P Lavancier

Rochet

O Chernova

F Lavancier

P Rochet

Averaging of density kernel estimators

Keywords: Aggregation, Bandwidth selection, Non-parametric estimation

Introduction

Kernel estimation is an efficient and commonly used method to estimate a density from a sample of independent identically distributed random variables. It relies on the convolution of the empirical measure with a function K (the kernel), adjusted via a tuning parameter h (the bandwidth).

Several commonly used data-driven methods for bandwidth selection such as [START_REF] Silverman | Density estimation for statistics and data analysis[END_REF] or [START_REF] Sheather | A reliable data-based bandwidth selection method for kernel density estimation[END_REF] have stood the test of time although none can be recognized as objectively best. In the past decades, aggregation for kernel density estimators has been investigated as an alternative to bandwidth selection. Methods proposed in the literature include stacking ( [START_REF] Smyth | Linearly combining density estimators via stacking[END_REF]), sequential processes ( [START_REF] Catoni | The mixture approach to universal model selection[END_REF][START_REF] Yang | Mixing strategies for density estimation[END_REF]) and minimization of quadratic ( [START_REF] Ph | Linear and convex aggregation of density estimators[END_REF]) or Kullback-Leibler ([BDD + 17]) criteria. In these papers, the initial estimators are assumed nonrandom, which is generally achieved by dividing the sample to separate training and validation.

The aim of the present article is to propose a new procedure to combine several competing density kernel estimators obtained from different, possibly data-driven, bandwidths. The method, in the spirit of model averaging, aims at minimizing the integrated square error of a linear combination of the kernel estimators. In this particular context, the first order asymptotic of the error is known up to a single parameter γ equal to the integrated squared second derivative of the density. The easily tractable error is precisely what makes kernel estimation a good candidate for averaging procedures, as we discuss in Section 2. Furthermore, the estimation of γ can be made from the same data used to estimate the density so that no sample splitting is needed. The method is detailed in Section 3, where it is proved to be asymptotically as efficient as the best possible combination, referred to as the oracle. Our simulation study demonstrates that our method compares favorably to other existing procedures and confirms that sample splitting may lead to poorer results in this setting.

Some facts on kernel estimators

Let X 1 , ..., X n be a sample of independent and identically distributed real random variables with density f with respect to the Lebesgue measure. Given a kernel K : R → R and a bandwidth h > 0, the kernel estimator of f is defined

as fh (x) := (nh) -1 n i=1 K h -1 (X i -x) , x ∈ R.
Henceforth, we assume that K is a bounded, symmetric around zero, density function on R such that

K 2 := K 2 (u)du < ∞ and c K := u 2 K(u)du < ∞. (H K ) 2
Concerning f , we assume it is twice continuously differentiable on R and f , f and f are bounded and square integrable. (H f )

Under (H K ) and (H f ), [START_REF] Hall | Limit theorems for stochastic measures of the accuracy of density estimators[END_REF] showed that the Integrated Square Error (ISE) of a kernel estimator fh satisfies

ISE( fh ) := fh -f 2 = K 2 nh + γ h 4 c 2 K 4 + o p 1 nh + h 4 , (1)
where γ := f (x) 2 dx. In the typical case where (nh) -1 and h 4 balance out, meaning that h = O(n -1/5 ) and h = o(n -1/5 ), or for short h n

h i -h * i = o p (n -2/5
), then

Σ = A + γB + o p (n -4/5 ), (3)
where

A ij = 1 n K(u/h i )K(u/h j )du and B ij := 1 4 h 2 i h 2 j c 2 K , i, j = 1, ..., k.
Proof. Following [START_REF] Hall | Extent to which leastsquares cross-validation minimises integrated square error in nonparametric density estimation[END_REF], let ∆(h) = ISE( fh ) and consider the first-order ex-

pansion ∆(h i ) -∆(h * i ) = (h i -h * i )∆ ( hi )
, for some hi between h i and h * i . From Section 2 and Lemma 3.2 in [START_REF] Hall | Extent to which leastsquares cross-validation minimises integrated square error in nonparametric density estimation[END_REF], we know that ∆ ( hi

