Aurélien Massein

David Daney

Yves Papegay

Robust Design of Parameter Identification

Keywords: parameter identification, optimal design, observability index, uncertainties, localisation

Quality of results computed during parameter identification problems relies on the selection of system's states while performing measurements. This choice usually does not take into account the uncertainty of states and of measures. For identifiability, classical methods focus only on the contribution of model errors on the uncertainty of parameters. We present an alternative approach that tackles this drawback: taking into account influence of all uncertainty sources in order to improve parameter identification robustness to uncertainties. A robotic application example that showcases the differences between approaches is developed as well.

Introduction

Design of experiments [START_REF] Fedorov | Optimal Design for Nonlinear Response Models[END_REF] is a way to improve results of a generic parameter identification problem. It is applied namely in robotics [START_REF] Kelly | Precision dilution in triangulation based mobile robot position estimation[END_REF], for calibration [START_REF] Daney | Choosing Measurement Poses for Robot Calibration with the Local Convergence Method and Tabu Search[END_REF], and in GPS area through dilution of precision [START_REF] Langley | Dilution of precision[END_REF].

In such identification problem, unknown parameters are related with states of the system, and with measured outputs through a model prone to uncertainties. Inaccuracy sources of the model are model discrepancy, measurement errors and inexactitude of system's states. Observation of different sets of measurements for different states of the system provides a way to compute unknown parameters through Aurélien Massein Inria Bordeaux -Sud-Ouest, France e-mail: aurelien.massein@inria.fr David Daney Inria Bordeaux -Sud-Ouest, France e-mail: david.daney@inria.fr Yves Papegay Inria Sophia Antipolis Méditerranée, France e-mail: yves.papegay@inria.fr a regression analysis process. In this process choice of states is crucial to enhance parameters identification. It refers to identifiability of the model and identifiability criteria.

In the non-linear case [START_REF] Hollerbach | Model Identification[END_REF], unknown parameters are classically computed by an iterative optimisation algorithm, starting from an initial estimation. During this iterative process, states and measures remain unchanged and influence of their uncertainties are neglected. At the end of the process, all uncertainties have the same order of magnitude. Despite this last remark, in the literature, choice of states is done with the same assumption that uncertainties on states and measures are negligible, by evaluating identifiability criteria to the so-called identification matrix. In this paper, we promote the idea to take into account the different sources of inaccuracy -namely all uncertainty sources-when selecting states for parameter identification. Hence, we are applying similarly identifiability criteria to what we call the uncertainty matrix, a matrix describing the contribution of all uncertainties to errors of the model.

In the next section, we are describing with more details the parameter identification problem, the regression analysis process, the iterative optimisation process, and identifiability criteria. We carefully define the identification matrix and the uncertainty matrix, and how they are used for selection of identification states of the system. The last section is devoted to a pedagogical application, that clearly shows the difference between the classical and the proposed approach of states selection.

This application concerns the localisation of a source by a mobile robot. In this application, measures are taken at regular time step, and the selection of states correspond to a trajectory determination, that can be easily visualized. Another interest of this example is the ability to perform the states selection incrementally, that allows some enhancements of the identification method.

Parameter Identification

Model Definition

We consider a system in step k = 1 . . . K, depending on unknown parameters x, system's states u = u 1 . . . u k T , and measured outputs m = m 1 . . . m k T , through the following model:

f k (u k , x, m k ) = 0 (1)
As we consider that system's states and measured outputs are prone to uncertainties, we distinguish each variable x of its actual value x * , and denote uncertainty on it by ∆ x. Once plugged into the model, the k-th observation of the system provides a set of Dim( f ) equations:

f k (u * k , x * , m * k ) = f k (u k + ∆ u k , x * , m k + ∆ m k ) = 0 (2)
We need to acquire p system observations, such that the number of equations will be greater than the number of unknown parameters (K • Dim( f ) > Dim(x)). The resulting system is usually widely over-constrained.

Regression Analysis

Based on the set of these observations leading to residual errors,

f k (u k , x * , m k ) = ε k with k = 1 . . . K (3)
a regression analysis estimator provides a solution x minimizing a given criterion. In the classical case of a Non-Linear Least Squares estimator, the criterion is the sum of the squares of residual errors:

K ∑ k=1 f k (u k , x, m k ) T • f k (u k , x, m k ) (4)
Method of optimisation is based on a first-order linear approximation of the model:

f (u + ∆ u, x + ∆ x, m + ∆ m) ≈ f (u, x, m) + J u • ∆ u + J x • ∆ x + J m • ∆ m or ∆ f ≈ J u • ∆ u + J x • ∆ x + J m • ∆ m (5) with ∆ u = [∆ u 1 , . . . , ∆ u p ] T , ∆ m = [∆ m 1 , . . . , ∆ m p ] T and ∆ f = [∆ f 1 , . . . , ∆ f p ] T .
Starting from an initial estimate x 0 of x, the Non-Linear Least Squares method performs several solving steps to reduce ∆ x. As J u • ∆ u + J m • ∆ m is assumed to be negligible compared to J x • ∆ x, see [START_REF] Gayral | A sufficient condition for parameter identifiability in robotic calibration[END_REF], j-th step consists in solving the following linear system -with ∆ f and J x computed from previous estimation x j -in the unknown variables x j+1 :

∆ f [x j ] = J x [x j ] • ∆ x j+1 with ∆ x j+1 = x j+1 -x j (6)
Iterative process ends when ∆ x is sufficiently small, and is the same order of magnitude of ∆ u and ∆ m -see [START_REF] Gayral | A sufficient condition for parameter identifiability in robotic calibration[END_REF] for the stop condition.

At the end of this iterative process, one classically considers that the quality of the final estimation x relies on the numerical quality of the Jacobian matrix J x , denoted by Identification Matrix. Numerical quality is precisely defined in terms of identifiability criteria (see below). As identification matrix depends on system's states, poses of the system for measurements are selected by optimisation of these criteria.

Identifiability Criteria

Identifiability criteria have been widely studied in optimal design of experiments -being called optimality criteria-and in robotics calibration problems -being called observability criteria-. Sun and Hollerbach have synthesized in [START_REF] Sun | Observability index selection for robot calibration[END_REF] those criteria and showed their similarities as some observability indexes have an optimality criterion counterpart.

Given a rectangular matrix M such that Y = M • X, identifiability criteria quantifies how uncertainties on X propagate to uncertainties on Y through M. The objective is to minimize some observability -or optimality-indexes, denoted O, associated to M, by choosing the state variable û which parametrize the matrix M, such that:

û = arg min u O(M(u)) (7)
Three criteria are popular, related to the singular values of the matrix M, obtained and sorted by a Singular Value Decomposition (and denoted by σ L ≤ . . . ≤ σ 1 ): np) . This index corresponds to the determinant of

• D-Optimality: O 1 = ∏ L l=1 σ l 1/(
M T .M here, det(M T .M) = ∏ L l=1 σ l . The sensitivity of Y with respect to X is decreased when O 1 is minimized. • Inverse of the condition number: O 2 = σ L /σ 1 • E-Optimality: O 3 = σ L
Sun and Hollerbach argued that O 1 (D-optimality) is the best criterion for a parameter estimation of an unscaled model or a model without a convincing scaling approach has to be minimized [START_REF] Sun | Observability index selection for robot calibration[END_REF]. Fedorov and Leonov stated that D-optimal (O 1 ) designs are most popular among theoretical and applied researchers in optimal experimental designs [START_REF] Fedorov | Optimal Design for Nonlinear Response Models[END_REF].

We can denote briefly that O 2 which related to condition number is for balancing parameter uncertainties importance, whereas O 3 is for reducing the worst parameter's uncertainty.

Uncertainty Matrix

At the end of the iterative process, a linear approximation of f in the neighborhood of the final estimation x of x is given by the p following equations:

f k (u * k , x * , m * k ) ≈ f (u k , x, m) + J u • ∆ u k + J x • ∆ x + J m • ∆ m k (8)
that can be written synthetically :

∆ f ≈ J u • ∆ u + J x • ∆ x + J m • ∆ m (9)
or by introducing what we denote by U, the Uncertainty Matrix such that:

∆ x = -J + x • J u J m -Id •   ∆ u ∆ m ∆ f   = U •   ∆ u ∆ m ∆ f   (10)
J +

x being the pseudo-inverse of J x .

We then claim that it is interesting to apply identifiability criteria to this uncertainty matrix U -minimizing O(U)when selecting states. To be efficient, at this point, we would benefit of normalizing the uncertainties as done in [START_REF] Hollerbach | Advances in robot calibration[END_REF][START_REF] Hollerbach | The calibration index and taxonomy for robot kinematic calibration methods[END_REF] or alternatively [START_REF] Gayral | Étalonnage d'un instrument d'observation spatial actif. phdthesis[END_REF].

Source Localisation Application

In this section, we illustrate the difference between using the identifiability matrix and using the uncertainty matrix in the previously described states selection process, on a pedagogical two-dimensional source localisation application.

Problem Statement

We aim to localize accurately and step-efficiently a fixed source with the help of a mobile robot.

The mobile robot we consider has an embedded sensor measuring its direction with respect to a source x = x S y S T , with a fixed sampling frequency and prone to bounded uncertainties. Our workspace is two dimensional and free of obstacle: wherever we are we can get a measure at each sampling step k and move anywhere. Speed of the robot is supposed constant and a constant distance r separates two consecutive measurement positions. We describe the robot motion in polar coordinates. To move robot from position u k = x k y k T at step k onto the next position u k+1 , we need a leading direction α k+1 as written belowf:

u k+1 = r cos α k+1 sin α k+1 + u k (11)
At each measurement step, the goniometric sensor on the robot provides the azimuth m k -with respect to a fixed reference frame-such that

sin m k -cos m k • (x -u k ) = 0 (12)
In the equations of the model, the position of the source x, the position u k and the measured azimuth m k play respectively the roles of unknown parameters x, states u and measures m. Expressed at the k-th step, the model f k is precisely given by the previous equations. Please note that u k depends on α k .

Hence first-order linear approximation of the model expressed as:

sin m k -cos m k • ∆ x + -sin m k cos m k • ∆ u k + cos m k -sin m k • (x -u k ) • ∆ m k (13)
The difference between robot and source position xu k is equal to ρ k , the distance between u k and x, modulo cos(m * km k ) which approximation is considered equal to 1.

States Choices

The expressions of the identifiability matrix and of the uncertainty matrix are respectively (at the k-th step):

J x k =   jx 1 . . . jx k   with jx k = sin m k , -cos m k (14)
U k = J + x k • N k (15)
where

N k =       n 1 0 . . . 0 0 n 2 0 . . . . . . 0 . . . 0 0 . . . 0 n k       with n k = sin m k -cos m k -ρ k 1 (16)
Then, the optimal next direction α k+1 for the mobile robot, is defined with the help of the identifiability criterion O 1 using either J x k+1 or U k+1 .

Experiment and Results

In our experiment, source x is at position x = 0 1 T , and the initial mobile robot position is u 1 = 0 0 T . The fixed motion step is r = 0.01, with a relative error lower than 1% and we have the same error on the direction α k about 0.1 • . Finally, the uncertainty of the measured angle ∆ m is uniformly distributed and bounded by ∆ m = ±10 • . In a first experiment, we select each step α k+1 according to the maximization of O 1 (J x k+1 ). In the second, we select it to minimize O 1 (U k+1 ).

The obtained results are presented in figures (1a) and (1b).

The first one displays the motion of the mobile robot into the euclidean workspace step-by-step: blue motion curve stands for a gradient determinant optimisation on identification matrix J x and orange motion curve for one on uncertainty matrix U.

