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Saturday, 28 February 2015

Energy Distance Theory / Note 3 / Energy and Functional


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
Sekinan Research Field of Language

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