Research report: "The w - the germ of function defined on the manifold r-reticular" ppsx

ω
r
ω f ∈
ε(n) Diff
r
(n) r
f ∈ ε(n) ω
Diff
r
(n)
1. Introduction
“ f
0 0
f 0 ”
[5]
[2]
f ∈ ε(n) ω Diff (n)
ω
K|
x
1
|
x
1
K 
0 {x
1
= 0}
f ∈ ε(n) ω Diff
r
(n)

ω
f ∈ ε(n) ω Diff
r
(n)
2. Preliminaries
ε(n, p) R
n
R
p
ε(n) ε(n, 1) r ∈ N, 0 ≤ r ≤ n I = {1, , r}
1
- Rec eived 30/09/2005, in revised form 20/01/2006.
X
i
{(x
1
, x
2
, , x
n
) ∈ R
n
: x
i
= 0} i ∈ I P (I)
I
X = {X
σ
: X
σ

=

i∈σ
X
i
, X
φ
= IR
n
, σ ∈ P(I)}
N
o
Diff
r
(n) = {Φ : (R
n
, 0) → (R
n
, 0) : Φ(X
σ
) = X
σ
, σ ∈ P(I)},
Φ
2.1. Definition. s f g ∈ ε(n)
s D
γ
f(0) = D
γ
g(0) γ ∈ N

o
| γ |≤ s
s f s f J
s
(f)
s ε(n) J
s
n
J
s
J
s
(f)
2.2. Definition. s f ∈ ε(n) s
Diff
r
(n) g ∈ ε(n) J
s
(f) = J
s
(g)
Φ ∈ Dif f
r
(n) g = f ◦ Φ
2.3. Definition. f ∈ ε(n) ω Diff
r
(n)
g ∈ ε(n) J
∞
(f) J

∞
(g)
Φ ∈ Dif f
r
(n) g = f ◦ Φ
3. The main results
3.1. Theorem. f : (R
n
, 0) → (R, 0) R
n
R
(a) f ω Diff
r
(n)
(b) f ω Diff (n) f

f |
X
σ
f X
σ
ω Dif f(n) ε(n−| σ |) | σ |
σ
(c) m
∞
n
⊂< x
1
∂f
∂x

1
, . . . , x
r
∂f
∂x
r
,
∂f
∂x
r+1
, . . . ,
∂f
∂x
n
> ε(n)
(d) C, α, δ > 0
r

i=1
x
2
i
(
∂f
∂x
i
)
2
+
n


i=r+1
(
∂f
∂x
i
)
2
≥ C x 
α
,  x < δ.
Proof. ⇒ f ∈ ε(n) f ω
Diff
r
(n) g ∈ ε(n) J
∞
(f) J
∞
(g)
Φ ∈ Dif f(n)
f = g ◦ Φ. (3.1)
f ω Diff
r
(n) g ∈ ε(n)
J
∞
(f) J
∞
(g) Φ ∈ Diff
r

(n)
g = f ◦ Φ Φ ∈ Dif f(n) f ω
Diff (n) f

= f |
X
σ
ω
Diff (n) ε(n− | σ |)
| σ |= k k ≤ r f

= f|
x
1
= =x
k
=0
ω
Diff
r
(n) ε(n−k) x = (x
1
, , x
k
) x

= (x
k+1
, , x
n

)
f

(x

) = f

(x
k+1
, , x
n
) = f (0, , 0, x
k+1
, , x
n
) g ∈ ε(n−k)
J
∞
(f)(x

) J
∞
(g)(x

) g(x

) = g(x
k+1
, , x
n

) f

(x

) = f(0, x)
g(x, x

) = f(x, x

) + ϕ(x

)
ϕ(x

) = g(x

) − f

(x

). (3.2)
g ∈ ε(n) g(x
k+1
, , x
n
) ∈ ε(n−k) J
∞
f

(x


) = J
∞
g(x

)
ϕ(x

) ∈ m
∞
n−k
J
∞
(g) = J
∞
(f)
Φ(x, x

) ∈ Dif f
r
(n) f(x, x

) = g ◦ Φ(x, x

)
f(0, x

) = g ◦ Φ(0, x

)

