SRFL News: March 2013

Saturday, 30 March 2013

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 x

M
Neighborhood of x U
Orthogonal frame field over U {e_i}^m_i₌₁

is constant over U

(e_i ) = 0
E(

) = 0
From <line 5>, geodesic is considered.

is harmonic map

' = 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 = M

R 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     wA certain constant     cCohomology class of w     2πc₁(X)c₁(X)>0
Kähler metric     gRealC^∞ function     f∫_X (e^f- 1)wⁿ= 0Ric(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_2Rand 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_BσRD²u

Cσ^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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Friday, 29 March 2013

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