Friday, 5 August 2022

What is signal? / Total Edition / Till 7 Understandability 2 / 10 January 2019

 Thursday, 10 January 2019

What is signal? / Total Edition / Till 7 Understandability 2 / 10 January 2019




What is signal? 
The existence that generates language

TANAKA Akio
SRFL Paper
Tokyo

21 November - 8 January 2019




Original Title
 What is signal? A mathematical model of nerve  
0
Preparation 1-15
For father and mother 



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Preface



This short paper would finally write on signal through the relation between language and nerve, for which using mathematical method at the way.
This paper's one of kernels is energy which is naturally accepted at the side of verve, but at the side of language, it may be not accepted widely till now.
In this paper, I probably do not refer to the language's energy, that has been written several times in the papers before.
If necessary refer to the next essays.


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Around 2008, I was thinking on energy in language related with distance in language. Distance is one of the kernel themes of my study of language universals in those days. In simply saying, distance is the result of movement and for movement there must inevitably needs energy. So I had thought that if language have distance, there must be energy or its alike in language that is supposed in mathematical models. But in my ability in 2008, I could not develop the deep and wide range of language from the theme, energy and distance. So Energy Distance Theory was still now unfinished.


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Preparation

Distance


1. 
Language

Next definition for language shows my simple image to language and energy.
.................................................................................................

Language and Spacetime 
Language
Definition for the Child who Lost the World 
0 The world spreads around the human being.
1 Language divides the world.
2 Language connects the world.
3 Language bends the world.
4 Language stretches the world.
5 Language shrinks the world.
6 Language extinguishes the world.
7 Language creates the world.
8 Language gives despair.
9 Language gives hope.
10 Language is pasting on spacetime with energy.
Postscript
[Referential note / November 29, 2007]
[Definition added / November 3, 2008]
Definition 10, the part of is newly added.
Tokyo March 3, 2007
Tokyo November 3, 2008 Added
..........................................................................

2. 
Human sense

On language and human sense, I ever wrote simply at the next paper.
Macro Time and Micro time
TANAKA Akio
24 July 2013
atbankofdam
1. Through natural language, in human being, occurred the electrical signal by eye or ear. These complex situations are beyond this paper’s limits.
2. Language is a physical object as signal and its transmission. At this circumstances, language must be recognised to be the existence that has finite time.
3. An apple on the desk gradually becomes rotten by passing the time very after the crop in the orchard. #0
4. Like an apple, language has passing physical time in oneself.
5. Language is metamorphosed  by the time progressing.  #1
6. Language includes the outer world from human being to universe. At this declaration, I recall Blaise Pascal’s Pensées. XXXIII. PROOFS OF JESUS CHRIST 308 The infinite distance between body and mind symbolizes the infinitely more infinite distance between mind and charity, for charity is supernatural.(Translated by A.J. Krailsheimer, 1966) #2
7. Language’s time goes freely from the present to the future or the present to the past. #3
8. Language symbolises the time from finiteness to infinity. #4
9. Human being recognises this vast language world perfectly. #5
References
#0 For WITTGENSTEIN Ludwig Position of Language / December 10, 2005 – August 3, 2012 / Sekinan Research Field of Language
#1 Time of Word / Complex Manifold Deformation Theory / January 1, 2009 / sekinanlogos
#2 PASCAL PENSÉES. Translated with an introduction by A.J. Krailsheimer. PENGUIN BOOKS 1966.
#3 Escalator language and Time For SHINRAN’s Idea and BOHDISATTVA / Escalator Language Theory / December 16, 2006 / Sekinan Research Field of Language
#4 From Finiteness to Infinity on Language / Topological Group Theory / February 1, 2009 / sekinanlogos
#5 Understandability of Language / Complex Manifold Deformation Theory /January 9, 2009 / sekinanlogos
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3. 
Signal

Signal Goes to meaning.
Traffic signal shows Go, Attention and Stop to people by three colour lights
Morse code sends a message by short and long signals.
Signal has structure that resembles language.
Morse code can regards as a written language.
What is signal's peculiarity?
It depends on signal's generation that has simple on-off phenomenon.
This phenomenon combines with the other on-off phenomena and become complex structure that has meaning like language.
Signal seems a primordial form of language.
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4. 
Nerve

Is nerve signal?
I have any knowledge on physiological nerve.
So at this paper I prepare a mathematical model of nerve comparative with signals.
At mathematical model, on-off system is not simply fit with the expression of mathematical space.
Here on-off system is arranged for more simple form.
I ever wrote a paper below.
The paper is a very intuitive one but there are some hints on signal.
Now I need one chapter for my new trial paper.
The chapter is " 6. Basic principle of quantum theory"
The original text is the following.
.................................................................................
６　量子理論の基本原理
　量子理論は、伝統的には実辞とされる陽性量子が1階層で進行することが基本である。したがってある種の量子群すなわち文において、陰性量子が冒頭に立つときには、その前に陽性量子または量子群が省略されたと考えることができる。
陰性量子が陽性量子の進行方向に変化を与えて、新しい階層に移行させることは、陰性量子が受けている被圧迫エネルギーによるとしたが、より正確には、以下のように説明できる。
一般に断定の虚辞とされる「也」は、その実辞としての意味は現代においては不分明であり、陽性量子としてはほぼ消滅したと考えることができる。その代わりに陰性量子としての虚辞機能が台頭して現在に至るが、その機能を細分すれば、断定、主題提示、呼びかけ、詠嘆、疑問、反語等の極めて多様である。
たとえば、「回也不愚」（『論語』為政篇）においては、「回」という人物が、（階層が変わり）孔子の心中において、（また階層が変わり）否定される存在であり、（さらに階層が変わり）「おろかものの類」が提示される。」
「階層が変わる」ということは、「回」という人物が、「也」と出現によって実在の人物から、孔子の心中における考察対象へと変換されたことを意味し、さらにその考察対象が「不」によって抹消されることを意味し、さらに新しく「愚」という概念が登場することを意味する。
すなわち陰性量子は、陽性量子が有する言語世界への直接的な意味を行うのではなく、実辞としてはもはや空白となった自らの領域へ、陽性量子を導くjことによって、一種の真空無重力の状態を現前させ、その方位を転換することであると仮定する。その転換に必要とされるエネルギーは、領域が受けている被圧迫のエネルギーから生ずるものと仮定する。　

Read more: https://srfl-lab.webnode.com/products/manuscript-of-quantum-theory-for-language-with-preface-note-and-note-2-2003-2018/
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At this chapter, the most important concept is positive quantum and negative quantum.
Negative quantum changed to positive quantum by energy.
At the linear space, quantum changes positive to negative and also negative to positive.
By this model, on-off phenomena are mapped at the linear space's quanta's situation.
In the models, quantum is led from V.G.Drinfel'd and M.Jinbo's Quantum group originated in 1985.
Reference
JIMBO Michio. Quantum group and Yang-Baxter equation. Maruzen Shuppan. Tokyo. 2012

5. 
Method

Manuscript of Quantum Theory for Language written in spring 2003 was roughly designed at a days I was in
hospital by pneumonia half a month in October 2002, when I always saw the river and the mountains west end of Tokyo. I was the very  reviewing life time for my research work.
My poor study was restricted in a narrow field of Chinese classical linguistics mainly developed in the late Qing dynasty the latter half of the 19th century,represented by DUAN Yucai, WANG Niansun, WANG Yingzhi, my favourite WANG Guowei and so forth.
So I determined that my approach to language was only in it and it was the most intimate for me at that time and probably herein after considering my tiny accumulation of study.
Next essay shortly showed the situation in 2002.


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1970s' dream, writing clear description on language universals by mathematics
From The Days of Ideogram

​​​4. Time property in characters
In Autumn 2002 I got pneumonia and was hospitalized about 2 weeks, where I thought of 1970s' dream, writing clear description on language universals by mathematics. The theme was as hard as ever. So, at the bed I thought the basis of language from the side of Chinese character’s classical approach which had vast heritage till Qing dynasty. I directed my attention to the character's figure which had compound meanings containing time elements continuing from Yin dynasty's hieroglyphic characters left on bones and tortoise carapaces some 2400 years ago. I thought that Chinese characters had containing time and its structure could be written by geometric approach once I had abandoned for difficulty. After leaving hospital, I wrote a paper titled On Time Property Inherent in Characters*3-1.

Read more: https://srfl-paper.webnode.com/news/the-days-of-ideogram/

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6. 
Quantum group

I think that language's foundation is in word.
Recently I started the new approach to word that has more simply at form and more wide-range-covered at usage.
Trial paper is the next.
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Quantum Group Language   
Word Indexed and Word Synthesized 
TANAKA Akio
1.
Word Synthesized is the word that is synthesized meanings in a word by arithmetic or algebraic  geometrical method.
For detailed refer to the next.
2.
Word Indexed is the word that has index in a word, by which meanings are separated in a word at waiting situation. In a word meanings are combined by algebraic axioms and theorems starting from group theory.
Word Indexed seems to be more simply structure than Word Synthesized.
Word indexed basis is at the below.
The details of Word Indexed will be shown the papers after this.
3.
Meanings contained in Word Indexed is deeply related with quantization and discreteness.
Refer to the next.
4.
All the basis of Word Indexed is generated at the next.
Reference
This paper is unfinished.
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Chinese belongs to the isolated language.
In Chinese one character has one meaning and becomes a word in classical usage. Modern Chinese has many words that contain two or over two characters for one meaning, but basically almost all the characters have still classical one-character-one-meaning usage at the root of language.
Shuowenjezi Zhu written by DUAN Yucai typically shows some 2,000 year history of characters and their meanings.
Modern Chinese precisely said by Hanyu is one language of the over 50 languages officially recognized at the research, and I only know several language's grammars by the field work by linguists.

National Minority Languages in China   2004

......................................................
National Minority Languages in China
Zhongguo Shaoshu Minzu Yuyan Jianzhi Congshu
Minzu Chubanshe
1 Buyi yu jianzhi    YU Cuirong edited    1980   
2 Dongbu yugu yu jianzhi    ZHAONA Situ edited    1981
3 Dulong yu jianzhi    SUN Hongkai edited    1982
4 Donsiang yu jianzhi    LIU Zhaoxiong edted    1981
5 Gelaoyu jianzhi    HE Jiashan edited   1983
6 Maonan yu jianzhi    LIANG MIin edited    1980
7 Menggu yu jianzhi    DAO Bu edited    1983
8 Li yu jianzhi    OUYANG Jueya  ZHENG Taiqing edited    1980
9 Pumi yu jianzhi    LU Shaozun edited    1983
10 Qiang yu jianzhi    SUN Hongkai edited    1981
11 Mulao yu jianzhi    WANG Jin  ZHENG Guoqiao edited    1980
12 Tai yu jianzhi    YU Cuirong  LUO Meizhen edited    1980
13 Tuzu yu jainzhi     ZHAONA Situ edited    1981
14 Wa yu jianzhi    Zhou Zhizhi  YAN Qixiang edited    1984
15 Yaozu yuyan jianzhi    MAO Zongwu  MENG Chaoji  ZHENG Zongze    edited    1982
       