= f (Φ(0, x

)) + g(Φ(0, x

)) − f

(Φ(0, x

))
= g(Φ(0, x

)).
f(Φ(0, x

)) = f

(Φ(0, x

)) f(0, x

) = f

(x

) f

(x

) =
g ◦ Φ(0, x


) f

(x

) ω Diff (n) ε(n−k)
⇒ f ω Dif f(n)
ε(n)
m
∞
n
⊂<
∂f
∂x
1
, ,
∂f
∂x
n
> ε(n), (3.3)
f

= f |
x
1
= =x
k
=0
= f (0, x


)
ω Diff (n) ε(n−k) 0 ≤ k ≤ r
m
∞
n−k
⊂<
∂(0, x

)
∂x
k+1
, ,
∂(0, x

)
∂x
n
> ε(n − k). (3.4)
ϕ(x, x

) ∈ m
∞
n
ϕ(x, x

) ∈< x
1
∂f
∂x
1

, , x
r
∂f
∂x
r
,
∂f
∂x
r+1
, ,
∂f
∂x
n
> ε(n). (3.5)
ϕ(x, x

) ∈ m
∞
n
ϕ(x, x

) =
k

i=1
∂f(x, x

)
∂x
i

ξ
i
(x, x

) +
n

j=k+1
∂f(x, x

)
∂x
j
η
j
(x, x

), (3.6)
ξ
i
(x, x

) η
j
(x, x

) ε(n)
ξ
i
(x, x


) ∈ m
∞
n
, i = 1, 2, . . . , k η
j
(x, x

) ∈ m
∞
n
, j = k + 1, . . . , n.
m
∞
n
⊂<
∂f
∂x
k+1
, ,
∂f
∂x
n
> ε(n) + x
1
x
k
.ε(n). (3.7)
ξ(x, x


) ∈ m
∞
n
g(x, x

) = ξ(x, x

) − ξ(0, x

).
g(x, x

) ∈ m
∞
n
g(0, x

) = 0
g(x, x

) = x
1
x
2
x
k
Q(x, x

) ∈ x
1

x
k
.ε(n).
ξ(x, x

) = ξ(0, x

) + x
1
x
2
x
k
Q(x, x

). (3.8)
ξ(0, x

) ∈ ε(n−k)
ξ(0, x

) =
n

i=k+1
∂f(0, x

)
∂x
i

η
i
(0, x

), (3.9)
f(0, x

) ∈< f(x, x

) > ε(n)
ξ(0, x

) ∈<
∂f
∂x
k+1
, ,
∂f
∂x
n
> ε(n) + x
1
x
k
.ε(n). (3.10)
ξ(x, x

) ∈<
∂f
∂x

k+1
, ,
∂f
∂x
n
> ε(n) + x
1
x
k
.ε(n)
ϕ(x, x

) ∈ m
∞
n
ξ
i
(x, x

)
ξ
i
(x, x

) =
n

l=k+1
∂f
∂x

l
h
l
(x, x

) + x
1
x
k
.h(x, x

), i = 1, 2, . . . , k. (3.11)
ϕ(x, x

) =
k

i=1
∂f
∂x
i

n

l=k+1
∂f
∂x
l
h
l

(x, x

) + x
1
x
k
.h(x, x

)

+
n

j=k+1
∂f
∂x
j
ξ
j
=
k

i=1
x
i
∂f
∂x
i

k


j=1,j=i
x
j

h(x, x

) +
n

j=k+1
∂f
∂x
j

h
j
(x, x

)
k

i=1
∂f
∂x
i
+ ξ
j

.

ϕ(x, x

) ∈< x
1
∂f
∂x
1
, , x
r
∂f
∂x
r
,
∂f
∂x
r+1
, ,
∂f
∂x
n
> ε(n).
⇒ g(x
1
, , x
n
) = (x
2
1
+ +x
2

