Complex Manifold Deformation Theory Additional Paper 1 Map between Words
Complex Manifold Deformation Theory
Additional Paper
Conjecture
Words has map.
[View]
(Theorem)
<line 1>Compact Riemann manifolds (M, g), (N, h)
<line 2>Harmonic map from (M, g) to (N, h) f
<line 3>Sectional curvature of (N, h) everywhere non-positive
<line 4>If Ricci curvature of (M, g) is positive, f is constant map.
<line 5>If Ricci curvature of (M, g) is non-positive, f is all geodesic map.
[Impression]
1
Theorem is assumptively considered for words.
From <line 1>, words are assumed as compact Riemann manifolds.
From <line 2>, grammar is assumed as harmonic map.
From <line 3>, for instance, m-dimensional real hyperbolic type space has everywhere -1 sectional curvature.
From <line 4>, orthogonal frame field is considered.
Arbitrary point x
M
Neighborhood of x U
Orthogonal frame field over U {ei}mi=1
is constant over U 
d (ei ) = 0
E() = 0
From <line 5>, geodesic is considered.
is harmonic map 
M
'' = 0
2
Manifold that Ricci curvature of (M, g) is positive is defined as notional word.
Manifold that Ricci curvature of (M, g) is non-positive is defined as functional word.
On notional word and functional word, refer to the next.
#1 Quantification of Quantum / Tokyo May 21, 2004 / Sekinan Research Field of Language
Also refer to the next.
#2 Property of Quantum / Tokyo May 21, 2004 / Sekinan Research Field of Language
Tokyo January 5, 2009
Sekinan Research Field of language
Back to sekinanlogoshome[View]
(Theorem)
<line 1>Compact Riemann manifolds (M, g), (N, h)
<line 2>Harmonic map from (M, g) to (N, h) f
<line 3>Sectional curvature of (N, h) everywhere non-positive
<line 4>If Ricci curvature of (M, g) is positive, f is constant map.
<line 5>If Ricci curvature of (M, g) is non-positive, f is all geodesic map.
[Impression]
1
Theorem is assumptively considered for words.
From <line 1>, words are assumed as compact Riemann manifolds.
From <line 2>, grammar is assumed as harmonic map.
From <line 3>, for instance, m-dimensional real hyperbolic type space has everywhere -1 sectional curvature.
From <line 4>, orthogonal frame field is considered.
Arbitrary point x
MNeighborhood of x U
Orthogonal frame field over U {ei}mi=1
is constant over U d (ei ) = 0E(
) = 0From <line 5>, geodesic is considered.
is harmonic map M'' = 02
Manifold that Ricci curvature of (M, g) is positive is defined as notional word.
Manifold that Ricci curvature of (M, g) is non-positive is defined as functional word.
On notional word and functional word, refer to the next.
#1 Quantification of Quantum / Tokyo May 21, 2004 / Sekinan Research Field of Language
Also refer to the next.
#2 Property of Quantum / Tokyo May 21, 2004 / Sekinan Research Field of Language
Tokyo January 5, 2009
Sekinan Research Field of language
Complex Manifold Deformation Theory 6 Orbit of Word
Complex Manifold Deformation Theory
6 Orbit of Word
Conjecture
Word has orbit.
[View¶]
¶Mathematics is a view in which I freely appreciate objects as if I see flowers, mountains and vigorous port towns at dawn.
1
Complex vector space that has Hermitian inner product V
Lie group of V U(V)
Connected closed Lie group G
Complexification of G GC
Coordinate ring of V A(V)
All the invariant of A(V) 's action to GC A(V)GC
Manifold defined by A(V)GC V//GC
2
Lattice M= Hom(GC, C*)
Dual lattice M*
x = t(1, ..., 1)V
(Theorem)
Necessary and sufficient condition that orbit GCx is closed is that σ = MR is satisfied.
σ is rational polyhedral convex cone defined by one-parameter subgroup and character of GC.
(Theorem)
Topological space of V//GC is quotient space that all the closed GC orbit is divided by equivalent relation.