Guojia minwei minzu wenti 5 zhong congshu zhi 1
Zhongguo Shaoshu Minzu Yuyan Jianzhi Congshu
Minzu Chubanshe
16 Achang yu jianzhi    DAI Qingxia  SUI Zhichao edited    1985
17 Bulang yu jianzhi    LI Daoyong  NIE Xiizhen  QIU Efeng edited    1986
18 Chaoxian yu jianzhi    XUAN Dewu  JIN Xiangyuan  ZHAO Xi edited    1985
19 Cuonamen yu jianzhi    LU Shaozun edited    1986
20 Deang yu jianzhi    CHEN Xiangmu WANG Jingliu LEI Yongliang edited   1986
21 Elunchun yu jianzhi    HU Zengyi edited    1986
22 Ewenke yu jianzhi    HU Zengyi  CHAO Ke edited    1986
23 Gaoshanzu yuyan jianzhi (Ameisi yu)     HE Rufen  ZENG Siqi  TIAn Zhongshan  LIN Dengxian edited    1986
24 Gaoshanzu yuyan jianzhi (Bunen yu)     HE Rufen  ZENG Siqi  LI Wensu  LIN Qingchun edited    1986
25 Gaoshanzu yuyan jianzhi (Peiwan yu)    CHEN Kang  MA rongsheng edited    1986
26 Heni yu jianzhi    LI Yongsui  WANG Ersong edited    1986
27 Hesake yu jianzhi    GENG Shimin  LI Zengxiang edited    1985
28 Heze yu jianzhi    AN Jun edited    1986
29 Jing yu jianzhi    OUYANG Jueya  CHENG Fang  YU Cuirong edited    1984
30 Jinuo yu jianzhi    GAI Xingzhi edited    1986
31 Keerkezi yu jianzhi    HU Shenhua edited    1986
32 Lahu yu jianzhi    CHANG Hongen  mainly edited
33 Lisu yu jianzhi     CHU Lin  MU Yuzhang  GAI Xingzhi edited    1986
34 Luoba zu yuyian jianzhi (Bengni-Bogaer yu)    OUYANG Jueya edited    1985
35 Naxi yu jianzhi    HE Jiren JIANG Zhu yi edited    1985  
36 Nu zu yuyan jianzhi (Nuban yu)    SUN Hongkai  LIU Lu edited    1986
37 Cangluo menba yu jianzhi    ZHANG Jichuan edited     1986
38 Sala yu    LIN Lianyu edited    1985
39 She yu jianzhi    MAO zongwu  MENG Chaoji edited    1986
40 Xibu yugu yu jianzhi    CHEN Zongzhen LEI Xuanchun edited    1985
41 Tajike yu jianzhi    GAO erjiang edited    1985
42Tayaer yu jianzhi     CHEN Zongzhen  YI Liqian edited    1986
43 Tujia yu jianzhi    TIAN Desheng  HE Tianzhen deng edited    1986
44 Weiwuer yu    ZHAO Xiangru  SHU Zhining edited    1985
45 Wuzubieke yu jianzhi   CHENG Shiliang  ABUTURE Heman edited    1987
46 Xibo yu jianzhi    LI Shulan  ZHONG Qian edited    1986
47 Yi yu jianzhi    CHEN Shilin  BIAN Shiming  LI Xiuqing dited    1985
TOKYO
December 31, 2004
Sekinan Research Field of Language


Read more: https://geometrization-language.webnode.com/news/national-minority-languages-in-china/
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 Xixiayu is a very interesting for me resembling the Hanyu and in the late 20th century the language was deciphered by Japanese linguist NISHIDA Tatsuo.
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The Days of Decipherment
On 20 July 2016 I went Tokyo National Museum, Ueno Park, Tokyo to see the exhibition A JOURNEY TO THE IMMORTALS: TREASURES OF ANCIENT GREECE, where I saw the linear A and B. It reminds me the youth days, so to say, the days of decipherment.

1960s -1970s is the age of decipherment in a sense. I was age 20 in 1967 and was learning language and literature at university. In 1958 John Chadwick's THE DECIPHERMENT OF LINEAR B was published from Cambridge University Press. At the preface of the book he wrote that the decipherment of linear B was told  at Documents in Mycenaean Greek (Cambridge University Press, 1956) and Michael Ventris that deciphered the Linear B.

In the same age in Japan, Xixia wenzi (Xixia characters) in China was deciphered by NISHIDA Tatsuo (1928-2012) who wrote the analysis and grammar of Xixia characters through the paper Seikamoji no bunseki narabini Seikago bunpou no kenkyuu in 1962.
In almost the same time, Inca characters were studying to decipher. I frequently heard that Russian team developed largely.

In early 1970s I frequently went to Kanda, Tokyo where old bookshops were selling vast Oriental books at the Hakusan street and Yasukuni Street. I bought Chinese classics, especially linguistic classics written in the Qing dynasty and I read them almost every day containing the comparison with the western linguistic results. The Qing dynasty's heritage were DUAN Yucai, WANG Niansun, WANG Yingzhi and WANG Guowei and so forth. DUAN Yucai's Showenjezi zhu and WANG Guowei's Guantang jilin  were the most important for me.

In France, 1960s was the days of Bourbaki that was one of the decipher of geometry by algebra, at least I thought so at that time. I sought and bought several Bourbaki's books at the old bookshops in Kanda, Tokyo,which is the largest old bookshop streets in Japan. But from my ability to mathematics Bourbaki was too much difficult to read on. From the days the long and winding road began to mathematics and its applicable study for language universals.

At the exhibition of ancient Greece I confirmed in particular that the stability of language was  kept by letters and characters from the Linear A and Linear B. 
These language or character's situation especially of ideogram has become my study's foundation.
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These language or character's situation, especially of ideogram has become my study's foundation.
For ideogram, refer to the next papers.
....................................................................
Ideogram Paper   2005-2018
1. Ideogram 2005
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7.
Quantum group 2

In 2008 I wrote on Quantum group at studying Kac-Moody-Lie Algebra.
.....................................................................
Note 2
Quantum Group
1
Base field     K
Finite index set     I
Square matrix that has elements by integer     A = ( aij )i, j ∈ I
Matrix that satisfies the next is called Cartan matrix.
i, j ∈ I
(1) aii = 2
(2) aij ≤ 0  ( i ≠j )
(3) aij = 0 ⇔ aji = 0
2
Cartan matrix     A = (aij)i, j ∈I
Family of positive rational number    {di}i∈I
Arbitrary i, j∈I    diaij = djaji
A is called symmetrizable.
3
Finite dimension vector space     h
Linearly independent subset of h     {hi}i∈I
Dual space of h     h*= HomK (h, K )
Linearly independent subset of h*     {αi} i∈I
Φ = {h, {hi}i∈I, {αi} i∈I }
Cartan matrix A = {αi(hi)} I, j∈I
Φis called fundamental root data of A that is Cartan matrix.
4
Symmetrizable Cartan matrix    A = (aij)i, j ∈I
Fundamental root data     {h, {hi}i∈I, {αi} i∈I }
E = αi ⊂h*
Family of positive rational number     {di}i∈I
diaij = djaji
Symmetry bilinear form over E     ( , ) : E×E → K     ( (αi ,αj ) = diaij )
The form is called standard form.
5
n-dimensional Euclid space    Rn
Linear independent vector     v1, …, vn
Lattice of Rn     m1v1+ … +mnvn     ( m1, …, mn ∈ Z )
Lattice of h     hZ
6
From the upperv3, 4 and 5, the next three components are defined.
(Φ, ( , ), hZ )
When the components satisfy the next, they are called integer fundamental root data.
 i ∈ I
(1)  ∈ Z
(2) αi ( hz ) ⊂ Z
(3) ti :=  hi ∈ hz
7
Vector space over K     A
Bilinear product over K     A×A → A
When A is ring, it is called associative algebra.
8
Integer     m
t similarity of m    [m]t
[m]t = tm-t-m / t- t-1
Integer   m, n   m≧n≧0
Binomial coefficient     (mn)
t similarity of m!     [m]t! = [m]t! [m-1]t!...[1]t
t similarity of (mn)    [mn]t = [m]t! / [n]t! [m-n]t!
[m0] = [mm]t = 1
8
Integer fundamental root data that has Cartan matrix A = ( aij )i, j ∈ I
      Ψ = ((h, {hi}i∈I, {αi} i∈I ), ( , ), hz )
Generating set     {Kh}h∈hz ∪{Ei, Fi}i∈I
Associative algebra U over K (q), that is defined the next relations, is called quantum group associated with Ψ.
(1) khkh’ = kh+h’     ( h, h’∈hZ )
(2) k0 = 1
(3) KhEiK-h = qαi(h)Ei    ( h∈hZ , i∈I )
(4) KhFiK-h = qαi(h)Fi   ( h∈hZ , i∈I )
(5) Ei Fj – FjEi = ij  Ki - Ki-1 / qi – qi-1     ( i , j∈I )
(6) p [1-aijp]qiEi1-aij-pEjEip = 0     ( i , j∈I , i ≠j )
(7) p [1-aijp]qiFi1-aij-pFjFip = 0     ( i , j∈I , i ≠j )
[Note]
Parameter q in K is thinkable in connection with the concept of at the paper Place where Quantum of Language exists / 27 /.
Refer to the next.
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In 2004 I wrote the paper named Place where Quantum of Language exists.
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Place where Quantum of Language Exists
1            Quantum of language is the smallest unit of language.
2            Quantum of language moves linearly on the floor of language.
3            Linear movement is the properties of quantum.
4            Floor of language is on the space of language.
5            The space of language is two dimensions.
6            Two dimensions are horizontal and vertical.
7            Horizontal movement makes word, #1
8            Vertical movement makes sentence. #1
9            The space of language is electrical digitized place.
10        Chinese /jiao shi/ means classroom in English. 
11        /Jiao/ is a quantum of language.
12        /Shi/ is a quantum of language.
13        /Jiao shi/ is a word.
14        /Jiao/ sends a quantum to /shi/ quantum.
15        /Shi/ quantum receives a quantum from /jiao/ quantum.
16        What sends quantum is called positive.
17        What receives quantum is called negative.
18        Quantum has positive energy in original condition.
19        Quantum changes negative in the situation of quanta set.
20        Quantum change occurs in two situations in general.
21        One situation is what quanta stand side by side on a floor and neighboring quanta connect well. #2
22        The other situation is what quanta change oneself by the non-use of quanta meaning in language history progress. #2
23        Word has a positive- negative construction.
24        Positive-negative construction occurs on a floor.
25        Sentence has a positive-positive construction.
26        Positive-positive construction occurs on different floors.
27        The latter quantum transfers on a different floor. This transfer is called .†
28        Quantum has electrical energy which flows to the electrical zero level.
29        Electrical zero level is a sentence end where quantum of language ideally accord with the real world. #3
30        A floor of language is a non-branches electrical circuit.
31        Word is a non-branch circuit.
32        Sentence is a branch circuit.
33        The meaning of word and sentence is a compound system of electrical signals.
#1 Definition of word and sentence can be seen in the paper “Method of Linguistics” and other papers on the site of Sekinan Research Field of Language /www.sekinan.org/.
#2 Definition of connect well can be seen in the paper of “Quantum Theory for Language Synopsis” and other papers on the site of SRFL.
#3 Definition of sentence end can be seen in the paper “Mirror Theory”, “Mirror Language” and other papers on the site of SRFL.
Tokyo July 18, 2004
Postscript 
[Referential note November 9, 2007]
 is related with the next papers on the concept.
________________________________
________________________________
[Referential note February 9, 2008]
† Concept of /27/ is thinkable in connection with q> at the paper Kac-Moody Lie Algebra / Quantum Group /.
Refer to the next.


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At this paper, 23-33 is the most important for signal's intuitive image.
23        Word has a positive- negative construction.
24        Positive-negative construction occurs on a floor.
25        Sentence has a positive-positive construction.
26        Positive-positive construction occurs on different floors.
27        The latter quantum transfers on a different floor. This transfer is called .†
28        Quantum has electrical energy which flows to the electrical zero level.
29        Electrical zero level is a sentence end where quantum of language ideally accord with the real world. #3
30        A floor of language is a non-branches electrical circuit.
31        Word is a non-branch circuit.
32        Sentence is a branch circuit.
33        The meaning of word and sentence is a compound system of electrical signals.
........................................................................................

