SRFL News : 2025

1 Language is expressed as structure of spacetime.

2 Spacetime is expressed as manifold.

3 Affine algebraic variety is selected for description of spacetime.

Algebraic variety is pasted together from affine algebraic variety.

Affine algebraic variety is irreducible affine algebraic set.

Affine algebraic set is the set that consists of common zero point of finite polynominal

Polynominal is in n-dimensional complex affine space.

4 Now n-dimensional projective space P ⁿ is presented.

C ⁿ⁺¹ \ {O} / ~

O is the coordinate’s origin of complex affine space.

~ is mathematical equivalence on elements of set.

5 Projective space P ⁿ is covered by n+1 affine space.

6 Now abelian category and derived category are presented.

Abelian category and algebraic variety is placed together.

Derived category is abelian category’s coherent sheaf’s complex that is composition of successive arrows becomes 0.

7 From derived category, distinguished triangle is presented.

8 Here time conjecture of language is presented.

(1)Distinguished triangle makes the model for shift of time on language.

(2)Time on language is closed, successive and circular in word.

Circulation is worked between starting point and ending point of word.

The origin of shift of time is derived from the following.

On Time Property Inherent in Characters Hakuba March 28, 2003

The concept of time in language is taught from the following.

SAPIR Edward LANGUAGE An Introduction to the Study of Speech Harcout, Brace & Co. Inc

Special thanks to KAWAMATA Yujiro for the mathematical approach on language research, especially from the following.

KAWAMATA Yujiro Daisukikagaku to doraiken Sugaku 58-1, January 2006

Tokyo April 20, 2007

Language and Spacetime Construction of Spacetime Especially on Transformation with Boundary for Dimensions 2003-2007

Language and Spacetime

Construction of Spacetime

Especially on Transformation with Boundary for Dimensions

1 Spacetime is putted to manifold.

2 When spacetime is expressed by sphere, spacetime becomes sphere symmetry.

Refer to the following paper.

Word Containing Time and 4 Dimensional Sphere Tokyo April 5, 2007

3 The symmetry is special orthogonal transformational group.

4 Generally n-dimensional SO(n) is compact Rie group.

5 On symmetry of spacetime, geometric invariant is presented.

Now on invariant, metric tensor g is presented.

g = Σg_ijdx_idx_j

g’s invariant transformation is isometry group.

When g_ij→ g_ji is established, infinitesimal motion (Killing vector field) is 0.

6 Here Ricci flow is expressed by the following.

g(t) = g_ij(t)

∂g_ij / ∂t = -2R_ij

R_ijis Ricci curvature tensor.

7 M defined by a time interval [0, T] has a unique g(0) = g_o on Ricci flow.

8 (S⁴,g₀) is 4 dimensional sphere surface with radius 1.

g(t) = r(t)²g₀

g(t) has finite time T = 1/2(n-1).

Ricci flow becomes a point after finite time.

Phenomenon what manifold becomes smaller dimensional space is called collapse.

9 Now word has meaning and time.

10 Here word is expressed by manifold.

Refer to the following paper.

Topological Tolerance Tokyo June 7, 2006

Super Fluidity Tokyo June 17, 2006

Also refer to the following paper.

Structure of Word April 6, 2007

On time of language, refer to the following paper.

On Time Property Inherent in Characters Hakuba March 28, 2003

On time principle of language, refer to the following paper.

Quantum Theory for Language Synopsis Tokyo January 15, 2004

10 Here time of word is defined by manifold’s time of Ricci flow’s collapse.

11 By Thurston conjecture, when compact 3 dimensional manifold is cut and separated by 2 dimensional sphere and torus, while generated compact manifold has sphere boundary, sphere is put together there. As the result new generated manifold has locally uniform structure.

Refer to the following paper.

Cut and Glue Cut-tube-ring generates multidimensional world Tokyo July 7, 2006

Also refer to the following paper.

Torus Chain Tokyo June 10, 2006

12 Peculiar solution of Thurston conjecture is Poincaré conjecture that is what simply connected 3 dimensional compact manifold is homomorphic to 3 dimensional sphere surface S³.

13 According by Thurston conjecture, what put together newly generated 3 dimensional manifold is called connected sum.

Connected sum is expressed by M₁#M₂. M₁ and M₂ are 3 dimensional manifolds.

14 Here sentence is defined by manifold’s connected sum.

Tokyo April 24, 2007

Language and Spacetime Stability of Language 2007

Language and Spacetime

Stability of Language

Language has stability.

Stability is defined by the following.

From KAWAMATA Yujiro, Stab X that consists of the set having all <locally finite> and <numerical> <stability condition> on derived category D(X) has finite dimensional complex manifold’s structure. Stab (X) is conjectured to be connected and simply connected. In derived category, exact sequence is extinct and distinguished triangle is generated.

Refer to the next paper. KAWAMATA Yujiro Daisukikagaku to doraiken Sugaku 58-1 Iwanami Shoten 2006

Now blow up and blow down is presented.

φ: Q_Y(M) → M

M is nonsingular variety. Y is M’s submanifold.

Q_Y(M) is blow up along Y.

φis arrow of algebraic variety

φis called Q_Y(M)’s blow down.

Also birational mapping that is extension of blow up and blow down is presented.

Birational mapping is what rational mappingφ: V→W has mapping ψ: W→V. Here existsφ∘ψ= id_V ψ∘φ=id_W .

By birational mapping, algebraic variety V, W and X is presented. These three are formed to be new distinguished triangle that is supposed to be equivalent with V, W and X.

Word that consists of category makes sentence that consists of distinguished triangle by birational mapping with algebraic method.

