Saturday 28 February 2015

Distance Theory Algebraically Supplemented / Bend

Distance Theory Algebraically Supplemented
Brane Simplified Model <Continuation of Escalator Language Theory>


1
Bend

TANAKA Akio

1
Language is expressed by< a pair of quanta> that consists of <quantum> and <anti-quantum>.
2
Quantum and anti-quantum have reverse direction.
3
Quantum and anti-quantum are in the shape of strings that are separated parallel longitude L apart.
4
Quantum and anti-quantum are in <AdS5 ( 5-dimensional anti-deSitter ) space>.
5
Left side of quantum and anti-quantum is on <D3 brane>.
6
Quantum starts from ra in AdS5 space and returns rb in AdSspace.
rand rare equal longitude from D3 brane.
7
<Classical solution> of D3 brane in AdSspace gives quantum and anti-quantum <a line of string> bending near r = 0 that is the location of D3 brane.
Quantum’s nearest point to D3 brane is rz.
rz’ is 1/2 of the longitude from ra to rb.
Sketch is below.




8
At the paper Spacetime Symmetry and Escalator Brane in Escalator Language Theory, language goes from ra to rz and in reverse movement to rb.
9
At the paper Actual Language and Imaginary Language, <real language> is from ra to rz and <imaginary language> is from rz to rb.
10
We see real language that is upper side of rz-rz’. Imaginary language is hidden under side of rz-rz’ in our life.
11
At the paper Mirror Language, imaginary language is mirror language of real language.
12
Distance of language is regarded as the longitude from ra ( or rb ) to D3 brane in AdSspace. 

[Reference]
Escalator Language Theory / 2 Turning Point of Time / Tokyo December 22, 2006

Tokyo October 17, 2007

Sekinan Research Field of Language

Distance Theory Algebraically Supplemented / Distance / Direct Succession of Distance Theory


Distance Theory Algebraically Supplemented
Brane Simplified Model <Continuation of Escalator Language Theory>

2
Distance
Direct Succession of Distance Theory

TANAKA Akio

1
Metric model of 5-dimensional spacetime is expressed below from Randall and Sundrum (1999). <RS model>
ds= e2U(y)ηmndxmdxn + dy2
Branes exist at = 0 and y = d.
Our world is regarded as brane y = 0.
U(y) is called <warp factor>.
2
Using <orientifold> of circle ( radius R )’s identification, y is expressed by <line segment> that scales from 0 to ±πR.   
Distance is defined in <line segment>.
According to <line segment>, <warp factor> of <RS model> is measured in bulk spacetime of 5 dimensional Anti-deSitter space.
3
Distance in <line segment> is expressed from <= –πR> to <y = 0> and from <y = 0> to <= +πR>.
Now “from <= –πR> to <y = 0>” is called <minus side> and “from <y = 0> to <= +πR>” is called <plus side>.
Values of <warp factor> are same at <minus side> and <plus side>.
4
In <Distance Theory Algebraically Supplemented> (abbreviation; DTAS), word is regarded by <warp factor>’s value.



5
Word has distance at <minus side> and <plus side>.
6
Now distance at <plus side> is called distance of <real language> and distance at <minus side> is called distance of <mirror language>.

[References]
<Quantum Theory for Language Map>
2.1.3 Distance Theory
2.1.4 Reversion Theory
2.2.1 Prague Theory
2.3  Warp Theory
2.4  Time Theory
4.3.1 Guarantee of Language
4.3.2 Place where Quantum of Language Exists
4.3.3 Actual Language and Imaginary Language
4.5.1 Mirror Language
4.5.2 Mirror Theory
[AT11] Distance and Time  Language Multi-Time Conjecture 3  5th Time for KARCEVSKIJ
<Aurora Time Theory>
[ATT3] Enlarged Distance Theory
<Algebraic Linguistics Linguistic Result>
Deep Fissure between Word and Sentence
From Cut and Glue Dimension to Krull Dimension  8th For KARCEVSKIJ Sergej

Tokyo October 26, 2007
Sekinan Research Field of Language

Energy Distance Theory / Conjecture 2 / Geometry of Word


Energy Distance Theory

Conjecture 2
Geometry of Word

TANAKA Akio


[Conjecture]
Word is infinite cyclic group.

[Explanation]
(Preissmann’s theorem)
When (Mg) is connected Riemann manifold and sectional curvature of M is always KM < 0, non-trivial commutative subset of functional group of Mπ1(M) always becomes infinite cyclic group.
Preparatory proposition for Preissmann’s theorem
(Proposition 1)
When (Mg) and (Nh) are compact Riemann manifold and N is non-positive curvature KN0, arbitrary continuous map f C0(MN) is free homotopic with harmonic map uC(M, N).
(Proposition 2)
When M is compact Riemann manifold, Ricci tensor of M is positive semidefinite RicM≥0 , is non-positive curvature KN0, and harmonic map is u : MN the next is concluded.
When N is negative curvature KN<0, u is constant map or map of u coincides with map of closed geodesic line.
Consideration for the theorem and propositions
1
m-dimensional C class manifold     M
Point of M     x
Tangent space of x     TxM
Inner product of TxM   gx
Coordinate neighborhood of     U
Local coordinate system of U     (x1, …, xm)
Function     gij : gx ( (/xi)x, (/xj)x), 1i, jm
gij is C class function over U.
Family of inner product     g = {gx}xM
g is called Riemannian metric.
When M has g, (Mg) is called Riemannian manifold.
2
Riemann manifold      (Mg)
M’s C class vector field    (M)   
Linear connection of M     
XYZX(M)
What  and XYuniquely satisfy the next is called Levi-Civita connection.
(i) Xg(YZ) = g(XYZ) + g(YXZ)
(ii) XY -YX = [XY]
3
m-dimensional Riemann manifold (Mg)    M
Levi-Civita connection of M     
XYX
R(XY) : = XY - YX - [XY]
Map R : = X(M×X(M)×X(M X(M)
R(XYZ) : = R(XY)Z
R is called curvature tensor of M.
4
xM
2-dimensional subspace of tangent space TxM     σ
σ’s normal orthogonal basis on gx     {vw} {v’w’}
K(vw) = R(x)(vwwv) = gx(R(x)(vw)wv)
v’ = cosθv + sinθww’ = sinθv±cosθw  (double sign directly used)
K(σ) : = R(x)(vwwv) = R(x)(v’w’w’v’)
K(σ) is called sectional curvature.

