Duality of Language
TANAKA Akio
1. Theorem(Poincaré duality)
For oriented compact n-dimension topological manifold M,
biliear form
is exact,
and 
become dual space each other.
♦
For oriented compact n-dimension topological manifold M,
biliear form
is exact,
and become dual space each other.♦
2. Interpretation
Word := MBoundary of language := pDual language := and Language := 
Word := MBoundary of language := pDual language :=
and Language :=
3. Explanation
Word is compact and has dimension.
Language has dimension.
Language has boundary.
Boundary makes dual language.
Word is compact and has dimension.
Language has dimension.
Language has boundary.
Boundary makes dual language.
4. References
Distance Theory
Mirror Theory
Guarantee of Language
Distance Theory
Mirror Theory
Guarantee of Language
Tokyo
25 May 2012
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