Christel Métivier
email: christel.metivier@ujf-grenoble.fr

Chérif Nouar
email: cherif.nouar@ensem.inpl-nancy.fr

Stability of a Rayleigh-Bénard poiseuille flow for yield stress fluids -Comparison between Bingham and regularized models

Keywords: Fluid mechanics, linear stability, yield stress fluids

Stability of a Rayleigh-Bénard poiseuille flow for yield stress fluids -Comparison between Bingham and regularized models

Introduction

Different models are used in the literature to describe a yield stress fluid. The simplest one is the Bingham model. It is commonly used to describe the rheological behaviour of a large range of fluids as muds of drilling for instance. It assumes that the material moves as a rigid solid prior to yielding and behaves as a viscous fluid afterwards. By definintion, the sol-gel transition for a Bingham fluid is not continuous in terms of material behaviour. The "gel-like" region, called plug or unyielded zone, and the "liquid-like" region are separated by a distinct yield surface. Furthermore, except few simple configurations [START_REF] Bird | The rheology and flow of viscosplastic material[END_REF], the determination of the yield surface location is the major difficulty in the numerical resolution of the Bingham fluid flow. To avoid these limitations, several authors use regularized models. These models consider a viscous behaviour in the whole flow domain and replace the unyielded zone by an extremely viscous fluid past a transition in the shear rate. The aim of our study is to compare the different models and show the relevance of using either regularized models or Bingham model in a particular situation: stability problems. In this paper, we investigate the Rayleigh-Bénard Poiseuille (RBP) flow for yield stress fluids. This configuration has already been studied for a Bingham fluid in [START_REF] Métivier | Linear stability involving the Bingham model when the yield stress approaches zero[END_REF] and [START_REF] Métivier | On linear stability of Rayleigh-Bénard Poiseuille flow of viscoplastic fluids[END_REF] performing linear stability analyses.

On the other hand, the usage of regularized models in stability analysis has been studied recently by [START_REF] Frigaard | On the usage of viscosity regularisation method for visco-plastic fluid flow computation[END_REF] and [START_REF] Caton | Linear stability of circular Couette flow of inelastic viscoplastic fluids[END_REF]. In [START_REF] Frigaard | On the usage of viscosity regularisation method for visco-plastic fluid flow computation[END_REF], the authors treat a linearly stable flow: the plane Poiseuille flow of a Bingham fluid. The authors show that the regularized models can exhibit spurious behaviour and can give rise to unstable eigenmodes. The eigenvalues are called spurious in the sense that they depend on the small parameter introduced in the regularized model. These values can be detected by varying the number of nodes in calculations. In [START_REF] Caton | Linear stability of circular Couette flow of inelastic viscoplastic fluids[END_REF], the linear stability of the circular Couette flow of viscoplastic fluid leads to critical conditions for both the Bingham and a regularized models. The author indicates that the regularization has practically no influence on critical conditions. Actually, it is not surprising that in the case of plane Poiseuille flow, the regularized model leads to an instability at finite Reynolds number. Indeed, replacing a rigid plug zone by a highly viscous zone (whatever, the large value of the viscosity, i.e. the small is the regularization parameter), leads to a problem fundamentally different from that of a true plane Bingham Poiseuille flow. However, it is not clear in Frigaard and Nouar [START_REF] Frigaard | On the usage of viscosity regularisation method for visco-plastic fluid flow computation[END_REF] how the critical conditions depend on the regularization model and on the regularization parameter. The analysis of the viscosity profiles combined with the work of Govindarajan et al. [START_REF] Govindarajan | Stabilization of hydrodynamic flows by small viscosity variations[END_REF] where it is shown that the critical conditions depend mainly on the viscosity stratification in the critical layer (region where the energy of fluctuations is produced by interactions with the mean flow), may lead to the conclusion that the critical conditions are mainly independent on the regularization parameter, when it is sufficiently small.

In the case of the Taylor Couette problem, where the instability is driven by the centrifugal force, it can be understood that here, replacing a rigid solid (if it exists) by a hyghly viscous zone could be equivalent. Here, we want to clarify this point by considering an other situation. In this respect, we propose to consider the linear stability of the RBP flow involving a yield stress fluid whether a Bingham or regularized models are used. The framework of this study concerns dominant buoyancy-driven source of instability compared to the shear-driven one, i.e. we consider weak Reynolds values. In outline, our paper proceeds as follows. In Section 2, the basic state of the RBP flow is given for the Bingham model and the regularized models. A comparison between the different models is performed. The linear stability analysis is developed in § 3. The numerical results are presented in Section 4 as well as a comparison between the viscoplastic models.

Fully developed Rayleigh-Bénard Poiseuille flow

We consider the plane Poiseuille flow of a yield stress fluid between two horizontal walls separeted by a distance L. The system of coordinates as well as the shape of the axial velocity component are represented on Fig. [START_REF] Bird | The rheology and flow of viscosplastic material[END_REF]. As classical, we assume that the lower wall is heated at temperature T 0 + δT /2 while the upper wall is at T 0 -δT /2. Using the Boussinesq approximation, the problem is governed by the following equations:

∇ • U = 0 , (1)
Re U t + Re 2 P r (U • ∇) U + ∇P -RaT e y -Re P r∇ • τ = 0 , (2) T t + P r Re (U • ∇) T -∇ 2 T = 0 . (3)
They are rendered in dimensionless form using the centerline velocity U 0 as velocity scale, the width L of the plane channel as lengthscale, then (μ 0 U 0 )/L as stress scale, the thermal diffusion time L 2 /a between the two walls as time scale, where a is the thermal diffusivity, and (μ 0 a)/L 2 as the pressure scale.

The modified temperature is used T = T -T 0 δT . The Reynolds Re, the Prandtl P r, and the Rayleigh Ra numbers are defined by: Re = ρ U 0 L μ 0 , P r = μ 0 aρ and Ra = gρ β δT L 3 μ 0 a . U is the velocity field, p the pressure field, τ the deviatoric stress tensor, T the temperature, ρ the density, β the thermal expansion and g the gravitational acceleration.

The velocity vector U is of the form U = U e x + V e y + W e z , where U, V, W are the velocity components and e x , e y , e z are unit vectors in the streamwise x, transversal y and spanwise z directions respectively. We write the no-slip and imposed temperature conditions at the walls as follows:

y U b (y) x y 0 /2 -y 0 /2 1/2 -1/2 ÔÐÙ ÞÓÒ ÔÐÙ ÞÓÒ
U (±1/2) = 0 and T (1/2) = -1/2, T (-1/2) = 1/2.

Bingham model

Employing the von Mises yield criterion, the Bingham model, in dimensionless form can be written [START_REF] Bird | The rheology and flow of viscosplastic material[END_REF]:

τ = μ γ if τ > B , (4) γ
= 0 if τ ≤ B . (5)
The parameter μ represents the effective viscosity and is defined by :

μ = 1 + B γ
, where B = τ 0 L/ (μ 0 U 0 ), is the Bingham number. The second invariant of the rate of strain tensor γ and that of the deviatoric stress tensor are respectively γ and τ .

Equations ( 1)-( 3) are satisfied in the whole flow domain. However, in the region where the yield stress is not exceeded, the rate of strain tensor is identically zero. This region moves then as a rigid solid and is called plug zone. The interface separating the "liquid like" and the "gel like" domains is a yield surface and is defined by the criterion τ = B, then γ = 0.

For a one dimensional shear flow and fully developed thermal field, we have V = W = 0 and U B = U B (y):

U B (y) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 ; |y| ≤ 1 2 y 0 , 1 - 2 |y| -y 0 1 -y 0 2 ; 1 2 y 0 <| y |< 1 2 , (6)
T b = T 0 δT -2y and P b (x, y) = P 0 -Ra y 2 -8 Re P r (1y 0 ) 2 x, with ± (1/2) y 0 , the position of the yield surfaces. The velocity of the basic flow depends only on B. The relation between y 0 and B is given by: (2y 0 -1) 2 B -4y 0 = 0.

regularized models

In this paper, we consider three models which are commonly used in the literature: Simple model with a regularized parameter, for example used by Beris et al. [START_REF] Beris | Creeping motion of a sphere through a Bingham plastic[END_REF]:

τ = μ s γs = 1 + B 1 γs + ε γs = (1 + B F s ( γs , ε)) γs . (7)
Papanastasiou model [START_REF] Papanastasiou | Flows of materials with yield[END_REF]:

τ = μ p γp = 1 + B 1 -e -γp ε γp γp = (1 + B F p ( γp , ε)) γp . (8)
Bercovier model [START_REF] Bercovier | A finite element method for incompressible non-Newtonian flows[END_REF] :

τ = μ be γ be = 1 + B 1 γ be 2 + ε 2 γ be = (1 + B F be ( γ be , ε)) γ be , (9)
μ s , μ p and μ be represent the effective viscosity for each model respectively.

Here the Bingham model is recovered when the parameter ε tends to zero. The velocity profile of the fully developed flow is completely defined by Eqs. ( 2) and ( 7), [START_REF] Papanastasiou | Flows of materials with yield[END_REF] or [START_REF] Bercovier | A finite element method for incompressible non-Newtonian flows[END_REF]. For a unidirectional shear flow, U reg (y) satisfies:

2B y 0 y = (1 + B F ) DU reg , (10)
with D = d dy . The velocity profile is determined numerically using finite difference scheme.

Comparison between the different models

This section compares the main characteristics of the basic flow whether the Bingham or the regularized models are used. For the axial velocity profiles, the numerical results indicate that the convergence is verified when ε → 0. However the effective viscosity, firstly is not defined for the Bingham model in the plug zone, and secondly for the regularized models, it behaves as B/ε near the axis. Actually, the analysis of effective viscosity profile for the regularized models (Fig. [START_REF] Métivier | Linear stability involving the Bingham model when the yield stress approaches zero[END_REF]) suggests that three zones can be distinguished: the zone 1 around the axis, where τ << B and for which the effective viscosity is very high. The zone 3 close to the wall, where τ >> B and for which the effective viscosity is lower than in the previous zone. Finally, the intermediate zone, zone 2, where τ ∼ B and for which the effective viscosity varies drastically.

A characteristic scale δ y of the intermediate zone thickness is estimated using an asymptotic method. This method is detailed in Appendix A. It consists on determining an approximation of γ in the regions 1 and 3. Then, approximations can be matched in the zone around y 0 /2. The results are

τ << B γs ∼ ε τ (B -τ ) γp ∼ -ε ln(1 - τ B ) γbe ∼ (ε 2 τ ) Zone 2: τ ∼ B γs ∼ √ εB γp ∼ δ p ∼ -ε ln(1 - γp B ) γbe ∼ (ε 2 B) Zone thickness: δ y δ ys ∼ 2 y 0 B √ εB δ yp ∼ 2 y 0 B δ p δ y be ∼ 2 y 0 B (ε 2 B) 1 3 Zone 3: τ >> B γs ∼ τ -B γp ∼ τ -B γ be ∼ τ -B
summarized in Table 1. It is noticeable that the transition zone thickness depends on the model used, moreover for all the models, δ y increases with ε and B. Moreover, it is noticeable that for ε and B given, we have: δ yp ≤ δ y be ≤ δ ys .