[References]
<Example of word’s infinite cycle is shown by the bellow.>
On Time Property Inherent in Characters / Hakuba March 28, 2003
Prague Theory / Tokyo October 2, 2004
Prague Theory 3 / Tokyo January 28, 2005
TOMONAGA’s Super Multi-Time Theory / Tokyo January 25, 2008
<On minimum unit of meaning, refer to the next.>
Cell Theory / From Cell to Manifold / Tokyo June 2, 2007
Reversion Analysis Theory / Tokyo June 8, 2008
Reversion Analysis Theory 2 /Tokyo June 12, 2008
Holomorphic Meaning Theory / Tokyo June 15, 2008
Holomorphic Meaning Theory 2 / Tokyo June 19, 2008
Energy Distance Theory / Conjecture 1 / Word and Meaning Minimum / Tokyo September 22, 2008

To be continued
Tokyo November 23, 2008
Sekinan Research Field of Language


Postscript
[Reference November 30, 2008]
Distance of Word / November 30. 2008 / Sekinan.wiki.zoho.com

Energy Distance Theory / Conjecture 1 / Word and Meaning Minimum

Energy Distance Theory

Conjecture 1
Word and Meaning Minimum

TANAKA Akio


1
Word is expressed by arbitrary figure W in space Rn.
Meaning minimum is expressed by n-dimensional sphere M that has diameter below δ.
is covered by sequence of M.1, M2, M3, .
Lower limit of all the covering ∑k (diam ( Mk ) ) α is expressed by Hα,δ ( W ).
Hα,δ ( W ) = inf diam ( Mk ) <δ (diam ( Mk ) ) α
2
Hα W ) : = limδ Hα,δ ( W )
3
Word has non-positive real number or  by Hα W ).
Hα W ) is restricted by measurable sets.
Hα W ) is treated as Hausdorff measure.
4
Word is expressed by Hausdorff measure.
Meaning minimum is expressed by limδ M .
5
Word has α dimension.
Meaning minimum has n dimension.
6
Language consists of word and meaning minimum.
7
Language has dimension.

[References]
<On meaning minimum>
Cell Theory / From Cell to Manifold / Tokyo June 2, 2007
<On place of meaning>
Stochastic Meaning Theory 3 / Place of Meaning / Tokyo July 11, 2008
<On confirmation of meaning>
Stochastic Meaning Theory 2 / Period of Meaning / Tokyo June 27, 2008

Tokyo September 22, 2008
Sekinan Research Field of Language


 [Reference December 22, 2008]

Energy Distance Theory / Note 4 / Finsler Manifold and Distance


Energy Distance Theory

Note 4
Finsler Manifold and Distance

TANAKA Akio

1
Banach space     E
Ck manifold       M
Point of M     p
Banach space     TxM
Norm of TxM     ||  ||x
Finsler metric is defined by the next.
(i) Topology by ||  ||is equal to topology by norm of Banach space.
(ii) Tangent vector bundle     T (M)
Point     pM
Coordinate neighborhood of p     (Uαα),  α UαE
Ψα : Uα×→ π-1(UαT (M)
||| v |||x : = ||Ψα (xv)||xUα , vE
> 0
1/C ||| v ||| ||| v |||x C ||| v |||,  xUα , vE
2
Banach manifold M that has Finsler metric     Finsler manifold M
Longitude of M     L (σ) : = ∫ba ||σ’(t)||dt
p, qM
Distance    ρ ( pq ) : = inf { L (σ) }
Distance space     ( M, ρ )
When ( Mρ ) is complete distance space, Finsler manifold is called complete.
3
Finsler Ck manifold     M
Cfunction over M     M  R
Condition (C) is defined by the next.
(i) Subset of M     S
is boundary over S.
infS ||df || = 0
Closure of S     S-
df = 0 at point p of S- 
4
Complete Finsler C2 manifold     M 
Cclass function     M → satisfies condition ( C ).
Theorem
Connected component of M     M0
When f is boundary from below, f has minimum value at M0.
5
1 > m/p , m = dim M
Banach space     L1,p MRN )
C manifold     L1,p MN )
Distance of L1,p MRN )     ρ0
ρu, v ) = ||  v ||1,u, v ∈ L1,p MRN )
Proposition
Finsler manifold (L1,p MN ) , ||  ||1,p ) is complete.

[Note]
Word is expressed by closed manifold in Banach space.
Distance is expressed by Finsler metric.
[References]
Distance Theory / Tokyo May 4, 2008
Reversion Theory / Tokyo September 27, 2008

To be continued
Tokyo November 7, 2008
Sekinan Research Field of Language


Postscript
[Reference November 30, 2008]
Distance of Word / November 30. 2008 / Sekinan.wiki.zoho.com