N Debergh

J.-P Petit

On spacetime algebra and its relations with negative masses

Keywords: geometric algebra, spacetime algebra, Dirac equation, negative mass

We consider four subsets of the complexified spacetime algebra, namely the real even part, the real odd part, the imaginary even part and the imaginary odd part. This naturally leads to the four connected components of the Lorentz group, supplemented each time by an additional symmetry. We then examine how these four parts impact the Dirac equation and show that four types of matter arise with positive and negative masses as well as positive and negative charges.

Introduction

Geometric algebra was discovered jointly by Clifford and Grassmann in the late 19th century ( [START_REF] Grassmann | Die Ausdehnungslehre[END_REF], [START_REF] Clifford | [END_REF]). Despite its innovative and promising aspects, this algebra did not get the attention it deserved at that time. In fact, most researchers have only exploited the purely algebraic aspects of this concept and have obscured the other facets.

It was not until the work of Hestenes [START_REF] Hestenes | Spacetime algebra[END_REF] that the richness of geometric algebra was fully realized. Since then, some scientists have made significant contributions and firmly believe (an opinion we share) that geometric algebra is the simplest and most coherent language that can reconcile physics with mathematics. The subject itself is covered in detail in ( [START_REF] Hestenes | Spacetime algebra[END_REF], [START_REF] Hestenes | Clifford algebra to geometric calculus[END_REF], [START_REF] Doran | Geometric algebra for physicists[END_REF]).

Geometric algebra can be considered in any physical discipline but where it probably takes a particular importance is the physics of the electron [START_REF] Doran | Spacetime algebra and electron physics[END_REF]. Generalizing the concept of complex numbers, it allows to highlight an interesting alternative to the Dirac theory. This other option is referred to as the 1 real formulation of the Dirac equation or the Dirac-Hestenes equation ( [START_REF] Hestenes | [END_REF], [8], [9]).

This proposition is based on the geometric algebra related to the Minkowski spacetime (namely the spacetime algebra) and, surprisingly enough, only one part of this algebra (the part referred to as the real even part) is necessary to have equivalence with the Dirac formulation.

On a different level, the acceleration of the expansion of the Universe has been observed by astronomers [10]. This discovery implies the existence of unknown matter and energy (i.e. the dark matter and the dark energy). If their natures are still a mystery, there is more and more evidence to say that particles of negative masses [11] could be the best candidates to solve this enigma ( [12], [13], [14], [15], [16], [17], [18]).

The negative mass matter interacting only by gravity with positive mass matter (and not through electromagnetic phenomena), it is not possible to directly experiment it. Its detection is done indirectly by analysing the alteration of the light emitted by positive mass matter. We refer the interested reader to [START_REF] Petit | Janus cosmological model[END_REF] for further details.

It is natural to wonder about a possible relation between spacetime algebra and negative masses and it is precisely the goal of this paper.

In Section 1, we will show, after a brief recall on the basics of the geometric algebra, that if only a part of the content (and by this we mean the even real part) of the spacetime algebra has been really exploited in most of the contributions on the geometric algebra of the electron, it is also possible to consider the other parts (odd real part, even imaginary part and odd imaginary part). These supplementary parts are connected to the usual one by applying to it the discrete unitary operators P ,T ,P T .

This will naturally leads us, in Section 2, to revise the relativistic invariance where not only the restricted Lorentz group SO + (1, 3) will be reached but also its three connected components. When the odd part is included, we will show that the Lie superalgebra gl(2|2) arises naturally. Section 3 will be devoted to the highlighting of the four types of solutions of the Dirac equation, both with positive and negative masses.

In Section 4, we will put in evidence the corresponding equations and solutions within the geometric algebra formalism.

The relativistic invariance of these equations as well as the transformation laws on their solutions will be discussed in Section 5. The remarkable point is the appearance of an additional symmetry compared to the traditional theory, symmetry which brings a transformation law on the mass.

We will conclude in the last Section.

Preliminary: the spacetime algebra and its four components

First, we briefly recall the fundamentals of a geometric algebra.

Let u and v be two vectors of a vector space of dimension n

u = u j e j ; v = v j e j (1)
(here and in the following, summation on repeated indices is understood).

The geometric product of these vectors is defined according to

u v = u.v + u ∧ v (2)
In the first term, we recognize the -symmetric-inner product (equivalent to the scalar product when dealing exclusively with vectors). It gives rise to a scalar (0-vector). In the second term, we find the -antisymmetric-outer product leading to a bivector (2-vector or oriented portion of plan).

In other words

u.v = 1 2 (u v + v u) , u ∧ v = 1 2 (u v -v u) (3)
Applied to the (orthogonal) vector basis, the geometric product is such that

e j e k = -e k e j , j ̸ = k (4)
It also generates a 2 n -dimensional algebra made of one 0-vector, n 1-vectors, n(n-1) 2

2-vectors and so on until one n-vector, respectively given by 1, e j , e j e k , ..., I n ≡ e 1 e 2 ...e n [START_REF] Doran | Geometric algebra for physicists[END_REF] This n-vector I n is also known as the pseudo-scalar unit. Second, we particularize to spacetime algebra (STA) for which n = 4 and

e 2 0 = -e 2 1 = -e 2 2 = -e 2 3 = 1 (6)
From ( 4) and ( 6), it is obvious to see that the basis vectors of the STA are nothing but the generators of the Clifford algebra Cl [START_REF] Grassmann | Die Ausdehnungslehre[END_REF][START_REF] Hestenes | Spacetime algebra[END_REF] and can be realized through the Dirac matrices (in the Weyl realization, for instance):

e 0 = γ 0 = ( 0 I I 0 ) , e j = γ j = ( 0 σ j -σ j 0 ) ( 7
)
where I is the identity matrix of dimension 2 and σ j are the Pauli matrices.

A remarkable fact about STA is that its real even part

E = a 0 I + a j γ j γ 0 - 1 2 ϵ kl j b j γ k γ l + b 0 I 4 , a 0 , a j , b 0 , b j ∈ R (8)
(where ϵ kl j is the completely antisymmetric Levi-Civita tensor) is a geometric algebra in itself, namely the one associated with the 3D Euclidian space (SA). This is easily seen by identifying the first three 2-vectors of the STA with the 1-vectors of the SA e ′ j ≡ γ j γ 0

It is then straightforward to see that

e ′ j e ′ k = -γ j γ k , I 3 = e ′ 1 e ′ 2 e ′ 3 = γ 0 γ 1 γ 2 γ 3 = I 4 (10)
Moreover, we can write

E = a 0 I + a j e ′ j + I 4 (b 0 I + b j e ′ j ) = a 0 I + ⃗ a + I 4 (b 0 I + ⃗ b) = a + I 4 b (11)
Here and in the following, we refer to the concept of relative vector [START_REF] Hestenes | [END_REF] (written with a lowercase letter)

u ≡ u 0 I + u j e ′ j = u 0 I + ⃗ u ( 12
)
by opposition to the one of proper vector (written with a capital letter)

U ≡ u e 0 = u µ e µ ; µ = 0, 1, 2, 3 (13)
In addition to this real even part, we will consider three other contributions. In total, we will have four types of elements of the (complexified) STA:

• the real even part

E = a + I 4 b (14a)
• the real odd part

O ≡ γ 0 E = a 0 γ 0 -a j γ j - 1 2 ϵ kl j b j γ 0 γ k γ l + b 0 γ 1 γ 2 γ 3 (14b)
• the imaginary even part

IE ≡ -iI 4 E = ib 0 I -ib j γ 0 γ j + i 2 ϵ kl j a j γ j γ k -ia 0 I 4 = i(b -a I 4 ) (14c)
• the imaginary odd part

IO ≡ -iγ 1 γ 2 γ 3 ; E = ib 0 γ 0 -ib j γ j + i 2 ϵ kl j a j γ 0 γ k γ l -ia 0 γ 1 γ 2 γ 3 (14d)
Notice that the great majority of the authors focus on the real even part, only. The reason is that this part is sufficient to recover the usual Dirac equation (see Section 4). The only reference we found dealing with other contributions is [START_REF] Sobczyk | Spacetime algebra of Dirac spinors[END_REF] but in a different context than the one involved here (the accent was there put on idempotent matrices).

If we follow the realization ( 7), these four contributions respectively read

E = ( N 0 0 N ) , O = ( 0 N N 0 ) , IE = ( N 0 0 - N ) , IO = ( 0 - N N 0 ) (15)
Here the matrix N is given by

N = (a 0 + ib 0 )I + (a j + ib j )σ j ( 16
)
while N refers, in agreement with the usual notation to

N = (a 0 -ib 0 )I -(a j -ib j )σ j ( 17
)
This involutive automorphism corresponds to the fact that each of the 3D basis vector has been reversed. The incoming calculations will be really simplified by remembering that

N = ( a b c d ) → N = ( d * -c * -b * a * ) ( 18
)
where * is the usual complex conjugate and a, b, c, d four complex numbers. Let us also mention that the operators in (14b)-(14d) are not chosen at random. In fact, as is well known, γ 0 is the parity operator P for the Dirac equation while -iI 4 actually is the chirality operator γ 5 . However we prefer to see this operator as the product of P and the last one -iγ 1 γ 2 γ 3 which has been identified [START_REF] Debergh | [END_REF] as the unitary time-reversal operator T for the Dirac equation.

