ΟΙ ΕΠΤΑ ΣΤΟΙΧΕΙΩΔΕΙΣ ΚΑΤΑΣΤΡΟΦΕΣ
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ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ ΤΜΗΜΑ ΜΑΘΗΜΑΤΙΚΩΝ Διατριβή Μεταπτυχιακού Τίτλου Σπουδών ΟΙ ΕΠΤΑ ΣΤΟΙΧΕΙΩΔΕΙΣ ΚΑΤΑΣΤΡΟΦΕΣ ΚΑΙ Η ΘΕΩΡΙΑ ΤΗΣ ΚΑΘΟΛΙΚΗΣ ΕΚΔΙΠΛΩΣΗΣ Σταύρος Αναστασίου Επιβλέπων Καθηγητής: Σπύρος Ν.Πνευματικός Εξεταστική Επιτροπή: Αναστάσιος Μπούντης, Καθηγητής Τμήματος Μαθηματικών Πανεπιστημίου Πατρών Σπύρος Πνευματικός, Καθηγητής Τμήματος Μαθηματικών Πανεπιστημίου Πατρών Ανδρέας Αρβανιτογεώργος, Λέκτορας Τμήμ. Μαθηματικών Πανεπιστημίου Πατρών. Πάτρα 2007
Transcript of ΟΙ ΕΠΤΑ ΣΤΟΙΧΕΙΩΔΕΙΣ ΚΑΤΑΣΤΡΟΦΕΣ
: .
: , , , . .
2007
. Henri Poincar.
2
. Ren Thom 1960 Stabilit Structurelle et Morphognse (, 1972) Modles Mathmatiques de la Morphognse (, 1980), [ , , 1985]. (+, ), , , , . , . , , , : . , , . , .1 R.Thom : , , , Waddigton . (deploiment universel/versal unfolding) . Thom, (deploument/unfolding) . , , , Taylor: , , Taylor, , . , . Thom 4: . , Thom, Whitney . Whitney Morse 1930. , Morse . : 2 2 f ( x1 ,.., xn ) = x12 + .. + xk xk2+1 ...xn . . , (jets) , , Ehresmann, , .1
. , Ren Thom. [ , , 1985.]
3
, Thom, , , . , , . - . , : . , , , , , . , , . Thom . , , . Thom. , , , . C. Zeeman, Warwick, . , , V.Arnold, Ren Thom.
4
: . 1.1 ................................................................6 1.2 ....................................................................10 1.3 ...............................................................................20 : . 2.1. .. 2.2. .......................................................................................50 2.3. .. 2.4. ......................................................................................56 2.5. Thom.............................................................................60 : . 3.1. .................................................................................................................. 3.2. Malgrange-Mather............................................. 3.3. ............................................... V 4.1 ..........................................................................................................89 ...95
5
1:
1. . , , , , . , .. 1 Pierre Simon Laplace(1749-1827)
, , , , , , . , , , , . , , , , . n . , .. , . , . , . . n . , xi (t ) , i = 1,..., n , , , x(t ) = ( x1 (t ),..., xn (t )) n . n :dxi = f i ( x1 ,..., xn ) , i = 1,..., n . dt
fi : U n
, i = 1,..., n ,
1 . , . , , 2007: . ; , . Heisenberg ; , . , , . Laplace .
6
. , :U:Un
,
:dx = U( x), x U. dt , .. , :xo : I Un
, xo (to ) = xo U .
, t , : gt : U n U n , gt ( xo ) = xo (t ) . :gt +t = gt o gt , t , t .
{ gt } t t = 0 . :g: U U , g(t , xo ) = gt ( xo ) .
xo U : O xo = {gt ( xo ) U / t }
:gt ( xo ) = xo , t
.
, :U:KUk
n
:dx = U(u , x), x U, u K . dt
, Ren Thom, , .. Ren Thom.
7
, 3 , 3 3 . 1 . , N 3 , 3N 3N :
F: :
3N
3N
3N
3N
x:I
,3N
& x (to ) x (to ) : d2x & = F ( x, x ) . dt 23N
,
: & & x = y , y = F ( x, y ) , , , , - 3 3 . ( ) 3N . , n- M Ta M , aM , . , :TM =aM
} U {{a} T M { .a
:T M =aM
} U {{a} T M { . a
, , :H( x, y ) = U( x) + K( y ) .
, , :n H H X H ( x, y ) = xi yi . i =1 yi xi
.1
. , . : , . , 2006. :
m
d2x & = F( x, x) . dt 2
8
..: S1 S1 . :U( x) = 2 (1 cos x) .
-.
, 1 2 . , , , .1
- .1
. , . : , . , 2006. Morse.
9
2. 1 f: U nn
, y = f(x1 ,..., x n ) .
, :d a f:n
,
d a f = xi f(a) dx i ,i =1
n
: r f(a) = x1 f(a),..., x n f(a)
(
)
.
: , . , .n
a U ::n a
, -
n a
,
(x1 ,..., x n ) = ( u1 = 1 (x1 ,..., x n ), ..., u n = n (x1 ,..., x n ) ) ,
:f:n a n a
, y = f(x1 ,..., x n ) ,) , y = f(u1 ,..., u n ) ,
:
) f:
:2 a
f
) f
2 a
) f(u1 ,..., u n ) = fo 1 (u1 ,..., u n ) :
) f(u1 ,..., u n ) = f(x1 ,..., x n ) .
.
1
.
10
, a f:n a
n
, :
, y = f(x1 ,..., x n ) ,
::n a
n a
, (x1 ,..., x n ) = (u1 ,..., u n ) ,
:f o 1 (u1 ,..., u n ) = i (u1 ,..., u n ) = u i , i = 1,..., n .1
n = 2 . : ) ) f1 (c) = x 2 / f(x) = c f1 (c) = u 2 / f(u) = c .
{
}
{
}
f: U n
, y = f(x1 ,..., x n ) .
2 , : Ha f : n , v = (v1 ,..., v n ) n
, :Ha f(v) = 1 2
1 2x x f(a)vi v j i, j=i j
n
1 , , , . , y b = f(a) :
u1 = y o f : :
n a
n
d a u1 = d b y o d a f : , n-1 : ui :n
,
, i = 2,..., n ,
( n ) , , : u i : n , i = 1,..., n . a
11
1 :Ha f(v) = [ v1 21x1 f(a) L 21x n f(a) v x x 1 L vn ] M O M M 2 f(a) L 2 f(a) v xn xn x n x1 n
.
