Distance Theory Algebraically Supplemented Note 2 Polydisk Bridge between Ring and Brane / October 31, 2007

Distance Theory Algebraically Supplemented

Note 2

Polydisk

Bridge between Ring and Brane

TANAKA Akio

1

For algebraic approach toward language, analytic space is prepared

Polydisk Dⁿ that is useful* to identification of some different figures is defined by the following.

Dⁿ = { ( x₁, …, x_n ) ∈ For all i, x_i < 1 }

2

Dⁿ has real topology that is stronger than Zariski topology.

On real topology, refer to the next.

Distance Theory Algebraically Supplemented / Preparatory Consideration 1 Distance / Tokyo October 8, 2007

On Zariski topology, refer to the next.

Algebraic Linguistics / Linguistic Note 13 Zariski Topology / Tokyo August 24, 2007

3

Dⁿ has structure sheaf O^h_Dn that has complex analytic function.

On sheaf and structure sheaf, refer to the next.

Algebraic Linguistics / Linguistic Premise / Premise of Algebraic Linguistics 4-1 / Tokyo September 29, 2007

4

Finite complex analytic function h₁, …, h_m ∈ ( D_n, O^h_Dn )

5

Sheaf of ideal I = ( h₁, …, h_m ) O^h_Dn

On ideal, refer to the next.

Algebraic Linguistics / Linguistic Premise / Premise of Algebraic Linguistics 2-1 / Tokyo September 20, 2007

6

Subset M = V ( I ) = { P ∈ Dⁿ ; All of j, h_j ( P ) = 0 }

7

Sheaf O^h_M = O^h_Dn / I

8

Local ringed space ( M, O^h_M ) is the model of complex analytic space.

On local ringed space, refer to the next.

Algebraic Linguistics / Linguistic Premise / Premise of Algebraic Linguistics 4-2 / Tokyo September 29, 2007

On complex analytic space, refer to the next.

Algebraic Linguistics / Linguistic Note 9 Homomorphism / Tokyo July 28, 2007

9

Neighborhood ( U, O^h_M | U ) is isomorphism of the model.

On isomorphism, refer to the next.

Algebraic Linguistics / Linguistic Premise / Premise of Algebraic Linguistics 2-1 / Tokyo September 20, 2007

[Note]

*Usefulness of polydisk will be developed at Distance Theory Algebraically Supplemented Brane Simplified Model on extension of circle’s identification as line segment.

On circle, refer to the next.

Algebraic Linguistics / Linguistic Note 4 Grassmann Manifold / Tokyo July 21, 2007

Tokyo October 31, 2007

Sekinan Research Field of Language