8.
Quantum group 3

Quantum group is defined by JIMBO Michio as the following.
Definition of quantum group 
Generating element: a, b, c, d
Relation: ba = qab,  ca = qac
              bd = q-1db,  cd = q-1dc
              bc = cb, ad - q-1bc = da - qbc = 1
Coproduct:  Δa   Δb           a     b             a     b
                (            )   = (             ) (χ)  (           )
                  Δc   Δd           c      d              c      d
      Δa = a (x）a + b (x) c
                                             JIMBO Michio                                       
                                             Bussei Kenkyu (1992), 57(5): 628-634
9.
Symmetry

Quantum group would be the fundamental concept on symmetry. 
JIMBO Michio early wrote at the end of the paper at Bussei Kenkyu 1992.
I ever wrote the symmetry of language several times, which seems to be one of the fundamental elements for language.
Refer to the recent paper.
............................................................
The comparison between 2003 and 2017
...............................................................
At the end of paper 2017, I wrote as the following, where I showed the early intuitive papers related with symmetry or mirror. This concept has succeeded till now and a little developed a new direction towards mathematical based concept especially of quantum group.
The concept called symmetry is very important to describe the complex situation of natural language.
Symmetry contains undifferentiated factors in itself, for example mirror, distance,ant-world and so forth.
I ever tried to cultivate this fantastic field to resolve the hardship on language universals one more step up.
My trying paper is the following.
Mirror Theory   2004
Mirror Language   2004
................................................................

10.
Quantum

Language based on quantum emerged from thinking the simplest model for containing the finite essential elements of language in summer 2003 at Hakuba, Nagano, Japan at the skirts of Japan Alps. Details are the next.
Inspiration   
The Time of Quantum
In August  2003, I went to Hakuba in Nagano prefecture for the summer vacation with my family.  At that time I had been thinking on the form of language for which I wrote the paper,  that connects with time inherent in characters, in March 2003 also at Hakuba.
At night of August 23 in cottage, I casually saw the advertising paper of electric dictionary.  The paper was brought from the convenience store near  the cottage in the evening. The dictionary  on the paper was Seiko’s English-Japanese dictionary that has additionally consultation for Chinese or French language with large scale. I vaguely considered that after this dictionaries are necessarily taken these multi-lingual way.
At the time I suddenly  realized that the form of language may be spherical style in which language contains all the information in itself.That was rather satisfied solution for the tough problem of language that I had been carrying in my life from my twenties.
I wrote the sketch-like paper of the theoretical approach after returning home of Tokyo. The paper was read at the international symposium of UNESCO opened in winter 2003 at Nara. In the paper, the spherical substance of language is seemed to be quantum in DELBRUCK’s image-like physical world. After 5 years from the inspiration at summer of Hakuba,  now I consider that spherical essence is manifold in infinite dimensional world.
Now I also realize that the toughest problem of language is minutely solvable in mathematical approach that has structurally definable terms.
Tokyo
September 29, 2008
[January 23, 2012]
The title changed.
The former title is “From Quantum to Manifold”.
[Postscript. January 25,2012]
On quantization of Language.
Refer to the next.

11.
Symmetry 2

Symmetry seems to be related with two elements of language universals, distance and time.
Refer to the  next.
...............................................................................
Distance Theory Algebraically Supplemented
Brane Simplified Model
1
Bend
TANAKA Akio
1
Language is expressed by< a pair of quanta> that consists of and .
2
Quantum and anti-quantum have reverse direction.
3
Quantum and anti-quantum are in the shape of strings that are separated parallel longitude L apart.
4
Quantum and anti-quantum are in <AdS5 ( 5-dimensional anti-deSitter ) space>.
5
Left side of quantum and anti-quantum is on .
6
Quantum starts from ra in AdS5 space and returns rb in AdS5 space.
ra and rb are equal longitude from D3 brane.
7
of D3 brane in AdS5 space gives quantum and anti-quantum bending near r = 0 that is the location of D3 brane.

8
At the paper Spacetime Symmetry and Escalator Brane in Escalator Language Theory, language goes from ra to rz and in reverse movement to rb.
9
At the paper Actual Language and Imaginary Language, is from ra to rz and is from rz to rb.
10
We see real language that is upper side of rz-rz’. Imaginary language is hidden under side of rz-rz’ in our life.
11
At the paper Mirror Language, imaginary language is mirror language of real language.
12
Distance of language is regarded as the longitude from ra ( or rb ) to D3 brane in AdS5 space. 
[Reference]

Tokyo October 17, 2007
Sekinan Research Field of Language
www.sekinan.org
...............................................................
In 2006 I wrote Escalator Language series.
Paper, Turning Point of Time is a intuitive paper for the three papers of  Distance Theory Algebraically Supplemented 2007.

........................................................................
Escalator Language Theory
Turning Point of Time
For SINGER Isaac Bashevis, WHEN SHLEMIEL WENT TO WARSAW AND OTHER STORIES
TANAKA Akio
1 Time has symmetry.
2 Time consists of the two, real time and imaginary time.
3 Real time is appeared time.
4 Imaginary time is hidden time.
5 Time is recognized by square.
Unit of real time is expressed by t2 = 1.
Unit of imaginary time is expressed by t2 = -1.
6 Real time makes future.
Imaginary time makes past.
Present is expressed by t2 = 0.
7 Time is on escalator belt.
8 Escalator belt has one Turing point of time.
9 Turning point of time is expressed by t2 + x2 + y2 + z2 = r2. Here r is radius of turning point.
10 Turning point of time is smooth and continuously differentiable.
11 Escalator belt has not self-intersection.
12 When time passes, real time and imaginary time are counted.
Real time is visible. Imaginary time is invisible.
13 The belt is in bulky space.
14 Real time is at real number’s coordinate in the space.
Imaginary time is at imaginary number’s coordinate in the space.
15 Real time and imaginary time is symmetrical for coordinate axis.
Tokyo December 22, 2006
Sekinan Research Field of Language
www.sekinan.org


Read more: https://srfl-collection.webnode.com/news/escalator-language-theory-turning-point-of-time-2006/

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

12.
Distance

Distance is one of the most important elements of my language model of language universals. For this element algebraic approach seems to be   clearer description to the model. Refer to the next paper.

......................................................................................
Distance Theory Algebraically Supplemented
Brane Simplified Model Escalator Language Theory>
2
Distance
Direct Succession of Distance Theory
TANAKA Akio
1
Metric model of 5-dimensional spacetime is expressed below from Randall and Sundrum (1999).
ds2 = e2U(y)ηmndxmdxn + dy2
Branes exist at y = 0 and y = d.
Our world is regarded as brane y = 0.
U(y) is called .
2
Using of circle ( radius R )’s identification, y is expressed by that scales from 0 to ±πR.   
Distance is defined in .
According to , of is measured in bulk spacetime of 5 dimensional Anti-deSitter space.
3
Distance in is expressed from <y = –πR> to <y = 0> and from <y = 0> to <y = +πR>.
Now “from <y = –πR> to <y = 0>” is called and “from <y = 0> to <y = +πR>” is called .
Values of are same at and .
4
In (abbreviation; DTAS), word is regarded by ’s value.

 
5
Word has distance at and .
6
Now distance at is called distance of and distance at is called distance of .
[References]
Tokyo October 26, 2007
Sekinan Research Field of Language
www.sekinan.org
........................................................................................
Before algebraic approach for distance, I early wrote the more intuitive paper for distance, its name is Distance Theory in 2004.
Refer to the next.
........................................................................................

Distance Theory

TANAKA Akio

1
Distance theory is an extension of Quantum Theory for Language.
2
Distance theory is an extension of strength rule in Quantum Theory for Language.
3
Distance theory is considered for the purpose of the guarantee to language.
4
What quanta of language propel to the end of the sentence is for the purpose of the guarantee to language, in which quanta of language finally unite the real world in the end of propelling.
5
The guarantee to the inherent signification of indicator in quantum of language is quantified by the distance which starts from the real world to the quantum of language.
6
A quantum consists of indicators.
An indicator has a signification and a period inherently.
The structure of quantum is indicated in Quantum Theory for Language.
7
An inherent signification is an element in a quantum.
An inherent time is an element in a quantum.
There are two types of elements, significant and periodical.
Element is defined.
8
A significant element gets a signification from the real world.
A periodical element gets a time from the real world.
9
An indicator gets a meaning and a period from elements.
10
An element emerges from the real world to the language world.
An indicator gets power from the elements in the language world.
A quantum moves in the language world by the power of indicators.
11
An element emerges to the language world, because each element has immanent perceptible area which works upon visual sensation and auditory sensation of the human beings.
12
An indicator gets energy in the language world, because each indicator has a tendency which will approach and finally coincide with the real world.
This continuous tendency guarantees the trust in language for the human beings.
13
A quantum moves in the language world toward the real world.
A quantum is not guaranteed in the situation of cessation.
A quantum is guaranteed by the connection to the real world.
Therefore a quantum propels to the real world.
14
Indicators make meaning and connection rule in a quantum, both are derived from significant and periodical elements in an indicator.
15
Meaning is guaranteed by the tendency of coincidence with the real world.
Guarantee of the meaning is reduced by the remoteness of distance from the real world.
16
Connection rule is decided by periodical elements in indicators.
Details are indicated in Quantum Theory for Language.
17
Signification in an indicator and meaning in a quantum once emerged are occasionally transformed or expanded in the language world.
This alteration is called multiplication.
Multiplication is defined.
18
Multiplication generally occurs by the addition of signification in an indicator.
But multiplication in meaning of a quantum sometimes occurs without any addition oneself.
19
Multiplication in a quantum without addition occurs by situational transition in the language world.
20
Situational transition in a quantum is caused by difference of distance from the real world.
Difference of distance at a quantum is a proceeding of abstract thinking in human beings.
21
A quantum of language itself becomes in the language world.
Word is defined.
Therefore each word has a distance toward the real world.
A distance immanent in a word does not emerge itself.
Distance emerges in the linear situation of words gathering.
This situation is called .
Sentence is defined.
Therefore sentence is an emergence of distance in words gathering.
Words form a line, thereafter one arrangement is determined.
Sentence is realized in our world.
22
In Chinese language, /lai/ come has a larger distance than /liao/completion.
Words are arranged from the end of a sentence, according to the own- possessing- distance.
Therefore /lai le/ having come is realized.


Tokyo 
May 5, 2004
For the memory of Kusatsu Shiranesan March 30, 2004
Sekinan Research Field of Language


Postscript 
[Referential note / October 14, 2007]
early work.>
On Time Property Inherent in Characters   Hakuba March 28, 2003
Quantum Theory for Language Synopsis   Tokyo January 15, 2004
Reversion Theory Tokyo September 27, 2004
Prague Theory Dedicated to KARCEVSKIJ, PRAGUE and CHINO   Tokyo October 2, 2004
Mirror Theory For the Structure of Prayer   Dedicated to the Memory of CHINO Eiichi   Tokyo June 5, 2004
Mirror Language   Tokyo June 10, 2004
Guarantee of Language For LÉVI-STRAUSS Claude   Tokyo June 12, 2004
Actual Language and Imaginary Language To LÉVI-STRAUSS Claude    Tokyo September 23, 2004

[Referential note / December 25, 2007]
Algebraic Linguistics / From September 11, 2007
Distance Theory Algebraically Supplemented / From October 4, 2007
Noncommutative Distance Theory / From November 30. 2007

[Referential note / July 7, 2008]

For KARCEVSKIJ Sergej

Invitation by Theme-Time / Data first arranged at Tokyo January 6, 2008

Invitation by Theme-Distance / Data first arranged at Tokyo February 20, 2008

Holomorphic Meaning Theory / From Tokyo June 15, 2008
Stochastic Meaning Theory / From Tokyo June 22, 2008

[Referential note / December 7, 2008]
Distance of Word / Tokyo November 30, 2008 / sekinan.wiki.zoho.com
Reflection of Word / Tokyo December 7, 2008 / sekinan.wiki.zoho.com
Mirror Theory Group / Tokyo December 9, 2008

[Referential note / December 22, 2008]

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

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

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

Warp Theory
Quantum Warp Theory / Warp
Warp Theory Group
Quantum Warp Theory Group

Amplitude of Meaning Minimum / sekinan.wiki.zoho.com
Complex Manifold Deformation Theory / sekinan.wiki.zoho.com

[Referential note / December 23, 2008]
Time of Word / sekinan.wiki.zoho.com

[Referential note / January 1, 2009]

Orbit of Word / sekinan.wiki.zoho.com

[Referential note / January 31, 2009]

Word Problem of Word-hyperbolic Group / sekinan.wiki.zoho.com

Read more: https://geometrization-language.webnode.com/products/distance-theory/
................................................................................

