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The theory of the (E) question

© Thierry PERIAT : Texts, ideas and photos

© by Thierry PERIAT: The Theory of the (E) question - Semantic

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Products and their extensions

Tensor product

See any good book or, as first help:

the extern link on Wikipedia – GB

The operator is denoted Ä(…, …)

Projectile

First argument:  Ä(Projectile, …)

Target

Second argument: Ä(…, Target)

Cube

Within the theory of deformed tensor products, a cube should be understood as a mathematical object fulfilled with elements of a given set K placed at the different knots of a Euclidean cubic crystal. It may also be represented as a superposition of matrices disposed in a three-dimensional non-deformed Euclidean space. Although the analogy with a connection is obvious, it should not systematically and a priori be identified with.


Symmetric

A cube is symmetric when: Aijk = Ajik

Anti-symmetric

A cube is symmetric when: Aijk + Ajik = 0

Reduced

A cube is reduced when: Aiki = Aijk

Anti-reduced

A cube is anti-reduced when: Aijk + Aikj = 0

Symmetric and reduced

Anti-symmetric and anti-reduced

Null

Hypercube

A hyper-cube is a generalization of the concept of cube to a space with a physical dimension greater than three.

Deformed tensor product

A deformed tensor product is a classical tensor product that has been deformed by a cube

The cubes are deforming the classical tensor products acting on a vector space E(D, K) because they are modifying their usual definition as follows:

ÄA(a, b) = Aijk. ai. bj. ek

Where the ek are the basis vector of E(D, K).

 Deformed exterior product

Following an historical way of doing, a deformed exterior product is:

ÙA(a, b)

=

ÄA(a, b) - ÄA(b, a)

=

Aijk. (ai. bj - bi. aj). ek

Deformed Lie product

A deformed Lie product is a deformed exterior product built on an anti-symmetric cube

Elements of a decomposition

Intrinsic ingredients

 

Projectile, target and cube are the intrinsic elements in a mathematical problem (The so-called (E) question) asking for the existence and the formalism of pairs ([P], z) such that:

|ÄA(a, b) > = [P]. |b > + |z > Î E*(D, K)

Main part

([P], …) is the main part in a decomposition ([P], z).

This is an element of M(D, K)

Residual part

(…, z) is the residual part in a

decomposition ([P], z); this is an element in E(D, K)

Trivial

A decomposition is said to be trivial when:

|ÄA(a, b) > = [P]. |b > + |0 >

Non-trivial

A decomposition is non-trivial when its residual part doesn’t vanish.

Methods of decomposition

Intrinsic

An intrinsic method of decomposition is a mathematical method allowing the discovery of one or several pair(s) ([P], z) with the help of intrinsic ingredients only. Up to now, I have only done in in a three-dimensional context for deformed Lie products.

Extrinsic

An extrinsic method of decomposition is any mathematical method offering an answer to the (E) question with the help of ingredients which are not only intrinsic to the question.

Russian dolls

The Russian dolls method is inspired by the well-known traditional objects and describes any procedure allowing the discovery of decompositions when the (E) question is asked in E(D + 1, K) but has been answered in E(D, K).

© Thierry PERIAT, 25 November 2018.