Friday 5 December 2014

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




1 Cell is defined by the following.
 n-dimensional ball Dn has 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. hn is attaching map.
n-dimensional sphere      Sn ē0 hn  ēn   
n-dimensional ball          Dn = ( ē0 hn-1  ēn ) ēn
Torus                              T2 = ( ēh1  ( ē0 ēn ) h2 ē2
3 Grassmann manifold is defined by the following.
Grassmann manifold GR(m, n) is all of n-dimensional linear subspaces in m-dimensional real vector space.
                                        S1 = GR( 2, 1 )
4 Canonical vector bundle γ is defined by the following. E is all space. π is projection.
γ= ( E, π, GR(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 ( GR(3, 1) ).

Tokyo
June 2, 2007

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