Linear stability analysis

The linear stability analysis is developed for the Bingham model in Ref. [START_REF] Métivier | Linear stability involving the Bingham model when the yield stress approaches zero[END_REF]. For bidimensional perturbations, we introduce the stream function perturbation ψ, defined by u = ∂ y ψ and v = -∂ x ψ, where (u, v) corresponds to the velocity perturbation components. Furthermore, the perturbation solution is seeked in normal mode form as follows:

(ψ, T ) = (f (y), θ(y)) e ıα(x-ct) [START_REF] Ern | Stability analysis of a shear flow with strongly stratified viscosity[END_REF] where the parameters α and c are respectively a real axial wave number and a complex wave speed. Then, the system of linearised perturbation equations writes:

In the two yielded zones, 1 2 y 0 <| y |< 1 2 :

L 1 f + P r Ra θ = c L 2 f, (12)
L 3 θ + f = c θ, (13)
where the linear differential operators L 1 , L 2 and L 3 are defined as:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ L 1 ≡ P r Re U b (D 2 -α 2 ) -D 2 U b + i P r α (D 2 -α 2 ) 2 -4 i α P r B D [D [F D]] , L 2 ≡ D 2 -α 2 , L 3 ≡ P r Re U -iα + i α D 2 , (14)
In the unyielded zone, y ∈ -y 0 2 , y 0 2 we have:

f = 0 and L 3 θ = c θ. ( 15
)
The boundary conditions, in terms of stream function, are:

f (±1/2) = f y (±1/2) = f (±y 0 /2) = f y (±y 0 /2) = 0 and θ (±1/2) = 0.
Concerning the regularized models, the linear stability analysis is similar to the one developed in the yielded zone for the previous case. However two fundamental differences remain: (i) the Bingham model leads to zero velocity perturbation at the interface and in the plug zone while the regularized models do not lead to unyielded regions, (ii) compared with the Bingham model, an additional viscous term appears in the Orr-Sommerfeld equation. Finally we get :

(L 1 + L * 1 ) f + P r Ra θ = c L 2 f , (16)
L 3 θ + f = c θ , (17)
where:

L * 1 ≡ -i P r B 1 α D 2 + α 2 G D 2 + α 2 and G = F + γ dF d γ .
The boundary conditions are written at walls: f (±1/2) = f y (±1/2) = 0 and θ (±1/2) = 0.

In comparison with the Bingham model, an additional operator L * 1 appears in Orr-Sommerfeld equation. Actually, this latter induces numerical diverging results due to the function D 2 G, since abrupt variations are observed at y = 0 and y = ±y 0 /2 (Fig. 3). For convenience, logarithmic coordinates are adopted, therefore the points for which D 2 G ≤ 0 are not represented.

Results and discussion

The discretization of the differential equations set is performed, for the Bingham and the regularized models, by means of a second-order centered These figures highlight the stabilizing effect of the Bingham number. According to [START_REF] Métivier | On linear stability of Rayleigh-Bénard Poiseuille flow of viscoplastic fluids[END_REF], this is a consequence of the the unyielded region increasing, in which the velocity perturbation vanishes, as well as the increase in the effective viscosity in the flow domain. When y 0 → 0, there is an outstanding discrepancy, in terms of critical conditions, with those obtained for Newtonian fluids. Métivier et al. [START_REF] Métivier | Linear stability involving the Bingham model when the yield stress approaches zero[END_REF] deal with this particular case. It is shown that the discrepancy is due to the fact that, in the frame of the linear stability analysis, the plug zone remains intact even when y 0 → 0.

When regularized models are used, the abrupt variation of D 2 G near the interface generates difficulties in the eigenvalues computation. In this case and similarly to [START_REF] Frigaard | On the usage of viscosity regularisation method for visco-plastic fluid flow computation[END_REF], spurious eigenvalues, are obtained as well as incoherent results from physical point of view. In fact, these difficulties increase when the thickness δ y of the transition zone decreases. In order to improve con-verging computations, the operator L * 1 is artificially cancelled and the values of ε are restricted to ε ≥ 0.01. The resulting equation can be seen as a regularization of the Orr-Sommerfeld equation.

The regularization of the Orr-Sommerfeld operator leads to spectra shown in Fig. [START_REF] Caton | Linear stability of circular Couette flow of inelastic viscoplastic fluids[END_REF]. These spectra are obtained at criticality. Preliminary numerical tests have been performed on the resulting spectra by varying N from 201 to 401. Results show that spectra are not modified with N , except some eigenvalues. These eigenvalues are such that c i < 0 and min |c i | = O(10 2 ). In this sense, these spurious values can't be confused with the least stable eigenvalues which are negative and close to 0. According to Fig. [START_REF] Caton | Linear stability of circular Couette flow of inelastic viscoplastic fluids[END_REF], one observes that spectra are slightly different comparing each model. Actually, the choice of the model modifies fundamentally the nature of the fluid via its rheological properties comparing to another model. Concerning the Bercovier and Papanastasiou models, one observes that for ε = 0.01, the tendency of spectra are close to the Bingham one. Few modifications are observed using the simple model by varying ε between 0.1 and 0.01.

Concerning critical conditions, Fig. [START_REF] Govindarajan | Stabilization of hydrodynamic flows by small viscosity variations[END_REF] presents the Ra c values obtained for each model as a function of y 0 , for the case P r = 10, Re = 1, ε = 0.1 and ε = 0.01. As expected, one observes that considering y 0 → 0, regularized models allow to obtain the critical Rayleigh number corresponding to the Newtonian case, i.e. Ra cN ew = 1750 [START_REF] Luijkx | Linear stability of mixed convection flows in horizontal rectangular channels of finite transversal extension heated from below[END_REF]. In addition, one can observe that the increase in y 0 has a stabilizing effect on the flow field for each model. Indeed, increasing y 0 increases the mean effective viscosity as well as the thickness of zone 1 in which the viscosity is very large. It is also noticeable that the critical conditions obtained with the regularized models can lead to strongly different values from those obtained with the Bingham model. The model leading to the largest difference is the simple model while the Papanastasiou model leads to the smallest difference.

When y 0 is fixed, decreasing ε makes the thickness δ y smaller and the effective viscosity larger in the zone 1. The increasing effective viscosity tends to stabilize the flow which is in agreement with [START_REF] Métivier | On linear stability of Rayleigh-Bénard Poiseuille flow of viscoplastic fluids[END_REF]. Concerning the decrease of δ y , it seems that it tends to destabilize the flow. The same tendency was indicated in [START_REF] Ern | Stability analysis of a shear flow with strongly stratified viscosity[END_REF]. Figure ( 6) also shows the effect of ε on Ra c for the three regularized models. One observes that the variation in ε values do not lead to converging critical conditions since ε modifies the rheological properties of the fluid, in particular the mean viscosity as well as the viscosity stratification in the flow. Actually, the decrease in ε stabilizes the flow for the simple and Papanastasiou models, while it destabilizes the flow for the Bercovier model.

The representation of the perturbation modes are given for each regularized model in Fig. [START_REF] Beris | Creeping motion of a sphere through a Bingham plastic[END_REF]. One can notice that the stream function has a non zero flat profile in the very high viscosity region (zone 1) considering the simple model, figure 7(a). Furthermore, using the Bercovier and Papanastasiou models, the stream function tends to vanish in this region similarly to the critical modes obtained with the Bingham model as given in Fig. [START_REF] Papanastasiou | Flows of materials with yield[END_REF].

As a conclusion, the usage of regularized models in instability problems leads to different results according to the chosen model. Indeed, the variation in ε value can modify the rheological properties of the material. In order to obtain physical results, the Orr-Sommerfeld equation has to be regularized, neglecting the ε terms. The comparison between regularized models indicates (i) that these models permit to recover the Newtonian critical conditions for weak values of y 0 , (ii) that the Bercovier and Papanastasiou models can tend to similar behaviour as the Bingham case, for y 0 > O(10 -1 ) and ε = 0.01. In this last case, same tendencies in spectra and perturbation modes are observed.

Finally, the RBP flow is unstable for both Newtonian and (exact) Bingham cases such as the Taylor-Couette flow. It seems that in such a case, the regularization can tend to the exact Bingham results, when the values of ε are small enough. -The effective viscosity is not defined in the plug zone for the Bingham model.

-Regularized models lead to the Newtonian critical conditions for weak values of yield stress.

-Convergence of the critical conditions between the simple and the Bingham models is not obtained.

-The Bercovier and Papanastasiou models can tend to the exact Bingham results.

Figure 1 :

 1 Figure 1: Fully developed Poiseuille flow for a yield stress fluid

Figure 2 :

 2 Figure 2: Effective viscosity as a function of y for the different regularized models, with y 0 = 0.2, ε = 0.1 and ε = 0.01 (--and ----: simple model, and : Bercovier model , ♦ and : Papanastasiou model). The vertical long-dashed lines which bound the transition zone are drawn only to illustrate the different zones and don't characterize the effective thickness δ y .

2 GFigure 3 :Figure 4 :

 234 Figure 3: Second derivative of function G for y 0 /2 = 0.2 (B = 5.55), ε = 0.01 (• : simple model, : Bercovier model , : Papanastasiou model)

Figure 5 :Figure 6 :

 56 Figure 5: Spectra obtained at criticality for the regularized models and y 0 /2 = 0.14 Re = 1, P r = 10 (♦: Papanastasiou model ; O: Bercovier model and ∇: Simple model, with ε = 0.1 (white symbols) and ε = 0.01 (black symbols))

Figure 7 :

 7 Figure 7: Critical modes of the perturbation: temperature (left) and stream function (right) ( ------: Real part) ; ( ---: Imaginart part) ; y 0 = 0.14, Re = 1 and P r = 10.

Table 1 :

 1 Asymptotic behaviour of the rate of strain and the thickness of zone 2

	Simple model	Papanastasiou model	Bercovier model
	Zone 1:

Preprint submitted to Elsevier December 3, 2010

Appendix A. Asymptotic analysis for the different regularized models

Appendix A.1. Simple model Considering first the simple regularized model. This model is described by Eq. [START_REF] Beris | Creeping motion of a sphere through a Bingham plastic[END_REF]. Developing it, we get a polynomial equation in power of γs :

for which the solution is :

In the zone 1, where τ << B, Eq. (A.2) could be expanded as follow: Then, we get:

In the zone 3, where τ >> B, Eq. (A.2) could be also written as follow:

This equation gives asymptotically:

Finally we obtain:

In the transition zone, where τ ∼ B, we write:

In the zone 1, where τ << B, we apply the principle of least degeneration :

The zone where τ >> B concerns the case where γbe >> ε, then we write:

Finally, we obtain: γbe ∼ τ -B.

