Quantization of Language / 24 June 2009

​Floer Homology Language

TANAKA Akio

Note7 

Quantization of Language​

​​

Theorem​

1​

(Barannikov, Kontsevich 1998)​

<.,.>, ° defines structure of Frobenius manifold at neighborhood of H's origin.​

​

2​

(Kontsevich 2003)​

There exists φk : EkΠ2(Γ(M;Ω(M))) → Π2CD(A, A), k = 2, ... .​

 is L∞ map.​

​

Explanation​

1​

(Local coordinates of Poisson structure)​

{f, g} ​

​

2​

(Map)​

{.,.} : C∞ × C∞  →C∞​

The map  has next conditions.​

(i)   {.,.} is R bilinear,{f, g} = - {g, f}.​

(ii)  Jacobi law is satisfied.​

(iii) {f, gh} = g{f, h} + h{f, g}​

​

3​

(Gerstenharber bracket)​​

​

4

​

​

5

​

​

6

​

​

7

​

​

8​

(  )​

Manifold     M= R2n​

Coordinates     p, q​

Differential form     w = dqidpi​

Subset of C∞( R2n )        A​

Element of A       F    ​

Differential operator of R2n      D(F)

​

D({F, G}) ≡ [D({F}, D({G}]​

​

​[Image 1]​

Quantization of language is defined by theorem (Kontsevich 2003).​

​[Image 2]​

Complex unit  is seemed to be essential for mirror symmetry of language
by explanation 8.​

​​​

Tokyo ​

June 24, 2009​ 

Sekinan Research Field of Language

​

Language and Dimension / 6 December 2014

Language and Dimension

TANAKA Akio

1.
History
Language and dimension seem to have relation at mathematical based thinking. The clue is language's going round and round in circles like the Crete's Lie.

• True-false problem of the Crete / The example of what language has structure / 22 July 2013

I have thought on the relation probably from 2008 with the minimum unit of meaning, which I called "meaning minimum". One of the papers at those days are at the next.

• Word and Meaning Minimum / 22 September 2008

Clear writing on dimension is very hard for me at the defined lo level's mathematics. After several years I had some stock on writing tools and thinking basis by mathematics, especially algebraic geometry. Some results are the following.

• Dimension of Word / Arithmetic Geometry / 24 December 2012 

2.

Arithmetic Geometry

In 2013 I set to work in earnest to learn arithmetic geometry for solving the hard problem. The next are the  results of learning, titled Arithmetic Geometry Language (AGL)

• Arithmetic Geometry Language (AGL)

1. Dimension of Language (AGL 1)
2. Synthesis of Meaning and Transition of Dimension (AGL 2)
3. Birth of Word, Synthesis of Meaning and Dimension of New Word (AGL 3)
4. Dimension Conjecture at synthesis of Meaning (AGL 4)

3.
Conjectures
From my study of AGL, I arranged the next three conjectures for the next step. They are the most important themes of my study.

Three Conjectures

• Dimension Decrease Conjecture  (Conjecture 1) 
• Synthesis Conjecture  (Conjecture 2) 
• Reversion Conjecture (Conjecture 3)

For details, refer to the next.

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

4.

Prospects after this
Over dimension, synthesis and reversion, I have thought on some important concepts of language, potential, quantization, discreteness and so forth. These concepts would perhaps open the possibility of further understanding of language and its surroundings.
I have written several trial papers for studying the concepts like the following.

• Potential of Language / 29 April 2009 - 16 June 2009
• Quantization of Language / 24 June 2009
• Discreteness of Language / 19 July 2009

5.

Conclusion at present

Roger Penrose' said at THE ROAD TO REALITY A Complete Guide to the Laws of the Universe, 2005,  "We cannot get any deep understanding of the laws that govern the physical world without entering the world of mathematics." 

• Roger Penrose's THHE ROAD TO REALITY A Complete Guide to the Laws of the Universe, 2005 /  25 October 2012

In my youth, I was deeply impressed with Ludwig Wittgenstein, from whom I was strongly affected at thinking  and writing  style of the study.

• For WITTGENSTEIN Ludwig / 10 December 2005

Tokyo

6 December 2014

SIL

Word and Meaning Minimum "Language has dimension." / 22 September 2008

Energy Distance Theory

Conjecture 1

Word and Meaning Minimum

TANAKAAkio

1

Word is expressed by arbitrary figure W in space Rⁿ.

Meaning minimum is expressed by n-dimensional sphere M that has diameter below δ.

W is covered by sequence of M.1, M2, M3, ….

Lower limit of all the covering ∑_k (diam ( M_k ) ) ^α is expressed by H_α,δ ( W ).

H_α,δ ( W ) = inf _{diam ( Mk ) <δ} (diam ( M_k ) ) ^α

2

H_α ( W ) : = lim_{δ →0} H_α,δ ( W )

3

Word has non-positive real number or ∞ by H_α ( W ).

H_α ( W ) is restricted by measurable sets.

