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Energy Distance Theory

Note 4

Finsler Manifold and Distance

TANAKA Akio

Banach space E

C^kmanifold M

Point of M p

Banach space T_xM

Norm of T_xM || ||_x

Finsler metric is defined by the next.

(i) Topology by || ||_xis equal to topology by norm of Banach space.

(ii) Tangent vector bundle T (M)

Point p∈M

Coordinate neighborhood of p (U_α, α), α : U_α→E

Ψ_α : U_α×E → π^-1(U_α) ⊂T (M)

||| v |||_x : = ||Ψ_α (x, v)||_x, x∈U_α , v∈E

C > 0

1/C ||| v |||_p≤ ||| v |||_x ≤C ||| v |||_p, x∈U_α , v∈E

Banach manifold M that has Finsler metric Finsler manifold M

Longitude of M L (σ) : = ∫^b_a ||σ’(t)||dt

p, q∈M

Distance ρ ( p, q ) : = inf { L (σ) }

Distance space ( M, ρ )

When ( M, ρ ) is complete distance space, Finsler manifold is called complete.

Finsler C^kmanifold M

C^kfunction over M f : M → R

Condition (C) is defined by the next.

(i) Subset of M S

f is boundary over S.

inf_S ||df || = 0

Closure of S S^-

df = 0 at point p of S^-

Complete Finsler C² manifold M

C^kclass function f : M → R satisfies condition ( C ).

Theorem

Connected component of M M₀

When f is boundary from below, f has minimum value at M₀.

1 > m/p , m = dim M

Banach space L_1,p( M, R^N)

C^∞ manifold L_1,p( M, N )

Distance of L_1,p( M, R^N) ρ₀

ρ₀( u, v ) = || u – v ||_1,p, u, v ∈ L_1,p( M, R^N)

Proposition

Finsler manifold (L_1,p( M, N ) , || ||_1,p ) is complete.

[Note]

Word is expressed by closed manifold in Banach space.

Distance is expressed by Finsler metric.

[References]

Distance Theory / Tokyo May 4, 2008

Reversion Theory / Tokyo September 27, 2008

To be continued

Tokyo November 7, 2008

Sekinan Research Field of Language

www.sekinan.org

Postscript
[Reference November 30, 2008]

Distance of Word / November 30. 2008 / Sekinan.wiki.zoho.com

Quantum Language between Quantum Theory for Language 2004 and Generation of Word 2008. 2018. Note on David Marr and Grisha Perelman. 2020

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
2018

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

TANAKA Akio
23 January - 26 January 2018
Tokyo
Sekinan Library

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.
In July 2017,  The comparison between 2003 and 2017 From Chinese character's meaning structure to Homological algebraic model of language universals was written looking back the days of quantum.
10.
Quantum Theory's time series representative is the following.
(1)
On Time Property Inherent in Characters 2003
Manuscript of Quantum Theory for Language 2003
Quantum Theory for Language 2004
Reversion Theory 2004
Prague Theory 2004
Distance Theory 2005
Generation Theorem 2008
Generation of Word 2008
Homology Generation of Language 2009
Homology Structure of Word 2009
(2)
von Neumann Algebra 1-4 2008
Functional Analysis 2008
Complex Manifold Deformation Theory 2008-2009
Floer Homology Language 2009
(3)
The Time of Quantum 2008
The Days of von Neumann Algebra 2015
The Days between von Neumann Algebra and Complex Manifold Deformation Theory2015
Connection between early paper's quantum and recent paper's geometry 2017
The comparison between 2003 and 2017 From Chinese character's meaning structure to Homological algebraic model of language universals 2017
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/#.XrS_K2bat-U.blogger

Read more: https://geometrization-language.webnode.com/news/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-2018/

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

Note
On David Marr and Grisha Perelman
5 June - 7 June 2020
Geometrization Language

David Marr's Vision 1982, Marr refers to possibility for the existence of modular design at the upper place in language. He also says that there needs multiple description to research meaning.
The book gives me a bit different way to approach to the root of language aparting from ordinary mathematical method of mine, but if anything, nearly to geometry, in my way, ,Perelman's.
Refer to the next two paper groups, especially at Heat equation and Geometrization Conjecture.
Heat and Geometrization has been the most important target of my research since about 2018.
Perelman's Approach 2
31/01/2016 20:25
Geometry of Geometrization Conjecture09/08/2015 19:08Canonical resoresolution09/08/2015 19:01Geometrization Conjecture09/08/2015 18:53Poincare Conjecture09/08/2015 18:46Parabolic neibourhood09/08/2015 18:41Open metric ball09/08/2015 13:15Maximum principle09/08/2015 13:14Uhlenbeck's trick09/08/2015 13:09Developmental equation09/08/2015 12:55Epsilon-delta method07/08/2015 13:25Heat equation07/08/2015 12:07Statistical mechanics system07/08/2015 11:55Ricci flow's curvature tensor07/08/2015 11:44Ricci flow06/08/2015 22:35Ricci soliton06/08/2015 22:23

Read more: https://srflnote.webnode.com/news-/newscbm_573562/20/
Read more: https://geometrization-language.webnode.com/news/perelmans-approach-2/

What is signal? Total Edition 21 November 2018-23 April 2019
23/04/2019 19:02
What is signal? The existence that generates language
TANAKA Akio
SRFL Paper
Tokyo
21 November - 23 April 2019
Original TitleWhat is signal? A mathematical model of nerve
Read more: https://srfl-theory.webnode.com/news/what-is-signal-total-edition-21-november-2018-23-april-2019/

Sunday, 21 June 2026

Prague Theory Prague Theory 2 About Language Machine 2004

Energy Distance Theory Note 4 Finsler Manifold and Distance 2008

Quantum Language between Quantum Theory for Language 2004 and Generation of Word 2008. 2018. Note on David Marr and Grisha Perelman. 2020

Quantum Language between Quantum Theory for Language 2004 and Generation of Word 2008. 2018. Note on David Marr and Grisha Perelman. 2020