The second one presents the error x *x as a function of the number of steps in the two experiments with the same color code. Note that the blue motion favours a circle approach centered on the source position whereas the orange motion favours a spiral approach on it: By neglecting uncertainties on measures (blue case), we intuitively want to change the angle m as much as possible between two measurement positions, see [START_REF] Takeda | A dbb-based kinematic calibration method for in-parallel actuated mechanisms using a fourier series[END_REF], for a better conditioning of J x k . On the orange trajectory, we also take into account the measurement errors, so we try to become closer to the source to minimize the influence of such errors.

Conclusion

We have introduced a new matrix to improve parameter estimation robustness to all uncertainties, in complement to classical identification optimisation. We applied our matrix optimisation in a source localisation application and demonstrated its possible application in a practical case. Results outperformed classical identification optimisation.

This application highlights that, to improve parameter estimation accuracy, we need to take into account more uncertainties than classically ones held in the identification matrix. Our proposed uncertainty matrix can be used in design of parameter identification problems, can improve significantly estimations accuracy and their robustness to uncertainties.

Fig. 1 :

 1 Fig. 1: Source Localisation Application


A Bensafi

M Benhamou

S Khaldi

H Benahmed

S Mansouri

S Souali

L Bouzina

I Abdellaoui

L Yakoubi

Scattered intensity from weakly charged cyclic polymer blends in solution

Keywords:

HAL is

I. INTRODUCTION

The phase behavior of ternary mixtures of two incompatible neutral linear polymers in solution has been the subject of several theoretical investigations using, first, the Random of blends of cyclic homopolymers towards phase separation as compared to systems of linear homologous, make them more attractive for some specific applications. From a theoretical point of view, cyclic macromolecules provide a model of polymers that are free from end effects. In some cases, these effects may have important implications, especially when one deals with short chains.

In comparison with linear charged polymers, few investigations have been devoted to similar properties of systems with other architectures. Consequently, this report aims to overcome this handicap by the study of the phase behavior of cyclic weakly charged polymers, and to compare the results with those obtained, in previous studies, for mixtures of linear chains with the same charge distributions [30]. By analogy with considerations mentioned below, the total structure factor of weakly charged cyclic homopolymers is expected to be different from that of linear chains mixtures.

The remainder of presentation proceeds as follows. Sec. 2 is devoted to a review of the theoretical formalism enabling us to apprehend the structural properties of systems under consideration, such as the total structure factor. A detailed discussion of our main results, by specifying the impact of the charge, composition and architecture effects on the total structure factor, is the aim of Sec. 3. Some concluding remarks are drawn in Sec. [START_REF] Amit | Field Theory, the Renormalization Group and Critical Phenomena[END_REF].

II. THEORETICAL BACKGROUNDS

The physical system we consider is a mixture of two weakly charged polymers A and B immersed in a good solvent. We are interested in the determination of the static structure matrix, S, of the considered ternary mixture. Within the framework of RPA, it has been found [30,31] that this structure matrix satisfies a generalized Zimm equation [32]

S -1 (Q) = S -1 0 (Q) + U (Q) . (1)
This equation gives the inverse total structure matrix for the interacting system as a sum of the inverse bare structure matrix S -1 0 (Q) and the interaction matrix U (Q), independently on the architecture of chains involved in the polymeric systems under investigation. There, the quantity Q = |q| stands for the module of the scattering wave-vector (or scattering wavevector amplitude), which is given by the standard relationship : Q = (4π/λ) sin (θ/2), where λ and θ are the wavelength of the incident radiation and the scattering angle, respectively.

We note that the bare structure matrix S 0 (Q) has zero off-diagonal elements, and we write

S 0 =    S 0 a 0 0 S 0 b    , (2)
where

S 0 i (Q) = Φ i Z i P i (Q) (i = a, b).
Here, Φ i is the concentration of polymer chains of type i, Z i their polymerization degree, and P i (Q) their form factor.

The interaction matrix in formula ( 1) is given by the sum

U (Q) = V + F . (3)
Here, V is the ordinary excluded volume matrix whose elements are the excluded volume

parameters v ij (i, j = a, b) V =    v aa v ab v ba v bb    , (4)
with

v aa = 1 Φ s -2χ as , (5a)
v bb = 1 Φ s -2χ bs , (5b)
v ab = v ba = 1 Φ s -χ as -χ bs + χ ab . (5c)
In these expressions, Φ s represents the volume fraction of solvent, while χ is and χ ab are the polymer i-solvent and polymer A-polymer B Flory-Huggins interaction parameters, respectively. For the sake of simplicity, we assume that the solvent has the same quality for A and B-polymers (χ as = χ bs ), and write

v aa = v bb = v ab -χ ≡ v . (6)
We have used the notation : χ ≡ χ ab .

The long-range electrostatic matrix F reads

F = α (Q)    f 2 a η ab f a f b η ab f a f b f 2 b    , (7)
with

α (Q) = 4πl B Q 2 + κ 2 . ( 8
)
In this expression, l B = e 2 /ǫk B T accounts for the Bjerrum length, of the order of 7

• A at room temperature.
Here, e is the electron charge, ǫ the dielectric constant of the medium, k B the Boltzmann's constant and T the absolute temperature. In Eq. ( 7), η ab = +1, if monomers

A and B have charges of the same sign, and η ab = -1, otherwise ; f a e and f b e represent the charges carried by A and B monomers, respectively. The Debye-Hückel screening length κ -1 is as follows

κ 2 = 4πl B i Φ ci + Φ salt . (9)
Here, Φ ci is the number density of counterions and Φ salt is that of the added salt (or electrolyte). The sum in the above relation runs over all free-point ions in the system.

With these considerations, the total interaction matrix writes

U =    U aa U ab U ba U bb    , (10)
with

U aa = v + α (Q) f 2 a , (11a)
U bb = v + α (Q) f 2 b , (11b)
U ab = U ba = v + χ + α (Q) η ab f a f b . (11c)
Relationships (11a) and (11b) clearly show that the repulsive electrostatic AA and BBinteractions increase the quality of the solvent. On the other hand, relationship (11c) tells us that, if η ab = +1, the electrostatic repulsion has tendency to increase the incompatibility of A and B polymers, and if η ab = -1, the electrostatic attraction has as effect to make the mixture more compatible.

Substituting Eqs. ( 11) and (2) into Eq. (1) yields

S -1 =    1 S 0 a + U aa U ab U ab 1 S 0 b + U bb    . (12)
Inverting this matrix gives

S aa (Q) = S 0 a (1 + U bb S 0 b ) D (Q) , (13a)
S bb (Q) = S 0 b (1 + U aa S 0 a ) D (Q) , (13b)
S ab (Q) = S ba (Q) = - U ab S 0 a (Q) S 0 b (Q) D (Q) , (13c)
with

D (Q) = 1 + U aa S 0 a (Q) + U bb S 0 b (Q) + U aa U bb -U 2 ab S 0 a (Q) S 0 b (Q) . (13d)
On the other hand, the total structure factor is defined by

S T (Q) = S aa (Q) + S bb (Q) + 2S ab (Q) . (14)
By using the above explicit forms of S 0 a , S 0 b , U aa , U bb and U ab , we find that

S T (Q) ZΦ = P (Q) 1 -2χ -α (Q) (f a -η ab f b ) 2 x (1 -x) ΦN P (Q) D (Q) . (15)
We assumed that polymers A and B have the same degree of polymerization, Z. Thus, we are concerned with a monodisperse ternary mixture. This implies, in particular, that the two polymers have the same form factor :

P (Q) ≡ P a (Q) = P b (Q). We have set Φ a = xΦ and Φ b = (1 -x) Φ
, where x is the composition and Φ = Φ a + Φ b the total monomer concentration. The denominator D (Q) appearing in formula (15) can be decomposed as follows

D (Q) = D n (Q) + D c (Q) , (16a)
where

D n (Q) = 1 + vΦZP (Q) -χ (2v + χ) x (1 -x) Φ 2 Z 2 P 2 (Q) (16b)
is the usual contribution, that is in the absence of charges, and

D c (Q) = α (Q) ΦZP (Q) xf 2 a + (1 -x) f 2 b + v 0 x (1 -x) ΦZP (Q) (16c)
accounts for the contribution of charges. In expression (16c), the parameter v 0 reads

v 0 = v (f a -η ab f b ) 2 -2η ab f a f b χ . ( 16d
)
Until now, the general formalism reviewed above applies to chains, whatever be their architecture. It is only at this stage that a difference appears in the expressions of their radii of gyration and form factors. For this purpose, we assume that the polyions behave as flexible polymers, and depending on weather chains are linear or cyclic, their form factors are respectively given by Debye [33] and Casassa [34] functions

g D (u i ) = 2 u 2 i e -u i + u i -1 , (17)
g C (u i ) = 2 √ u i e -u i /4 √ u i /2 0 e t 2 dt . (18)
In formulae ( 17) and ( 18), u i represents the dimensionless variable

u i = Q 2 R 2 gi = Q 2 Z i σ 2 i 6 , (19)
where σ i is the monomer size, and R gi represents the radius of gyration of a Gaussian linear chain of specie i (i = a, b), unperturbed by the excluded volume interactions. Moreover, the radii of gyration of linear chains and their cyclic counterparts are assumed to be governed by excluded volume effects. The calculation of the gyration radius of cyclic chains can be performed using some more recent model [29], according to which the mean-squared distance between two points separated by n monomers along the cyclic chain writes

r 2 n = n 1+ζ 1 - n Z i 1+ζ σ 2 i . (20)
Combining this relation and the general definition of the radius of gyration of a given polymer yields

R 2 gc = Z 1+ζ i σ 2 i 1 0 t 1+ζ (1 -t) 2+ζ dt . (21a)
Explicitly, we have

R 2 gc = Z 1+ζ i σ 2 i B (2 + ζ, 3 + ζ) , (21b)
where B (x, y) = Γ (x) Γ (y) /Γ (x + y) is the standard Euler's Beta function [START_REF] Gradshteyn | Table of Integrals, Series and Products[END_REF]. While in the case of their linear homologous, one obtains [36]

R 2 gl = Z 1+ζ i σ 2 i (2 + ζ) (1 + ζ) , (22)
with the notation ζ = 2ν -1, where ν ≃ 0.6 is the swelling critical exponent. This assumption, which is valid only in the case of weakly charged polymers, means that the conformational properties of chains are dominated by the excluded volume interactions, and the local rigidity due to the electrostatic potential can be neglected. Indeed, if such polymeric systems are highly charged, the chains become strongly extended, and thus, they take configurations similar to that of rods. In addition to the local rigidity, it is interestingly to recall the existence of the overall one due to electrostatic repulsions.

III. RESULTS AND DISCUSSION

A. The scattered intensity

In order to illustrate the angular dependence of the total structure factor S T (Q), we have plotted it as a function of the renormalized wavenumber QR g , in various conditions of charge distribution, using the following numerical parameters : Z = 104, σ = 2.5

• A , R gl = 135 • A , R gc = 93 • A , v = 30 • A 3 and l B = 7 • A .
It is possible to access directly to the total structure factor S T (Q), by choosing two polymers whose refraction indices or diffusion lengths are equal. The scattered intensity is then proportional to S T (Q) of expression [START_REF] Katchalsky | Chemical Physics of Ionic Solutions[END_REF], namely

I (Q) = (a -s) 2 S T (Q) , (23)
where a -s is the contrast factor with respect to solvent molecules.

At fixed Z and monomer concentration Φ, the scattered intensity curve depends on the charges values f a and f b and their relative sign (through η ab = -1, +1).