) = O p (n -3/5 ). Com- bined with the fact that ∆(h * i ) O p (n -4/5 ), the condition h i -h * i = o p (n -2/5 ) implies ISE( fhi ) -ISE( fh * i ) / ISE( fh * i ) = o p (1) for all i.
Using the arguments of Theorem 2 in [START_REF] Hall | Limit theorems for stochastic measures of the accuracy of density estimators[END_REF], we get Σ = A * + γB * + o p (n -4/5 ), where A * and B * are defined similarly as A and B with h * i in place of h i . The map (x, y) → K(u/x)K(u/y)du is continuous for x, y > 0, which implies its uniform continuity on every compact set in (0, +∞) 2 . Applying this function to the sequences (x * n , y * n ) = (n 1/5 h * i , n 1/5 h * j ), i = j, which are bounded away from zero, and (x n , y n ) = (n 1/5 h i , n 1/5 h j ), we deduce that n 4/5 (A -A * ) = o p (1).

Similarly, n 4/5 (B -B * ) = o p (1) yielding the result.

The most common bandwidth selection procedures do verify the condition

h i -h * i = o p (n -2/5
) for some deterministic h * (see [START_REF] Jones | A brief survey of bandwidth selection for density estimation[END_REF]), making the approximation (3) available. For instance, Silverman's rule of thumb approximates the deterministic bandwidth h * = c min{σ, iqr/1.34} n -1/5 where σ is the standard deviation, iqr the inter-quartile range and c is either equal to 0.9 or 1.06, based on empirical considerations, see [START_REF] Silverman | Density estimation for statistics and data analysis[END_REF]. The biased and unbiased least-square cross-validation bandwidths discussed in [START_REF] David | Biased and unbiased crossvalidation in density estimation[END_REF][START_REF] Hall | Extent to which leastsquares cross-validation minimises integrated square error in nonparametric density estimation[END_REF] and the plug-in approach of [START_REF] Sheather | A reliable data-based bandwidth selection method for kernel density estimation[END_REF] approximate the deterministic bandwidth

h * = K 2/5 (nc K γ) -1/5
. The latter achieves a rate h/h * -1 = O p (n -5/14 ) that can be improved up to O p (n -1/2 ) if γ is estimated following [START_REF] Hall | On optimal data-based bandwidth selection in kernel density estimation[END_REF].

Note finally that, as argued by several authors, a truncation argument allows to extend Proposition 2.1 to non compactly supported kernels K, see e.g. [START_REF] Hall | On the amount of noise inherent in bandwidth selection for a kernel density estimator[END_REF] or Remark 3.9 in [START_REF] Byeong | Comparison of data-driven bandwidth selectors[END_REF].

3 The average estimator

Let h = (h 1 , ..., h k ) ∈ R k
+ be a collection of (possibly data-driven) bandwidths and set f = ( fh1 , ..., fh k ) . Following [START_REF] Lavancier | A general procedure to combine estimators[END_REF], we consider an estimator of f expressed as a linear combination of the fhi 's,

fλ = λ f = k i=1 λ j fhi , (4)
where the weight vector λ = (λ 1 , ..., λ k ) is constrained to sum up to one, i.e. λ 1 = 1 for 1 = (1, ..., 1) . Under this normalizing constraint, the integrated square error of fλ has the simple expression ISE fλ = λ Σλ. If Σ is invertible (which we shall assume throughout), the optimal weight vector λ * minimizing the ISE under the constraint λ 1 = 1, is given by λ

* = 1 Σ -1 1 -1 Σ -1 1. The resulting average estimator f * = λ * f is called the oracle.
With all bandwidths h i of order n -1/5 , we know from Proposition 2.1 that

Σ = A + γB + o p (n -
C = o p (n -4/5 ). Therefore ΣΣ -1 = (A + γB)Σ -1 = I -CΣ -1 + (γ -γ)BΣ -1 and since BΣ -1 = O p (1), ||| I -ΣΣ -1 ||| ≤ |||CΣ -1 ||| + |γ -γ|O p (1). ( 6
)
The result follows from the fact that CΣ -1 = o p (1) and γ -γ = o p (1).