. . Morse , , . : HADAMARD: f: n , a n :Hi :n an
,
, H i (a) = xi f(a) , i = 1,..., n ,n
:f(x) = f(a) + (x i a i ) H i (x) .2i =1
. , a n , :f(x) f(a) = df(a + t(x a)) = (x i a i ) x i (a + t(x a)) dt1 1 0 i =1 0 n
:H i (x) = xi (a + t(x a)) dt , i = 1,..., n .0 1
, Hadamard :H i (x) = H i (a) + (x j a j ) hij (x)j=1 n
f(x) = f(a) + (x i a i ) xi f (a) +i =1 n
(x i a i )(x j a j ) Hi j (x) i, j 1=
n
Hi j (x) = hi j (x) + h ji (x)H i j (a) =
1 x x f(a) . 2 i jn i, j=1
1
Q :
n
Q(u1 ,..., u n ) = a iju i u j , a ij :
,
Q(u1 ,..., u n ) = 2 Q(u1 ,..., u n ),
.
2 Taylor .
12
MORSE: a f:n an
C 2 - :
, y = f(x1 ,..., x n ) ,
::n a
n a
, (x1 ,..., x n ) = (u1 ,..., u n ) ,
:2 fo 1 (u1 ,..., u n ) = f(a) + u1 + ... + u 2 u 2 +1 ... u 2 .1 p p n
: Morse.
n=1, . n=2, Hadamard a = (a1 ,a 2 ) 2 :Hi :2 a
, H i (a) = xi f(a) , i = 1, 2 ,
:
f(x) = f(a) + (x1 a1 ) H1 (x) + (x 2 a 2 ) H 2 (x) .
Hadamard : H1 (x) = (x1 a1 ) H11 (x) + (x 2 a 2 ) H12 (x)H 2 (x) = (x1 a1 ) H 21 (x) + (x 2 a 2 ) H 22 (x)
f(x) = f(a) + (x1 a1 ) 2 H11 (x) + (x 2 a 2 ) 2 H 22 (x) + (x1 a1 )(x 2 a 2 ) H12 (x) + H 21 (x) H11 (a) =
(
)
1 2 1 1 x x f(a) , H 22 (a) = 2 2 x 2 f(a) , H o (a) = 21x 2 f(a) x x 2 11 2 2H o (x) =
1 H12 (x) + H 21 (x) . 2
(
)
:H11 (a) H 22 (a) H2 (a) 0 . o
H11 (a) 0 , :H11 (x) H 22 (x) H2 (x) 0 o
= 1 = 1 : H11 (x) > 0 H11 (x) H 22 (x) H2o (x) > 0
(
)
:f(x) = f(a) + (x1 a1 ) H11 (x) + (x 2 a 2 ) H o (x) / H11 (x)1
(
)
2
+
Marston Morse 1939 : , , . p . p = n p = 0 .
13
+
(x 2 a 2 ) H11 (x) H 22 (x) H2 (x) / H11 (x) o
(
(
)
)
2
: u1 = (x1 a1 ) H11 (x) + (x 2 a 2 ) H o (x) / H11 (x)u 2 = (x 2 a 2 ) H11 (x) H 22 (x) H2 (x) / H11 (x) o
(
)
:2 f(u) f(a) = u1 u 2 . 2
H 22 (a) 0 , H11 (a) = H 22 (a) = 0 H o (a) 0 : x1 = x1 + x x 2 = x1 x . 2 2
Hadamard Hi j (x) = H ji (x) , i, j = 1,..., n , :f(x) = f(a) +
(x i a i )(x j a j ) Hi j (x) i, j 1=
n
1 2 x x f(a) , i, j = 1,..., n , 2 i j H i j (a) = 2 x f(a) . xi j ij
:2 2 f(u) = f(a) u1 ... u l 1 +
(u i a i )(u j a j ) Hi j (u) i, j > l
n
: Hi j (u) = H ji (u) , i, j = l,..., n , , , : H ll (a) 0 . :u = ui , i l , in H (u) jl u = ul + u j H ll (u) , l j > l H ll (u)
: f(u) = f(a) u1 2 ... u 2 + l : f(u) = f(a) u1 2 ... u 2 . _ n
i, j > l
(ui a i )(uj a j ) Hi j (u )
n
14
MORSE
2 f( x1 , x2 ) = x12 + x2
2 f( x1 , x2 ) = x12 x2
2 f( x1 , x2 ) = x12 x2
f( x1 , x2 ) = ( 2 x1 , 2 x2 )
f( x1 , x2 ) = ( 2 x1 , 2 x2 )
f( x1 , x2 ) = ( 2 x1 , 2 x2 )
.
. .
, . , :2 f(x) = x1 ... x 2 n
:
x i f(x) = 2x i
. Morse . Morse 1. Morse, : o (f) 1 (f) + 2 (f) = M
i (f) i = 0,1, 2, M Euler . .. : o (f) 1 (f) + 2 (f) = 2
, , , : + = +2.1
. Morse Theory, J.Milnor.
15
MORSE: a f:n an
C 2 - :
, y = f(x1 ,..., x n ) ,
n r , ::n a
n a
, (x1 ,..., x n ) = (u1 ,..., u n ) , f:n r
:
:2 2 fo 1 (u1 ,..., u n ) = f(a) u1 ... u r + f (u r +1 ,..., u n ) .1
. Sylvester, , , :1 .. .. .. .. .. .. .. O .. .. .. .. .0. .. .. 1 .. .. .. .. .. .. .. 1 .. .. .. .. .. .. .. O .. .. .. .. .. .. .. 1 .. .. .0. .. .. .. .. 0 .. .. .. .. .. .. .. .. .. .. .. .. .. .. 0
n , , : u 1 f(u ) = ... = u r f(u ) = 0
:g:nr
r
.
{( g1 (ur +1 ,.., un ),..., g r (ur +1 ,..., un ), ur +1 ,..., un )}(u1 ,.., un ) = (u1 g1 (ur +1 ,.., un ),.., ur g r (ur +1 ,.., un ), ur +1 ,.., un )
:
:F(ur +1 ,...,un ) :r
, F(ur +1 ,...,un ) (u1 ,.., ur ) = F(u1 ,.., ur , ur +1 ,.., un ) ,r
F = fo f (ur +1 ,.., un ) = F(0,.., 0, ur +1 ,.., un ) : f = f + F( ur +1 ,..,un ) . Morse F(u rr +1 ,.., un
.
