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
June 2, 2007
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