Energy in language
2008
-----------------------------------------------------------------------------------------------------------------------Energy in language is now preparatory description til now.
vide:
- Energy of Language / Stochastic Meaning Theory
- Energy and Distance / Energy Distance Theory
- Energy and Functional / Energy Distance Theory
- Potential of Language / Floer Homology Language
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Stochastic Meaning Theory 4
Energy of Language
For ZHANG Taiyan and Wenshi 1908
TANAKA Akio
1
Domain Λ∈R3
Substantial particles N-number m-mass
Particles are assumed to Newton dynamics.
Place coordinate of particle i in N-number particles ri∈Λ
Momentum of particle pi∈R3
State at a moment γ = (r1, …, rN, p1, …, pN)
Set of state γ PΛ, N ≃ΛN ×R3N⊂R6N
PΛ, N is called phase space.
2
Volume V
Particles n- mol
Energy U
Parameter space E
Point of E ( U, V, n )
3
Subspace PΛ, N ( U )
Volume of PΛ, N ( U ) WΛ, N ( U )
4
Adiabatic operation ( U, V, n ) → ( U’, V’, n’ )
Starting state of γ∈PΛ, N
Ending state of γ∈PΛ’, N
Map of time development f
5
Volume of PΛ’, N ( U’ ) WΛ’, N ( U’ )
Volume of f (PΛ, N ( U ) ) is equivalent to WΛ, N ( U ).
f (PΛ, N ( U ) ) is subspace of PΛ’, N ( U’ )
WΛ, N ( U ) ≤ WΛ’, N ( U’ )
6
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 WΛ, N ( U ) WΛ’, N’ ( U’ )
Operation of logarithm of equilibrium state at phase space S ( U, V, n ) = k log WΛ, N ( U ) , (k ; arbitrary constant)
7
Phase space 2n- dimension
Differential 2-form ω
Local coordinate qi, pi
ω = ∑ni=1d qi, ∧dpi
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.
8
Hamiltonian system
Coordinate ( q, p ) = (q1, …, qn, p1, …, pn )
Phase space R2n
C1 class function H = (q, p, t )
=
( 1≤i ≤n )
=
( 1≤i ≤n )
9
An assumption from upper 8
H : = Sentence
q : = Place where word exists
p : = Momentum of word
t : = Time at which sentence is generated
10
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
1
Curve in 3-dimensional Euclidian space l : [0, 1] → R3
Longitude of l L ( l ) = 
dt
dt
2
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 )
3
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 = 
dt
dt = dt
gij is Riemann metric.
4
Longitude is defined by the next.
L ( x, xˑ ) = 
dt
dt
5
Energy is defined by the next.
E ( x, xˑ ) =
∑I,j gi,j (x(t))xˑi(t)xˑj(t)dt
∑I,j gi,j (x(t))xˑi(t)xˑj(t)dt
6
2 E ( x, xˑ ) ≥ (L ( x, xˑ ) )2
7
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.
8
What longitude is the minimum in curve is equivalent what energy is the minimum in curve.
9
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
1
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 )
2
Norm |du|
|du|2 =∑mi,j=1 ∑nαβ=1 gijhαβ(u)(δuα/δxi)( δuβ/δxj)
Energy density e(u)(x) = 1/2 |du|2(x), x∈M
Measure defined on M from Riemannian metric g μg
Energy E(u) = ∫M e(u)dμg
3
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
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).
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]
First designed on <energy of language> at
Tokyo April 29, 2009
Newly planned on further visibility at
Tokyo June 16, 2009
Newly planned on further visibility at
Tokyo June 16, 2009
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Tokyo
30 April 2019