)
, -
, , :
f(v1 ,..., vr , ur +1 ,.., un ) = v12 .. vr2 + f (ur +1 ,.., un ) . _ . f f . .1
16
:
40. 06 0
30. 05 0 0. 04 0
0.0004
0.0003
20. 03 00.0002
1
0. 02 00.0001
0. 01 0
2
1
1
2
- .4 0
- .2 0
0 .2
0 .4
-0.4
-0.2
0.2
0.4
f( x) = x
2
f( x) = x
4
f( x) = x
6
0.000075
0.020.00005
0.010.000025
-1
0.5
0.5
1
-0.4
-0.2 -0.000025
0.2
0.4
-0.0 1-0.00005
-0.0 2-0.000075
f( x) = x 3
f( x) = x 5
1 0.75 0.5 0.25 0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
f
0 -1 -2 -3 -4 -2 -1 0 x 1 2 -2 -1
2 1 0 y
f( x1 , x2 ) = x12
2 f( x1 , x2 ) = x12 x2
f( x1 , x2 ) = x12
1 -. 05
02 1 0 1 2 1 -. 05 0 05 . 1 1 -. 05 05 . 1
1 0.5 0 -0.5 -10
1
05 .0.5
-1 -0.5
0
1 20 0.5 1 -1 -0.5
1 0 1 -. 05 0 05 . 1
3 f( x1 , x2 ) = x13 + x2
2 f( x1 , x2 ) = x13 x1 x2
2 4 f( x1 , x2 ) = x1 x2 + x2
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1:
2 2 f: R 2 R , f(x1 , x 2 , x 3 ) = 8x1 + 13x 2 + 4x 3 + 4x1 x 2 + 40x1 x 3 + 44x 2 x 3 , 2
. , Gauss, . : 8 2 20 1 C = (A + t A) = 2 13 22 2 20 22 4
:f(x) : x,Cx , x R 3 .
1 = 36, 2 = 27, 3 = 0 , : 1 2 2 2 1 2 2 2 1 ( , , ),( , , ),( , , ) . 3 3 3 3 3 3 3 3 3
1/ 3 2 / 3 2 / 3 P = 2 / 3 1/ 3 2 / 3 2 / 3 2 / 3 1/ 3
:t
36 0 0 PCP = 0 27 0 0 0 0
, :0 0 1 0 0 1/ 36 0 CP 1/ 27 0 = 0 1 0 0 0 0 0 0 0
:2 f( y1 , y2 , y3 ) = y12 + y2 .
2:
f: R 2 R , f( x, y ) = x 2 + 2 xy 2 + x 2 y 2 ,
1:1 0 D 2 f(0, 0) = 2 0 0
.
: x f( x, y ) = 2 x + 2 y 2 + 2 xy 2 x f(h( y ), y ) = 0
h(y) y . h( y ) = y 2 (1 + y 2 ) 1 , , f ( y ) = f( h( y ), y ) = y 4 (1 + y 2 ) 1 :
18
f(u , y ) = u 2
y4 . 1 + y2
. : k ( x, y ) = f( x + h( y ), y ) f ( y ) = x 2 (1 + y 2 ) 1 ( x, y ) = x 1 + y 2 , (u = x 1 + y 2 , y ) :f( u 1+ y2 y2 y4 , y) = u 2 . 1+ y2 1+ y2
u 2 v 4 :v=( y 4 1/ 4 ) . (1 y 2 )
:
f: U 2
, f( x, y ) = 1 x 2 y 2 ,x dx y 1 x2 y2 dy
Df ( x, y ) = 1 x y2 2
(0,0) . , :2
( x0 , y0 ), x0 0
2
,
( x, y ) = (f( x, y ), y ) = (u, v) ,
( x0 , y0 ) D f ( x0 , y0 ) D2 f ( x0 , y0 ) D( x0 , y0 ) = 1 0 1
::2
2
, (u, v) = ( (1 u 2 v 2 ), v) ,
x0 . f ( (u , v )) = u .
2) .4
2
0
- 2
- 4 - 7 . 5 - 5 - 2 . 5 0 2 . 5 5 7 . 5
-. F U ( x) = (1 cos x ) , 2
U '( x) = 2 sin x x = 0, ,... .
19
, U ''( x) = cos x , , Morse U(x) . , Taylor U(x) 0 x x2 x3 x2 x3 + ( 2 sin t )dt = 2 + 2 (cos x + 1) . U ( x) = 2 2 6 2 0 6 :R R2
xax (
22
+
26
x(cos x + 1))
0,
U ( 1 ( x)) = ( x ') 2 . U ( x ) = x 0 , , . U(x) , 2
:R R
x a x (
22
+
26
x(cos x + 1))
,
,
U ( x ') = ( x ') 2 , .: , , Taylor 2 . , , . Taylor, 2, : R R , , :U R R
, R R % U % U(x) U ( x ) . Morse , , : R R Taylor .:
:
&& = x 2 . x
& x= y
& y = x2
,
, H ( x, y ) := -:
1 2 y + x 3 , 2
20
1
0.5
0
-0.5
-1 -1 -0.5 0 0.5 1
x3
U ( x ) = x , 0. : R R
( x) := x x 0.
, ,
21
:1 , , . f = ai xi2 , , ai 0 i = 1,.., n .i =1 n
, , f 0. :
: R n R , f + 0 f + t , t [0,1] . n n
gt : R R ,
( f + t )( gt ( x)) = f ( x), g 0 ( x ) = x, gt (0) = 0 , t, : . (1)
u ( g ( x)) = (
d g t ( x)) . dt t =(2)
(1) t ut :
( gt ( x)) + grad ( f + t ) ut ( gt ( x)) = 0 .ut = ut ,ii =1 n
, , :
, xi
(2) :
( gt ( x)) + ut ,i (2ai xi + tx )i
= 0.gt ( x )
2ai xi + txi , , yi .
:
yui =1
n
i t ,i
( gt ( x)) = ( gt ( x)) . (3)
t det(y / x)
x
i
, x=0.
t
( y, t ) .
t ,
Hadamard n
= yi i , i (0, t ) = 0 .i =1
(3) ut ,i = i .
gt ( x) : d gt ( x) = ut ( gt ( x)), g 0 ( x) = x . dt1
R.Thom, . Singularities of Differentiable maps V.I.Arnold et.al.
