Jean Dolbeault

Giuseppe Toscani

Best matching Barenblatt profiles are delayed

Keywords: numbers: Primary: 02.30.-f, 02.30.Jr, 02.30.Sa. Secondary: 02.30.Xx, 02.60.Cb entropy -entropy production method, fast diffusion equation, second moment, intermediate asymptotics, self-similar solutions, Barenblatt profiles, delay

The growth of the second moments of the solutions of fast diffusion equations is asymptotically governed by the behavior of self-similar solutions. However, at next order, there is a correction term which amounts to a delay depending on the nonlinearity and on a distance of the initial data to the set of self-similar Barenblatt solutions. This distance can be measured in terms of a relative entropy to the best matching Barenblatt profile. This best matching Barenblatt function determines a scale. In new variables based on this scale, which are given by a self-similar change of variables if and only if the initial datum is one of the Barenblatt profiles, the typical scale is monotone and has a limit. Coming back to original variables, the best matching Barenblatt profile is delayed compared to the self-similar solution with same initial second moment as the initial datum. Such a delay is a new phenomenon, which has to be taken into account for instance when fitting experimental data.

In 1905, A. Einstein established in [1] that the diffusion coefficient D in Brownian motion is determined by the number of atoms and can be measured by considering the second moment, which linearly grows with respect to time and can be experimentally measured. More precisely, if v is a solution to the heat equation

v τ = D ∆v (τ, x) ∈ R + × R d with nonnegative initial datum v 0 , then R d |x| 2 v(τ, x) dx = R d |x| 2 v 0 (x) dx + D τ R d v 0 (x) dx
for any positive time τ .

In case of the porous medium (m > 1) or fast diffusion (m < 1) equation, that is

v t = D ∆v m (τ, x) ∈ R + × R d (1)
it turns out that the self-similar solutions found by G.I. Barenblatt in [2] v ∞ (τ, x) = 1 R(D τ

) d C + 1 -m 2 m x R(D τ ) 2 1 m-1 +
with R(τ ) = (τ /α) α , 1/α = d (m -m c ) and m c := (d -2)/d, play the role of Green's function in the nonlinear case, at least as far as large time asymptotics are concerned. Moreover, it holds that

I(τ ) := R d |x| 2 v(τ, x) dx ∼ R d |x| 2 v ∞ (τ, x) dx =: J(τ ) = J(1) τ 2α (2)
as τ → +∞. In other words, the second moment of any solution asymptotically grows for large time like the second moment of a self-similar solution.

In this paper, we will consider the fast diffusion case m ∈ (m 1 , 1) with m 1 := (d-1)/d and prove that there is a correction to this asymptotic behavior, which amounts to a time delay. It can be briefly described as follows. Let τ 0 be such that

I(0) = J(τ 0 ) ,
that is, let us define τ 0 by τ 2α 0 = I(0)/J (1). If the nonnegative initial datum v 0 is a self-similar profile, i.e. if v 0 (x) = v ∞ (τ 0 , x), then it is straightforward to realize that I(τ ) = J(τ + τ 0 ) for any positive time τ . For any other initial datum, we will prove that I(τ ) < J(τ + τ 0 ) for any τ > 0 and I(τ ) = J(τ + τ 0 -δ) + o(τ 2α-1 ) as τ → +∞ (3) for some delay δ > 0. A precise statement goes as follows.

Theorem 1 Let d ≥ 1 and m ∈ (m 1 , 1). Assume that v solves (1) for some nonnegative initial datum v 0 such that v 0 |x| 2 and v m 0 are both integrable. Then the delay δ is nonnegative and δ = 0 if and only if v 0 = v ∞ (τ 0 , •) for some τ 0 > 0.

In the range m 1 < m < 1, we will provide estimates of δ in terms of a distance of the initial datum v 0 to the set of Barenblatt profiles. With our method, however, the lower bound on δ goes to 0 as m approaches either 1 or m 1 and in particular nothing is known in the range m c ≤ m ≤ m 1 . This distance to the set of Barenblatt profiles can be measured in terms of a relative entropy with respect to the best matching Barenblatt profile and determines a scale. The notion of best matching will be discussed in Section 3. We will introduce new variables based on this scale, which are self-similar variables if and only if the initial datum is one of the Barenblatt profiles. In the new variables, we observe two effects:

(i) The relative entropy is decaying faster than when measured in self-similar variables.

In other words, there is an initial time layer during which convergence towards the set of self-similar variables is faster than in the asymptotic regime, in which it goes exponentially.

(ii) In the new variables, the moment of the best matching Barenblatt profile is decaying towards a finite, positive limit. Denote by ρ the ratio by which it is decreased: we will prove that ρ is positive but less than 1 which corresponds to the fact that δ is positive. Even if its value is not known accurately, it can be estimated in terms of the relative entropy with respect to the best matching Barenblatt profile.

The behavior of the solution during the initial time layer and the ratio ρ are interdependent. Interpreting these two effects in the original setting is tricky, as not only the space scale is changed, but also the time scale, and the change of variables heavily depends on the solution itself. However, by undoing the change of variables and considering the asymptotic regime for large times τ , it is possible to estimate δ.

The positivity of the delay δ is a new phenomenon and should be taken into account whenever experimental data are used to estimate exponents in nonlinear diffusions. It is indeed only in the asymptotic regime that the growth of the self-similar solutions is recovered and can be used to estimate the exponent m and the diffusion coefficient D in (1). By taking into account the delay δ, we obtain the next term in the asymptotic expansion for large time scales.

Our last remark is that this delay is a purely nonlinear effect. All our estimates lose their meaning by pushing the exponent m to 1 in the nonlinear diffusion equation (1). In the linear case indeed, the growth of the second moment of any solution depends only of the initial value of the second moment itself, and it is identical to the growth of the second moment of the self-similar solution (a Gaussian density).

Numerous mathematical papers have been devoted to the qualitative description of the solution to nonlinear diffusion equations. In the porous medium case (m > 1), the large-time behavior (2) of the second moment has been established in [3]. In the fast diffusion case (m < 1), we shall give a quick justification of this rate in Section 1 and refer to J.L. Vázquez' books [START_REF] Vázquez | The porous medium equation Oxford Mathematical Monographs[END_REF][START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF] for more details. We will also provide refined estimates when m < 1. The importance of the knowledge of the second moment in nonlinear diffusion equations has been outlined in [START_REF] Carrillo | [END_REF]. Large time asymptotics of (1) have been intensively studied from the mathematical point of view, with essentially two main techniques: comparison methods starting with [7], and relative entropy methods based on the functional introduced by J. Ralston and W.I. Newman in [8,9] are the two main approaches. As for the second one, connection with optimal functional inequalities in [10] allowed to characterize best possible rates. F. Otto gave an interpretation of (1) in terms of gradient flows which a posteriori justifies the exponential rate of convergence in self-similar variables. Asymptotic rates of convergence have been connected with optimal constants in a family of Hardy-Poincaré inequalities in [11] and a properly linearized regime in [START_REF] Denzler | Proc. Natl. Acad. Sci[END_REF][START_REF] Blanchet | [END_REF][START_REF] Denzler | Memoirs of the American Mathematical Society[END_REF]. Based on this method, improved convergence rates have been obtained in [START_REF] Bonforte | Proc. Natl. Acad. Sci[END_REF][START_REF] Dolbeault | [END_REF] for well prepared initial data. However, the most striking result up to now is an improved functional inequality which has been established in [17]. For completeness, let us quote [18,19,20] in case of time-dependent diffusion coefficients. It is also worthwhile to indicate that the approach based on Rényi entropies is to some extent a parallel approach to the one developed in [START_REF] Dolbeault | [END_REF]17] (also see [21,22,23], and [24,25] for earlier related results) except that the emphasis is then put on the second moment rather than on the relative entropy term.

