What is signal?
The existence that generates language
TANAKA Akio
SRFL Paper
Tokyo
21 November - 10 January 2019
SRFL Paper
Tokyo
21 November - 10 January 2019
Original Title
What is signal? A mathematical model of nerve
Preparation 1-15
For father and mother
-----------------------------------------------------------------------------------------------------------------------------
Preparation 2
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
Here I show the two of them
1.
2.
------------------------------------------------
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
| )
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
-------------------------------------------------------------
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
f
2
Monge-Ampère equation
(Equation 1)
Use continuity method
(Equation 1-2)
Kähler form w' = w + 
f
f
Ric(w') = tw' + (1-t)w'
δ>0
I = {
}
}
3
is differential over t.
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
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
Read more: https://srfl-paper.webnode.com/news/complex-manifold-deformation-theory-conjecture-a-5-time-of-word/
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
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
---------------------------------------------------------------------------
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.
TANAKA Akio
[DRAFT]
Language, Word, Distance, Meaning and Meaning Minimum
by Riemann-Roch Formula
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, < , >) =
2
Reduced integer ring R
Total quotient ring of R K
Z free basis of R {w1, ..., wn}
DR:= det(TrK/Q (wi ・wj))
3
(Proposition)
vol(R, < , >
hcan =
4
Hermitian R module (H, h)
5
(Theorem)
Order r
Hermitian R module (H, h)
6
Metric of V
7
Impression
Language := (H, h)
Word :=
Distance := hV
Meaning := V
Meaning minimum := H
[Reference]
June 2, 2007
Tokyo August 15, 2009
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