(A.25)

The method is similar as previous and permits to lead to:

Finally, connecting the solutions, we get:


Thomas Simon

Positive stable densities and the bell-shape

Keywords: 2000 Mathematics Subject Classification, 60E07, 62E15 Bell-shape -Exponential mixture -Exponential sum -Positive stable density -Total positivity, 1

 in the positive case.

Introduction

Consider a smooth probability density defined on some open interval I ⊂ R and such that all its derivatives vanish at both ends of I. Rolle's theorem shows that the n-th derivative vanishes at least n times on I. The density is said to be bell-shaped if the n-th derivative vanishes exactly n times on I for every n ≥ 1. For n = 1 this amounts to strict unimodality. For n = 2 this means, as for the familiar bell curve, that there is one inflection point on each side of the mode and that the second derivative is successively positive, negative, and positive. Rolle's theorem also entails that the zeroes of the successive derivatives of a bell-shaped density strictly interlace, and hence that each derivative has an alternating sign sequence, starting positive. The sign sequence of the third derivative can be observed on a curve in comparing the aberrancy and the slope, as shown by Transon's formula [START_REF] Schot | Geometrical properties of the penosculating conics of a plane curve[END_REF], but contrary to the second derivative it requires an educated eye to guess what this sequence must be at first glance. The fourth derivative of a density has also a kinematic interpretation in terms of penosculating conics -see [START_REF] Schot | Geometrical properties of the penosculating conics of a plane curve[END_REF] and the references therein for a complete account, which however cannot be understood through the sole sequence of its signs.

For a given density whose derivatives vanish at both ends of its definition interval, the usual way to show the bell-shape is to factorize its n-th derivative by a positive function and a polynomial function of degree n. This entails that this n-th derivative vanishes at most, and hence exactly, n times. Basic examples -whose details can be checked by the reader and serve as an exercise in calculus for sophomores -are the Gaussian, Gumbel and centered Student densities on R, and the inverse Gamma densities on (0, +∞). When the density is not explicit, such factorizations are not appropriate anymore. Another criterion for the bell-shape is the ETP character of the associated additive convolution kernel -see Chapter 6.11.C in [START_REF] Karlin | Total positivity[END_REF] -but the latter is rather difficult to check for non-explicit densities, and also quite stringent since it is e.g. fulfilled neither by the Student nor by the inverse Gamma densities. In this paper we are interested in the positive α-stable random variables X α and their densities f α , 0 < α < 1. The latter are characterized by their Laplace transform and we choose the normalisation

E[e -λXα ] = ∞ 0 e -λx f α (x) dx = e -λ α , λ ≥ 0.
It is known -see e.g. the expansions (14.31) and (14.35) in [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF] -that f α is real-analytic on (0, +∞) and that all its derivatives vanish at zero and at infinity. By Proposition 7.1.3 in [START_REF] Karlin | Total positivity[END_REF] and (14.31) in [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF], the kernel f α (xy) is not TP 2 and hence not ETP. The function f 1/2 is explicit thanks to the identity in law

X 1/2 d = 1 4Γ 1/2
where here and throughout Γ a denotes the standard Gamma random variable of parameter a > 0, and is hence bell-shaped. The functions f 1/3 and f 2/3 can be written down in terms of a modified Bessel resp. a confluent hypergeometric function -see e.g. (2.8.31) and (2.8.33) in [START_REF] Zolotarev | One-dimensional stable distributions[END_REF] -but it does not seem to the author that the associated second order ODE's can provide any substantial information on their bell-shape.

The strict unimodality of f α was first proved by Ibragimov and Chernin -see the proof of the Theorem in [START_REF] Ibragimov | On the unimodality of stable laws[END_REF] -and then extended to all real stable densities by Sato and Yamazato -see Theorem 1.4 in [START_REF] Sato | On distribution functions of class L. Z. Wahrsch[END_REF]. Since the latter are real-analytic, this actually follows from Yamazato's original theorem [START_REF] Yamazato | Unimodality of infinitely divisible distribution functions of class[END_REF] on self-decomposable distributions. See also [START_REF] Simon | A multiplicative short proof for the unimodality of stable densities[END_REF] for a short proof. The graphic simulations made by Holt and Crow [START_REF] Holt | Tables and graphs of the stable probability density function[END_REF], partly reproduced in [START_REF] Zolotarev | One-dimensional stable distributions[END_REF], display bell-shaped curves for all real stable densities, at least at the level of the second derivative. A consequence of Theorem 5.1.(ii) in [START_REF] Sato | On distribution functions of class L. Z. Wahrsch[END_REF] is that f α has only one inflection point on the left side of its mode, and more generally that each n-th derivative of f α vanishes only once before the first zero of the (n -1)-th derivative. The bell-shape property was claimed in [START_REF] Gawronski | On the bell-shape of stable densities[END_REF] for all stable densities but a serious mistake on the TP character of the kernel f α (xy -1 ) -see Remark (f) in [START_REF] Simon | A multiplicative short proof for the unimodality of stable densities[END_REF] for an explanation -invalidates this result, also at the unimodal level, except for a particular class of stable densities which will be discussed in Section 3.3 and does not include the positive case. Apart from [START_REF] Holt | Tables and graphs of the stable probability density function[END_REF][START_REF] Simon | A multiplicative short proof for the unimodality of stable densities[END_REF][START_REF] Gawronski | On the bell-shape of stable densities[END_REF] the bell-shape of stable densities seems to have escaped investigation, even in the most visual case n = 2. In this note, we show the full property in the positive case.

Theorem. The densities f α are bell-shaped.

Our main argument comes from Schoenberg's variation-diminishing property and the total positivity of certain infinitely convoluted exponential kernels. The little known fact that X α admits infinite exponential sums as additive factors had been noticed in [START_REF] Yamazato | On strongly unimodal infinitely divisible distributions[END_REF]. It is used here in conjunction with a precise analysis of the other factor, which contains a certain exponential mixture with no atom at zero and can be chosen in order to have a weak bell-shape property of arbitrary order.

In the case when 1/α is an integer, another proof of the bell-shape is obtained quickly by a multiplicative factorization with inverse Gamma laws -see Section 3.1 below, but the argument does not extend to the other cases. With this method, the bell-shape of the density of log X 1/n on R can also be established. In Section 3.2, we raise two natural conjectures on the bell-shape of positive self-decomposable densities.

Proof of the theorem

We first establish an additive factorization of X α via standard exponential variables. Recall that a positive random variable X ∼ µ is said to be an exponential mixture (ME) if its law has the form µ(dx) = cδ 0 (dx) + f (x)dx with c ∈ [0, 1] and f a completely monotone function on (0, +∞). When c = 0 we will use the notation X ∈ ME * , and in this case Bernstein's theorem shows that the density f writes

(1) f (x) = ∞ 0 θe -θx µ(dθ)
over (0, +∞), where µ is some probability measure on (0, +∞). In other terms

X d = Exp (1) × X -1 µ
where here and throughout Exp (λ) stands for the exponential law with parameter λ, and X µ ∼ µ.

The following lemma is mostly due to Yamazato -see [START_REF] Yamazato | On strongly unimodal infinitely divisible distributions[END_REF] pp. 601-602 -but we rephrase it and give a proof for the reader's convenience.

Lemma (Yamazato). One has the independent factorization

X α d = Y α + n≥1 Y α,n , (2)
where Y α,n ∼ Exp((nπ/ sin(πα)) 1/α ) for every n ≥ 1 and Y α ∈ ME * .

Proof. We first write

λ α = α Γ(1 -α) ∞ 0 (1 -e -λx ) dx x α+1 = ∞ 0 (1 -e -λx ) ∞ 0 (c α u) α e -xu du dx
with the notation c α = sin(πα)/π. This yields the further decomposition

λ α = ∞ 0 (1 -e -λx )l α (x)dx + ∞ 0 (1 -e -λx ) ∞ 0 [(c α u) α ] e -xu du dx
where [.] stands for the integer part and

l α (x) = ∞ 0 ((c α u) α -[(c α u) α ])e -xu du
is a completely monotone function. By Theorem 51.10 in [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF] there exists a random variable Y α belonging to the Bondesson class such that

-log E[e -λYα ] = ∞ 0 (1 -e -λx )l α (x)dx, λ ≥ 0. Because 0 ≤ (c α u) α -[(c α u) α ] ≤ 1 for all u ≥ 0 and u -1 ((c α u) α -[(c α u) α ]
) is integrable at zero, Steutel's criterion -see Theorem 51.12 in [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF] -shows that Y α is actually ME. We next prove that Y α has no atom at zero, which will entail Y α ∈ ME * . A change of variable and an integration by parts yield

l α (x) = x c α ∞ 0 e -xc -1 α u u 0 (t α -[t α ]) dt du
for every x > 0, and it is a bit tedious but elementary to see that

u 0 (t α -[t α ]) dt ∼ [u α ] 1/α 2 ∼ u 2 as u → +∞. Theorem XIII.5.4 in [2] entails (3) l α (x) ∼ c α 2x as x → 0,
so that Y α has infinite Lévy measure and, by Theorem 27.4 in [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF], no atom at zero. On the other hand, setting κ α,n = (nπ/ sin(πα)) 1/α for every n ≥ 1, one has

∞ 0 [(c α u) α ] e -xu du = n≥1 ∞ κα,n e -xu du = n≥1 e -xκα,n x • Hence, ∞ 0 (1 -e -λx ) ∞ 0 [(c α u) α ] e -xu du dx = n≥1 ∞ 0 (1 -e -λx ) e -xκα,n x dx = n≥1 (log(λ + κ α,n ) -log(κ α,n )) = -log E[exp[-λ n≥1 Y α,n ]],
where the first equality comes from Fubini's theorem and the second from Frullani's. Putting everything together completes the proof.

Remarks 1. (a) Because α ∈ (0, 1), the sum on the right-hand side of (2) has finite expectation.

This shows that X α and Y α have the same L p -integrability index which is α, excluded. The latter can also be seen from Theorem 25.3 in [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF] and the fact that

l α (x) ∼ (α/Γ(1 -α))x -(α+1) as x → +∞.
(b) Since the infinite exponential sum has a PF 2 viz. log-concave density -see the proof of the theorem below for a stronger property -and since the density of f α is not PF 2 , one can interpret Y α as the factor which breaks down the PF 2 property for f α . Recall indeed that f α is log-concave until its second inflection point -see Theorem 1.3 (vii) in [START_REF] Sato | On distribution functions of class L. Z. Wahrsch[END_REF]. On the other hand Y α has a log-convex density by Hölder's inequality. It is plausible that log f α has only one inflection point, which can be readily checked for α = 1/2.