3
gGC
Orbit of x gx
px(g) := ||gx||2
4
2n dimensional rational manifold X
Lie ring from X to G g
Dual space of g g*
Map: X 
g*
5
(Theorem)
All the closed GC orbit of V is corresponded with
(0) / G one to one.
[Impression¶]
¶ Impression is developed from the view.
1
Language is supposed to be V.
Word is supposed to to be GC.
Meaning minimum is supposed to be x.
Meaning is supposed to be orbit of x.
Meaning has distance ||gx||2 .
2
and 
are supposed to be grammar of language.
3
Under the supposition, meaning and grammar are corresponded one to one by the theorem.
[References]
Orbit of language is an essential concept of Quantum Theory for Language group.
Especially refer to the next.
#1 Quantum Theory for Language Synopsis / Tokyo January 15, 2004
#2 Quantification of Quantum / Toky May 29, 2004
#3 Mirror Language / Tokyo June 10, 2004
#4 Prague Theory 3 / Tokyo January 28, 2005
Related with orbit, distance is also essential concept from early work on Quantum Theory for Language.
Espetially refer to the next.
#5 Distance Theory
Tokyo December 23, 2008
Sekinan Research Field of language
Back to sekinanlogoshome
[View¶]
¶Mathematics is a view in which I freely appreciate objects as if I see flowers, mountains and vigorous port towns at dawn.
1
Complex vector space that has Hermitian inner product V
Lie group of V U(V)
Connected closed Lie group G
Complexification of G GC
Coordinate ring of V A(V)
All the invariant of A(V) 's action to GC A(V)GC
Manifold defined by A(V)GC V//GC
2
Lattice M= Hom(GC, C*)
Dual lattice M*
x = t(1, ..., 1)
V(Theorem)
Necessary and sufficient condition that orbit GCx is closed is that σ = M
R is satisfied.σ is rational polyhedral convex cone defined by one-parameter subgroup and character of GC.
(Theorem)
Topological space of V//GC is quotient space that all the closed GC orbit is divided by equivalent relation.
3
g
GCOrbit of x gx
px(g) := ||gx||2
4
2n dimensional rational manifold X
Lie ring from X to G g
Dual space of g g*
Map
: X g*5
(Theorem)
All the closed GC orbit of V is corresponded with
(0) / G one to one.[Impression¶]
¶ Impression is developed from the view.
1
Language is supposed to be V.
Word is supposed to to be GC.
Meaning minimum is supposed to be x.
Meaning is supposed to be orbit of x.
Meaning has distance ||gx||2 .
2
and are supposed to be grammar of language.3
Under the supposition, meaning and grammar are corresponded one to one by the theorem.
[References]
Orbit of language is an essential concept of Quantum Theory for Language group.
Especially refer to the next.
#1 Quantum Theory for Language Synopsis / Tokyo January 15, 2004
#2 Quantification of Quantum / Toky May 29, 2004
#3 Mirror Language / Tokyo June 10, 2004
#4 Prague Theory 3 / Tokyo January 28, 2005
Related with orbit, distance is also essential concept from early work on Quantum Theory for Language.
Espetially refer to the next.
#5 Distance Theory
Tokyo December 23, 2008
Sekinan Research Field of language
Back to sekinanlogoshome
Complex Manifold Deformation Theory 5 Time of Word
Complex Manifold Deformation Theory
5 Time of Word
Conjecture
Word has time.
[View¶]
¶Mathematics is a view in which I freely appreciate objects as if I see flowers, mountains and vigorous port towns at dawn.
1
Kähler manifold X
Kähler form w
A certain constant c
Cohomology class of w 2πc1(X)
c1(X)>0
Kähler metric g
Real C∞ function f
∫X (ef- 1)wn = 0
Ric(w) -w =f
2
Monge-Ampère equation
(Equation 1)
Use continuity method
(Equation 1-2)
Kähler form w' = w + f
Ric(w') = tw' + (1-t)w'
δ>0
I = { }
3
is differential over t.