Remark 3.2. In our setting, the number k of initial estimators is assumed fixed although the result remains valid if k = k n increases slowly with n. As seen in the proof, the ISE of fAV approaches that of the oracle f * provided that ||| I -ΣΣ -1 ||| = o p (1). This can still be achieved if k n increases sufficiently slowly with n, e.g. logarithmically. In practice however, the numerical study shows that the results are less satisfactory with a too large number of initial estimators, due to Σ being close to singular. For better performances, we suggest to use no more than four initial estimators, obtained from different methods, in order to reduce linear dependencies (see the discussion in Section 4).

One may be interested in setting additional constraints on the weights λ i , restricting λ to a proper subset Λ ⊂ {λ : λ 1 = 1}. A typical example is to impose the λ i 's to be non-negative, a framework usually referred to as convex averaging. In fact, the same result as in Theorem 3.1 holds for any such subset Λ, using the corresponding oracle and average estimator, the proof being identical. A reason for considering additional constraints on λ is to aim for a more stable solution, which may be desirable in practice especially when working with small samples (see e.g. Table 1 in Section 4). However, since the oracle is necessarily worse (in term of integrated square error) for a proper subset Λ, the result lacks a theoretical justification for using a smaller set. Note that, on the contrary, the constraint λ 1 = 1 is necessary for the equality ISE( fλ ) = λ Σλ to hold true.

The next proposition establishes a rate of convergence in the case where the bandwidths h i used to build the experts fhi are deterministic and of the order

h i n -1/5 . The additional assumption γ -γ = o p (n -2/5
) is mild as the best known convergence for an estimator γ is γ -γ = O p (n -1/2 ), see for instance Proof. Under the assumptions, Theorem 2.1 applies with second order asymp-

totic expansion C = Σ -A -γB = O p (n -6/5
). In view of ( 5) and ( 6), the rate The result of Proposition 3.3 improves on the residual term O(n -1 ) obtained in [START_REF] Ph | Linear and convex aggregation of density estimators[END_REF] where the initial estimators, or experts, are built from a training sample of size n tr , while the aggregation is performed on an independent validation sample of size n va with n = n tr + n va . In fact, [START_REF] Ph | Linear and convex aggregation of density estimators[END_REF] show that conditionally to the training sample (making the experts built once and for all), their aggregation procedure reaches the minimax rate O(n -1 va ), which is at best of the order O(n -1 ). In our setting, the rate of the residual term is improved due to the initial kernel estimators contributing a factor O p (n -4/5 ).

Simulations

Based on a sample of n independent and identically distributed observations, we consider the estimation of the following density functions, depicted in Figure 4:

the standard normal distribution N (0, 1); the Gamma distribution with shape parameter 2 and scale parameter 1; the Cauchy distribution; the equiprobable mixture of N (-1.5, 1) and N (1.5, 1); and the mixture of N (-1.5, 1) with probability 0.7 and N (1.5, 1) with probability 0.3.

The initial kernel estimators are built with Gaussian kernel and data-driven bandwidths nrd0 (Silverman's rule of thumb), nrd (its variation with normaliz- This was confirmed by simulations (not displayed here), where the inclusion of these estimators did not improve the performances described below. The three kernel estimators are then combined by our method where γ is estimated as in [START_REF] Hall | On optimal data-based bandwidth selection in kernel density estimation[END_REF]. We also assess convex averaging where, in addition, the weights λ i are restricted to non-negative values. For the sake of comparison with existing techniques, we implement the linear and convex aggregation methods considered in [START_REF] Ph | Linear and convex aggregation of density estimators[END_REF] who also use a quadratic loss function. In their setting, the experts fhi are computed from a training sample of half size, independent from the remaining validation sample on which the weights λ i are estimated.

In the same spirit, we have also tested this splitting scheme for our average estimator, where γ is estimated from the validation sample. For robustness, it is advised in [START_REF] Ph | Linear and convex aggregation of density estimators[END_REF] to average different aggregation estimators obtained over multiple sample splittings. We followed this recommendation and we considered 10 independent splittings into two samples of equal size. The mean integrated square errors of the aforementioned estimators are summarized in Table 1, depending on the sample size n. These errors are approximated by the average over 10 3 replications of the integrated square errors.

It shows that our averaging procedure (AV in the table) outperforms every single initial kernel estimators when the sample size is large (n ≥ 500) and the gain becomes significant when n ≥ 1000. On the contrary, our averaging procedure is inefficient for small sample sizes (n = 50), which is probably explained by a poor use of the asymptotic expansion of Σ in this case. In fact, the convex averaging procedure (AVconv in the table) seems preferable for small n although it also fails to achieve the same efficiency as the best estimator in the initial collection.

