Energy in language 2008-2009

30/04/2021 13:10

Energy in language 2008-2009

01/05/2019 19:32

Energy in language

2008
 
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Energy in language is now preparatory description til 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

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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     riΛ
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     UVn )
3
Subspace     PΛ, N ( U )
Volume of PΛ, N ( U )     WΛ, N ( U )
4
Adiabatic operation      ( UVn ) →  ( U’V’n’ )
Starting state of γPΛ, N
Ending state of γPΛ’, N
Map of time development    f
5
Volume of PΛ’, N U’ )     WΛ’, N ( U’ )
Volume of (PΛ, N ( U ) ) is equivalent to WΛ, N ( U ).
(PΛ, N ( U ) ) is subspace of PΛ’, N U’ )
WΛ, N ( U ) ≤ WΛ’, N ( U’ )
6
Equilibrium state     ( UVn )
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 ( UVn ) = k log WΛ, N ( U ) , (k ; arbitrary constant)
7
Phase space     2n- dimension
Differential 2-form    ω
Local coordinate     qipi
ω = ∑ni=1d qi, ∧dpi
ω is called symplectic form.
2n- dimensional manifold     M
Pair    (Mω)
(Mω) that satisfies the next is called symplectic manifold.
(i) dω = 0
(ii) ω≠0
Phase space is expressed by symplectic manifold.
8
Hamiltonian system
Coordinate    ( qp ) = (q1, …, qnp1, …, pn )
Phase space     R2n
C1 class function     = (qpt )
 = ( 1≤n )
 = ( 1≤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     : [0, 1] → R3
Longitude of l     L ( ) = dt
2
Surface     S
Curve combines A and B in S     l
Coordinate of     φ : U → S
Coordinate of     x1x2
φ = (φ1, φ2, φ3 )
=φ ( x0 )
=φ x1 )
3
Curve in S     : [0, 1] → R3
Curve on U    x ( )
Ω(x0x1) = { l : [0, 1] → R(0 ) = x0l (1 ) = x}
x(t)∈Ω(x0x1)
l ( ) =φ ( ( t ) )
x ( 0 ) = x0
( 1 ) = x1
L ( ) = dt   dt
gij is Riemann metric.
4
Longitude is defined by the next.
L ( x, xˑ )  =  dt
5
Energy is defined by the next.
E ( x, xˑ )  = I,j gi,j (x(t))i(t)j(t)dt
6
2 E ( x, xˑ ) ≥ (L ( x, xˑ ) )2
7
Theorem
For xΩ(x0x1), 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     (Mg) , (Nh)
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 : → 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),  xM
Measure defined on from Riemannian metric g    μg
Energy     E(u) = e(u)dμg
3
is compact.
Space of all u     . C(MN)
Functional     E : C(MN) → 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     (MwM)
Poincaré duality     < . , . >
Product     <V1°V2V3> = V1V2V3)
(MwM) 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

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Tokyo
30 April 2019

 



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