Energy Distance Theory
Energy and Functional
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-1TN
Tangent space of N Tu(x)N
Cotangent vector bundle of M TM*
Map du : M → TM*⊗ u-1TN
Section du ∈Γ(TM*⊗ u-1TN )
|du|2 =∑mi,j=1 ∑nαβ=1 gijhαβ(u)(δuα/δxi)( δuβ/δxj)
Energy density e(u)(x) = 1/2 |du|2(x), x∈M
Measure defined on M from Riemannian metric g μg
Energy E(u) = ∫M e(u)dμg
M is compact.
Space of all u . C∞(M, N)
Functional E : C∞(M, N) → R
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.
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