Operator Algebra
Conjecture 3
Recognition
TANAKA Akio
1
Total of words is finite at a certain time.
2
Total of being generated sentences is infinite at a certain time.
3
Recognition of total words is practical for its finiteness.
4
Recognition of total of being generated sentences is uncertain for its infinity.
5
For guarantee of recognition of words and sentences, finiteness is indispensable condition.
6
From the viewpoint of guarantee of recognition, uncertainty of total’s infinity at sentences is demanded.
7
From the correspondence with classification of von Neumann algebra, total of sentences is concerned with <properly infinite>, not concerned with <purely infinite>.
[Reference]
Kac-Moody Lie Algebra / Conjecture 1 / Finiteness in Infinity on Language / Tokyo February 10, 2008
Tokyo March 23, 2008
Sekinan Research Field of Language
Operator Algebra
Conjecture 2
Grammar
TANAKA Akio
1
Word E and F are seemed to be complex local vertex space.
2
Set of all continuous linear map from E to F L ( E, F )
3
E and F are norm space.
4
E and F are Banach space.
5
Linear map x : E → F is bounded.
6
Now bounded linear map x is called operator.
7
Operator x is seemed to be grammar between words E and F.
Tokyo March 2, 2008
Sekinan Research Field of Language
Operator Algebra
Conjecture 1
Order of Word
TANAKA Akio
1 Root of Language is word.
2 Word is a function. It is called word function, abbreviated to WF.
3 WF is holomorphic.
4 In WF, differential and integral is commutative on order relation.
5 Word has meaning.
6 Meaning is led by differential of word. This situation is called <horizontal†>.
7 Word makes sentence.
8 Sentence is led by integral of word. This Situation is called <vertical†>.
9 Word has order of operation on differential and integral.
[Reference]
†On <horizontal> and <vertical>, refer to the next.
Place where Quantum of Language Exists / Tokyo July 18, 2004
Tokyo February 16, 2008
Sekinan Research Field of Language
Operator Algebra
Note 4
Frame Operator
TANAKA Akio
1
Hilbert space H
Sequence of points in H {xn}
Certain constants 0 < A ≦ B < ∞
x ∈ H
A ||x||2 ≦∑n |<x, xn>|2 ≦ B ||x||2
{xn} is frame.
A is lower bound.
B is upper bound.
2
x ∈ H
∑n <x, xn>||xn|| is convergent.
S : x ↦ ∑n <x, xn>xn satisfies A ≦S≦ B.
S is frame operator.
[References]
Frame / Tokyo February 27, 2005
More details, Quantum Theory for Language Map 1
Frame-Quantum Theory / Tokyo March 12, 2005
More details, Frame-Quantum Theory Map 2
Tokyo April 2, 2008
Sekinan Research Field of Language
Operator Algebra
Note 3
Self-adjoint and Symmetry
TANAKA Akio
Hilbert space H, K
Operator from H to K A
Domain of A dom A
Graph of A G ( A ) : = { x ⊕ Ax ; x ∈ A }
Operators A, B
A ⊂ B : = G ( A ) ⊂ G ( B )
Minimum of B containing A Closure of A, described by Ā
Now closure of dom A = H
Operator from H to H Operator over H
x ∈ H <x, Ay> = <x’, y>
A*x = x’
A* that is operator over H A* is adjoint operator of A
When A ⊂ A* A is symmetric operator.
When A = A* A is self-adjoint operator.
When Ā = A** A is essentially self-adjoint.
[References]
Distance Theory Algebraically Supplemented / Distance / Tokyo October 26, 2007
Theme / Peak Symmetry and Infinity / Tokyo February 3, 2008
Tokyo April 1, 2008
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