Theorem / Eliashberg​, Tomita, Borchers

Symplectic Language Theory 

TANAKA Akio 

     

​

Note 1 

Symplectic Topological Existence Theorem​

​

​

[Theorem]​

(Eliashberg)​

Symplectic homeomorphism   is C0 convergent to differential 

homeomorphism .​

Under the upper condition, φ is symplectic homeomorphism.​

​

​

[Note]​

1​

For language's understandability, differential homeomorphic C0 convergence is related with 

the finiteness and infinity of language. ​

2​

For the finiteness and infinity of language, next theorem is eficient to solve the problem.​

​

(Tomita's fundamental theorem)​

H       Hilbert space        ​

B(H)  Banach space B(H, H)​

N       B(H)'s *subalgebra that contains identity operator and closes for τuw topology​

J        Conjugate linear equidistance operator​

Δ       Unbounded positive self-adjoint operator​

Δit     τs-continuous 1 parameter unitary group​

(1) ​

(2) ​

​

(Borchers' theorem 1992)​

The theorem is deeply connected with Tomita's theorem.​

​

​

[Impression]​

Symplectic geometric structure is seemed to be solvable for language's understandability that 

simultaneously connotes finiteness and infinity within. ​

​

​

[References]​

<Topological approach>​

#1 Topological Approach​

<Language's understandability>​

#2 Deep Fissure between Word and Sentence​

#3 Finiteness in Infinity on Language​

<Related theorems>​

#4 Tomita's Fundamental Theorem​

#5 Borchers' Theorem​

<Related twitter site>​

#6 sekinanlu​

​

​

​

To be continued​

Tokyo February 27, 2009​

Sekinan Research Field of language​

Source:

  

   
                   https://writer.zoho.com/public/sekinan/SLTNote1SymplecticTopologicalExistenceTheorem1

Borchers’ Theorem

von Neumann Algebra 4

Note 2

Borchers’ Theorem  

TANAKA Akio

[Theorem]

von Neumann algebra     N

Cyclic and separate vector of N     Ω

Continuous 1 coefficient group of unitary operator     U (λ)   

U (λ) has next condition.

U (λ)Ω = Ω 

U (λ)N U (λ)* ⊂N   

Generation operator of U (λ)       H

Modular operator on (N, Ω)     Δ

Modular conjugation on (N, Ω)     J

Next 2 conditions are equivalent.

(i) H ≧ 0

(ii) Δ^it U (λ) Δ^-it = U (e^-2πtλ)     J U (λ) J = U (-λ)  

[Preparation]

<1 Cyclic vector>

Representation of C*algebra A     {H, π}

x∈H

{π(A)x} ^- = H

x is called cyclic vector.

<2 separate vector>

Norm space     V

Subset of V     D

sup{||x|| ; x∈D} < ∞

D is called bounded.

Linear operator from norm space V to norm space V₁      T

D ( T ) = V

||Tx||≦γ (x∈V )  γ > 0

T is called bounded linear operator.

||T || := inf {γ : ||Tx||≦γ||x|| (x∈V)} = sup{||Tx|| ; x∈V, ||x||≦1} = sup{; x∈V,  x≠0}

||T || is called norm of T.

Hilbert space     H , K

Bounded linear operator from H  to K     B (H, K )

B ( H ) : = B ( H, H )

B⊂B (H)

x∈H

Q⊂B

Qx = 0 → Q = 0

x is called separate vector.

