From Cell to Manifold, On Meaning Minimum hinted by Roman Jakobson / 2 June 2007

Cell Theory

  

From Cell to Manifold

Continuation of Quantum Theory for Language

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