Tokyo April 30, 2007

Language and Spacetime Structure of Word From KARCEVSKIJ to MACLANE 2006-2007

Language and Spacetime

Structure of Word

From KARCEVSKIJ to MACLANE

1 Word has structure.

On one case of the structure, refer to the following papers.

Cube Theory Dimension Tokyo March 22, 2006

Cube Theory Structure of Cube Tokyo March 25, 2006

Cube Theory Cube Language Hakuba April 1, 2006

2 Now the simplest structure example is presented.

Word has starting point of meaning. The meaning is called m_s.

Word has time for shifting meaning. The time is called t_s.

Word has ending point of meaning. The meaning is called m_e.

3 By category, word’s m_s, t_s and m_e are expressed below.

t_s : m_s → m_e

Here t_s is replaced by arrow f. m_s and m_e are replaced by objects a and b. The relation is expressed by diagram below.

f : a → b

4 Here word X and word Y are presented as category that consists of topological space or group.

Word X is expressed by f : a → b.

Word Y is expressed by g : b → c.

Composition X and Y is expressed by composite f ∘g.

f ∘g : a → c

5 Here f ∘g is putted to h.

The diagram that consists of objects a, b, c and arrow f, g, h is presented.

6 Now category’s morphism is called functor.

Functor that is commutatively transformed to another functor is called natural transformation.

Natural transformation is called component.

On component, refer to the following paper’s concept <passage>.

Cube Theory Structure of Cube A 10 Tokyo March 25, 2006

Natural transformation means functor’s morphism.

7 Product of two categories B and C is defined by the following.

B ×C’s arrow <b, c> →<b’, c’> is <f, g>. Here, f : b → b’ and g : c → c’

<f’, g’>∘<f, g> = <f’∘f, g’∘g>

8 C×2 consists of C×0, C×1 and arrows combined C×0, C×1.

Here when any natural transformation has unique functor, it is called universal natural transformation.

9 Product category C×2 is expressed by concept <garden> in <Cube Theory>.

10 By upper rough sketch, <Cube Theory>’ concepts are transplanted to category.

11 Word is reconstructed by topological space or group.

12 Structure of word that is presented by KARCEVSKIJ has the chance affirmatively verified in accordance with the concepts of MACLANE.

Tokyo April 6, 2007

Language and Spacetime Generation of Sentence For WANG Guowei and CELAN Paul 2007

Language and Spacetime

Generation of Sentence

For WANG Guowei and CELAN Paul

1 Word consists of meaning and time.

Refer to the following paper.

Structure of Word Tokyo April 6, 2007

2 Now word is expressed by cylinder.

In category, cylinder is made by the following.

X is topological space that has base point x₀.

I is unit closed interval. 　I = {t ｜0≦t≦1}　

Cylinder is expressed by X×I.

Here t is time of word.

X × 0 is meaning of starting point. X × 1 is meaning of ending point.

3 Now Suspension ΣX of topological space X is presented.

ΣX is made by shortening of upper surface X × 0 and under surface X × 1 of cylinder X × I.

4 Now Reduced suspension SX is presented.

SX= S₁ ×X

S₁ is circumference that is stuck together of point 0 and 1 of closed interval I.

Here SSⁿ =Sⁿ⁺¹ Sⁿ is n-dimensional sphere.

SX is made by shortening of x₀ × I from ΣX.

SX is expressed by point.

The point is base point of SX.

5 SX defines group in category.

6 SX has operation of group.

7 Now word follows operation of group in algebra.

8 Connection of words generates sentence.

9 Connection is made by the following.

Now H (I, X) is presented.

H ( I, X) is a road space of topological space X.

H ( I, X) is a set that consists of continuous mapping ψ: I → X at interval I = [0, 1].

Road isψ∈ H (I, X).

Ifψisψ(0) =ψ(1),space X becomes loop space ΩX.

10 If a topological space R is presented, H ( X, ΩR) becomes group.

11 If an Abelian group A is presented, space K_n(n = 1,2,3….), a set H(X, K_n) becomes Abelian space.

This space is identified with connection of words, namely, sentence.

Tokyo April 13, 2007

Language and Spacetime Description of Meaning For KARCEVSKIJ Sergej 2007

Language and Spacetime

Description of Meaning

6^th Time
For KARCEVSKIJ Sergej

1 Word consists of meaning and time.

Refer to the following paper.

Structure of Word Tokyo April 6, 2007

2 Sentence consists of connection of words.

Refer to the following paper.

Generation of Sentence Tokyo April 13, 2007

3 Now presheaf is presented by MACLANE Saunders.

Presheaf is contravariant functor F on small category C. F: C^op → Set (Set is all of small sets and functions.)

Category has duality principle by axiom.

Category C has opposite category C^op.

Arrow f has opposite arrow f ^op.

Object of C^op is object C.

Arrow of C^op is arrow f ^op.

4 Presheaf F gives homomorphism by the following.

V ⊂U ( V and U are arbitrary open sets.)

ρ : F (U) → F (V)

ρ gives restrictive mapping.

5 From category, word is given intuitively as below.

F (V) is starting point of meaning.

F (U) is ending point of meaning.

Presheaf F is time for shifting meaning.

6 Grp that consists of groups and homomorphisms are presented.

Now here exist a group G and other groups H.

Arrow’s set Hom_Grp( H,G) makes the relation between H and G clear. ( Hom_Grp( H,G) is all the arrows’ set connecting objects H and G in category Grp.)

Here G is meaning of word. H is elements of meaning G.

7 Meaning of word is clarified by all the arrow’s set.

8 The clarification is called description of meaning.

Tokyo April 16, 2007