Sunday 29 May 2022

Sekinan Data: Farewell to Language Universals Revised 2021

Sekinan Data: Farewell to Language Universals Revised 2021: Farewell to Language Universals 18/03/2020 19:40 ; First uploaded date to Sekinan View I ever thought of language from the vie point of ...

Monday 23 May 2022

In the 21st century, language will definitely become one of the most important themes and targets of mathematics for us all 2016-2019

20/05/2019 19:01

In the 21st century, language will definitely become one of the most important themes and targets of mathematics for us all

TANAKA Akio

The main reasons are the following.

1.Language is the most convenient tool for human being.

2.Language has the vast variations in natural language and technical language.

3.Natural language has speech language, written language and their recorded language, for example books, records, CD and so on.

4.Technical language has many computer languages and their related devices.

5.At natural language, there exists many meanings not to confirm clear definitions, for example finite, infinite, discrete, continuity, universal and super.

6.Technical language has naturally defines the meanings by its discrete arithmetical basic using of discreteness.

7.Mathematics is probably belonged to natural language, that uses speech and written language by usual conversations and papers or books.

8.But mathematics can clearly define the meaning through a few axioms and derived theorems. for example boundary, continuity, distance, finite, infinite and space.

9.Mathematics has many strong tools for description, for example mapping, projection and identification.

10.Mathematics has solid structure through long historical verifications from ancient Greece.

References
Boundary:
Why is boundary necessary in language?
Description
THE ROAD TO REALITY A Complete Guide to the Laws of the Universe
Dimension
Language and Dimension
Distance
Distance of Word
Geometry
Arithmetic Geometry Language

Language

Presupposition on Natural Language
Meaning
Meaning Minimum
Space
Quantization of Language
Structure
Homology Structure of Word
True and False
The True-false problem of the Crete

Tokyo

8 February 2016

Geometrization Language

Important theme and target is now at

What is signal? The existence that generates language

Tokyo

3 January 2019

SRFL Paper

Hurrying up to library Note added 2015-2016

Hurrying up to library Note added

20/05/2019 19:03

Hurrying up to library

TANAKA Akio

When I was a high school student, my dream of near future was to go to library everyday for reading, learning and researching my themes at that time unknown. In the school I liked solving easy questions of mathematics and physics, but those were all rudimentary questions for general students aiming university’s entrance.

I thought that there would be clear themes of my own life somewhere in my learning world, which never came up to the surface. So above all things I wanted to go library to find my themes for my study life, in which I probably would satisfy in my youth time. I dreamed my figure hurrying up to library with carrying books under my arm being bothered nothing perfectly.

At university my dream surely came to true. But a new more difficult problem emerged up in my front. It was a talent or gift for keeping study deeply. I was a common person having nothing peculiar gift. From that time my true long winding road to language study started towards hard field or precipitous mountain.

Tokyo
27 January 2015
SIL

[Note added, 20 January 2016]
In June 2016, I become 69 years old, that is the time to remember the life to this day. How has my youth time dream realised? What has I obtained the objects I ever hoped?. Some are realised, some are not. I obtained the real aim to keep studying on. It is in language I ever dreamt. But I have not made enough contribution to the people around me and social itself. I only walked my narrow road for my study, not paid attention to others.

I met many splendid people leading me generally to my study way. I was too happy to my tiny research results ever made. I sincerely thank to all and I truly love this small road. It was a long and winding road as the Beatles sang in my youth time and I feel that I surely got fine strawberry from language field. It is For Ever.

Read more: https://srfl-essay.webnode.com/news/hurrying-up-to-library1/

Sekinan Data: Letter to O. again. When I could not find any object at the campus while I ruminated KIYOOKA's stanza

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Sekinan Data: The first paper on Inherent time in word at SRFL. 2014

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Thursday 19 May 2022

Energy in language 2008-2009

01/05/2019 19:32

Energy in language

2008

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

Energy in language is now preparatory description til now.
vide:

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Stochastic Meaning Theory 4

Energy of Language

For ZHANG Taiyan and Wenshi 1908

TANAKA Akio

Domain Λ∈R³

Substantial particles N-number m-mass

Particles are assumed to Newton dynamics.

