Energy Distance Theory / Note 3 / Energy and Functional

 

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^-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 )

2

Norm     |du|

|du|² =∑^m_i,j=1 ∑ⁿ_αβ=1 g^ijh_αβ(u)(δu^α/δxⁱ⁾( 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^-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