A transition seems to occur for moderate sample sizes around n = 100, where the results of the average estimator are comparable to the best kernel estimator.

In all cases, our averaging procedure outperforms the alternative aggregation method of [START_REF] Ph | Linear and convex aggregation of density estimators[END_REF]. Finally, according to the numerical results, a splitting scheme for our method (AVsplit in the table) is not to be recommended, suggesting that all the available data should be used both for the initial estimators and for γ, which is in line with our theoretical findings.

  4/5 ). Because both A and B are known, approximating Σ is reduced to estimating γ = f (x) 2 dx. This problem has been tackled in the literature, see for instance[START_REF] Hall | Estimation of integrated squared density derivatives[END_REF][START_REF] Hall | On optimal data-based bandwidth selection in kernel density estimation[END_REF][START_REF] Sheather | A reliable data-based bandwidth selection method for kernel density estimation[END_REF]. Hence, given an estimator γ of γ, one obtains an approximation of Σ by Σ = A + γB. Replacing Σ by its approximation Σ yields the average density estimator fAV = fλ =1 Σ -1 1 -1 1 Σ -1 f .Theorem 3.1. Under the assumptions of Proposition 2.1, if Σ and Σ are invertible and γ -γ = o p (1), then ISE fAV = ISE( f * ) 1 + o p (1) . Proof. Write ISE( fAV ) = λ Σ λ = λ Σ λ + λ Σ -Σ λ. By construction, λ Σ λ ≤ λ * Σλ * = λ * Σλ * + λ * Σ -Σ λ * . Moreover, denoting by |||.||| the operator norm, |||A||| = sup ||x||=1 ||Ax||, we have for all λ ∈ R k , |λ Σ -Σ λ| ≤ ||| I -ΣΣ -1 ||| λ Σλ, see the proof of Lemma A.1 in [LR16]. Applying the above inequality to λ and λ * , we get 1 -||| I -ΣΣ -1 ||| ISE( fAV ) ≤ 1 + ||| I -ΣΣ -1 ||| ISE( f * ) (5) where we recall ISE( f * ) = λ * Σλ * . It remains to show ||| I -ΣΣ -1 ||| = o p (1). By Proposition 2.1, Σ = A + γB + C with A = O p (n -4/5 ), B = O p (n -4/5 ) and

[

  HSJM91].Proposition 3.3. Assume (H K ) and (H f ). If the bandwidths h i are deter-ministic with h i n -1/5 , Σ and Σ are invertible and γ -γ = o p (n -2/5 ), ISE( fAV ) = ISE( f * ) + O p (n -6/5 ).

  of convergence for ISE( fAV ) -ISE( f * ) follows from investigating |||CΣ -1 ||| and |γ -γ|. Here, |||CΣ -1 ||| = O p (n -2/5 ) while |γ -γ| is negligible in comparison by assumption.

Figure 1 :

 1 Figure 1: Densities functions considered in the numerical examples

Table 1 :

 1 Estimated MISE (based on 10 3 replications) of the kernel estimators with bandwidths

	n	Law	nrd	nrd0 SJ	. AV	AVsplit RT	AVconv RTconv
	50	Norm	1936 1737 1902 . 1788 1698	2480 1844	2030
		Gamma 1822 1903 1864 . 1897 2088	2685 1841	2173
		Cauchy 1214 1289 1233 . 1292 1493	1778 1218	1452
		Mix05	999	1043 1056 . 1239 1397	1393 1063	1164
		Mix03	1086 1145 1155 . 1238 1401	1582 1159	1267
	100	Norm	1057 957	1026 . 962	947	1325 998	1113
		Gamma 1129 1239 1146 . 1153 1305	1615 1135	1376
		Cauchy 748	848	756	. 775	906	1059 750	938
		Mix05	634	699	677	. 796	922	888	705	772
		Mix03	669	751	703	. 728	852	898	717	823
	200	Norm	628	578	616	. 568	556	767	597	660
		Gamma 701	795	705	. 701	772	944	696	839
		Cauchy 462	540	454	. 452	531	613	454	585
		Mix05	391	453	409	. 474	559	501	452	492
		Mix03	398	470	406	. 395	467	486	412	510
	500	Norm	310	286	299	. 276	274	346	292	321
		Gamma 388	462	370	. 367	411	459	369	459
		Cauchy 232	283	220	. 208	240	294	221	293
		Mix05	210	253	209	. 223	258	227	240	264
		Mix03	223	271	218	. 199	222	234	222	282
	1000 Norm	183	171	177	. 163	163	196	174	191
		Gamma 231	285	216	. 211	240	262	215	272
		Cauchy 145	182	133	. 121	138	178	134	181
		Mix05	126	158	120	. 120	138	120	134	159
		Mix03	132	165	125	. 108	117	124	127	164
	2000 Norm	111	104	106	. 99	98	113	105	114
		Gamma 146	183	132	. 130	147	160	132	167
		Cauchy 84	108	76	. 66	73	110	76	101
		Mix05	79	100	73	. 68	74	68	79	95
		Mix03	77	98	72	. 59	61	66	72	92