Diffusion processes described by (1) are known to arise in various fields of physics such as plasma physics, kinetic theory of gases, solid state physics, filtration models and transport in porous media. In many metals and ceramic materials the diffusion coefficient can, over a wide range of temperatures, be approximated as v -ν for some exponent ν such that 0 < ν < 2, see, e.g., [26], which leads to an exponent m < 1 in (1). Also, the special case ν = 1 emerges in plasma physics as a model of the crossfield convective diffusion of plasma including mirror effects, and in the central limit approximation to Carleman's model of the Boltzmann equation, according to [27,28]. In the range m > 1, the porous medium equation has been used by J. Boussinesq in the study of groundwater infiltration and in the description of the flow of an isentropic gas through a porous medium. It also applies to the theory of heat radiation in plasmas developed by Ya.B. Zel'dovich and his coworkers. In particular, point source, self-similar solutions have been obtained by Ya.B. Zel'dovich and A.S. Kompaneets, G.I. Barenblatt, and also R.E. Pattle; see [2,29]. In mathematics, the fast diffusion case is better known than the porous medium case for its properties in connection with functional analysis. However, both cases have been generalized in the setting of filtration equations and are now considered as building blocks for various models in mathematical biology, in mechanics of viscous fluids, particularly for boundary layers or thin films, etc. when the diffusion coefficient depends on the density. In this paper, we will not try to further justify the interest of the equation for applications, but focus on some qualitative aspects by which it differs from the linear theory of the heat equation. The emphasis will be put on new effects by which generic solutions differ from self-similar ones.

Asymptotic growth of the second moment

In this section, we introduce some notation, recall known results and quickly establish some of them which are of crucial importance for the new results of our paper. We shall quote relevant references for the reader interested in further details. In the rest of the paper, we shall assume as in Theorem 1 that the initial datum v 0 is a nonnegative function of bounded second moment such that v m 0 is integrable. Let us first establish that Relation (2) corresponds to the generic growth for the second moment. We start by rewriting (1) in self-similar variables, i.e. for any

(τ, x) ∈ R + × R d , v(τ, x) = µ d R(D τ + α) d u 1 2 log R(D τ + α), µ x R(D τ + α) (4)
for some µ > 0 to be fixed later. We recall that R(τ

) = (τ /α) α , 1/α = d (m -m c ) and m c := (d -2)/d. We assume that m > m c , so that R(D τ + α) ∼ R(τ ) as τ → +∞.
Rescaling by R(D τ + α) instead of rescaling by R(τ ) has the advantage that the parameter D is scaled out and the initial datum is preserved by the change of variables, up to a simple scaling: u 0 (x) := u(t = 0, x) = µ -d v(τ = 0, x/µ) ≥ 0. As in [10], we may notice that the problem of intermediate asymptotics, that is, showing that v(τ, x) ∼ v ∞ (τ, x) as τ → +∞, now amounts to prove the convergence of u to the Barenblatt profile

B 1 (x) = (C M + ǫ |x| 2 ) 1 m-1 +
where µ = (|1 -m|/(2 m)) α and ǫ = ±1 has the sign of (1 -m). Here C M > 0 is uniquely determined by the condition

R d B 1 dx = M := R d u 0 dx > 0. An elementary computation shows that C M = (M/M * ) 2 (m-1) d (m-mc ) with M * = R d (1 + ǫ |x| 2 ) 1 m-1 + dx. It is also straightforward to check that R(D τ ) d v ∞ (τ, R(D τ ) x) = µ d B 1 (µ x) and C M = µ 2-1/α C
with the notations of the introduction. From now on, we shall assume that we are in the fast diffusion case, that is, m < 1 and ǫ = 1.

In self-similar variables, the equation satisfied by u is

u t + ∇ • u ∇u m-1 -2 x = 0 (5)
whose unique radial stationary solution of mass M is B 1 . Note that the mass M is preserved along the evolution for any m ∈ (m c , 1). We refer to [30] for a proof.

Next we consider the relative entropy, or free energy, functional of J. Ralston and W.I. Newman, that can be defined as

F 1 [u] := 1 m -1 R d u m -B m 1 -m B m-1 1 (u -B 1 ) dx and observe that, if m ∈ [m 1 , 1)
where

m 1 := d -1 d ,
then for any t ≥ 0 we have

d dt F 1 [u(t, •)] ≤ -4 F 1 [u(t, •)]
according to [10], so that, for any t ≥ 0,

0 ≤ F 1 [u(t, •)] ≤ F 1 [u 0 ] e -4t (6)
if u is a solution to [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF]. Moreover, this rate is sharp as can be checked by taking as initial datum u 0 a non-centered Barenblatt profile, B 1 (• -x 0 ) for some x 0 = 0. As a third step, we introduce the relative moment and the entropy, respectively

K 1 [u] := R d |x| 2 (u -B 1 ) dx and S 1 [u] := R d (u m -B m 1 ) dx , so that F 1 [u] = 1 m-1 S 1 [u] -m m-1 K 1 [u]
. By Jensen's inequality, we know that F 1 [u] ≥ 0 with equality if and only if u = B 1 . If u is a solution to (5), using the identity

R d |x| 2 ∇ • u ∇u m-1 dx = -2 d 1 -m m R d u m dx ,
an elementary computation shows that

d dt K 1 [u(t, •)] + 4 K 1 [u(t, •)] = 2 d m (1 -m) S 1 [u(t, •)] that is, d dt K 1 [u(t, •)] + 2 d (m -m c ) K 1 [u(t, •)] = - 2 d m (1 -m) 2 F 1 [u(t, •)] .
Using [START_REF] Carrillo | [END_REF], this shows that

0 ≥ d dt e 2t α K 1 [u(t, •)] ≥ 1 -m m d dt e 2t α -4t F 1 [u 0 ] .
and hence, after an integration from 0 to t, 1), so that α ∈ ( 1 2 , 1], and finally lim sup t→+∞ e

0 ≤ 1 m S 1 [u 0 ] e -2t α + 1-m m F 1 [u 0 ] e -4t ≤ K 1 [u(t, •)] ≤ K 1 [u 0 ] e -2t α with m ∈ [m 1 ,
2t α K 1 [u(t, •)] < ∞.
By undoing the self-similar change of variables (4), if v is a solution to (1) with initial datum v 0 (x) = µ d u 0 (µ x), we end up with the estimate lim sup

τ →+∞ 1 + D α τ µ 2 R d |x| 2 v(τ, x) dx (1 + D τ /α) 2α - R d |x| 2 B 1 dx < ∞ .
As we shall see next, a better adjustment of the Barenblatt profile allows us to give more accurate estimates.

Large time asymptotic refinements

Let us assume that m ∈ (m 1 , 1). Detailed mathematical justifications needed for the computations of this section can be found in [START_REF] Blanchet | [END_REF][START_REF] Bonforte | Proc. Natl. Acad. Sci[END_REF][START_REF] Dolbeault | [END_REF]. Large time asymptotics are determined by the time evolution of the relative entropy. If u is a solution to (5), then

d dt F 1 [u(t, •)] = -J 1 [u(t, •)] := - m 1 -m R d |∇u m-1 -∇B m-1 1 | 2 u dx .
It has been established in [10] that

4 F 1 [u] ≤ J 1 [u] ,
which is the key inequality to prove [START_REF] Carrillo | [END_REF]. As t → +∞, if we define w such that

u = B 1 1 + B 1-m 1 w , then B 1-m 1 w(t, •) uniformly converges to 0, F 1 [u(t, •)] ∼ m 2 R d |w| 2 B 2-m 1 dx := F 1 [w(t, •)] , J 1 [u(t, •)] ∼ m (1 -m) R d |∇w| 2 B 1 dx := I 1 [w(t, •)] ,
and, if m ∈ [m 1 , 1), it has been established in [11] that the following Hardy-Poincaré inequality holds