Consequently, the four contributions read

E, O = P E, IE = P T E, IO = T E ( 19
)
This highly suggests to associate them to the four connected components of the Lorentz group. That's precisely the aim of the next Section.

The relativistic invariance

Following the standard STA approach, we define the position vector by

X = x µ γ µ = ( 0 x x 0 ) (20)
where

x = x 0 I + x j σ j , x = x 0 I -x j σ j .
Let us now consider the transformations that this position vector can undergo. The general relation is

X ′ = M X M ( 21
)
The notation M refers to a particular conjugation (called reversal) which consists in totally swapping the order of the vectors (γ j γ k → γ k γ j , γ j γ k γ l → γ l γ k γ j and so on). It has been chosen so because, in general, M M = diag(αI, α * I), a property rather similar to the unitarity of transformations. Moreover, when matrices of dimension 2 are concerned, this conjugation is identical to the hermitic conjugation

N = ( a b c d ) → N = N † = ( a * c * b * d * ) (22)
We will also need a third conjugation combining the first two ones

N = ( a b c d ) → N ≡ ( N ) = ( d -b -c a ) (23)
The transformations can then be of four types as suggested by ( 19)

X ′ = M E X M E = ( N 0 0 N ) ( 0 x x 0 ) ( N 0 0 N ) → x ′ = N x N (24a) X ′ = M O X M O = ( 0 N N 0 ) ( 0 x x 0 ) ( 0 N N 0 ) → x ′ = N x N (24b) X ′ = M IE X M iE = ( N 0 0 -N ) ( 0 x x 0 ) ( N 0 0 -N ) → x ′ = -N x N (24c) X ′ = M IO X M IO = ( 0 -N N 0 ) ( 0 x x 0 ) ( 0 N -N 0 ) → x ′ = -N x N
(24d) Each of these transformations (24) can be rewritten in a more usual way as

x ′ = Rx (25)
Besides, in all cases, the three rotations are given by

N rotations = e i 2 θ σ j ( 26
)
(we write the same parameter θ for the simplicity of the notation but it is evidently subtended that each transformation has its own parameter) while the boosts are obtained from

N boosts = e 1 2 θ σ j (27)
If we allow the determinant of N to differ from 1 (and it is without any consequence on R whose determinant still is 1), we also have to add a supplementary transformation characterized by

N supplementary = e i 2 θ I (28)
In the case of the transformation (24a), it is the well-known homomorphism between the special linear group SL(2, C) (related to the matrix N when we restrict ourselves to the cases where det N = 1 -a restriction that we will give up in the following-) and the restricted Lorentz group SO + (1, 3) (associated with R). It has to be mentioned that the transformations recovered here -namely three rotations and three boosts-are proper and orthochronous (component L ↑ + of the Lorentz group). In other words, they preserve the orientation of space (a fact that can be seen from det R = 1) and the direction of time (the element at the intersection of the first row and the first column of R is positive). The explicit form of the transformation matrices coming from the geometric algebra point of view is given by

M rotations = e -1 2 θ γ j γ k , M boosts = e -1 2 θ γ 0 γ k , ; M supplementary = e 1 2 θ I 4 (29a)
At the algebra level, we thus have to consider

J jk ≡ - 1 2 γ j γ k , J 0j ≡ - 1 2 γ 0 γ j , J ≡ 1 2 I 4 (29b)
As well known, the six first operators generate the Lorentz algebra so [START_REF] Grassmann | Die Ausdehnungslehre[END_REF][START_REF] Hestenes | Spacetime algebra[END_REF] [J αβ , J λρ ] = g βλ J ρα + g αρ J λβ + g βρ J αλ + g αλ J βρ ; g = diag(1, -1, -1, -1) (30) while J (still with a vanishing trace) commutes with each of these operators. To find back the other three connected components, the three other cases (24b)-(24d) are necessary. The proper retrochronous transformations (i.e. L ↓ + for which the time is reversed, a fact that is marked through a change of sign in R 00 ) are obtained through (24c). Indeed the result (24c) just means that R has been replaced by -R. At the level of the rotations, the interpretation is that not only the time but also one of the spatial coordinates has been reversed. The rotations themselves, in each of the planes subtended by two spatial coordinates, are no longer rotations of angle θ in the clockwise direction but rotation of angle π -θ in the trigonometric direction. At the boosts, all the coordinates have been reversed but the velocity is unchanged. This total inversion will be also found at the additional symmetry:

x ′ = -x.
The explicit form of the corresponding transformation matrices are obtained from (29a) on which P T = -iI 4 is applied. Obviously, they cannot be written as exponentials anymore. However the relations (30) are still valid if we consider the seven operators (29b) on which P T is applied on the left. The improper orthochronous component (i.e. L ↑ -for which the orientation of space has been changed or, in other words, the determinant of R is now equal to -1) is obtained from the odd transformation (24b). In fact, with respect to the original matrix R, assigned to the proper orthochronous transformations, only the first row is unchanged. The other rows have changed sign. In what concerns the rotations, the time is unchanged while a spatial coordinate is reversed. Once again, in each of the plane determined by the two other spatial coordinates, there is no longer a rotation of θ in the clockwise direction but a rotation of π -θ in the trigonometric sense. For the boosts, we notice that the time remains unchanged while the spatial coordinates have been inversed and so is the velocity. The explicit form of the corresponding transformation matrices is obtained from Eqs (29) on which P = γ 0 is applied. Let us also mention that the supplementary transformation is nothing but this P operation. The corresponding matrix R gets the well-known form diag(1, -1, -1, -1). The improper retrochronous transformations (L ↓ -) correspond to (24d). The situation is the reverse of the previous one: only the first row of the matrix R has changed sign. So the rotations are the ones of the proper orthochronous transformations and only time has been reversed. The boosts are, as in the previous context, characterized by an inversion of the velocity due, in this case, to the reversal of time (the spatial coordinates remain unchanged with respect to the proper orthochronous boosts).

As expected, the explicit form of the corresponding transformations is obtained from Eqs (29) on which T = -iγ 1 γ 2 γ 3 is applied. It is thus not surprising that the supplementary transformation reduces to this unitary time-reversal transformation and the corresponding R is diag(-1, 1, 1, 1). Let us end this Section by mentioning that gl(2, C) is usually recognized as the Lie algebra associated with the even part of the STA. It is due to the presence of the matrix N on which no constraint is required. In particular, the determinant (= ad -bc) can take any complex value, say re i θ . We have focused here on the case where r = 1. Nevertheless, there remains an additional parameter (with respect to the Lorentz group) in the presence of the angle, which is associated with this operator J.

If we consider the odd part in addition (as it is the case when we deal with the improper transformations), we speak about a Lie superalgebra, that is precisely gl(2|2). This superalgebra is characterized by the relations [23] [E pq , E rs ] = δ qr E ps -(-1) ((p)+(q))((r)+(s)) δ ps E rq (31)

with the following rules : if p + q and r + s are odd numbers, the Lie bracket is an anticommutator while it reduces to a commutator in all other cases. To be convinced of this link, we realize the Cartan subalgebra generators with

H 1 = E 11 + E 33 = 1 2 I - i 2 I 4 , H 2 = E 22 + E 33 = 1 2 I - i 2 γ 1 γ 2 (32a)
H 3 = E 22 + E 44 = 1 2 I + i 2 I 4 (32b)
H 4 = E 11 + E 22 -E 33 -E 44 + α (E 11 + E 22 + E 33 + E 44 ) = -γ 0 γ 3 + α I (32c)
We complete the identification by realizing the simple raising operators (two even operators and one odd operator) as

E 13 = - i 4 γ 2 γ 3 + 1 4 γ 0 γ 1 + 1 4 γ 3 γ 1 + i 4 γ 0 γ 2 (33a) E 24 = - i 4 γ 2 γ 3 - 1 4 γ 0 γ 1 - 1 4 γ 3 γ 1 + i 4 γ 0 γ 2 (33b) E 32 = - 1 4 γ 0 + 1 4 γ 3 + i 4 γ 0 γ 1 γ 2 - i 4 γ 1 γ 2 γ 3 (33c)
together with their lowering counterparts

E 31 = E † 13 , E 42 = E † 24 , E 23 = E † 32 (33d)
Finally, we add the following odd generators

E 12 = E 13 E 32 , E 34 = E 32 E 24 , E 14 = E 12 E 24 ( 34
)
where the product involved here is the geometric one.

Let us now turn our attention to the Dirac equation.

