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

Energy Distance Theory / Note 2 / Heat and Diffusion

Energy Distance Theory

Note 2

Heat and Diffusion

TANAKA Akio

1 Heat equation

Time     t

Situation     x

Temperature of s

²u / x²     (k ; constant)

2 High dimensional heat equation

 = ku     (k ; constant)

 is Laplacian.

3 Diffusion equation

Time     t

Situation     x

Density of minute particles

 = div ( ku )     (k ; constant)

4 Assumption of heat equation

Assumption     k = 1

 = u

5 Initial value problem

Space     Rⁿ

Heat equation      = u

Initial time     t = 0

Temperature distribution of initial time     u₀ ( x )

Transition of temperature distribution is expressed by the next.

Initial condition  = u  x∈Rⁿ , t > 0 )

Initial value     x, 0 ) = u₀ ( x )  (x∈Rⁿ )

The upper two formulas are called initial value problem.

6 Delta function

(i) δ (x) = 0

(ii) dx = 1

7 Fundamental solution of initial value problem

Function     U ( x, y, t )

  = _xU

lim_t↘0 U ( x, y, t ) =δ (x-y)

_x is Laplacian of variable x.

8 Probability density

Particle is situated by the next.

t = 0, probability 1, point y

Probability density of the particle that has Brownian motion over x- axis, time t and point x     U ( x, y, t )

9 Heat kernel

U ( x, y, t ) = K ( x-y, t )

Function K ( x, t ) is called heat kernel.

10 Hausdorff dimension

Arbitrary figure in space Rⁿ     S

Sequence of n-dimensional sphere     B₁, B₂, B₃, …    

S is covered by the sequence B_k that diameter is below δ.

H_α, δ( S ) : = inf _{diam ( Bk ) <δ} (diam(B_k))^α

H_α( S ) : = lim_k→0 H_α, δ( S )

H_α( S ) is called figure S’s α dimensional Hausdorff outer measure.

To be continued

Tokyo September 15

Sekinan Research Field of Language

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     l : [0, 1] → R³

Longitude of l     L ( l ) = dt

2

Surface     S

Curve combines A and B in S     l

Coordinate of S     φ : U → S

Coordinate of U     x₁, x₂

φ = (φ₁, φ₂, φ₃ )

A =φ ( x₀ )

B =φ ( x₁ )

3

Curve in S     l : [0, 1] → R³

Curve on U    x ( t )

Ω(x₀, x₁) = { l : [0, 1] → R³ | l (0 ) = x₀, l (1 ) = x₁ }

x(t)∈Ω(x₀, x₁)

l ( t ) =φ ( x ( t ) )

x ( 0 ) = x₀

x ( 1 ) = x₁

L ( l ) = dt  =  dt

g_ij 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 g_i,j (x(t))xˑⁱ(t)xˑ^j(t)dt

6

2 E ( x, xˑ ) ≥ (L ( x, xˑ ) )²

7

Theorem

For x∈Ω(x₀, x₁), 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

At least three elements for language universals

Supposition 1:
Three elements for language universals

For language universals, now I suppose at least three elements being  based from mathematical description.

Three elements for language universals are energy, dimension and distance.

The most fundamental element is energy. By this energy, all the movements and changes occur in language.

vide: 

• Edward Sapir, Language, 1921
• Substantiallity

All the languages are located at a certain dimension in space. By this dimension, confusions in language are averted.

vide:

• True-false problem of the Crete

In language, all the movements and changes inevitably make distance occurred. By this distance, important phases of language are clearly defined.

vide:

• Distance Theory 

...........................................................................................................................

Supposition 2:

Mathematical description for three elements of language universals

Energy, dimension and distance can be describe by mathematical writing.

Energy in language is now preparatory description til now.

vide:

• Energy of Language / Stochastic Meaning Theory
• Energy and Distance / Energy Distance Theory
• Energy and Functional / Energy Distance Theory
• Potential of Language / Floer Homology Language

For dimension, definite results are presented being aided by arithmetic geometry.

vide:

• Three Conjectures for Dimension, synthesis and Reversion with Root and Supplement

For distance, its vast and vagueness of the concept can not be grasped up. But related papers of mine are probably the most in number.

vide:

• Distance / Direct Succession of Distance Theory / Distance Theory Algebraically Supplemented
• Distance of Word / Complex Manifold Deformation Theory

............................................................................................................................

This paper is not finished.

Tokyo

27 February 2015

SIL

The days of Sekinan Library

TANAKA Akio

Sekinan Library was opened at Tachikawa, Tokyo in April 1986. Till March of that year I was a student of the post graduate course for cultural sciences and also worked at a night course of the high school for keeping daily life. At university I studied the history of Buddhism of ancient Japan mainly reading Japanised Chinese writings while learning the history of European linguistics typically represented by The Linguistic Circle of Prague in 1920s. Buddhism was taught from Prof. KAWASAKI Tsuneyuki and linguistics was from Prof. CHINO Eiichi. In those days I had been concerned with the presentation of historical text by Chinese characters which was not any inflections in sentence, so usually called isolated language, which grammar only controlled by word order.

I wondered that what is word order, how the grammar changes in the non-inflation word. All were in the utterly dark world because European linguistics had not made up the developed isolated language like classical Chinese. But in China especially in the era of Qing dynasty, many linguists were studied Chinese classics using vast methods historically formed since circa 500 B.C.

At March 1986, I determined that study theme was limited in the linguistic study of ideogram like Chinese character, for the study of which a new perspective would be appeared from our Oriental region still now using ideogram like China and Japan. From April I started my study while teaching modern and classical Chinese at Sekinan Library. At night frequently called as the lecturer of Buddhism and Chinese of civic education. At that time the most interesting theme of the study was ancient Chinese characters inscribed on oracle bones and tortoise shells, which already showed the perfect system of ideogram and had been studied in the Qing dynasty's linguists, in which I was fascinated by WANG Guowei's work Quantangjilin. 

Over decade after, in 2003 I could first written a paper on diagram focused on inherent time in Chinese characters. The title was " On Time Property Inherent in Characters" and its direct succession, "Quantum Theory for Language" which was presented at a conference opened at Nara in December 2003.

References
1.

1. Meaning Minimum On Roman Jakobson, Sergej Karcevskij and CHINO Eiichi
2. 40 years passed from I read WANG Guowei

2.

1. On Time Property Inherent in Characters
2. Quantum Theory for Language

3.

1. Early Work. Sekinan Library
2. Chronicle. Sekinan Library
3. Work. SIL

Tokyo
26 February 2015
SIL

Early Work from Sekinan Library

Early Work 
from Sekinan Library

1. On Time Property Inherent in Characters
2. Quantum Theory for Language
3. Distance Theory
4. Prague Theory
5. Reversion Theory

Sekinan Library from 1986

Sekinan Library from 1986

Sekinan Library is the research site on language from 1986 at Tokyo, where I started language study mainly related with Chinese classical philology called "Small Study", the most fundamental study base of Chinese characters using texts of Chinese philosophical classics.
After the deepening down the classical study, from about millennium days, I started the youth's dream, investigating language universals inherited from the Linguistic Circle of Prague in 1920s' results.

Milestone Paper
On Time Property Inherent in Characters
Distance Theory
Prague Theory
Meaning Minimum On Roman Jakobson, Sergej Karcevskij and CHINO Eiichi

Referential Essay
Road to Language Universals
Tachikawa, Youth Days

Sekinan Library: sekinanlibrary.weebly.com

Nihonbashi Tokyo