Dimension of Language
TANAKA Akio
Tokyo
4 September 2013
1.
Here language means language model that is described by mathematics.
Here language means language model that is described by mathematics.
2.
Description is done by arithmetic geometry.
Description is done by arithmetic geometry.
3.
Language model has not any relationship with natural language.
Language model has not any relationship with natural language.
4.
Language model adopts the themes derived from some phenomena of natural language.
Language model adopts the themes derived from some phenomena of natural language.
5.
The model dealt here is related with the next two themes.
(1) Parts and whole at the construction of word.
(2) Dimension of word constructed from parts to whole.
The model dealt here is related with the next two themes.
(1) Parts and whole at the construction of word.
(2) Dimension of word constructed from parts to whole.
6.
(Theorem)
The next i and j are regular closed immersion of codimension c, d.
(Theorem)
The next i and j are regular closed immersion of codimension c, d.
i :X→Y, j :Y→Z
Here is given Cartesian diagram of the next scheme.
X'—i'→Y'— j'→Z'
↓ ↓ ↓
X—i→Y— j →Z
↓ ↓ ↓
X—i→Y— j →Z
At this time the next equality sign is set up.
(j O i)! = i! O j! : CHr(Z') → CHr–c–d(X')
7.
Consideration
Here i is supposed to be one meaning minimum and j is the another one meaning minimum.
Also X is supposed to be old word and Z is supposed to be constructed new word and Y is supposed to be internal word that is located between X and Z.
(1) The whole of word is constructed from the parts of word.
(2) The dimension of new word has lower dimension than the old word. The new word's dimension's decrease is equal to the r minus the sum of codimensions of i and j.
That is to say, the dimension of new word decreases than the dimension of old word. The degree of decrease is related with the codimensions of additional meanings.
Consideration
Here i is supposed to be one meaning minimum and j is the another one meaning minimum.
Also X is supposed to be old word and Z is supposed to be constructed new word and Y is supposed to be internal word that is located between X and Z.
(1) The whole of word is constructed from the parts of word.
(2) The dimension of new word has lower dimension than the old word. The new word's dimension's decrease is equal to the r minus the sum of codimensions of i and j.
That is to say, the dimension of new word decreases than the dimension of old word. The degree of decrease is related with the codimensions of additional meanings.
♦ Here ends the paper.
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