Among the many recommendations that can be made to future researchers, one is to become familiar with past and current work related to their subject of study.
This is illustrated on this page through the example of what I somewhat randomly named "the (E) question," which refers to the decomposition of deformed vector products.
At the beginning of this presentation, I mentioned the abstract link to the notion of torsion. Here, it is a matter of studying the variations of one or more functions of several variables.
Let F(V = E(D, K) ; K) be the set of functions from a vector space (v.s.) V of integer dimension D built over a commutative field K to this field.
Let u be any element of this space V. It is customary to refer this v.s. to a canonical basis Ω which is in reality an element (e1, …, eα, …, eD) in V^D = V x ... x V (D times) and to note the vector u :
u = ∑ uα. eα
Let f be any element of F(V ; K), it acts in such a way that:
u ∈ V → f(u) ∈ K
When the vector u describes a part of V, the function f causes its image to traverse a part of K. In general, this image is not constant, and it is necessary to know how to mathematically describe its variations. This is where the concept of Taylor expansion for numerical functions depending on several variables comes into play. Indeed, the action of the function f can also be described by:
u ≡ (…, uα, …) ∈ KD → f(…, uα, …) ∈ K
The study of these variations has been known for a very long time, notably thanks to the work of Taylor, Mac Laurin, and Lagrange.
Taylor's formula
As this introduction is not intended to provide an exhaustive overview of Taylor series expansions to any given order p, I will limit myself to presenting Taylor's formula for the case of second-order expansions (p = 2), drawing inspiration from references [01; § 7.3, p. 43], [02; chapter 5, p. 41], [03; p. 143]:
f(u + δu)
=
f(u) + < ∇uf(u), δu > + ½. < δu, [Hessu f(u)]. | δu > + || δu ||2. ε(δu)
A reminder regarding Hessians
I remind you that the Hessian appearing in this famous formula contains information about the second-order variations with respect to the components of the vector u of the function being studied f(u).
Hessian and Pythagorean table
I note in passing that any Hessian acts on any element f(u) of K just as a very specific form of "square" Pythagorean table does, built around the "function composition" operation and acting on the gradient operator with respect to the components of the vector u of the function f(u):
∀ f ∈ F(V ; K), ∀ u ∈ V : Hessu ≡ T2(o)(∇u, ∇u)
Reformulation of the Taylor expansion as a Euclidean dot product
But that is not the subject of this introduction. I am talking about the variations of the image of the function f. A few manipulations of Taylor's formula and the assumed commutativity of the field K allow us to write:
df(u)
=
f(u + δu) - f(u)
=
< ∇uf(u), δu > + ½. < δu, [Hessu f(u)]. | δu > + || δu ||2. ε(δu)
=
< δu |. {| ∇uf(u) > + ½. {[Hessu f(u)]. | δu >} + || δu ||2. ε(δu)
The final part of the second-order Taylor expansion is sometimes called the "third-order zero." I will neglect it.
|| δu ||2. ε(δu) = 0(3) ∼ 0
Thus, up to third-order terms, the variations of the function f(u) can be understood as the result of a Euclidean scalar product between the sum of vectors:
| ∇uf(u) > + ½. [Hessu f(u)]. | δu >
... and δu.
Suggestion for a formal link with the question of the decompositions of deformed tensor products
The formalism of the sum of two vectors evokes the way in which I generically defined the decompositions of a family of deformed tensor products:
| ⊗A(a, b) > = [P]. | b > + | z > ∈ E*(D, K)
This formal analogy therefore raises the question of the existence of a pair (A, a) allowing the following relationship to be written:
∃ ? (A, a) ∈ ⊞(D, K) x V :
| ⊗A(a, δu) > = ½. [Hessu f(u)]. | δu > + | ∇uf(u) >
With, to ensure the connection with question (E):
[P] = ½. [Hessu f(u)]
b = δu
z = ∇uf(u)
Explorations conducted elsewhere within the framework of an algebraic resolution of the decomposition question show that this suggestion is not entirely devoid of meaning; e.g.: discover the extrinsic methods.
[01] Fonctions de plusieurs variables ; Université de Rennes 1, L2, MIEE 2014-2015, 55 p.
[02] Ley, O. : Fonctions à plusieurs variables ; INSA, Analyse 3, STPI – 2ème année, 60 p.
[03] Encyclopédie Bordas : 50/51 mathématiques, Bordas-Éditeur 1972.