B. Mixture of a charged and a neutral polymers in solution

We consider a ternary mixture made of a charged polymer A (f a = f ), a neutral one B (f b = 0) and a good solvent. The corresponding total structure factor is obtained from general formula [START_REF] Katchalsky | Chemical Physics of Ionic Solutions[END_REF] setting f a = f and f b = 0. We find that

S T (Q) ZΦ = P (Q) 1 -[χ -α (Q) f 2 /2] χ -1 c (Q) D (Q) , (24)
with the notation

χ -1 c (Q) = 2x (1 -x) ΦZP (Q) . ( 25
)
The angular dependences of the total structure factor, defined in Eq. ( 24), are plotted in Figs. 1a and1b, for f a = f and f b = 0. The first one illustrates the effects of the charge parameter f with a composition fixed to the value x = 0.8. Analysis of this figure reveals that, even for a relatively important charge parameter, that is f = 0.3, the presence of 20% in neutral polymer is sufficient to provoke practically the disappearance of the scattering peak, whatever be the architecture of chains.

This observation is in agreement with those of Refs. [13] and [24], according to which a peak collapse occurs due to adding a small content in neutral polymer. This phenomenon appears more clearly in Fig. 1b which represents the angular dependence of the total structure factor for f = 0.1, vΦZ = 1, χ/v = 0.1, and several values of the composition x in monomers of specie 1. On the other hand, we observe clearly a peak in the case of a single polyion (x = 1), regardless its architecture, but the scattering maximum almost disappears due to adding of only 5% in neutral polymer. Contrary to linear chains systems, for which the scattering peak disappears in the presence of only 5% in neutral polymer, the one corresponding to systems made of their cyclic homologous is significant only for a composition of value x = 10%. These both figures show that the height of the scattering maximum is higher, as compared to that observed in the case of their linear counterparts.

C. Mixture of two charged polymers of the same sign in solution

We consider now two charged polymers A and B, and denote by f a and f b the respective non-vanishing charge parameters. The corresponding total structure factor can be obtained substituting η ab = +1 into general expression (15)

S T (Q) ZΦ = P (Q) 1 -2χ -α (Q) (f a -f b ) 2 x (1 -x) ΦZP (Q) D (Q) , (26)
where the denominator D (Q) is given by Eqs. (16a)-(16d), with η ab = +1.

Firstly, if we fix the composition to the value x = 0.8 and neutralize the charge of the second polymer, that is f b = 0, we will have a behavior similar to that illustrated in Fig. 1a. In the second case, the charge parameter of polymer A is maintained constant, namely f a = 0, but that of the second one is increased from 0 to 0.1, with vΦZ = 1. The behavior of the polymeric system under consideration is depicted in Fig. 2a. This latter shows a progressive reemergence of the peak as the charge parameter f b increases, and we recover the behavior almost similar to that shown in the single polyion limit, with a small χ-parameter as compared to the electrostatic part.

However, it is interestingly and worthwhile to examine the total structure factor S T (Q)

in the case where both charge parameters f a and f b are similar, that is f b = f b = f . With these considerations, Eq. ( 26) becomes

S T (Q) ZΦ = P (Q) 1 -2χx (1 -x) ΦZP (Q) 1 + v ef f (Q) ΦZP (Q) -χ [2v ef f (Q) + χ] x (1 -x) Φ 2 Z 2 P 2 (Q) , (27a)
with the notation

v ef f (Q) = v + α (Q) f 2 . ( 27b
)
At this stage, we remark that, with a modified excluded volume parameter, the expression governing the variations of the total structure factor S T (Q), has a form similar to that obtained in the neutral limit.

We note that, if χ = 0, Eq. (27a) reduces to that of a single polyion in solution, whose behavior is shown in Fig. 2b. The above observations remain valid for linear chains as well as for their cyclic homologous.

D. Mixture of two charged polymers of different sign in solution

In this case, we find for the total structure factor the expression

S T (Q) ZΦ = P (Q) 1 -2χ -α (Q) (f a + f b ) 2 x (1 -x) ΦZP (Q) D (Q) , (28)
where D (Q) is given by Eqs. (16a)-(16d), with η ab = -1.

For this particular system, we have represented, in Fig. 3a, the variations of the total structure factor S T (Q), as a function of QR g , for f a = 0.1, χ/v = 0.1, x = 0.8, and vΦZ = 1, and several values of the charge parameter f b ranging from 0 to 0.3. This figure

shows that, as the charge parameter f b increases, the total structure factor S T (q) increases substantially at small values of Q. In fact, this originates from strong electrostatic attraction force between polymers of different species, which induces an important scattering signal in that Q's domain.

Likewise, we show, in Fig. 3b, the variations of the total structure factor versus QR g for f a = f b = 0.1 (η ab = -1), vΦZ = 1, and χ/v = 0.1, and several values of composition

x ranging from 0 to 0.8. Analysis of this figure reveals that the peak disappears rapidly when only 5 % of chains are unlikely charged with respect to the remainder of chains. As expected, we observe that a small quantity in neutral polymer induces a collapse of the scattering peak, because of the significant reduction of the electrostatic repulsion between different polymeric species.

IV. CONCLUSIONS

This paper gives a theoretical framework for the study of structural properties of mixtures of two weakly charged cyclic homopolymers, in solution and their comparison with those corresponding to linear chains systems. This study has been stimulated by several recent reports. However, the present report is voluntarily limited to the examination of the only angular dependence of the total structure factor of polymeric systems under investigation. Various conditions of charge distributions were considered. At this point of view, we note that this latter quantity is readily accessible by radiation scattering measurements, if the contrast factors, with respect to solvent molecules (when neutrons are used) or the increments of refractive indices (in the case of light-scattering) of two monomeric species A and B, are the same.

On one hand, the interactions between two charged monomers are assumed to be governed by an ordinary screened Debye-Hückel potential of the form exp {-κd} /d, where d is the distance which separates the centers of two monomeric species. Due to this fact, we are in agreement with the assumptions that the charges are not in a near neighboring each other, and with the fact that the electrostatic interaction is relatively weak. However, in the case where the first hypothesis is not fulfilled, only the excluded volume interaction determines the phase behavior of the considered polymeric system. These features, illustrated clearly by the scattering curves, show that, at small values of the wavenumber Q, the electrostatic interaction predominates, and then, the thermodynamic interaction becomes more and more pronounced as Q increases.

Elsewhere, we have used the Debye and Casassa functions to describe the form factors of Gaussian linear and cyclic chains unperturbed by the excluded volume effects, respectively.

Moreover, the radii of gyration of linear chains and cyclic counterparts are assumed to be governed by the excluded volume interactions. For this purpose, the concept of persistence length is not introduced, in such a way that we remain in agreement with our description restricted only to weakly charged polymers. We have also assumed that the small ions, as the counterions, are point-like without any specific contribution to scattering curves or molecular interactions. However, their presence enables to ensure the overall electroneutrality condition and contributes to the screening of the electrostatic interactions.

In summary, the present model is based mainly on two assumptions. The first one is the RPA which allows to relate the scattering functions for the total system made of a mixture of identical homopolymers in solution. This is introduced through the interaction matrix which, in this case, corresponds to the sum of the bare excluded-volume matrix and the longrange Coulombian one. Meanwhile, we make use of the second approximation according to which the latter matrix is modeled through the Debye-Hückel potential. Strictly speaking, these two approximations apply only far from the overlap polymer concentration (threshold),

where the fluctuations of matter are not very strong, so-that the RPA is expected to be reliable. Furthermore, the Debye-Hückel approximation assumes that, even in the absence of salt, it occurs a substantial screening of the electrostatic forces between charged monomers.

This requirement is also satisfied in the semi-dilute regime where the counterions, coming from the dissociation of the polymer, are in large number, because of the electroneutrality condition which needs a concentration of counterions equal to f Φ.

In the light of this study, it appears the following facts : (i) in the neutral limit, the total structure factor S T (Q) was found to be identical to the one obtained for a homopolymer with a form factor P (Q) and an excluded-volume parameter v+χ/2. Several cases of distributions of charge were examined. (ii) In the case of a mixture of partially charged homopolymers in which the neutral one is isorefractive with respect to solvent molecules, the structure factor shows the same features than those observed for a single polyion system (ordinary polyelectrolyte behavior). This observation was explained by the fact that the whole chains undergo the same repulsions due to the long-range potential. Consequently, the interaction of the polyion is decreased in the presence of polymer B. In other words, this means that an addition of a small quantity of neutral polymer implies a disappearance of the scattering peak. However, if the charges, carried by monomers of species A and B, are of the same sign, the counterions of the second polymer induce a screening of the interaction due to the first one. This implies that, if a small percentage in chains do not contribute to the scattering, the scattering maximum tends to disappear. It has been also observed that, when one passes from a neutral polymer to a charged polymer, the scattering curves decrease substantially, due to the electrostatic interactions which predominate at small values of Q. On the other hand, these curves tell us that cyclic weakly charged polymers scatter better than their linear homologous. This originates from their architectural differences and the fact that the cyclic chains are characterized by a higher segmental density. Elsewhere, we have remarked that the peak location, when it exists, is slightly shifted towards the lower values of Q ; in particular, one observes that the peak, corresponding to cyclic chains, disappears at a value of the charge parameter higher than that referring to the case of their linear counterparts.

In the case of similarly charged monomer species, we essentially found that the results of neutral polymers remain valid with a modified excluded-volume parameter, which becomes Q-dependent, in order to include the effect of long-range electrostatic repulsions. If the two monomer species are oppositely charged, we again found that formulae for neutral polymers could be used, but both the excluded-volume v and the interaction parameter χ should be modified, in order to take into account the long-range electrostatic interactions between

FIGURE CAPTIONS

Fig. 1a : Variations of the reduced total structure factor, S T (Q)/ZΦ, upon the renormalized wave-number QR g , for a ternary system made of charged (f a = 0) and neutral (f b = 0) homopolymers in solution, at several values of the charge parameter f a , with no added salt.

The solid curves correspond to cyclic chains, whereas the dashed ones represent their linear homologous. From the top, each series of curves corresponds to f a = 0 (neutral polymeric system), 0.075, 0.1, 0.15, 0.2 and 0.3, respectively. In plotting of these curves, we have used the following numerical parameters : x = 0.8, vΦZ = 1 and χ/v = 0.1.

Fig. 1b : Variations of S T (Q)/ZΦ upon QR g , for a polymeric system, similar to that considered in Fig. 1a, for various values of the composition x. The solid curves correspond to cyclic chains, whereas the dashed ones represent their linear homologous. From the bottom, each series of curves corresponds to x = 1 (single polyion), 0.95, 0.90, 0.85, 0.80 and 0.50, respectively. The used parameters are : v = 1, vΦZ = 1 and Φ salt = 0 (no added salt).

Fig. 2a :

 2a Fig. 2a : Angular dependence of S T (Q)/ZΦ upon the variable QR g , for a ternary system made of neutral (f a = 0) and charged (f b = 0) homopolymers in solution, in the absence of salt and for several values of the charge parameter f b . The solid curves correspond to cyclic chains, whereas the dashed ones represent their linear homologous. From the top, each series of curves corresponds to f b = 0 (neutral polymeric system), 0.01, 0.03, 0.05, 0.075 and 0.1, respectively. The used parameters are similar to those of Fig. 2a

Fig. 2b :

 2b Fig. 2b : Angular dependence of S T (Q)/ZΦ upon the variable QR g , for a ternary system made of two similarly charged homopolymers (f a = f b ≡ f = 0, η ab = +1), with vΦZ = 1, χ/v = 0.1, in the absence of added salt and for several values of the common charge parameter f . The solid curves correspond to cyclic chains, whereas the dashed ones represent their linear homologous. From the top, each series of curves corresponds to f a = f b = 0 (neutral polymeric system), 0.075, 0.1, 0.15 and 0.2, respectively.

Fig. 3a :

 3a Fig.3a: Angular dependence of S T (Q)/ZΦ upon the variable QR g , for a ternary system made of two unlikely charged homopolymers (η ab = -1) for f a = 0.1, χ/v = 0.1, and several values of the charge parameter f b , with vΦZ = 1 and no added salt. The solid curves correspond to cyclic chains whereas the dashed ones represent their linear homologous. From the bottom, each series of curves corresponds to f b = 0 (polymer 2 is neutral), 0.075, 0.1, 0.15, 0.2 and 0.3, respectively.