The Dirac equation with positive and negative masses

A remark to begin with: when Dirac established his famous equation, in the free case, he only imposed conditions on the squares of the matrices (see equation ( 6) of [24]) leaving the field open to two possibilities of signs before the mass term. On page 615 of his paper he specifies that he marks one possible choice to satisfy these conditions. So, at the start, the possibility of having negative masses was already present. Now, let us consider the Dirac equation when the electron is in interaction with an electromagnetic field

(γ µ (∂ µ + i e A µ ) + i m) |ϕ > + = 0 (35)
With the realization [START_REF] Hestenes | [END_REF] and |ϕ > + ≡

( ξ + η + )
, this equation can be written as

{ (∆ + i e A) η + + i m ξ + = 0 ( ∆ + i e A) ξ + + i m η + = 0 ( 36
)
where the usual gradient ∆ issued from STA has been used as well as its conjugate ∆

∆ ≡ ∂ 0 -σ j ∂ j ; ∆ ≡ ∂ 0 + σ j ∂ j ( 37
)
To be coherent, an analogue of the electromagnetic field has also been defined

A ≡ A 0 -σ j A j ; A ≡ A 0 + σ j A j (38)
By taking the complex conjugate of the system (36) and then multiplying it, on the right, by -iσ 2 , we obtain

{ ( ∆ -i e A) (-iσ 2 η * + ) -i m (-iσ 2 ξ * + ) = 0 (∆ -i e A) (-iσ 2 ξ * + ) -i m (-iσ 2 η * + ) = 0 (39)
Here we have made use of

-iσ 2 ∆ * = ∆ (-iσ 2 ) ; -iσ 2 A * = A (-iσ 2 ) ( 40
)
We are now ready to highlight four types of matter by looking closely at the systems (36) and (39). Indeed, the system (36) shows us that if the electron of positive matter

|ϕ > e,+ = ( ξ + η + ) ≡     ϕ 1 ϕ 2 ϕ 3 ϕ 4     (41a)
is the usual solution, there is an additional solution corresponding to a change of sign in the last term and thus to an electron of negative mass ; it is given by |ϕ > e,-=

( ξ + -η + ) ≡     ϕ 1 ϕ 2 -ϕ 3 -ϕ 4     (41b)
In what concerns the system (39), it tells us that there is a solution associated with a positron of positive mass

|ϕ > p,+ = ( -iσ 2 η * + iσ 2 ξ * + ) ≡     -ϕ * 4 ϕ * 3 ϕ * 2 -ϕ * 1     (41c)
as well as a solution corresponding to a positron of negative mass

|ϕ > p,-= ( -iσ 2 η * + -iσ 2 ξ * + ) ≡     -ϕ * 4 ϕ * 3 -ϕ * 2 ϕ * 1     (41d)
We notice easily that

|ϕ > p,+ = iγ 2 |ϕ > * e,+ ; |ϕ > p,-= -iγ 2 |ϕ > * e,- (42)
The first of these two relations is what is expected from the well-known charge conjugation operator. A change of sign with no consequence (a ket being defined up to a phase factor) is introduced within the negative masses counterpart.

We also see that

|ϕ > e,-= -iI 4 |ϕ > e,+ ; |ϕ > p,-= -iI 4 |ϕ > e,- (43)
Now, the matrix -iI 4 , belonging to the imaginary extension of the STA, is nothing else than the chirality operator γ 5 .

The Dirac-Hestenes equation with positive and negative masses

We owe it to Hestenes to have proposed the counterpart of the Dirac equation in the context of geometric algebra. This (real) equation is

γ µ (∂ µ ϕ γ 2 γ 1 -e A µ ϕ) = m ϕ γ 0 (44)
It can also be proposed on the form

{ -i∆ Φ σ 3 + e A Φ + m Φ = 0 -i ∆ Φ σ 3 + e A Φ + m Φ = 0 (45) with ϕ = ( Φ 0 0 Φ ) .
It is then usually said that the whole Dirac formalism is contained in the first 2 by 2 equation of (45) as the second one is simply its conjugate in the sense σ j → -σ j . Hestenes' demonstration of his equation is as follows: he proposed to consider a bispineur |u > such that

< u|u >= 1 ; γ 0 |u >= |u > ; γ 2 γ 1 |u >= i|u > (46)
It is then sufficient to put

|ϕ > + = ϕ |u > (47)
to recover (35).

If we are interested in the precise form of the ket |u > satisfying (46) with the realization [START_REF] Hestenes | [END_REF], we obtain

|u >= 1 √ 2     1 0 1 0     (48)
up to a constant C such that |C| = 1. Both equations ( 45) and ( 47)-( 48) lead to

ϕ ≡ ϕ +1 = √ 2     ϕ 1 -ϕ * 4 0 0 ϕ 2 ϕ * 3 0 0 0 0 ϕ 3 -ϕ * 2 0 0 ϕ 4 ϕ * 1     = ( Φ 0 0 Φ ) (49)
At this stage, we have to make a few remarks. First, the STA approach allows us to see unusual facts. By this we mean that the first block on the diagonal contains all the information of the traditional theory. But what do we see there? The simultaneous consideration of the electron of positive mass and of what seems to be, at first sight, the positron of positive mass (cf. the first two components of (41a) and (41c)). But if we look at it more closely and consider the second block on the diagonal, we see that, if we confirm the presence of the electron with positive mass, the antiparticle is, on the other hand, characterized by a negative mass (cf. (41d))! Second, this combination of positive matter-negative antimatter is typical to have considered the real even part of the STA only. Indeed, (49) is of type E in ( 15)). So the question arises naturally: is it possible to put in evidence the other solutions (41c) (we restrict ourselves to even matrices)? For this purpose, we modify (47) and consider, instead

|ϕ > -= -iI 4 |u > ( 50
)
where ∆ has been defined in (37). Second, we introduce the transformation (54) in (53a) and ask the result to be equivalent to

M -1 E (γ µ ∂ ′ µ ± i m ′ ) M E |ϕ > ± (55)
If so, by applying M E on the left of Eq (55), this implies, at least for the six Lorentz transformations, that

m ′ = m, |ϕ ′ > ± = M E |ϕ > ± ( 56
)
The key is to ensure that

γ µ M E ∂ ′ µ = M E γ µ ∂ µ ( 57
)
with ∂ ′ µ being given by (54). As easily checked, this is satisfied with M E given by (29a). As an example, let us consider the rotation

       x ′0 = x 0 x ′1 = x 1 x ′2 = cos θ x 2 + sin θ x 3 x ′3 = -sin θ x 2 + cos θ x 3 (58)
The constraint (57) leads then to

{ γ 0 M E = M E γ 0 , γ 1 M E = M E γ 1 γ 2 M E = M E (γ 2 cos θ + γ 3 sin θ), γ 3 M E = M E (-γ 2 sin θ + γ 3 cos θ) ( 59
) These relations are indeed satisfied with

M E = cos θ 2 I -sin θ 2 γ 2 γ 3 (60)
The same results hold true for each of the six Lorentz transformations. The case of the supplementary operator M E = e 1 2 θ I 4 is a bit trickier. Indeed, in this case, as x ′ = x, ∆ ′ = ∆, we have two possibilities: either m ′ = m and

|ϕ ′ > ± = |ϕ > ± or m ′ ̸ = m and |ϕ ′ > ± = M E |ϕ > ± .
The first possibility is the only one allowed by the traditional way of seeing while the second possibility is typical of the geometric algebra point of view which is richer compared to the usual one because the subtended group GL(2, C) has more parameters than the Lorentz one.

It seems to us much more logical to favor this second possibility for reasons of consistency as, for the Lorentz transformations, we precisely have |ϕ ′ > ± = M E |ϕ > ± each time. We therefore opt for, in all cases

|ϕ ′ > ± = M E |ϕ > ± (61)
We also notice that, for M E = e 1 2 θ I 4 , we have

γ µ M E = M -1 E γ µ (62)
and finally obtain

(γ µ ∂ ′ µ ± im ′ ) |ϕ ′ > ± = ±i(-M -1 E m + m ′ M E ) |ϕ > ± (63)
So, the invariance leads to

m ′ = m ( e -i θ 0 0 e i θ ) (64)
What this tells us is that the mass itself undergoes a transformation, which consists of a rotation in the complex plane. For θ = π, these rotations lead to an inversion of the sign of the mass. As noticed in [START_REF] Daviau | New insights in the standard model of quantum physics in Clifford Algebra[END_REF], the relations (62) lead to different transformation laws for each spinor of |ϕ

> ± |ξ ′ > ± = N |ξ > ± ; |η ′ > ± = N |η > ± (65)
In what concerns the version coming from the geometric algebra, it is easy to convince ourselves that

ϕ ′ ϵ = M E ϕ ϵ ; ϵ = ±1 (66)
Let us now turn, in a similar way, to the other cases (24c), (24b) and (24d). First, we consider the proper retrochronous transformations. At the start, we observe a simple change of sign through (24c) and

∆ = -N ∆ ′ N (67)
This leads to anticommutation relations where there were commutation relations and vice versa. In particular, (59) now reads

{ γ 0 M IE = -M IE γ 0 , γ 1 M IE = -M IE γ 1 γ 2 M IE = -M IE (γ 2 cos θ + γ 3 sin θ), γ 3 M IE = M IE (γ 2 sin θ -γ 3 cos θ) (68)
and is satisfied with

M IE = -i sin θ 2 γ 0 γ 1 -i cos θ 2 γ 0 γ 1 γ 2 γ 3 = -i I 4 M E ( 69
)
as expected. This change of sign is also recovered for the supplementary symmetry in the sense that we now have In what concerns the results (65) and (66), they become respectively

γ µ M IE = -M -1 IE γ µ ( 70
|ξ ′ > ± = N |ξ > ± ; |η ′ > ± = -N |η > ± (71) and ϕ ′ ϵ = i M IE ϕ ϵ γ 1 γ 2 (72)
We now come to the improper orthochronous transformations characterized by (24b). The transformation law on the derivatives is given by Usually, when one study the spacetime algebra, one focus to its even part, isomorphic to the space algebra. This is because this even part allows to find the whole content of the Dirac equation. The relativistic invariance is then that of the restricted Lorentz group. We wanted to extend these considerations here by proposing three other types of elements of this spacetime algebra, i.e. the odd and the imaginary parts. This allowed us to highlight all connected components of the Lorentz group.

