Hyperbolic Language. Connection of Words 2012

Hyperbolic Language

Connection of Words

TANAKA Akio

0.
[Preface]
From Distance to Pseudo-Kobayashi-Distance

1.
C is complex plane.
 is unit disk which centre is the origin of C.
z, w are the two points of  .
Hyperbolic distance  between z and w are defined by the next. ,  .

2.
M is complex manifold.
x, y are arbitrary points of M.
f_v is finite sequence of regular curve.
Point z_v is  .
 ,  .
 .
{  } is called regular chain.
Kobayashi pseudodistance d_M is defined by the next.

 .

3.
[Interpretation on 2.]
 := Meaning minimum of word.
d_M := Distance of word.
M:= Word.

4.
[Definition]
When d_M becomes distance function, M is called Kobayashi hyperbolic.
When d_M becomes complete distance, M is called complete Kobayashi hyperbolic.

5.
When M =  is satisfied at d_M , d_M 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.

12.
[References]
<On meaning minimum of word>From Cell to Manifold / Cell Theory / Tokyo June 2, 2007
Amplitude of Meaning Minimum / Complex Manifold Deformation Theory / Tokyo December 17, 2008
<On distance of word>
Distance Theory /Tokyo May 5, 2004 
Distance of Word / Complex Manifold Deformation Theory / Tokyo November 30, 2008
<On connection of words>
Quantum Theory for Language / Tokyo January 15, 2004
<On hyperbolicity>
Reflection of Word / Complex Manifold Deformation Theory / Tokyo December 7, 2008
Boundary of Words / Topological Group language Theory / Tokyo February 12, 2009

13
[History]
Distance of language Historical Review

14.
[Appendix]
The Time of Language / Tokyo January 10, 2012 

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