Λ F 1 [w] ≤ I 1 [w] (7)
with optimal constant Λ = 4 for any w

∈ L 2 (R d , B 2-m 1 dx) such that ∇w ∈ L 2 (R d , B 1 dx) and R d w B 2-m 1 dx = 0. Consider on L 2 (R d , B 2-m 1 dx) the self-adjoint operator L 1 such that L 1 w := -2 (1 -m) B m-2 1 ∇(B 1 ∇w) .
Its kernel is generated by the constants, and the eigenspace corresponding to the first non-zero eigenvalue, Λ = 4, is spanned by x i , for any i = 1, 2. . . d, so that Λ in ( 7) is actually the spectral gap of L 1 . If we further assume that R d x w B 2-m 1 dx = 0, then the spectral gap in ( 7) is improved and the best possible Λ on this restricted space is Λ = 8+4 d (m-1). See [START_REF] Bonforte | Proc. Natl. Acad. Sci[END_REF][START_REF] Dolbeault | [END_REF] for details. For a solution u of (5), it is straightforward to check that R d x u dx is preserved along the evolution. Hence, by enforcing the condition R d x u 0 dx = 0, we asymptotically obtain an improved convergence rate compared to [START_REF] Carrillo | [END_REF]:

lim sup t→+∞ e (8+4 d (m-1)) t F 1 [u(t, •)] < ∞ .
The second non-zero eigenvalue of L 1 , in the range m ∈ (m 1 , 1), is associated with dilations and the corresponding eigenspace is spanned by |x| 2 . However, according to (5), R d |x| 2 u dx is not preserved along the evolution. Well prepared initial data are therefore not sufficient to get rid of the associated degree of freedom and a more detailed analysis is required. Our analysis is now based on [START_REF] Dolbeault | [END_REF]. To a solution v of (1), we may now associate a function u given by

v(τ, x) = √ σ µ R (D τ + α) d u t, √ σ µ x R(D τ + α) for any (τ, x) ∈ R + × R d , with t = 1 2 log R(D τ + α) and R(τ ) = (τ /α) α
, so that the equation satisfied by u becomes

u t + ∇ • σ d 2 (m-mc) u ∇u m-1 -2 x u = 0 . ( 8
)
Here σ is a real, positive, time-independent parameter, to be adjusted. When σ = 1, we recover [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF]. For a general σ > 0, the function

B σ (x) := σ -d/2 B 1 (x/ √ σ)
is the unique radial stationary solution to (8) with mass M = R d u 0 dx. Assume further that R d x u 0 dx = 0 and write u = B σ (1 + B 1-m σ w) where σ is chosen such that, asymptotically as t → +∞, w is orthogonal to the eigenspace associated to the second non-zero eigenvalue of L σ defined by

L σ w := -2 (1 -m) B m-2 σ ∇ (B σ ∇w) .
The value σ is uniquely defined and this can be justified in the framework of [START_REF] Dolbeault | [END_REF] (see in particular [16, Section 4]) but we will give a simpler evidence for this result in Section 3. The relative entropy is now defined with respect to B σ :

F σ [u] := 1 m -1 R d u m -B m σ -m B m-1 σ (u -B σ ) dx ,
and we look for improved bounds on the functional

K σ [u] := R d |x| 2 (u -B σ ) dx
when u is a solution to (8), as t → +∞. According to [START_REF] Dolbeault | [END_REF], we obtain the improved asymptotic rate lim sup

t→+∞ e Λt F σ [u(t, •)] < ∞ with, for any d ≥ 2, Λ =        (d-4-m (d-2)) 2 2 (1-m) if m c < m ≤ d+4 d+6 , 8 (d + 2) m -8 d if d+4 d+6 ≤ m ≤ d+1 d+2 , 8 if d+1 d+2 ≤ m < 1 . Note that m 1 ≥ d+4 d+6 if d ≥ 6,
K σ [u(t, •)]) 2 = O(F σ [u(t, •)]) as t → +∞, hence proving that lim sup τ →+∞ 1 + D α τ 1+γ µ 2 R d |x| 2 v(τ, x) dx (1 + D τ /α) 2α - R d |x| 2 B 1 dx < ∞
where we have used the fact that σ

R d |x| 2 B σ dx = R d |x| 2 B 1 dx and γ = 1 4 α Λ -1 .
It is straightforward to check that γ is positive for any m ∈ (m 1 , 1), thus improving the estimate given in Section 1. Details are left to the reader. Note that improved convergence rates of the second moment can also be achieved if m ∈ (d/(d + 2), m 1 ) and d > 2. The case m < d/(d + 2) is far less interesting. In the range [m c , d/(d + 2)] and d ≥ 2, the asymptotic rate of convergence is determined by the continuous spectrum of L σ . If m < m c , all solutions with finite mass extinguish in finite time and L σ has only continuous spectrum: see [START_REF] Blanchet | [END_REF] for details. The case m = m c has been considered in [31]. The limit case m = 1 (heat equation) is covered by the standard decomposition into Hermite functions: see for instance [32]. All above improvements on the rate of decay of the relative entropy are achieved only in the asymptotic regime, by considering the best matching Barenblatt profile as t → +∞. This suggest to do the same for any finite time t.

Best matching Barenblatt functions and the second moment

A time dependent rescaling

In this section we assume that m ∈ (m 1 , 1) and consider a solution v of (1) with

R d v 0 dx = M.
The key idea of [START_REF] Dolbeault | [END_REF]17] is to find the best matching Barenblatt function by minimizing w.r.t. v C,y,λ the functional

1 m -1 R d v m -v m C,y,λ -m v m-1 C,y,λ (v -v C,y,λ ) dx
where v C,y,λ is a generic Barenblatt function depending on the parameters (C, y, λ) ∈ (0, +∞) × R d × (0, +∞):

v C,y,λ (x) = λ -d 2 C + 1 -m 2 m |x -y| 2 λ 1 m-1 .
It turns out that the best matching Barenblatt function is obtained by choosing C, y and λ as follows:

C M = µ 2-1/α C i.e. such that R d v C,y,λ dx = M, y = 1 M R d x v 0 dx and R d |x -y| 2 v C,y,λ dx = R d |x -y| 2 v(τ, x) dx.
According to Sections 1 and 2, if v is a solution to (1), it is clear that λ = λ(τ ) explicitly depends on τ and is such that τ -2α λ(τ ) converges as τ → +∞ to σ ∞ . Actually, we can say much more on τ → λ(τ ), but for this purpose, it is more convenient to introduce a time-dependent change of variables as in [START_REF] Dolbeault | [END_REF]. Let u be such that

v(τ, x) = µ d R(D τ ) d u 1 2 log R(D τ ), µ x R(D τ ) with τ → R(τ ) now given as the solution to 1 R dR dτ = µ 2 K M R d |x| 2 v(τ, x) dx -d 2 (m-mc) , R(0) = 1 ,
where

K M := R d |x| 2 B 1 dx = d (1 -m) (d + 2) m -d M C M .
Note that the initial condition is still R(0) = 1, which was previously taken into account by considering R as a function of (D τ + α) instead of a function of D τ . Then the equation for u is given by ( 8) where σ now depends on t according to

σ(t) = 1 K M R d |x| 2 u(t, x) dx .
As it has already been observed in [START_REF] Dolbeault | [END_REF], σ(t) is also characterized as the unique minimizer of σ → F σ [u(t, •)]. Hence our change of variables is given by

1 R dR dτ = σ(t) R 2 (τ ) -d 2 (m-mc) , R(0) = 1 ( 9
)
where σ is defined as a function of

t = 1 2 log(R(D τ )) .
According to [17], if m ∈ (m 1 , 1), with

f (t) := F σ(t) [u(•, t)] , σ(t) = 1 K M R d |x| 2 u(x, t) dx , j(t) := J σ(t) [u(•, t)] , J σ [u] := m σ d 2 (m-mc) 1 -m R d u ∇u m-1 -∇B m-1 σ 2 dx ,
we can write a system of coupled ODEs

         f ′ = -j ≤ 0 σ ′ = -2 d (1-m) 2 m K M σ d 2 (m-mc) f ≤ 0 j ′ + 4 j = d 2 (m -m c ) j -4 d (1 -m) f σ ′ σ -r (10)
with initial data (f 0 , σ 0 , j 0 ), where

r := σ d 2 (m-mc) 2 (1 -m) m R d u m |∇z| 2 -(1 -m) (∇ • z) 2 dx ≥ 0
by the arithmetic-geometric inequality, and

z := σ d 2 (m-mc) ∇u m-1 -2 x. Asymptotically as t → +∞, we know that lim t→+∞ f (t) = lim t→+∞ j(t) = 0 and lim t→+∞ σ(t) = σ ∞ > 0 .
Here σ ∞ takes the same value as σ in Section 2 and we shall give below an explicit estimate showing that σ ∞ > 0. As an easy consequence of the last identity in (10), we get that f (t) ≤ f 0 e -4t and j(t) ≤ j 0 e -4t for any positive time t, if m ∈ (m 1 , 1). An integration of j ′ + 4 j ≤ 0 on (t, +∞) gives

4 f (t) ≤ j(t) (11)
for any t ≥ 0, which turns out to be equivalent to a Gagliardo-Nirenberg inequality according to [10]. See [33,[START_REF] Carrillo | [END_REF]34,17] for further details on the entropy-entropy production method. From (10), we can do better and deduce improved decay estimates.