Fig. 3b :

 3b Fig.3b: Angular dependence of S T (Q)/ZΦ upon the variable QR g , for a polymeric system similar that considered in Fig.3a, with f a = f b = 0.1 (η ab = -1) and several values of the composition x of the mixture. The solid curves correspond to cyclic chains, whereas the dashed ones represent their linear homologous. From the bottom, each series of curves corresponds to x = 1 (single polyion system), 0.95, 0.90, 0.85 and 0.8, respectively. The used parameters are : χ/v = 0.1, vΦZ = 1 and Φ salt = 0 (no added salt).

ACKNOWLEDGMENTS This paper was completed during a brief stay of the first author at the Laboratoire de Physique des Polymères et Phénomènes Critiques of the Faculté des Sciences Ben M'sik of Casablanca (Morocco). One of us (A.B) is much indebted to Professor M. Benhamou for his warm hospitality. The authors express their thanks to Prof. M. Benmouna for useful discussions and Dr. H. Benahmed for helping in computation.

charges within the same species (repulsion) and between different ones (attraction).


Hamza Rihani

Claire Maiza

Matthieu Moy

Efficient Execution of Dependent Tasks on Many-Core Processors

I. INTRODUCTION

The increasing performance requirements of safety-critical real-time embedded systems made traditional single-core architectures obsolete. Moving to more complex many-core systems requires new techniques and tools for the certification of the embedded software. Timing and functional behaviours are subject to specific requirements of certification guidelines such as D-178B/C for avionics and ISO26262 for automotive systems. Determinism and predictability of such systems are a major challenge. Running tasks should be known in advance which makes the static and non-preemptive scheduling a suitable approach to reach an optimal execution with a guarantee of a certain degree of determinism. Recent work on mapping and scheduling problems [START_REF] Giannopoulou | Scheduling of mixed-criticality applications on resource-sharing multicore systems[END_REF], [START_REF] Puffitsch | Mapping a multi-rate synchronous language to a many-core processor[END_REF], [START_REF] Walter | Energy-aware mapping and scheduling of large-scale macro data-flow applications[END_REF] consider a known value (or a set of possible values) of the Worst-Case Response Time (WCRT) and computes a mapping that optimizes a predefined cost function. When the response time analysis is too pessimistic, the static scheduling may introduce an idle time which reduces the core utilization.

Scheduling techniques must rely on tight estimations of the WCRT which in turn depends on co-runner tasks. However, in order to obtain a tight upper-bound on the response time, a mapping and scheduling should be known in advance. Indeed, the response time is highly influenced by the co-runner tasks. Concurrent accesses to the same shared resource may introduce interferences that should be accounted for in the response time analysis. The search for an optimal scheduling with a tight WCRT analysis that includes the shared resource interferences is a challenging open problem.

II. THE OPEN PROBLEM

We illustrate the open problem on a Synchronous Data-Flow (SDF) application. SDF languages such as Lustre [START_REF] Halbwachs | The synchronous data flow programming language lustre[END_REF] offer an efficient programming paradigm that, using a certified compiler, can produce deterministic sequential code. An SDF application is represented with a task dependency graph, where the amount of exchanged data among the tasks is deterministic and known in advance. We consider the application to be running on a multi/many core architecture with a banked shared memory. Figure 1 represents an example of a node in an SDF application. The node increases the values of an input array by 1. Among the challenges while parallelizing such applications, we need to i) specify a task mapping that optimizes a certain cost function; ii) specify a scheduling per processing element that respects dependencies; iii) find a tight estimate of the response time that takes into account the interferences from co-runners. There exists several solutions for the mentioned points when taken individually. However, the connexion and the interaction among them remains as an open problem. § ¤ In this context, Figure 2 represents a potential solution for an efficient execution of such applications. The proposed technique relies on an iteration between a scheduling/mapping function and the WCRT analysis. Existing approaches may be used in each step but must interact altogether. In the following, we highlight open problems regarding such composability:

1) Code Generation: A code generator transforms the example given in Figure 1 into a low level language (such as C language) which in turn is compiled to run on a target architecture. We consider that in_data (produced by a previous node) is present in the local memory bank. After computation, out_data is copied to the next nodes' local memory banks. Several execution models can be applied: 1) A single phase execution model where the output is "sent" to the next node as soon as it is ready. execution phase uses only the local memory bank to compute and store the output data. Then, a replication phase is added to copy the values of out_data into their memory destinations. The number of shared resource accesses may be significantly increased, but with a good scheduling of the execution and replication phases the interference may be reduced. The choice of the execution model affects the results in the next steps. It is not clear whether it is possible to recognize at this stage which execution model is more efficient by analyzing the structure of the application. 2) Task Mapping: We are interested in the optimization of the overall response time according to an estimated WCRT of each task. In memory intensive applications, the scheduler may consider minimizing the interference by avoiding concurrent accesses to the same shared resource. Each task is modeled by its release date, response time, deadline, and period. The single phase execution model may use traditional scheduling techniques to respect all the deadlines. The structured execution model offers more flexibility by co-scheduling communications and computations such that the interference is reduced [START_REF] Melani | Memory-processor co-scheduling in fixed priority systems[END_REF]. However, we still need to investigate the efficiency of this method w.r.t a single-phase execution model where the number of accesses is not increased but the interference can be pessimistically overestimated. 3) WCRT Analysis: The WCRT analysis must be able to derive tight upper-bounds on the interference due to concurrent accesses to shared resources, and therefore estimate the response times and release dates for all tasks. This requires a task mapping and scheduling as well as the constraints on the task dependencies. The execution model given by the code generation step can highly influence the analysis. In the case of a structured execution model, the upper bounds on the shared resource interferences in the execution and replication phases may be too pessimistic due to pessimistic assumptions on the arbitration policy. This may as well result in a larger estimation on the overall interference per task depending on the application's topology. As a result, one may run the analysis on both execution models and retain the one with the best overall response time. 4) WCRT/Mapping Iteration: Depending on the chosen task mapping and scheduling per processing elements, the response times must be re-estimated by taking into account the interference in order to check that all constraints are respected and potentially reduce idle times. However, the optimality obtained from step (2) may be lost due to the updated timing information. A possible solution relies on constraint programming languages. A solver finds a satisfiable task mapping that optimizes the cost function. The WCRT Analysis and a schedulability test can then compute a tighter estimation by taking into account potential interferences. New timing constraints are added to the model and the search stops when no better solution can be found. Since the mapping problem is NP-hard, it is hard to prove that the iteration done in 4) converges toward an absolute optimal solution (assuming its existence). In the absence of such guarantee, one can iterate for a certain amount of time and choose the best solution. This interaction remains as an open problem.

Fig. 1 :

 1 Fig. 1: Example of a Lustre node and its high level graphical representation

Fig. 2 :

 2 Fig. 2: High level work flow of a complete WCRT/Scheduling interaction

ACKNOWLEDGMENT

This work was funded by the grant CAPACITES (PIA-FSN2 n • P3425-146798) from the French Ministère de l'économie, des finances et de l'industrie.


L E Pair, L'impair

Les Superalgebres

Leurs Derivations

Claude Roger

We give an introduction to superalgebra, founded on difference between even (commuting) and odd (anti commuting) variables. We give a sketch of the work of Grassmann, and show how derivations of those structures induce various superalgebra structures, even derivations giving Lie superalgebras of Cartan type, while odd derivations give Jordan type superalgebras.

Résumé

Ce texte constitue une introduction à la superalgèbre, fondée sur la distinction entre variables commutantes, dites paires, et les anticommutantes, dites impaires. Nous donnons un aperçu historique sur l'oeuvre de Grassmann, et nous montrons comment les dérivations de ces superalgèbres permettent d'engendrer de nouvelles structures de superalgèbres de Lie. Les dérivations d'algèbres associatives et commutatives permettent d'engendrer des algèbres de Lie du type algèbres de champs de vecteurs; le résultat s'étend au cas commutatif gradué, la construction produisant alors des superalgèbres de Lie, pourvu que la dérivation de départ soit paire, c'est à dire respecte le degré. Nous explorons les structures algébriques très différentes (algèbres de Jordan) obtenues à partir d'une dérivation impaire (qui modifie le degré).

x j x i , tandis que les variables impaires anticommutent, vérifiant θ α θ β = -θ β θ α .

Les exemples standards en sont respectivement les algèbres symétriques et antisymétriques sur un espace vectoriel de dimension finie, que l'on peut représenter respectivement par des algèbres de polynômes et des algèbres extérieures, dites aussi "de Grassmann". Ce sont les algèbres commutatives (resp. anticommutatives) libres sur un nombre fini de générateurs. La référence de base pour les questions de supergéométrie est l'article de D. Leites [START_REF] Leȋtes | Introduction to the theory of supermanifolds[END_REF], voir aussi [START_REF] Deligne | Notes on supersymmetry (following Joseph Bernstein)[END_REF].

2

Aperçu historique C'est ici l'endroit d'évoquer l'oeuvre d'un mathématicien quelque peu méconnu: Hermann Günther Grassmann (1809-1877). Formé à Stettin (aujourd'hui Sczczecin, Pologne), puis à Berlin, son principal titre de gloire en mathématiques est d'avoir été le premier à jeter les bases de l'algèbre linéaire et bilinéaire. Il présente ses résultats sous forme de mémoire de doctorat en 1840, le rapporteur le rejette. En 1844, il en publie une version étendue à compte d'auteur: Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (La théorie de l'extension linéaire, une nouvelle branche des mathématiques); le livre finira au pilon, faute d'acheteurs. Le saut épistémologique aussi fondamental qu'audacieux fut de considérer les êtres géométriques comme pouvant être soumis à des opérations algébriques, comme des nombres. Comme le dit Dieudonné : "A partir de là, toute une série d'idées nouvelles vont élargir considérablement ces conceptions, et conduire peu à peu à l'idée que l'Algèbre est la science des opérations sur des objets arbitraires"([2] p.93).

Citons Grassmann en traduction française: La première impulsion est venue de considérations sur la signication des nombres négatifs en géométrie. Habitué à voir AB comme une longueur, j'étais néanmoins convaincu que AB = AC + CB, quelle que soit la position de A, B et C sur une droite.

Il décrit ensuite comment il a prolongé cette réflexion aux rectangles, à leurs aires orientées, etc., aux parallélépipèdes et à leurs volumes. Il anticipe d'un même mouvement le calcul barycentrique de Möbius, allant plus loin que lui dans la formalisation de la multiplication d'un scalaire par un vecteur. Il développe une 'analyse géométrique', allant plus loin que le calcul vectoriel de A. Cayley en calculant sur des grandeurs orientées ("extensive Grössen" ) de dimension quelconque, ce qui l'a amené à construire l'algèbre extérieure, que l'on peut donc qualifier "de Grassmann" de façon tout à fait légitime.

Il a ainsi jeté les bases de l'algèbre linéaire et multilinéaire, sans toutefois donner une définition générale et axiomatique des espaces vectoriels et applications linéaires, faute d'avoir à sa disposition la théorie des ensembles; ce sera le travail de Giuseppe Peano, quelque trente ans plus tard. Il donne à cette époque la démonstration du célèbre théorème permettant de déterminer la dimension de la somme de deux sousespaces en fonction de la dimension de ceux-ci et de celle de leur intersection. Le travail de Grassmann reste cependant méconnu; sa candidature à un poste universitaire en 1847 est rejetée suite à un rapport négatif d'Ernst Kummer.

