Energy in language 2008

01/05/2019 19:30

Energy in language

2008

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Energy in language is now preparatory description til now.
vide:

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Stochastic Meaning Theory 4

Energy of Language

For ZHANG Taiyan and Wenshi 1908

TANAKA Akio

Domain Λ∈R³

Substantial particles N-number m-mass

Particles are assumed to Newton dynamics.

Place coordinate of particle i in N-number particles r_i∈Λ

Momentum of particle p_i∈R³

State at a moment γ = (r₁, …, r_N, p₁, …, p_N)

Set of state γ P_{Λ, N}≃Λ^N ×R^3N⊂R^6N

P_{Λ, N}is called phase space.

Volume V

Particles n- mol

Energy U

Parameter space E

Point of E ( U, V, n )

Subspace P_{Λ, N} ( U )

Volume of P_{Λ, N} ( U ) W_{Λ, N} ( U )

Adiabatic operation ( U, V, n ) → ( U’, V’, n’ )

Starting state of γ∈P_{Λ, N}

Ending state of γ∈P_{Λ’, N}

Map of time development f

Volume of P_{Λ’, N}( U’ ) W_{Λ’, N} ( U’ )

Volume of f (P_{Λ, N} ( U ) ) is equivalent to W_{Λ, N} ( U ).

f (P_{Λ, N} ( U ) ) is subspace of P_{Λ’, N}( U’ )

W_{Λ, N} ( U ) ≤ W_{Λ’, N} ( U’ )

Equilibrium state ( U, V, n )

Another equilibrium state ( U’, V’, n’ )

Two volume of equilibrium states are seemed to be one state at phase space W_{Λ, N} ( U ) W_{Λ’, N’} ( U’ )

Operation of logarithm of equilibrium state at phase space S ( U, V, n ) = k log W_{Λ, N} ( U ) , (k ; arbitrary constant)

Phase space 2n- dimension

Differential 2-form ω

Local coordinate q_i, p_i

ω = ∑ⁿ_i₌₁d q_i, ∧dp_i

ω is called symplectic form.

2n- dimensional manifold M

Pair (M, ω)

(M, ω) that satisfies the next is called symplectic manifold.

(i) dω = 0

(ii) ωⁿ≠0

Phase space is expressed by symplectic manifold.

Hamiltonian system

Coordinate ( q, p ) = (q₁, …, q_n, p₁, …, p_n )

Phase space R²ⁿ

C¹ class function H = (q, p, t )

( 1≤i ≤n )

An assumption from upper 8

H : = Sentence

q : = Place where word exists

p : = Momentum of word

t : = Time at which sentence is generated

Equilibrium state of sentence H

Another equilibrium state of sentence H’

Adiabatic process of language H → H’

Entropy of language S

H → H’ ⇔ S (H ) ≤ S (H’ )

[References]

Warp Theory / Tokyo October 24, 2004

Quantum Warp Theory Warp / Tokyo December 31, 2005

To be continued

Tokyo July 24, 2008

Sekinan Research Field of Language

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Energy Distance Theory

Note 1

Energy and Distance

TANAKA Akio

Curve in 3-dimensional Euclidian space l : [0, 1] → R³

Longitude of l L ( l ) =

Surface S

Curve combines A and B in S l

Coordinate of S φ : U → S

Coordinate of U x₁, x₂

φ = (φ₁, φ₂, φ₃ )

A =φ ( x₀ )

B =φ ( x₁ )

Curve in S l : [0, 1] → R³

Curve on U x ( t )

Ω(x₀, x₁) = { l : [0, 1] → R³| l (0 ) = x₀, l (1 ) = x₁}

x(t)∈Ω(x₀, x₁)

l ( t ) =φ ( x ( t ) )

x ( 0 ) = x₀

x ( 1 ) = x₁

L ( l ) =

dt =

g_ijis Riemann metric.

Longitude is defined by the next.

L ( x, xˑ ) =

Energy is defined by the next.

E ( x, xˑ ) =

∑_I,jg_i,j(x(t))xˑⁱ(t)xˑ^j(t)dt

2 E ( x, xˑ ) ≥ (L ( x, xˑ ) )²

Theorem

For x∈Ω(x₀, x₁), the next two are equivalent.

(i) E takes minimum value at x.

(ii) L takes minimum value at x.

What longitude is the minimum in curve is equivalent what energy is the minimum in curve.

Longitude L is corresponded with distance in Distance Theory.

[References]

Distance Theory / Tokyo May 4, 2004

Property of Quantum / Tokyo May 21, 2004

Mirror Theory / Tokyo June 5, 2004

Mirror Language / Tokyo June 10, 2004

Guarantee of Language / Tokyo June 12, 2004

Reversion Theory / Tokyo September 27, 2004

Tokyo August 31, 2008

Sekinan Research Field of Language

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Energy Distance Theory

Note 3

Energy and Functional

TANAKA Akio

Riemannian manifold (M, g) , (N, h)

C^∞ class map u : M → N

Tangent vector bundle of N TN

Induced vector bundle on M from TN u^-1TN

Tangent space of N Tu(x)N

Cotangent vector bundle of M TM*

Map du : M → TM*⊗ u^-1TN

Section du ∈Γ(TM*⊗ u^-1TN )

Norm |du|

|du|² =∑^m_i,j₌₁∑ⁿ_αβ₌₁ g^ijh_α_β(u)(δu^α/δxⁱ⁾( δu^β/δx^j⁾

Energy density e(u)(x) = 1/2 |du|²(x), x∈M

Measure defined on M from Riemannian metric g μ_g

Energy E(u) =　∫_Me(u)dμ_g

M is compact.

Space of all u . C^∞(M, N)

Functional E : C^∞(M, N) → R

[Additional note]

1 Vector bundle TM*⊗ u^-1TN is compared with word.

2 Map du is compared with one time of word.

3 Norm |du| is compared with distance of tome.

4 Energy E(u) compared with energy of word.

5 Functional E is compared with function of word.

[Reference]

Substantiality / Tokyo February 27, 2005

Substantiality of Language / Tokyo February 21, 2006

Stochastic Meaning Theory 4 / Energy of Language / Tokyo July 24, 2008

Tokyo October 18 2008

Sekinan Research Field of Language

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Floer Homology Language

TANAKA Akio

Note1
Potential of Language

¶ Prerequisite conditions
Note 6 Homology structure of Word

1
(Definition)
(Gromov-Witten potential)

2
(Theorem)
(Witten-Dijkggraaf-Verlinde-Verlinde equation)

3
(Theorem)
(Structure of Frobenius manifold)
Symplectic manifold     (M, wM)
Poincaré duality     < . , . >
Product     <V1°V2, V3> = V1V2V3(

)
(M, wM) has structure of Frobenius manifold over convergent domain of Gromov-Witten potential.

4
(Theorem)
Mk,β (Q1, ..., Qk) =

N(β) expresses Gromov-Witten potential.

[Image]
When Mk,β (Q1, ..., Qk) is identified with language, language has potential N(β).

[Reference]

Quantum Theory for language / Synopsis / Tokyo January 15, 2004

First designed on <energy of language> at

Tokyo April 29, 2009
Newly planned on further visibility at
Tokyo June 16, 2009

Sekinan Research Field of language

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Tokyo

30 April 2019

ENSILA

Saturday, 21 May 2022