Relative entropy: improved estimates

Another estimate can indeed be derived by observing that

j -4 d (1 -m) f = d (1 -m) (j -4 f ) + d (m -m 1 ) j ≤ d (m -m 1 ) j
by (11) and that the last equation of (10) implies

j ′ + 4 j ≤ κ j σ ′ σ with κ := 1 2 (m -m c ) (m -m 1 ) d 2 .
Note that 0 < κ < 1 if m ∈ (m 1 , 1) and κ = κ(m) is such that lim m→m 1 κ(m) = 0, lim m→1 κ(m) = 1. A Gronwall estimate then gives σ(t) ≥ σ 0 j(t) e 4t j 0 1/κ for any t ≥ 0. Even if this estimate is rough, it proves that σ(t) is always positive. We shall get a better estimate in Section 3.3. We can also use the equation for σ ′ in (10) to get that σ(t) is decreasing and

j ′ + 4 j ≤ 8 a j f = -4 a (f 2 ) ′ with a := d 4 (1 -m) 2 m K M σ -d 2 (1-m) 0 κ .
By integrating this last inequality from t to +∞, we find that

j -4 f ≥ 4 a f 2 .
Using f ′ = -j and integrating once more, we get

f (t) ≤ f 0 (1 + ε) e 4t -ε =: f ⋆ (t) with ε := a f 0 .
Note that ε depends on M and f 0 σ

-d 2 (1-m) 0
. Hence, as t → +∞, we have

f (t) f 0 1 + ε e -4t
and we improve the standard estimate f (t) ≤ f 0 e -4t by a factor 1/(1 + ε) without requiring orthogonality conditions as in Section 2.

Second moment: asymptotic estimates

Assume again that m ∈ (m 1 , 1). As a function of t, σ is non-increasing with initial value σ(0) = σ 0 > 0, and a slightly more precise estimate is achieved by writing that

- d dt σ d 2 (1-m) = d 2 (1 -m) 3 m K M f ≤ d 2 (1 -m) 3 m K M f 0 e -4t
which provides the estimate

σ d 2 (1-m) ∞ ≥ σ d 2 (1-m) 0 - d 2 (1 -m) 3 4 m K M f 0 . (12)
Since u(x, t) and B σ(t) have the same mass and second moment, we know that

f (t) = 1 1-m R d (B m σ(t) -u m (t)) dx. By observing that d R d B m 1 dx = - R d x • ∇B m 1 dx = 2 m 1 -m R d |x| 2 B 1 dx ,
we can write f 0 in the form

f 0 = 2 m K M d (1 -m) 2 σ d 2 (1-m) 0 - 1 1 -m R d u m 0 dx .
Hence we end up with the positive lower bound

σ d 2 (1-m) ∞ ≥ d 2 (m -m c ) σ d 2 (1-m) 0 + d 2 (1 -m) 2 4 m K M R d u m 0 dx , (13)
which is just a rewriting of [START_REF] Denzler | Proc. Natl. Acad. Sci[END_REF]. Note that f 0 and σ 0 satisfy the constraint

f 0 ≤ 2 m K M d (1 -m) 2 σ d 2 (1-m) 0 .
As a consequence the factor 1/(1 + ε) which appears in Section 3.2 is constrained by the condition

ε ≤ 1 4 (m -m c ) (m -m 1 ) d 2 = κ 2 < 1 2 .
Notice that ε → 0 as m → m 1 , with m > m 1 .

Second moment: improved asymptotic estimates

With the notations of Section 3.2, if we use the estimate f ≤ f ⋆ instead of the estimate f (t) ≤ f 0 e -4t , a slightly better estimate of σ can be given, namely,

σ(t) d 2 (1-m) ≥ σ d 2 (1-m) 0 - d 2 (1 -m) 3 m K M 1 4 ε log (1 + ε) e 4t -ε - t ε f 0
and, as a consequence,

σ d 2 (1-m) ∞ ≥ σ d 2 (1-m) 0 - d 2 (1 -m) 3 4 ε m K M log(1 + ε) f 0 .
This also gives an estimate of ρ := σ ∞ /σ 0 , namely

ρ ≥ 1 - d 2 (1 -m) log(1 + ε) ε 2 d (1-m) . (14)
Notice that ρ → 1 as m → 1, with m < 1. 0 (e 2t α -1) and, as t → +∞,

τ 0 (t) -τ (t) = t 0 2 D e 2s α σ 1 2α 0 -σ 1 2α (s) ds ∼ α D σ 1 2α 0 -σ 1 2α ∞ e 2t α = τ 0 (t) -τ 0 (t -t ∞ )
for some delay t ∞ > 0 such that e -2t∞ = √ ρ = σ ∞ /σ 0 . This also proves that

τ (t) ∼ τ 0 (t -t ∞ ) ∼ ρ 1 2α τ 0 (t) as t → +∞ .
It is however not so easy to reinterpret t ∞ in terms of the original solution of (1) and this is what we are going to study next. Notice that, according to Sections 3.3 and 3.4, our estimates of t ∞ converge to 0 as either m → m 1 or m → 1.

v(τ, x) ∼ µ d R(τ ) d B σ∞ µ x R(τ ) ∼ µ d R 0 (τ ) d B σ 0 µ x R 0 (τ )
.

The asymptotic profile of the solution is not affected by the delay, at least at main order. Similarly, I(τ ) = R d |x| 2 v(τ, x) dx = K M µ -2 R 2 (τ ) σ(t(τ )) ∼ K M µ -2 e 4t σ ∞ ∼ J(τ ) as τ → +∞, which is consistent with the results of Sections 1 and 2. A more careful analysis is therefore needed to observe the counterpart of the delay t ∞ or of the factor ρ in the original variables.

It is straightforward to observe that δ is nonnegative and δ = 0 occurs if and only if the initial datum is a Barenblatt profile.

Conclusion

The asymptotic regime of solutions to the fast diffusion equation is determined by the scaling properties of the equation. However, as a purely nonlinear effect, the convergence when measured in relative entropy is faster when solutions are far from the self-similar solutions than in the asymptotic regime. The second moment, when written in variables corresponding to best matching asymptotic profiles, is monotone decreasing, a new and important feature of solutions to fast diffusion equations, that is, of (1) with m < 1. By undoing the change of variables, we are able to translate this decay into a delay of the asymptotic profile of the solution with respect to the self-similar solution with same initial second moment, which can moreover be estimated using [START_REF] Bonforte | Proc. Natl. Acad. Sci[END_REF] and macroscopic quantities (second moment, relative entropy and relative Fisher information) governed by (10).

  and Λ > 4 if m > m 1 , for any d ≥ 2. The result applies in the whole range m ∈ (m c , 1). If d = 1, then Λ = 8 for any m ∈ (0, 1). Using [17, Theorem 4], we get that (

4 .

 4 Best matching Barenblatt profiles are delayed4.1. A delay in the new time scaleWith σ(0) = σ 0 and τ (0) = τ 0 (0) = 0, we deduce from(9) and R(D τ ) = e 2t

4. 2 .)- 1 )

 21 Back to the original time scale, at main order For simplicity, assume that D = 1. The change of variables R(τ ) = e 2t , R 0 (τ ) α and τ (t) ∼ τ 0 (t -t ∞ ) = α σ allow us to get thate 2t ∼ e 2t∞ R 0 (τ ) = σ 0 σ ∞ R 0 (τ ) ,thus proving that, as τ → +∞,

4. 3 . 1 2α 1 -f

 311 A delay, at lower order, on the moment Now, let us come back toI(τ ) = R d |x| 2 v(τ, x) dx = K M µ -2 R 2 (τ ) σ(t(τ )).For simplicity, we keep assuming that D = 1. Using (9),(10) and dt dτ = 1 2R dR dτ > 0, we get d dτ R 2 (τ ) σ(t) = 2 R(τ ) dR dτ (τ ) σ(t) + R(τ ) 2 σ ′ (t) dt dτ = 2 R 2 (τ ) σ(t) 1ζ f (t) σ(t) -d 2 (1-m) with t = t(τ ) and ζ := d (1-m) 2 m K M . Because f (t) = O(e -4t ) = O(τ -2α)) is integrable, we obtain that R 2 (τ ) σ(t) = σ (t(s)) σ(t(s)) -d 2 (1-m) ds .