Il va alors enseigner au lycée de Stettin sa ville natale, et se tourne vers d'autres recherches, d'abord en physique avec une théorie des couleurs, mais la célébrité allait venir de ses travaux en linguistique génórale; en contact avec Franz Bopp, pionnier de la linguistique comparée indo-européenne, il se lance dans l'étude du sanskrit et la réalisation d'un dictionnaire de cette langue, puis dans la traduction du Rig-Veda. Sa principale découverte, dite "dissimilation des aspirées" reste connue des linguistes sous le nom de loi de Grassmann: sous certaines conditions on observe une mutation des consonnes occlusives aspirées en occlusives sourdes; cela se produit notamment en grec ancien et sanskrit; pour les hellénistes la mutation consiste en : (φ, θ, χ) → (π, τ, κ) 1 .Pour une introduction accessible au comparatisme indo-européen, on poura se reporter à l'ouvrage d'André Martinet [START_REF] Martinet | Des steppes aux océans, L'indo-européen et les "Indo-Européens[END_REF](remarque: les chapitres non purement linguistiques de cet ouvrage sont fortement contestés par les spécialistes).

La vérification de l'identité de Jacobi est un calcul algébrique très simple qui permet d'obtenir la famille des algèbres de Lie dites 'de Cartan': pour toute variété différentiable V , l'algèbre de Lie des dérivations de l'algèbre C ∞ (V ) s'identifie à l'algèbre de Lie V ect(V ) des champs de vecteurs tangents. Il est remarquable que ce résultat ne dépende pas d'hypothèses topologiques ou analytiques sur les dérivations considérées. Les algèbres de Cartan sont alors des sous algèbres de Lie de V ect(V ) liées à des structures géométriques, unimodulaires, symplectiques, de contact...En particulier si V = S 1 , on obtient l'algèbre de Virasoro sans son terme central.Voir par exemple [START_REF] Guieu | L'algèbre et le groupe de Virasoro[END_REF] pour les propriétés algébriques et géométriques de cette algèbre de Lie.

La construction de l'algèbre de Virasoro peut se généraliser de la façon suivante: pour toute dérivation δ d'une algèbre commutative et associative A (par exemple A = C ∞ (V ), mais pas nécessairement), l'espace des dérivations du type aδ pour a ∈ A forment une sous-algèbre de Lie de Der(A) pour le commutateur des dérivations, et le crochet obtenu est encore donné par la célèbre formule:

[aδ, bδ] = (aδ(b) -bδ(a))δ.
Cette construction est appelée "virasorisation"(cf. [START_REF] Ovsienko | Looped cotangent Virasoro algebra and non-linear integrable systems in dimension 2 + 1[END_REF], appendice). On montre ensuite facilement que le crochet donné par le commutateur gradué:

[δ 1 , δ 2 ] = δ 1 • δ 2 -(-1) |δ 1 ||δ 2 | δ 2 • δ 1 .
vérifie l'identité de Jacobi et définit donc une structure de superalgèbre de Lie; cf. encore [1, vol.1, part.1] pour les structures de superalgèbres de Lie. . On obtient ainsi une superalgèbre de Lie notée V(A) que nous appellerons virasorisation de A; on détermine immédiatement la parité: V(A) i est isomorphe à A i pour i = 0, 1 modulo 2. On retrouve ainsi l'algèbre de Virasoro ainsi que certaines de ses partenaires supersymétriques, notamment la célèbre algèbre de Neveu-Schwarz et la famille des algèbres dites superconformes, utilisées en théorie des champs, dont on trouvera une présentation détaillée dans [3, chap. 9].

Notre construction est cependant beaucoup plus générale car elle fonctionne pour toute superalgèbre associative et supercommutative.

Le cas où la dérivation est impaire

Ici la construction inverse la parité: V(A) i est isomorphe à A i+1 pour i = 0, 1 modulo 2.

La condition que les produits des aδ soient des éléments du même type nous impose le choix des signes, comme le montre un calcul élémentaire (en langage physicien, il faut que l'algèbre "ferme"), il suffit d'annuler les termes en δ 2 . On doit poser:

aδ.bδ = (aδb + (-1) (|a|+1)(|b|+1) bδa)δ.

On a donc un commutateur si |a| = |b| = 0 , un anticommutateur dans tous les autres cas, et il s'agit d'étudier la nature des structures algébriques obtenues.

V(A) 0 × V(A) 0 → V(A) 0 V(A) 0 × V(A) 1 → V(A) 1 V(A) 1 × V(A) 1 → V(A) 0
On remarque immédiatement que ces multiplications vérifient les mêmes conditions de symétrie-antisymétrie que les Antigèbres de Lie définies et étudiées par Valentin Ovsienko et al. dans [START_REF] Ovsienko | Lie antialgebras: prémices[END_REF].

Nous allons tout d'abord considérer un cas particulier important, celui où A est l'algèbre des fonctions sur le supercercle S 1|1 en les variables t, θ, et où δ = D θ = θ ∂ ∂t -∂ ∂θ . Cet opórateur vérifie une relation remarquable D θ • D θ = ∂ ∂t , en quelque sorte D θ est la racine carrée du temps. Cette algèbre sera notée V(A( 1)). On a alors la :

Proposition:

Le produit . associé à δ = D θ munit V(A(1)) d'une structure d'antigèbre de Lie isomorphe à celle notée AK(1) dans [START_REF] Ovsienko | Lie antialgebras: prémices[END_REF].

Cette proposition va résulter d'un calcul direct, le produit de deux éléments de V(A(1)) s'écrit :

(u + θφ)D θ .(v + θψ)D θ = (uψ + vφ + θ(uv -vu + 2φψ))D θ
En prenant des bases trigonométriques appropriées on retrouve les formules de [START_REF] Ovsienko | Lie antialgebras: prémices[END_REF] p.4.

Il est maintenant naturel d'étudier la généralisation pour une dimension impaire N arbitraire: à toute dérivation impaire δ de la superalgèbre A on associe une algèbre graduée notée V(A, δ). Ici nous n'avons plus de structure d'antigèbre de Lie dès que N > 2, par contre: Proposition: La partie paire V(A, δ) 0 est une Algèbre de Jordan, et si de plus δ 2 = 0 la partie impaire V(A, δ) 1 est un module de Jordan.

Il est maintenant naturel d'étudier la généralisation pour une dimension impaire N arbitraire: à toute dérivation impaire δ de la superalgèbre A on associe une algèbre graduée notée V(A, δ). Ici nous n'avons plus de structure d'antigèbre de Lie dès que N > 2, par contre: Proposition: La partie paire V(A, δ) 0 est une Algèbre de Jordan, et si de plus δ 2 = 0 la partie impaire V(A, δ) 1 est un module de Jordan.

Pour les résultats de base sur les Algèbres de Jordan, voir [START_REF] Mccrimmon | A taste of Jordan algebras[END_REF]. Pour démontrer cette proposition la définition nous suffira; une algèbre commutative est de Jordan si pour tous x, y, on a: -les structures orthosymplectiques ou supersymplectiques paires. Pour des coordonnées p i , q i , θ j , i = 1....n, j = 1...N sur la variété de dimension 2n|N , la forme s'écrit ω = n i=1 dq i ∧ dp i + N j=1 L'action adjointe de a ∈ A pour ces crochets de Poisson respectifs définit une dérivation impaire de A, si a est impair (cas orthosymplectique), ou pair (cas périplectique). On peut interpréter géométriquement la condition δ 2 = 0 comme l'annulation du crochet d'un champ impair avec lui même, voir le formalisme type 'Master Equation' pour les structures BV.

Essai de classification en N = 2

Considérons donc A(2) l'algèbre des fonctions sur le supercercle S 1|2 en les variables (t, θ 1 , θ 2 ),et cherchons un opérateur D = A∂ t + U 1 ∂ θ 1 + U 2 ∂ θ 2 impair, tel que D 2 = 0. On en déduit 3 équations différentielles en les fonctions coefficients, soit 6 équations en 6 fonctions sur le cercle S 1 , et on aboutit facilement à la forme générale:

D = λ(u 2 θ 1 -u 1 θ 2 )∂ t + (u 1 + λu 1 θ 1 θ 2 )∂ θ 1 + (u 2 + λu 1 θ 1 θ 2 )∂ θ 2

4

  Ces considérations s'étendent sans difficultés majeures au cas gradué. Soit A une algèbre associative Z/2Z-graduée, commutative-graduée; si on note par |a| le degré d'un élément a ∈ A,la commutativité graduée s'écrit:ab = (-1) |a||b| ba.Une application linéaire f est de degré |f | si pour tout a ∈ A, on a |f (a)| = |f | + |a|, les applications paires conservent le degré, les impaires l'échangent. On va maintenant s'intéresser aux dérivations graduées de A. Une application δ : A → A est une dérivation graduée de degré |δ| si pour tous a, b ∈ A, on a: δ(ab) = δ(a)b + (-1) |a||δ| aδ(b).

5

  La virasorisation graduéeConsidérons le supercercle unité S 1|N , qui est la supervariété dont la variété sousjacente est S 1 , et l'algèbre de fonctions est donnée par C ∞ (S 1 ) ⊗ Λ * (N ) où Λ * (N ) désigne l'algèbre extérieure sur N générateurs impairs. Nous allons donc généraliser la construction de l'algèbre de Virasoro à partir de l'algèbre commutative et associative des fonctions sur le supercercle unité S 1|N . Si δ est une dérivation fixée, et a ∈ A, aδ définie par [aδ](b) = aδ(b), est une dérivation de degré |aδ| = |a| + |δ|; en particulier pour δ paire |aδ| = |a|. Dans ce cas on définit naturellement le commutateur gradué de deux telles dérivations: [aδ, bδ] = (aδb -(-1) |a||b| bδa)δ.

  (x.y).(x.x) = x.(y.(x.x)).(J 1 ) Calculons d'abord avec les termes pairs, de la forme aδ où |a| = 1; dans ce cas le produit est commutatif suivant aδ.bδ = (aδb + bδa)δ. On trouve: aδ.aδ = 2aδaδ bδ.(aδ.aδ) = (2aδaδb + 2bδaδa + 2abδ 2 a)δ aδ.(bδ.(aδ.aδ)) = (6aδaδbδb)δ + (2bδaδaδa)δ En particulier les termes en δ 2 disparaissent miraculeusement. aδ.bδ = (aδb + bδa)δ (aδ.aδ)(aδ.bδ) = (6aδaδbδb)δ + (2bδaδaδa)δ Et on a donc bien (aδ.aδ)(aδ.bδ) = aδ.(bδ.(aδ.aδ)), ce qui achève la démonstration de la première assertion. Pour la suite, il convient de définir la notion de module de Jordan: on dira que M est un module sur l'algèbre de Jordan A s'il est muni d'actions à droite et à gauche telles que a.m = m.a pour tout a ∈ A et m ∈ M , et telles que l'espace A + M muni de la multiplication (a, m).(b, n) = (a.b, a.n + m.b) soit une algèbre de Jordan. On développe suivant les formules J 1 et on en déduit deux conditions indépendantes: (aδ.aδ).(aδ.nδ) = aδ.((aδ.aδ).nδ)(J 2 ) (aδ.aδ).(bδ.mδ) + 2(aδ.bδ).(aδ.mδ) = (bδ.(aδ.aδ)).mδ + 2aδ.(bδ.(aδ.mδ))(J 3 ) On peut tout calculer explicitement en prenant garde aux parités: |a| = |δn| = 1 |n| = |δa| = 0. On trouve: (aδ.aδ).(aδ.nδ) = (6aδaδaδn + 2nδaδaδa)δ = aδ.((aδ.aδ).nδ) (aδ.aδ).(bδ.mδ)+2(aδ.bδ).(aδ.mδ) = 12aδaδaδn+6bδmδaδa+6mδaδaδb = (bδ.(aδ.aδ)).mδ+2aδ.(bδ.(a Conjecture Pour δ 2 = 0, l'algèbre commutative graduée V(A, δ) est une superalgèbre de Jordan (voir [6] pour les définitions). Pour δ 2 = 0 ,comme par exemple pour δ = D θ i = θ i ∂ ∂t -∂ ∂θ i sur A(N ) l'algèbre des fonctions sur le supercercle S 1|N en les variables t, θ i , i = 1....N la démonstration est en défaut . 7 Zoologie des dérivations impaires 7.1 Dérivations des structures (super)symplectiquesOn distingue deux types de structures supersymplectiques suivant la parité de la forme correspondante:

  -les structures périplectiques ou symplectiques impaires, sur des variétés de dimension N |N , et où la forme s'écrit suivant ω = N i=1 dx i ∧ dθ i le crochet de Poisson

	s'écrit comme	{f, g} =	N i=1	(	∂f ∂x i	∂g ∂θ i	-	∂f ∂x i	∂g ∂θ i	)
					1 2 dθ 2 j , et le crochet de Poisson correspondant s'écrit
	suivant la formule
		{f, g} =	n i=1	(	∂f ∂p i	∂g ∂q i	-	∂f ∂q i	∂g ∂p i	) +	N j=1	∂f δθ j	∂g δθ j	.

