Grassmann Language
TANAKA Akio
1.
is (N+1) dimensional complex space.
is Grassmann manifold that is all the spaces which are made from (k+1) dimensional linear subspace of 
.
Element of (k+1) dimensional exterior space
of 
is called (k+1) vector.
Elements of is given by (k+1) vector of decomposable is givenby the below.
.
is one fixed base of .
is also able to written.
is called Cayley-Plücker-Grassmann coordinates.
is unitary group.
.
.
Hermitian inner product is defined as the below.
.
.
is called norm of 
.
Over
.
.
.
.
.
.
Kähler form is
.
The upper measuring is Kähler measuring.
.
.
is (N+1) dimensional complex space. is Grassmann manifold that is all the spaces which are made from (k+1) dimensional linear subspace of .Element of (k+1) dimensional exterior space
of is called (k+1) vector.Elements of
is given by (k+1) vector of decomposable is givenby the below. . is one fixed base of . is also able to written. is called Cayley-Plücker-Grassmann coordinates. is unitary group. . .Hermitian inner product is defined as the below.
. . is called norm of .Over
... . . .Kähler form is
.The upper measuring is Kähler measuring.
. .
2.(Theorem)
Grassmann manifold has invariant Kähler measuring by operation of .
Its Kähler form is equal to
times of hyperplane section bundle's curvature form over 
that is determined by immersion of Cayley-Plücker-Grassmann coordinates and hermitian norm 
.
Grassmann manifold has invariant Kähler measuring by operation of
.Its Kähler form is equal to
times of hyperplane section bundle's curvature form over that is determined by immersion of Cayley-Plücker-Grassmann coordinates and hermitian norm .
3.Grassmann language model is supposed as the below.
Word is expressed by grassmann manifold.
Word has location and distance, that are expressed by Cayley-Plücker-Grassmann coordinates and hermitian norm .
Word is expressed by grassmann manifold.
Word has location and distance, that are expressed by Cayley-Plücker-Grassmann coordinates and hermitian norm
.
4.
References are below.
<On distance of word>Distance Theory /Tokyo May 5, 2004
Distance of Word / Complex Manifold Deformation Theory / Tokyo November 30, 2008
<On manifold>
From Cell to Manifold/ Cell Theory / June 2, 2007
[ Cell Theory / Continuation of Quantum Theory for Language / For LEIBNIZ and JAKOBSON ]
References are below.
<On distance of word>Distance Theory /Tokyo May 5, 2004
Distance of Word / Complex Manifold Deformation Theory / Tokyo November 30, 2008
<On manifold>
From Cell to Manifold/ Cell Theory / June 2, 2007
[ Cell Theory / Continuation of Quantum Theory for Language / For LEIBNIZ and JAKOBSON ]
Tokyo
March 6, 2012Sekinan Research Field of Language
March 6, 2012Sekinan Research Field of Language
No comments:
Post a Comment