We then linked these developments to the Dirac equation, with the two signs of mass and confirmed that the four propositions were associated with four types of matter, of positive or negative mass, of positive or negative charge. Each time, the Lorentz invariance was confirmed but also completed by a new transformation: a rotation in the complex plane concerning the mass and thus allowing the inversion of the sign of this mass. This supplementary transformation is due to the presence of the unit pseudoscalar of the STA. We are convinced that this pseudo-scalar unit could bring other interesting developments. For example, we know that the mass term of the usual Dirac Lagrangian, m φϕ, is purely scalar. But what happens to this mass term in its geometric algebra version? We just have to take again (49) and (51) simultaneously on the form We believe that a mass acquisition process "à la Higgs" applied to (85) could bring interesting developments and this is what we will work on in the future.

ϕ ϵ = ( Φ 0 0 ϵ Φ ) (83)

)

  Despite this change of sign, the relation (64) still holds true.

  ϕ 1 ϕ * 3 + ϕ * 1 ϕ 3 + ϕ 2 ϕ * 4 + ϕ * 2 ϕ 4 c2 ≡ i(-ϕ 1 ϕ * 3 + ϕ * 1 ϕ 3 -ϕ 2 ϕ * 4 + ϕ * 2 ϕ 4 )(86)What is remarkable in (85) is the combination of both scalar and pseudoscalar contributionsm φϵ ϕ ϵ = c1 + c2 I 4(87)a fact that was absent from traditional considerations since m φϕ = m < ϕ|γ 0 |ϕ >= m c1 (88)

Acknowledgments

Warm thanks are due to the members of the Support au Modèle Cosmologique Janus for their financial support.

The new matrix -iI 4 ϕ is then of IE-type and writes explicitly

So we now have the combination of negative matter-positive antimatter and the Dirac-Hestenes equation is slightly modified to write

Compared to the original one (44), we notice a change of sign in what concerns the mass term. The four types of matter (antimatter) can thus be embedded in a system made up of the equations ( 44) and (52).

Let us now have a look on the transformations of the wave functions.

Relativistic invariance of the Dirac equation with positive and negative masses

We concentrate here on the free case since it is implied that the transformations of the derivatives and components of the electromagnetic field are identical in form.

The question is the following: how can we characterize the transformation

if ∂ ′ µ is specified according to one of the transformations (24)? This question has already been answered for the transformation (24a) in [START_REF] Daviau | New insights in the standard model of quantum physics in Clifford Algebra[END_REF]. We resume the results by presenting them from a point of view that can be applied to other transformations (24). First, we notice that the transformation (24a) x ′ = N x N implies ∆ = N ∆ ′ N (54)


Jonathan Jaglin
email: jonathan.jaglin@gipsa-lab.inpg.fr

Carlos Canudas-De-Wit
email: carlos.canudas-de-wit@gipsa-lab.inpg.fr

Cyrille Siclet

Adaptive Quantization: a comparative study

. We show that D-ZIZO generates a smoother quantization noise than ZIZO algorithm and that D-ZIZO allows a faster convergence for same given properties. Moreover in the multivariable case, D-ZIZO allows us to use the smallest number of quantization bits in order to achieve stability.

I. INTRODUCTION

In a networked controlled system, the output signals have to be digitalized before transmission. Our objective is then to use the minimum quantization bits necessary to maintain stability on the closed loop system, that is to say the minimum bandwidth.

Several quantization methods have already been proposed during the last decade. In [START_REF] Elia | Stabilization of linear systems with limited information[END_REF], authors use a logarithm quantization of the signal which only ensures local stability and does not minimize the necessary channel bandwidth. In [START_REF] Brockett | Quantized feedback stabilization of linear systems[END_REF], authors propose an adaptive algorithm based on a ∆ modulation and which allows a global stability. The key point of this method is to increase the value of the quantization steps when the signal is large ("Zoom In") and to decrease this value otherwise ("Zoom Out"). This method is improved in [START_REF] Sharon | Input to state stabilization with number of quantization regions[END_REF] in order to reduce the communication bandwidth. Thus, authors divide the "Zoom In" step into two steps ("Zoom In Escape" and "Zoom In Measurements"). We will call this method ZIZO in the rest of the paper. In [START_REF] Hespanha | Towards the control of linear systems with minimum bit-rate[END_REF], [START_REF] Tatikonda | Control under communications constaints[END_REF] authors prove that semi global stability is possible in a NCS provided that the communication rate R is greater than a minimum communication rate depending on the eigenvalues of the open loop matrix of the system. Recent works [START_REF] Canudas-De-Wit | Differential coding in networked controlled linear systems[END_REF] and [START_REF] Jaglin | Delta modulation for multivariable centralized linear networked controlled systems[END_REF] based on ∆ modulation provide local stability with a better characterization of the attraction domain than [START_REF] Hespanha | Towards the control of linear systems with minimum bit-rate[END_REF] and [START_REF] Tatikonda | Control under communications constaints[END_REF]. In [START_REF] Jaglin | Delta modulation for multivariable centralized linear networked controlled systems[END_REF], authors generalize their work to non diagonalizable systems. This also permits to tune the quantization steps in order to get a better signal reconstruction quality. In [START_REF] Jaglin | Adaptive quantization for linear systems[END_REF], we expose an adaptive algorithm which allows us to use the theoretical minimum bandwidth. Previously, in [START_REF] Rubio | Adaptive delta-modulation coding in networked controlled systems[END_REF] authors have presented a quantization method based on a one-bit-adaptive ∆ modulation. The interest of this technic is that it permits global stability in the scalar case provided that the open loop eigenvalue λ is such that: |λ| < 1.3. Nevertheless it can be shown in [START_REF] Tatikonda | Control under communications constaints[END_REF] that in this case (one bit quantization) it is theoretically possible 1 Networked Controlled Systems 2 Zoom In Zoom Out 3 ZIZO with Dwell Time to just have |λ| < 2. In [START_REF] Liberzon | Input to state stabilization of linear systems with quantized state measurements[END_REF] authors propose an adaptive quantization method robust to any bounded perturbation. This method also works for multivariable class of linear systems. However, the necessary bandwidth is generally greater than the minimum one exposed in [START_REF] Tatikonda | Control under communications constaints[END_REF], even if it may be very close in certain cases. At the contrary, the algorithm proposed in [START_REF] Sharon | Input to state stabilization with number of quantization regions[END_REF] needs a bandwidth which is always only slightly greater than, or even sometimes equal to, the minimum bandwidth. Nevertheless, this algorithm (ZIZO) does not guaranty that the reconstruction error signal is smooth, but it is subject to frequency pics. This issue is solved in [START_REF] Jaglin | Adaptive quantization for linear systems[END_REF] thanks to the introduction of a Dwell Time phase between Zoom In and Zoom Out. That is why we call this algorithm D-ZIZO. Moreover an improvement of the quantization itself permits to use the minimum bandwidth in the scalar case as well as in the multivariable one.

The paper is organized as follows. Firstly, we present the problem formulation in Section II. We go on in Section III with the description of the two Adaptive quantization algorithms. These two algorithms are described without and with bounded exogenous inputs. We continue with a comparison between the performances of ZIZO and D-ZIZO algorithms in Section IV. With some analysis and simulation results, we point out three results. Firstly, ZIZO generates high frequencies though the D-ZIZO signals are smoother. Secondly, for same constraints on regulation detection, D-ZIZO is faster than ZIZO. Moreover, with the same convergence rate, the regulation detection is faster for D-ZIZO. Finally, the change of coordinates enables D-ZIZO to achieve theoretical minimum channel bandwidth.

II. PROBLEM FORMULATION

The problem considered is the stabilization of a multivariable system in which sensor signals are centralized, and then transmitted through a digital communication link to the controller.

We assume the following:

• the coding process is centralized : a single encoder can be used to encode all the sensed states of the system, • the encoded information is transmitted through a noiseless perfect transmission channel. Hence delay, errors due to the transmission are not considered, • the encoder and decoder clocks are assumed to be synchronized, and samples are assumed to occur at each T s . The following notations will be used:

• x k = [x 1 k , . . . , x n k ] T ∈ R (n×1)
is the n-dimensional sensed state vector at instant kT s (each x i k corresponds to the i -th sensor) ; m×1) , is m-dimensional control input vector at instant kT s .

• u k = [u 1 k , . . . , u m k ] T ∈ R (
• M i number of words by signal zi k . The discretized system is described by:

x k+1 = Ax k + Bu k (1)
u k = -K xk (2)
with K such as A -BK is Schur (eigenvalues have their module strictly inferior than 1). xk is an estimation of x k , and xk denotes the estimation error :

xk = x k -xk , (3)
and, more generally, for a given signal s k , ŝk represents an estimated value of s k and sk represents the error s k -ŝk .

