Energy Distance Theory
Note 1
Energy and Distance
TANAKA Akio
1
Curvein 3-dimensional Euclidian space l : [0, 1] → R 3
Longitudeof l L ( l ) = dt
2
Surface S
Curvecombines A and B in S l
Coordinateof S φ : U → S
Coordinateof U x 1 , x 2
φ = ( φ 1 , φ 2 , φ 3 )
A = φ ( x 0 )
B = φ ( x 1 )
3
Curvein S l : [0, 1] → R 3
Curveon U x ( t )
Ω ( x 0 , x 1 )= { l : [0,1] → R 3 | l (0) = x 0 , l (1 ) = x 1 }
x ( t ) ∈ Ω ( x 0 , x 1 )
l ( t )= φ ( x ( t ) )
x ( 0 )= x 0
x ( 1 ) = x 1
L ( l ) = dt = dt
g ij isRiemann metric.
4
Longitudeis defined by the next.
L ( x, xˑ ) = dt
5
Energyis defined by the next.
E ( x, xˑ ) = ∑ I,j g i,j ( x ( t )) xˑ i ( t ) xˑ j ( t ) dt
6
2 E ( x, xˑ ) ≥ ( L ( x, xˑ ) ) 2
7
Theorem
For x ∈ Ω ( x 0 , x 1 ), the nexttwo are equivalent.
(i) E t akesminimum value at x .
(ii) L takes minimum value at x .
8
What longitudeis the minimum in curve is equivalent what energy is the minimum in curve.
9
Longitude L is corresponded with distancein 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
Source: Energy Distance Theory Note 1 Energy and Distance / 31 August 2008
Source 2: SIT Paper 2
Source 2: SIT Paper 2
[Note, 20 December 2014]
