Preparation 2
Energy
1.
Generation
From where language is born?
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}
; 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.
..............................................................................................
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.
12.
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
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
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.
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)
Distance Theory 2005
Generation Theorem 2008
Generation of Word 2008
(2)
Functional Analysis 2008
Complex Manifold Deformation Theory 2008-2009
(3)
The Time of Quantum 2008
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.
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.
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.
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.
- Potential of Language / Floer Homology Language / 16 June 2009
- Homology structure of Word / Floer Homology Language / Tokyo June 16, 2009
- Amplitude of meaning minimum / Complex Manifold Deformation Theory / 17 December 2008
- Time of Word / Complex Manifold Deformation Theory / 23 December 2008
Read more: https://geometrization-language.webnode.com/news/preparation-for-the-energy-of-language/
.......................................................................
Floer Homology Language Note 1 Potential of Language
Note 1
Potential of Language
¶ Prerequisite conditions
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]
First designed on <energy of language> at
Tokyo April 29, 2009
Newly planned on further visibility at
Tokyo June 16, 2009
Read more: https://geometrization-language.webnode.com/news/floer-homology-language-note-1-potential-of-language/?utm_source=copy&utm_medium=paste&utm_campaign=copypaste&utm_content=https%3A%2F%2Fgeometrization-language.webnode.com%2Fnews%2Ffloer-homology-language-note-1-potential-of-language%2F
......................................................................
4
Energy 2
Signal and language send message, for which they need energy directly or indirectly.
What is the relation between meaning and energy.
I
I ever wrote several trial papers.
Here I show two of them.
............................................................................
Complex Manifold Deformation Theory
Conjecture A
4 Amplitude of Meaning Minimum
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
.
ampBσRD2u
Cσe(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
[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
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>
#2<For meaningminimum>
#3<For meaning minimum's finiteness>
Tokyo January 1, 2009
......................................................................
From thee two papers, Amplitude of Meaning Minimum and Time of Word, I suppose that energy,
distance, meaning and time are closely related.
5
Time
distance, meaning and time are closely related.
5
Time
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