13.
Distance  2
Hoph algebra

The upper section 12.'s algebraic defined distance is also related with Hoph algebra.
Refer to the next.
S3 and Hoph Map
......................................................................
Distance Theory Algebraically Supplemented
Brane Simplified Model
3
S3 and Hoph Map
TANAKA Akio
1
From RS model, and are abstracted.
Refer to the next.
2
s from to at and are both seemed as circle S1.
3
3-dimensional sphere S3 = { ( x1, y1, x2, y2 ) | x12 + x22 + y12 + y22 = 1 }
Point of S3     ( x1, y1, x2, y2 )
( x1, y1 ) ≠ ( 0, 0 )     π ( x1, y1, x2, y2 ) = ( x2 + i y2 ) / ( x1 + i y1 ) ∈ C
( x1, y1 ) = ( 0, 0 )    π ( x1, y1, x2, y2 ) = ∞
Hopf map π : S3 → Riemann Sphere, C ∪ {∞} 
Inverse image of a point p     π -1 (p) is S1.
Hoph map is fiber bundle that derived from fiber S1.
On fiber bundle, refer to the next.
4
Now at is identificated as S1.
Two points a ( xa, ya ), b (xb, yb ) at Gauss plane is objected to Riemann sphere.
At Riemann Sphere, two points a, b is marked by a’ ( x’a, y’a ), b’ (x’b, y’b ) .
On Riemann sphere, refer to the next.
Point of S3 is marked by ( x’a, y’a , x’b, y’b ) .
On S3, refer to the next.
5
at is also marked on S3.
6
and are algebraically considered by S3.
Tokyo November 12, 2007
Sekinan Research Field of Language
www.sekinan.org


Read more: https://srfl-collection.webnode.com/news/distance-theory-algebraically-supplemented-brane-simplified-model-3-s3-and-hoph-map-2007/

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

14
Symmetry 3

On symmetry I wrote Symmetry Flow Language and Symmetry Flow language 2 in 2007.
Contents are the next.
Symmetry Flow Language
On Symmetry of Language and Time

1 Premise for Symmetry Flow in Language
2 Riemannian Metric, Flow and Entropy
3 Leaf of Language
Pourparlers>
4 Homology on Language
5 Simplex, Simplicial Complex and Polyhedron
6 Meaning Variation and Time Shift in Word As Homotopy
   


Symmetry Flow Language 2
On Symmetry of Language and Time 2

1 Boundary, Deformation and Torus as Language
2 Time Shift of Meaning in Moduli Space

...........................................................................
Related with Hoph algebra, next papers are referential.


Symmetry Flow Language

5 Simplex, Simplicial Complex and Polyhedron
6 Meaning Variation and Time Shift in Word As Homotopy



Symmetry Flow Language 2

2 Time Shift of Meaning in Moduli Space

Texts are the next.
..........................................................................
Symmetry Flow Language
5
Simplex, Simplicial Complex and Polyhedron
TANAKA Akio
1 Language is given by homology group in topological space.
Homology group is given by module’s chain complex.
{Ck}k=0n, {∂k } k=1n
Ck is module.
∂k: Ck → Ck—1  is homomorphism.
∂k-1o∂k = 0   It is meant that set’s boundary’s boundary becomes null set.
2 Meaning is divided by boundary.
3 Word is given by simplex.
4 Simplex has m+1 vertex and m-dimension. It is called m-simplex.
5 Simplex has orientation that is given by permutation.
6 Orientation gives sentence to vector that consists of words.
7 Vector’s element is given by simplex’s vertex.
8 Sentence is given by simplicial complex.
9 Language is given by polyhedron that consists of union of all simplexes.
10 Polyhedron has triangulation.
11 Sphere’s triangulation is given tetrahedron by homeomorphism.
12 Riemann sphere obtains information from Gauss plane.
Refer to the following papers’ group.
13 Minimum model of Symmetry Flow Language is testified by tetrahedron being led from polyhedron’s triangulation.
Tokyo May 17, 2007


Read more: https://srfl-paper.webnode.com/products/symmetry-flow-language-simplex-simplicial-complex-and-polyhedron/



Symmetry Flow Language
6
Meaning Variation and Time Shift in Word as Homotopy
TANAKA Akio
1< Meaning> in word is given by homotopy.
2< Time> in word is given by continuous map of homotopy.
3 Time is expressed by t of path α(t).
4 In path from x0 to x1 of closed interval [0, 1] , α(0) is called initial point and α(1) is called terminal point.
5< Time shift> in word given by interval of path
5 When path has α(0)= α(1) = x0, it is called loop. In this case, initial point is called base point.
6 Initial point or base point makes
in word.
7< Meaning variation> is generated from    
Refer to the following paper.
8 Loop has group’s structure that is called fundamental group.
9 In homotopy, 3 laws are secured. (1) Reflection law   (2) Symmetry law   (3)transition law
10 Laws in homotopy secures construction of word’s meaning.
11 Loop’s equivalence class makes homotopy class.
12 Homotopy class [α] and [β] has product.   [α] * [β] = [α * β]
13 Product makes in word.
14 Now 2-dimensional square I2 is presented. Continuous map α  to space X is given by the following.
α : I2  →X
α is called 2-loop that expresses dimensional extension of loop concept.
15 2-dimensional sphere is 2-loop.
16 Riemann sphere has information from Gauss plane.
Refer to the following papers’ group.
17 Meaning variation and concomitant time shift in Symmetry Flow Language are testified by 2-loop sphere.
Tokyo May 17, 2007
Sekinan Research Field of Language
Symmetry Flow Language 2
2
Time Shift of Meaning in Moduli Space
TANAKA Akio
1 Word has variation of meaning in accordance with time shift in space.
Refer to the following papers.
2 Space that is deformed successively is described by parameter that is called moduli.
3 Set M consists of all of moduli.
4 Moduli space M (M) is presented for variation of word’s meaning.
5 Parameter t is presented for time shift of meaning in word.
6 Calabi-Yau manifold K has two moduli that are deformation of complex structure and Kähler manifold. Moduli have symmetry that is called Mirror symmetry.
7 K’s Ricci tensor is the following.
Rij¯ = 0   i is regular coordinate. j¯ is non-regular coordinate.
Refer to the following paper.
8 Replaced two moduli in Calabi-Yau manifold K is called reflection Calabi-Yau manifold K~.
Refer to the following paper.
9 In K, Kähler manifold, n-dimensional toric manifold Pn is presented.
Complex torus of Kähler’s form ω described by polar coordinates is the following.
ω = ∑i dzi ∧ dz⁻i = ∑i d|zi|2 ∧ dθi
P1 with line bundle’s direct sum becomes resolved conifold.
10 Conifold’s toric graph has line segment that has the longitude | z1 |2+ | z2 |2 = r. r is parameter of conifold.
11 Parameter r becomes parameter t that is described as complex.
12 Resolved conifold that has parameter t = 0 and is deformed becomes deformed conifold that has S3.
13 Parameter t of deformed conifold S3 means time of word’s meaning variation in language.
14 Deformed conifold S3 means word in language.
15 P1× P1 becomes square that is connected by world sheets.
16 P1× P1 means sentence in language.
Refer to the following paper.
Tokyo May 23, 2007
Sekinan Research Field of Language
.............................................................................................................
15.
Derived category　Time conjecture of language
At language universals, distance and time are the kernel concepts for my language models.
Measure of distance is deeply related with time which is transcendental unit still now.
Category theory is probably very useful for the relation between distance and time.
For these concepts, I ever wrote several trial papers. Refer to the next one.

 


1 Language is expressed as structure of spacetime.
2 Spacetime is expressed as manifold.
3 Affine algebraic variety is selected for description of spacetime.
Algebraic variety is pasted together from affine algebraic variety.
Affine algebraic variety is irreducible affine algebraic set.
Affine algebraic set is the set that consists of common zero point of finite polynominal
Polynominal is in n-dimensional complex affine space.
4 Now n-dimensional projective space P n is presented.
C n+1 \ {O} / ~
O is the coordinate’s origin of complex affine space.
~ is mathematical equivalence on elements of set.
5 Projective space P n is covered by n+1 affine space.
6 Now abelian category and derived category are presented.
Abelian category and algebraic variety is placed together.
Derived category is abelian category’s coherent sheaf’s complex that is composition of successive arrows becomes 0.
7 From derived category, distinguished triangle is presented.
8 Here time conjecture of language is presented.
(1)Distinguished triangle makes the model for shift of time on language.
(2)Time on language is closed, successive and circular in word.
Circulation is worked between starting point and ending point of word.
The origin of shift of time is derived from the following.
The concept of time in language is taught from the following.
SAPIR Edward   LANGUAGE  An Introduction to the Study of Speech   Harcout, Brace & Co. Inc
Special thanks to KAWAMATA Yujiro for the mathematical approach on language research, especially from the following.
KAWAMATA Yujiro   Daisukikagaku to doraiken   Sugaku 58-1, January 2006
Tokyo April 20, 2007

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

At this paper, now I extract the next phrases.
Derived category is abelian category’s coherent sheaf’s complex that is composition of successive arrows becomes 0.
7 From derived category, distinguished triangle is presented.
8 Here time conjecture of language is presented.
(1)Distinguished triangle makes the model for shift of time on language.
(2)Time on language is closed, successive and circular in word.
Circulation is worked between starting point and ending point of word.
Paper group, Language and Spacetime is especially focused at time in space.
Every paper of Language and Spacetime, Symmetry Flow Language and Symmetry Flow Language 2
can be seen at the at the site, SRFL Paper  Top pager's right column's  Paper 2003-2007 .


Paper 2003-2007


Read more: https://srfl-paper.webnode.com/

Language and Spacetime Contents

Papers at Time and Spacetime, next two papers are also important to time concepts for me.
Especially No.2 paper is dedicated to Mac Lane for his Category  Theory's work.
No.9 Paper is dedicated Kohari Akihiko and his days for his early death at unhappy accident in 1970s.

2 Word Containing Time and 4 Dimensional Sphere
 

9 Time Flow in Word
 

Two papers text are the next.