A. Architecture of ZIZO

In [START_REF] Sharon | Input to state stabilization with number of quantization regions[END_REF], the architecture of ZIZO is composed of two main components

• The vector quantizer block transforms the error xk , into a finite codeword set, each signal is coded with the same granularity ∆. • The predictor, that transforms back the codeword into a system state prediction xk with the equation

xk+1 = Ax k + Bu k (4)
This predictor uses input and signal information at the encoder: this hypothesis is less realistic than the one in [START_REF] Jaglin | Adaptive quantization for linear systems[END_REF] where only signal information is used at the encoder. In ZIZO algorithm, we have the error equation:

xk+1 = A(x k -xk ) (5)

B. Architecture of D-ZIZO

Figure 1 shows the architecture of the proposed differential coding algorithm. It is composed of three main components:

• The vector quantizer block transforms the error zk , into a finite codeword set • The predictor , that transforms back the codeword into a system state prediction xk • The rotation matrix T k transforms the estimation error xk between the signal x k and its estimated (reconstructed) value xk into a new set of coordinates zk , i.e.

zk = T -1 k xk (6)
In [START_REF] Jaglin | Delta modulation for multivariable centralized linear networked controlled systems[END_REF], we have proven that the change of coordinates with a dynamic matrix T k permits to reduce the study to the following class of systems:

zk+1 =   |λ| 1 0 0 |λ| 1 0 0 |λ|   F (|λ|)∈R µ * µ (z k -ẑk )

with

• ẑk is the quantized signal of zk with its associated quantization steps ∆ k .

• µ is the readjusted size of zk .

• ∆ k = [∆ 1 k , . . . , ∆ µ k ]
T are the quantization steps associated to ẑk

III. ADAPTIVE DELTA MODULATION: ALGORITHM PRINCIPLES

A. Presentation of the algorithm introduced in [10]: ZIZO (Results without noise)

Let us begin with the following lemma: Lemma 1: Important item to create ZIZO. Assuming that xk is computed thanks to the quantization procedure given in [START_REF] Liberzon | On stabilization of linear systems with limited information[END_REF], and suppose that

z0 ∈ Ω ext = {z ∈ R µ : |z i | M ∆ 2 , 1 i µ}
with ∆ the quantization step. And M the number of words per signal chosen such that:

A ∞ M Then i) Ω ext is an invariant set ii) zk ∈ Ω int , ∀k 1
where

Ω int = {z ∈ R µ : z ∞ A ∞ ∆/2}
In the case of ZIZO and D-ZIZO algorithms, we introduce the set Ω ext k defined by

Ω ext k = {z ∈ R µ : |z i | M i ∆ i k 2 , 1 i µ} It is worthwhile noting that ∆ i = ∆ and M i = M ∀1 i µ.
We emphasize the fact that contrary to the set Ω ext , this set is dynamic. We will go on with the main phases of ZIZO algorithm:

• "Zoom In measurement" strategy: The quantization steps decrease during p -1 sample times. • "Zoom In escape" strategy: After the "Zoom In measurement" process, quantization steps increase such

that if xk ∈ Ω ext k then xk+1 ∞ (M -2)∆ k+1 /2. So that xk+1 ∈ Ω ext k+1 . Else xk / ∈ Ω ext k
and we switch to a "Zoom Out" process.

• "Zoom Out" strategy: The signal is outside Ω ext , the quantization steps increase with the target to ensure that the signal will be inside Ω ext after a finite time. To realize this strategy, ZIZO needs different coefficients such that Θ in,m , Θ in,e and Θ out and p which respectively are the compression ratio for "Zoom In measurement" strategy, the expansion factor for "Zoom In escape" strategy and the expansion factor for "Zoom Out" strategy, p corresponds to the number of necessary sample times that ZIZO needs to prove that the signal belongs to Ω ext k . Those parameters are constrained by

A ∞ M < Θ in,m < 1 (7) Θ out > A ∞ (8) Θ in,e > A ∞ M -2 (9)
Θ p-1 in,m Θ in,e < 1 (10) ∆ k+1 = (Θ in,m or Θ in,e or Θ out )∆ k (11)
With regards to these inequalities, the convergence depends on p. Bigger is p, faster is the convergence but worse is the regulation detection.

B. Presentation of the algorithm introduced in [7]:D-ZIZO (Results without noise)

Though in ZIZO, each signal is quantized with the same quantization step, in D-ZIZO each signal has its own quantization step but they are constrained by: Lemma 2: Important item to obtain D-ZIZO. Assuming that ẑk is computed thanks to the quantization procedure given in [START_REF] Jaglin | Delta modulation for multivariable centralized linear networked controlled systems[END_REF], and suppose that

z0 ∈ Ω ext = {z ∈ R µ : |z i | M i ∆ i 0 2 , 1 i µ}
and the quantization steps satisfy the equations

|λ| + ∆ i+1 0 ∆ i 0 M i , 1 i µ -1 (12) Then i) Ω ext is an invariant set ii) zk ∈ Ω int , ∀k 1
where

Ω int = {z ∈ R µ : |z i | |λ|∆ i 0 /2 + ∆ i+1 0 /2 ∀i : 1 i µ -1 and |z µ | |λ|∆ µ 0 /2} Remark 1:
In a semi-global stabilization context, item ii) of 1 and 2 allows us to obtain the maximal convergence rate. In D-ZIZO algorithm (global stabilization), we can choose the quantization steps. If we choose ∆ i+1 0 /∆ i 0 close to 0, convergence rate max

1 i n-1 (|λ| + ∆ i+1 0 /∆ i 0 )/M i of D-ZIZO is near |λ|/M that
could be quantitatively least than the maximal shrinkage rate A ∞ /M of ZIZO. Since in ZIZO there is no change of coordinates (with T k matrix) and no quantization steps tuning methods, the necessary number of words to stabilize the system is also more important.

In what follows, we briefly explain the principle of the algorithm. Initially, the signal z0 ∈ Ω ext 0 , the quantization steps have to increase faster than the signal to ensure that there exists a k 0 such that zk0 ∈ Ω ext k0 . Our aim is to determine whether zk0 belongs to Ω ext k0 . When this situation is obtained, the quantization steps related to the set Ω ext k decrease.

The algorithm proposed in [START_REF] Jaglin | Adaptive quantization for linear systems[END_REF] is presented in 3 parts:

• "Zoom Out" strategy: zk / ∈ Ω ext k , the quantization steps increase with the target to ensure that the signal will belong to Ω ext k after a finite time.

• Dwell time phase: the algorithm does not have enough information to know if the signal zk0 at a precise moment k 0 belongs to Ω ext k0 or not. Quantization steps do not change.

• "Zoom In" strategy: zk ∈ Ω ext k , the quantization steps decrease. The dwell time phase is the key point of the algorithm. In the scalar case, we define m ⋆ the necessary dwell time to decide whether |z m | M ∆ m /2: m ⋆ is the smallest value of m verifying this constraint (considering that m = 0 at the beginning of the dwell time phase). After m ⋆ iterations, a criterion exposed in [START_REF] Jaglin | Adaptive quantization for linear systems[END_REF] allows us to determine whether D-ZIZO enters in "Zoom Out" or "Zoom In" phase. To extend to multivariable class of linear systems, we use the cascade structure of F (λ). Thus we use the same analysis but for zµ and then zµ-1 and so on. To obtain the algorithm, we must introduce two scalars C out , C in which respectively correspond to the expansion factor for a "Zoom Out" procedure and the compression ratio for a "Zoom In" procedure. Those two parameters are constrained by

max 1 i n-1 ( |λ| + ∆ i+1 0 ∆ i 0 M i ) < C in < 1 (13) max 1 i n-1 (|λ| + ∆ i+1 0 ∆ i 0 ) < C out (14)
∆ k+1 = (C in or 1 or C out )∆ k (15)
Remark 2: In ZIZO algorithm, authors do not use the change of coordinates exposed in the section concerning the architecture of D-ZIZO. To prove the stability results, the analysis is realized on |x k |. The positivity of the eigenvalues and the cascade structure of F (|λ|) provide some properties on the sign of each signal.

C. Results with noise

In the previous section, we have presented the case of noiseless systems. In what follows, we will extend this result to a bounded noisy systems. Equation (1) becomes, with noise:

x k+1 = Ax k + Bu k + s k (16) xk+1 = (A + BK)x k+1 + A xk (17) xk+1 = A(x k -xk ) + s k (18)
1) D-ZIZO:

zk+1 = T -1 k+1 AT k (z k -ẑk ) + T -1 k+1 T k s k Since T k ∞ is bounded, T -1 k+1
T k s k is also bounded. We can then write, with w k a bounded noise:

zk+1 = F (λ)(z k -ẑk ) + w k (19)
The initialization of quantization steps is very important for D-ZIZO. When there are some exogenous inputs, the initialization of quantization steps in the algorithm depends on the maximal value of each exogenous input. Each initial value of quantization steps have to be tuned by a new method. Nevertheless in [START_REF] Sharon | Input to state stabilization with number of quantization regions[END_REF], the quantization steps of each signal are the same, thus there is no condition on the quantization steps.

Let us define two dynamic scalars C out (k), C in (k) which respectively correspond to the expansion factor for a "Zoom Out" procedure and the compression rate for a "Zoom In" procedure. Those two parameters are constrained by

max 1 i n-1 ( |λ| + ∆ i+1 0 ∆ i 0 + 2W i ∆ i k M i ) < C in (k) < 1 max 1 i n-1 (|λ| + ∆ i+1 0 ∆ i 0 + 2W i ∆ i k ) < C out (k)
The principle of the algorithm is the same except two rules.

