Thursday 11 April 2024

Hyperbolic Language Connection of Words 2012

 


Connection of Words
1.
C is complex plane.
 is unit disk which center is the origin of C.
zw are the two points of  .
Hyperbolic distance  between z and w are defined by the next.
 ,  .
2.
M is complex manifold.
xy are arbitrary points of M.
fv is finite sequence of regular curve.
Point zv is  .
 ,  .
 .
 } is called regular chain.
Kobayashi pseudodistance dis defined by the next.

 .
3.
[Interpretation on 2.]
 := Meaning minimum of word.
dM := Distance of word.
M:= Word.
4.
[Definition]
When dM becomes distance function, M is called Kobayashi hyperbolic.
When dM becomes complete distance, M is called complete Kobayashi hyperbolic.
5.
When M =  is satisfied at ddM is equal to Poincaré distance.
6.
X is complex manifold.
M is contained in X as relative compact.

7.
[Definition]
What embedding  is hyperbolic embedding is defined by the next.
M is KObayashi hyperbolic.
Arbitrary boundary points  .
 .
 .
8.
[Theorem,Kwack 1969]
When M is hyperbolicly embedding in X,
What arbitrary regular map  \{0}  is regularly connected to  .
9.
[Interpretation on 6,7,8,9]
X:= Language.
M:= Word.
:= Distance of word.
:= Connection of words.
10.
[Conjecture, Kobayashi]
(i) If d is  , degree d's general hypersurface X of  is Kobayashi hyperbolic.
(ii)If d is  ,  \  is hyperbolicly embedded in 
 .
11.
[Interpretation on 10.]
X:= Language.
d:= Hierarchy of language.


Tokyo
February 3, 2012
At the Last Wintry Day of Classical Calendar in Japan
Sekinan Research Field of Language

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