Language and Spacetime Word Containing Time and 4 Dimensional Sphere Dedicated to MAC LANE Saunders

Word Containing Time and 4 Dimensional Sphere
Dedicated to MAC LANE Saunders
1 Word contains time inside.
Refer to the following paper.
2 Word has construction that is expressed by category.
On category, refer to Categories for the Working Mathematicians by Saunders Mac Lane, 1998 Springer-Verlag New York, LLC.
3 Time has starting point and ending point in word.
4 Meaning has starting point and ending point in word.
5 Generally word shifts meaning from starting point to ending point by time.
Refer to the following paper.
6 Primitive word does not shift time and stays in the same point.
7 In category, the names of language are called by following terms.
Meaning is called object.
Shift is called arrow.
Starting point is called domain or source.
Ending point is called codomain or target.
No shift or stay is called identity.
Arrow can be composed. The composed arrow is called composite.
In category two axioms are presupposed.
One is associativity.
The other is unit law.
These terms are all complied with above quoted Mac Lane.
8 Word with starting point is expressed on Gauss plane.
9 Word with ending point is also expressed on Gauss plane.
10 Shift time from starting point to ending point is expressed by coordinates’ transference.
11 Here Riemann sphere is putted with having the same origin to Gauss plane.
12 Word with starting point is projected toward the North Pole of Riemann sphere.
13 Word with starting point has coordinates ws (xs ys, zs) on Riemann sphere’s surface.
Refer to the following paper.
14 Word with ending point has also coordinates we (xe, ye, ze) on Riemann sphere.
15 Here 4 dimensional sphere is putted with having the same origin to Riemann sphere.
16 A indefinite strait line determined by coordinates ws and coordinates we makes coordinates wt (xt, yt, zt, tt) on 4 dimensional sphere’s surface.
17 Coordinates wt on 4 dimensional sphere’s surface is uniquely reflected in word with starting point and word with ending point.
18 Word containing time is uniquely expressed on 4 dimensional sphere’s surface.
Tokyo April 5, 2007

Language and Spacetime Time Flow in Word For KOHARI Akihiro and His Time

Time Flow in Word
For KOHARI Akihiro and His Time
1
Word is expressed by phase space in the situation on dynamical system.
Phase space M has a point x.
Continuous time t is expressed by group R that consists of real number.
Homomorphism φt(x)expresses the system.
φs+t(x) =φs (φt(x)), φ0 (x) = x
2
In thermodynamics, equilibrium is presented.
Equilibrium situation is expressed by (U, V, n).
U is energy in the system. V is volume. n is material.
Here new equilibrium is expressed by (U’, V’, n).
3
Now a category is presented.
(U, V, n) →f (U’, V’, n)
f is function of entropy.
Here (U, v, n) is expressed by P. (U’, V’, n) is expressed by P’.
The situation of equilibrium is expressed by the following.
f : P → P’
4
The situation of equilibrium is rewritten by phase space.
Each situation of equilibrium is expressed by a point x.
After time t, point x and x’ is presented by the following.
φ : φs(x) →φs+t(x)
5
Now equilibrium system is expressed by {φt} that is called flow.
Here phase space M can take tangent vector X.
Point x has tangent vector Xx = dφt(x)/dtt=0
The flow has solution as equation φt(x)/dt = Xφt(x)
6
Here set {φt(x) | t∈R } is presented. The set is called orbit.
7
In word, meaning is expressed by flow and time is expressed by orbit.
Tokyo May 3, 2007



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What is signal? A mathematical model of nerve  Preparation 1-15
is over 
29 December 2018

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Tokyo
29 December 2018
                    



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



Preparation 2

Energy




1.
Generation
From where language is born?
Signal generates language as the Morse code generates letters and language.
But language does not generate signal code because language has not electric energy.
It maybe that signal is the root of language.Truly or not?
What is signal?
What is generation?
I  once wrote a trial paper, Generation Theorem in 2008.
Text is the below.



