Reversion Analysis Theory 2 / June 12, 2008

Reversion Analysis Theory 2

TANAKA Akio

1

Open set of Cⁿ     Ω

Closed subset of Ω     X

Arbitrary point of X     x₀

Neighborhood of x₀ at Cⁿ     U

Set of all the holomorphic functions over Ω     A (Ω)

System of functions     {f_α}_α∈Λ⊂A (U)

X ∩U = {z∈U | f_α (z) = 0, α∈Λ}

X is called analytic subset

{f_α}_α∈Λ is called local defining functions over U.

Element of A (U)     F

When F satisfies F | U ∩X = f | U ∩X |, function f over X is called holomorphic function†.

2

r graded differential form over Ω

Element of C^∞₀ (Ω)’     u

∑_I, J u_IJdz_I∧d_J, u_IJ = sgn( )sgn( ) u_I’J’

I, J   Multiple index from natural number 1 to n

When longitude of I, J is constant p, q, u is called (p, q) type differential form.

Set of (p, q) type differential form is notated C ^p, q (Ω).

(1, 0) type complex exterior differentiation operator ∂ : C ^p, q (Ω) → C ^p+1, q (Ω)

(0, 1) type complex exterior differentiation operator  : C ^p, q (Ω) → C ^p, q+1 (Ω)

∂ ( ∑’_I, J u_IJdz_I∧d_J) = ∑’_I, J∑_k dz_k∧dz_I∧d_J

( ∑’_I, J u_IJdz_I∧d_J) = ∑’_I, J∑_k d_k∧dz_I∧d_J

3

L : = {z∈Cⁿ | z | = … = z_n-m = 0}

Holomorphic function over Ω∩L     f = {z_n-m+1, …, z_n}

W : = {z∈Cⁿ | ( 0, …, 0, z_n-m+1, …, z_n ) ∈Ω∩L}

Holomorphic function over W      (z) : = f ( 0, …, 0, z_n-m+1, …, z_n )

C^∞ class function ρ_W : W → [0, 1]

supp (ρ_W – 1 ) ∩L = Ø

supp ρ_W ∩∂Ω= Ø     

ρ_W ‘s trivial expansion to Ω    

 ∈C^∞ (Ω)

 | Ω∩L = f　　　　

Therefore

u = 

u | Ω∩L = 0 .     (1)

H ^p, q (Ω) : = Ker ∩C ^p, q (Ω) / Im ∩C ^p, q (Ω)

H ^p, q (Ω) is called  cohomology of type (p, q).

4

(i)

Serre’s condition

H ^0, q (Ω) ={ 0 }  ( 1 ≤ q ≤ n-1 )

(ii)

Arbitrary z₀∈∂Ω

(iii)

Sequence p_μ in Ω that is convergent to z₀, there exists f ∈A (Ω).

From (i) (ii) (iii)

_μ→∞ | f (p_μ) | = ∞      (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 p_μ that is convergent to x₀ in Ω

Additional meaning† : =  sequence p_μ

Word and sentence, i.e. language : = f ∈A ( Ω )

Language in LS is considered at _μ→∞ | f (p_μ) | = ∞.

Tokyo June 12, 2008

Sekinan Research Field of Language

[Postscript June 19]

On holomorphic, refer to the next.

Holomorphic Meaning Theory / Tokyo June 15

Holomorphic Meaning Theory 2 / Tokyo June 19