nrd, nrd0 or SJ (by default in R) and the combinations of these estimators by our method (AV), our method with sample splitting (AVsplit), the linear method in

[START_REF] Ph | Linear and convex aggregation of density estimators[END_REF]

(RT), our convex method (AVconv) and the convex method in [RT07] (RTconv).


Mohamad Houda

Paul Lescot
email: paul.lescot@univ-rouen.fr.

ON SOME BERNSTEIN PROCESSES SIMILAR TO COX-INGERSOLL-ROSS ONES

Keywords: Bernstein processes, Bessel processes, interest rate models, heat equation

We present some results on Bernstein processes, which are Brownian diffusions that appear in Euclidean Quantum Mechanics. We express the distributions of these processes with the help of those of Bessel processes. We then determine two solutions of the dual equation of the heat equation with potential.

These results first appeared in the first author's PhD thesis (Rouen, 2013).

Introduction

Bernstein processes or reciprocal diffusions combine two dynamic behaviors : forward and backward.These processes are useful tools of stochastic quantum mechanics, and also find applications in various other fields.

We give in this paper an explicit expression for the distribution density of a special Bernstein process. This process is similar to the Cox-Ingersoll-Ross process of financial mathematics. Indeed, in this work, we show that each one-factor affine interest rate model (in the sense of Leblanc-Scaillet [START_REF] Leblanc | Path dependent options on yields in the affine term structure model[END_REF]) can be described using such a Bernstein process. See also [START_REF] Lescot | Solving stochastic differential equations with Cartans exterior differential systems[END_REF].

The groundbreaking idea of replacing the complex Schrödinger equation by forward and backward heat equations in duality goes back to Schrödinger ( [START_REF]Schrödinger Über die Umkerhrung der Naturgesetze[END_REF], pp. 144-153).

Two solutions of the dual equation have been computed by Lescot, Quintard and Zambrini [START_REF] Lescot | Solving stochastic differential equations with Cartans exterior differential systems[END_REF] by using the Gaussian character of Ornstein-Uhlenbeck process and Brownian motion. In this paper, this goal is reached by using another class of diffusion which is that of Bessel processes. Indeed, in our case, the Bernstein process defined by a square root of a Cox-Ingersoll-Ross process (CIR) can be represented in terms of a Bessel process ( [START_REF] Göing-Jaeschke | A survey and some generalizations of Bessel processes[END_REF], p.314 (2)), and the density of this Bessel process is well known [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], p.441 (1.4)). So, by using the link between the two families of diffusions, we are able to determine the density of the law of this Bernstein process.

Let us first give some definitions and recall some preliminary results. and, for each given t > 0, the law of z(t) is η(t, q)η * (t, q)dq. The function η is assumed to be an everywhere positive solution to the backward heat equation with potential V :

θ 2 ∂η ∂t = - θ 4 2 ∂ 2 η ∂q 2 + V η (C (V ) 1 ) .
Similarly, η * is assumed to be everywhere positive and a solution to

-θ 2 ∂η * ∂t = - θ 4 2 ∂ 2 η * ∂q 2 + V η * (C (V )
2 ) .

Definition 1.2.

[1],p.454 (2.1) The Bessel function J λ with index λ ∈ C is defined by

J λ (z) = ( z 2 ) λ ∞ n=0 (-z 2 ) n 2 2n n! Γ(n + λ + 1)
.