We now introduce our notations concerning the sequence of signs of a real function defined on (0, +∞). We set (n/2 -1) times on (0, +∞) if n is even and either (n -1)/2 or (n -3)/2 times on (0, +∞) if n is odd. For example, the n-th derivative of a completely monotone function is of type ± n 0 for every n ≥ 0. The density function f α is of type 0+0 and, by the strict unimodality of X α , its derivative

sign(x) =    + if x > 0 0 if x = 0 -if x < 0 and ± n = + if n is even -if n is odd for every x ∈ [-∞, +∞], n ∈ N. For {u n } = {ε 1 , . . . ,
f ′ α is of type 0+0-0. Introducing the sequences {a n } = 0± 0 0± 1 0 • • • ± n 0 and {b n } = ± 0 0± 1 0 • • • ± n 0
for every n ≥ 1, we can now state the central definition of this paper.

Definition. For every n ≥ 0, a smooth function f : (0, +∞) → R is said to be weakly bell-shaped

of order n (WBS n ) if f (i) ∼ {a i } ∀ i = 0 . . . n and (-1) n+1+i f (i) ∼ {b n+1 } ∀ i ≥ n + 1.
The bell-shape property of a function f means that f (i) ∼ {a i } ∀ i ≥ 0, so that weakly bell-shaped functions are never bell-shaped. Notice also that WBS p ∩ WBS q = ∅ if p = q. The following result provides our key-argument.

Proposition. Let X ∈ ME * and 0 < λ 1 ≤ λ 2 ≤ . . . ≤ λ n ≤ . . . be some sequence. For every n ≥ 1,

the independent sum X + Exp(λ 1 ) + • • • + Exp(λ n ) has a WBS n-1 density.
Proof. Set f 0 for the density of X and recall from (1) that it is completely monotone. In particular f 0 is real-analytic on (0, +∞). We denote by f n the density of X + Exp(λ 1 ) + • • • + Exp(λ n ) for every n ≥ 1. The latter are connected to one another by the formula (4)

f n (x) = λ n e -λnx
x 0 e λny f n-1 (y)dy, and hence solutions on (0, +∞) to the linear ODE's

(5)

f ′ n + λ n f n = λ n f n-1 .
An induction hinging on (4) and [START_REF] Holt | Tables and graphs of the stable probability density function[END_REF] show that all densities f n are real-analytic on (0, +∞), that all derivatives f (i) n vanish at infinity, and that f

(i) n (0+) = 0 for all n ≥ i + 1 ≥ 1.
In particular one has f n ∼ {a 0 } for every n ≥ 1. Using the complete monotonicity of f 0 for the initial step, another induction shows (-1) n+i f (i) n (0+) > 0 for all i ≥ n ≥ 0, possibly with infinite values. This entails f (n) n (0+) > 0 for all n ≥ 0, so that 0+ is an isolated zero of f (i) n for all n ≥ i + 1 ≥ 1 (notice that this cannot be proved directly by analyticity because 0+ is also a singular zero). The weak bell-shape property will now be established through a more elaborate induction on n.

We first show that f 1 is WBS 0 . Since f 1 ∼ {a 0 }, we know that f ′ 1 vanishes at least once on (0, +∞). Set

x 1 = inf{x > 0, f ′ 1 (x) = 0} > 0, the strict inequality coming from f ′ 1 (0+) > 0. Differentiating (5) yields f ′′ 1 (x) = λ 1 f ′ 0 (x) < 0 for every x > 0 such that f ′ 1 (x) = 0. This shows that f ′ 1 vanishes only at x 1 , whence f ′ 1 ∼ {b 1 }. Again, this entails that f ′′ 1 vanishes at least once on (0, +∞). Set x 2 = inf{x > 0, f ′′ 1 (x) = 0} > 0, the strict inequality coming from f ′′ 1 (0+) < 0. Differentiating (5) further entails f ′′′ 1 (x) = λ 1 f ′′ 0 (x) > 0
for every x > 0 such that f ′′ 1 (x) = 0. This shows that f ′′ 1 vanishes only at x 2 , whence -f ′′ 1 ∼ {b 1 }. Using the complete monotonicity of f 0 , the same argument yields (-1) i+1 f (i) 1 ∼ {b 1 } for every i ≥ 1, as required.

We next show the induction step and suppose that f n is WBS n-1 for some n ≥ 1. We have already seen that f n+1 ∼ {a 0 } and we will first prove by induction on i that f (i) n+1 ∼ {a i } for every 1 ≤ i ≤ n. We begin with f ′ n+1 for the sake of clarity. Since f n+1 ∼ {a 0 }, we know that f ′ n+1 vanishes at least once on (0, +∞). Set

x 1 = inf{x > 0, f ′ n+1 (x) = 0} > 0,
where the strict inequality comes from the fact that 0+ is an isolated zero of f ′ n+1 . Differentiating (5) as above entails

λ n f ′ n (x 1 ) = f ′′ n+1 (x 1
) ≤ 0, where the inequality comes from the initial profile 0+0 of f ′ n+1 . Suppose that f ′′ n+1 (x 1 ) = 0. Then by analyticity x 1 is an isolated zero of f ′′ n+1 , so that lim

x↑x 1 f ′′ n+1 (x) f ′ n+1 (x) = -∞.
By [START_REF] Holt | Tables and graphs of the stable probability density function[END_REF], this entails f ′ n (x) < 0 as x ↑ x 1 . But the induction hypothesis yields f ′ n ∼ {b 1 } if n = 1 and f ′ n ∼ {a 1 } if n > 1, so that one cannot have f ′ n (x 1 ) = 0, a contradiction. Hence f ′′ n+1 (x 1 ) < 0 and if there exists x 2 > x 1 such that f ′ n+1 (x 2 ) = 0, then again the induction hypothesis entails f ′ n (x 2 ) < 0 so that f ′′ n+1 (x 2 ) < 0, a contradiction. This shows that f ′ n+1 vanishes only once on (0, +∞) and, all in all, that f ′ n+1 ∼ {a 1 }. We now suppose f (i) n+1 ∼ {a i } for some i < n and show f (i+1) n+1 ∼ {a i+1 }. We know that f (i+1) n+1 vanishes at least i + 1 times on (0, +∞), and also from the above that f (i+1) n+1 has initial profile 0+0. Set {x j , j ≥ 1} for the ordered sequence of its zeroes on (0, +∞). Again, one gets from ( 5)

f (i+2) n+1 (x j ) = λ n f (i+1) n (x j ) and lim x↑x j f (i+2) n+1 (x) f (i+1) n+1 (x) = lim x↑x j f (i+1) n (x) f (i+1) n+1 (x) = -∞ (6)
for every j ≥ 1. For clarity we consider two separate cases.

(i) If i + 1 = n, the induction hypothesis entails f (i+1) n ∼ {b n } = ± 0 0 • • • ± n 0. Set I k n ↔ {± k 0} for the k-th half-closed interval corresponding to each part ± k 0 of the graph of f (i+1) n , k = 0 . . . n.
The last equality in (6) with j = 1 and the initial profile 0+0 of f (i+1) [START_REF] Ibragimov | On the unimodality of stable laws[END_REF] with j = 1, 2 and the induction hypothesis show that

n+1 entail that x 1 ∈ I k 1 n with k 1 odd. If f (i+1) n (x 1 ) < 0, then
x 2 ∈ I k 2 n with k 2 even. If f (i+1) n (x 1 ) = 0, then x 2 ∈ I k 2 n with k 2 > k 1 .
Iterating the procedure shows that there are at most, and hence exactly, n = i + 1 zeroes and that x k ∈ I k n for every k = 1 . . . n. All in all this shows that f We finally show that (-1) n+1+i f (i) n+1 ∼ {b n+1 } for every i ≥ n + 1. We only sketch the proof since the arguments are the same. Consider first the case i = n + 1. Since f

(n) n+1 ∼ {a n }, we know that f (n+1)
n+1 vanishes at least n + 1 times on (0, +∞) and set {x k , k ≥ 1} for the ordered sequence of its zeroes. Recall f (n+1) n+1 (0+) > 0 and, from the induction hypothesis, that f

(n+1) n ∼ ± 1 0 • • • ± n+1 0. Set again I k n ↔ {± k 0} for the k-th half-closed interval corresponding to each part ± k 0 of the graph of f (n+1) n , k = 1 . . . n + 1.
The same reasoning as above shows that necessarily x k ∈ I k n for all k and hence that f (n+1) n+1 vanishes exactly n + 1 times on (0, +∞), in other terms f

(n+1) n+1 ∼ {b n+1 }. It is then easy to show by induction on i that (-1) n+1+i f (i) n+1 ∼ {b n+1 } for every i ≥ n + 1, in using (-1) n+1+i f (i) n+1 (0+) > 0 and the induction hypothesis (-1) n+1+i f (i) n ∼ ± 1 0 • • • ± n+1 0.
Remarks 2. (a) The complete monotonicity of f 0 is crucial in the above argument, initializing all induction steps. It seems difficult to extend the proposition to broader classes than ME.

(b) When X itself has an exponential law and all parameters are different, the density of X + Exp(λ 1 ) + • • • + Exp(λ n ) is a linear combination of n + 1 distinct exponentials functions. Since the latter define a Chebyshev system -see [START_REF] Karlin | Total positivity[END_REF] p. 24, the property WBS n is much easier to prove.

We leave the details to the reader.

(c) The method of examining the sign of the (n+1)-th derivative at the zero of the n-th derivative is used in [START_REF] Ibragimov | On the unimodality of stable laws[END_REF] for n = 1 in order to show the strict unimodality of f α , after some explicit contour integration. The involved computations to extend this argument directly on the further derivatives of f α seem however extremely heavy, and the author does not believe that they could yield the full bell-shape. See [START_REF] Gawronski | Asymptotic forms for the derivatives of one-side stable laws[END_REF] for an application of the contour integration of [START_REF] Ibragimov | On the unimodality of stable laws[END_REF] to the asymptotic behaviour of these further derivatives.

End of the proof. Fix n ≥ 1 and set f α,n resp. g α,n for the density of

X α + n+2 k=1 Y α,k resp. ∞ k=n+3 Y α,k
with the notations of the Lemma, so that one has

f α (x) = ∞ 0 f α,n (x -y)g α,n (y)dy.
By the Proposition, the function f α,n has a C n+1 extension on R and one obtains

f (n+1) α (x) = ∞ 0 f (n+1) α,n (x -y)g α,n (y)dy = ∞ 0 f (n+1) α,n (y)g α,n (x -y)dy.
The Laplace transform of g α,n reads

∞ 0 e -λx g α,n (x) dx = ∞ k=n+3 1 1 + (sin(πα)/kπ) 1/α λ
and its reciprocal is of the Pólya-Laguerre class E * 1 defined in [START_REF] Karlin | Total positivity[END_REF] p. 336. By Theorem 7.3.2. (b) in [START_REF] Karlin | Total positivity[END_REF] p. 345, this shows that the kernel g α,n (xy) is TP ∞ on R × R. Notice that g α,n (xy) is however not STP ∞ , because of the indicator function. Using the notation (3.1) p. 20 in [START_REF] Karlin | Total positivity[END_REF] with I = (0, +∞), Theorem 3.1. (a) p. 21 in [START_REF] Karlin | Total positivity[END_REF] and the Proposition entail

S -(f (n+1) α ) ≤ S -(f (n+1) α,n ) = n + 1. Now since f (n) α
has isolated zeroes on (0, +∞) and vanishes at zero and infinity, Rolle's theorem yields

S + (f (n) α ) ≤ S -(f (n+1) α ) -1 ≤ n
where we have used the notation (3.2) p. 21 in [START_REF] Karlin | Total positivity[END_REF] with I = (0, +∞). This shows that f (n) α vanishes at most n times, and hence exactly n times, on (0, +∞). The proof is complete.

