Cell Theory. Continuation of Quantum Theory for Language. From Cell to Manifold. For LEIBNIZ and JAKOBSON

      

Cell Theory　

Continuation of Quantum Theory for Language

From Cell to Manifold

For LEIBNIZ and JAKOBSON

TANAKA Akio

1 Cell is defined by the following.

 n-dimensional ball Dⁿhas interior that consists of cells. Cell is expressed by Dⁿ - δDⁿ and notated to eⁿ that has no boundary.

δis boundary operator.　

Homomorphism of Dⁿ is notated to ēⁿ.

ēⁿ  - δēⁿ = eⁿ

2 Set of no- boundary-cells becomes cell complex.

3 Some figures are expressed by cell. h_n is attaching map.

n-dimensional sphere      Sⁿ =  ē⁰ ∪h_n  ēⁿ   

n-dimensional ball          Dⁿ = ( ē⁰ ∪h_n-1  ēⁿ ) ∪ ēⁿ

Torus                              T² = ( ē⁰  ∪h₁  ( ē⁰ ∪ēⁿ ) )∪h₂ ē²

3 Grassmann manifold is defined by the following.

Grassmann manifold G^R(m, n) is all of n-dimensional linear subspaces in m-dimensional real vector space.

                                        S¹ = G^R( 2, 1 )

4 Canonical vector bundle γ is defined by the following. E is all space. π is projection.

γ= ( E, π, G^R(m, n) )

5 Here from JAKOBSON Roman ESSAIS DE LINGUISTIQUE GÉNÉRALE, <semantic minimum> is presented.

Now <semantic minimum> is expressed by cell ē³.

6 <Word> is expressed by D².

7 <Sentence> is expressed by Grassmann manifold’s canonical vector bundle γ¹ ( G^R(3, 1) ).

Tokyo June 2, 2007

Sekinan Research Field of Language

[Reference note / December 23, 2008]

Time of Word / sekinan.wiki.zoho.com