Energy in language 2008

Energy in language is now preparatory description till now.

vide:

1. Energy of Language / Stochastic Meaning Theory
2. Energy and Distance / Energy Distance Theory
3. Energy and Functional / Energy Distance Theory
4. Potential of Language / Floer Homology Language

1

Domain     Λ∈R³Substantial particles     N-number m-mass Particles are assumed to Newton dynamics.Place coordinate of particle i in N-number particles     r_i∈ΛMomentum of particle     p_i∈R³State at a moment     γ = (r₁, …, r_N, p₁, …, p_N)Set of state γ     P_{Λ, N} ≃Λ^N ×R^3N⊂R^6NP_{Λ, N} is called phase space.

2

Volume     VParticles     n- molEnergy     U  Parameter space     EPoint of E     ( U, V, n )3Subspace     P_{Λ, N} ( U )Volume of P_{Λ, N} ( U )     W_{Λ, N} ( U )4Adiabatic 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    f5Volume 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’ )6Equilibrium 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)7Phase space     2n- dimensionDifferential 2-form    ωLocal coordinate     q_i, p_iω = ∑ⁿ_i=1d q_i, ∧dp_iω is called symplectic form.2n- dimensional manifold     MPair    (M, ω)(M, ω) that satisfies the next is called symplectic manifold.(i) dω = 0(ii) ωⁿ ≠0Phase space is expressed by symplectic manifold.8Hamiltonian systemCoordinate    ( q, p ) = (q₁, …, q_n, p₁, …, p_n )Phase space     R²ⁿC¹ class function     H = (q, p, t ) = ( 1≤i ≤n ) = ( 1≤i ≤n )9An assumption from upper 8H : = Sentenceq : = Place where word existsp : = Momentum of wordt : = Time at which sentence is generated10Equilibrium state of sentence     HAnother equilibrium state of sentence     H’Adiabatic process of language     H → H’Entropy of language     SH → H’ ⇔ S (H ) ≤ S (H’ ) [References]Warp Theory / Tokyo October 24, 2004Quantum Warp Theory Warp / Tokyo December 31, 2005 

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1

Curve in 3-dimensional Euclidian space     l : [0, 1] → R³Longitude of l     L ( l ) = dt2Surface     SCurve combines A and B in S     lCoordinate of S     φ : U → SCoordinate of U     x₁, x₂φ = (φ₁, φ₂, φ₃ )A =φ ( x₀ )B =φ ( x₁ )3Curve in S     l : [0, 1] → R³Curve 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 ) = x₀x ( 1 ) = x₁L ( l ) = dt  =  dtg_ij is Riemann metric.4Longitude is defined by the next.L ( x, xˑ )  =  dt5Energy is defined by the next.E ( x, xˑ )  = ∑_I,j g_i,j (x(t))xˑⁱ(t)xˑ^j(t)dt62 E ( x, xˑ ) ≥ (L ( x, xˑ ) )²7TheoremFor x∈Ω(x₀, x₁), the next two are equivalent.(i) E takes minimum value at x.(ii) L takes minimum value at x.8What longitude is the minimum in curve is equivalent what energy is the minimum in curve.9Longitude L is corresponded with distance in Distance Theory. 

[References]Reversion Theory / Tokyo September 27, 2004

1

Riemannian manifold     (M, g) , (N, h)C^∞ class map u : M → NTangent vector bundle of N     TNInduced vector bundle on M from TN     u^-1TNTangent space of N     Tu(x)NCotangent vector bundle of M     TM*Map      du : M → TM*⊗ u^-1TN    Section     du ∈Γ(TM*⊗ u^-1TN )2Norm     |du||du|² =∑^m_i,j=1 ∑ⁿ_αβ=1 g^ijh_αβ(u)(δu^α/δxⁱ⁾( δu^β/δx^j)Energy density     e(u)(x) = 1/2  |du|²(x),  x∈MMeasure defined on M from Riemannian metric g    μ_gEnergy     E(u) =　∫_M e(u)dμ_g3M 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 

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

 

Tokyo

30 April 2019
ENSILA