H_α ( W ) is treated as Hausdorff measure.

4

Word is expressed by Hausdorff measure.

Meaning minimum is expressed by lim_{δ →0} M .

5

Word has α dimension.

Meaning minimum has n dimension.

6

Language consists of word and meaning minimum.

7

Language has dimension.

[References]

<On meaning minimum>

Cell Theory / From Cell to Manifold / Tokyo June 2, 2007

<On place of meaning>

Stochastic Meaning Theory 3 / Place of Meaning / Tokyo July 11, 2008

<On confirmation of meaning>

Stochastic Meaning Theory 2 / Period of Meaning / Tokyo June 27, 2008

 

Tokyo September 22, 2008

SekinanResearch Field of Language

 [Reference December 22, 2008]

Amplitude of Meaning Minimum / sekinan.wiki.zoho.com

Complex Manifold Deformation Theory / sekinan.wiki.zoho.com

From Cell to Manifold, On Meaning Minimum hinted by Roman Jakobson / 2 June 2007

Cell Theory

  

From Cell to Manifold

Continuation of Quantum Theory for Language

TANAKA Akio

1 Cell is defined by the following.

 n-dimensional ball Dⁿhas interior that consists of cells. Cell is expressed by Dⁿ - δDⁿ and notated to eⁿ that has no boundary.

δis boundary operator.　

Homomorphism of Dⁿ is notated to ēⁿ.

ēⁿ  - δēⁿ = eⁿ

2 Set of no- boundary-cells becomes cell complex.

3 Some figures are expressed by cell. h_n is attaching map.

n-dimensional sphere      Sⁿ =  ē⁰ ∪h_n  ēⁿ   

n-dimensional ball          Dⁿ = ( ē⁰ ∪h_n-1  ēⁿ ) ∪ ēⁿ

Torus                              T² = ( ē⁰  ∪h₁  ( ē⁰ ∪ēⁿ ) ∪h₂ ē²

3 Grassmann manifold is defined by the following.

Grassmann manifold G^R(m, n) is all of n-dimensional linear subspaces in m-dimensional real vector space.

                                        S¹ = G^R( 2, 1 )

4 Canonical vector bundle γ is defined by the following. E is all space. π is projection.

γ= ( E, π, G^R(m, n) )

5 Here from JAKOBSON Roman ESSAIS DE LINGUISTIQUE GÉNÉRALE, semantic minimum is presented.

Now semantic minimum is expressed by cell ē³.

6 Word is expressed by D².

7 Sentence is expressed by Grassmann manifold’s canonical vector bundle γ¹ ( G^R(3, 1) ).

Tokyo
June 2, 2007

Sekinan Research Field of Language

Time in Word with Note and Note 2 / 24 October 2013, 17 November 2014 and 3 December 2014

Time in word

TANAKA Akio

Time inherent in word has been one of the most fantastic theme for me for a long period. WANG Guowei’s paper was strongly hinted to the theme towards first writing a trial paper on it. Roman Jakobson was also gave me the basic notion related with the theme. Still now the problem has not solved as the starting point. But being aided by mathematical method and description,  learning on it has become more attractive in these days.

• Trial paper: On Time Property Inherent in Characters
• Basic notion: From Cell to Manifold
• Mathematical description: Dimension Decrease Conjecture

Tokyo
24 October 2013
SIL

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

[Note]

When I wrote this essay, "Time in Word" in the autumn of 2013, the relation between time and dimension cannot be thought clearly but now in 2014 I distinctly think that the relation exists at the situation of synthesis of the new meaning on the old meaning. The four papers of Arithmetic Geometry Language written between 4 September 2013 and 9 September 2013 are indicating the affinity with time and dimension. Now I show the compact paper in the four, titled " Synthesis Meaning and Transition of Dimension".

• Synthesis of Meaning and Transition of Dimension, 6 September 2014 

Four papers of Arithmetic Geometry Language are the following.

Arithmetic Geometry Language (Abbreviated AGL)

• Dimension of Language (AGL1)
• Synthesis of Meaning and Transition of Dimension (AGL 2)
• Birth of Word, Synthesis of Meaning and Dimension of New Word (AGL 3)
• Dimension Conjecture at synthesis of Meaning (AGL 4)

Tokyo
17 November 2014
SIL

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

[Note 2]

The relation between time and dimension is now arranged at the following paper.

• Three Conjectures for Dimension, Synthesis and Reversion with Root and Supplement / 19 November 2014

The overview of time in language is now shown at the following paper.

• How is the time alive in language?

On WANG Guowei and Roman Jakobson, refer to the next.

• 40 years passed fro I read WANG Guowei / 16 November 2013
• Roman Jakobson /16 July 2012
• Meaning minimum On Roman Jakobson, Sergej Karcevskij and CHINO Eiichi / 6 April 2013
• From Cell to Manifold, On Meaning Minimum hinted by Roman Jakobson / 2 June 2007

Tokyo
3 December 2014

SIL