La reconnaissance mathématique allait venir, mais bien tardivement; une réédition de l'"Ausdehnunglehre" en 1867 facilite la diffusion de ses idées et en 1871, il est admis à l'Académie des Sciences de Göttingen grâce à l'appui de Clebsch et de Klein. Vers la même époque, le jeune Sophus Lie lui rend visite à Stettin pour des explications à propos d'espaces vectoriels.La véritable reconnaissance sera posthume, avec les travaux de J.C. Maxwell (axiomatisation de l'électromagnétisme) et de W.K.Clifford vers 1880, lui reconnaissant la paternité des notions de produit scalaire et produit vectoriel. Enfin, vers 1900, en partant de l'algèbre extérieure de Grassmann et généralisant les formes de Pfaff, Elie Cartan introduisit les formes différentielles extérieures et l'opérateur d de différentiation extérieure qu'il utilisa pour établir les équations de structure des groupes de Lie, des " groupes infinis ", pour formuler la théorie du repère mobile et de la géométrie de ce qu'on appelle maintenant les variétés riemanniennes. 3 Les dérivations d'une algèbre associative et commutative A constituent naturellement une algèbre de Lie Der(A) pour le commutateur des dérivations; une application δ : A → A est une dérivation si pour tous a, b ∈ A, on a: δ(ab) = δ(a)b + aδ(b).1 Suivant la prononciation restituée du grec ancien selon Antoine Meillet[START_REF] Meillet | Aperçu d'une histoire de la langue grecque[END_REF],(φ, θ, χ) correspondaient respectivement aux phonèmes (p, t, k) aspirés, donc différents de la prononciation dite "érasmienne" du grec ancien, ou de celle du grec moderne


Claude Zuily

REAL ANALYTICITY OF RADIATION PATTERNS OF RESONANT STATES ON ASYMPTOTICALLY HYPERBOLIC MANIFOLDS

We show that resonant states in scattering on asymptotically hyperbolic manifolds that are analytic near conformal infinity, have analytic radiation patterns at infinity. On even asymptotically hyperbolic manifolds we also show that smooth solutions of Vasy operators with analytic coefficients are also analytic. That answer a question of M.Zworski ([14] Conjecture 2). The proof is based on previous results of Baouendi-Goulaouic and Bolley-Camus-Hanouzet and for convenience of the reader we present an outline of the proof of the latter.

Introduction and statement of the main results.

In this note we consider the question of analyticity of suitably renormalized resonant states on asymptotically hyperbolic manifolds. Referring to [START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF], [START_REF] Guillarmou | Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds[END_REF], [START_REF] Vasy | Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov)[END_REF], [START_REF] Zworski | Mathematical study of scattering resonances[END_REF] (section 3.1), [START_REF] Dyatlov | Mathematical theory of scattering resonances[END_REF] (Chapter 5) and [START_REF] Zworski | Resonances for asymptotically hyperbolic spaces: Vasy's method revisited[END_REF] for detailled presentations and for issues of geometrical invariance, we briefly recall the set-up in the case where the metric is analytic near infinity. 1.0.1. Radiation patterns. Let M be a compact n + 1 dimensional manifold with boundary ∂M = ∅ and let M := M \ ∂M. We assume that M is a real analytic manifold near ∂M . The Riemanian manifold (M, g) is said to be asymptotically hyperbolic and analytic near infinity if there exist functions y ′ ∈ C ∞ (M , ∂M ) and y 1 ∈ C ∞ (M , (0, 2)) such that

y 1 | ∂M = 0, dy 1 | ∂M = 0, M ⊃ y -1
1 ([0, 1)) ∋ m → (y 1 (m), y ′ (m)) ∈ [0, 1) × ∂M is a real analytic diffeomorphism and near ∂M the metric g has the form

(1.1)

g| y 1 ≤ε = dy 2 1 + h(y 1 ) y 2 1
where [0, 1) ∋ t → h(t) is an analytic family of real analytic Riemanian metrics on ∂M.

We recall now the following results of Mazzeo-Melrose [START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF] and Guillarmou [START_REF] Guillarmou | Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds[END_REF].

(*) Supported in part by Agence Nationale de la Recherche project ANA É ANR-13-BS01-0010-03.

1

For Im λ > 0, the operator R g (λ) = -∆ gλ 2 -n 2 2 -1 can be defined from L 2 (M ) to

H 2 (M ). Then (1.2) R g (λ) : C ∞ 0 (M ) → C ∞ (M )
continues to a meromorphic family of operators for λ ∈ C \ (-i 2 N).

Moreover for λ ∈ C \ (-i 2 N),

(1.3) u ∈ R g (ζ) dζ C ∞ 0 (M ), (-∆ g -λ 2 - n 2 2 )u = 0, =⇒ F (y) := y iλ-n 2 1 u ∈ C ∞ (M ).
Here the integral is over a small circle enclosing λ and no other singularity of R g .

The function F (or F | ∂M ) can be considered as the radiation pattern of the resonant state u.

In the analytic case it is natural to ask if the radiation patterns are real analytic. This is indeed the case as shown by the following result.

Theorem 1.1. Suppose that M and g are real analytic near the ∂M . Then, with the above terminology and for λ ∈ C \ (-i 2 N), radiation patterns, F , of resonant states u are real analytic near ∂M . Remarks 1.2. 1. The result is local in the sense that analyticity needs to be assumed only at m ∈ ∂M with the corresponding local conclusion.

2. An equivalent conclusion would be to say that F | ∂M is real analytic. That is true in view of Theorems 0.2 and 0.3 which come from [START_REF] Baouendi | Cauchy problems with characteristic initial hypersurface[END_REF]. That restriction is what we would normally call the radiation pattern. 1.0.2. Vasy operators. Motivated by analysis of the wave equation for Kerr-de Sitter black hole metrics Vasy [START_REF] Coddington-N.Levinson | Theory of ordinary differential equations[END_REF] introduced a microlocal approach to the meromorphic continuation (1.2). The key component are radial propagation estimates first obtained by Melrose [START_REF] Melrose | Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces[END_REF].

Vasy's approach works for even asymptotically hyperbolic metrics. This means that in the notation of (1.1) the metric is given by (1.4) g|

y 1 ≤ǫ = dy 2 1 + h(y 2 1 ) y 2 1
, where [0, 1) ∋ t → h(t), is an analytic family of real analytic Riemannian metrics on ∂M . In that case (1.5)

y iλ-n 2 1 (-∆ g -λ 2 -( n 2 ) 2 )y -iλ+ n 2 1 = x 1 P (λ), x 1 = y 2 1 , x ′ = y ′ , , where, near ∂M , (1.6) P (λ) = 4(x 1 D 2 x 1 -(λ + i)D x 1 ) -∆ h + iγ(x) 2x 1 D x 1 -λ -i n-1 2 .
This is the Vasy operator [START_REF] Coddington-N.Levinson | Theory of ordinary differential equations[END_REF].

The operator (1.6) can be considered locally as a special case of the following class of operators:

(1.7)

L := xD 2 x -(λ + i)D x + γ(x, y)xD x + Q(x, y, D y ), D = 1 i ∂,
where

(-1, 1) ∋ x → Q(x, y, D y ) = |α|≤2 a α (x, y)D α y , y ∈ U ⊂ R n
is a an analytic family of (positive) elliptic second order differential operators with analytic coefficients.

Theorem 1.3. Assume that the coefficients of L are real analytic in a neighborhood of a point m 0 = (0, y 0 ) ∈ (-1, 1) × R n and that λ / ∈ -iN * .

Then if u is a C ∞ function near m 0 such that Lu is real analytic in a neighborhood of m 0 then u itself is real analytic near m 0 .

Remarks 1.4. (i) Notice that the operator L is elliptic for x > 0 and hyperbolic for x < 0.

(ii) When γ = 0 and Q = ∆ M is the Laplace-Beltrami operator on a compact manifold M the analytic regularity has recently been proved by Lebeau and Zworski [START_REF] Lebeau | Remarks on Vasy's operator with analytic coefficients To appear[END_REF].

(iii) Vasy's adaptation of Melrose's radial estimates shows that to have u ∈ C ∞ we only need to assume that u ∈ H s+1 near m 0 , where s + 1 2 > -Im λ, see [START_REF] Zworski | Resonances for asymptotically hyperbolic spaces: Vasy's method revisited[END_REF] §4, Remark 3.

Acknowledgements. Warm thanks are due to Maciej Zworski for providing informations on the mathematical background of this note, upon which this introduction was written and for his great generosity.

Proofs

2.1. Preliminaries. In this section we recall some results by Baouendi-Goulaouic [START_REF] Baouendi | Cauchy problems with characteristic initial hypersurface[END_REF] and Bolley-Camus-Hanouzet [START_REF] Bolley-C. Camus-B | Étude de l'analyticité et de la régularité Gevrey pour une classe de problèmes aux limites elliptiques et dégénérés[END_REF].

We begin by a "Cauchy-Kovalevska" type theorem which is a particular case of Theorem 1 in [START_REF] Baouendi | Cauchy problems with characteristic initial hypersurface[END_REF].

Consider a "Fuchs type" operator with analytic coefficients near m 0 = (0, y 0 ), of the form (2.1)

P = x∂ 2 x + a(y)∂ x + Q(x, y, ∂ y ) + |β|≤1 x∂ x b β (x, y)∂ β y
where Q is a second order differential operator in y (non necessarily elliptic). To this operator we associate the caracteristic equation,

(2.2) C(µ, y) = µ(µ -1) + µa(y) = µ(µ -1 + a(y)).
Then we have the following result.

Theorem 2.1. The following properties are equivalent.

(i) For any integer µ ≥ 1we have C(µ, y 0 ) = 0.

(ii) For any analytic functions v 0 near y 0 and f near m 0 there exists a unique function v which is analytic near m 0 such that

Pv = f, v(0, •) = v 0 .
The second result is a "Holmgren type" theorem which is also a particular case of Theorem 2 of [START_REF] Baouendi | Cauchy problems with characteristic initial hypersurface[END_REF].

Theorem 2.2. Let P be defined by (2.1) and h ∈ N. Assume that the roots of the equation

C(µ, y 0 ) = 0 satisfy Re µ k < 1 + h, k = 1, 2.
Then any function U of class C 1+h in (x, y) near m 0 satisfying

PU = 0, ∂ j x U (0, •) = 0, for 0 ≤ j ≤ h vanishes identically near m 0 .
In the sequel we shall denote by C ω the space of analytic functions.

