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 r_a in AdS₅ space and returns r_b in AdS₅ space.

r_a and r_b are equal longitude from D3 brane.

7

<Classical solution> of D3 brane in AdS₅ space 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 r_z.

r_z’ is 1/2 of the longitude from r_a to r_b.

Sketch is below.

8

At the paper Spacetime Symmetry and Escalator Brane in Escalator Language Theory, language goes from r_a to rz and in reverse movement to r_b.

9

At the paper Actual Language and Imaginary Language, <real language> is from r_a to r_z and <imaginary language> is from r_z to r_b.

10

We see real language that is upper side of r_z-r_z’. Imaginary language is hidden under side of r_z-r_z’ 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 r_a ( or r_b ) to D3 brane in AdS₅ space. 

[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>

ds² = e^2U(y)η_mndx^mdxⁿ + dy²

Branes exist at y = 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 <y = –πR> to <y = 0> and from <y = 0> to <y = +πR>.

Now “from <y = –πR> to <y = 0>” is called <minus side> and “from <y = 0> to <y = +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

<Aurora Theory>

[AT11] Distance and Time  Language Multi-Time Conjecture 3  5^th Time for KARCEVSKIJ

<Aurora Time Theory>

[ATT3] Enlarged Distance Theory

[ATT4] Opened Time and Closed Time

<Algebraic Linguistics Linguistic Result>

1 Deep Fissure between Word and Sentence

2 From Cut and Glue Dimension to Krull Dimension  8^th 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]

Ⅰ

(Preissmann’s theorem)

When (M, g) is connected Riemann manifold and sectional curvature of M is always K_M < 0, non-trivial commutative subset of functional group of M, π₁(M) always becomes infinite cyclic group.

Ⅱ

Preparatory proposition for Preissmann’s theorem

(Proposition 1)

When (M, g) and (N, h) are compact Riemann manifold and N is non-positive curvature K_N≤0, arbitrary continuous map f ∈C⁰(M, N) is free homotopic with harmonic map u_∞∈C^∞(M, N).

(Proposition 2)

When M is compact Riemann manifold, Ricci tensor of M is positive semidefinite Ric^M≥0 , N is non-positive curvature K_N≤0, and harmonic map is u : M→N,  the next is concluded.

When N is negative curvature K_N<0, u is constant map or map of u coincides with map of closed geodesic line.

Ⅲ

Consideration for the theorem and propositions

1

m-dimensional C^∞ class manifold     M

Point of M     x

Tangent space of x     T_xM

Inner product of T_xM   g_x

Coordinate neighborhood of M     U

Local coordinate system of U     (x¹, …, x^m)

Function     g_ij : g_x ( (∂/∂xⁱ)_x, (∂/∂x^j)_x), 1≤i, j≤m

g_ij is C^∞ class function over U.

Family of inner product     g = {g_x}_x∈M

g is called Riemannian metric.

When M has g, (M, g) is called Riemannian manifold.

2

Riemann manifold      (M, g)

M’s C^∞ class vector field    X (M)   

Linear connection of M     ∇

X, Y, Z∈X(M)

What ∇ and X, Y, Z uniquely satisfy the next is called Levi-Civita connection.

(i) Xg(Y, Z) = g(∇_XY, Z) + g(Y, ∇_XZ)

(ii) ∇_XY -∇_YX = [X, Y]

3

m-dimensional Riemann manifold (M, g)    M

Levi-Civita connection of M     ∇

X, Y∈X

R(X, Y) : = ∇_X∇_Y - ∇_Y∇_X - ∇[X, Y]

Map R : = X(M) ×X(M)×X(M) → X(M)

R(X, Y, Z) : = R(X, Y)Z

R is called curvature tensor of M.

4

x∈M

2-dimensional subspace of tangent space T_xM     σ

σ’s normal orthogonal basis on g_x     {v, w} {v’, w’}

K(v, w) = R(x)(v, w, w, v) = g_x(R(x)(v, w)w, v)

v’ = cosθv + sinθw, w’ = ∓sinθv±cosθw  (double sign directly used)

K(σ) : = R(x)(v, w, w, v) = 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.

W is covered by sequence of M.1, M2, M3, ….

Lower limit of all the covering ∑_k (diam ( M_k ) ) ^α is expressed by H_α,δ ( W ).

H_α,δ ( W ) = inf _{diam ( Mk ) <δ} (diam ( M_k ) ) ^α

2

H_α ( W ) : = lim_δ →0 H_α,δ ( W )

3

Word has non-positive real number or ∞ by H_α ( W ).

H_α ( W ) is restricted by measurable sets.

H_α ( W ) is treated as Hausdorff measure.

4

Word is expressed by Hausdorff measure.

Meaning minimum is expressed by lim_δ →0 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]

Amplitude of Meaning Minimum / sekinan.wiki.zoho.com

Complex Manifold Deformation Theory / sekinan.wiki.zoho.com

Energy Distance Theory / Note 4 / Finsler Manifold and Distance

Energy Distance Theory

Note 4

Finsler Manifold and Distance

TANAKA Akio

1

Banach space     E

C^k manifold       M

Point of M     p

Banach space     T_xM

Norm of T_xM     ||  ||_x

Finsler metric is defined by the next.

(i) Topology by ||  ||_x is equal to topology by norm of Banach space.

(ii) Tangent vector bundle     T (M)

Point     p∈M

Coordinate neighborhood of p     (U_α, α),  α : U_α→E

Ψ_α : U_α×E → π^-1(U_α) ⊂T (M)

||| v |||_x : = ||Ψ_α (x, v)||_x , x∈U_α , v∈E

C > 0

1/C ||| v |||_p ≤ ||| v |||_x ≤C ||| v |||_p ,  x∈U_α , v∈E

2

Banach manifold M that has Finsler metric     Finsler manifold M

Longitude of M     L (σ) : = ∫^b_a ||σ’(t)||dt

p, q∈M

Distance    ρ ( p, q ) : = inf { L (σ) }

Distance space     ( M, ρ )

When ( M, ρ ) is complete distance space, Finsler manifold is called complete.

3

Finsler C^k manifold     M

C^k function over M     f : M → R

Condition (C) is defined by the next.

(i) Subset of M     S

f is boundary over S.

inf_S ||df || = 0

Closure of S     S^-

df = 0 at point p of S^- 

4

Complete Finsler C² manifold     M 

C^k class function     f : M → R 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     L_1,p ( M, R^N )

C^∞ manifold     L_1,p ( M, N )

Distance of L_1,p ( M, R^N )     ρ₀

ρ₀ ( u, v ) = || u – v ||_1,p , u, v ∈ L_1,p ( M, R^N )

Proposition

Finsler manifold (L_1,p ( M, N ) , ||  ||_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