Place coordinate of particle i in N-number particles r_i∈Λ

Momentum of particle p_i∈R³

State at a moment γ = (r₁, …, r_N, p₁, …, p_N)

Set of state γ P_{Λ, N}≃Λ^N ×R^3N⊂R^6N

P_{Λ, N}is called phase space.

Volume V

Particles n- mol

Energy U

Parameter space E

Point of E ( U, V, n )

Subspace P_{Λ, N} ( U )

Volume of P_{Λ, N} ( U ) W_{Λ, N} ( U )

Adiabatic operation ( U, V, n ) → ( U’, V’, n’ )

Starting state of γ∈P_{Λ, N}

Ending state of γ∈P_{Λ’, N}

Map of time development f

Volume of P_{Λ’, N}( U’ ) W_{Λ’, N} ( U’ )

Volume of f (P_{Λ, N} ( U ) ) is equivalent to W_{Λ, N} ( U ).

f (P_{Λ, N} ( U ) ) is subspace of P_{Λ’, N}( U’ )

W_{Λ, N} ( U ) ≤ W_{Λ’, N} ( U’ )

Equilibrium state ( U, V, n )

Another equilibrium state ( U’, V’, n’ )

Two volume of equilibrium states are seemed to be one state at phase space W_{Λ, N} ( U ) W_{Λ’, N’} ( U’ )

Operation of logarithm of equilibrium state at phase space S ( U, V, n ) = k log W_{Λ, N} ( U ) , (k ; arbitrary constant)

Phase space 2n- dimension

Differential 2-form ω

Local coordinate q_i, p_i

ω = ∑ⁿ_i₌₁d q_i, ∧dp_i

ω is called symplectic form.

2n- dimensional manifold M

Pair (M, ω)

(M, ω) that satisfies the next is called symplectic manifold.

(i) dω = 0

(ii) ωⁿ≠0

Phase space is expressed by symplectic manifold.

Hamiltonian system

Coordinate ( q, p ) = (q₁, …, q_n, p₁, …, p_n )

Phase space R²ⁿ

C¹ class function H = (q, p, t )

( 1≤i ≤n )

An assumption from upper 8

H : = Sentence

q : = Place where word exists

p : = Momentum of word

t : = Time at which sentence is generated

Equilibrium state of sentence H

Another equilibrium state of sentence H’

Adiabatic process of language H → H’

Entropy of language S

H → H’ ⇔ S (H ) ≤ S (H’ )

[References]

Warp Theory / Tokyo October 24, 2004

Quantum Warp Theory Warp / Tokyo December 31, 2005

To be continued

Tokyo July 24, 2008

Sekinan Research Field of Language

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

Energy Distance Theory

Note 1

Energy and Distance

TANAKA Akio

Curve in 3-dimensional Euclidian space l : [0, 1] → R³

Longitude of l L ( l ) =

Surface S

Curve combines A and B in S l

Coordinate of S φ : U → S

Coordinate of U x₁, x₂

φ = (φ₁, φ₂, φ₃ )

A =φ ( x₀ )

B =φ ( x₁ )

Curve in S l : [0, 1] → R³

Curve on U x ( t )

Ω(x₀, x₁) = { l : [0, 1] → R³| l (0 ) = x₀, l (1 ) = x₁}

x(t)∈Ω(x₀, x₁)

l ( t ) =φ ( x ( t ) )

x ( 0 ) = x₀

x ( 1 ) = x₁

L ( l ) =

dt =

g_ijis Riemann metric.

Longitude is defined by the next.

L ( x, xˑ ) =

Energy is defined by the next.

E ( x, xˑ ) =

∑_I,jg_i,j(x(t))xˑⁱ(t)xˑ^j(t)dt

2 E ( x, xˑ ) ≥ (L ( x, xˑ ) )²

Theorem

For x∈Ω(x₀, x₁), the next two are equivalent.

(i) E takes minimum value at x.

(ii) L takes minimum value at x.

What longitude is the minimum in curve is equivalent what energy is the minimum in curve.

Longitude L is corresponded with distance in Distance Theory.