<3 Continuous 1 coefficient group of unitary operator >

Self-adjoint operator     A

Spectrum measure     {E_λ}

A = ∫^∞_-∞ λdE_λ

Unitary operator over H       U = ∫^∞_-∞ e^iλE_λ

U_t = e^itA = ∫^∞_-∞ e^itλ E_λ

Continuous 1 coefficient group of unitary operator     {U_t ; t ∈R}

U₀ = I

U_s+t = U_s + U_t   ∀s,t ∈R

U_t* = U_-i

To be continued

Tokyo May 2, 2008

Sekinan Research Field of Language

www.sekinan.org

Meaning minimum / On Roman Jakobson, Sergej Karcevskij and CHINO Eiichi

Meaning minimum
On Roman Jakobson, Sergej Karcevskij and CHINO Eiichi

TANAKA Akio
ifbetrue
6 April 2013

1.
Meaning minimum is one of the kernel concepts for the model of language universals on my study. The concept was at first thought from Roman Jakobson's semantic minimum on which I first read at his book, ESSAIS DE LINGUISTIGUE GENERALE, 1973.　His concept was yet intuitive at the linguistic study  history in the latter half of the 20 th century.Comparison with his concept, my definition of meaning minimum was a certain basis prepared in the learning of mathematics, especially on algebraic geometry, that is the most fantastic approach to the construction of the language model. But the contents of meaning minimum is vacant. This concept shows the minimum unit of one big constructive meaning of word. meaning minimum seems to be correspondent with element of set theory, which theory and foundations of mathematics had been my favourite mathematical basis in 1970s, my youth time. Bourbaki was always echoed around us. Grothendieck was a highest star in this world.
2.
Time went vast from at that time. Set theory became one of the premise field of mathematics. But in my part, set theory's agenda was put at a another point. Because language has a certain expanded world that seems to be continuous. Set theory's atomic discreteness does not match in my primary learning level.  So, in my  age 30s, I had sank in the philosophically intuitive thinking often referring the tradition of 1920s, especially of the Linguistic Circle of Prague. On the circle my teacher CHINO Eiichi had taught me from time to time on the campus of university or coffee shop near the station we used. CHINO had gone to the Czechoslovakia Republic from 1959 to 1967. I first met him in 1969 at his Russian class at my  third year of university student. I was the age 21 and he was 37.
3.
At the age 23's 1971 spring,  I graduated university and  once became a high school's teacher and again returned back to university in 1979 after 8 year job of the school. At that time I thought of characters' distinctive features on Written  Chinese classics. I mainly read WANG Guowei, ZHANG Binglin, DUAN Yucai, WANG yinzhi guiding by Japanese modern scholoar KANO Naoki. Expressly I had attracted to  WANG guowei and his book Guantangjilin. Besides reading these China's Qing dynasty's linguistic peaks, I had always thought on Ludwig Wittgenstein for his endless pursuit on language. So I resigned school and came back to the campus where I again met with CHINO. I was age 30s and he was 50s. He was already the big scholar at the linguistic field  but I was a poor return student. But I dare to say we were colleagues for language study. He taught me the detailed and strict tradition of the linguistic Circle of Prague. He frequently talked on Sergej Karcevskij and his eminent discernment on language. In the later year's masterpiece, Janua Linguisticae Reserata, 1994, he wrote only Karcevskij as genius in the great linguists.
4.
Being led by CHINO, I again started linguistic learning on meaning that I had been interested in from my 20s but too hard to approach by my talent. This time I had Karcevskij's fine insight to meaning enough absorbing the fertile tradition of Prague, where also exist Jakobson and Mathesius. Through the learning I gradually lean to desire to write clear definitions on language. I again remembered the little learning of my 20s age's mathematics. Bourbaki, Godel, TAKEUCHI and their set theory, foundations of mathematics and that Incompleteness theorem. I had learnt mathematics little by little, inch by inch.
5.
CHINO Died in 2002 at age 70 and I became 55. The next year 2003, I wrote a short paper titled "Quantum Theory for Language". This paper was showed at a international symposium on Silk road for dealing with language from Chinese characters on linguistic viewpoint. I knew  that Asian civilisation and history had great concern not only from Asia but also European continents. At the symposium some 400 researchers gathered in the various scholarly fields. It was a awesome encounter for my study, namely, East meet West. Probably Chinese character's agenda will be written by Europe oriented mathematics. WANG Guowei will meet with karcevskij mediated through mathematics' description. The target confronted at that time was time inherent in characters, or time in word.  In Chinese, particularly in classical written Chinese, all the characters show enough independent meaning in one character probably including even time. It was my first conjecture taught from Karcevskij and CHINO. Meaning minimum is on the boat going across to the opposite shore. This metaphor was derived from WANG Guowei's famous paper, "Yin- bu zhong suojian xiangong xianwang kao"
6.
The concept of time inevitably led to the distance of distance. In 2004, I wrote a paper titled "Distance Theory". But the paper was yet intuitive and not clear for descriptive definition. So hereinafter I learnt algebra inch by inch assisting with the rich imagine of geometry. In the centre of learning, always exist time that connotes finiteness and infinity. But infinity is not easily obtained without probably only loop space at the present. And again returns back to meaning minimum as the passenger of the boat named time property inherent. This time the passenger on the boat  is called operad or algebraic language.
7.
After all I came back to the very dream that I had embraced since the high school days. It was  a fundamental ask on language related with mathematics and physics. The root of language would be able to describe by mathematics and physics. In my mind language is always put at the centre of the pursuit that was what anyone can clearly understand. Description by mathematics, but physics why. Physics treats with substance that constructs the world in which I had desire to let language enter. Substantial language. It was my dream and will be hereafter.

