Monday 24 June 2013

From Cell to Manifold For LEIBNIZ and JAKOBSON

Continuation of Quantum Theory for Language

From Cell to Manifold


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 Dis notated to ēn.
ēn  - δēen
2 Set of no- boundary-cells becomes cell complex.
3 Some figures are expressed by cell. hn is attaching map.
n-dimensional sphere      S=  ēhn  ēn   
n-dimensional ball          D= ( ēhn-1  ēn )  ēn
Torus                              T2 = ( ē0  h ēēn ) )hē2
3 Grassmann manifold is defined by the following.
Grassmann manifold GR(mn) is all of n-dimensional linear subspaces in m-dimensional real vector space.
                                        S1 = GR2)
4 Canonical vector bundle γ is defined by the following. E is all space. π is projection.
γ= ( EπGR(mn) )
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(31) ).

Tokyo June 2, 2007

[Reference note / December 23, 2008]

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