[References]

Distance Theory / Tokyo May 4, 2004

Property of Quantum / Tokyo May 21, 2004

Mirror Theory / Tokyo June 5, 2004

Mirror Language / Tokyo June 10, 2004

Guarantee of Language / Tokyo June 12, 2004

Reversion Theory / Tokyo September 27, 2004

Tokyo August 31, 2008

Sekinan Research Field of Language

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

Energy Distance Theory

Note 3

Energy and Functional

TANAKA Akio

Riemannian manifold (M, g) , (N, h)

C^∞ class map u : M → N

Tangent vector bundle of N TN

Induced vector bundle on M from TN u^-1TN

Tangent space of N Tu(x)N

Cotangent vector bundle of M TM*

Map du : M → TM*⊗ u^-1TN

Section du ∈Γ(TM*⊗ u^-1TN )

Norm |du|

|du|² =∑^m_i,j₌₁∑ⁿ_αβ₌₁ g^ijh_α_β(u)(δu^α/δxⁱ⁾( δu^β/δx^j⁾

Energy density e(u)(x) = 1/2 |du|²(x), x∈M

Measure defined on M from Riemannian metric g μ_g

Energy E(u) =　∫_Me(u)dμ_g

M is compact.

Space of all u . C^∞(M, N)

Functional E : C^∞(M, N) → R

[Additional note]

1 Vector bundle TM*⊗ u^-1TN is compared with word.

2 Map du is compared with one time of word.

3 Norm |du| is compared with distance of tome.

4 Energy E(u) compared with energy of word.

5 Functional E is compared with function of word.

[Reference]

Substantiality / Tokyo February 27, 2005

Substantiality of Language / Tokyo February 21, 2006

Stochastic Meaning Theory 4 / Energy of Language / Tokyo July 24, 2008

Tokyo October 18 2008

Sekinan Research Field of Language

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

Floer Homology Language

TANAKA Akio

Note1
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

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

Tokyo

30 April 2019

ENSILA

Additional meaning and embedding 2016

Additional meaning and embedding

1.Derived category

Category theoretic mirror symmetry conjecture

Fukaya category is
Fuk(X, ω).

2.
General symplectic manifold is

.
3.

Derived category of A_∞

4.
Conjecture

When

and

have physical mirror relation,

there exists the next triangle category's equivalence

2.Embedding

Derived Category Language 2

Kawamata Conjecture

Conjecture

is birational map between smooth objective algebraic manifolds.

And

At This condition,

there exists next fully faithful embedding.

3.Stable and mobile

Notes for KARCEVSKIJ Sergej

Condition of Meaning
TANAKA Akio
September 11, 2011

[Preparation]

Graded differential algebra

Minimal model of graded differential algebra

Degree of homogenious element x of graded differential algabra |x|

Basis of linear space is given by homogenious and elements x₁, ....., x_n

Λ (V) = Λ(V)_k =Λ (x₁, ....., x_n )

Operation of minimal model

Spherical surface Sⁿ, n≥2

de Rham complex *(Sⁿ)

When n iseaven number,

Volume element of S_n

M_n = Λ (x), |x| = n, dx = 0,

M_2n-1gives minimal model S_n to de Rham complex .

When n is odd number,

M_ngives minimal model S_n to de Rham complex .

[Interpretation]

Word is given by spherical surface.

Meaning of word is given by elements x₁, ....., x_n.

Word has minimal model.

Word becomes formal.

Fundamental group of word contains free group of rank b₁(M).

Here KARCEVSKIJ's "stable part" is identified to fundamental group and " mobile part" is identified to free group.

Refer to the paper Notes for KARCEVSKIJ Sergej, "Dusualisme asymétrique du signe linguistique" .

