Abstract from Wellspring, Energy and Brownian Motion of Language 2008. Related Papers added 2020

20/06/2020 22:55

Abstract from Wellspring, Energy and Brownian Motion of Language 2008. Related Papers added 2020

4
From Orient, WANG Yinzhi lead me the meaning of word  especially by analysis of functional words in Chinese classics.
Jingzhuan shici is a compilation of his analysis by which I tried the concept of
positive and negative on the generation of word. The result is shown by early work Quantification of Quantum, from which the concepts of actual language and imaginary language are emerged. Early work of showing the concepts are Guarantee of Language, Mirror Language and Reversion Theory and the like.
5
ZHANG Bingling is the another peak of China.
Wenshi is the work I have tried to get understanding over and over again from my youth. My paper is tried as Energy of Language and Brown Motion as Language. Two papers are targeted to the energy inherent in language.

Quantification of Quantum

TANAKA Akio

1         /Hanyu/ Chinese in English what is classified in isolating language by linguistic typology has characters in each which there are one syllable and one meaning in principle.
2         Here I assume that one Chinese character is a quantum of language which has a unit of one meaning.
3         In Chinese language /shici/notional word in English is assumed a positive quantum which has a positive unit in general.
4         In Chinese language /xuci/ functional word in English is assumed a negative quantum which has a negative unit in general.
5         A positive quantum has a propelling energy, by which a quantum can go forward to a next quantum.
6         A negative quantum has an absorbing energy, by which a quantum can absorb, stop or change a next quantum.
7         When a positive quantum stands close by a negative quantum, it is forced to stop or to change own propelling course.
8         Summing up above mentioned, a quantum is the smallest unit of language.
9         A quantum appears to the language world for being requested to send a new meaning.
10     A quantum has an energy which maintains the position in the language world. If a quantum has not energy, a new meaning can never appears to the language world.
11     An energy is united to a meaning inseparably.
12     A quantum disappears from the language world being in the no used situation.
13     /Bu/ not in English is originally meant a part of a flower. But this meaning disappears early days in the Chinese language. Later /bu/ is used as a functional word. 
14     In Chinese language a notional word and a functional word is not clearly classified.
There are mutual extension and conversion in the Chinese language.
15     When a positive quantum ends a role of investing a notional meaning to the language world, an energy what a quantum has originally disappears.
16     Non-energy quantum is pressurized from other energetic positive quantum. As a consequence non-energy quantum accepts a negative energy internally. A negative quantum comes into existence.
17     A negative quantum has a negative energy, therefore a positive quantum is forced to stop or change the own propelling course.
18     Two or over two quantum form a new construction.
19     A positive quantum transfers a meaning to the next quantum.
20     A negative quantum stops or changes the propelling course of the front quantum.
In other words, a negative quantum accepts the energy of a front positive quantum.
21     A positive quantum propels on an orbit.
When a negative quantum changes an orbit of a positive quantum, new construction come into the language world.
22     An orbit exists on a floor of multilayer construction.
23     A negative quantum changes the flour of a positive quantum to another floor.
24     A positive quantum propels on an orbit of a floor of a multilayer construction.
25     Therefore when a negative quantum stands at first on an orbit, a positive quantum is originally standing front of the first-standing negative quantum.
26     A negative quantum is assumed that it does not invent a meaning to the language world, but it leads a positive quantum to the own area that is vacant in non-energy situation.
27     A negative quantum newly has energy of absorption, which was born by the vacant and continually being pressured situation among the positive quanta.

Tokyo May 29, 2004
For the Memory of Hakuba August 23, 2003
Sekinan Research Field of Language

Postscript
[References / January 29, 2008]

Property of Quantum /Tokyo May 21, 2004
Prague Theory / Tokyo October 2, 2004

Quantum Theory for Language Map
[Note for basis]
In this paper, on binary relations of , <notional word – functional word> and <positive quantum –negative quantum>, refer to the next.
Clifford Algebra / Note 7 / Creation Operator and Annihilation Operator / January 29, 2008

[Reference / December 22, 2008]

Reflection of Word / Sekinan.wiki.zoho.com

[Reference / January 1, 2009]

Orbit of Word / sekinan.wiki.zoho.com

[Reference / January 5, 2009]

Map between Words / sekinan.wiki. zoho.com

Read more: https://srfl-paper.webnode.com/news/quantification-of-quantum/

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 =∑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
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


Read more: https://srfl-paper.webnode.com/products/energy-distance-theory-note-3-energy-and-functional/

Preparation for the energy of language

TANAKA Akio

The energy of language seems to be one of the most fundamental theme for the further step-up  study on language at the present for me. But the theme was hard to put on the mathematical description. Now I present some preparatory  papers written so far.

  1. Potential of Language / Floer Homology Language / 16 June 2009
  2. Homology structure of Word / Floer Homology Language / Tokyo June 16, 2009
  3. Amplitude of meaning minimum / Complex Manifold Deformation Theory / 17 December 2008
  4. Time of Word / Complex Manifold Deformation Theory / 23 December 2008

Tokyo
3 April 2015
Sekinan Library

Read more: https://srfl-paper.webnode.com/news/preparation-for-the-energy-of-language1/