Ding's functional Fw
4
(Lemma)
There exists constant that is unrelated with t.
When utis the solution of equation 1-2, the next is satisfied.
Fw(ut)
C
5
Proper of Ding's functional is defined by the next.
Arbitrary constant K
Point sequence of arbitrary P(X, w)K {ui}
(Theorem)
When Fw is proper, there exists Kähler-Einstein metric.
[Impression¶]
¶ Impression is developed from the view.
1
If word is expressed by u , language is expressed by Fw and comprehension of human being is expressed by C, what language is totally comprehended by human being is guaranteed.
Refere to the next paper.
#Guarantee of Language
2
If language is expressed by being properly generated, distance of language is expressed by Kähler-Einstein metric and time of language is expressed by t, all the situation of language is basically expressed by (Equation1-2).
Refer to the next paper.
#Distance Theory
3
If inherent time of word is expressed by t's [δ, 1], dynamism of meaning minimum is mathematically formulated by Monge-Ampère equation.
Refer to the next papers.
#1<For inherent time>
On Time Property Inherent in Characters
#2<For meaningminimum>
From Cell to Manifold
#3<For meaning minimum's finiteness>
Amplitude of Meaning Minimum
Tokyo December 23, 2008
Sekinan Research Field of language
[View¶]
¶Mathematics is a view in which I freely appreciate objects as if I see flowers, mountains and vigorous port towns at dawn.
1
Kähler manifold X
Kähler form w
A certain constant c
Cohomology class of w 2πc1(X)
c1(X)>0
Kähler metric g
Real C∞ function f
∫X (ef- 1)wn = 0
Ric(w) -w =
f2
Monge-Ampère equation
(Equation 1)
Use continuity method
(Equation 1-2)
Kähler form w' = w +
fRic(w') = tw' + (1-t)w'
δ>0
I = {
}3
is differential over t.Ding's functional Fw
4
(Lemma)
There exists constant that is unrelated with t.
When utis the solution of equation 1-2, the next is satisfied.
Fw(ut)
C5
Proper of Ding's functional is defined by the next.
Arbitrary constant K
Point sequence of arbitrary P(X, w)K {ui}
(Theorem)
When Fw is proper, there exists Kähler-Einstein metric.
[Impression¶]
¶ Impression is developed from the view.
1
If word is expressed by u , language is expressed by Fw and comprehension of human being is expressed by C, what language is totally comprehended by human being is guaranteed.
Refere to the next paper.
#Guarantee of Language
2
If language is expressed by being properly generated, distance of language is expressed by Kähler-Einstein metric and time of language is expressed by t, all the situation of language is basically expressed by (Equation1-2).
Refer to the next paper.
#Distance Theory
3
If inherent time of word is expressed by t's [δ, 1], dynamism of meaning minimum is mathematically formulated by Monge-Ampère equation.
Refer to the next papers.
#1<For inherent time>
On Time Property Inherent in Characters
#2<For meaningminimum>
From Cell to Manifold
#3<For meaning minimum's finiteness>
Amplitude of Meaning Minimum
Tokyo December 23, 2008
Sekinan Research Field of language
Back to sekinanlogoshome
Complex Manifold Deformation Theory 4 Amplitude of Meaning Minimum
Complex Manifold Deformation Theory
4 Amplitude of Meaning Minimum
TANAKA Akio
Conjecture
Meaning minimum has finite amplitude.
[View*]
*Mathematics is a view in which I freely appreciate objects as if I see flowers, mountains and vigorous port towns at dawn.
1
Bounded domain of Rm Ω
C∞ function defined in Ω u, F
u, F satisfy the next equation.
F(D2u) = Ψ
D2u is hessian matrix of u.
F is C∞ function over Rm×m .
Open set that includes range of D2u U
U satisfies the next.
(i) Constant λ, Λ
(ii) F is concave.
