From Cell to Manifold
From Cell to Manifold
Cell Theory
Continuation of Quantum Theory for Language
For LEIBNIZ and JAKOBSON
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 = ( ē0 ∪h1 ( 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
Sekinan Research Field of Language