Acknowledgments

The authors acknowledge support both by the ANR projects NoNAP, Kibord and STAB (JD) and by MIUR project "Optimal mass transportation, geometrical and functional inequalities with applications" (GT). c 2014 by the authors. This paper may be reproduced, in its entirety, for noncommercial purposes.


Julien Fade
email: julien.fade@univ-rennes1.fr

Muriel Roche

Mehdi Alouini

Computational polarization imaging from a single speckle image

We experimentally demonstrate that imaging of the degree of polarization (DOP) can be achieved from a single intensity image acquired under coherent illumination. This computational technique is based on the analysis of the speckle characteristics in a statistically homogeneous neighborhood of the pixel. Over a variety of samples

obtained experimentally, we show that a simple calibration step allows this method to quickly and simply provide correct estimated values of the DOP, with precision in agreement with theoretical predictions. It is shown that unlike linear polarimetric contrast imaging, this method remains valid on birefringent samples. c 2014 Optical Society of America OCIS codes: 120.5410, 030.4280, 120.6150, 030.6600.

Polarimetric imaging is a promising and widely investigated technique in many different fields of application such as military, medical, or robotic domains [1,2]. In particular, imaging the degree of polarization (DOP) of the light backscattered by a scene can reveal contrasts that do not appear 1 He:Ne Laser in standard intensity images [2]. Four polarimetric measurements are actually needed to determine the polarization state of light, but simplified imaging modalities have been proposed to reduce both costs and acquisition times, such as the Orthogonal States Contrast (OSC) imaging, requiring only two measurements [2]. Recently, theoretical studies [3][4][5][6][7] have shown that an alternative DOP imaging technique, using a single image acquisition under coherent active illumination and without polarizing element, could lead to estimation performances compatible with potential applications. This computational approach is based on the analysis of the speckle statistics in the image, which contain information on the DOP of the reflected light [8]. In addition to its simplicity, this technique is applicable at any wavelength and is not affected with image registration issues which are in general detrimental to the quality of polarimetric imaging devices. In this Letter, we experimentally demonstrate the applicability of such a technique over a variety of samples, and we analyze its precision in comparison to theoretical characterizations [5] and to standard methods.

The experimental setup used is sketched in Fig. 1.a. The sample is illuminated with a 5 cm diameter spot from a collimated 633 nm He:Ne laser beam of 15 mW power. A polarizer (P 1 ) ensures a vertical direction of the illumination polarization. Imaging of the sample is operated with a 12-bits Basler A312f CCD camera (782 × 582 pixels) and a 50 mm objective, whose line of sight is almost aligned with the illumination beam axis (to avoid undesirable specular reflections). The camera has been selected for its good noise properties under low illumination, when the objective aperture is shrunk to obtain at least a speckle grain per pixel in average. Four Stokes images can be recorded with appropriate configurations of a polarization analyzer (P 2 ) and a quarter-wave plate (see Fig. 1.a). The image of the DOP in each pixel of the scene can then be classically obtained from these 4 Stokes measurements [9]. In addition, the OSC image can also be derived from OSC=(I // -I ⊥ )/(I // + I ⊥ ), where I // and I ⊥ denote the first two Stokes images acquired.

OSC is often used in polarimetric imaging applications since it is equal to the DOP over a majority of materials (purely depolarizing materials) [2].

In addition to these 4 Stokes measurements, a fifth image denoted I T is recorded without any polarimetric component in front of the camera objective. As detailed in [3][4][5][6][7], the square DOP, denoted P 2 , can be retrieved very simply from this bare speckle intensity image, by computing the following estimator:

P 2 = 2 I 2 T χ / I T 2 χ -1 /κ -1,
where . χ denotes statistical averaging over a homogeneous neighborhood χ of the considered pixel. Previous theoretical work [5] has shown that statistically homogeneous neighborhoods of typically a few hundred pixels are needed for such a method to provide a satisfactory estimation precision (about 10%). In the previous equation, the parameter κ is directly linked to the speckle order L = 1/κ, which characterizes the average number of speckle grains contained in one pixel of the imaging system [5]. This computational method must therefore be calibrated to characterize the value of the parameter κ, which will be detailed in the following.

In the first experiment reported here, we have considered 8 different polarimetric scenes (see Fig. and smaller regions ensure good illumination uniformity. Lastly, it allows a statistical analysis of the estimation precision to be carried out from the 60 realizations.

As mentioned previously, a precise characterization of the parameter κ is needed before the estimator P 2 can be applied on a single intensity image. For a fixed wavelength and experiment geometry, κ is inversely proportional to the objective aperture and does not depend on the sample considered. Computing the ratio of the variance to the square mean intensity of a fully polarized speckle pattern is a simple way to determine this parameter experimentally [8]. We thus calibrated the estimator P

(Ref) without polarimetric component. Estimation results are plotted with blue diamond symbols in Fig. 2.a for the reference sample and for the 8 test samples. It can be immediately checked that except for test sample 8, the estimator P 2 provides a correct value of the square DOP, in agreement (at least within the estimated error-bar) with the results obtained with standard Stokes measurements (red squares) or OSC (green triangles). This experimentally demonstrates the applicability of the computational technique addressed here. Furthermore, this result shows that the value of κ estimated on the reference sample remains valid on other materials. Unfortunately, this might not be always the case, for instance with samples showing a non uniform reflectivity distribution or with textured materials. We suspect such a situation to occur with sample 8 for which P 2 leads to irrelevant values ( P 2 = 1.2). It has been checked that the parameter κ was equal to 0.95 on this black plastic slab, which can be interpreted as follows: since the sample is very dark, light mainly undergoes surface backscattering (and hence very low depolarization [10]) on the smooth, glossy plastic surface, making it difficult to avoid detrimental contribution from specular backscattering.

As stated previously, the precision of the single-image computational estimation method can be evaluated from the 60 homogeneous regions of 400 pixels on which the estimation was operated. In Fig. 2.a, one can see that a larger standard-deviation is of course the price to pay for a simplification of the imaging system. It must however be noticed that, even on samples of limited size (20 × 20 pixels), the computational estimation method allows materials of distinct polarimetric properties to be discriminated, with a 10% precision on P 2 . Previous theoretical results can be used to assess the variance of P 2 for various speckle models [5]. This can be checked in Fig. 2.b, where the experimental variance of P 2 is compared with theoretical predictions obtained when a so-called standard (resp. compound) speckle model is considered, i.e., when the intensity distribution for P=1 is assumed to follow a Gamma distribution (resp. a K-distribution) [8], with an appropriate shape parameter equal to L=1/κ. As can be seen from the linear fit parameters, a fair agreement between experiments and theory can be observed, especially with a compound speckle model, which seems to be more relevant in the conditions of our experiment. To check the validity of the method when the speckle order varies, a second experiment has been conducted with a green 532 nm frequency-doubled Nd:YVO laser of 10mW power. Two samples have been imaged (a metal slab and a cardboard sheet) for two distinct objective apertures (f/16 and f/4), leading to different parameters κ (see Table 2). Both measurements of κ and P 2 have been carried out on 100 homogeneous regions of 10 × 10 pixels. In this experiment, the calibration procedure has been performed on the cardboard reference sample. The results are summarized in we were thus able to obtain an almost fully-polarized backscattered speckle field (P 2 = 0.93 from Stokes measurements), with a polarization state varying between a vertical linear polarization and a circular polarization, when the BS compensator retardation angle ϕ was varied between 0 and π/2. In Fig. 3, we plotted the estimation results obtained with the computational method (blue diamonds) and with the OSC (green triangles). It can first be checked on this figure that the OSC is unable to correctly evaluate the DOP in the presence of birefringent samples while estimator P 2 remains valid for any polarization state. It can be also noted that the estimation precision does not depend on the polarization state ellipticity, in agreement with theoretical predictions plotted with a dashed error-bar for a compound speckle model.

In this Letter, a computational technique involving a single image acquisition has been tested successfully on a variety of non-textured materials to achieve DOP estimation, with precision in agreement with theory [5]. The applicability of such a technique at different wavelengths, on birefringent materials, and for distinct speckle orders has also been demonstrated. As a perspective

Fig. 1 .