Remarks and open questions

3.1. The case when 1/α is an integer. In this situation the bell-shape follows easily from the independent factorization ( 7)

X 1/n = n -n Γ -1 1/n × • • • × Γ -1 (n-1)/n ,
which was pointed out in [START_REF] Williams | Some representations of stable random variables as products[END_REF] and also basically shown in [START_REF] Karlin | Total positivity[END_REF] pp. 121-122. We discard the explicit case n = 2 and set g n resp. h n for the density of n

-n Γ -1 1/n resp. Γ -1 2/n × . . . × Γ -1 (n-1)/n . The multiplicative convolution yields f 1/n (x) = ∞ 0 g n (xy -1 )h n (y)
dy y and one can clearly differentiate under the integral. This entails

f (i) 1/n (x) = ∞ 0 g (i) n (xy -1 )h n (y) dy y i+1 = 1 x i ∞ 0 h n (xy -1 )g (i) n (y)y i-1 dy for every x > 0, i ≥ 1. The kernel h n (xy -1
) is the composition of (n -2) kernels that are STP ∞ on (0, +∞) × (0, +∞) by Theorem 2.1 p. 18 in [START_REF] Karlin | Total positivity[END_REF], and is hence itself STP ∞ on (0, +∞) × (0, +∞) by the Binet-Cauchy formula. With the above notation for S -and S + , Theorem 3.1 (b) p. 21 in [START_REF] Karlin | Total positivity[END_REF] and the aforementioned bell-shape property of g n yield

S + (f (i) 1/n ) ≤ S -(g (i) n ) = i
for every i ≥ 1 so that f 1/n is bell-shaped, too. The above factorization [START_REF] Karlin | Total positivity[END_REF] shows also easily that log X 1/n has a bell-shaped density on R, a property which we believe to be true for all α ∈ (0, 1).

When α is rational, another multiplicative factorization of X α in terms of inverse Beta and inverse Gamma random variables was obtained in Lemma 2 of [START_REF] Simon | Multiplicative strong unimodality for positive stable laws[END_REF]. But the latter seems useless for the bell-shape of X α or log X α , because inverse or log Beta laws have neither bell-shaped densities nor TP ∞ convolution kernels.

3.2.

Positive self-decomposable densities. A positive random variable X is self-decomposable when its Laplace transform reads

E[e -λX ] = exp -γ 0 λ + ∞ 0 (1 -e -λu ) k(u) u du , λ ≥ 0
for some γ 0 ≥ 0 and k : (0, +∞) → R + non-increasing. For example, the positive stable law is self-decomposable with γ 0 = 0 and k(x) = α/Γ(1α)x α . It is known and easy to see -see Theorem 27.7 in [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF] -that X has a density f when k = 0. The unimodality of f was shown by Wolfe [START_REF] Wolfe | On the unimodality of L functions[END_REF].

In the following we will suppose that γ 0 = 0 w.l.o.g. and set f k for the positive self-decomposable density associated with the non-degenerate spectral function k.

When k(0+) = +∞, Theorem 28.4 (ii) in [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF] and Lemma 2.5 in [START_REF] Sato | On distribution functions of class L. Z. Wahrsch[END_REF] show that f k is smooth and that all its derivatives vanish at zero and at infinity, and Theorem 1.4 in [START_REF] Sato | On distribution functions of class L. Z. Wahrsch[END_REF] entails that f k is strictly unimodal. When k(0+) < +∞, it is also known -see Remark 28.6 in [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF] -that f k is not smooth at with the same method seems however not quite obvious, because of the memory involved in this integro-differential equation.

Let us now consider the case where k(0+) is finite. When k(0+) < 1, Theorem 1.3 in [START_REF] Sato | On distribution functions of class L. Z. Wahrsch[END_REF] shows that f k is decreasing. When k(0+) = 1, the situation is complicated because of the three different types I 2 , I 3 and I 4 of [START_REF] Sato | On distribution functions of class L. Z. Wahrsch[END_REF]. In particular f k may have a non-trivial modal interval -see Lemma 53.2 in [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF]. When k(0+) > n + 1 for some n ≥ 0, then Theorem 28.4 (i) in [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF] and Lemma 2.5 in [START_REF] Sato | On distribution functions of class L. Z. Wahrsch[END_REF] show that f k is C n with derivatives vanishing at zero and at infinity, and again Theorem 1.4 in [START_REF] Sato | On distribution functions of class L. Z. Wahrsch[END_REF] entails that f k is strictly unimodal. The following is hence quite natural. entails that f a,b satisfies the required properties.

3.3. Two-sided stable densities. As mentioned in the introduction, stable densities are all visually bell-shaped on Holt and Crow's graphics, and it is an interesting problem to show the property rigorously. It is known that two-sided stable densities are real-analytic on R, never vanish, and that all their derivatives tend to zero at infinity -see Remarks 14.8 and 28.8 in [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF]. Hence, their n-th derivative vanishes at least n times on R by Rolle's theorem (a more complicated proof of this fact is also given in [START_REF] Gawronski | On the bell-shape of stable densities[END_REF] pp. 234-235). The strict unimodality follows from Yamazato's theorem [START_REF] Yamazato | Unimodality of infinitely divisible distribution functions of class[END_REF] by analyticity. The article [START_REF] Gawronski | On the bell-shape of stable densities[END_REF] had claimed the full bell-shape property for all stable densities with an argument relying on a certain integral representation through a Student-type kernel, and a multiplicative TP property. The integral representation is correct and basically amounts to Bochner's subordination and Zolotarev's duality -see Lemma 2 in [START_REF] Gawronski | On the bell-shape of stable densities[END_REF]. However the multiplicative TP property claimed in Lemma 1 (iv) of [START_REF] Gawronski | On the bell-shape of stable densities[END_REF] and crucial to bound the number of zeroes -see pp.

236-237 in [START_REF] Gawronski | On the bell-shape of stable densities[END_REF], is false in general. Indeed, Schoenberg's theorem shows -see pp. 121-122 and p.

390 in [START_REF] Karlin | Total positivity[END_REF] -that the kernel K α (x, y) = f α (e x-y ) is STP ∞ if and only if 1/α is the reciprocal of an integer. Actually, K α is not even TP 2 for α > 1/2 -see the main result of [START_REF] Simon | Multiplicative strong unimodality for positive stable laws[END_REF]. In a forthcoming paper, we also show that K α is not TP n as soon as α > 1/n and 1/α is not the reciprocal of an integer. In [START_REF] Gawronski | On the bell-shape of stable densities[END_REF] the skewed Cauchy case is also treated separately via Zolotarev's representationsee (2.8) therein. All in all the main result of [START_REF] Gawronski | On the bell-shape of stable densities[END_REF] must hence be reduced to the Theorem (Gawronski). Two-sided α-stable densities are bell-shaped when α = 1, 2 or when 1/α is an integer.

Notice that this result does not extend directly to the one-sided case in spite of the affirmation made in (iv) p. 239 of [START_REF] Gawronski | On the bell-shape of stable densities[END_REF], because the number of zeroes of a function sequence might clearly increase at the continuous limit. In order to cover all stable densities, the multiplicative method of [START_REF] Gawronski | On the bell-shape of stable densities[END_REF] seems not appropriate because of the poor variation-diminishing properties of K α -see however [START_REF] Simon | A multiplicative short proof for the unimodality of stable densities[END_REF] for a somehow related multiplicative point of view, which works at the unimodal level. On the other hand, Yamazato's additive factorization extends to all two-sided stable densities and delivers yet another proof of their unimodality -see again [START_REF] Yamazato | On strongly unimodal infinitely divisible distributions[END_REF] pp. 600-601. Unfortunately the complete monotonicity of the factor Y α which is crucial in our argument to get the full bell-shape -see the above Remark 2 (a), is clearly lost in the two-sided case.

Acknowledgements. Ce travail a bénéficié d'une aide de l'Agence Nationale de la Recherche portant la référence ANR-09-BLAN-0084-01.

  i+1 }, as required. (ii) If i + 1 < n, one has f (i+1) n ∼ {a i+1 } = 0± 0 0 • • • ± i+1 0 by the induction hypothesis. But adding a 0 at 0+ in the profile does not change anything in the above analysis, and the fact that f (i+1) n+1 ∼ {a i+1 } follows exactly along the same lines as in (i).

Conjecture 1 .

 1 zero and hence not bell-shaped. In view of the main result of the present paper, one might raise the The density f k is bell-shaped if and only if k(0+) = +∞.Apart from the positive stable, examples of densities verifying the above conjecture are the inverse Gamma, or more generally the densities of Γ -a t with t > 0, a ≥ 1. Indeed, the latter are self-decomposable by the HCM criterion[START_REF] Bondesson | Generalized Gamma convolutions and related classes of distributions and densities[END_REF], and bell-shaped by a direct computation. Other selfdecomposable examples are infinite exponential sums, whose bell-shape property can be proved exactly in the same manner as for f α . As pointed out in the introduction, when k(0+) = +∞ Theorem 5.1.(ii) in[START_REF] Sato | On distribution functions of class L. Z. Wahrsch[END_REF] already shows that f (n) k vanishes only once before the first zero of f (n-1) k , for every n ≥ 2. The proof of this latter property relies on a repeated use of Steutel's equationsee e.g. Theorem 51.1 of [8]. Analyzing the further zeroes of f (n) k

Conjecture 2 .

 2 If k(0+) > n + 1 for some n ≥ 0, then f (i) k ∼ {a i } for all i = 0 . . . n. An example supporting this conjecture is the density f a,b oflog β a,b , where β a,b is the Beta random variable with parameters a > 0, b > 1. Indeed, one can show by a direct computation that f a,b is self-decomposable with spectral function k a,b (x) = e -ax (1e -bx ) (1e -x ) so that k a,b (0+) = b, and the explicit formula f a,b (x) = Γ(a + b) Γ(a)Γ(b) e -ax (1e -x ) b-1

  ε n } some finite sequence in {-, 0, +}, we say that a continuous function f : (0, +∞) → R is of type ε 1 . . . ε n if it has limits (finite or infinite) at

zero and at infinity, vanishes on a finite set, and if the ordered sequence of its signs on [0, +∞] is given by {ε 1 , . . . , ε n }. For brevity, we will write f ∼ {u n } or f ∼ ε 1 . . . ε n to express this property.

