-p N -+1 K p

(Ki-1 )

(K

N -1

).

0 K = 0 q > 0 , KN = KN q = 0

N -i+1

= 0 , i > N ,

0 DN = K1 + § § § + KN . 3 (. [1]). > 0 , |K1 | + . . . + |KN | < , BN (K1 , . . . , KN ) N , bN (K1 , . . . , KN ) . , n cN (K1 , . . . , KN ) bN (K1 , . . . , KN ) = n=1

U G ½ . . . ½ G ( N ), U = = {(K1 , . . . , KN ) : |K1 | + § § § + |KN | < } , G , eK1 . . . eKN = eBN (K1 ,...,KN ) (K1 , . . . , KN ) U .




X (t) = K (t)X (t)

185

ç 5. . ()

H (t) . , adX g : adX Y = [X, Y ] .
5.1.






) H (t) X (t) = eH (t) X (0) , X (t) X (t) = K (t)X (t) , : adH (t) dH (t) = ad K (t), dt e H (t) - 1 ) H (t) X (t) = X (0)eH (t) , X (t) X (t) = X (t)K (t) , : adH (t) dH (t) = - ad (-K (t)). H (t) - 1 dt e . , K (t) . adH (t) . , , , pi § adi (t) , H
i=0

K (t) , adi (t) i - H () adH (t) . , . , (. , ). 1. m(z ) = (ez - 1)/z . () (): ( G(t) ) d = M1 (t)eG(t) = eG(t) M2 (t), dt e M1 (t) = m(adG )G (t) M2 (t) = m(- adG )G (t) . G(t) -- .


186

. .

. : e
G(t+c)

-e c

G(t)

1 = c

1 ( de
0

sG(t+c) e(1-s)G(t)

) ds =

ds 1 =
0

( e
sG(t+c)

) G(t + c) - G(t) (1 e c

-s)G(t)

ds.

c , d( e dt
G(t)

)

[ 1 =
0

] e
sG(t)

G (t)e



-sG(t)

ds e

G(t)

.

s = 0 , e
sG(t)

G (t)e

-sG(t)

= G (t) + (adG )G (t)s + (adG )2 G (t)(s/2)2 + . . . ,

, , (). () , s : e
5.2.
(1-s)G(t+c) sG(t)

e

.







, exp(H ) . . k (z ) = 1 +
p=1

k2 p z

2p



k (z ) =

z 1-e

-z

z -, 2

, m(z ) : k (z ) = 1 z -. m(-z ) 2




X (t) = K (t)X (t)

187

, , (. [3]). , (1) ( K (t) ). (1) X (t) = X (t)K (t)

,

X = X (0)eH . Ht =
- ad

e

adH (-K ). H -1

2. X = X K so(3), su(2), e(2), sl(2, R), sl(2, R)+ . 1) R3 -- (3 ½ 3) . 2) so(3) -- SO(3) = SU(2)/Z2 3- E 3 . 3) su(2) -- SU(2) (2 ½ 2) - . so(3) su(2) . 4) e(2) -- E(2) () 2- E 2 . 5) sl(2, R) -- SL(2, R) (2 ½ 2) - . SL(2, R) 2-, .. . 6) sl(2, R)+ -- SL(2, R)/Z2 , . , SL(2, R)+ 3- I. sl(2, R) sl(2, R)+ .


188

. .

7)

dX (t) = X (t)K (t) -- , dt X (t) - K (t) , G , G , .

8) K (t) ( -) E 3 , E 3 : R3 , so(3), su(2), e(2), sl(2, R), sl(2, R)+ . K (t) . E 3 G , | § | . 9) , K K (t) = (t)k (t) , k t , (t) = ‘|K (t)| k (t) = ‘K (t)/|K (t)| , K (t) = 0 . 10) s k (s) , 2- (, 2- ). 11) H (t) H (t) = (t)h(t) , X (t) = X (0)eH (t) , H (t) G , |h(t)| = 1 , = |H | . 12) dH adH = - ad (-K ) -- H -1 dt e X = X K .