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Language as Brown Motion
 
 
[A]
1
Abstractive space     Ω
σ additive family that consists of subset of Ω     F
Measure that is defined over F     P
P satisfies P (Ω ) = 1. 
P      probability measure over ( Ω, F )
Ω      sample space     
( Ω, F , P )     Probability space
Element of Ω     sample ω
Element of F     event A
Probability that event A occurs     probability P ( A )
Real number valued Borel measurable function over Ω     random variable X = X ( ω )
Random variable is integrable.
Mean (Expectation) of X     E[X] = ∫Ω X ( ω ) P ( dω )
2
Measurable space     ( S, S )
X : ( Ω, F ) → ( S, S )
X is measurable.
X      S-value random variable.
Random variable     X1, …, Xd
X : = (X1, …, Xd )     Rd-value random variable
3
Rd-value random variable     X
E[X i 2] < ∞
E[(X - E[X])2]     variance
4
S-value random variable.     X
PX : = P ( X ∈A ), A∈S     distribution
5
Real number space     R
Borel set family over R    B ( R )
Probability measure over ( R, B ( R ) )     μ
6
Rd-value random variable     X
ψX (ξ ) : = E[eiξ・X], ξ∈Rd　　　　　characteristic function
7
Lebesgue measure     dx
Mean     m∈R
Variance     v >0
Measure over R     μ ( dx ) = e -(x-m)2 / 2v dx /     Gauss distribution ( normal distribution)
8
(2p – 1 ) !! : = (2p – 1 ) ・(2p – 3 ) … 3・1
E[X2p] = (2p – 1 ) !! v p     moment of X
9
Event     A, B∈F
When (A∩B) = P(A) P(B), A and B are independent each other.
10
Integrable and independent random variable     X, Y
Product XY     integrable
E[XY] = E[X]E[Y]
11
Time     t
t ∈[0, ∞)
Family of Rd-value random variable ≥    X = ( Xt ) t ≥ 0     d-dimensional stochastic process
∀ω∈Ω
When Xt (ω) is continuous as function of t., d-dimensional stochastic process is called to be continuous.
12
σ additive family     Ft
Ft  ⊂F
0 ≤ s ≤ t
F s  ⊂Ft
(Ft  ) = (Ft  ) t ≥ 0   increase information system
13
d-dimensional stochastic process     X = ( Xt ) t ≥ 0    
∀t ≥ 0
Xt : Ω → Rd  is Ft – measurable.
X = ( Xt ) t ≥ 0 is (Ft ) – adapted.
14
Mapping ( t, ω) ∈([0, ∞)×Ω, B([0, ∞)]×F) ↦ Xt ( ω) ∈( Rd, B ( Rd ) )
When the mapping is measurable, X = ( Xt ) t ≥ 0  is called to be measurable.
15
X = ( Xt )
Ft0 = Ft0,X : = σ ( XS ; s≤t )
16
Probability space      ( Ω, F , P )    
Stochastic process defined over  ( Ω, F , P )      (Bt)t ≥ 0 = (Bt(ω)) t ≥ 0
(Bt)t ≥ 0 that satisfies the next, it is called Brownian motion.
(i) P ( B0 = 0  ) = 1
(ii) For ∀ω∈Ω, Bt (ω) is continuous on t.
(iii) For 0 = t0<∀t1<…<tn, ∀n∈N, {Bti-Bti-1} satisfies the next.
a) {Bti-Bti-1} are independent each other.
b) {Bti-Bti-1} are followed by mean 0 and variance ti-ti-1 of Gauss distribution.
17
(Existence theorem)
Over adequate probability space, there exists Brownian motion.
18
Ω = W0
F = B ( W0 )
Brownian motion has the next.
(i)Bt ( w ) = Wt
(ii) w = ( wt ) t ≥0 ∈W 0
Measure over ( W0, B ( W0 ) )      P
P is called Wiener measure.
19
d-dimensional Brownian motion     B = ( Bt ) t ≥ 0
d×d orthogonal matrix     A
ABt     d-dimensional Brownian motion
Sphere     S : = δ B (0, r),  B (0, r) = {|x| ≤ r }
Hitting time     σS (ω) : = inf{t >0; Bt ∈S }
Hitting place    BσS (ω)
Distribution of BσS (ω)      uniform stochastic measure
20
d-dimensional Brownian motion     B = ( Bt ) t ≥ 0
x∈Rd
Brownian motion from x     ( x + Bt ) t ≥ 0
W  d = B ( W d )
Space     (W d, W d )
Distribution over  (W d, W d )     Px
Mean on Px     Ex [ ・ ]
Probability space     (W d, W d , Px )
Stochastic process over (W d, W d , Px )    Bt ( w ), w∈W d ; Bt ( w ) = wt
Sub σ additive family of W d     Ft0 =σ (Bs ; s≤t ) , Ft = Ft0 ⋁ N, t≥0 ; N : = {N∈W d ; Px (N) = 0, ∀x ∈Rd }
Ft* = Ft+ : = ∩s>t Fs
Shift operator over W d     θs : W d → W d , s≥0 ; (θs (w) ) t : = wt+s
Bt ∘  θs  = Bt+s
21
(Markov property)
∀x∈Rd
∀s≥0
∀Y = Y (w) : W d –measurable bounded function over W d
Ex[Y∘θs ・1A] = Ex[EBs(w)[Y]∘θs ・1A] , ∀A∈F s
By conditional mean
Ex[Y∘θs | Fs] (w) = EBs(w)[Y
Px-a.s.w
22
(Blumenthal’s 0-1 law)
When A∈F0 ( = F0* ), Px (A) = 0 or 1
23
Random variant of 1-dimensional Brownian motion starting from the origin     B
σ (0,∞) : = inf {t >0; Bt∈(0,∞) }
A = {σ(0,∞) = 0 }
A ∈F0*
P (σ(0,∞) = 0 ) = 0 or 1
t↓0
P (σ(0,∞) = 0 ) = 1
From symmetry of Brownian motion Bt = -Bt
 
[B]
Language that has Brownian motion     LB
LB has actual language and imaginary language.
 
[References]
 
 
To be continued
Tokyo August 12, 2008



Read more: https://srfl-paper.webnode.com/news/stochastic-meaning-theory-5-language-as-brown-motion/

 

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Tokyo

20 June 2020

GeometrizationLanguage

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