 1 Fig. 1. (a) Experimental setup; (b) Example of an intensity image of the reference sample and sample 5 (green paint).

  1.b), comprising an identical reference material (beige painted cardboard slab) in the upper part

2

 2 on the reference sample by computing κ on the 4 Stokes images acquired (which are fully polarized). Very similar results were obtained on each Stokes image and along the 8 experiments, finally leading to the following estimated value κ = 0.808 ± 0.003, which corresponds to a speckle order of L = 1.24.Once characterized the parameter κ, the estimator P 2 can be applied on the image I T acquired

Fig. 2 .

 2 Fig. 2. (a) Values of the square DOP estimated on homogeneous regions of 400 pixels on the 9 samples. (b) Comparison between experimental variance of P 2 and theoretical predictions for distinct speckle models. Sample 8 leading to an erroneous estimated value of P is not represented in this figure. The error-bar on experimental variance is estimated from the 8 experiments conducted on the reference sample.

Table 1 .

 1 Description of the samples. Bracketted figures correspond to the value of P 2 estimated from Stokes measurements.

	Reference: Beige cardboard (0.61)
	1: Black paper (0.72)	2: White paper (0.04)
	3: Cream plastic (0.20)	4: Gray paint (0.60)
	5: Green paint (0.98)	6: Aluminium (0.98)
	7: Sandblasted Alum. (0.70) 8: Black plastic (0.91)

of the image, and 8 different test materials described in Table

1

in the lower part of the image. To avoid illumination inhomogeneities, regions of interest (ROIs) of 240 × 100 pixels are selected in the upper and lower parts of the images. These regions are then further partitioned in 60 sub-regions of 400 pixels each (see Fig.

1

.b). Indeed, performing the estimation on such regions of limited size appears more realistic (homogeneous regions > 10 4 pixels are unlikely to occur in real conditions),

Table 2 .

 2 Estimated values of the square DOP for metal and cardboard samples for different speckle orders.

	Bracketted number is the estimation standard-deviation.
	Sample	f/.	κ	P 2 OSC	P 2
	Metal	f/16	0.7 [0.1]	0.94 [0.01]	0.9 [0.3]
		f/4 0.12 [0.02] 0.96 [0.01]	0.8 [0.3]
	Card-	f/16 0.66 [0.07] 0.05 [0.02] 0.04 [0.1]
	board	f/4 0.14 [0.02] 0.06 [0.01] -0.03 [0.1]

Table 2 .

 2 It can be checked that the single-image computational technique remains valid for different values of the parameter κ, provided it has been carefully characterized.A last experiment is reported in this Letter to check the applicability of the single-image computational technique on non purely depolarizing materials (e.g., birefringent samples). The experimental configuration is similar to the one described in the previous paragraph, but we added a tunable Babinet-Soleil (BS) compensator within the path of the backscattered light, with slow and fast axes oriented at a 45 • angle from the illumination polarization direction. With the metallic sample,

to this work, other experiments should be conducted to validate previous theoretical predictions which showed that this computational technique should be quite robust to photon noise [6] and detector noise [7], thus enabling low-flux real-time polarimetric imaging.

The authors thank Pr. Ph. Réfrégier for fruitful discussions.


S E Pomares Hernandez

The Minimal Dependency Relation for Causal Event Ordering in Distributed Computing

Keywords: Happened-Before Relation, Event Ordering, Partial Ordering, Distributed Systems

Several algorithms of different domains in distributed systems are designed over the principle of the Happened-Before Relation (HBR). One common aspect among them is that they intend to be efficient in their implementation by identifying and ensuring the necessary and sufficient dependency constraints. In this pursuit, some previous works talk about the use of a transitive reduction of the causality. However, none of these works formally prove in a broad manner that such transitive reduction is the minimal expression of the HBR. In this paper, a formal study of the minimal binary relation (transitive reduction) of the HBR is presented, which is called the Immediate Dependency Relation (IDR).

The study shows that since the transitive closure of the HBR is antisymmetric and finite, it implies that the IDR is unique. This is important because it means that all of the works that deal with a minimal expression of the HBR discuss the same minimal binary relation. In addition, an extension to the IDR to identify causal immediate dependencies only among a subset of relevant events is presented. Finally, as case of study, the extension of the IDR is applied to the causal delivery of messages.

Introduction

The Happened-Before Relation (HBR) introduced by Lamport [START_REF] Lamport | Time, Clocks and the Ordering of Events in Distributed Systems[END_REF], denoted by "→", without using global references establishes the conditions to determine for any pair of single events a, b in a system if the event a causally occurs before the event b (denoted by a → b). Several solutions in different domains are designed over this principle. For example, the HBR was applied to ensure temporal and causal dependencies among heterogeneous data in multimedia distributed systems such as telehealth systems [START_REF] Hernandez | An Intermedia Synchronization Mechanism for Multimedia Distributed System[END_REF]. One common aspect among works based on the HBR is that most of them intend to be efficient in their implementation by identifying and ensuring the necessary and sufficient dependency constraints among events. In this pursuit, the present paper analyzes the minimal binary relation (transitive reduction) of the HBR that is called the Immediate Dependency Relation (IDR). The IDR, denoted in this paper by "↓", identifies the smallest set of causally related pair of events in a given distributed computation Ê = (E, →), such that for every causal path between a pair of events established with the HBR, there exists a causal path between those events established by the IDR. This property means in graph theory that the (E, →) and the (E, ↓) ⊂ Ê have the same reachability.

Some previous works for a particular domain deal with a transitive reduction of the HBR; nevertheless, none of these works formally prove in a general way that such transitive reduction is the minimal expression of the HBR. Some of the most important works are: [START_REF] Anceaume | A Note on the Determination of the Immediate Predecessors in a Distributed Computation[END_REF][START_REF] Helary | Minimal size of Piggybacked Information for tracking causality: a graph-based characterization[END_REF] in causality tracking for relevant events, [START_REF] Peterson | Preserving and Using Context Information in Interprocess Communication[END_REF] for context graphs, [START_REF] Prakash | An Adaptive Causal Ordering Algorithm Suited to Mobile Computing Environments[END_REF] and [START_REF] Hernandez | The Immediate Dependency Relation: an Optimal way to Ensure Causal Group Communication[END_REF] for multicast and group communication, respectively, and [START_REF] Hernandez | From the Happened-Before Relation to the Causal Ordered Set Abstraction[END_REF] for a consistent and compact representation of a distributed system. As far as I know, the first work that indirectly talked about the transitive reduction of the causality by considering only immediate causal predecessors was the work presented by Peterson in [START_REF] Peterson | Preserving and Using Context Information in Interprocess Communication[END_REF].

In this paper, an abstract and general study of the IDR with the objective of being independent of a particular domain is presented. The IDR is proven to be the transitive reduction of the HBR. In particular, it is proven that the IDR has the same transitive closure as does the HBR. Moreover, it is shown that since the transitive closure of the HBR is antisymmetric and finite, it implies that the IDR is unique. This property is important because it means that all present, past or future works that deal with or will deal with a minimal expression or a transitive reduction of the HBR discuss the same binary relation.

In addition, an extension to the IDR in order to identify causal immediate dependencies only among a subset of relevant events1 is presented. This is important since for a given distributed computation, usually only a subset of events is taken into account according to the problem to be solved. For example, for snapshot algorithms, a relevant event corresponds to the modification of a local variable involved in a global predicate; and for checkpointing algorithms, a relevant event is the definition of a local checkpoint [START_REF] Anceaume | A Note on the Determination of the Immediate Predecessors in a Distributed Computation[END_REF].

The extension of the IDR applied to the set of relevant events for the causal delivery of messages is presented as case of study. It is shown that ensuring the IDR dependencies among the set of relevant events is necessary and sufficient in order to ensure the causal delivery of messages in the system. This paper proceeds as follows. In Section 2, the system model is presented, as well as some order theory concepts and the Happened-Before Relation. In Section 3, the immediate dependency relation is presented, along with its extension for relevant events. Next, in Section 4, the IDRs minimality proof is given. Finally, in Section 5, some conclusions are presented.

Preliminaries

System Model

Processes. The system (see Figure 1) is composed of a set of processes P = {p 1 , p 2 , ..., p n }. The processes present an asynchronous execution and communicate only by message passing.

Messages. There is a finite set of messages M , where each message m ∈ M is sent considering an asynchronous reliable network that is characterized by no transmission time boundaries, no order delivery, and no loss of messages. The set of destinations of a message m is identified by Dest(m).

