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Saturday, 28 February 2015

Energy Distance Theory / Note 1 / Energy and Distanc


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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