Observe that by the intermediate value theorem, the zero set of f ∼ ε 1 . . . ε n corresponds to a subsequence of period 2 in {ε 1 , . . . , ε n }. In particular, a function of type ε 1 • • • ε n vanishes exactly


Samson Abramsky
email: samson.abramsky@comlab.ox.ac.uk

Jean ; Krivine
email: jean.krivine@pps.univ-paris-diderot.fr

Michael W Mislove

David Balduzzi

Ross Duncan

Jérôme ; Feret

Ferdinanda ; Camporesi

Vincent ; Danos

Feret

Jonathan ; Hayman

Walter ; Fontana

Russell ; Harmer

Chris ; Thompson-Walsh

Glynn Winskel

Information Flow and Its Applications

Keywords: Classification H.1.1 Systems and Information Theory Keywords and phrases Information flow, semantics of computation, quantum computing, systems biology, information theory, informatics Digital Object Identifier 10.4230/DagRep.2.8.555 and distributed computation 5 Structured Data Analysis Quantum Computing: Circuits, Entanglement, and MBQC

This report documents the program and the outcomes

concepts. It is understood that such substrate can be artificial, for instance in the case of an electrical circuit, or natural, as in the complex signaling pathways that govern cellular fate. Moreover, information may be treated by systems in a designed manner, for instance a computer that processes its inputs according to a determined program, or be the result of evolution, like the internet which is a perfect example of a system that carries and processes information in spite of the absence of a pre-existing specification.

Although traditionally information processing is studied by distinct communities, scattered along the Artificial-Natural and Designed-Evolved axes, it is noteworthy that this separation is, to some extent, a historical artifact in the sense that artificial systems may be the fruit of evolution (as the internet) while natural ones may be used in a purely specified manner (as in synthetic biology). It is therefore natural to expect that tools and techniques developed in one field may be also relevant to others.

Another unifying scheme of the seminar was the emphasis on the use of formal languages in the representation of information flows. Indeed once "a real world" computing system, such as the cell or a quantum circuit, is abstracted as a formal programming language, one may then start to apply techniques imported from theoretical computer science. In the study of evolved systems, these techniques may be used to extract a specification of what is being observed, while in the context of systems where a specification is a priori at disposal, one may use these techniques to verify that the way information is processed conforms to the expectation.

Over the 4 days of talks, which gave rise to feedback that went beyond the expectation of the organizers, the participants of the seminar "Information Flow and its Applications" have had the opportunity to listen to talks ranging from Systems Biology to Theoretical Physics, from Quantum Computing to the study of Ecological systems. As organizers, we believe that the original guess that Information Flow should be a topic of its own was largely a good one.

Computations implemented in physical systems can be described at many different spatiotemporal granularities. For example, the work performed by the brain can be described at the level of atoms, molecules, neurons and potentially higher-order structures. I present an information-theoretic approach to coarse-graining distributed computations. The first step is to introduce effective information, which can be shown to incorporate Kolmogorov complexity, mutual information and VC-entropy (an important measure of complexity in statistical learning theory) as special cases. A second measure, excess information, provides a geometric characterization of indecomposable computations. The two measures are then applied to study coarse-grainings in Conway's Game of Life and Hopfield networks.

A Hypothesis Test For Bell's Inequality

Peter Bierhorst (Tulane University, US) A cell's ability to adapt to constantly changing environmental conditions is derived from its signaling networks. Despite there importance, there is currently no consensus regarding the nature of the protein complexes such networks employ. One prominent view involves signaling machines, while the inherent combinatorial complexity of such networks has led to the more recent proposal of pleiomorphic ensembles. In this work, we use rule-based modeling techniques to explore this question in the case of the yeast pheromone MAPK cascade. We constructed a model of this cascade based on current understanding of the interactions in the pathway. We found that, despite exhibiting considerable ensemble character, this model can replicate existing experimental data for the cascade. We also considered a model designed to Modelers of molecular signaling networks must cope with the combinatorial explosion of protein states generated by post-translational modifications and complex formation. Rulebased models provide a powerful alternative to approaches that require an explicit enumeration of all possible molecular species of a system. Such models consist of formal rules stipulating the (partial) contexts for specific protein-protein interactions to occur. These contexts specify molecular patterns that are usually less detailed than molecular species. Yet, the execution of rule-based dynamics requires stochastic simulation, which can be very costly. It thus appears desirable to convert a rule-based model into a reduced system of differential equations by exploiting the lower resolution at which rules specify interactions.

In this talk, we present a formal framework for constructing coarse-grained systems. We track the flow of information between different regions of chemical species, so as to detect and abstract away some useless correlations between the state of sites of molecular species.

The result of our abstraction is a set of molecular patterns, called fragments, and a system which describes exactly the concentration (or population) evolution of these fragments. The method never requires the execution of the concrete rule-based model and the soundness of the approach is described and proved by abstract interpretation.

Bell's (1964) theorem is popularly supposed to establish the non-locality of quantum physics as a mathematical-physical theory. Building from this, observed violation of Bell's inequality in experiments such as that of Aspect and coworkers (1982) is popularly supposed to provide empirical proof of non-locality in the real world. My talk reviews recent work on Bell's theorem, linking it to issues in causality as understood by statisticians. The talk starts with a new proof of a strong (finite sample) version of Bell's theorem which relies only on elementary arithmetic and (counting) probability. This proof underscores the fact that Bell's theorem tells us that quantum theory is incompatible with the conjunction of three cherished and formerly uncontroversial physical principles, nicknamed here locality, realism, and freedom. I will argue that (accepting quantum theory) Bell's theorem should lead us to seriously consider relinquishing not locality, but realism, as a fundamental physical principle. In the talk I hope also to discuss statistical issues, in the interpretation of state-of-the-art Bell type experiments, related to post-selection in observational studies. Finally I state an open problem concerning the design of a quantum Randi challenge: a computer challenge to Bell-deniers. Can we prove useful probabilistic Bell inequalities for the situation that Alice and Bob's measurement stations receive all their measurement settings at once, and all signals from the source at once, and then (disconnected from the one another and the source) generate all their outputs at once. A "useful" Bell inequality in this context is one where the In this talk we describe how the Cantor Fan can be used as a basis for approximating continuous channels with discrete ones. The Cantor Fan is our term for the order-ideal compilation of the rooted full binary tree, with the path order, and is so named because the set of maximal elements is homeomorphic to the middle-third Cantor set, when the completion is regarded as an algebraic domain. We describe the effect of applying the Probability monad to this structure, and how it then gives a setting to describe how a channel with the Cantor set as input/output set can be approximated by a sequence of channels, C n , in a hierarchy, where C n has 2 n inputs/outputs for each n. We discuss how entropy and capacity of the We review the meta-theoretical situation of the scientific discipline of ecology, taken as a typical example of the family of 'complex system sciences'. We argue that its current relation to classical physics as paradigmatic supplier of concepts, and to computer science as operational supplier of tools is outdated. We propose to regard computer science on the same level as physics, and to explore the scientific potential of concepts imported from theoretical computer science. In particular, we predict a major impact of the richer CS concept of behavior, seen as primary target of empirics and formalization rather than a mechanistic consequence of state. We exemplify this claim by putting the concepts of safety and liveness into ecological context. We show how they can be used as both metaphors and mathematical entities, informing scientific discourse, methodology and modeling.

Coalgebraic Infinite Games without Discounting -Towards Reflexive Economics

Viktor Winschel (Universität Mannheim, DE)

License

Creative Commons BY-NC-ND 3.0 Unported license © Viktor Winschel After a short introduction into the economic field of optimal currency areas and the current problems of the European Monetary Union we sketch the many reflexive issues in economic theory that result from the fact that economic modeling takes place within the modeled system. As a first example towards reflexive structures in economics we present coalgebraic infinite games. The technical part introduces the predicate coinduction proof principle that allows to proof subgame perfect equilibria defined as predicates on the carrier of coalgebras. The economic important result shows that this proof principle does not depend on the usual discounting that essentially transforms infinite structures into finite ones where backwards induction can be applied.

Probabilistic event structures

Glynn Winskel (University of Cambridge, GB)

License

Creative Commons BY-NC-ND 3.0 Unported license © Glynn Winskel This talk presented a new definition of probabilistic event structures, extending existing definitions, and characterized as event structures together with a continuous valuation on their domain of configurations. Probabilistic event structures possess a probabilistic measure on their domain of configurations. This prepares the ground for a very general definition of a probabilistic strategies, which are shown to compose, with probabilistic copy-cat strategies as identities. The result of the play-off of a probabilistic strategy and counter-strategy in a game is a probabilistic event structure so that a measurable pay-off function from the configurations of a game is a random variable, for which the expectation (the expected pay-off) is obtained as the standard Legesgue integral.

Eric Deeds (

Participants


Feng Zhang

Peggy Zwolinski

TOWARD UNIFIED ENVIRONMENTAL ACTIVITIES IN ENTERPRISE

Keywords: environmental strategies, eco design, environmental management, activities

Introduction

The organization today must respond to an increasing rate of environmental challenges. At first, the industry should respect the legislative regulations which demand the "ecologically necessary", a set of minimum environmental requirements [START_REF] Manzini | Limits and Possibilities of Eco-design[END_REF]. These regulations focus on either the performance of offers (such as ERP, RoHS, and WEEE etc.) or the daily activities of enterprise. (Ex: local antipollution standards and WEEE -recycling framework, etc.) The second, with the "sustainable development" concept being gradually understood and accepted by the public, implies the growing diversity of the green requirements to improve human's "life quality". These requirements take attention into two categories, "green product by eco design" (some surveys show that more than a third of US consumers are willing to pay more for eco-friendly products, and in other countries, such as in China, this is even higher) and "green company" (Ex: 44% of UK consumers want more information on what companies are doing to be green) [START_REF]Global Eco label Monitor 2010[END_REF].

Till now, to answer these above requirements, the "eco design" concept had been focused on to contribute to the corporate environmental improvement. The large number of available eco design methods and tools leads to a grand challenge: how to identify the suitable eco methods. "Design", as one of corporate core activities, should coordinate with other corporate functions to implement the daily acts. This paper, analyzes and classifies some environmental improvement activities (named as "Eco activities" below) in different corporate functions (eco design, eco purchasing, eco manufacturing, and eco communication), to expatiate that the suitable identification of the "eco design" methods and tools should not only consider the technical targets and the related criteria, It has also to consider the coherence with other parallel executing eco design methods and the coherence with eco activities of other corporate functions. Finally, this paper proposes a systematic model, which treats about the relationships among different eco activities and related methods on global level. This framework needs to be developed to support the selection of suitable eco design methods.