This function satisfies the Bessel equation with parameter λ ∈ C

z 2 ω + z ω + (z 2 -λ 2 ) ω = 0 . (1.1)
Definition 1.3. The Bessel modified function I ν with index ν ∈ R is defined by

I ν (z) = i -ν J ν (iz) .
This function satisfies the linear differential equation of the second order

z 2 Ïν (z) + z İν (z) -(z 2 + ν 2 )I ν (z) = 0 , thatis (1.2) Indeed, we have I ν (z) = i -ν J ν (iz) , İν (z) = i -ν i Jν (iz) et Ïν (z) = i -ν i 2 Jν (iz) . Equation (1.1) gives (iz) 2 i ν i -2 Ïν (z) + iz i ν i -1 İν (z) + (-z 2 -ν 2 ) i ν I ν (z) = 0 z 2 Ïν (z) + z İν (z) -(z 2 + ν 2 )I ν (z) = 0 , which yields (1.2).
We refer to [START_REF] Dieudonné | Calcul infinitésimal[END_REF] for details on Bessel functions.

Definition 1.4. For all δ ≥ 0 and x 0 ≥ 0, the unique solution of the stochastic differential equation

Y t = x 0 + δt + 2 t 0 |Y s |dw(s)
(starting from x 0 ) is called squared Bessel process of dimension δ. This process will be denoted by BESQ δ x0 . We refer to [START_REF] Göing-Jaeschke | A survey and some generalizations of Bessel processes[END_REF] and [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] for a survey on Bessel processes.

Definition 1.5. The Cox-Ingersoll-Ross (CIR) family of diffusions, also known as square root diffusions, solve

dX t = (a + bX t )dt + c X t dw t ,
where X 0 = x 0 , a ≥ 0, b ∈ R, c > 0, and (w t ) t≥0 is a standard Brownian motion.

Proposition 1.6. [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], p.441 (1.4)) If x 0 = 0, then the density q δ t of the law of Y t is given by

q δ t (0, y) = (2t) -δ 2 Γ( δ 2 ) -1 y δ 2 -1 exp(- y 2t ) , t > 0 . (1.3)
If x 0 > 0, then the density q δ t of the law of Y t is given by

q δ t (x 0 , y) = 1 2t ( y x 0 ) ν 2 exp(- x 0 + y 2t ) I ν ( √ x 0 y t ) , t > 0, (1.4)
where ν: = δ 2 -1.

Affine interest rate models

Let (Ω, F, (F t ) t≥0 , Q) be a filtered probability space, and let (w(t) t≥0 ) be an (F t ) t≥0 standard Brownian motion.

A one-factor affine interest rate model is characterized by the instantaneous rate r(t), satisfying the stochastic differential equation

dr(t) = αr(t) + β dw(t) + (φ -λr(t)) dt under the risk-neutral probability Q [3], p.351).
Assuming α = 0, let us set, with the notations of Lescot-Quintard-Zambrini [START_REF] Lescot | Solving stochastic differential equations with Cartans exterior differential systems[END_REF],

φ = φ + λβ α , δ = 4 φ α , A = α 4 128 (δ -1)(δ -3) , and
B = λ 2 8 .
The following result of Lescot-Quintard-Zambrini [START_REF] Lescot | Solving stochastic differential equations with Cartans exterior differential systems[END_REF] shows that the diffusion term of a one-factor affine interest rate model is a Bernstein process. So this means that we are able to describe the law of the interest rate process by using this Bernstein process.

Proposition 2.1. (Lescot-Quintard-Zambrini [START_REF] Lescot | Solving stochastic differential equations with Cartans exterior differential systems[END_REF]) Let X t be defined by

X t = αr(t) + β .
The process X is then Cox-Ingersoll-Ross diffusion in the sense of Definition 1.5. Define Z(t) by

Z(t) = αr(t) + β .
Then Z is a Bernstein process with θ = α 2 and the potential

V (t, q) = A q 2 + Bq 2 . Lemma 2.2. [2] p.314 (2) Let X 0 = x 0 , then X(t) = e -λt Y (s) ,
where s = α 2 (e λt -1)

4λ
and Y is BESQ δ x0 .

