Energy Distance Theory Note 1 Energy and Distance

30/01/2016 10:44

Energy Distance Theory Note 1 Energy and Distance

 
Energy Distance Theory
 
Note 1
Energy and Distance
 
TANAKA Akio
 
1
Curve in 3-dimensional Euclidean space   : [0, 1] → 3
Longitude of   ) = dt
2
Surface   S
Curve combines and in   l
Coordinate of     φ → S
Coordinate of   2
φ = ( φ , φ , φ )
φ )
φ )
3
Curve in   : [0, 1] → 3
Curve on   )
Ω )= { : [0,1] → (0) = (1 ) = }
∈ Ω )
)= φ ) )
( 0 )= 0
( 1 ) = 1
) = dt     dt
ij is Riemann metric.
4
Longitude is defined by the next.
x, xˑ     dt
5
Energy is defined by the next.
x, xˑ   =   ∑ I,j i,j )) xˑ xˑ dt
6
x, xˑ ≥ ( x, xˑ ) ) 2
7
Theorem
For ∈ Ω ), the next two are equivalent.
(i) takes minimum value at .
(ii) takes minimum value at .
8
What longitude is the minimum in curve is equivalent what energy is the minimum in curve.
9
Longitude 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