13) K = K + K -- K E 3 . K h(t) S2 , K -- K h , K -- K , S2 h .
ç 6. . . so(3)

so(3) . su(2) . so(3) (3 ½ 3) -. , , E 3 . .




X (t) = K (t)X (t)
3

189

E dX (u) = X (u)K (u), du

(2)

0 u t , K (u) - so(3) . so(3) E 3 , X (u)K (u) K (u) X (u) . -- 3- . (2) X (t) = X (0)eH (t) , H (t) . , Rn (2)
t

K (u)du

X (t) = X (0) e H (t) = t
0

0

,

K (u)du H (t) H =
- ad

e

adH (-K ), H -1

adA : so(3) so(3) : adA (B ) = = [A, B ] . , () (2) -- . , , . ( ) , ( ). (2). , K (t) , H (t) . , ( ) H , K (t) = exp(-H (t)) eH (t) t . () P so(3) P = p , -- , p -- . ep E 3 , p . so(3) | § | , () E 3 .


190

. .

: = ‘|P | , P = 0 , p = P /|P | . P t , p(t) S2 ( ), E 3 . , .. K (t) = (t)k (t) , , (t) k (t) t . H (t) , .. H (t) = (t)h(t) . , = ‘|H | = ‘|K | . , h(t) k (t) , . t , h(t) k (t) 2- S2 , S2 E 3 (. 1).

K(t) K(0) k(t) k(0) O S ²(0) h(t) k(t) k(0)=h(0)
. 1.
2

h(t) H(t)

, H (t) 0 3-. s k (t) S2 , s t . s t k |t=0 s = 0 . . . . . .




X (t) = K (t)X (t)

191

ç 7. . X = X K ,

1. X = X K ( K (t) ) ( 0 t T ), K (t) K0 = const . k (t) , .. . , k (t) k0 t . 2. X = X K ( K (t) ) ( 0 t T ), K (s) = 0 k (s) , 0 = 0 k (s) 2- (.. ). , r(s) ( ) . , r(s) ( E 3 ). , k (s) K (s) (. 2).

0

O

. 2. k(s)

3. X = X K ( K (t) ) -, dt(s) r(t) = (t) , ds r(t) -- k (t) , > 0 t(s) s . , , , K ,


192

. .

(. 3).



k

k(0)

)

k(0)

)

. 3.

t = s , - K (s) : r(s) = (s). . K (t) : K (t)t 3- k (t) (t)t . -, ds(t) K (t) = r(t) k (t), dt r(t) -- k (t) 2- ( 3-!) s = s(t) -- , .. . , k (t) ( k ). k (s) 1 . 2. X = X K = X k - , ) t s ( k r ) (s) K (s) r(s) , r(s) -- , ) , K , k (s) .




X (t) = K (t)X (t)

193

, -, , X = X K , Xs = X t k = s = X rk . 22. a) A(t) 3 - (, , t , .. X (t) = Y (t)A(t) ), (2) , - = 0 . ) A(t) . . , E 3 , (2) Y = 0 . X (t) = Y (t)eH (t) . , , , (2). , , . , ( ) (2) , -. , . . X = Y A X = X K Y A = Y AK - Y A , Y = Y G G = (AK - A )A-1 . , A(t) , G -. : ds g (t), dt g (t) = G(t)/|G(t)| , r(t) -- g (t) . r(t) g (t) , , - G G(t) = r(t) |G|2 |g |4 = 2 (|[g , g ]|2 - |g |6 ), G = (AK - A )A-1 , K , A(t) . .


194

. .

. . 23. g (t) -- . r(t) r2 = |[g , g ]|2 |g |-6 - 1. . g (t) t s s , -- t . r : |g - g , g g | = r. è è §,§ R3 . , |g | = 1 , ds(t) dg (s) dt(s) . , |g | = , = g dt ds ds d2 g dt = g 2 ds ds ,
2

+g



d2 t . ds2

dt d2 t = |g |-1 = -|g | |g |-3 . ds ds2 2 dg d2 g - ,g g = r ( t) = ds2 ds2 = |g |g |-2 - g |g | |g |-3 - g , g g |g |-2 + g , g |g | |g |-3 g | = = |g |-4 § |g |g |2 - g g , g - g , g |g |2 g + g , g g , g g | = = |g |-4 § |g |g |2 - g g , g - g |g |2 g , g |.