22
ut (0) = 0 , , 0,
t [0,1] . g1 ( x) f + f = ai xi2 , ai 0 . Sylvester i =1 n
f ._: . f : M R ,
. f p M
Hess p ( f ) : Tp M Tp M R
U,W u,w p U(W(f))(p) f p. ( x1 ,.., xm ) m U = U i , W = Wi xi xi i =1 i =1 m
(u , w) a U (W ( f ))( p)
,
, U i ( p ) = ui Wi ( p ) = wi , i 1 m. :
Hess p ( f )(u, w) =
i , j =1
[U i
m
W j f 2 f ]( p) + U iW j xi x j xi x j
(*).
, (*) :
2 f ui w j x x ( p) , i , j =1 i jm
u,w. .
23
3. :U:KUk
n
.
Ren Thom, . , , , Morse, , . , . , V(x). , gradV(x), V, gradV. , , . , . Morse , , . , . f : R n R , F : R n R k R F ( x, 0) = f ( x) . . , Ren Thom. , 4 , 7 Thom :f (u, x) = x3 + ux ()
f (u1 , u2 , x) = x 4 (u1 x 2 + u2 x) ()f (u1 , u2 , u3 , x) = x5 + (u1 x3 + u2 x 2 + u3 x) ()
f (u1 , u2 , u3 , u4 , x) = x 6 (u1 x 4 + u2 x3 + u3 x 2 + u4 x) ()3 f (u1 , u2 , u3 , x1 , x2 ) = ( x13 + x2 ) + u1 x1 x2 + u2 x1 + u3 x2 ( ) 2 2 f (u1 , u2 , u3 , x1 , x2 ) = ( x13 x1 x2 ) + u1 ( x12 + x2 ) + u2 x1 + u3 x2 ( ) 4 2 f (u1 , u2 , u3 , u4 , x1 , x2 ) = ( x12 x2 + x2 ) (u1 x12 + u2 x2 u3 x1 u4 x2 ) ( )
f : R R , , n
24
, , . , , 7 . . , . Ren Thom, . , R R . , . Ren Thom 4 . : 4 f : 4 n , , , .n k
25
1 : (FOLD)U(a, x) = x3 ax
, : : -. . , . , .
0=
d 3 (x + ax) = 3x 2 + a , dx
, (u, x) = ( 3x 2 , x) . ,
0=
d2 3 (x + ax) = 6x d2 x
(a, x) = (0,0) . a = 0 , . , , : , ,
26
& x= V ( x, y ) x V ( x, y ) & y= y2
(1)
V V: ( x, y ) a x 3 + cx + y 2 c. . , c. , V : r V ( x, y ) V ( x, y ) V = ( , ) = (3x 2 + c, 2 y ) x y
, c , (
c 3
,0),(
c 3
,0) ,
c=0, (0,0) c, 3x 2 + c = 0 . r V . , c 0 . D = 0 , : D = 0 a 0 b 0 , D = 0 a = b = 0 . , , Va , b , (a, b) 2 4a 3 + 27b 2 = 0 . : , , B1 B2 P.
.
(,b) 4a3 + 27b 2 < 0 4a 3 + 27b 2 > 0 . :
29
(,b) (,b) (,b) B1 B2 , 2 .
B1 , B2 . (,b)=(0,0) 0. .
Va ,b ( x) = x 4 + ax 2 + bx .
1 4
1 2
(,b).
, Va , b (a, b) I , (a, b) E , (a, b) B1 B2 (a, b) = (0,0) . Va ,b = x 4 , x 2 , . , V , . , . ( x, a, b) . ( x, a, b) x3 + ax + b = 0 .1 4
30
.
3 . : Y M , Y (x,), ( x, a) = ( x, a, b) , x3 + ax + b = 0 , b ( x, a ) = ( x, a, x 3 ax) . 2 . ( x, a, x 3 ax) (a, b) . M d 2Va ,b x3 + ax + b = 0, 0 = = 3x 2 + a . dx 2 , a = 3x 2 , b = 2 x 3 . x ( x, 3x 2 , 2 x3 ) , x. (3 x 2 , 2 x3 ) . a = 3 x 2 , b = 2 x3 4a3 + 27b 2 = 0 , , , (,b) . . (,b) x (,b) M. , (,b) , 3 . 2 . . ( x, 3x 2 , 2 x3 ) . .
31
3 : (Swallowtail)U (a, b, c, x) = 1 5 c 3 b 2 x + x + x + ax 5 3 2
M (a, b,c, x) 0= d U (a,b,c) (x) = x 4 + cx 2 + bx + a dx
3
(x, c, b, bx cx 2 x 4 ) . , 0= d2 U (a,b,c) (x) = 4x 3 + 2cx + b . dx 2
c.
, , ,
32
.
.
33
4 : (Butterfly)
U (a, b, c, d , x) = x 6 + (dx 4 + cx3 + bx 2 + ax)
, , . 5, . c d. (,) c,d . (,) (c,d).
34
5 : (Hyperbolic Umbilic)f (a, b, c, x, y ) = x3 xy 2 + a ( x 2 + y 2 ) + bx + cy
. , (x,b,c) .
35
6 : (Elliptic Umbilic)f (a, b, c, x, y ) = x3 + y 3 + (axy + bx + cy )
, . ,x , . , , , .
36
7 : (Parabolic Umbilic)
f (t , w, u, v, x, y ) = x 2 y + y 4 (+ty 2 + wx 2 ux vy )
4, , . t (u,v), w, :
t (u,v,w) :
37
, w, (u,v)
38
2:
1. p . ~ : n
n
F1 : U1 R F2 : U 2 R , U1 ,U 2 , p U1 U 2 U1 U 2 p. p.
f : R n R p R n f : (R n , p ) R . f : R n R n En . , , n . x = x1 ...x , 0 n 0 n .1 n
m 0 n 0 n En . :2 En mn mn ... m n En , En : k mn = I mn .