Our main result is the following : we determine two solutions of the dual equation (C

(V )
2 ), when the potential V is written as

V (t, q) = A q 2 + Bq 2 .
Firstly we compute the density ρ t (q) of the law of Z t . Here, two cases arise : x 0 = 0 or x 0 > 0.

Proposition 2.3. If x 0 = 0, then the density ρ t (q) of the law of Z t is written as

ρ t (q) = α -δ 2 δ 2 +1 λ δ 2 (e λt -1) -δ 2 Γ( δ 2
) -1 q δ-1 exp( λδ 2 t) exp( -2λe λt q 2 α 2 (e λt -1) )1 {q>0} .

Proof. For fixed t and for all g bounded continuous,

E (g(X t )) = E [g (exp(-λt)Y (s))] = (1.3) ∞ 0 g(exp(-λt)y)(2s) -δ 2 Γ( δ 2 ) -1 y δ 2 -1 exp(- y 2s )dy = (2s) -δ 2 Γ( δ 2 ) -1 ∞ 0 g(exp(-λt)y)y δ 2 -1 exp(- y 2s )dy .
By the change of variable x = exp(-λt)y, we get

E (g(X t )) = (2s) -δ 2 Γ( δ 2 ) -1 ∞ 0 g(x)[x exp(λt)] δ 2 -1 exp(- e λt x 2s ) exp(λt)dx = ∞ 0 g(x)(2s) -δ 2 Γ( δ 2 ) -1 x δ 2 -1 exp( λδ 2 t) exp(- e λt x 2s )dx.
Then the density Xt of the law of X t is given by

Xt (x) = α -δ (2λ) δ 2 (e λt -1) -δ 2 Γ( δ 2 ) -1 x δ 2 -1 exp( λδ 2 t) exp( -2λe λt x α 2 (e λt -1) )1 {x>0} .
For fixed t and for all bounded continuous ϕ, we have

E[ϕ(Z t )] = E[ϕ( X t )] = ∞ 0 ϕ( √ x) Xt (x)dx .
By the change of variable q = √ x , we get

E[ϕ(Z t )] = ∞ 0 ϕ(q)α -δ (2λ) δ 2 (e λt -1) -δ 2 Γ( δ 2 ) -1 q δ-2 exp( λδ 2 t) exp( -2λe λt q 2 α 2 (e λt -1) ) 2qdq = ∞ 0 ϕ(q)α -δ 2 δ 2 +1 λ δ 2 (e λt -1) -δ 2 Γ( δ 2 ) -1 q δ-1 exp( λδ 2 t) exp( -2λe λt q 2 α 2 (e λt -1) ) dq .
This proves the result.

Let η(t, q) be defined by

η(t, q) = exp( λδt 4 - λq 2 α 2 ) q δ-1 2 .
The function η is a solution of

θ 2 ∂η ∂t = - θ 4 2 ∂ 2 η ∂q 2 + V η (C (V ) 1 ) , with V (t, q) = A q 2 + Bq 2 .
Proposition 2.4. The function η * (t, q) defined by

η * (t, q) = ρ t (q) η(t, q) = 2 δ 2 +1 α δ λ δ 2 Γ( δ 2 ) (e λt -1) -δ 2 q δ-1 2 exp( λδt 4 - λq 2 α 2 tanh( λt 2 )
) ,

satisfies the dual equation

-θ 2 ∂η * ∂t = - θ 4 2 ∂ 2 η * ∂q 2 + V η * (C (V ) 2 ) . Proof. Let C = α -δ 2 δ 2 +1 λ δ 2 Γ( δ 2 ) -1
and

C 1 = C(e λt -1) -δ 2 . We have ∂η * ∂t = - δ 2 (λe λt ) + λδ 4 - λq 2 α 2 (- 2λe λt (e λt -1) 2 ) C(e λt -1) -δ 2 q δ-1 2 exp( λδt 4 - λq 2 α 2 tanh( λt 2 ) ) , ∂η * ∂q = ( δ -1 2 ) q δ-3 2 + q δ-1 2 (- 2λq α 2 tanh( λt 2 ) ) C 1 exp( λδt 4 - λq 2 α 2 tanh( λt 2 ) ) , ∂ 2 η * ∂q 2 = ( δ -1 2 )( δ -3 2 ) q δ-5 2 + ( δ -1 2 ) q δ-3 2 (- 2λq α 2 tanh( λt 2 ) ) + q δ-1 2 (- 2λ α 2 tanh( λt 2 ) ) + ( δ -1 2 ) q δ-3 2 (- 2λq α 2 tanh( λt 2 ) ) + q δ-1 2 (- 2λq α 2 tanh( λt 2
)