Let us recall a particular case of a result by Bolley-Camus-Hanouzet [START_REF] Bolley-C. Camus-B | Étude de l'analyticité et de la régularité Gevrey pour une classe de problèmes aux limites elliptiques et dégénérés[END_REF].

Theorem 2.3. Let y 0 ∈ R n and P 1 (resp. P 2 ) be a differential operator of order 1 (resp. 2) with analytic coefficients on [0, 1) × V y 0 , (2.3)

P 1 = |α|+k≤1 a α,k (t, y)D k t D α y , P 2 = |α|+k=2 b α,k (t, y)D k t D α y .
Assume that P 2 is uniformly elliptic in [0, 1) × V y 0 . Define L by

Lu = P 2 (tu) + P 1 u. Then (2.4) u ∈ C ∞ ([0, 1) × V y 0 ), Lu ∈ C ω ([0, 1) × V y 0 ) =⇒ u ∈ C ω ([0, 1) × V y 0 ).
For the reader's convenience we shall sketch briefly the proof of this theorem at the end of this note.

Remarks 2.4. 1. A previous result of this type is due to Baouendi-Goulaouic [START_REF] Baouendi | Étude de l'analyticité et de la régularité Gevrey pour une classe d'opérateurs elliptiques dégénérés[END_REF].

2. Notice that (2.4) holds without condition on the boundary data. This fact is due to the degeneracy on the operator L on the boundary t = 0 and to the particular form of L.

3. It follows that an operator (with analytic coefficients) of the form

(2.5) L = t D 2 t + R 2 (t, y, D y ) + R 1 (t, y, D t , D y ) where R j is of order j and σ(R 2 )(t, y, η) ≥ c|η| 2 satisfies (2.4).
2.2. Proof of Theorem 1.1. Using the special form of the metric (1.1) we compute (in the coordinates valid near the boundary),

-∆ g = (y 1 D y 1 ) 2 + i(n + y 1 γ 0 (y 1 , y ′ ))y 1 D y 1 -y 2 1 ∆ h(y 1 ) , γ 0 (t, y ′ ) := -1 2 ∂ t h(t)/ h(t), h(t) := det h(t), D := 1 i ∂. (2.6)
To reduce the presentation to a special case of the results by Bolley-Camus-Hanouzet, we perform the conjugation which transforms an equation for u into an equation for F (in the notation of (1.3)). We can write (2.7)

0 = y iλ-n 2 1 (-∆ g -λ 2 -( n 2 ) 2 )u = y iλ-n 2 1 (-∆ g -λ 2 -( n 2 ) 2 )y -iλ+ n 2 1 F = (y 1 D y 1 ) 2 + (in + iy 1 γ 0 -2λ)y 1 D y 1 + y 1 γ 0 ( 1 2 n -iλ) -y 2 1 ∆ h(y 1 ) F = y 1 y 1 (D 2 y 1 -∆ h(y 1 ) ) + f (y)D y 1 + g(y)
F, where f and g are analytic functions. The operators in brackets on the right hand side is precisely an operator of the form (2.5). We can therefore apply the point 3. in Remark 2.4 to conclude.

2.3.

Proof of Theorem 1.3. Let L be the operator defined by (1.7). Then

-L = x∂ 2 x + (1 -iλ)∂ x + iγ(x, y)x∂ x -Q(x, y, D y ). It is of the form (2.1) with a 1 (y) = 1 -iλ. Therefore C(µ, y 0 ) = µ(µ -iλ).
Since by hypothesis we have iλ / ∈ N * we see that the condition in Theorem 2.1 is satisfied.

Let u be the C ∞ solution in Theorem 1.3. Assume that we can prove that u(0, •) is analytic near y 0 . Let v be the analytic solution of the problem

Lv = f, v(0, •) = u(0, •)
given by Theorem 2.1. Setting U = vu we see that U is C ∞ and satisfy (2.8)

x∂ 2 x + (1 -iλ)∂ x + iγ(x, y)x∂ x -Q(x, y, D y ) U = 0, U (0, •) = 0.
We claim that, when iλ / ∈ N * , we have (2.9) ∂ j x U (0, •) = 0, for all j ∈ N. This is true for j = 0. Assume this is true for 0 ≤ j ≤ k. We differentiate k times the equation in (2.8) with respect to x and and we take the trace on x = 0. Using the induction we see that (k + 1iλ)∂ k+1

x U (0, •) = 0 which ends the induction. Now the roots of the equation C(µ, y 0 ) = 0 are µ 1 = 0, µ 2 = iλ. Taking h ∈ N such that 1 + h > Re µ 2 = -Im λ we see that the condition in Theorem 2.2 is satisfied. Using then (2.9) we see that Theorem 2.2 implies that U = 0. Thus u = v and u is analytic near m 0 which completes the proof. Therefore we are left with the proof of the following claim.

"Under the conditions of Theorem 1.3 u(0, •) is analytic near y 0 ."

Consider now an operator on [0, 1) × R n with analytic coefficients of the form

P = xD 2 x + c(x, y)D x + d(x, y) + Q(x, y, D y ) where σ 2 (Q)(x, y, η) ≥ c|η| 2 .
The goal is to prove that if u ∈ C ∞ on [0, 1) × V y 0 and P u = f is analytic on [0, 1) × V y 0 then u is analytic in [0, 1) × V y 0 . In particular u(0, •) is analytic.

One can reverse the calculation (1.5) and (2.7) to see that, with x = t 2 , if u is a solution in [0, 1) of the equation P u = f then w defined by w(t, y) = u(t 2 , y) satisfy an equation on [0, 1) × V y 0 of the form P w = g where g(t, y) = tf (t 2 , y) and P w = tP 2 (t, y, D t , D y ) + P 1 (t, y, D t , D y ), where P 2 is an elliptic operator of order two and P 1 is a first order operator -both with analytic coefficients. Using the point 3. in Remarks 2.4 we see that w is analytic in [0, 1) × V y 0 .

To prove that u(0, •) is analytic we use the following lemma.

Let u : [0, 1) × V y 0 → C be a C ∞ function. Set (2.10) w(t, y) = u(t 2 , y) 0 ≤ t ≤ 1, y ∈ V y 0 .
Lemma 2.5. Assume that there exist positive constants A, B 1 , B 2 such that for all α ∈ N n , j ∈ N we have

(2.11) |∂ α y ∂ j t w(t, y)| ≤ AB |α| 1 B j 2 α!j!, 0 ≤ t ≤ 1, y ∈ V y 0 . Then (2.12) |∂ α y ∂ j x u(x, y)| ≤ AB |α| 1 B 2j 2 α!j!, 0 ≤ x ≤ 1, |y ∈ V y 0 . Proof. By definition w is C ∞ in [0, 1) × V y 0 and we have (2.13) ∂ α y ∂ 2ℓ+1 t w(0, y) = 0, ∀α ∈ N n , ∀ℓ ∈ N.
We claim that we have for all j ∈ N, all α ∈ N n and all t ∈ [0, 1)

(2.14) (∂ j x ∂ α y u)(t 2 , y) = 1 2 j 1 0 1 0 • • • 1 0 j-1 ℓ=1 λ 2(j-ℓ) ℓ (∂ 2j t ∂ α y w)(t j ℓ=1 λ ℓ , y) j ℓ=1 dλ ℓ .
where the first product in the integral is equal to 1 if j = 1.

We shall prove this formula for all α ∈ N n by induction on j. We begin by j = 1. Differentiating both members of (2.10) with respect to t and y we obtain ,y). Using the Taylor formula and (2.13) we can write

(2.15) 2t(∂ x ∂ α y u)(t 2 , y) = ∂ t ∂ α y w(t
(2.16) (∂ t ∂ α y w)(t, y) = t 1 0 (∂ 2 t ∂ α y w)(λ 1 t, y) dλ 1 .
From (2.15) and (2.16) we obtain for t = 0

(∂ x ∂ α y u)(t 2 , y) = 1 2 1 0 (∂ 2 t ∂ α y w)(λ 1 t, y) dλ 1
and the later formula extends to t = 0 by continuity. This proves (2.14) for j = 1. Assume that (2.14) is true up to the order j and differentiate this equality with respect to t. We obtain

(2.17) 2t(∂ j+1 x ∂ α y u)(t 2 , y) = 1 2 j 1 0 1 0 • • • 1 0 j ℓ=1 λ 2(j-ℓ)+1 ℓ ∂ 2j+1 t ∂ α y w (t j ℓ=1 λ ℓ , y) j ℓ=1 dλ ℓ .
Using (2.13) we can write

∂ 2j+1 t ∂ α y w (t j ℓ=1 λ ℓ , y) = t j ℓ=1 λ ℓ 1 0 ∂ 2j+2 t ∂ α y w (t j+1 ℓ=1 λ ℓ , y) dλ j+1 .
Plugging the right hand side into the integral (2.17) and dividing both members by t for t = 0 we obtain (2.14) for j + 1 and t = 0 and also for t = 0 by continuity.

We can now prove the lemma. Indeed using (2.14) and (2.11) we obtain for x ∈ [0, 1)

|(∂ j x ∂ α y u)(x, y)| ≤ AB |α| 1 B 2j 2 α!(2j)! 1 2 j j-1 k=1 1 2k + 1 ≤ AB |α| 1 B 2j 2 α!(2j)! 1 2 j 2 j j! (2j)!
which proves (2.12).The proof is complete.

An even simpler proof of the analyticity of the trace of u would be to consider the Taylor series of w and to use (2.13).

2.4. Sketch of the proof of Theorem 2.3. We follow closely [START_REF] Bolley-C. Camus-B | Étude de l'analyticité et de la régularité Gevrey pour une classe de problèmes aux limites elliptiques et dégénérés[END_REF]. We shall be using the following version of Hardy's inequality.

Lemma 2.6.

For u ∈ C ∞ , with compact support in [0, +∞) × R n k ∈ N, β ∈ N n we have (2.18) D k t D β y u L 2 (R + ×R n ) ≤ 2 2k + 1 D k+1 t D β y (tu) L 2 (R + ×R n ) .
Proof. Set v = D β y u, w = ∂ t (tv). We have v(t, y) = 1 t t 0 w(s, y) ds = 1 0 w(tx, y) dx. Differentiating this equality k times with respect to t we get

∂ k t v(t, y) = 1 0 x k (∂ k s w)(tx, y) dx from which we deduce ∂ k t v L 2 (R + ×R n ) ≤ 1 0 x k (∂ k s w)(x•, •) L 2 (R + ×R n ) dx ≤ 1 0 x k-1 2 dx ∂ k t w L 2 (R + ×R n )
which completes the proof.

The first step of the proof is the following a priori estimate.

Proposition 2.7. Let y 0 ∈ R n . There exists p 0 ∈ N such that for all p ∈ N with p ≥ p 0 there exist C p > 0, ε p > 0 such that

(2.19) tu H p+2 (R + ×R n ) ≤ C p Lu H p (R + ×R n ) + tu H p+1 (R + ×R n )
for every u such that supp u ⊂ {(t, y) : 0 ≤ t < ε p , |yy 0 | < ε p } and the right hand side is finite.

Proof. This inequality is the estimate (1.7) in [START_REF] Bolley-C. Camus-B | Étude de l'analyticité et de la régularité Gevrey pour une classe de problèmes aux limites elliptiques et dégénérés[END_REF] since, with their notations we have χ p = 0.

To explain this last point we follow [START_REF] Bolley | Sur une classe d'opérateurs elliptiques et dégénérés à plusieurs variables[END_REF] (see conditions H 1 (p; Ω), H 2 (p; Ω)). We have χ p = 1-r p where r p is the number of roots of the characteristic equation ib 0,2 (0, y)ρ + a 0,1 (0, y) = 0 (with the notations in (2.3)) such that Re ρ(y) > -p -3 2 . Taking p large enough we find that r p = 1 so that χ p = 0. Therefore no boundary condition is required.