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



von Neumann Algebra 2
Note
Generation Theorem 
TANAKA Akio
[Main Theorem]
<Generation theorem>
Commutative von Neumann Algebra N is generated by only one self-adjoint operator.
[Proof outline]
N is generated by countable {An}.
An = *An
Spectrum deconstruction       An = ∫1-1  λdEλ(n)
C*algebra that is generated by set { Eλ(n) ; λ∈Q∩[-1, 1], n∈N}     A
A’’ = N
A is commutative.
I∈A
Existence of compact Hausdorff space Ω = Sp(A  )
A   = C(Ω)
Element corresponded with f∈C(Ω)     A∈A
N is generated by A.
[Index of Terms]
|A|Ⅲ7-5
|| . ||Ⅱ2-2
||x||Ⅱ2-2
<x, y>Ⅱ2-1
*algebraⅡ3-4
*homomorphismⅡ3-4
*isomorphismⅡ3-4
*subalgebraⅡ3-4
adjoint spaceⅠ12
algebraⅠ8
axiom of infinityⅠ1-8
axiom of power setⅠ1-4
axiom of regularityⅠ1-10
axiom of separationⅠ1-6
axiom of sumⅠ1-5
B ( H )Ⅱ3-3
Banach algebraⅡ2-6
Banach spaceⅡ2-3
Banach* algebraⅡ2-6
Banach-Alaoglu theoremⅡ5
basis of neighbor hoodsⅠ4
bicommutantⅡ6-2
bijectiveⅡ7-1
binary relationⅡ7-2
boundedⅡ3-3
bounded linear operatorⅡ3-3
bounded linear operator, B ( H )Ⅱ3-3
C* algebraⅡ2-8
cardinal numberⅡ7-3
cardinality, |A|Ⅱ7-5
characterⅡ3-6
character space (spectrum space), Sp( )Ⅱ3-6
closed setⅠ2-2
commutantⅡ6-2
compactⅠ3-2
complementⅠ1-3
completeⅡ2-3
countable setⅡ7-6
countable infinite setⅡ7-6
coveringⅠ3-1
commutantⅡ6-2
D ( )Ⅱ3-2
denseⅠ9
dom( )Ⅱ3-2
domain, D ( ), dom( )Ⅱ3-2
empty setⅠ1-9
equal distance operatorⅡ4-1
equipotentⅢ7-1
faithfulⅡ3-4
Gerfand representationⅡ3-7
Gerfand-Naimark theoremⅡ4
HⅡ3-1
Hausdorff spaceⅠ5
Hilbert spaceⅡ3-1
homomorphismⅡ3-4
idempotent elementⅡ9-1
identity elementⅡ9-1
identity operatorⅡ6-1
injectiveⅢ7-1
inner productⅡ2-1
inner spaceⅠ6
involution*Ⅰ10
linear functionalⅡ5-2
linear operatorⅡ3-2
linear spaceⅠ6
linear topological spaceⅠ11
locally compactⅠ3-2
locally vertexⅠ11
NⅢ3-8
N1Ⅲ3-8
neighborhoodⅠ4
normⅡ2-2
normⅡ3-3
norm algebraⅡ5
norm spaceⅡ2-2
normalⅡ2-4
normalⅡ3-4
open coveringⅠ3-2
open setⅠ2-2
operatorⅡ3-2
ordinal numberⅡ7-3
productⅠ8
product setⅡ7-2
r( )Ⅱ2
R ( )Ⅱ3-2
ran( )Ⅱ3-2
range, R ( ), ran( )Ⅱ3-2
reflectiveⅠ12
relationⅢ7-2
representationⅡ3-5
ringⅠ7
Schwarz’s inequalityⅡ2-2
self-adjointⅡ3-4
separableⅡ7-7
setⅠ7
spectrum radius r( )Ⅱ2
Stone-Weierstrass theoremⅡ1
subalgebraⅠ8
subcoveringⅠ3-1
subringⅠ7
subsetⅠ1-3
subspaceⅠ2-3
subtopological spaceⅠ2-3
surjectiveⅢ7-1
system of neighborhoodsⅠ4
τs topologyⅡ7-9
τw topologyⅡ7-9
the second adjoint spaceⅠ12
topological spaceⅠ2-2
topologyⅠ2-1
total order in strict senseⅡ7-3
ultra-weak topologyⅢ6-4
unit sphereⅡ5-1
unitaryⅡ3-4
vertex setⅡ3-3
von Neumann algebraⅡ6-3
weak topologyⅡ5-3
weak * topologyⅡ5-3
zero elementⅡ9-1
[Explanation of indispensable theorems for main theorem]
ⅠPreparation
<0 Formula>
0-1 Quantifier
(i) Logic quantifier  ┐ ⋀  ⋁  → ∀ ∃
(ii) Equality quantifier  =
(iii) Variant term quantifier
(iiii) Bracket  [  ]
(v) Constant term quantifier
(vi) Functional quantifier
(vii) Predicate quantifier
(viii) Bracket  (   )
(viiii) Comma  ,
0-2 Term defined by induction
0-3 Formula defined by induction 
<1 Set>
1-1 Axiom of extensionality     ∀x∀y[∀z∈x↔z∈y]→x=y.
1-2 Set     a, b
1-3 a is subset of b.    ∀x[x∈a→x∈b].Notation is a⊂b. b-a = {x∈b ; x∉a} is complement of a.
1-4 Axiom of power set     ∀x∃y∀z[z∈y↔z⊂x]. Notation is P (a).
1-5 Axiom of sum     ∀x∃y∀z[z∈y↔∃w[z∈w∧w∈x]]. Notation is ∪a.
1-6 Axiom of separation     x, t= (t1, …, tn), formula φ(x, t)     ∀x∀t∃y∀z[z∈y↔z∈x∧φ(x, t)].
1-7 Proposition of intersection     {x∈a ; x∈b} = {x∈b; x∈a} is set by axiom of separation. Notation is a∩b.
1-8 Axiom of infinity     ∃x[0∈x∧∀y[y∈x→y∪{y}∈x]].
1-9 Proposition of empty set     Existence of set a is permitted by axiom of infinity. {x∈a; x≠x} is set and has not element. Notation of empty set is 0 or Ø.
1-10 Axiom of regularity     ∀x[x≠0→∃y[y∈x∧y∩x=0].
<2 Topology>
2-1
Set     X
Subset of power set P(X)     T
T that satisfies next conditions is called topology.
(i) Family of X’s subset that is not empty set     <Ai; i∈I>, Ai∈T→∪i∈I Ai is belonged to T.       
(ii) A, B ∈T→ A∩B∈T
(iii) Ø∈T, X∈T.
2-2
Set having T, (X, T), is called topological space, abbreviated to X, being logically not confused.
Element of T is called open set.
Complement of Element of T is called closed set.
2-3
Topological space     (X, T)
Subset of X     Y
S ={A∩Y ; A∈T}
Subtopological space     (Y, S)   
Topological space is abbreviated to subspace.
<3 Compact>
3-1
Set     X
Subset of X     Y
Family of X’s subset that is not empty set     U = <Ui; i∈I>
U is covering of Y.     ∪U = ∪i∈I ⊃Y
Subfamily of U 　　V = <Ui; i∈J > (J⊂I)
V is subcovering of U.
3-2
Topological space     X
Elements of U     Open set of X
U is called open covering of Y.
When finite subcovering is selected from arbitrary open covering of X, X is called compact.
When topological space has neighborhood that is compact at arbitrary point, it is called locally compact.
<4 Neighborhood>
Topological space     X
Point of X     a
Subset of X     A
Open set    B
a∈B⊂A
A is called neighborhood of a.
All of point a’s neighborhoods is called system of neighborhoods.
System of neighborhoods of point a     V(a)
Subset of V(a)     U
Element of U     B
Arbitrary element of V(a)     A
When B⊂A, U is called basis of neighborhoods of point a.
<5 Hausdorff space>
Topological space X that satisfies next condition is called Hausdorff space.
Distinct points of X     a, b        
Neighborhood of a     U
Neighborhood of b     V
U∩V = Ø
<6 Linear space>
Compact Hausdorff space     Ω
Linear space that is consisted of all complex valued continuous functions over Ω     C(Ω)
When Ω is locally compact, all complex valued continuous functions over Ω, that is 0 at infinite point is expressed by C0(Ω).
<7 Ring>
Set     R
When R is module on addition and has associative law and distributive law on product, R is called ring.
When ring in which subset S is not φ satisfies next condition, S is called subring.
a, b∈S
ab∈S
<8 Algebra>
C(Ω) and C0(Ω) satisfy the condition of algebra at product between points.
Subspace     A ⊂C(Ω) or A ⊂C0(Ω)
When A is subring, A is called subalgebra.
<9 Dense>
Topological space     X
Subset of X     Y
Arbitrary open set that is not Ø in X     A
When A∩Y≠Ø, Y is dense in X.
<10 Involution>
Involution * over algebra A over C is map * that satisfies next condition.
Map * : A∈A ↦ A*∈A
Arbitrary A, B∈A, λ∈C
(i) (A*)* = A
(ii) (A+B)* = A*+B*
(iii) (λA)* =λ-A*
(iiii) (AB)* = B*A*
<11 Linear topological space>
Number field     K
Linear space over K     X
When X satisfies next condition, X is called linear topological space.
(i) X is topological space
(ii) Next maps are continuous.
(x, y)∈X×X ↦ x+y∈X
(λ, x)∈K×X ↦λx∈X
Basis of neighborhoods of X’ zero element 0     V
When V⊂V is vertex set, X is called locally vertex.
<12 Adjoint space>
Norm space     X
Distance     d(x, y) = ||x-y|| (x, y∈X )
X is locally vertex linear topological space.
All of bounded linear functional over X    X*
Norm of f ∈X*      ||f||
X* is Banach space and is called adjoint space of X.
Adjoint space of X* is Banach space and is called the second adjoint space.
When X = X*, X is called reflective.
ⅡIndispensable theorems for proof
<1 Stone-Weierstrass Theorem>
Compact Hausdorff space     Ω
Subalgebra     A ⊂C(Ω)
When A ⊂C(Ω) satisfies next condition, A  is dense at C(Ω).
(i) A  separates points of Ω.
(ii) f∈A →　f－∈A
(iii) 1∈A
Locally compact Hausdorff space        Ω
Subalgebra     A ⊂C0(Ω)
When A ⊂C0(Ω) satisfies next condition, A  is dense at C0(Ω).
(i) A  separates points of Ω.
(ii) f∈A → f－∈A
(iii) Arbitrary ω∈A ,  f∈A ,  f(ω) ≠0
<2 Norm algebra>
C* algebra     A
Arbitrary element of A     A
When A is normal, limn→∞||An||1/n = ||A||
limn→∞||An||1/n  is called spectrum radius of A. Notation is r(A).
[Note for norm algebra]
<2-1>
Number field     K = R or C
Linear space over K     X
Arbitrary elements of X     x, y
< x, y>∈K satisfies next 3 conditions is called inner product of x and y.
Arbitrary x, y, z∈X, λ∈K
(i) <x, x> ≧0,  <x, x> = 0 ⇔x = 0
(ii) <x, y> = 
(iii) <x, λy+z> = λ<x, y> + <x, z>
Linear space that has inner product is called inner space.
<2-2>
||x|| = <x, x>1/2
Schwarz’s inequality
Inner space     X
|<x, y>|≦||x|| + ||y||
Equality consists of what x and y are linearly dependent.
||・|| defines norm over X by Schwarz’s inequality.
Linear space that has norm || ・|| is called norm space.
<2-3>
Norm space that satisfies next condition is called complete.
un∈X (n = 1, 2,…), limn, m→∞||un – um|| = 0
u∈X   limn→∞||un – u|| = 0
Complete norm space is called Banach space.
<2-4>
Topological space X that is Hausdorff space satisfies next condition is called normal.
Closed set of X     F, G
Open set of X     U, V
F⊂U, G⊂V, U∩V = Ø
<2-5>
When A  satisfies next condition, A  is norm algebra.
A  is norm space.
∀A, B∈A
||AB||≦||A|| ||B||
<2-6>
When A is complete norm algebra on || ・ ||, A is Banach algebra.
<2-7>
When A is Banach algebra that has involution * and || A*|| = ||A|| (∀A∈A),  A is Banach * algebra.
<2-8>
When A is Banach * algebra and ||A*A|| = ||A||2(∀A∈A) , A is C*algebra.
<3 Commutative Banach algebra>
Commutative Banach algebra     A
Arbitrary A∈A
Character X
|X(A)|≦r(A)≦||A||
[Note for commutative Banach algebra]  (   ) is referential section on this paper.
<3-1 Hilbert space>
Hilbert space     inner space that is complete on norm ||x||      Notation is H.
<3-2 Linear operator>
Norm space     V
Subset of V     D
Element of D     x
Map T : x → Tx∈V
The map is called operator.
D is called domain of T. Notation is D ( T ) or dom T.
Set A⊂D
Set TA     {Tx : x∈A}
TD is called range of T. Notation is R (T) or ran T.
α , β∈C,   x, y∈D ( T )
T(αx+βy) = αTx+βTy
T is called linear operator.
<3-3 Bounded linear operator>
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 )
Subset K ⊂H
Arbitrary x, y∈K, 0≦λ≦1
λx + (1-λ)y ∈K
K  is called vertex set.
<3-4 Homomorphism>
Algebra A  that has involution*       *algebra
Element of *algebra     A∈A
When A = A*, A is called self-adjoint.
When A *A= AA*, A is called normal.
When A A*= 1, A is called unitary.
Subset of A     B
B * := B*∈B
When B = B*, B is called self-adjoint set.
Subalgebra of A     B
When B is adjoint set, B is called *subalgebra.
Algebra     A, B
Linear map : A →B  satisfies next condition, π is called homomorphism.
π(AB) = π(A)π(B) (∀A, B∈A )
*algebra    A
When π(A*) = π(A)*, π is called *homomorphism.
When ker π := {A∈A ; π(A) =0} is {0},π is called faithful.
Faithful *homomorphism is called *isomorphism.
<3-5 Representation>
*homomorphism π from *algebra to B ( H ) is called representation over Hilbert space H of A .
<3-6 Character>
Homomorphism that is not always 0, from commutative algebra A  to C, is called character.
All of characters in commutative Banach algebra A  is called character space or spectrum space. Notation is Sp( A ).
<3-7 Gerfand representation>
Commutative Banach algebra     A
Homomorphism ∧: A →C(Sp(A))
∧is called Gerfand representation of commutative Banach algebra A.
<4 Gerfand-Naimark Theorem>
When A is commutative C* algebra, A  is equal distance *isomorphism to C(Sp(A)) by Gerfand representation.
[Note for Gerfand-Naimark Theorem]
<4-1 equal distance operator>
Operator     A∈B ( H )
Equal distance operator A     ||Ax|| = ||x|| (∀x∈H)
<4-2 Equal distance *isomorphism>
C* algebra      A
Homomorphism π
π(AB) = π(A)π(B) (∀A, B∈A )
*homomorphism   π(A*) = π(A)*
*isomorphism     { π(A) =0} = {0}
<5 Banach-Alaoglu theorem>
When X is norm space, (X*)1 is weak * topology and compact.
[Note for Banach-Alaoglu theorem]
<5-1 Unit sphere>
Unit sphere X1 := {x∈X ; ||x||≦1}
<5-2 Linear functional>
Linear space     V
Function that is valued by K     f (x)
When f (x) satisfies next condition, f is linear functional over V.
(i) f (x+y) = f (x) +f (y)   (x, y∈V)
(ii) f (αx) = αf (x)   (α∈K, x∈V)
<5-3 weak * topology>
All of Linear functionals from linear space X to K     L(X, K)
When X is norm space, X*⊂L(X, K).
Topology over X , σ(X, X*) is called weak topology over X.
Topology over X*, σ(X*, X) is called weak * topology over X*.
<6 *subalgebra of B ( H )>
When *subalgebra N of B ( H ) is identity operator I∈N , N ”= N is equivalent with τuw-compact.
[Note for *subalgebra of B ( H )]
<6-1 Identity operator>
Norm space     V
Arbitrary x∈V
Ix = x
I is called identity operator.
<6-2 Commutant>
Subset of C*algebra B (H)     A
Commutant of A     A ’
A ’ := {A∈B (H) ; [A, B] := AB – BA = 0, ∀B∈A }
Bicommutant of A     A ' ’’ := (A ’)’
A ⊂A ’’
<6-3 von Neumann algebra>
*subalgebra of C*algebra B (H)     A
When A  satisfies A ’’ = A  , A  is called von Neumann algebra.
<6-4 Ultra-weak topology>
Sequence of B ( H )     {Aα}
{Aα} is convergent to A∈B ( H )
Topology     τ
When α→∞, Aα →τ A
Hilbert space     H
Arbitrary {xn}, {yn}⊂H
∑n||xn||2 < ∞
∑n||yn||2 < ∞
|∑n<xn, (Aα- A)yn>| →0
A∈B ( H )
Notation is Aα →uτ A
[ 7 Distance theorem]
For von Neumann algebra N over separable Hilbert space, N1 can put distance on τs and τw topology.
[Note for distance theorem]
<7-1 Equipotent>
Sets     A, B
Map     f : A → B
All of B’s elements that are expressed by f(a) (a∈A)     Image(f)
a , a’∈A
When f(a) = f(a’) →a = a’, f is injective.
When Image(f) = B, f is surjective.
When f is injective and surjective, f is bijective.
When there exists bijective f from A to B, A and B are equipotent.
<7-2 Relation>
Sets     A, B
x∈A, y∈B
All of pairs <x, y> between x and y are set that is called product set between a and b.
Subset of product set A×B     R
R is called relation.
x∈A, y∈B, <x, y>∈R     Expression is xRy. 
When A =B, relation R is called binary relation over A.     
<7-3 Ordinal number>
Set     a
∀x∀y[x∈a∧y∈x→y∈a]
a is called transitive.
x, y∈a
x∈y is binary relation.
When relation < satisfies next condition, < is called total order in strict sense.
∀x∈A∀y∈A[x<y∨x=y∨y<x]
When a satisfies next condition, a is called ordinal number.
(i) a is transitive.
(ii) Binary relation ∈ over a is total order in strict sense.
<7-4 Cardinal number>
Ordinal number    α
α that is not equipotent to arbitrary β<α is called cardinal number.
<7-5 Cardinality>
Arbitrary set A is equipotent at least one ordinal number by well-ordering theorem and order isomorphism theorem.
The smallest ordial number that is equipotent each other is cardinal number that is called cardinality over set A. Notation is |A|.
When |A| is infinite cardinal number, A is called infinite set.
<7-6 Countable set>
Set that is equipotent to N     countable infinite set
Set of which cardinarity is natural number     finite set
Addition of countable infinite set and finite set is called countable set.
<7-7 Separable>
Norm space     V
When V has dense countable set, V is called separable.
<7-8 N1>
von Neumann algebra     N   
A∈B ( H )
N1 := {A∈N; ||A||≦1}
<7-9 τs and τw topology>
<7-9-1τs topology>
Hilbert space     H
A∈B ( H )
Sequence of B ( H )  {Aα}
{Aα} is convergent to A∈B ( H )
Topology     τ
When α→∞, Aα →τ A
|| (Aα- A)x|| →0 ∀x∈H
Notation is Aα →s A
<7-9-2 τw topology>
Hilbert space     H
A∈B ( H )
Sequence of B ( H )  {Aα}
{Aα} is convergent to A∈B ( H )
Topology     τ
When α→∞, Aα →τ A
|<x, (Aα- A)y>| →0 ∀x, y∈H
Notation is Aα →w A
<8 Countable elements>
von Neumann algebra N over separable Hilbert space is generated by countable elements.
<9 Only one real function>
For compact Hausdorff space Ω,C(Ω) that is generated by countable idempotent elements is generated by only on real function.
<9-1>
Set that is defined arithmetic・     S
Element of S     e
e satisfies a・e = e・a = a is called identity element.  
Identity element on addition is called zero element.
Ring’s element that is not zero element and satisfies a2 = a is called idempotent element.
To be continued
Tokyo April 20, 2008
Sekinan Research Field of Language
www.sekinan.org
2
Generation 2

I also wrote a overview paper on generation in 2018.
Text is the following.