• The value m ⋆ i,k ( ∆ i k Wi
) is dynamic • In the Dwell Time phase, we fix a maximal time for this phase. If the dwell time is over this period, the algorithm enters in "Zoom Out" phase. Remark 3: The smaller is Q, the bigger is the asymptotical value of quantization steps, so the set Ω ext ∞ will be larger. However regulation is faster. If the value of Q is big, asymptotical quantization steps are near their minimal value. But if a disturbance occurs, the system slowly regulates.

2) ZIZO: To cope with an exogenous input, ZIZO is constrained by a maximal value of quantization. This maximal value depends on the matrix A and the amplitude of the noise. The conception of this algorithm is simpler than D-ZIZO.

IV. PERFORMANCE COMPARISON BETWEEN ZIZO AND D-ZIZO

In this section, we will compare with some examples and by simulations those two strategies in term of regulation performances. We also show that D-ZIZO always can reach the minimum bit constraint though ZIZO can reach it only for special cases. The simulation results are presented here to emphasize the differences between the two algorithms.

A. High frequency generation

1) Case study: Suppose the system (1) with A = 2.8, B = 0.2. The controller gain is K = 8, so A -BK is Schur. The initial conditions are z1 0 = 100, x0 = 0, M 1 = 3. The initialization of the quantization step is ∆ 0 = 0, 6. The parameters of ZIZO are Θ in,m = 2.81/3, Θ in,e = 2.81 and Θ out = 3 and p = 22. The parameters of D-ZIZO are C out = 3,C in = 2.81/3 and m ⋆ = 6. Those parameters are used for Figure 2 and Figure 3.

In this part, we highlight that for the "Zoom In" process the evolution of zk for D-ZIZO is smother than the evolution of xk with ZIZO. And we introduce the Lyapunov function V k = z2 k . Without loss of generalization, we assume that the initial condition is in Ω ext (0). So zk is in Ω ext (k), hence we have

V k = z2 k V k ( M ∆ k 2 ) 2 = W k
In Figure 2, we see the evolution of V k , the coefficients in the two algorithms are chosen such that V k = W k , we easily find out that the Lyapunov function is piecewise decreasing for ZIZO and that is smoothly decreasing for D-ZIZO. Since in that example V k = W k , the Lyapunov function depends on ∆ 2 k . So the evolution of V k is only driven by the algorithms ZIZO and D-ZIZO. In the "Zoom In Measurement" procedure of ZIZO, the quantization step begins to decrease and after a certain number of sample times p (coefficient relied to regulation detection), that depends on [START_REF] Jaglin | Adaptive quantization for linear systems[END_REF], algorithm enters in "Zoom In escape" process for only one sample time. In this procedure, quantization steps increase. This method is reproduced each p times. In D-ZIZO, quantization step only decreases. In Figure 3, we are interested in the spectral density of zk = xk . We remark that ZIZO generates a lot of frequency pics though D-ZIZO not. It is very important for closed loop system to be robust to noise generation (here, quantization noise). So D-ZIZO is better than ZIZO for the non-generation of frequency pics.

B. Convergence rate

Moreover, in the noiseless adaptive process, we assure for the scalar case that the convergence to 0 is faster with same regulation performance p = m ⋆ . Let us define the convergence coefficient C R with an initialization at z0 ∈ For processes which are p cyclic, we have

Ω ext 0 C R = lim n→∞ n i=1 n ∆ i ∆ i-1 = lim n→∞ n ∆ n ∆ 0
C R = p ∆ p ∆ 0 . ZIZO is p cyclic so we have C R = p Θ p-1
in,m Θ in,e . When these coefficients are close to their minimal constraints exposed in [START_REF] Jaglin | Adaptive quantization for linear systems[END_REF], we obtain

C R = p |λ| M p-1 |λ| M -2 with p such that C R < 1.
In D-ZIZO, when coefficients are close tho their minimal constraints exposed in (13), we have

C R ( |λ| M ) ln(|λ|) ln( M +2-|λ| M -|λ| )
In that follows, we interest in the closed loop system equations. The following analysis is realized with a c = a -bK and when |x k | M ∆ k . Then we obtain

x k+1 = a n c x 0 + bK k i=0 a i-1 c xk-i |x k+1 | |a c | k |x 0 | + M |bK| k i=0 |a c | i-1 ∆ k-i |x k+1 | |a c | k |x 0 | + M |bK| C k R |a c | 1 -( |ac| C R ) k+1 1 -|ac| CR ∆ 0
In Fig 4, we present the convergence behavior represented by C R for the regulation constraint on detection p = m ⋆ . If there is an impulse disturbance or a change of initial conditions, algorithms react at the same moment, i.e algorithm enters in a "Zoom Out" process.

In Figure 5, we present for same mean convergence rate (same C R ), the evolution of detection rapidity. That means the necessary number of samples to decide to enter in a "Zoom Out" process while an impulsive disturbance occurs. It is related to m ⋆ for D-ZIZO and p for ZIZO.

C. Distance to theoretical stabilization rate bound

Firstly, we introduce the theorem given in [START_REF] Tatikonda | Control under communications constaints[END_REF] Theorem 1: Suppose the system [START_REF] Brockett | Quantized feedback stabilization of linear systems[END_REF] Then, there exists a coding structure ensuring that the system (1) is globally stable.

With D-ZIZO, the bound is achieved for all linear systems. ZIZO could verify this theorem for certain class of systems. If the matrix A of the system is diagonal and its eigenvalues are greater than 3, ZIZO verifies the theorem. However, since this algorithm needs at least 3 words M = 3. That means for |λ| < 2, there is a loss of one word by signal (though D-ZIZO needs 2 words). If the matrix A is not diagonalizable, each signal has the same quantization step ∆ in ZIZO. The analysis realized in [START_REF] Sharon | Input to state stabilization with number of quantization regions[END_REF] is too much conservative to achieve the bound exposed in the theorem. The algorithm ZIZO needs more words by signal than the minimum number of words exposed in the theorem. With D-ZIZO, the tuning of quantization steps permits to achieve the constraint for all linear systems. In the table IV-C, we present the necessary number of words by signal to assure global stability for ZIZO and D-ZIZO .

V. CONCLUDING REMARKS

In this paper, we have firstly recalled two recent algorithms of adaptive quantization namely ZIZO and D-ZIZO. Those two algorithms permit to obtain global stability. When the signal is too important it needs that the quantization steps increase otherwise, quantization steps decrease. In "Zoom In Measurements" and "Zoom In Escape" (ZIZO) phases, quantization step respectively decrease and increase though, only for the "Zoom In" phase (D-ZIZO) quantization steps decrease and remain identical in Dwell Time period. Even if the essence of the algorithms are the same, it seems that the dwell time phase permits to have some improvements. This comparison could be summarized with these following items.

• This change of coordinates and the quantization steps tuning method enable D-ZIZO to obtain the minimum bandwidth on the communication channel for all linear systems whereas ZIZO may sometimes reach the theoretical limit for certain classes of linear systems. • ZIZO generates high frequencies on the quantization noise though this signal is smoother for D-ZIZO. • For same regulation detection, D-ZIZO is faster than ZIZO. Moreover, with the same convergence rate, the detection is faster for D-ZIZO. • One drawback for D-ZIZO is the generalization to non-linear systems that seems difficult. Without the change of coordinates which is the key point of D-ZIZO, it is impossible to obtain an adaptive algorithm in D-ZIZO case. With ZIZO, the issue is solved. In future works, we will investigate the study of uncertain models with fixed and adaptive quantization.

Fig. 1 .

 1 Fig. 1. Schematic representation of the system in our study.

Fig. 2 .

 2 Fig. 2. Comparison between Lyapunov functions. Figure (a) is obtained with ZIZO Algorithm result and Figure (b) represents D-ZIZO algorithm.

Fig. 3 .

 3 Fig. 3. Comparison realized with the power signal densities of quantization noise with T s = 1s.

Fig. 4 .

 4 Fig.[START_REF] Rubio | Adaptive delta-modulation coding in networked controlled systems[END_REF]. Comparison of C R between ZIZO and D-ZIZO with p = m ⋆ (same constraints on regulation detection). We see that if λ > 2, C R > 1 so the algorithm can not be used because it diverges. Moreover with λ < 2 we can use only 2 words.

Fig. 5 .