n
)
2
g ∈ m
4
n
g ∈ m
∞
n
g =
k

i=1
x
i
∂f
∂x
i
ξ
i
+
n

i=k+1
∂f
∂x
i
η
i
≤


k

i=1

x
i
∂f
∂x
i

2
+
n

i=k+1

∂f
∂x
i

2

1
2

k

i=1
ξ

2
i
+
n

i=k+1
η
2
i

1
2
≤ c

k

i=1

x
i
∂f
∂x
i

2
+
n

i=k+1


∂f
∂x
i

2

1
2
,
c

k

i=1
ξ
2
i
+
n

i=k+1
η
2
i

1
2
 x ≤ δ δ > 0

k


i=1

x
i
∂f
∂x
i

2
+
n

i=k+1

∂f
∂x
i

2

1
2
≥
1
c
(x
2
1
+ + x

2
n
)
2
⇔

k

i=1

x
i
∂f
∂x
i

2
+
n

i=k+1

∂f
∂x
i

2

1
2

≥
1
c
 x 
4
⇔

k

i=1

x
i
∂f
∂x
i

2
+
n

i=k+1

∂f
∂x
i

2

≥

1
c
2
 x 
8
.
C =
1
c
2
, α = 8
⇒ g ∈ ε(n) ϕ = g − f ϕ ∈ m
∞
n
f
t
(x) = f(x) + tϕ(x), t ∈ [0, 1]
C, α, δ ≥ 0
r

i=1

x
i
∂f
t
∂x
i

2

+
n

i=r+1

∂f
t
∂x
i

2
≥ C  x 
α
,
 x < δ t ∈ [0, 1]
r

i=1

x
i
∂f
t
∂x
i

2
+
n


i=r+1

∂f
t
∂x
i

2
= G
−ϕ =
G
2
(−ϕ)
G
2
=

k

i=1

x
i
∂f
t
∂x
i

2
+

n

i=k+1

∂f
t
∂x
i

2

2
−ϕ
G
2
.
ξ =

r

i=1

x
i
∂f
t
∂x
i

2

+
n

i=r+1

∂f
t
∂x
i

2

−ϕ
G
2
,
−ϕ =

r

i=1

x
i
∂f
t
∂x
i

2

+
n

i=r+1

∂f
t
∂x
i

2

ξ,
ξ x t η(x, t) =
(η
1
, , η
n
)
η
i
(x, t) = x
2
i

∂f
t
∂x
i


ξ
i
, i = 1, 2, , r
η
j
(x, t) =

∂f
t
∂x
i

ξ
j
, j = r + 1, , n.
−ϕ =
∂f
t
∂x
η(x, t) η
i
x t

∂h
∂t
= η(h(x, t), t),
h(x, 0) = x,
h(x, t) η(0, t) = 0 h(0, t) = 0
h(x, t) : (IR
n

, 0) → (IR
n
, 0) t
h ∈ Dif f
r
(n) f
t
(x) = f (x) + tϕ(x)
∂f
t
∂t
= ϕ(x)
∂
∂t
(f
t
◦ h) =
∂f
t
∂x
◦ h
∂h
∂t
+
∂f
t
∂t
◦ h
=
∂f

t
∂x
◦ h.η(h(x, t)) +
∂f
t
∂t
◦ h
= −ϕ(h(x, t)) +
∂f
t
∂t
◦ h(x, t)
= 0.
f
t
◦ h t
f ◦ h(x, 0) = f(x) = f
1
◦ h(x, 1) = g(h(x, 1)).
f ω Dif f
r
(n) 
3.2. Remark.
(1) r = 0
(2) r = 1 f
(3) r = 1
Remark. ω
Diff
r
(n)

ω K|
x
1
, ,x
r
K|
x
1
, ,x
r
{x
1
= 0}, . . . , {x
r
= 0}.
Acknowledgement
References
[1] V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenca,
, Birkh¨auser, Boston-Base l-Stuttgart, 1985.
[2] N. T. Cuong, N. H. Duc, N. S. Minh, H. H. Vui,
, C. R. Aced. Sc. Paris. 285 (1977).
[3] Michael Demazure, , Printed in Germany, Springer-Verlag,
Berlin Heidelberg 2000.
[4] Nguyen Tien Dai, Nguyen Huu Duc et Fr´ed´eric Pham, `e `e
`e , Memoire de la societe Mathematique de France, Nouvelle
s´erie N
o
6, Suppl´ement au Bulletin de la Soci´et´e Mat´ematique de France, (3) 109 (1981).
[5] John N. Mather, C
∞

, Publica-
tions math´ematiques de l
´
I.H.
´
E.S., 35 (1968), 127-156.
[6] John N. Mather, C
∞
, Ann.
Math., 89 (1969), 254-291.
[7] D. Siersma, , Quart. J. Math. Oxford (2)
32 (1981), 119 - 127.
[8] H. H. Vui, , Thesis of doctorate, 1980.
ω r
ω
f ∈ ε(n) Diff
r
(n) r
f ∈ ε(n) ω
Diff
r
(n)