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



Quantum Language 
between Quantum Theory for Language 2004 and Generation of Word 2008
adding their days and after
 A conclusion for the present on early papers of Sekinan Library
23 January - 26 January 2018
Tokyo


1.
I wrote a paper titled Quantum  Theory  for Language in 2004.
This paper was read at the international symposium on Silk Road at Nara, Japan in December 2003.
The encounter with this time's persons and thoughts are written at The Time of Quantum  in September 2008.

2.
This paper's concept was prepared at Hakuba, Nagano, Japan in March 2003.
This concept was jotted down at the hotel of Hakuba so the publication became late till 2015. The title was named Manuscript of Quantum Theory for Language.
3.
In Autumn 2002 I was hospitalized by pneumonia for two weeks, when I thought to put the linguistic research on old Chinese characters so far in order. The result was arranged as a paper titled On Time Property Inherent in Characters also at Hakuba in March 2003.
4.
Quantum Theory for Language was added proviso, Synopsis, because the paper was thought at that time as a role of a rather long mathematical paper's preface on quantum theory on language.
5.
In 2005 Distance Theory was written as a successive paper of Quantum  Theory  for Language. In those days Reversion Theory in September 2004, Prague Theory in October 2004 were successively written.
6.
Time passed by rapidly.
After some preparations of mathematics, I wrote successive papers related with Quantum Theory for Language. von Neumann Algebra was put at the centre of preparation. The days at that time was simply wrote titled as The Days of von Neumann Algebra and The Days between von Neumann Algebra and Complex Manifold Deformation Theory in 2015.
7.
von Neumann Algebra succeeds from 1 to 4. 2's Generation Theorem was written in April 2008. After von Neumann Algebra, Functional Analysis was written. 2's Generation of Word 's result is directly connected to Quantum Theory for Language's mathematical background.
8.
From 2008 I frequently used Zoho site because of easily writing by mathematical equation system. Complex Manifold Deformation Theory was the first result of Zoho. At Floer Homology Language some complementary fruits were gotten for Quantum Theory, Homology Generation of Language in June 2009 and Homology Structure of Word also in June 2009.
9.
Algebraic geometry had been consistently flowing in Quantum Theory.
Recently written Connection between early paper's quantum and recent paper's geometry, November 2017 summarizes the situation at that time concisely.
10.
Quantum Theory's time series representative is the following.
(1)
(2)
(3)
11.
Basis of On Time Property Inherent in Characters 2003, Manuscript of Quantum Theory for Language 2003 and Quantum  Theory  for Language 2004 are all led by Qing Dynasty's linguistic (Xiaoxue) tradition, especially from WANG Guowei, whose influence is written at The Time of WANG Guowei in December 2011.

12.
In 1970s at my age 20s, while I had read WANG Guowei, also read Ludwig Wittgenstein, from whom I narrowly learnt writing style that was maintained through early papers. On Wittgenstein I wrote The Time of Wittgenstein in January 2012. Especially written essayFor WITTGENSTEIN Ludwig Position of Language intermittently wrote from December 2005 to August 2012.    
13.
WANG Guowei taught me the micro phase of language and Edward Sapir taught me the macro phase of language. His book,  Language 1921 shows us the conception of language's change system, Drift. I ever wrote 
some essays on him and his book titled Flow of Language in September 2014.
On Edward Sapir I recently wrote a essay titled Edward Sapir gave me a moment to study language universals together with Sergej Karcevskij  in July 2017. 
14.
I met again with CHINO Eiichi in 1979, from whom I learnt almost all the contemporary linguistics' basis, because of my bias to Chinese historical linguistics ( Xioxue) and Japanese classical phonology in characters. Reunion with CHINO was written at a essay titled Fortuitous Meeting What CHINO Eiichi Taught Me in the Class of Linguistics in December 2004. Also wrote Under the Dim Light in August 2012, CHINO Eiichi and Golden Prague in June 2014, Coffee shop named California in February 2015 and Prague in 1920s in April 2016.

15.
CHINO Taught me the existence of Linguistic Circle of Prague and Sergej Karcevskij  at Prague in 1920s. I wrote Linguistic Circle of Prague in July 2012 and also wrote on Karcevskij, Gift from Sergej Karcevskij in October 2005, Sergej Karcevskij, Soul of Language in November 2012, , Follower of Sergej Karcevskij in November 2012  Meaning Minimum On Roman Jakobson, Sergej Karcevskij and CHINO Eiichi in April 2013 and For KARCEVSKIJ Sergej from time to time.

16.
In 1970s, I also learnt mathematics for applying to describe language's minute situation. I had thought that language had to be written clear understanding form for free and precise verification going over philosophical insight. When set theory led by Kurt Godel was raised its head to logical basis, I was also deeply charmed by it. But even if  fully using it, language's minute situation seemed to be not enough to write over clearly by my poor talent. The circumstance was written titled ​Glitter of youth through philosophy and mathematics in 1970s in March 2015. .

17.
One day when I found and bought Bourbaki's series Japanese-translated editions, which were seemed to be possibility to apply my aim to describe language's situation. But keeping to read them were not acquired  at that time. So I was engrossed  in Chinese classical linguistics achieved in Qing dynasty, typically DUAN Yucai, WANG Niansun, WANG Yinzhi and so forth. The days were written as  The Time of Language Ode to The Early Bourbaki To Grothendieck.

18.
Algebraic geometry began from von Neumann Algebra. After these days, Zoho time came to me. Its first result is shown as the title Complex manifold Deformation Theory in 2008. Distance of Word in November 2008 is a mathematical conclusion of Distance Theory in May 2005. Zoho's main papers were seen at the site Sekinan Zoho.

19.
Distance Theory has some derivations towards physical phases in my thought. Distance Theory Algebraically Supplemented Brane Simplified Model was written in October-November 2007. Each paper is the following.

Distance Theory Algebraically Supplemented
Brane Simplified Model
Bend
Distance <Direct Succession of Distance Theory>
S3 and Hoph Map
Physics was one of the most fantastic fields in high school days. I ever wrote the days of yearning for physics and after that. Perhaps Return to Physics in April 2014, Winding road to physics in January 2015, Thanks to physics about which I ever dreamt in my future in April 2015,

20.
After 2008 at Zoho sites, mathematics based language papers were successively written aiming clearer definition. Zoho's annual papers are shown at Sekinan Zoho's Zoho by year from 2008 to 2013. While I continued writing papers, my aim was gradually changed to confirm language's basis through mathematical, especially algebraic geometrical description by language models a little parting from natural language. The circumstances behind confirmation was written at Half farewell to Sergej Karcevskij and the Linguistic Circle of Prague in October 2013 and 40 years passed from I read WANG Guowei in November 2013.


Read more: https://geometrization-language.webnode.com/products/quantum-language-between-quantum-theory-for-language-2004-and-generation-of-word-2008-adding-their-days-and-after/



3
Energy

Signal needs energy for dispatching messages to the world by hand power of flag semaphore, hand power and electricity of Mores code and light and electricity of  lighthouse.  
Language also needs energy for dispatching by human voice and hand of speaking and writing. But there is not  human energy, there is not language.
Signal's energy is more diverse than language's.
There is energy's diversity at the root of distinction between signal and language.
The question, what is signal is also meant what is energy.
Now I cannot describe accurate explanation to this question clearly.
Little by little it may be able to writing using mathematics hereafter.
Now I would show several trial papers on the relation between language and energy.
Refer to the below. 

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

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://geometrization-language.webnode.com/news/preparation-for-the-energy-of-language/

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


Floer Homology Language ​ ​ ​ Note 1 ​ Potential of Language

Floer Homology Language 
TANAKA Akio  
   ​
 ​
 ​
  
Note 1 ​
Potential of Language ​
 ​
 ​
 ​
¶ Prerequisite conditions ​
Note 6 Homology structure of Word​
 ​
 ​
1 ​
(Definition) ​
(Gromov-Witten potential) 
 ​
 ​
2 ​
(Theorem) ​
(Witten-Dijkggraaf-Verlinde-Verlinde equation) 
 ​
 ​
 ​
3 ​
(Theorem) ​
(Structure of Frobenius manifold) ​
Symplectic manifold     (M, wM) ​
Poincaré duality     < . , . > ​
Product     <V1°V2, V3> = V1V2V3( ) ​
(M, wM) has structure of Frobenius manifold over convergent domain of Gromov-Witten 
potential. ​
 ​
4 ​
(Theorem) ​
Mk,β (Q1, ..., Qk) =  ​
 ​
N(β) expresses Gromov-Witten potential. ​
 ​
  ​
[Image] ​
When Mk,β (Q1, ..., Qk) is identified with language, language has potential N(β). ​
 ​
 ​     
[Reference] 
Quantum Theory for language / Synopsis / Tokyo January 15, 2004  
 ​  
First designed on <energy of language> at ​
Tokyo April 29, 2009 ​
Newly planned on further visibility at ​
Tokyo June 16, 2009  
Sekinan Research Field of language 

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


4
Energy 2

Signal and language send message, for which they need energy directly or indirectly.
What exists between message and energy?
I ever wrote several trial papers on this theme.
Here I show the two of them

1.
2.

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

Complex Manifold Deformation Theory 
​
Conjecture A 
4 Amplitude of Meaning Minimum 
​
TANAKA Akio 
     
​
Conjecture 
Meaning minimum has finite amplitude.​
​
[View*]​
*Mathematics is a view in which I freely appreciate objects as if I see flowers, mountains 
and vigorous port towns at dawn.  ​
1​
Bounded domain of Rm      Ω​
C∞ function defined in Ω     u, F​
u, F satisfy the next equation.​
F(D2u) = Ψ​
D2u is hessian matrix of u.​
F is C∞ function over Rm×m .​
Open set that includes range of D2u     U​
U satisfies the next.​
(i) Constant λ, Λ     ​
(ii) F is concave.​
2​
(Theorem)​
Sphere that has radius 2R in Ω       B2R​
Sphere that has same center with B2R and has radius σR in Ω      BσR​
Amplitude of D2u     ampD2u​
ampBσRD2u = supBσRD2u – infBσRD2u​
0<σ<1​
C and e are constant that is determined by dimension m and .​
ampBσRD2uCσe(ampBRD2u +  supB2R|D| + supB2R |D2| )​
​
​
[Impression]​
1 Meaning minimum is the smallest meaning unit of word. Refer to the reference #2 and 
#2′.​
2 If meaning minimum of word  is expressed by BσR, it has finite amplitude in adequate 
domain.​
​
​
[References 1 On meaning minimum]​
#1 Holomorphic Meaning Theory / 10th for KARCEVSKIJ Sergej​
#2 Word and Meaning Minimum​
#2′ From Cell to Manifold​
#3 Geometry of Word​
​
[References 2 On generation of word]​
#4 Growth of Word​
#5 Generation Theorem​
#6 Deep Fissure between Word and Sentence​
#7 Tomita’s Fundamental Theorem​
#8 Borchers’ Theorem​
#9 Finiteness in Infinity on Language​
#10 Properly Infinite​
#11 Purely Infinite​
​
[References 3 on distance and mirror on word]​
#12 Distance Theory / Tokyo May 5, 2004 / Sekian Linguistic Field​
#13 Quantification of Quantum / Tokyo May 29, 2004 / Sekinan Linguistic Field​
#14 Mirror Theory / Tokyo June 5, 2004 / Sekinan Linguistic Field​
#15 Mirror Language / Tokyo June 10, 2004 / Sekinan Linguistic Field​
#16 Reversion Theory / Tokyo September 27, 2004 / Sekinan Linguistic Field​
#17 Mirror Theory Group / Tokyo December 9, 2008 / Sekinan Linguistic Field​
​
To be continued​
Tokyo December 17, 2008​
Sekinan Research Field of language​
​
[References 4 / December 23, 2008 / on time of word]​
#18 Time of Word / Tokyo December 23, 2008 / sekinan.wiki.zoho.com