Events. There are two types of events under consideration: internal and external events. An internal event is a unique action that occurs at a process p in a local manner (denoted in this paper by internal(p)) and which changes only the local process state. The finite set of internal events is denoted as E i . On the other hand, while an external event is also a unique action that occurs at a process, it is seen by other processes, thus, affecting the global state of the system. The external events considered in this paper are the send and delivery events. Let m be a message. send(m) denotes the emission event, while delivery(p, m) represents the delivery event of m to participant p ∈ P . The set of events associated to M is the set

E m = {send(m) : m ∈ M } ∪ {delivery(p, m) : m ∈ M ∧ p ∈ P }.
The whole set of events in the system is the finite set E = E i ∪ E m . Each event e ∈ E is identified by a tuple e = (p, x), where p ∈ P is the producer of e, and x is the local logical clock for events of p, when e is carried out.

Order Theory Concepts

Transitive Closure. The transitive closure in our domain establishes the reachability between events. For a pair of events in the system, it is said that an event a is reachable from an event b if a causal path exists between them. The transitive closure is defined in general as follows [START_REF] Lidl | Applied abstract algebra[END_REF]. Covering Relation. In the order theory, a covering relation is a binary relation which holds between two comparable elements in a partially ordered set if they are immediate neighbors [START_REF] Aho | The Transitive Reduction of a Directed Graph[END_REF]. The covering relation is commonly used to graphically express the partial order by means of the Hasse diagram. Its definition is as follows: Definition 3. Let u and v be elements of a partially ordered set W . Then v covers u, written as u <: v, if u < v and there is no element w ∈ W such that u < w < v. Property 3. If a partially ordered set (W, R) is finite, then its covering relation R is the transitive reduction of the partial order relation R.

Only if Property 3 is accomplished, a partially ordered set (W, R) is completely described by its Hasse diagram. On the other hand, for example in a dense order, such as in the case of the rational numbers, no element covers another.

Happened-Before Relation

The Happened-Before Relation (HBR) was defined by Lamport [START_REF] Lamport | Time, Clocks and the Ordering of Events in Distributed Systems[END_REF]. It establishes logical precedence dependencies over a set of events. The HBR is a strict partial order (transitive, irreflexive and antisymmetric) defined as follows: Definition 4. The causal relation "→" is the smallest relation on a set of events E satisfying the following properties:

1. If a and b are events belonging to the same process, and a was originated before b, then a → b. The poset Ê = (E, →) constitutes the formal model adopted in this paper for a distributed computation.

The Happened-Before Relation for Relevant Events (HBR-R). Usually for a given distributed computation Ê, only a subset of events R ⊆ E is relevant. The HBR for relevant events denoted in this paper by " R →" has been defined in [START_REF] Fromentin | On-the-fly Analysis of Distributed Computation[END_REF] in the following way:

∀(a, b) ∈ R × R : (a R → b) ⇔ (a → b)
The poset R = (R, R →) ⊆ (E, →) constitutes the abstraction considered in this paper of the distributed computation for the relevant events.

Immediate Dependency Relation

The Immediate Dependency Relation (IDR) is known in order theory as a covering relation (see Definition 3). According to Property 3, if a partially ordered set is finite, its covering relation is the transitive reduction of the partial order relation. In this context, the IDR is then the covering relation of the HBR. Moreover, according to Property 2, since the poset (E, →) is finite and the HBR is a strict partial order, the IDR is unique. In this paper, the IDR is denoted by "↓", and its formal definition is as follows:

Definition 5. Two events a, b ∈ E have an immediate dependency relation "a ↓ b" if the following restriction is satisfied. a ↓ b if a → b and ∀c ∈ E, ¬(a → c → b)
Thus, an event a causal immediately precedes an event b, if and only if no other event c belonging to E exists (E is the set of events of the system), such that c belongs to the causal future of a and to the causal past of b. In Section 4 it is proved that the IDR is the transitive reduction of the HBR.

Based on the IDR, the following property is presented. This means that for every pair of events a, b ∈ E with common IDR dependencies, the events are concurrently related. This property is leveraged in [START_REF] Hernandez | From the Happened-Before Relation to the Causal Ordered Set Abstraction[END_REF] in order to achieve a compact and consistent representation of a distributed system.

Finally, it is noted that (E, ↓) ⊂ (E, →).

The Hasse diagram for the IDR of the scenario in Figure 1 is shown in Figure 2.

Immediate Dependency Relation for Relevant Events (IDR-R). As for the HBR-R, the IDR must only reflect the IDR among the relevant events that belong to R ⊆ E. For this case, the IDR is referred as IDR-R, and it is denoted by "↓ R ". It is defined over R as follows:

a ↓ R b if a R → b and ∀c ∈ R, ¬(a R → c R → b) Remark 1. (R, ↓ R ) ⊂ (R, R →) ⊆ (E, →), but (R, ↓ R ) ⊂ (E, ↓)
This means that the IDR-R is no longer a transitive reduction of the HBR. Instead, the IDR-R is the transitive reduction of the HBR-R (the proof is similar as for the IDR).

Case of Study: Message Causal Delivery

The selection of the set of relevant events must be determined according to the problem to be solved. For message causal delivery, there are two possible types of relevant events which are the send and the delivery events. It has been shown in [START_REF] Hernandez | The Immediate Dependency Relation: an Optimal way to Ensure Causal Group Communication[END_REF] and [START_REF] Prakash | An Adaptive Causal Ordering Algorithm Suited to Mobile Computing Environments[END_REF] for group and multicast communication, respectively, that in order to ensure the causal delivery of messages in the system, it suffices to ensure the causal delivery of immediately related send events. Therefore, in general, to ensure message causal delivery for group and multicast communication, the set of relevant events is determined to be R = {send(m) : m ∈ M } (see Figure 3). Formally, the message causal delivery based on the IDR-R can be defined as follows:

Theorem 1. If ∀((send(m), send(m )) ∈ R, send(m) ↓ R send(m ) ⇒ ∀p ∈ Dest(m) Dest(m ) : delivery(p, m) → delivery(p, m ) then ∀((send(m), send(m )) ∈ R, send(m) R → send(m ) ⇒ ∀p ∈ Dest(m) Dest(m ) : delivery(p, m) → delivery(p, m )
The proof relies on the fact that for any pair send(m) R → send(m ) if send(m) ↓ R send(m ) does not hold, then a message m exists such that

p 1 p 2 p 3 (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) (2,4) (3,1) (3,2)
(1,2) → send(m ). Using inductive reasoning and the fact that the event send(m ) may only have a finite number of "causes" or predecessors for the causal relation, (at least) one sequence (send(m i ), i = 0, 1, ..., h) can be found, such that m = m 0 , m = m h and send(m i ) ↓ R send(m i+1 ), for all i = 0, 1, ..., h -1. For any participant p, we have delivery(p, m i ) → delivery(p, m i+1 ), and by transitivity, the required property is obtained.

Clearly, the causal delivery of messages ensured by the IDR-R is not only a sufficient but also a necessary condition for the causal delivery of all causally related messages. From an algorithmic point of view, if the reference of some message m IDR-R related to a message m is not piggy-backed with m, the causal delivery of m with respect to m may fail. Theorem 1 shows that this information is sufficient. The Hasse diagram for the relevant events for message causal delivery of the scenario example is shown in Figure 4.

Minimality Proof of the IDR

In this section a proof to demonstrate that the IDR is the transitive reduction (minimal relation) of the HBR is given. In order to prove this, it must be demonstrated that, according to Definition 2, the IDR must have the same transitive closure as the HBR. By Property 1, which says that if the original binary relation (in this case the HBR) is transitive, then the transitive closure will be the same, one can conclude that the only property to demonstrate is that the transitive closure of the IDR is the HBR. By using the graph theory, the proof of this property is as follows.

Let Ê be a poset with strict partial order →. Then Ê can be viewed as a directed graph where the vertex set is the ground set E, and the edge set is defined by →.

Conclusions

A formal study of the minimal binary relation of the HBR (Happened-Before Relation) which is called the Immediate Dependency Relation (IDR) is presented. In this paper, it is shown that the IDR identifies the smallest set of causally related pair of events in a given distributed computation. One important aspect is that because the HBR is a strict partial ordering, it implies that the IDR is unique. In addition, the IDR-R relation to identify causal immediate dependencies only among a subset of relevant events is introduced. As case of study, the IDR-R was applied to the particular problem of causal delivery of messages. The IDR-R has shown that it suffices to ensure the causal delivery of messages with IDR-R related send events in order to ensure the causal delivery of all messages in the system.

p 1 p 2 p 3 Figure 1 :

 231 Figure 1: A Distributed Computation Scenario

Definition 1 .

