Energy Distance Theory
Note 3
Energy and Functional
TANAKA Akio
1
Riemannian manifold (M, g) , (N, h)
C^{∞} class map u : M → N
Tangent vector bundle of N TN
Induced vector bundle on M from TN u^{-1}TN
Tangent space of N Tu(x)N
Cotangent vector bundle of M TM*
Map du : M → TM*⊗ u^{-1}TN
Section du ∈Γ(TM*⊗ u^{-1}TN )
2
Norm |du|
|du|^{2} =∑^{m}_{i,j}_{=1 }∑^{n}_{αβ}_{=1} g^{ij}h_{α}_{β}(u)(δu^{α}/δx^{i}^{)}( δu^{β}/δx^{j}^{)}
Energy density e(u)(x) = 1/2 |du|^{2}(x), x∈M
Measure defined on M from Riemannian metric g μ_{g}
Energy E(u) = ∫_{M }e(u)dμ_{g}
3
M is compact.
Space of all u . C^{∞}(M, N)
Functional E : C^{∞}(M, N) → R
[Additional note]
1 Vector bundle TM*⊗ u^{-1}TN 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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