Eco activities in enterprise 2.1 Eco-engineering

Eco-design is an approach that takes into account environmental impacts in the design and product development and integrates environmental aspects throughout its life cycle (raw material, at the end of life, from manufacturing, logistics, distribution and use). After 20 years of development, we see many of the major steps of ecological design. According to the analysis of methods and tools, the eco design approaches are classified by three main categories: [START_REF] Brezet | Ecodesign -A promising approach to sustainable production and consumption[END_REF], [START_REF]Ecodesign trajectories: a strategy for environmental integration in design process at SMEs[END_REF] and [START_REF] Hallstedt | An approach to assessing sustainability integration in strategic decision systems for product development[END_REF] -Partial eco design: the environment has been considered as a new "constraint" for the product.

The eco design activities modify and improve the product by certain environmental indicators (the energy consumption of final product, the recycling rate of components and the GHG emission during the manufacturing phase, etc.). To identify and track these changes, environmental requirements are noted into the specifications of the product. The typical methods and tools of this area are "DfX" series, for example, "Design for recycling", "Design for disassembly" and the regulation compliant tools.

-Classic eco design: the environment has been considered as a new "criterion" during the design phase. This type of eco design activities considers the entire life cycle of the product from a multi-criteria perspective. According to this global view, eco design improves the environmental impacts and avoids the transfer of impacts to other phases of life cycle of product.

In this area, we find a series of LCA and LCA-related tools, such as simplified LCA, LCAbased metrics and LCA in combination with economic tools. [START_REF] Baumann | Mapping the green product development field: engineering, policy and business perspectives[END_REF], [START_REF] Hur | Simplified LCA and matrix methods in identifying the environmental aspects of a product system[END_REF].

-Innovative eco design: the environment has been considered as a new "value" for the product.

The improvement of environmental impacts has been focused on as a "key" of development for the product and the enterprise. The innovative eco design does not analyze the used components and materials, nor the entire product; it creates and develops the new service, functionality and concepts of offers which bring some positive influence on environment [START_REF]Ecodesign trajectories: a strategy for environmental integration in design process at SMEs[END_REF]. "Sustainable product Design" is an example of the eco-tools for this area.

To contribute to each eco activities in the area mentioned above, a vast range of methods, tools and concepts have been developed [START_REF] Brezet | Ecodesign -A promising approach to sustainable production and consumption[END_REF], [START_REF] Unger | A review of eco-design and environmental assessment tools and their appropriateness for electrical and electronic equipment[END_REF], [START_REF] Baumann | Mapping the green product development field: engineering, policy and business perspectives[END_REF], [START_REF] Siegenthaler | LCA Software Guide 2005 -Market Overview -software Portraits[END_REF] and [START_REF] Hallstedt | An approach to assessing sustainability integration in strategic decision systems for product development[END_REF]. Even for a same environmental topic, according to different characteristics of each eco method or tool, such as the output nature, the main function, the working manners and quality of inputs, these methods require different inputs, related-to competences and information system. The methods and factors to integrate the environmental aspects with other product characteristics had been studied by number of authors. De Benedetto [START_REF] Benedetto | The Environmental Performance Strategy Map: an integrated LCA approach to support the strategic decision-making process[END_REF] created an environmental strategic map, a new tool with some new indicators for analyzing the product impacts combining environment and economy. Sakao [START_REF] Sakao | Quality engineering for early stage of environmentally conscious design[END_REF] and Kobayasi [START_REF] Kobayasi | A systematic approach to eco-innovative product design based on life cycle planning[END_REF] tried to combine some selected eco tools with quality tools to solve the problem at the earlier design phase. Bovea M.D. [START_REF] Bovea | A taxonomy of ecodesign tools for integrating environmental requirements into the product design process[END_REF] classified and analyzed some eco tools by some indicators: The contribution area, the life cycle aspect, when it might be integrated, the nature of results, etc. But these articles did not define quantifiable methods to identify the suitable tools. Besides, as usually, the designer might treat several environmental issues at the same time, such as the recycling rate, the energy efficiency and the product's material etc. The augmentation of topics increases the complexity of dealing with them. But, till now, it's not found that the studies focused on the selection of several methods to deal with some combined environmental problems (ex: deal with recycling rate, energy efficiency and CO2 emission at same time, etc.).

The growing number of environmental methods, concepts and tools really leads to some new challenges for product designer: which inputs should be collected, what competences will be developed and how to select the suitable methods to build an "Eco product" [START_REF] Hallstedt | An approach to assessing sustainability integration in strategic decision systems for product development[END_REF], [START_REF] Pardo | Contribution to the characterisation of eco-design projects[END_REF]. As mentioned above, the first condition, on project level, is to deliver customer's values in a timely efficient manner (combine all issues at same time) to assure that continual profitable growth occurs. [START_REF] Sissel | Re-considering product design: a practical "road-map" for integration of sustainability issues[END_REF]. Secondly, if we consider the "Eco design" as a part of the enterprise, a same understanding of "Eco design" among different levels of the enterprise is very important to deploy all related-activities [START_REF] Hallstedt | An approach to assessing sustainability integration in strategic decision systems for product development[END_REF], [START_REF] Zhang | Optimized navigation system for eco design management[END_REF]. The selection of eco methods for each project should be also restricted by the strategic decision from management level. These strategic decisions comply with the principal developmental-related issues, (such as the financial performance, the long-term relationship with client and suppliers, the mark or brand management etc.) and the balance between the global direction, the speed and the economics of all product segments. They pilot the global competence development in the enterprise and the establishment of all related information systems [START_REF] Hallstedt | An approach to assessing sustainability integration in strategic decision systems for product development[END_REF].

All previous illustrations just consider the analysis and classification of the technique axis. In general, as a classic design, "eco design" should be seen as a set of internal functions in a company to achieve the balance of product between economic, technical and sustainable aspects [START_REF]Ecodesign trajectories: a strategy for environmental integration in design process at SMEs[END_REF], [START_REF] Brezet | Ecodesign -A promising approach to sustainable production and consumption[END_REF]. So it's not enough to just focus on the eco-engineering: the actions of "eco design" should be tailored according to the specifications of product which integrate the contribution from all support-services (supply chain, manufacturing, and marketing etc.) [START_REF] Baumann | Mapping the green product development field: engineering, policy and business perspectives[END_REF].

Eco purchasing and supply management

Supply chain is a system of organizations, people, technology, activities, information and resources involved in moving a product or service from supplier to customer. The green supply chain is to indicate the supply management activities that attempts to contribute to the sustainable performance of all traditional and related-extended supply activities [START_REF] Beamon | Designing the green supply chain[END_REF].

Greening purchasing and supply management relies on the deployment of relevant supply management capabilities and methods. Two main categories of green supply might be identified to classify them [START_REF] Bowen | The role of supply management capabilities in green supply[END_REF], [START_REF] Seuring | From a literature review to a conceptual framework for sustainable supply chain management[END_REF]:

 Supplier management for risks and performance based on "Greening the supply process"

To answer the external pressures, the first manner is to set up an environmental management system of supply chain to avoid the environmental related risks [START_REF]Dirty Laundry" report[END_REF]. This management system requires some minimum criteria (in accordance with the external pressures) to monitor, evaluate and report the supplier's process and its global performance [START_REF] Seuring | From a literature review to a conceptual framework for sustainable supply chain management[END_REF]. Then, these environmental performances are seen as the prerequisites for suppliers that range and allow them to provide materials or service(s) to operate as part of the supply chain [START_REF] Lamming | The environment as a supply chain management issue[END_REF], [START_REF] Min | Green purchasing practices of US firms[END_REF] In this category, the regulation and normal standards play an important role. They mainly focus on the certification of ISO 140001 and EMAS. To obtain these certifications, some activities, such as the monitoring of daily processes and the evaluation of supplier's environmental management system, are established. Then, the corporate private questionnaire which is sent to suppliers for self-audit -are usually mentioned into the papers [START_REF] Baumann | Mapping the green product development field: engineering, policy and business perspectives[END_REF]. This selfaudit indicates how the supplier deals with environmental issues and how it collects data and declares the results. Thirdly, enterprises have to take a longer part of the supply chain into account to allow an improvement in the supply relations as well as in the performance. A company-overlapping communication and a strategic training to improve the supplier's performances are usually proposed. [START_REF] Seuring | From a literature review to a conceptual framework for sustainable supply chain management[END_REF].

Besides of above mentioned traditional activities of supply chain, according to environmental challenges, some extended activities are appearing; such as the relationship with recyclers and remanufacturers. Beamon B.M. [START_REF] Beamon | Designing the green supply chain[END_REF] indicated that these activities and related material flows should be embedded into management system.

 Supply chain management for sustainable products based on the "Product based green supply"

A manner to contribute to the needs for "Eco design" is to set up a product life cycle management to require, train and cooperate with suppliers. In order to realize a "sustainable product", the competence of suppliers plays an important role. Supplier developments are required before enterprises are even able to offer sustainable products to their customers. For example, textile and apparel producers/retailers have to make sure there is an organic cotton supplier before they are able to offer such products [START_REF] Kogg | Greening a cotton-textile supply chain: a case study of the transition towards organic production without a powerful focal company[END_REF], [START_REF] Goldbach | Coordinating sustainable cotton chains for the mass marketthe case of the German mail order business OTTO[END_REF]. To keep a long term relationship with suppliers, a training and cooperation on product level are considered as a key part. In this case, these communications not only require an exchange among supply chain on process level, but also on product level, when some deeper technical information should be translated [START_REF] Handfield | Integrating environmental management and supply chain strategies[END_REF].

In this category, the LCA or LCA-related methods are usually mentioned to support the enterprise to address the eco design related issues and necessary information. The additional condition is the corporate definition of "Eco design" which pre-indicates the goal and boundaries of the cooperation. The project's buyer, delegate the project team, keeps the real contact and cooperation with suppliers to achieve specific project's goal.

Figure 1 Relationships among the eco-activities of supply chain

The figure 2 summaries the relationships as above mentioned between the enterprise and its suppliers.

Commodity purchasing department delegates the enterprise to direct contact with suppliers to keep a long term strategic relationship. This figure shows that the two categories are not mutually exclusive. There is a positive relationship between the two above categories to accelerate and complete the environmental performance [START_REF] Bowen | The role of supply management capabilities in green supply[END_REF], [START_REF] Seuring | From a literature review to a conceptual framework for sustainable supply chain management[END_REF]. The selection of eco design methods should involve the consideration of the acts of two categories and their internal relationships to make the coherence.

Eco manufacturing

The Environmental management system (EMS) is usually used to manage and resolve the environmental problems of manufacturing facilities. ISO 14001EMS standard and related ISO 14011 play the most important role in this area. The European EMAS standard and Canadian "Responsible Care", which focuses on the activities of chemic industry, is also often mentioned. These above three standards require the manufacturing facilities to monitor, register, manage and improve all environmental related data. EMS also intends that the enterprise should set up their management organization to continuously deal with their environmental issues.