 5 Fig. 5. Comparison of p between ZIZO (a) and D-ZIZO (b) with the same C R for λ ∈ [1, 2.999].

  with the pair (A, B) controllable. And a channel rate R bounded by

	1.6
			ZIZO
	1.4		D-ZIZO
	1.2
	1
	C R
	0.8
	0.6
	0.4
	0.2	1	1.2	1.4	1.6	1.8	2 λ	2.2	2.4	2.6	2.8	3
	n
	⌈|λ l |⌉ < 2 R
	l=1,|λ l |>1

TABLE I COMPARISON

 I RESULT WITH OPTIMAL CODING, D-ZIZO AND ZIZO. A*B MEANS THAT THE COMPONENT 1 NEEDS A WORDS AND THECOMPONENT 2 NEEDS B WORDS.

	condition	Optimal result	ZIZO	D-ZIZO
	a < 2	2	3	2
	3 < a < 4	4	4	4
	A =	a 10 0 a	4*4	14 * 14	4 * 4
	with 3 < a < 4
	A =	a 1 0 a	4*4	5 * 5	4 * 4
	with 3 < a < 4


Kevin Loquin
email: kevin.loquin@lirmm.fr

Olivier Strauss
email: olivier.strauss@lirmm.fr

Parzen Rosenblatt

Imprecise functional estimation: the cumulative distribution case

In this paper, we propose an adaptation of the Parzen Rosenblatt cumulative distribution function estimator that uses maxitive kernels. The result of this estimator, on every point of the domain of F, the cumulative distribution to be estimated, is interval valued instead of punctual valued. We prove the consistency of our approach with the classical Parzen Rosenblatt estimator, since, according to consistency conditions between the maxitive kernel involved in the imprecise estimator and the summative kernel involved in the precise estimator, our imprecise estimate contains the precise

Introduction

The probability density function (pdf) f and the cumulative distribution function (cdf) F of a random variable X on Ω ⊆ IR are fundamental concepts for describing and representing real data in statistics. These representations are linked by ∀ω ∈ Ω , F(ω) = ω -∞ f (u)du. When they cannot be specified, estimates of these functions may be performed by using a sample of n observations independent and identically distributed (X 1 , ..., X n ) of X. These observations are summarized by the empirical distribution defined by e n = 1 n ∑ n i=1 δ X i , where δ X i is the Dirac distribu- tion on X i or by the empirical cumulative distribution function defined on Ω by

E n (x) = 1 n ∑ n i=1 1l [X i ≤x]
, where 1l A is the characteristic function on A. Different methods have been proposed in the literature for estimating or manipulating the pdf or the cdf underlying a sample of observations. The Parzen Rosenblatt method is one of the most efficient non-parametric techniques [START_REF] Parzen | On estimation of a probability density function and mode[END_REF][START_REF] Rosenblatt | Remarks on some nonparametric estimates of a density function[END_REF]. It belongs to the class of functional estimation methods.

Generally speaking, functional estimation [START_REF] Bosq | Théorie de lestimation fonctionnelle[END_REF] consists of estimating, for all x ∈ Ω , a function h : Ω -→ IR from another function g : Ω -→ IR related to h. The nature of this relation between h and g can take different form: g can be replaced by a sequence (g n ) n>0 , such that g n -→ h, when n -→ +∞, h can be a modification of g (or h is a filtered signal obtained from the signal g), or g can be a discretization of h that has to be recovered by interpolation. So, the estimate of h, at x ∈ Ω , is function of g and x, which can be expressed as ĥ(x) = ϕ(g, x).

For the Parzen Rosenblatt pdf estimator, the function g is the empirical distribution e n , h is the pdf to be estimated f . The estimator ĥ is defined for all x ∈ Ω , by: ĥ

(x) = f nκ ∆ (x) = 1 n n ∑ i=1 κ ∆ (x -X i ), (1)
with κ the kernel used to perform this estimate and ∆ the bandwidth. Note that κ ∆ (x) = 1 ∆ κ( x ∆ ). When g is the empirical cumulative distribution E n , the cdf F is the function h to be estimated and the estimator ĥ is defined for all x ∈ Ω , by: ĥ

(x) = F nκ ∆ (x) = x -∞ f nκ ∆ (u)du. (2)
In the Parzen Rosenblatt like methods, and more generally in all the functional estimation methods, the particular role of the kernel is to define a neighborhood that can be shifted to any location of Ω . The classical (precise) approach makes use of summative kernels. A summative kernel can be seen as a probability distribution, defining a probabilistic neighborhood around each location x of Ω . This paper considers a new approach (imprecise) that makes use of maxitive kernels. A maxitive kernel can be seen as a possibility distribution, defining a possibilistic neighborhood around each location x of Ω . The main consequence of replacing a summative kernel by a maxitive kernel is that the estimated value is an interval [h(x), h(x)], instead of a single value ĥ(x). We are interested in the relation between the point estimate obtained with the classical approach and the interval estimate obtained with our approach.

The paper is organized as follows. In section 2 we present the classical functional estimation using a summative kernel. In section 3, functional estimation with maxitive kernels is exposed. In section 4, the imprecise functional estimation is presented and mathematically justified. In section 5, we apply our method to the Parzen Rosenblatt cdf estimator. Before concluding, we discuss in section 6 of the choice of the involved maxitive kernel. The method is illustrated by an experiment.

Functional estimation with summative neighborhoods

In functional estimation, a summative kernel can be considered as a weighted neighborhood of a given location, called its mode, formally similar to a probability distribution.

Definition 1 Summative kernels are IR + -valued functions κ defined on a domain Ω , verifying the summativity property: Ω κ(x)dx = 1.

Note that any given monomodal summative kernel κ, can be the basis for a family of summative kernels tuned by a location-scale parameter θ = (u, ∆ ), with u a translation factor and ∆ > 0 its bandwidth. Any element of this family is obtained, for u ∈ Ω and ∆ > 0, by

κ u ∆ (ω) = 1 ∆ κ( ω -u ∆ ), ∀ω ∈ Ω . (3)
When seen as a probability distribution, a summative kernel κ has a relevant meaning in the scope of uncertainty theories. It induces a probability measure given by P κ (A) = A κ(ω)dω, ∀A ⊆ Ω . The value P κ (A) can be interpreted as the degree of probability for a realization of the underlying uncertain phenomenon to fall in A.

Estimation of a given function of h : Ω → IR in a summative neighborhood κ x ∆ of a given location x with bandwidth ∆ is given by the expectation of its related function g according to the probability measure P κ x ∆ : ĥ

(x) = E κ x ∆ (g). (4)
This approach can be found in [START_REF] Bosq | Théorie de lestimation fonctionnelle[END_REF] for functional estimation in statistics. [START_REF] Jiri | Digital Signal Filtering, Analysis and Restoration[END_REF] presents digital signal processing methods that can be reformulated as functional estimators (4).

Functional estimation with maxitive neighborhoods

A maxitive kernel is also a weighted neighborhood of a given location, called its mode, formally similar to a possibility distribution or membership function of a normalized fuzzy subset [START_REF] Dubois | Theorie des possibilités : Applications 'a la représentation des connaissances en informatique[END_REF].

Definition 2 A maxitive kernel is a [0, 1]-valued function π, defined on a domain Ω , verifying the maxitivity property: sup ω∈Ω π(ω) = 1.

Note that any given monomodal maxitive kernel π, defined on Ω , can be the basis for a family of maxitive kernels tuned by a location-scale parameter θ = (u, ∆ ), with u a translation factor and ∆ its bandwidth. Any element of this family is obtained, for u ∈ Ω and ∆ > 0, by

π u ∆ (ω) = π( ω -u ∆ ), ∀ω ∈ Ω . (5)
A possibility distribution π has a relevant meaning in the scope of uncertainty theories. π induces a possibility measure given by Π π (A) = sup ω∈A π(ω), ∀A ⊆ Ω . The value Π π (A) can be interpreted as the degree of possibility for a realization of the underlying uncertain phenomenon to fall in A.

Now, when the summative neighborhood κ x ∆ is replaced by a maxitive neighborhood π x ∆ of a given location x with bandwidth ∆ , the Lebesgue integral in estimator (4) has to be replaced by the Choquet integral [START_REF] Denneberg | Non Additive Measure and Integral[END_REF][START_REF] Marichal | Aggregation Operations for Multicriteria Decision Aid[END_REF] of g.

Imprecise functional estimation

A possibility measure is a special case of concave Choquet capacity ν [START_REF] Walley | Towards a unified theory of imprecise probability[END_REF]. The conjugate ν c of such a capacity, defined by ν c (A) = 1 -ν(A c ), ∀A ⊆ Ω , is a convex capacity. A concave capacity ν can encode a special family of probability measures, noted core(ν) and defined by

core(ν) = {P κ , | ∀A ⊆ Ω , ν c (A) ≤ P κ (A) ≤ ν(A)}. ( 6
)
David Schmeidler and Dieter Denneberg proved the following theorem ( [START_REF] Schmeidler | Integral representation without additivity[END_REF] proposition 3 and [2] proposition 10.3) for capacities.

Theorem 3. The capacity ν is concave if and only if for all g such that

C ν (|g|) < +∞, then ∀κ|P κ ∈ core(ν), C ν c (g) ≤ E κ (g) ≤ C ν (g).
From Theorem 3, since a maxitive kernel defines a possibility measure, a maxitive kernel-based estimation of h, generalizing expression ( 4) is interval valued. The upper and lower bounds are the Choquet integrals of g computed with respectively Π π x ∆ and N π x ∆ , which are capacities (or non additive measures) associated to π x ∆ a maxitive neighborhood of x, with bandwidth ∆ . N π x ∆ is the conjugate of the possibility measure Π π x ∆ , called a necessity measure. These remarks leads to the following corollary of Theorem 3.