 1 The transitive closure of a binary relation R on a set W is the smallest transitive relation on W that contains R. Property 1. If the original relation is transitive, the transitive closure will be that same relation; otherwise, the transitive closure will be a different relation. Transitive Reduction. The transitive reduction of a binary relation is the minimal binary relation that expresses the same behavior (in this case, distributed computation) with the smallest set of related pair of elements. Its definition is as follows [1]. Definition 2. A transitive reduction of a binary relation R on a set W is a minimal relation R on W , such that the transitive closure of R is the same as the transitive closure of R. Property 2. If the transitive closure of R is antisymmetric and finite, then R is unique.However, neither existence nor uniqueness of transitive reduction are generally guaranteed .

Figure 2 : 2 . 3 .

 223 Figure 2: Hasse Diagram for the IDR of the scenario in Figure 1 (the partial order is established from left to right)

Property 4 .

 4 For all a, b ∈ E, a = b if ∃c ∈ E such that (a ↓ c and b ↓ c) or (c ↓ a and c ↓ b) then a||b

Figure 3 :

 3 Figure 3: Relevants Events for Message Causal Delivery

Figure 4 : 3 send

 43 Figure 4: Hasse Diagram for the IDR-R of the Relevant Events Example in Figure 3

Proposition 1 .

 1 Suppose every interval of Ê has a finite height. Then → is the transitive closure of ↓. Proof. This is proven by induction on height. By definition of ↓, if a → b and the height of [a, b] is 1, then a ↓ b. Assume by induction that whenever a → b and the height of [a, b] is at most n, then (a, b) is in the transitive closure of ↓. Suppose that a → b and that the height of [a, b] is n + 1. Since every chain in [a, b] is finite, it contains an element c which is strictly larger than a and minimal with respect to this property. Therefore [a, c] = {a, c}, from which it is concluded that a ↓ c. Since the interval [c, b] is a proper subinterval of [a, b], it has a height of at most n, so by the induction assumption one can conclude that (c, b) is in the transitive closure of ↓ . Since (a, c) and (c, b) are in the transitive closure of ↓, so is (a, b). Hence, whenever a → b and the height of [a, b] is at most n + 1, then (a, b) is in the transitive closure of ↓.

Email address: spomares@inaoep.mx (S.E. Pomares Hernandez)

In general, the relevant events are also referred as observable events.


Adel Belattar
email: adel.belattar@insa-rouen.frlakhdar.taleb@insa-rouen.fr

Lakhdar Taleb

Said Taheri
email: said.taheri@edf

AXIAL CYCLIC BEHAVIOR AND FATIGUE AFTER TORSIONAL PRE-HARDENING FOR A 304L SS

HAL is

INTRODUCTION:

To determine the required parameters for the reliable and accurate prediction of fatigue life, the effect of prior straining on the subsequent cyclic behavior and fatigue, with or without mean stress and strain, must be considered. This work is a continuation of a previous research performed by Kpodekon and al. [2010]; where, fatigue tests were executed under strain or stress control on virgin and prehardened specimens using monotonic and cyclic pre-hardening. In this previous study, four configurations of the last point of the pre-hardening in the axial stress-strain loops were considered. The mean stress may be positive or negative depending on the location of these breakpoints. The obtained results show that fatigue life is strongly influenced by prehardening; it appears beneficial when the fatigue test is executed under stress control but detrimental in strain-control. It appears for both prehardening modes that the presence of mean stress in the fatigue cycles has a significant effect on fatigue life. Therefore, our objective here is to study the effect of a pre-hardening in the shear direction on the fatigue in the axial direction at zero shear stress and strain. The comparison with the axial pre-hardening will be given later in a full length paper.

PROCEDURES, RESULTS AND DISCUSSION:

The cyclic behavior is investigated through cyclic shear strain control experiments carried out in four loading sequences with increasing/decreasing amplitude on the same specimen. The first sequence is composed of (applied cycles/equivalent strain amplitude): (50/0.2%)  (40/0.3%)  (30/0.4%)  (20/0.5%)  (20/1%)  (10/1.5%). The second sequence is a progressive come back to the origin: (20/1%)  (20/0.5%)  (30/0.4%)  (40/0.3%)  (50/0.2%). The third and the fourth sequences are the same as the first and the second ones respectively. The effect of pre-hardening on fatigue is studied through fully-reversed axial strain controlled tests conducted at different amplitudes (0.22%, 0.36% and 0.5%) on specimens initially subjected to the four sequences described above. The chosen pre-hardening process has the main advantage to ensure a progressive come back to the origin which ensure zero mean stress and strain during the subsequent fatigue tests. The fatigue life of prehardened specimens is compared to the one of specimens without previous loading history.

For a given loading sequence, the shearing cyclic curve represents the evolution of Von Mises equivalent stress amplitude obtained at the steady state (or at the last cycle) versus the Von Mises equivalent strain amplitude. Cyclic curve for the second sequence (BC) was higher than that of the first sequence (AB) (Fig. 1). This observation confirms the results obtained previously (Colin and al. [2009], Taheri [1996]) performed in tensioncompression. However, we want through the application of another sequence loading to see the evolution of the cyclic behavior. The results show that the difference between the third and the fourth cyclic curves with the second one is relatively small. Therefore, it seems that the behavior of the material has a tendency to stabilize. Cyclic strain hardening obtained during sequential cycles is analyzed in terms of kinematic and isotropic hardening (see Fig. 2). The procedure adopted (Taleb, Hauet [2009]) is described in Fig. 3. The hardening obtained was mainly of the kinematic type (X); especially for the higher strain amplitudes. The isotropic component (R) presented a quasi-linear evolution versus the number of cycles; it appeared to be independent of the loading path. It should also be noted that the saturation levels of the kinematic variable for each strain amplitudes obtained during the second sequence (High-Low amplitudes) is higher than the one recorded in the first (High-Low amplitudes). The difference is about 40 Mpa for 0.2% stain amplitude.

Fatigue results test (Fig. 3) showed that prehardening significantly reduced the fatigue life of the material, which was approximately 144000 cycles in the absence of prehardening. Furthermore, It seems that the increase in the fatigue amplitude decrease the effect of pre-hardening. Life reduction rate is about 52% in case of 0.22% fatigue strain amplitude however, it is of the order of 37% and 27% in the case of 0.36% and 0.5% respectively. The pre-hardened specimens loaded at 0.22% and 0.36% fatigue strain amplitude represent a cyclical softening to feature. Secondary hardening has been observed on virgin specimens loaded at 0.5% and at 0.36%. It is also observed on pre-hardened specimen loaded at 0.5%.

Obtained fatigue life will be compared to the case where both pre-hardening and fatigue tests are carried out in the same loading direction. Such studies with microstructural investigations are in progress and will be presented in a future full length paper.

CONCLUSIONS:

After an initial sequence of preloading, the 304L SS displayed a cyclic hardening. However, as the loading was continued, the hardening tended to stabilize. The strain hardening obtained during the cycles was mainly of the kinematic type. The isotropic component presented a quasi-linear evolution. Torsional prehardening reduces significantly the fatigue life; such affect seems smaller when the strain amplitude applied during the fatigue tests is higher.

REFERENCES:

Colin, J., Fatemi, A. and Taheri, S. 2010, "Fatigue behavior of stainless steel 304l, including strain hardening, pre-straining, and mean stress effects". J. Eng. Mater. Technol. 132/021008-13. Kpodekon, C., Taleb, L., Taheri, S. 2010, "Effect of pre-deformation on the fatigue and isotropic/kinematic cyclic hardening of 304L anstenitic stainless steel". Submitted to Inter. J. Pressure Vessels and Piping. Taheri, S. 1996, Multiaxial andFatigue Design, ESIS 21. London 283-299. Taleb, L., Hauet, A. 2009, "Multiscale experimental investigations about the cyclic behavior of the 304L SS". Int. J. Plast. 25, 1359-1385.

Fig. 1 .

 1 Fig. 1. Shearing cyclic curves corresponding to each applied loading sequence