k =1
: Hadamard m En ,
xi , m k En xi k. Borel En / m R[[ x1 ,.., xn ]] . A Gn n . , . Gn En . g Gn , f En , Gn En . ( f , g) a f (g) En . . , . .En Gn En
39
. f , g En s [0,1] yi En +1 , i = 1,.., n : yi (0, t ) = 0
, [ f ( x) + ( s + t )( g ( x) f ( x))] xi i =1 (x,t) R n +1 . . f,g U R n . F :U R R , F ( x, t ) = (1 t ) f ( x) + tg ( x) , , Ft : U R . Ft ( x) = F ( x, t ) , , F0 = f , F1 = g . f,g , s [0,1] , I s , s I s , , [ Ft ] ~ [ Fs ], t I s . , A = {s [0,1] :[ Fs ] ~ [ f ]} , [0,1], I s [0,1] A, s A . , , s I s A , [ Fs ] ~ [ f ] , s . [0,1] , 1 , . I s V U, , I R , 0 , : V I R n : ) (x,0)=x x V. ) (0,t)=0 t ( x, t ) U , ) . F ( ( x, t ), s + t ) = Fs ( x), ( x, t ) V I . , t : V R n . t ( x) = ( x, t ), t I . , % det( D t (0)) 0, t I I ,g ( x) f ( x) = yi ( x, t )n
0 I% . , t , 0, t I% . ) [ Fs + t ][ t ] = [ Fs ] , , [ Ft ] ~ [ Fs ], t I s = s + I% . ),) (V I ) U , ) d F ( ( x, t ), s + t ) = 0 , , F ( ( x,0), s ) = Fs ( x) . ) dt
( x, t ) U
')
, F (( x, t ), s + t ) + Di F ( ( x, t ), s + t ) ( x, t ) = 0 t t
( x, t ) V I . ),),) : U J R n , J R 0, , , ) (0, t ) = 0, t J , n . ) F ( x, s + t ) + i ( x, t ) Di F ( x, s + t ) = 0, ( x, t ) U J t i =1 , x U ( x, t ) = ( ( x, t ), t ) , t
40
(x,0)=x. V U, 0, , J, 0, , x V, I R n , t a ( x, t ) , (x,t) U (x,0)=x, : V I R n . ). ) ), ) ) . , = ( 1 ,.., n ) ) ), . Taylor . g : R n R n :f R n R
g
id
Tk f R n R
id Tk f Taylor k f. Taylor k k- .
:
n
k
.
f , g En
k-
D f (0) = D g (0), N , k . k- f k f j k [ f ] . k- En , k R n , J n . J nk En / m k +1 k k- En En / mn +1 . Taylor k k , , P k = { p R[ x1 ,.., xn ] : deg p k} : J k Pk , j k [ f ] a T fk T fk Taylor k f, .n P k , , N = dim P k = (k + k ) , R N , J k : J k Pk E Jk , j k [ f ] a T fk [ f ] a jk [ f ] . , k En / m k +1 J n , [ f ] + m k +1 a j k [ f ] n , dim( E / m k +1 ) = (k + k ) . J k R .
: J nk (R n , R) k- n , : dim J 0 (R n , R) = n + 1, dim J 1 (R n , R ) = n + 1 + 1n,...
41
{x1 ,.., xn } y U U x U R n f ( x) U ' , f : R n R , j k f : U J k (U , R ) J nk (R n , R) :{x1 ,.., xn },{ y},{ f 2 f k f },{ },...,{ k } . xi xi x j x
, f : R n R , k- , , k. :
: k . j k [ ] = ( j k [1 ],.., j k [n ]) k- Gn , (0) = 0 . k- Gnk . Gn k k k Gn Gn Gn
k
, j k [id ] j k [ ] j k ([ ]1 ) .
( j k , j k ) a j k [ ( )]
,
: . U J nk , M(U), , k- U. {M(U)}, U J nk , C k (R n , R) . Whitney, C k (R n , R) , , Baire. , Taylor , , , .
2. k- . k (determinacy) det[f]. k.
. f En k- g En f, g k.
: To g f Gn , g=f . h En k g. j k [ f ] = j k [ g ( 1 )] = j k [h( 1 )] .
f k- y Gn , f = h( ) y , 1
42
g = h( 1 ) y ,
g k-.
1: 1. , .
2: 2 . J. Mather1 1960, (modules), Nakayama.
Nakayama. M . G P, G P, P, G . . A AG
={0}.
: B A : A B U{0} G , =cardB, . =0, =. B = {b1 ,.., bk }, k 1 . 0 b1 = rk bk , rk G . (1 r1 )b1 = rk bk , (1 r1 )b1 .k =1 k =2
1 r1 G, G , G=P. , 1 r1 :b1 = (1 r1 ) 1 rk bk {b2 ,.., bk ,0}k =2
G
,
, .
. ,L , . , K L + K K L .M
: A K , K = A P . K % A := {a + L / a A} ( AM
M
= A
M
, A L + A
M
,
+ L) / L .
( A
M
+ L) / L
1
J.Mather . , . :
Stability of C
mappings I-VI.
43
r aj j
j
+ L = rj ( a j + L ) ,j
( A
M
% + L) / L = A
M
% % , A A
M
% . A = ,
. En . :J [ f ] := D1 f ,.., Dn f
.
:
. f En - m +1 m 2 J [ f ] .
: g En , j [ f ] = j [ g ] . g-f m +1 . s [0,1] . h(x,t)=f(x)+(s+t)(g(x)-f(x)), , En +1 . Di f = Di h + ( s + t )( Di f Di g ) .
Di f Di g mn , 2 mn J [ f ] En+1
mn {D1h,.., Dn h}
En +1
+ mn +1
mn +1
.
,
mn +1
n +1
,
Nakayama mn +1 mn {D1h,.., Dn h}En +1
,
g ( x) f ( x) = yi ( x, t )i =1 n
h ( x, t ) , xi
. . , m k +1 mJ [ f ] . .
. c J k , k-jet z = j k [ f ] f En , zG k . C G k , , :c '(0) = d [ zC (t )]t = 0 . dt
44
: , , . Mather, Lie . .
. f En . f - , m +1 mJ [ f ] .
: g m +1 . j +1[ g ] j +1 ( mJ [ f ] ) ,
m +1 mJ [ f ] + m + 2 , K L+ KM
K L,
m +1 mJ [ f ] + m + 2 m +1 mJ [ f ] + m +1 m +1m
,
mJ [ f ]
. t . f+tg - f, c(t ) = j +1 ( f + tg )
z = j +1[ f ] G +1 . C : J G +1 ,
J 0, :C (0) = j k +1[id ], . d [ zC (t )]t = 0 = c '(0) = j +1[ g ] dt
zC(t). , , J +1 P +1 , G +1 H +1 , :z = T f +1 C (t ) = (Ci (t ))1 i n , Ci (t ) =0 < < +1
i (t ) x ,
i : J
. zC (t ) = T fo+1(t ) , Cn d f (C (t )( x))t = 0 = Di f ( x)Ci' (0)( x) dt i =1
mJ [ f ] . ,
45
d +1 T f oC ( t ) t = 0 = T d +1 t = 0 , f oC (t ) dt dt
j +1[ g ] j +1 ( mJ [ f ] ) .
.
. f En , , , m mJ [ f ] . , det[f]
-1.
