Complex Manifold Deformation Theory Additional Paper 1 Map between Words

Complex Manifold Deformation Theory

Additional Paper

1 Map between Words

TANAKA Akio  

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      xM
Neighborhood of x     U
Orthogonal frame field over U     {e_i}^m_i=1
 is constant over U d (e_i ) = 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

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Complex Manifold Deformation Theory 6 Orbit of Word

Complex Manifold Deformation Theory

6 Orbit of Word

TANAKA Akio   

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     G^C
Coordinate ring of V     A(V)
All the invariant of A(V) 's action to G^C    A(V)^GC
Manifold defined by A(V)^GC     V//G^C
2
Lattice     M= Hom(G^C, C*)
Dual lattice     M*
x = ^t(1, ..., 1)V
(Theorem)
Necessary and sufficient condition that orbit G^Cx is closed is that σ = MR is satisfied.
σ is rational polyhedral convex cone defined by one-parameter subgroup and character of G^C.
(Theorem)
Topological space of V//G^C  is quotient space that all the closed G^C orbit is divided by equivalent relation.
3
g G^C
Orbit of x     gx
p_x(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 G^C 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 G^C.
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

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Complex Manifold Deformation Theory 5 Time of Word

Complex Manifold Deformation Theory

5 Time of Word

TANAKA Akio  

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 (e^f- 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     F_w

4
(Lemma)
There exists constant that is unrelated with t.
When u_tis the solution of equation 1-2, the next is satisfied.
F_w(u_t)C
5
Proper of Ding's functional is defined by the next.
 Arbitrary constant     K 
Point sequence of arbitrary P(X, w)_K     {u_i}

(Theorem)
When F_w 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 F_w 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

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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 R^m      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 R^m×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       B_2R
Sphere that has same center with B_2R and has radius σR in Ω      B_σR
Amplitude of D²u     ampD²u
amp_BσRD²u = sup_BσRD²u - inf_BσRD²u
0<σ<1
C and e are constant that is determined by dimension m and .
amp_BσRD²uCσ^e(amp_BRD²u + sup_B2R|D| + sup_B2R |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 B_σR, 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

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Saijiki of Haiku

Sekinan Research Field of Language

Saijiki of Haiku 

TANAKA Akio 





                                        

Saijiki is the glossary of seasonal words for Haiku composers with illustrative verses. I have read it from my youth and recently I have realised one important fact that the reading style of the saijiki is one of the best ways to read, understand and enjoy the books, in which one is interested. The way is the following mentioned.
The saijiki is classified by the five seasons; the new year, spring, summer, autumn and winter. The each season is further classified by the three small seasons; early, middle and late season. Under the small seasons the glossary of seasonal words are arranged by the classification, such as plants, animal, foods, folklore and so on.
The reader read the most concerned words freely and easily. at that place word's seasonal meaning and illustrative Haiku are describing. This example Haiku is so interesting and sometimes fantastic, because one can fly the different season, time and folklore by one's interest.
So recently I realised that I can read mathematical books alike the saijiki. Many theorems, corollaries and lemmas are alike the glossary of the saijiki. I read them freely and adopt one of them for my study just as one seasonal word is adopted for the composition of Haiku at that time.

Tokyo
7 July 2012
Sekinan Research Field of Language

KURATA Reijiro’s Phrase

Sekinan Research Field of Language

KURATA Reijiro’s Phrase

TANAKA Akio

Impressive phrase in KURATA Reijiro's Mathematics for Everyone, 1970.

            "History of mathematics exists for everyone”

Tokyo

19 October 2012

Sekinan Research Field of Language

Complex Manifold Deformation Theory 3 Uniqueness of Word

Complex Manifold Deformation Theory 

3 Uniqueness of Word

TANAKA Akio  

Conjecture

Word has uniqueness.

[Explanation]
1
Smooth manifold     M
Hermitian metric of M     h    
Tangent bundle of      T_M    
positive definite Hermitian inner product over T_M      h

Local coordinate system     z₁, ..., z_n 

Hermitian symmetric positive definite function's matrix     h_ij    

h = h_ijz_ij
Correspondence differential form     w

w = h_ijdzⁱd_j

When dw = 0, h is called Kller metric and w is called Kllerform.

Complex manifold that has Kller metric     Kller manifold

2

n-dimensional compact Kller manifold     X

Kller metric of X     g

Correspondence Kller form     w

Local coordinate system z₁, ..., z_n 

Ricci curvature of X     R = - log det (g)

Ricci form    Ric(w) = Rdzd

A certain constant     c

When Ric(w) = cw, g is called Kller-Einstein metric.

When c = 0, c₁(X) = 0

When c = -1, Linear bundle ⁿTX 's cobundle is ample. The situation is briefly abbreviate expressed by c₁(X) <0.

3

Compact Kller manifold     X

X satisfies c₁(X) = 0 or c₁(X) <0.
When Kller form's cohomology class is fixed, Kller-Einstein metric exists uniquely.

[Comment]

When word is expressed by compact Kller manifold in adequate condition, word has uniqueness.

[References]

Quantum Theory for Language Group / Sekinan Linguistic Field

Mirror Theory Group / Tokyo December 9, 2008 / Sekinan Linguistic Field

Especially the next is important.

Uniformity / Tokyo December 18, 2004 / Sekinan Linguistic Field

Fixation / Tokyo February 20, 2005 / Sekinan Linguistic Field

To be continued
Tokyo December 11, 2008
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