This paper has been published by Sekinan Research Field of Language.
All rights reserved.
© 2011 by The Sekinan Research Field of Language

4.Additional meaning

Stable and Unstable of Language

For the Supposition of KARCEVSKIJ Sergej

Completion of Language

TANAKA Akio

September 23, 2011

[Preparation]

n dimensional complex space Cⁿ

Open set

Whole holomorphic function over U

Ring sheaf for

U →O^an(U)

Complex analytic manifold C_anⁿ

Algebraic manifold Aⁿ multinomial of C_anⁿ

Ideal of multinomial ring a

[x₁, x₂, ..., x_n]

V(a) = {(a₁, a₂, ..., a_n)

Cⁿ f (a₁, a₂, ..., a_n) = 0,

a }

Whole closed set of V(a)

Fundamental open set D(f) = {(a₁, a₂, ..., a_n)

Cⁿ | f (a₁, a₂, ..., a_n) ≠ 0}

Arbitrary family of open set {Ui}

Easy sheaf F

Zariski topological space

Ring sheaf O

Affine space Aⁿ = (

, O)

Ring R

Set of whole maximum ideal Spm R1

Spm R Spectrum of R

Spm R is Noether- like.

◊

R is integral domain.

Whole of open sets without null set U_x

Quotient field K

Mapping from U_xto whole partial set of K O

O(V(a)^c) =

R_f

^c expresses complementary set.

O is easy sheaf of ring over Spm R that is whole set K.

◊

R is finite generative integral domain over k.

Triple (i) (ii) (iii) is called affine algebraic variety.

(i) Set Spm R

(ii) Zariski topology

(iii) Ring's sheaf O

O is called structure sheaf of affine algebraic variety.

◊

Ring homomorphism between definite generative integral domains

Upper is expressed by

Ring holomorphism O_X(U) → O_Y((^t

)^-1U)

Morphism from affine algebraic variety Y to X ( O_X(U) → O_Y((^t )^-1U), X→Y )

When

is surjection, ^t

is isomorphism overclosed partial set defined by p= Ker

Upper is called to closed immersion.

Ring holomorphism

Morphism between affine algebraic varieties

Kernel of

Image of

It is called that when

is injection

is dominant.

◊

R is medium ring between S and its quotient field K.

When

that is given by natural injection

is isomorphism over open set,

is called open immersion.

◊

When X is algebraic variety, longitude of maximum chain is equal to transcendental dimension of function field k(X).

It is called dimension of algebraic variety X, expressed by dim X.

◊

Defined generative field over k K

Space ( X, O_x )added ring that is whole sets of K that has open covers {U_i}

satisfies next conditions is called algebraic variety.

(i) Each U_iis affine algebraic variety that has quotient K .

(ii) For each i, j

I, intersection

is open partial set of

◊

Tensor product between ring and itself becomes ring by each elements products.

Elements

that defines surjective homomorphism is expressed by

Image

of closed embedding defined by

is called diagonal.

◊

Field K

Ringed space that have common whole set K (A, O_A) (B, O_B)

Topological space C

Open embedding

A and B have common partial set C.

Topological space glued A and B by C

Easy sheaf over W O_W

ahere, arbitrary open set Ø ≠

Ringed space

is called glue of A and B by C.

◊

Intedgral domains that have common quotient field K R, S

Element R a_m ≠ 0

Element S b_n ≠ 0

Spm T

Spm R, Spm T

Spm S

Glue defined by the upper is called simple.

◊

Affine algebraic varieties U¹, U²

Common open set of U¹, U²U^C

Diagonal embedding

When the upper is closed set, glue is called separated.

◊

For simple glue

, next is equivalent.

(*) It is separated.

(**) Ring

is generated by R and S.

◊

R and S are integral domains that have common quotient field K.

For partial ring T=RS generated by R and S, when simple is satisfied, it is called "Spm R ad Spm S are simple glue."

◊

Projective space Pⁿ is simple glue.

◊

Algebraic Variety's morphism is glue of affine algebraic variety's ring homomorphism image.

Algebraic direct product is direct product of affine algebraic variety.

◊

Affine algebraic variety X

Ring over k R

is called R value point of X.

Whole

is called set of R value point of X, expressed by X(R).

Ring homomorphism over k

X(f) := X(R)

X(S)

Ring homomorphism

is function from ring category over k to category of set.

◊

Functors from ring category to set category F, G

Ring R

Family of

over ring R {

}

{

}

has functional morphism.

Functors F,G have isomorphism ( or natural transformation).

Functor from ring's category to set's category that is isomorphic to algebraic variety, is called representable or represent by X, or fine moduli.

◊

Functor from ring's category to set's category F

When