: f - m +1 m 2 J [ f ] , m mJ [ f ] . f (-1)- , , , m mJ [ f ] , , , .
. 1) ax k det=k, Taylor . 2)f : R2 R ( x, y ) a x 3 + y 3
.
3- , J [ f ] = x2 , y2E
x2 , y 2 =
m2 J [ f ] =
x 2 , xy, y 2E
E
= x 4 , x 2 y 2 , x3 y, xy 3 , y 4
,
m m J [ f ] .4 2
3) f : R2 R ( x, y ) a x 3 + xy 2
,
3- . T f2 = 0 , 2, J [ f ] = 3 x 2 xy 2 , x 2 y,3 x 2 y y 3 , xy 2 = x 3 , x 2 y, xy 2 , y 3E E
=
= m3
,
. 4) f : R2 R
( x, y ) a x 2 y + y 4
,
det[f]=4. T f3 = 0 , [f] 3- . ,
46
J [ f ] = 2 xy, x 2 + 4 y
, E
mJ [ f ] = 2 x 2 y, 2 xy 2 , x3 + 4 xy 3 , yx 2 + 4 y 4
,
m 4 . Taylor, . .
2. , . .
: 1) P,L, R 2 , a,b c.
2) L P a,b c.
3) P,L R a.
3
a , Ta R R . , 3 3
.
47
: P,L : 1) 2) . P L P,L. P,L , P,L . P,L M, P,L , . , P,L , P,L . , . .
: L q P, q . F P . TF ( x ) L + Dx F (Tx P ) = TF ( x ) M , F L F(x) .
: 1) L (1,2,z) R 3 , P R 2 . F : R 2 R3 . 2 2 ( x, y ) a (2 x,3 y, e ( x + y ) ) F(P) L =(1,2, e )1 R 3 . L u3 = (0,0,1) . F(P) 2 2 (2 x,3 y, e ( x + y ) ) a = (2,0, e ) x . 2 2 4 u3 = (2 x,3 y, e ( x + y ) ) a = (0,3, e ) y 3
u1 =
2 0 e 4 det 0 3 e = 6 0 , 3 0 0 1
R 3 , F L. 2) L x R 2 F : P = R R2 . t a (t , t )
L x, F(P) (0,0) , (1,1). R 2 , F L. 3) L x R 2 , P F : R R2 t a (t + 1,(t 1) 4 )
.
, =(2,0). L (1,0), F( R ) d (t + 1,(t 1) 4 )t =1 = (1, 4(t 1)3 )t =1 = (1,0) , dt
. , F L. .
1
0,694.
48
3. , : m n m-n .
: f m 2 . m/J[f] f codim[f]. : O f f. En dim( En / T f O f ) , T f O f O f f o f . O co dim(O f ) Milnor f , Milnor f [f]. . , , .
. [ f ] m 2 , . [ g ] m3 [f]. codim[f]=codim[g]. : , [f] [g], [g] . , codim[f]=codim[g]. . , f En , Gn , J ([ f ][ ]) = ( J [ f ])[ ] ,
( J [ f ])[ ] = {h , h J [ f ]} . . g = f ( ) , Di g = ( Di f ) Di j .j =1 n
y = ( ) 1 . Di j = ( Di j ) o y o , J [ g ] ( J [ f ]) . , J [ f ] ( J [ g ])[ y ] , ( J [ f ])[ ] J [ g ] .
.
. f m2 . g E f codim[f]=codim[g]. : Gn g = f ( ) . h a h( )
m, m m . , ( J [ f ])[ ] = J ( f ( )) , J[f] J[g]. m / J[ f ] m / J[g] h + J [ f ] a h( ) + J [ g ] , .
49
, . Morse .
. f m2 . codim[f]=0 , , , , . : codim[f]=0, m=J[f], m3 = m 2 J [ f ] , [f] 2-, , , . , [f] , , codim[f]=0. . .
. . / , , , m k A . , > c0 > c1 > ... > ck = 0 , c j := dim( A + m j ) / A .
: m k A dim E / A dim E / m k < , n dim E / m k = ( k + k ) . , c0 = dim E / A < . E A + m A + m 2 .. A + m j .. A , > c0 ... c j ... .
, A + m k = A + m k +1 , , {c j } , . A + m k = A + m k +1 m k A , ck = 0 .
. f m 2 , , . : [f] . m k J [ f ] k, , /J[f] . . . [ f ] m 2 . det[f]codim[f]+2. : c1 = cod [ f ] < . k m k J [ f ] , k , c1 k 1 . m k +1 mJ [ f ] , det[f]k+1, .
: 1)
f :R R x a ax k
.E
J [ f ] = x k 1 2)
= m1k 1 . dim E1 / m1k 1 = k 1 codim[f]=k-2.f : R2 R
( x, y ) a x 3 + y 3
J [ f ] = x 2 , y 2
E
, , m = x, y, xy + J [ f ] , [x],[y],[xy],
m/J[f]. H ax + by + cxy + g ( x, y ) x 2 + h( x, y ) y 2 = 0 ,
50
a,b,c g,h . ax+by m 2 , , a=b=0. 0, c=0, codim[f]=3 3) f : R2 R
( x, y ) a x3 xy 2
.
J [ f ] = 3x 2 y 2 , xy E , m = x, y, x 2 + y 2 + J [ f ] , [x],[y], [ x 2 + y 2 ] m/J[f], , codim[f]=3. 4) f : R2 R
( x, y ) a x 2 y + y 4
,
J [ f ] = 2 xy, x 2 + 4 y 3
E
, m = x, y, x 2 , y 2 + J [ f ] . m/J[f]
[x],[y], [ x 2 ],[ y 2 ] codim[f]=4. .
. f m k . cod [ f ] k ( k + 1) .2
1 2
: =0 codim[f]= . 1 codim[f] 4 , . ) 5, . Thom. 6. , 1-moduli1 , .
1 . V.I.Arnold, Classification of Unimodal Critical Points Of Functions.
54
3:
1. , . f :R R
1 3, x 3 0. x, , f ':R R , 1 x a x 3 + ax 3 , , a , a . , , . Ren Thom1, .xa
: f En . F En + r r- f F(x,0)=f(x) x R n . x1 ,.., xn R n , , u1 ,.., ur R r , . F f , : Rn R . Fu ( x) a F ( x, u ) , R n , R r . .