r2 |g |8 = |g |2 |g |4 + |g |2 g , g 2 + |g |2 |g |4 g , g 2 - 2g , g 2 |g |2 - - 2g , g 2 |g |4 + 2g , g g , g g , g |g |2 . , g , g = 0 ( ), |g | = 1 g , g = -|g |2 . d2 g dt 2 , g , 2 = g , g ds ds . , r2 = |g |-6 (|g |2 |g |2 - |g |6 - g , g 2 ) = = |[g , g ]|2 |g |-6 - 1 .




X (t) = K (t)X (t)

195

ç 8.

4.
Xt = X (t)K (t)



Y = Y ( )P ( )

(3)

, s (.. ) :
Xs = X (s)K (s)



Ys = Y (s)P (s),

K (s) = m(s)P (s), m(s) -- s . . , k (s) = p(s) , .. 2- . . 5.
Xt = X (t)K (t)



Y = Y ( )P ( )

, t = t( ) , K (t( ))t = m( )P ( ) m( ) . 3. 4 5 . .
Xt = X t = X (t( ))K (t( ))



X = X (t( ))K (t( ))t .

X (t( )) = Y ( ) , Y = Y ( )K (t( ))t




Y = Y ( )P ( ).

K (t( ))t = m( )P ( ) , s . .


196

. .

- t , X S2 X K (t) , S 2 , K (t) . X K 2- ( X K ). X K , . X K = 0 , , , . , v (t) , (. 4). , K (t) . v (t) t . t , v (t) .

. 4.

4. (3) 1 - () t . m(t) X (t) 2 -. . . .
ç 9. X =X K , -

t

-

1. X = X K , k (t) = K (t)/ (t) t . ( ) , - ( ).




X (t) = K (t)X (t)

197

. Xt = X (t)K (t) = X (s)K (s) , K (s) = s . Xs = K (t(s))t . : s

Ys = Y (s)

K (s) |K (s)|

r(s),

r(s) -- k (s) = k (s) 2-. m(s) = r(s)/|K (s)| . -. .
ç 10. ( /) - 2-

(2), t ( ). 2- . - K (t) l E 3 , k (0) . 2- k (0) = N = ( ) . N . ë ¨ = ²(0) (. 5). ²(0) k (t) k (0) . ë ¨ l = k (0) . ( ) l (t)/ .
N=k(0)

l
O ²(0)
. 5.

, : l k (t) . , t l k (t) .


198

. .

, ( ), : 1) l k (t) , 2) k (t) l = k (t) (t)/ . . 6. ( /) - ( ( /) - ). t g (t) 2-. ( /) - . -. k (t) () k (ti ) ( ). , k (ti-1 ) . k (ti-1 ) k (ti ) , k (ti ) (ti )t/ . i = 1, . . . , n , , (. 6).

²(t) ²(0) k(0)

O

. 6.

. ( /) - . () ( ), . t




X (t) = K (t)X (t)

199

D(t) ( ). D(t) K (t) . X (0)g (t) -- X = X D(t) . ( /) - t = 0 t . g (t) 3-. ²(t) t . , ²(t) (ë¨) ²(0) g (t) . . . 1) g (t) K (t) ? 2) k (t) d(t) ? .
ç 11. so(3) . K (t) 2-, X = X K

2. K (t) k (t) h(t) k (0) = h(0) (. 1). X = X K so(3) - . 4. 1) K (t) K0 -- ( ), H (t) = tK0 , .. H (t) K0 . , X (t) = X (0)e
tK
0

t . 2) K (s) = 0 k (s) - , .. k (s) |k | = 1 ( t = s ) K 0 ,


200

. .

, X = X K (. . 1 ). . . . , () , k , K . X = X K so(3) K , 2-. 5. a) 2- K (t) (.. r(t) 0 2r = dt/ds ) k (t) = K (t)/ (t) -- . ( ) ²(0) ²(t) , k k (0) k (t) . -- h(t) (. 7). ²(0) ²(t) (t)/2 . b) , ( ) X (t) = X (0)e (t) h(t) .
k(t)
(t)h(t)

,

k(0) 1 (t) 2

h(t) ²(0) ²(t)

. 7.