We only review the main points of the proof of (2.19) referring to [START_REF] Bolley | Sur une classe d'opérateurs elliptiques et dégénérés à plusieurs variables[END_REF] for the details. In the first step (the main one) one consider a one dimensional constant coefficient operator of the form Lu = P 2 (D t )(tu) + P 1 (D t )u where P 2 is of order 2 and P 1 of order ≤ 1. Introduce for m ≥ 1 the space

H m = {u ∈ H m-1 (R + ), tu ∈ H m (R + )}
where H k (R + ) is the usual Sobolev space. Assuming that

P 2 (τ ) = 0, ∀τ ∈ R,
it is proved that there exists p 0 ∈ N such that for all p ≥ 0 the operator L is an isomorphism from H p+2 onto a subspace of H p (R + ) of codimension one. To prove this fact they apply the "Fuchs theory" (see [START_REF] Coddington-N.Levinson | Theory of ordinary differential equations[END_REF]) to the operator L = P 2 (τ )(-D τ ) + P 1 (τ ) obtained (in spirit) from L in taking the Fourier transform in t. This leads to an inequality in some appropriate spaces for L and then for L.

In a second step they consider a partial differential operator with constant coefficients of the form Lu = P 2 (D t , D y )(tu) + P 1 (D t , D y )u, where P 2 is assumed to be elliptic. Performing a Fourier transform with respect to y we reduce ourselves to a one dimensional operator to which we apply the first step.

Finally one consider the variable coefficient case where Lu = P 2 (t, y, D t , D y )(tu) + P 1 (t, y, D t , D y )u, where P 2 is assumed to be elliptic. We write Lu = L 0 u + L 1 u with L 0 u = P 2 (0, y 0 , D t , D y )(tu) + P 1 (0, y 0 , D t , D y )u, we use the second step to deal with L 0 , the smallness of the coefficients of L 1 and the Hardy inequality (2.18) to end the proof of (2.19).

Next we use the classical method of nested open sets introduced by Morrey-Nirenberg in [START_REF] Morrey | On the analyticity of the solutions of linear elliptic systems of partial differential equations[END_REF]. Let a ∈]0, 1] and ω be a neighborhood of y 0 such that

Ω =: [0, a[×ω ⊂ {(t, y) : 0 ≤ t < ε p , |y -y 0 | < ε p }.
For 0 < ε < a we set

ω ε = {y ∈ ω : d(y, ω c ) > ε}, Ω ε = [0, a -ε[×ω ε , N ε (u) = u L 2 (Ωε) .
For ε > 0 and ε 1 > 0 small enough there exists a function

ψ ∈ C ∞ 0 (Ω ε 1 ) such that (2.20) (i) ψ(t, y) = 1 if (t, y) ∈ Ω ε+ε 1 , (ii) D j t D α y ψ L 2 (Ωε 1 ) ≤ Cε -j-|α| , ∀j ∈ N, ∀α ∈ N n ,
with a constant C independent of ε, ε 1 .

Lemma 2.8. There exists C > 0 such that for all ε > 0, ε 1 > 0 such that ε + ε 1 < a, all j ∈ N, ν ∈ N n with j + |ν| ≤ p + 2 and all u ∈ C ∞ (R × R n ), we have

N ε+ε 1 (D j t D ν y (tu)) ≤ C ℓ+|β|≤p ε -p+ℓ+|β| N ε 1 (D ℓ t D β y Lu)+ ℓ+|β|≤p+1 ε -p-2+ℓ+|β| N ε 1 (D ℓ t D β y (tu))
Proof. We apply (2.19)

A ≤ Cε -|a|-1 D b u L 2 (Ωε 1 ) ≤ C ′ ε -|a|-1 D t D b (tu) L 2 (Ωε 1 ) ≤ C ′′ |c|≤p+1 ε -p-2+|c| N ε 1 (D c (tu)).
We begin now to estimate the derivatives of higher order.

Lemma 2.9. There exist C > 0, K > 0 such that for all ε > 0,

ε 1 > 0 such that ε + ε 1 < a, all j ∈ N, ν ∈ N n with j + |ν| ≤ p + 2 all α ∈ N n and all u ∈ C ∞ (R × R n ), we have N ε+ε 1 (D j t D ν+α y (tu)) ≤ C(A 1 + A 2 + A 3 ), A 1 = ℓ+|β|≤p ε -p+ℓ+|β| N ε 1 (D ℓ t D β+α y Lu), A 2 ℓ+|β|≤p+1 ε -p-2+ℓ+|β| N ε 1 (D ℓ t D β+α y (tu)), A 3 = |β|+ℓ≤p |δ|+r≤2 ε -p+ℓ+|β| α 1 ≤α α 1 =0 β 1 ≤β ℓ 1 ≤ℓ α α 1 β β 1 ℓ ℓ 1 K ℓ 1 +|α 1 |+|β 1 | (ℓ 1 + |α 1 | + |β 1 |)! ε -(ℓ 1 +|α 1 |+|β 1 |) 1 N ε 1 (D r+ℓ-ℓ 1 t D δ+β-β 1 +α-α 1 y (tu)).
Proof. We apply Lemma 2.8 to D α y u. and we have to prove that N ε 1 (D

|D α a| ≤ K |α| α!ε -|α| 1
(which follows from the analyticity of the coefficient a) and (2.18). All the other terms are estimated by the same way. Now we use the fact that our C ∞ solution u is such that Lu is analytic near the point (0, y 0 ). Corollary 2.10. For any integer p ≥ p 0 one can find M > 0 such that for every α ∈ N n , every integer ℓ ≤ p + 2 and every ε ∈]0, a ℓ+|α| [ we have

(2.21) N (ℓ+|α|)ε (D ℓ t D α y (tu)) ≤ M ℓ+|α|+1 ε -(ℓ+|α|) .
Proof. We use an induction on j = ℓ+|α|. Since u is C ∞ and ε < 1 we may assume that (2.21) is true for j ≤ p + 1. Assume it is true up to the order j and let (ℓ

′ , α ′ ) ∈ N × N n be such that ℓ ′ ≤ p+2, ℓ ′ +|α ′ | = j+1. We write α ′ = α+ν with |ν| = p+2-ℓ ′ , then |α| = j-p-1. Applying Lemma 2.21 with 0 < ε < a j+1 , ε 1 = jε we obtain N (j+1)ε (D ℓ ′ t D α+ν y (tu)) ≤ C 3 i=1 A i .
We have A 1 = ℓ+|β|≤p ε -p+ℓ+|β| N jε (D ℓ t D β+α y Lu). Using the analyticity of Lu and the fact that |α| = jp -1 and jε < a we obtain

A 1 ≤ ε -(j+1) ℓ+β≤p K j+ℓ+|β|-p 1 (j + ℓ + |β| -p -1)! ε j+ℓ+|β|-p+1 .
Since k! ≤ k k we have (j + ℓ + |β|p -1)! ε j+ℓ+|β|-p+1 ≤ (jε) j+ℓ+|β|-p+1 ≤ a j+ℓ+|β|-p+1 . It follows that there exists a constant M 2 such that (2.22)

A 1 ≤ K j+2 2 ε -(j+1) . To estimate A 2 we use the induction. We have ε 1 = jε. Moreover in the sum we have ℓ + |β| ≤ p + 1. Therefore ℓ + |β| + |α| ≤ p + 1 + jp -1 = j. By the induction A 2 ≤ ℓ+|β|≤p+1 ε -p-2+ℓ+|β| M ℓ+|β|+|α|+1 ε -(ℓ+|β|+|α|) . Since |α| + p + 2 = j + 1 we can write (2.23)

A 2 ≤ M j+1 ε -(j+1) ℓ+|β|≤p+1 1 ≤ C M M j+2 ε -(j+1) ,
where C depends only on p and n. Therefore ε N ≤ ε -(j+1) .

We will have a constant of the form K This constant is therefore bounded by M j+2 K 3

K 3 M ℓ 1 +|α 1 |+|β 1 | ≤ M j+2 K 3 K 3 M |α 1 | if M ≥ K 3 .
Since ℓ + |β| ≤ p + 2 the sums in ℓ 1 and β 1 are bounded by a fixed constant depending only on p and the dimension n. Therefore we are left with the sum in α 1 . Now since j = |α| + ℓ -1 we have

α α 1 |α 1 |!j -|α 1 | ≤ 1.

Consider A 3 .

 3 Since r + |δ| + ℓ + |β| -|β 1 | + |α| -|α 1 | ≤ 2 + p + |α| -|α 1 | ≤ p + 1 + |α| ≤ j (since α 1 = 0) we can use the induction. Recall |α| = jp -1 and ε 1 = jε. We obtain a term of the form ε N withN = -p + ℓ + |β| -(r + |δ| + ℓℓ 1 + |β| -|β 1 | + |α| -|α 1 |) -(ℓ 1 + |α 1 | + |β 1 |) = = -p -(r + |δ) -|α| = -(j + 1) + 2 -(r + |δ) ≥ -(j + 1).

ℓ 1 +|α 1 |+|β 1 |+1 3 M

 3 C withC = r + |δ| + ℓ + |β| + jp -1ℓ 1 -|α 1 | -|β 1 | + 1, = (r + |δ| -2) + (ℓ + |β|p) + j + 2 -(ℓ 1 + |α 1 | + |β 1 |) ≤ j + 2 -(ℓ 1 + |α 1 | + |β 1 |).

  to ψu. We write L(ψu) = ψLu + [L, ψ]u. There is only one term in the commutator which does not contain tu. With the notations in (2.3) it can be written [P 1 , ψ]u and has to be estimated in the H p norm. Essentially one has to estimate terms of the form A =: (D a Dψ)D b u L 2 (where D is a derivation and |a| + |b| ≤ p) by the right hand side of the inequality in the lemma. According to (2.20) and (2.18) we can write

  Now P 1 is a finite sum of terms of the form aD t + bD y j .

	ℓ t D β y [L, D α y ]u) can be bounded by the term A 3 appearing in the statement. According to (2.3) we have [L, D α y ]u = [P 2 , D α y ]tu + [P 1 , D α y ]u. We have
	D ℓ t D β y [D α y , aD t ]u = D ℓ t D β y [D α y , a]D t u. Then the desired estimate follows from the Leibniz
	formula, (2.20), the estimate
	sup
	ωε 1

Summing up we have proved that (2.24)

if M ≥ 2K 3 . Taking M still larger, compared to the fixed constants K 1 , K 2 , K 3 , C p,n , C ′ p,n , we deduce from (2.22),(2.23),(2.24) that (2.21) is satisfied with the same constant M when ℓ + |α| = j + 1.

Corollary 2.11. For any integer p ≥ p 0 one can find H > 0 such that for every α ∈ N n , every integer ℓ ≤ p + 2 we have

ℓ+|α| and we conclude using the Stirling formula.

The purpose now is to prove an estimate of type (2.25) for all derivatives of tu. The operator P 2 being elliptic we can assume that the coefficient of D 2

t is equal to one. Therefore one can write

Proposition 2.12. One can find M 1 > 0, M 2 > 0 such that for all (α, ℓ) ∈ N n × N we have

Proof. The proof is quite technical. Here we only give the main ideas; details can be found in [START_REF] Bolley-C. Camus-B | Étude de l'analyticité et de la régularité Gevrey pour une classe de problèmes aux limites elliptiques et dégénérés[END_REF]. By (2.25) the estimate (2.27) is true for ℓ ≤ p + 2 and every α ∈ N n . Assume it is true for every α ∈ N n and for ℓ ≥ p + 2 and let us prove it for (α, ℓ + 1) were p is very large. We apply the operator D ℓ-1 t D α y to both members of (2.26). Very roughly speaking N a 2 (D ℓ+1 t D α y (tu)) will be bounded by terms (besides derivatives of the coefficients which are under control) of the following type.