: F En + r f En . M F := {( x, u ) R n + r : DFu ( x) = 0} F. CF := {( x, u ) M F : det[ D 2 Fu ( x)] = 0} , Rr ,BF := {u R r / x R n : ( x, u ) CF }
F. M F . [ x 3 + ux 2 ] [ x3 ] , . ( x0 , u0 ) M F 1
CF ,
. R.Thom , 1985
55
( x0 , u0 ) U , U M F . , .
: [ F ] En + r [ f ] En . [G ] En + s [F] [ ] mn + s , n ,[ ] ms , r ,[ ] ms , :) ( y,0) = y ) G ( y, u ) = F ( ( y, u ), (u )) + (u ) ( y, u ) R n + s . [ ] [ ] . G f, G ( y,0) = F ( ( y,0), (0)) + (0) = f ( y ) . [G] [F] [f].
: G En + s F En + r En r=s G F, 0. . , , .
: . [ F ] En + r , [G ] En + s [ ] n + t [ f ] En . [] [G] [G] [F]. [] [F]. : G ( y, u ) = F ( ( y, u )) + (u ) H ( z , w) = G ( ( z , w)) + ( w) , [ ] mn + s , n + r ,[ ] ms ,[ ] mn + t , n + s ,[ ] mt . H ( z, w) = F ( ( ( z , w))) + ( 2 ( w)) + ( w) , 2 ( z , w) = ( 1 ( z, w), 2 ( z , w)) . H ( z , w) = F ( X ( z , w)) + a ( w) , a ( w) := ( 2 ( w)) + ( w), X = o . [ ] mt ,[ X ] mn + t , n + r . ( y, u ) = (1 ( y, u ),2 (u )) ,
X ( z , w) = (1 ( ( z , w)),2 ( ( w)))
1 ( ( z,0)) = 1 ( z,0) = z ,
. .
. [ F ] En + r , [G ] En + s [ f ] En . H ( x, u , v) := F ( x, u ) + G ( x, v) f ( x)
56
(r+s)- [ H ] En + r + s [f], [F] [G]. [F], [G] []. : [ H ] En + r + s H ( x,0,0) = F ( x,0) + G ( x,0) f ( x) = f ( x) , [] (r+s)- [f]. [ ] mn + r , n + r + s ( x, u ) := ( x, u,0) [ ] := 0 mr . H ( ( x, u )) + (u ) = H ( x, u ,0) = F ( x, u ) + G ( x,0) f ( x) = F ( x, u ) , [F] []. , [G] [].
, .
: [ f ] En [f]. , . [F] [f], q , [q]+[F] , , [q]+[f]. . [ f ],[ g ] En [ ] Gn [ g ] = [ f ][ ] , r- [F] [f] [G ] En + r G ( x, u ) := F ( ( x), u ) . G ( x,0) = F ( ( x),0) = f ( ( x)) = g ( x) , [G] r- [g] [ F ] . r- [f] r- [g]. , . , [ H ] En + s [ f ] En , [F]. H ( y, u ) = F ( ( y, u ), (u )) + (u ) [ ] mn + s , n , ( y,0) = y , [ ] ms , r ,[ ] ms . H ( ( y ), u ) = F ( ( ( y ), u ), (u )) + (u ) = = G ( ( y, u ), (u )) + (u )
( y, u ) = ( ( ( y ), u )) [ ] := [ ]1 . [ ] mn + s , n ( y,0) = y , [ H ] [ F ] . , , . .
2. f(z), U , P(z), k, U. Q(z), U, R(z), k-1, : f(z)=P(z)Q(z)+R(z). : U, P(z) k. , P(z)
57
, f P(z) k P(z), k . P(z) Pu ( z ) = z k + ui z i ,i =1 k 1
u P U. u k-1, P0 , Pu , u 0. pu , f ( z ) = Pu ( z )Q( z , u ) + pu ( f ) . Weierstrass pu ( f ) . P(x). P(x) P(x). P(x) P ( x) = x k + u j x j ,j =1 k 1
. pu u, P(x) . Malgrange u. : aij (t ) xi = c j (t ) ,i
a ij , c j t T , , t, xi (t ) , xi (t ) t. M aij , . , , g : T M , t a ij (t ) . Malgrange Mather, g M, x(t) t.1 , J.Mather . n,s , U R n , : U R s , (0) = 0 . : * : E s En [g] a [g o ]
, . * (module) En Es , Es M M
([ g ], a ) a [ g ]a
[ g ]a := ( *[ g ]) a .
Malgrange. En . Es , , M / ms M .
1
. L.Nirenberg A proof of the Malgrange Preparation Theorem, Liverpool Singularities Symposium, Springer, 1970
58
.
. k,s , Ek + s . Ek + s A + Es B + ms Ek + s , Ek + s A + Es B .% : E := Ek + s / A . % ( ms Ek + s + A) / A = ms E
, [h j ][ f j ] + A = [h j ]([ f j + A]) ,j j
[h j ] ms [ f j ] Ek + s . , % % % % E B + ms E , B := ( Es B + A) / A .
% E Es , Nakayama % % E B Ek + s A + E s B . % E Es . % n=k+s, M = E
: Rn Rs( x1 ,.., xk , xk +1 ,.., xk + s ) a ( xk +1 ,.., xk + s )
.
% * Es Ek + s . E Ek + s , % E = [1] + AEk + s
,
[1] Ek + s .% % , E / ms E
, Es B B + ms Ek + s
,
[h ][b ]j j j
Es B j j
h (0)[b ] + [hj j
h j (0)][b j ] .
, % % % % ms E = ( ms Ek + s + A) / A, E B + ms E
% % E ( B + A) / A + ms E , % % % ([b] + A) + ms E ,[b] B E / ms E .
Malgrange-Mather , % E Es . , .
. C En , C, C. C A + Es B + msC C A + Es B .
59
3. Ren Thom. , , .
. 2 k,r [f] k- mn . , r k- [f] . : r- [], j [ H ] , mn , H ( x,0) H (0,0) . u j u j
j [ H ] , j=1,..,r V[H]. [ H ] En + r [f] , k- [ F ] En + r [f] k- [ F '] En + r [f], [], j ( F ') j ( F ) J [ f ], j = 1,.., r . , [ F ] En + r k- [f]. c:=cod[f] r. [ g1 ] + J [ f ],..,[ g c ] + J [ f ]
mn / J [ f ] . a jl j ( F ) + J [ f ] = aij ([ gl ] + J [ f ]) .c l =1
c c (a jl ) , 1 ( F ) + J [ f ],.., r ( F ) + J [ f ]
mn / J [ f ] . (a jl ) , r-c ,
r r , . u R r x R n ,
H ( x, u ) := f ( x ) + u1 g1 ( x) + ... + uc g c ( x) , [f]. F '( x, u ) := H ( x, (u )) [F] [f], [],
j ( F ') = l ( H )a jl = [ g l ]a jl .l =1 l =1
c
c
j ( F ') j ( F ) J [ f ] ,j=1,..,r, V ( F ) J [ f ] + V [ F '] , k- [F]. [F] [F] k- [ f ] En + r , j ( F ') j ( F ) J [ f ], j = 1,.., r .