. 1) ²(t)2 k k (t) t . , ²(t) , k k (t) , -- ²(t)=2 ( /2) - k . ²(t)=2 ²(t) . 2) h(t) (t) k




X (t) = K (t)X (t)

201

. , ( t ). 1. 2 - K h . , k k (0) (. 7). , h(t) ²(0) ²(t) . 2. , k k (0) (. 7). h(t) k (.. k ).
ç 12. so(3) 12.1.


4

: X (t) = X (t)K0 , K0 = const (.. E 3 ). , (2) X (t) = X (0)eK0 t . k (t) k = K0 /|K0 | ( , K0 = 0 ). H (t) = K0 t , .. K0 , (t) = |K0 |t . 4 . 2 , 0 = 1 . X (s) = X (s)k (s) , .. (s) 1 k (s) . k (s) ( ) (= 1) , k (s) a E 3 . , k (s) , k (s) , k (s) = eAs K (s0 )e
-As

A . k (s0 ) P0 , k = eAs P0 e
-As

,

(4)

eAs , .. SO(3) . (4) P a . ,


202
Xs = X eAs P e -As

. .

X e

As

= X eAs P0 , (5)

(X eAs ) - X (eAs ) = X eAs P0 .

A const , (eAs ) = AeAs = eAs A . , eAs A ( A -- ). (5) (X eAs ) = X eAs A + X eAs P0 = (X eAs )(A + P0 ). Y = X e
As

, Y = Y (A + P0 ) , Y = Y B0 , B0 = A + P0 .

4 .
12.2.


2

k (t) , k (0) , h(t) , h(0) . , k (t) h(t) ( t 0 ) , .. k (0) = h(0) . , ²(0) k (t) h(t) t = 0 . K (t) t 0 t . K (t) , K (t) t , .. K0 K0 + K . K (0) = K0 . K (t) = K0 + tK0 /t , 0 t t . H (t) t . H (t) = H1 - H2 + . . . ( ), :
t

H1 =
0

K (u) du,

1 H2 = 2

] t[ u K (u), K (v ) dv du. u=0 v =0


) ( ) t( tK0 K0 uK0 du = K0 t + = K0 + t. H1 = K0 + t 2 2 0




X (t) = K (t)X (t)

203

1 H2 = 2 1 = 2 1 2
)] t[ u ( uK0 v K0 dv du = K0 + , K0 + t t u=0 t v =0

[

] uK0 u2 K0 K0 + , uK0 + du = t 2t ][ ][ ]) uK0 u2 K0 uK0 u2 K0 + , uK0 + , du = 2t t t 2t

u=0 t

([ K0 ,

=

1 = 2

u=0 t

(2 ) u u2 1 [K0 , K0 ] - du = - (t)2 [K0 , K0 ]. 2t t 12

u=0

, ( ) 1 1 H (t) = K0 + K0 t + (t)2 [K0 , K0 ] + . . . 2 12 ( ) K0 + 1 K0 K 1 t . 2 2 , , . 8: h(t) ( ) h(0) + k 1 t + [k (0), k (0)], 2 -- h(0) = k (0) . , = 1 = 12 (t)2 t .

h(t) k( 1 t) 2 k(0)=h(0)

k(t)

. 8.

2 . 2. , . ,


204

. .

