Wednesday 18 June 2014

Reversion Analysis Theory 2 / 12 June 2008



1
Open set of Cn     Ω
Closed subset of Ω     X
Arbitrary point of X     x0
Neighborhood of xat Cn     U
Set of all the holomorphic functions over Ω     A (Ω)
System of functions     {fα}αΛ(U)
= {zU | fα (z) = 0, αΛ}
X is called analytic subset
{fα}αΛ is called local defining functions over U.
Element of (U)     F
When F satisfies F | X f | |, function f over X is called holomorphic function†.
2
graded differential form over Ω
Element of C(Ω)’     u
IJ uIJdzIdJuIJ = sgn( )sgn( uI’J’
IJ   Multiple index from natural number 1 to n
When longitude of IJ is constant pqu is called (pq) type differential form.
Set of (pq) type differential form is notated C pq (Ω).
(1, 0) type complex exterior differentiation operator  : C pq (Ω)  C p+1, q (Ω)
(0, 1) type complex exterior differentiation operator  : C pq (Ω)  C pq+1 (Ω)
∂ IJ uIJdzIdJ) = IJk dzkdzIdJ
IJ uIJdzIdJ) = IJk dkdzIdJ
3
L : = {zC| = … = zn-m = 0}
Holomorphic function over ΩL     f = {zn-m+1, …, zn}
W : = {zC| ( 0, …, 0, zn-m+1, …, zΩL}
Holomorphic function over W      (z) : = f ( 0, …, 0, zn-m+1, …, z)
C class function ρ: W  [0, 1]
supp (ρ– 1 ) L = Ø
supp ρ∂Ω= Ø     
ρW ‘s trivial expansion to Ω    
 C∞ (Ω)
 | Ωf    
Therefore
u | Ω= 0 .     (1)
H pq (Ω) : = Ker pq (Ω) / Im pq (Ω)
H pq (Ω) is called  cohomology of type (pq).
4
(i)
Serre’s condition
H 0, q (Ω) ={ 0 }  ( 1 ≤ ≤ n-1 )
(ii)
Arbitrary z0∂Ω
(iii)
Sequence pμ in Ω that is convergent to z0, there exists f (Ω).
From (i) (ii) (iii)
μ→∞ | f (pμ) | = ∞      (2)
5
From (1) and (2), solution on the domain and the equation is expanded to mathematical formality of word, i.e. language.
Space in which word and sentence is generated : = Ω
The space is called language space. Notation is LS.
Base meaning that becomes root of word : = x0 and sequence pμ that is convergent to xin Ω
Additional meaning† : =  sequence pμ
Word and sentence, i.e. language : = f A ( Ω )
Language in LS is considered at μ→∞ | f (pμ) | = ∞.

Tokyo June 12, 2008


[Postscript June 19]
On holomorphic, refer to the next.

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