[F] [F] . , U R n ,V R r , F F U V . Ft := (1 t ) F + tF ' , U V , r- [f], [ F0 ] = [ F ] [ F1 ] = [ F '] . j ( Ft ) j ( Fs ) = (t s )( j ( F ') j ( F )) J [ f ] , , k- [F] k- [ Ft ] . [F] [F] k- [ f ] En + r ,
60
j ( F ') j ( F ) , , s [0,1]
I s R , s , [ Ft ] [ Fs ] t I s . s [0,1]. A U B V , 0, : A B I Rn : B I Rr :BI R
,
: (a ) ( x, u ,0) = x, (u ,0) = u , (u ,0) = 0 (b) ( x,0, t ) = x (c) B R r , u a (u , t ) , (0, t ) = 0 (d ) (0, t ) = 0 . (e)( ( x, u , t ), (u , t )) U V Fs + t ( ( x, u , t ), (u , t )) + (u , t ) = Fs ( x, u ) , , , t : A B R n , t ( x, u ) := ( x, u , t ) t , t . t I, [t ] mn + r , n ,[ t ] Gr ,[ t ] mr ,
t ( x,0) = x , Fs ( x, u ) = Fs + t (t ( x, u ), t (u )) + t (u )
(x,u) R n + r . [ Fs ] [ Fs + t ] t I, I s := s + I . (a),(b),(c),(d), ( A B I ) U , ( B I ) V . ( x, u , t ) := Fs + t ( x, u ) , (e) d ( ( x, u , t ), (u , t ), t ) + (u , t ) = 0 , dt t
( ( x, u ,0), (u ,0),0) + (u ,0) = ( x, u ,0) = Fs ( x, u ) .
(e) (e ')( ( x, u , t ), (u , t )) U Vn ( ( x, u , t ), (u , t ), t ) + D1i ( ( x, u , t ), (u , t ), t ) i ( x, u , t ) + t t i =1 r + D2 j ( ( x, u , t ), (u , t ), t ) j (u , t ) + (u , t ) = 0 t t j =1
(a),(b),(c),(d),(e) J, , % : U V J Rn % : V J Rr % : V J R
,
:% % () ( x,0, t ) = 0, (0, t ) = 0, % (0, t ) = 0
()
n r % % ( x, u , t ) + i ( x, u , t ) ( x, u , t ) + j (u , t ) ( x, u , t ) + % (u , t ) = 0 t xi u j i =1 j =1
% % % % i , j , .
61
% ( x, u , t ) = ( ( x, u , t ), (u , t ), t ) t % , (u , t ) = ( (u , t ), t ) t (u , t ) = % ( (u , t ), t ) t ( x, u,0) = x, (u,0) = u, (u,0) = 0 . A B U V , I J , 0, ( x, u ) A B
I U V R ,
t a ( ( x, u , t ), (u , t ), (u, t ))
( ( x, u, 0), (u, 0), (u, 0)) = ( x, u, 0) ,A B I Rn Rr R ( x, u, t ) a ( ( x, u , t ), (u, t ), (u , t ))
. (a) . (b),(c),(d) (0, t ) = 0 , () . (e) (), x ( x, u , t ) u (u , t ) . D1 (u ,0) = Er , ,
t
,
(c) . % % , , % En + r +1 = AEn + r +1 + BEr +1 , ,..., } B := { ,..., ,1} . x1 xn u1 ur % % ( x,0, t ) = 0 [i ] mr En + r +1 , i = 1,.., n . ,
A := {
% [ j ] mr Er +1 , j = 1,.., r [% ] mr Er +1 . = F ' F F '( x,0) F ( x,0) = f ( x) f ( x) = 0 , t [ / t ] mr En + r +1 . () ()
mr En + r +1 Amr En + r +1 + Bmr Er +1 ,
En + r +1 = AEn + r +1 + BEr +1 , mr . En + r +1 = AEn + r +1 + BEr +1 En + r +1 = AEn + r +1 + BEr +1 + mr +1En + r +1
,
. En + r +1 = AEn + B + mr +1En + r +1 ,
, , . [h] En + r +1 , h h R n {0} R n R r +1 , [h] En . [ Fs ] k- [f] k-, [hi ] En , i = 1,.., n b0 , b1 ,.., br :[h '] = [ Di f ][ hi ] + b0 + b j j ( Fs ) .j =1 r
c := b0 b j Fs (0,0) . u j j =1r
[ Di f ]
[ / xi ]
R n {0} R n R r +1 , j ( Fs )
62
[
Fs (0,0)] , u j u jr ][hi ] c b j [ ] xi u j j =1
[h] [i =1 n
R n {0} R n R r +1 , mr +1En + r +1 . [F] [G] k- [f], En + r , [],[F],[F] . [F] [G] , []. , [F] [F] [G] [G] , [F] [G]. k- k, , :
. 2 k [ f ] mn k- . , k- [f] . : [ F ] En + r k- [f], [f], [G ] En + s . [G] [F]. [] [F] [G]. , , [G] [], [] k-, V [F ] V [H ] . A [K] (r+s)- [f] K ( x, u , v) := F ( x, u ) . , [] k-. [] [] , [] [], [F]. , [G] [F]. k- . .
. 2 [F] r- [ f ] mn , [F] k k r + 1 . [f] (r+2)-, [f] r. : k mn = J [ f ] + mn +1 + V [ F ] . k +1 A := J [ f ] + mn , , dim(mn / A) dimV [ F ] r . l l, mn A dim( En / A) l . l +1 r r + 1 l mn A . Nakayama mn +1 J [ f ] , [f] k (r+2)-. mn = A + V [ F ] mn +1 J [ f ] , mn = J [ f ] + V [ F ] , , cod [ f ] r . , .
. 2 r- mn , , k- k>r+1. 2 : [ F ] En + r r- [ f ] mn , [F] . [F] k- k, k>r+1.
63
, , [F] k- k>r+1. [f] (r+2)-, (r+2)- . k=r+2 .
. 2 mn k- k. : r- [F] [f] k- k. [f] r- r