References
1.
<Meaning minimum>
Cell Theory Continuation of Quantum Theory for Language / From Cell to Manifold For LEIBNIZ and JAKOBSON / Tokyo June 2, 2007
<Bourbaki>
The Time of Language Ode to The Early Bourbaki To Grothendieck / January 10, 2012
2.
<CHINO Eiichi>
Fortuitous Meeting What CHINO Eiichi Taught Me in the CLass of Linguistics / Tokyo December 5, 2004
<The Linguistic Circle of Prague>
Linguistic Circle of Prague / Tokyo 13 July 2012, 19 July 2012 Added
3.
<WANG Guowei>
The Complete Works of WANG Guowei / Tokyo 24 May 2012
<Ludwig Wittgenstein>
The Time of Wittgenstein / Tokyo January 20, 2012
<Sergej Karcevskij>
Notes for KARCEVSKIJ Sergej / Note for KARCEVSKIJ Sergej's "Du dualisme asymetrique du signe linguistique" / Tokyo September 8, 2011
4.
<Set Theory>
Quantum Linguistics / Growth of Word Dedicated to TAKEUCHI Gaishi / Tokyo January 30, 2006
5.
<Quantum Theory for Language>
Quantum Theory for Language Synopsis / Tokyo January 15, 2004
<Time inherent in characters>
On Time Property Inherent in Characters / Hakuba March 28, 2003
6.
<Distance>
Distance Theory / Tokyo May 5, 2004
<Meaning minimum and distance>
sekinanlogos / Floer Homology Language / Supersymmetric Harmonic Oscillator / Tokyo May 6, 2009
<Loop space>
sekinanmodel / Infinite Loop Space Language / Word as Infinite Loop Space / 6 December 2012
<Operad>

ifbetruezoho / Operad Meaning Theory / From Cell to Operad / 4 March 2013

7.
<Description>
sekinanmetria / Notes for KARCEVSKIJ Sergej / Description of Language / September 9, 2011
<Substantiality>
sekinanlogos / Floer Homology Language / Potential of Language / Tokyo June 16, 2009
<Amalgation>
Language, amalgation of mathematics and physics / ifbetrue 2 April 2013

# Here ends the paper.

Simplicial Space Language -Composition of Word

Simplicial Space Language

Home Site: ifbetrue

Autor: TANAKA Akio

Theme: Language Universals

Method: Homotopy Theory

Title: Homotopy Language

Paper 1: Simplicial Space Language

-----------------------------------------

Simplicial Space Language

-Composition of Word

TANAKA Akio

20 January 2013

ifbetrue

1. 

 is topological space.

 is simplicial space.

 is represented by geometric realization of   as the next.

2.

 is topological space.

 is singular simplex.

 is free Abelian group generated by set .

 .

 is singular chain complex of  and has strongly convergence spectral sequence called Atiyah-Hirzebruch spectral sequence.

3.

Interpretation.

Word  is given by geometric realization of simplicial space  .

Meaning unit in word is given by free Abel group .

(Paper end)

Lang Model

Lang Model

TANAKA Akio

1. Lang conjecture (S.Lang 1974)
Algebraic field F ( finite dimensional overfield over Q ) is defined over projective algebraic variety V.
For a certain embedding F -> C, V is seen that V is seen as complex projective algebraic variety.
If V is Kobayashi hyperbolic, V ( F ) is finite set.

2. Shafarevich conjecture ( I.Shafarevich 1963)
F is algebraic field.
O_F is integer ring of F.
Finite subset  and integer  are fixed.
When the upper conditions are satisfied, g dimensional Abelian variety over F and genius g's algebraic curves that degenerate in only S at most only exist finite number.

3. Interpretation
3.1 Finiteness of words by Shafarevich conjecture
Language := FSentence := O_FWord := SWord's dimension := gWord's meaning := genius g's algebraic curves 3.2 Grammar (Connection of words) by Lang conjecture
Language := FGrammar (Words' connection rule) := Kobayashi hyperbolicity

4. References
4.1 On dimension and meaning
Quantum Theory for Language 2004
4.2 On Kobayashi hyperbolicity
Ditance Theory 2004
From Distance to Pseudo-Kobayashi-Distance 2012

This paper's description is unfinished.

Tokyo

17 June 2012

Sekinan Research Field of Language

Arithemetic Geometry 1 Dimension of Word

Arithemetic Geometry 1

Dimension of Word

TANAKA Akio
24 December 2012

ifbetrue / Tokyo

1. 

Theorem by C. Soulé, Lectures on Arakelov Geometry, 1992. 

 is morphism between regular arithmetic varieties.

 is pullback.

When  are morphism between regular arithmetic varieties, the next is concluded.

 .

2.

Interpretation of the upper theorem.

Word : .

Decomposition of word : , named pullback.

Decomposed meaning unit in word :  and .

3.

Pullback is defined by the next from Efton Park, Complex Topologicak K-Theory, 2008.

Let  and be topological spaces, let  be avector bundle over , and suppose  is a continuous map,

Define

4.

Interpretation of theupper theorem 2.

Word can be deposed to meaning units by pullback.

Meaning unit also become pullback.

5.

Pullback contains order function ord.

ord contains length.

Here length is longitude of composition series.

Here composition series is defined by the next.

A is commutative ring.

M is A module.

If A be field,  is M's dimension over A.

6.

From 5. leads the next supposition.

When word is decomposed to meaning unit, each unit has demension that determines word's dimension.

(Here ends the paper.)