18 Holomorphic Meaning Theory 2

Holomorphic Meaning Theory 2

11^th for KARCEVSKIJ Sergej

TANAKA Akio

1

Open set of Cⁿ     Ω

Holomorphic function over Ω     f

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

Open set     U ⊂ Ω

f・A ( U ) is called f ’s divisor class at U.

Divisor class is notated by D ( f, U ).

2

n-dimensional polydisk is defined by the next.

Open set {z | | z_j-a_j | < r,  1≤j≤n }

n-dimensional polydisk is notated by ∆(a, r) (r = (r₁, …, r_n))

∆(0, 1) is notated by ∆.

∆ⁿ = ∆×…×∆  (Number of ∆ is n.)

∆(a, r) and ∆ⁿ are biholomorphic equivalent.

Hartogs figure      T_ε = {(z₁, z₂) ∈∆ ² | |z₁| <ε}

When holomorphic map from Hartogs figure to Ω is always expanded to holomorphic map from ∆ ² to Ω, Ω is called Hartogs pseudo-convex.

3

C is Hartogs pseudo-convex.

Cⁿ is Hartogs pseudo-convex.

Holomorphic open set is Hartogs pseudoconvex.

4

Subharmonic function is defined by the next.

Open set at complex plane     Ω

Semicontinuous function that is valued at [-∞, ∞)     ψ : Ω → [-∞, ∞)

 ⊂ Ω

ψ(z ) ≤ (z + re^iθ)dθ

5

Plurisubharmonic function is defined by the next.

Open set at complex plane     Ω

Semicontinuous function that is valued at [-∞, ∞)     ψ : Ω → [-∞, ∞)

(z, ω) ∈Ω×Cⁿ

Function     ψ( z+ζω )

When  ψ( z+ζω ) is subharmornic as ζ ‘s function, ψ( z+ζω ) is called plurisubharmonic function.

Set of all the plurisubharmonic functions      PSH (Ω)

6

What Ω is pseudoconvex is defined by the next.

Continuous plurisubharmonic function     ψ : Ω → R

Arbitrary c∈R

Ω_ψc : = {z∈Ω |ψ(z) < c }

Ω_ψc is relatively compact in Ω .

7

Pseudoconvex open set     Ω

H² (Ω, Z) = {0}

Open subset of Ω     U

g ∈A(U)

Element of A(U)    f

When V(g) is closed set of Ω, there exists D ( f, U ).∋g.

8

Locally finite open ball     B_j = B (p_j, R_j)

Family of B_j     { B_j }^∞_{j = 1}

Ω = ∪^∞_{j = 1} B_j

B_j ∩V(g) ≠Ø ⇒ B_j ⊂ U

g_j ∈A(B_j) is defined by the next.

g_j = g | B_j  (B_j ∩V(g) ≠ Ø)

g_j = 1    (B_j ∩V(g) = Ø) 

g_jk ∈A(B_j∩B_k) : = g_j / g_k  (B_j∩B_k ≠ Ø)

B_j∩B_k is convex and simply connected.

g_jk has not zero point.

(j, k) has one to one correspond with branch u_jk of logg_jk

u_ijk over B_i∩ B_j∩B_k is defined by the next.

u_ijk : = u_ij + u_jk+ u_ki

9

Language is defined by the next.

Meaning minimum : = B_j   

Word : = g_jk

Sentence : = u_ijk

[References]

Reversion Analysis Theory / Tokyo June 8, 2008

Reversion Analysis Theory 2 / Tokyo June 12, 2008

Holomorphic Meaning Theory / Tokyo June 15, 2008

Tokyo June 19, 2008

Sekinan Research Field of Language

www.sekinan.org