In the "plan" phase of ISO 14001, the initial review or gap analysis between the facilities and all the related manufacturing processes of a global situation is recommended. The results support the identification of all the elements of the current operations (the inputs/outputs/discharges and noise, dust, heart and VOC, etc.), which might interact with the environment [START_REF] Martin | ISO 14001 Guidance Manual[END_REF]. According to this review, two categories of actions could be launched, [Tibor, 1996]:

 The improvement of global environmental performance

To answer the local environmental protection needs and to contribute to the global performance of enterprise, some topics are usually selected to be solved [START_REF]Ecodesign trajectories: a strategy for environmental integration in design process at SMEs[END_REF]

 The improvement of certain manufacturing processes

According to the global improvement planning and some specific requirements of production from eco design, there are certain actions which focus on the improvement of the product manufacturing process. Based on the LCA of products, the best available technologies are identified as new solutions to optimize the environmental impacts. [START_REF] Nieminen | EU COST Action 628: Life cycle assessment (LCA) of textile products, eco-efficiency and definition of best available technology (BAT) of textile processing[END_REF] Above topics or evaluation indicators of the first category could be used also for this improvement. But some operational topics and indicators of production process are decided by the designer [EuP directive, 2005], such as the selection of input materials, the definition of production manners, etc. So a relationship between the manufacturing faculties and eco design is necessary to accelerate this improvement. Inversely, the improvement of manufacturing process contributes also to the eco design results which are developed by design team.

Figure 2 the summary of relationships among the eco-activities of production facilities

Figure 3 summaries all above mentioned relationships between the improvement of facilities performance and the improvement of some production line, which has a direct relationship with eco design practices. On one side, the needs of facilities provide the general constraints and the requirements of environmental improvement in manufacturing area to each production line. With the deployment of BATs -Best Available Technologies at facilities level, it also modifies the environmental conditions for each eco design practices. On the other side, the eco design practices make same optimization of its production line, which directly changes the performance of facilities. So an analysis of all relationships has to be developed to have a suitable selection of eco design activities.

Eco communication

The eco communication is defined as all the communication activities, through dissemination of publications (booklets, marks, reports in all kinds of form: paper or electronic) or through the completion of events (seminars, conferences and exhibitions, etc.). It contributes significantly to various environmental improvements [START_REF]The guideline of "Eco communication[END_REF]. Those improvements might indicate the activities of enterprise, or their performances throughout the product life cycle [START_REF]Global Eco label Monitor 2010[END_REF].

The eco certification of enterprise's environmental performance is more and more visible. The certification report of ISO 14001 could be considered as an important example. Besides, there are other evaluation systems, which were established by some independent third party, to monitor, evaluate and support the improvement of the enterprise's activities. We launched an analysis showing that from 84 found eco labeling system in textile industry, there are 31 evaluation systems focused on the environmental performance on enterprise level. To obtain the certification, the enterprise might set up a set of criteria to comply with the requirements on certain activities, including engineering, purchasing, industry, etc. Those criteria might have an influence on the decision for the selection of the eco methods for designers.

Till now, the environmental product declaration systems had become a wide-spread method in order to fulfill the effective market communication around the eco consumption. The voluntary Eco labeling systems presented an important role. ISO divides the environmental labeling into three categories:

 Type I eco labeling was established by a third independent organization which identifies overall environmental performance of a product or service within a specific category based on life cycle considerations [START_REF]Global Eco label Monitor 2010[END_REF]. The criteria of each label were set up according to the common significant environmental topics of such product/service throughout entire life cycle of product. But even for same product's category, the criteria from different labeling systems are not similar. This distinction leading to these standards cannot be used directly by the enterprise to establish uniform standards for an international market. But on project level, in order to answer the extern requirements, some criteria of certain eco labeling system might be involved into the specification of the product as a reference of development. A method which supports the project to analyze and check the compliance of the product with the criteria of eco labeling, had been developed [START_REF] Houe | Assessing the compliance of a product with an eco-label: From standards to constraints[END_REF].

 Type II and type III eco labelling allow the enterprise to self-declare their private environmental claims. Similar with type I labelling system, type II system create an endorsement marks system to have an effect on the purchasing decisions of the customers. The type III system, which provides quantified environmental data of product, is intended for B-to-B business mode. These two private systems are feasible to set a normal standard for all products on international marketing. So the corporate definition of "eco design" and "eco product" should been considered and embedded into the establishment process of those systems.

As above mentioned in the section about "eco engineering", enterprise develops its eco competences and related support system according to its corporate definition of "eco design" and "eco product". The eco communication, as a part of support system, has to be identified following this definition. The product communication plan should consider this definition and integrate it with all real market needs (for example, type I eco labelling). We believe that an interactive relationship between those two levels could accelerate the improvement of the entire environmental performance for both the product and the enterprise.

Need for a unified environmental view to integrate the eco design activities

The above chapters tried to make a review of typical eco activities for principal functions of the enterprise -engineering, manufacturing, supply chain and communication. In each section, the profits and all related environmental objectives could be classified into two generic axes:

Axe 1: The eco activities which contribute to the global performance of the enterprise Axe 2: The eco activities which contributes to the needs for eco design throughout the entire life cycle of the product

The activities contributing to the needs of the two above axes are not exactly independent. On the contrary, they are mutually constrained, and sometimes, they are mutually supported. As above mentioned in the precedent chapters, the selection of eco method to answer one axe's needs, should consider the constraints from another one. For example, an enterprise's private definition of "Eco design" is defined focusing on the topics of material, such as the dematerialization, the decrease of hazardous substances. This strategic definition provides a corporate objective of competence development, the topics of internal training and the recruitment needs. But the lack of competences leads to a low level support to other needs coming from some projects, such as the energy efficiency or GHG emission. So the selection of methods to answer these topics for the project should be enough simple and easy to understand by no professional person. On the contrary, the establishment of corporate "eco design" definition and some global environmental improvement could also optimize the condition of the implementation of some eco design activities in the project. So we hypothesize that a systematic conversation channel between two axes could support the enterprise to make a suitable identification of eco design activities and related eco methods [START_REF]Dirty Laundry" report[END_REF], [START_REF] Hallstedt | An approach to assessing sustainability integration in strategic decision systems for product development[END_REF].

Transverse, as a united organization, the selection of eco design activities and related eco methods in enterprise need to consider also the coherence among different corporate functions, such as the competences and abilities of each function, the working flow and collaborative channel, and the actual database, etc. ISO 14000 series and other classic LCA methods presented that a classic and precise product life cycle improvement measured and realized into design team should be supported by some contributions of operational departments, such as the production process, the imported materials evaluation and the logistic optimization, etc. But these contributions could not only follow the requirements of LCA. They should be coherent with their departmental context (competence, habitude, location and finance situation etc.) and related development tendencies. To answer several environmental issues at the same time, the designer might launch some different eco activities using different eco methods. . [START_REF] Zhang | Optimized navigation system for eco design management[END_REF] presented the same points through an industrial review of the corporate eco design programs of electric and electronic enterprises. [START_REF] Zhang | Optimized navigation system for eco design management[END_REF] The growing number of used methods might lead to an additional challenge of method selection: How to optimize the network of eco design methods. In other word, does an implementable method affect the dynamic operational context to select another parallel one? At first, this optimization might take the sharability of data flow into account. The sharability means the maximum sharing of necessary inputs to each eco activity (by the common data) or making the data-bridge to transfer the data form to answer several needs. For example, if the supplier could provide a certification on a global level to attest there is no RoHS substance used into their process, the project could directly transfer this information on material's level to avoid double requirements for the same thing. This kind of input's number reduction might encourage the positivity of each stakeholder and reduce the negative influences on the actual system. Secondly, this optimization might also take the co-operability among different eco activities into account. The typical analysis might be to establish a chain of activities which plans all as an entire program (ex: the outputs of some eco activities could be used as the inputs of others). This chain of activities might reduce the complexity of the cooperation, the related cost and time use, and also it's positive to explain to all employees the signification of each activity which encourages the final achievement. So, a hypothesis is that there is a positive influence to take all relationships among eco activities and related methods into account according to the context of an enterprise. This relational analysis could support the identification of the suitable combination of eco methods. And this harmonization of selected methods might accelerate the corporate environmental optimization.

Although there are several available eco methods and tools to assist the realization of eco activities, [START_REF] Baumann | Mapping the green product development field: engineering, policy and business perspectives[END_REF] underlining that there was a lack of systematic perspective and relationship between strategic intents and eco design practices. [10] Indicated the similar point, a systematic view of enterprise and a common language which ensures the comprehension between different functions or between different levels in enterprise are necessary to be developed to support the realization of eco design. In 1996, a uniform framework of Strategic Sustainable Development (SSD framework) has been developed [START_REF] Robèrt | Strategic sustainable developmentselection, design and synergies of applied tools[END_REF], [START_REF] Hallstedt | An approach to assessing sustainability integration in strategic decision systems for product development[END_REF] to find out a general applicable definition. This framework has successfully identified the general frontier and the interface for different levels in the enterprise to deploy the environmental program. But it did not provide the practicable criteria or indicators to support the enterprise make the decision.

[Reyes T. 2007] proposed a "trajectory" methodology to guide the medium-small size enterprise to plan some eco design related activities. She did not take the axe of global performance into account.

Zhang F [START_REF] Zhang | Optimized navigation system for eco design management[END_REF] indicated that the adaptive eco design action or actions group should be prioritized and identified from the complex constraints of enterprise, such as the strategic position, the technical capacity and implementation situation etc. And he proposed an internal decision-communication system to describe all environmental issues/problems by a common language. But he did not touch the details on method level.

Figure 3 Identification of the suitable eco design methods

Finally, according to these above needs and the lack of related researches on systematic relationships among the eco activities in enterprise, a framework to identify the suitable eco design methods and the tools need be developed. Beside of the classic criterion, the requirements of customs, this framework could take a systematic analysis which needs to consider:

-the coordination with the corporate global direction, which might includes the corporate definition of "Eco design" and the related competence and source development situation, etc.;

-the cooperation with other corporate functions considering their complex context; -the cooperation with other executing eco design methods and related process to treat about several issues at the same time;

Conclusion

Based on a literature review of the different eco activities in different corporate functions, this paper tried to classify these activities into two contribution axes: those that contribute to the corporate global performance development and those that contribute at the product level. The literature review showed also that these two axes are not yet coordinated and indicated that this coordination has to be developed to harmonize the eco activities in the enterprise to increase the whole eco -efficiency. To accelerate the integration of the suitable eco design activities, a systematic analysis model has to be proposed. It will treat with the relationships among different eco activities and related methods on global level, to support the selection of suitable eco design methods.

  Water pollution : separate the rain water and waste water / interceptors, filtering recycling 4. Perturbation: Measure discharge to atmosphere / Land use 5. Other risks and hazards: various power usage / solvent storage

: 1. Local surroundings: Compliant with regulation / the complaints from neighbors 2. Waste : collection center / risks control / reduction program / encourage recycling 3.