Corollary 1. Imprecise functional estimation

Let π be a maxitive kernel, then ∀x ∈ Ω and ∀∆ > 0,

∀κ|P κ ∈ core(Π π x ∆ ), C N π x ∆ (g) ≤ E κ (g) ≤ C Π π x ∆ (g). (7)
Imprecise estimation of a given function of h : Ω → IR in a maxitive neighborhood π x ∆ of a given location x with bandwidth ∆ is given by the Choquet integrals of its related function g according to the possibility and necessity measures Π π x ∆ and

N π x ∆ : [h(x), h(x)] = [C N π x ∆ (g), C Π π x ∆ (g)]. (8)
According to Corollary 1, an estimate ĥ(x) of h obtained with a summative kernel κ, such that P κ belongs to core(Π π x ∆ ), belongs to the estimated interval [START_REF] Loquin | On the granularity of summative kernels[END_REF]. Besides, the estimation bounds are attained, i.e. there exist two summative kernels η and µ, whose associated probability measures P η and P µ are in core(Π

π x ∆ ), such that E η (g) = C N π x ∆ (g) and E µ (g) = C Π π x ∆ (g).
Replacing a summative kernel by a maxitive kernel for estimating a function h aims at taking into account the imperfect knowledge of the modeler to choose a particular κ. The specificity [START_REF] Yager | Entropy and specificity in a mathematical theory of evidence[END_REF][START_REF] Loquin | On the granularity of summative kernels[END_REF] of the maxitive kernel chosen by the modeler for performing this imprecise estimation reflects his knowledge. The most specific is the maxitive neighborhood, the smallest is the encoded set. Indeed, if π is more specific than π , some summative kernels encoded by π will not be encoded by π. The smaller is the encoded set of summative neighborhoods, the closer are the estimation bounds with this method.

Imprecise cumulative distribution function estimation

The Parzen Rosenblatt density estimator (1) can be expressed as the estimation of the pdf f , with the empirical distribution g = e n (summarizing the observations) according to a summative neighborhood κ x ∆ (see expression ( 3)):

f nκ ∆ (x) = E κ x ∆ (e n ). (9)
Corollary 1, associated with expression [START_REF] Marichal | Aggregation Operations for Multicriteria Decision Aid[END_REF] suggests that an imprecise estimation of the Parzen Rosenblatt pdf estimator should be performed by computing the Choquet integral of the empirical distribution e n according to a maxitive kernel (encoding a family of summative kernels). This direct approach is however not applicable here, since the Choquet integral of the empirical distribution does not exist. Indeed, the computation of this integral only exists for bounded functions. The empirical distribution is not bounded. Actually, the Dirac delta functions, forming e n , are not functions but mathematical constructions, called distributions.

Nevertheless, the Parzen Rosenblatt cdf estimator (2) involves the empirical cumulative distribution E n , which is a bounded function. Theorem 4 expresses the Parzen Rosenblatt cdf estimate at a point x, as the estimation of the cdf F with the cumulative empirical distribution g = E n according to a summative neighborhood of x, κ x ∆ .

Theorem 4. Let κ be a summative kernel and ∆ > 0 and n > 0, then ∀x ∈ Ω ,

F nκ ∆ (x) = E κ x ∆ (E n ). ( 10
)
Proof. First, note that

f nκ ∆ (x) = Ω κ ∆ (ω)e n (x -ω)dω. Indeed, Ω κ ∆ (ω)e n (x - ω)dω = 1 n ∑ n i=1 Ω κ ∆ (ω)δ X i (x -ω)dω = 1 n ∑ n i=1 κ ∆ (x -X i ). Thus, F nκ ∆ (x) = x -∞ Ω κ ∆ (ω)e n (u -ω)dω du, = Ω x -∞ e n (u -ω)du κ ∆ (ω)dω, E n is the cumulative distribution associated to the empirical distribution, i.e. E n (ω) = ω -∞ e n (u)du.
Then by successive changes of variable v := u -ω and t := x -ω, we obtain:

F nκ ∆ (x) = Ω E n (x -ω)κ ∆ (ω)dω, = Ω E n (t)κ ∆ (x -t)dt, = E κ x ∆ (E n ).
Since E n is bounded, an imprecise estimation of F at x can be obtained with a maxitive kernel π x ∆ .

Theorem 5. Let π be a maxitive kernel, then ∀x ∈ Ω , ∀n > 0 and ∀∆ > 0,

∀κ ∆ |P κ ∆ ∈ core(Π π x ∆ ), C N π x ∆ (E n ) ≤ F nκ ∆ (x) ≤ C Π π x ∆ (E n ). (11)
We now present the computation of the imprecise Parzen Rosenblatt cdf estimate. First, observe that E n is a simple function that can be expressed on

Ω by E n (ω) = ∑ n i=1 i n 1l [X (i) ,X (i+1) ]
, where (.) indicates a permutation of the observations such that X (i) ≤ X (i+1) . Thus, the Choquet integral of E n can be rewritten as

C Π π x ∆ (E n ) = 1 n ∑ n i=1 Π π x ∆ ({ω ∈ Ω : E n (ω) ≥ i n }). It can easily be observed that {ω ∈ Ω : E n (ω) ≥ i n } = {ω ∈ Ω : ω ≥ X (i) }.
Since the summation does not depend on the order of the summed elements,

C Π π x ∆ (E n ) = 1 n ∑ n i=1 Π π x ∆ ({ω ∈ Ω : ω ≥ X i }).
With similar developments on C N π x ∆ (E n ), we obtain:

C Π π x ∆ (E n ) = 1 n n ∑ i=1 1 -N π x ∆ ({ω ∈ Ω : ω < X i }) , C N π x ∆ (E n ) = 1 n n ∑ i=1 1 -Π π x ∆ ({ω ∈ Ω : ω < X i }) .
As exposed in [START_REF] Dubois | Theorie des possibilités : Applications 'a la représentation des connaissances en informatique[END_REF][START_REF] Dubois | Possibility theory and statistical reasonning[END_REF],

F π x ∆ (u) = N π x ∆ ({ω ∈ Ω : ω < u})
is the lower cdf of the set of cdf associated to the summative kernels of core(Π π x ∆ ). It is the lower cdf of a p-box [START_REF] Ferson | Constructing probability boxes and dempster-shafer structures[END_REF], whose upper cdf is given by F π x ∆ (u) = Π π x ∆ ({ω ∈ Ω : ω < u}). As shown in [START_REF] Dubois | Possibility theory and statistical reasonning[END_REF], we have:

F π x ∆ (u) = 0 if u < x, 1 -π x ∆ (u) otherwise, and
F π x ∆ (u) = π x ∆ (u) if u < x, 1 otherwise.
We thus obtain the imprecise cdf estimate:

C Π π x ∆ (E n ) = 1 n n ∑ i=1 π x ∆ (X i )1l [x≤X i ] + 1l [x>X i ] , (12)
C N π x ∆ (E n ) = 1 n n ∑ i=1 (1 -π x ∆ (X i ))1l [x≥X i ] . (13)
6 Experiment and choice of a maxitive kernel

As in the case of the summative kernel methods, the problem of the choice of a particular maxitive kernel for performing imprecise functional estimation can be discussed. The choice of the summative kernel shape κ is often considered as insignificant in the non-parametric statistics community. The main argument is that the asymptotic behavior (when n → +∞) of F nκ ∆ and f nκ ∆ depend more on ∆ than on the choice of κ [START_REF] Simonoff | Smoothing Methods in Statistics[END_REF][START_REF] Bosq | Théorie de lestimation fonctionnelle[END_REF]. However, the asymptotic conditions are barely fulfilled. In non-asymptotic conditions, the shape of the estimate strongly depend on the shape of κ. Moreover, the knowledge of the modeler is generally insufficient for choosing the appropriate kernel. Instead of choosing one particular summative kernel, we propose to the modeler to choose a family of summative kernels matching his knowledge via the choice of a maxitive kernel.

In such kernel methods, where a summative kernel is considered as a neighborhood, it seems sensible to assume that the chosen basic kernel to be shifted and dilated with expression (3) is centered, even and with a support included in [-1, 1]. Therefore, it naturally leads to choose a basic maxitive kernel π encoding these particular summative kernels. As shown in [START_REF] Dubois | Probability-possibility transformations, triangular fuzzy sets, and probabilistic inequalities[END_REF], the triangular maxitive kernel T is the most specific of such maxitive kernels. The triangular possibility distribution is defined on Ω by T (ω) = (1 -|ω|)1l [|ω|≤1] . We now illustrate Theorem 5 by per- forming the summative and maxitive estimates of the cdf underlying a set of 107 observations of the duration in minutes of the eruptions of the Old Faithful geyser in Yellowstone National Park. 1 Each precise estimate has been performed by using four different summative kernels κ ∆ : uniform, Epanechnikov, triweight and cosine kernels, with ∆ = 0.3. The definitions of the used kernels can be found in [START_REF] Loquin | On the granularity of summative kernels[END_REF]. The imprecise estimate is obtained by using a triangular maxitive kernel T with the same ∆ . As illustrated on Figure 1, every precise estimates of the cumulative distribution are included in the imprecise estimation interval.

Conclusion

In this paper, we proposed an extension of the Parzen Rosenblatt cdf estimate, which takes into account a possible lack of knowledge of the appropriate summative kernel to be involved. Compared to the classical method, our method results in an interval estimate instead of a point estimate. The imprecision of the obtained estimate consistently reflects the lack of knowledge of the modeler, quantified by the specificity of the involved maxitive kernel. We put this sensible imprecise cdf estimation into a wider framework of imprecise functional estimation. Now, the next significant step, in soft statistics, is likely to be the imprecise estimation of the pdf.

Fig. 1

 1 Fig. 1 Imprecise cumulative distribution estimate

This example, taken from[START_REF] Silvermann | Density Estimation for Statistics and Data Analysis[END_REF], is a popular benchmark in nonparametric estimation.