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

Complex Manifold Deformation Theory​
​
Conjecture A 
5 Time of Word 
​
TANAKA Akio 
     
​
Conjecture 
Word has time.​
​
[View]​
¶Mathematics is a view in which I freely appreciate objects as if I see flowers, mountains 
and vigorous port towns at dawn.  ​
1​
Kähler manifold     X​
Kähler form     w​
A certain constant     c​
Cohomology class of w     2πc1(X)​
c1(X)>0​
Kähler metric     g​
Real C∞ function     f​
∫X (ef- 1)wn = 0​
Ric(w) -w = f​
2​
Monge-Ampère equation ​
(Equation 1)​
Use continuity method​
(Equation 1-2)​
Kähler form     w' = w +  f​
Ric(w') = tw' + (1-t)w'​
δ>0​
I = {  }​
3 is differential over t.​
Ding's functional     Fw​
​
4​
(Lemma)​
There exists constant that is unrelated with t.​
When utis the solution of equation 1-2, the next is satisfied.​
Fw(ut)C​
5​
Proper of Ding's functional is defined by the next.​
Arbitrary constant     K  ​
Point sequence of arbitrary P(X, w)K     {ui}​
(Theorem)​
When Fw is proper, there exists Kähler-Einstein metric.​
​
[Impression]​
¶ Impression is developed from the view.​
1​
 If word is expressed by u , language is expressed by Fw and comprehension of human 
being is expressed by C, what language is totally comprehended by human being is 
guaranteed.​
Refere to the next paper.​
#Guarantee of Language​
2​
If language is expressed by being properly generated, distance of language is expressed by 
Kähler-Einstein metric and time of language is expressed by t, all the situation of language 
is basically expressed by (Equation1-2).​
Refer to the next paper.​
#Distance Theory​
3​
If inherent time of word is expressed by t's [δ, 1], dynamism of meaning minimum is 
mathematically formulated by Monge-Ampère equation.​
Refer to the next papers.​
#1<For inherent time>​
On Time Property Inherent in Characters​
#2<For meaningminimum>​
From Cell to Manifold​
#3<For meaning minimum's finiteness>​
Amplitude of Meaning Minimum​
​
​
Tokyo January 1, 2009​
Sekinan Research Field of language​

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

From upper two papers, Amplitude of Meaning Minimum and Time of Word, I think that energy, distance, meaning and time are closely related.



6
Understandability

Meaning is understood in the finite time by human or machine assisted by human. 
Understandability of meaning is closely related with time.
At this situation I ever wrote a following trial paper titled understandability of language in 2008.

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

Complex Manifold Deformation Theory


Conjecture B

2 Understandability of Language


TANAKA Akio
    

Conjecture
Language is understandable.


[View]
0
(Eells-Sampson Theorem)
Compact Riemannian manifolds (M, g), (N, h)
Section curvature of (N, h) everywhere non-positive
Arbitrary C∞ map     f : M → N
Equation 
Solution of the equation exists at .
When there exists ,  is convergent to harmonic map  and  is free homotopic with .
1
(Harmonic map)
Arbitrary variation of  { } 
2
(Section of )
3
(Levi-Civita connection)
Levi-Civita connection of (M, g) and (N, h)     

[Impression]
1
From Eells-Sampson Theorem, if language is supposed to be expressed by the equation and word is supposed to be expressed by  , language is understandable in finite time.The situation contributes guarantee of language. 
2
In infinite time, Language still can be understood by word's generation system  .  

[References]
For impression, refer to the next.
<Understandability of language>
#1 Finiteness in Infinity of Language / Kac-Moody Lie Algebra / Conjecture 1 / Tokyo February 10, 2008
#2 Properly Infinite / von Neumann Algebra 3 / Note 1 / Tokyo May 1, 2008
#3 Purely Infinite / von Neumann Algebra 3 / Note 2 / Tokyo May 1, 2008
<Guarantee of language>
#4 Guarantee of Language / Tokyo June 12, 2004
<Generation system>
#5 Generation Theorem /von Neumann Algebra 2 / Note / Tokyo April 20, 2008



Tokyo January 9, 2009
Sekinan Research Field of language


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

7
Understandability 2

Understandability of signal and language is dependant on time processing.
Finite time is generally understandable for human or human assisted machine.
In case of infinite time, what situation occur to understandability?
I once think about infinity's details at the next two papers at von Neumann Algebra 3 in 2007.


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


von Neumann Algebra 3

Note 1
Properly Infinite  

TANAKA Akio


[Theorem]
On von Neumann algebra N, next are equivalent.
(i) N is properly infinite.
(ii) There exist {En : n∈N}⊂P(N) and En~I, ∑nEn = I.
(iii)There exist E∈P(N) and E~E⊥~I.

[Explanation]
<1 Objection Operator>
<1-1>
Hilbert space     H
Linear subspace of H     Subspace
Subspace that is closed by norm || ・|| of H    Closed subspace
Arbitrary subspace of H         K
K⊥: = {x∈H ; <x, y> = 0, ∀y ∈K}     Orthogonal complement of K
Subspaces of H     K, L
<x, y> = 0 ∀x∈K  ∀y∈L     It is called that x and y are orthogonal each other. Notation is K⊥L.
Direct sum K⊕L : = {x+y ; x∈K, y∈L}
<1-2>
x∈H
d = dist(x, K) : = inf{||x-y|| ; y∈K}
z∈K
d = ||x-z||
z : = PKx
PK is called objection operator from H to K.
<1-3>
von Neumann algebra     N
All of objection operators that belong to N     P (N)
All of unitary operators that belong to N     U (N)

<2 Bounded operator>
<2-1>
Hilbert space      H, K
Subspace of H     D
Map    A
A(λx+μy) = λAx+μAy, ∀x, y∈D, ∀λ, μ∈C
A is called linear operator from H to K.
D     domain of A    Notation is dom A.
Set {Ax ; x∈D}     range of A    Notation is ran A.
<2-2>
dom A = H
Constant M>0
||Ax|| ≦M||x||  (∀x∈H)
A is called bounded operator from H to K
All of As     B(H, K)
H = K
B(H) := B(H, H)
<2-3>
A∈B(H)
A*∈B(H)
<x, Ay> = <A*x, y>
A* is called adjoint operator of A.
A = A*
A is called self-adjoint.
A*A = AA*
A is called normal operator.
A = A* = A2
A is called objection operator.
||Ax|| = ||x|| (∀x∈H)
A is called isometric operator.
A*A = AA* = I   ( I is identity operator.)
A is called unitary operator.
Ker A := {x∈H, Ax = 0}
A that is isometric over (Ker A)⊥ is called partial isometric operator.
<2-4>
von Neumann algebra     N
Commutant of N     N ‘
Center of N     Z := N∩N ‘      
Z = CI
N is called factor.
E∈P(N)
Central projection     E that belongs to Z    
All of central projections     P(Z)
<2-5>
Projection operator     E, F∈P(N)
Partial isometric operator     W∈N
F1∈P(N)
F1≦F
E ~ F1
Situation is expressed by E ≺ F.
≺ gives P(N) partial order relation.

<3 Comparison theorem>
<3-1>
[Theorem]
For E, F∈P(N), there exists P∈P(Z) , while EP≺FP and FP⊥≺EP⊥.

<4 Cardinality>
<4-1 Relation>
Sets     A, B
x∈A, y∈B
All of pairs <x, y> between x and y are set that is called product set between a and b.
Subset of product set A×B     R
R is called relation.
x∈A, y∈B, <x, y>∈R     Expression is xRy. 
When A =B, relation R is called binary relation over A.     
<4-2 Ordinal number>
Set     a
∀x∀y[x∈a∧y∈x→y∈a]
a is called transitive.
x, y∈a
x∈y is binary relation.
When relation < satisfies next condition, < is called total order in strict sense.
∀x∈A∀y∈A[x<y∨x=y∨y<x]
When a satisfies next condition, a is called ordinal number.
(i) a is transitive.
(ii) Binary relation over a is total order in strict sense.
<4-3 Cardinal number>
Ordinal number    α
α that is not equipotent to arbitrary β<α is called cardinal number.
<4-4 Cardinality>
Arbitrary set A is equipotent at least one ordinal number by well-ordering theorem and order isomorphism theorem.
The smallest ordinal number that is equipotent each other is cardinal number that is called cardinality over set A. Notation is |A|.
When |A| is infinite cardinal number, A is called infinite set.
<4-5 Countable set>
Set that is equipotent to N     countable infinite set
Set of which cardinarity is natural number     finite set
Addition of countable infinite set and finite set is called countable set.
<4-6 Zermelo’s well-ordering theorem>
If there exist Axiom of Choice, there exists well-ordering over arbitrary set.
<4-7 Order isomorphism theorem>
Arbitrary well-ordered set is order isomorphic to only one ordinal number.
<4-8 Axiom of choice>
∀x∃f [ f ∈Map(x, ∪x)∧∀y[y∈x∧y≠0 → f(y)∈y]]


To be continued
Tokyo May 1, 2008
Sekinan Research Field of Language
www.sekinan.org

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


von Neumann Algebra 3

Note 2
Purely Infinite  

TANAKA Akio


[Theorem]
The necessary and sufficient condition for what von Neumann algebra N is purely infinite ( Ⅲtype) is what semi-finite normal trace that is not 0 does not exist over N.

[Explanation]
<1 Trace>
<1-1>
Trace over von Neumann algebra N          τ : N+ → [0, ∞]  0∞ := 0
τ is the map that has next condition.
(i) τ ( A+B ) =τA +τB,   ∀A,B∈N
(ii) τ (λA ) = λτ ( A )      ∀A∈N+,   ∀λ∈[0, ∞)
(iii) τ ( A*A ) = τ ( AA* )   ∀A∈N
<1-2>
Trace over von Neumann algebra N          τ
(1) τ is faithful.     A∈N, τ (A) = 0 → A = 0
(2) τ is normal.     Increase net {An} ⊂N+   τ (supα Aα) = supα τ (Aα)
(3) τ is definite.    τ (I ) < ∞
(4) τ is semi-definite.     When A(≠0)∈N+,    there exist B(≠0) ∈N+  while B≦A and τ (B) ≠0.

To be continued
Tokyo May 1, 2008
Sekinan Research Field of Language
www.sekinan.org

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On infinity and finiteness in language, I have not total image through mathematical approach.
At present I wrote several papers related with meaning and time in language.
Several papers are the next.

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Arithmetic Geometry Language​
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TANAKA Akio​
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[DRAFT] 
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Language, Word, Distance, Meaning and Meaning Minimum​
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by Riemann-Roch Formula​
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1​
Finite generated Z module          M​
Positive Definite Hermitian form of M          <  ,   > : M × M → C​
Hermitian Z module           (M, <  ,  >)​
M's volume on <  ,  >          vol (M, <  ,  >) = exp(-
 (M, <  ,  >))​
M           free Z module.​
Free basis of M          e1, ..., er​
vol (M, <  ,  >) = 
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2​
Reduced integer ring          R​
Total quotient ring of R       K​
Z free basis of R                 {w1, ..., wn}​
DR:= det(TrK/Q (wi ・wj))​
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3​
(Proposition)​
vol(R, < , >
hcan = 
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4​
Hermitian R module          (H, h)​
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5​
(Theorem)​
Order        r​
Hermitian R module          (H, h)​
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6​
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Metric of V        
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7​
Impression​
Language := (H, h)​
Word       := 
  ​
Distance  := hV​
Meaning   := V​
Meaning minimum  := H​
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[Reference]​
Cell Theory / Continuation of Quantum Theory for Language / From Cell to Manifold / 
June 2, 2007​
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Tokyo August 15, 2009​
Sekinan Research Field of Language​
sekinan.org​
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