Vector Bundle Model by Grothemdieck's Theorem/ For SAEKI Shizuto and our youth days / 6 January 2012

Vector Bundle Model by Grothendieck's Theorem

For SAEKI Shizuto and our youth days

TANAKA Akio

Pⁿ is projective space.
P¹ is projective line.
E is vector bundle over P¹.

Theorem (Grothendieck)
Vector bundle E over projective line P¹ is isomorphic with direct sum of line bundles.

[Note 1]
Language is supposed to be projective space Pⁿ.
Word is supposed to be projective line P¹.
Meaning of word is supposed to be vector bundle E over projective line P¹.
Under the upper suppositions, by theorem (Grothendieck) meaning is identified with direct sum of line bundles.

[Note 2]
X is algebraic manifold.
O^X is module over X.
L is easy sheaf of O^X.
Line bundle satisfies the following at L.
(i) Stalk L_μ at generating point is one-dimensional vector space over functional field k(X).
(ii) L is locally isomorphic with structural sheaf O_X .

[Note 3]
(Replacement (i)) One-dimensional vector space over functional field k(X) is replaced to r-dimentional.
(Replacement (ii)) Locally structural sheaf O_X is replaced to to .
Replacement (i) and (ii) is called vector bundle of rank r or locally free sheaf.

[Note 4]
Language universal model is expressed by .

[Reference]
Vector bundle model is a description of the next paper of Aurora Theory.
Dictoron and Aurora < Language is aurora dancing above us> For SAEKI Shizuto. Tokyo, September24, 2006.

Tokyo
January 6, 2012
Sekinan Research Field of Language

[Reference added, January 10, 2012]
The Time of Language

The Time of Language Ode to The Early Bourbaki To Grothendieck / 10 January 2012

The Time of Language Ode to The Early Bourbaki To Grothendieck TANAKA Akio  In early 1970s, I had think of language from the side of mathematics, that level of mine is very low and primitive, moreover I never had any talent to mathematics. But my eager to trying the approach was going to overcome hard barriers before me. So the route had really fascinated my mind for long time. At that time I had read Chinese classics almost every day. WANG Guowei*, DUAN Yucai and WANG Yingzhi. They were giants on Chinese language historically and modernly. On the other hand I had thought of language generally, not defined by Chinese. But in front of the vast world of language, I had stood still lonely, not taking any method for approaching. Mathematics was the only gleam of hope in the wasteland. I never took the route of ordinary linguistics. I really dreamt a dream that time. There exists set theory before me. Probably there was the influence of Bourbaki**, that several translations to Japanese, shared from Tokyo Tosho Publisher, were on the desk of mine. My talent and endeavour were so low, so I had not any results at that time. My desire was deep but my hand was so shallow. The time passed by. In 1979 the meeting again with CHINO Eiichi*** made me the chance to learn on language, the object was clear and direct. Language universals by mathematics became the never-ending goal of the study hereafter. Sergej Karcevskij**** gave me the courage to the research. All the way to investigation were taught from CHINO, who was the genuine teacher on language. In mathematics I took the route from geometry, especially by projection. Now I stand at algebraic geometry. Grothendieck is in the northernmost at the end of Bourbaki. SAITO Takeshi said at the essay on Grothendieck***** that the object of mathematics for Bourbaki was the set of being attached by construction and the object of mathematics for Grothendieck was the object of category representing the presentable functor. The time has come for describing****** on language by mathematics despite my poor ability. Sincere thanks for the pioneers letting us make the fascinating route of modern mathematics. 1. *WANG Guowei Encounter in life / A Letter /2005 Influenced paper / On Time Property Inherent in Characters / 2003 , Quantum Theory for Language / 2004 2. **Bourbaki SAITO Takeshi. Bourbaki, Mathematics Seminar, vol.41 no.4 487. Nihonhyoronsya, Tokyo, 2002. 3.***CHINO Eiicji First met in 1969, again in 1979. Fortuitous Meeting 4. ****Sergej Karcevskij Note on Karcevskij's theme. Note for KARCEVSKIJ Sergej's "Dudualisme asymetrique du signe linguistique" 5. *****Grothendieck SAITO Takeshi. Grothendieck, Mathematics Seminar, vol.49 no.5 584. Nihonhyoronsya, Tokyo, 2010. 6. ******describing Note on Grothendieck's theorem. Vector Bundle Model Tokyo January 10, 2012 Sekinan Research Field of Language