H (t) . . 9. H (t) O = H (0) H (0) = K (0) . , h(t) k (0) , t . H (t) , , H (t) , .. h(t) . . 9.
H' (0)=K(0)

k(0)=h(0) h(t) S
2

H(t) O=H(0) h(t) t
. 9.

k(0) 0

4, . . . , . , X = X K , X (t) = X (0)eH (t) . H (t) : H (t) = H (0) + tH (0) + , H (0) = 0 . X = eH (t) = ( )( )2 t2 1 t2 2 2 =E + tH (0)+ H (0)+o(t ) + tH (0)+ H (0)+o(t ) +o(t2 )= 2 2 2 2 2 t t = E + tH (0) + H (0) + (H (0))2 + o(t2 ) = 2 2 t2 = E + tH (0) + (H (0) + (H (0))2 ) + o(t2 ). 2 X (0)
-1

t2 H (0) + o(t2 ). 2




X (t) = K (t)X (t)

205

, (X (0))-1 X = H (0) + t(H (0) + (H (0))2 ) + o(t). : K = K (0) + tK (0) + o(t) X (0)
-1

X K = (E + tH (0) + o(t))(K (0) + tK (0) + o(t)) = = K (0) + t(K (0) + H (0)K (0)) + o(t).

, X = X K , H (0) = K (0) H (0)+(H (0))2 = = K (0) + H (0)K (0) , .. H (0) = K (0) H (0) = K (0) . k . K, K = K (0) + tK (0) + o(t), K (0) + tK (0) + o(t) = = |K (0)|2 + 2tK (0), K (0) + o(t). |K | = |K (0)| + t|K (0)|-1 K (0), K (0) + o(t) k = K/|K | = = (K (0) + tK (0) + o(t))(|K (0)| + t|K (0)|-1 K (0), K (0) + o(t))-1 = = (K (0) + tK (0) + o(t))|K (0)|-1 (1 - t|K (0)|-2 K (0), K (0) + o(t)) = = K (0)|K (0)|-1+t(K (0)|K (0)|-1-|K (0)|-3 K (0), K (0)K (0))+o(t). H : H, H = tH (0) + t2 H (0) + o(t2 ), tH (0) + 2 = t2 |H (0)|2 + t3 t2 H (0) + o(t2 ) = 2 H (0), H (0) + o(t3 ),

|H | = t|H (0)| +

t2 |H (0)|-1 H (0), H (0) + o(t2 ), 2

h = H (0)|H (0)|+ t + (H (0)|H (0)|-1 - |H (0)|-3 H (0), H (0)H (0)) + o(t) = 2 = K (0)|K (0)|-1 + t + (K (0)|K (0)|-1 - K (0), K (0)K (0)|K (0)|-3 ) + o(t). 2 , H (0) = K (0) H (0) = K (0) . , k (0) = h(0) . .


206

. .

. [k (0), k (0)] k (0) k (0) (. 10), [§,§] 3-. , k (0) = h(0) , k (t) h(t) (. 8) k (0) - h(t) h(t) - k (t) .

k0+k k0+ 1 k 2

0

0

k0

[k 0 ,k 0 ]

O
. 10. [k(0), k(0)]

5 so(3) , su(2) .
12.3.


so(3)

su(2) . su(2)





, su(2) . so(3) . (3 ½ 3) - X , .. X = -X . so(3) su(2) SU(2) . su(2) , , , so(3) . su(2) (2 ½ 2) - . ( ) ( ) ( ) i0 01 0i e1 = 0 -i , e2 = -1 0 , e3 = i 0 su(2) . : [e1 /2, e2 /2] = e3 /2, ...




X (t) = K (t)X (t)

207

H su(2) h , h2 = -E , E -- . su(2) , , so(3) , . ( ) ( ) ( ) 000 0 01 0 10 e1 = 0 0 1 , e2 = 0 0 0 , e3 = -1 0 0 0 -1 0 -1 0 0 0 00 so(3) . : [e1 , e2 ] = e3 , ...

H so(3) H = h , -- 3- , h . su(2) so(3) , : ei /2 ei . h/2 h . , e : eh/2 exp h, exp -- SO(3) . e SU(2) , exp SO(3) . , SU(2) 2- SO(3) . , exp H = exp h SO(3) , -- E 3 , SU(2) /2 . su(2) , . su(2) so(3) , ( , ) 2. , () H su(2) : H = h , h2 = -E ( E -- ). | § | su(2) , = |H | |h| = 1 . t , h(t) S2 su(2) . Y su(2) (2 ½ 2) )- : )( ) ( ia u + iv ia z Y = -z -ia = -u + iv -ia , a, u, v -- .


208

. .