2
(Theorem)
Sphere that has radius 2R in Ω B2R
Sphere that has same center with B2R and has radius σR in Ω BσR
Amplitude of D2u ampD2u
ampBσRD2u = supBσRD2u - infBσRD2u
0<σ<1
C and e are constant that is determined by dimension m and.
ampBσRD2uCσe(ampBRD2u + supB2R|D| + supB2R |D2| )
[Impression]
1 Meaning minimum is the smallest meaning unit of word. Refer to the reference #2 and #2'.
2 If meaning minimum of word is expressed by BσR, it has finite amplitude in adequate domain.
[References 1 On meaning minimum]
#1 Holomorphic Meaning Theory / 10th for KARCEVSKIJ Sergej
#2 Word and Meaning Minimum
#2' From Cell to Manifold
#3 Geometry of Word
[References 2 On generation of word]
#4 Growth of Word
#5 Generation Theorem
#6 Deep Fissure between Word and Sentence
#7 Tomita's Fundamental Theorem
#8 Borchers' Theorem
#9 Finiteness in Infinity on Language
#10 Properly Infinite
#11 Purely Infinite
[References 3 on distance and mirror on word]
#12 Distance Theory / Tokyo May 5, 2004 / Sekian Linguistic Field
#13 Quantification of Quantum / Tokyo May 29, 2004 / Sekinan Linguistic Field
#14 Mirror Theory / Tokyo June 5, 2004 / Sekinan Linguistic Field
#15 Mirror Language / Tokyo June 10, 2004 / Sekinan Linguistic Field
#16 Reversion Theory / Tokyo September 27, 2004 / Sekinan Linguistic Field
#17 Mirror Theory Group / Tokyo December 9, 2008 / Sekinan Linguistic Field
To be continued
Tokyo December 17, 2008
Back to SekinanLogosHome[View*]
*Mathematics is a view in which I freely appreciate objects as if I see flowers, mountains and vigorous port towns at dawn.
1
Bounded domain of Rm Ω
C∞ function defined in Ω u, F
u, F satisfy the next equation.
F(D2u) = Ψ
D2u is hessian matrix of u.
F is C∞ function over Rm×m .
Open set that includes range of D2u U
U satisfies the next.
(i) Constant λ, Λ
(ii) F is concave.
2
(Theorem)
Sphere that has radius 2R in Ω B2R
Sphere that has same center with B2R and has radius σR in Ω BσR
Amplitude of D2u ampD2u
ampBσRD2u = supBσRD2u - infBσRD2u
0<σ<1
C and e are constant that is determined by dimension m and
.ampBσRD2u
Cσe(ampBRD2u + supB2R|D| + supB2R |D2| )[Impression]
1 Meaning minimum is the smallest meaning unit of word. Refer to the reference #2 and #2'.
2 If meaning minimum of word is expressed by BσR, it has finite amplitude in adequate domain.
[References 1 On meaning minimum]
#1 Holomorphic Meaning Theory / 10th for KARCEVSKIJ Sergej
#2 Word and Meaning Minimum
#2' From Cell to Manifold
#3 Geometry of Word
[References 2 On generation of word]
#4 Growth of Word
#5 Generation Theorem
#6 Deep Fissure between Word and Sentence
#7 Tomita's Fundamental Theorem
#8 Borchers' Theorem
#9 Finiteness in Infinity on Language
#10 Properly Infinite
#11 Purely Infinite
[References 3 on distance and mirror on word]
#12 Distance Theory / Tokyo May 5, 2004 / Sekian Linguistic Field
#13 Quantification of Quantum / Tokyo May 29, 2004 / Sekinan Linguistic Field
#14 Mirror Theory / Tokyo June 5, 2004 / Sekinan Linguistic Field
#15 Mirror Language / Tokyo June 10, 2004 / Sekinan Linguistic Field
#16 Reversion Theory / Tokyo September 27, 2004 / Sekinan Linguistic Field
#17 Mirror Theory Group / Tokyo December 9, 2008 / Sekinan Linguistic Field
To be continued
Tokyo December 17, 2008