Saturday, 28 April 2018

Holomorphic Meaning Theory 2 11th for KARCEVSKIJ Sergej



11th for KARCEVSKIJ Sergej


1
Open set of Cn     Ω
Holomorphic function over Ω     f
Set of all the holomorphic function over Ω     A Ω )
Open set     ⊂ Ω
fU ) is called ’s divisor class at U.
Divisor class is notated by D ( f).
2
n-dimensional polydisk is defined by the next.
Open set {z | | zj-aj | < r,  1j}
n-dimensional polydisk is notated by (ar) (r = (r1, …, rn))
(0, 1) is notated by .
××∆  (Number of ∆ is n.)
(ar) and are biholomorphic equivalent.
Hartogs figure      Tε = {(z1z2 2 | |z1| <ε}
When holomorphic map from Hartogs figure to Ω is always expanded to holomorphic map from ∆ 2 to ΩΩ 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 )  (+ re)
5
Plurisubharmonic function is defined by the next.
Open set at complex plane     Ω
Semicontinuous function that is valued at [-∞, ∞)     ψ : Ω  [-∞, ∞)
(z, ωΩ×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 cR
Ωψc : = {zΩ |ψ(z) < c }
Ωψc is relatively compact in Ω .
7
Pseudoconvex open set     Ω
H(ΩZ) = {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).g.
8
Locally finite open ball     Bj B (pjRj)
Family of Bj     { B}= 1
Ω = 1 Bj
BV(gØ  B U
gj A(Bj) is defined by the next.
g= g | Bj  (BV(g) ≠ Ø)
g= 1    (BV(g) = Ø) 
gjk A(BjBk) : = gj / gk  (BjBk ≠ Ø)
BjBis convex and simply connected.
gjk has not zero point.
(j, k) has one to one correspond with branch ujk of loggjk
uijk over Bi BjBis defined by the next.
uijk : = uij + ujk+ uki
9
Language is defined by the next.
Meaning minimum : = Bj   
Word : = gjk
Sentence : = uijk

[References]

Tokyo June 19, 2008

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