) 2 C 1 exp( λδt 4 - λq 2 α 2 tanh( λt 2 )
) , and

V (t, q) η * (t, q) = ( A q 2 + Bq 2 ) η * (t, q) = ( α 4 2 7 (δ -1)(δ -3) q -2 + λ 2 2 3 q 2 ) η * (t, q) = α -δ+4 2 δ 2 -6 λ δ 2 (δ -1)(δ -3)q δ-5 2 + α -δ 2 δ 2 -2 λ δ 2 +2 q δ+3 2 Γ( δ 2 )
-1

(e λt -1) -δ 2 × exp( λδt 4 - λq 2 α 2 tanh( λt 2 )
) .

It is then straightforward to see that, for all (t, q

) ∈ R * + × R * + , -θ 2 ∂η * ∂t = - θ 4 2 ∂ 2 η * ∂q 2 + V η * (C (V )
2 ) .

Proposition 2.5. If x 0 > 0, then the density ρ t (q) of the law Z t is given by

ρ t (q) = 4λ α 2 z ν 0 e λt( δ 4 + 1 2 )
(e λt -1) I ν ( 4λz 0 e λt 2 q α 2 (e λt -1)

) q δ 2 exp(-2λ α 2 ( z 2 0 + e λt q 2 (e λt -1)

))1 {q>0} .

Proof. We have

X t = e -λt Y (s) , s = α 2 4λ (e λt -1)
and

Z t = √ X t Z 0 = √ x 0 .
For fixed t, for all g bounded continuous, using (1.4), we get

E(g(Z t )) = E g( e -λt Y (s)) = ∞ 0 g( e -λt y) 1 2s ( y x 0 ) ν 2 exp(- x 0 + y 2s ) I ν ( √ x 0 y s )dy .
By the change of variable z = e -λt y, we obtain

E(g(Z t )) = ∞ 0 g(z) 1 2s ( e λt z 2 z 2 0 ) ν 2 exp(- z 2 0 + e λt z 2 2s ) I ν ( z 0 ze λt 2 s ) 2ze λt dz .
We then find the density ρ t (q) of the law de Z t by replacing s and ν by their values.

Proposition 2.6. The function η * (t, q) defined by

η * (t, q) = ρ t (q) η(t, q) = 4λ α 2 z ν 0 e λt 2
(e λt -1) I ν ( 4λz 0 e λt 2 q α 2 (e λt -1)

)

q 1 2 exp( λq 2 α 2 - 2λ α 2 ( z 2 0 + e λt q 2 (e λt -1) ))
satisfies the dual equation (C

(V )
2 ).

Proof. We have

∂η * ∂t = (- 2λ 2 α 2 z ν 0 e λt 2 (e λt + 1) (e λt -1) 2 q 1 2 + 8λ 3 α 4 z ν 0 e 3λt 2 (e λt -1) 3 q 5 2 + 8λ 3 α 4 z ν 0 e 3λt 2 (e λt -1) 3 z 2 0 q 1 2 )I ν ( 4λz 0 e λt 2 q α 2 (e λt -1) ) - 8z 0 λ 3 α 4 z ν 0 e λt e λt + 1 (e λt -1) 3 q 3 2 I ν ( 4λz 0 e λt 2 q α 2 (e λt -1) ) exp λq 2 α 2 - 2λ α 2 z 2 0 + e λt q 2 e λt -1 , ∂η * ∂q = ( 1 2 q -1 2 - 2λ α 2
(e λt + 1) (e λt -1) q

2 ) I ν ( 4λz 0 e λt 2 q α 2 (e λt -1) ) + 4λz 0 α 2 e λt 3
(e λt -1) q

1 2 I ν ( 4λz 0 e λt 2 q α 2 (e λt -1) ) × 4λ α 2 z ν 0 e λt 2 (e λt -1) exp λq 2 α 2 - 2λ α 2 z 2 0 + e λt q 2 e λt -1 , ∂ 2 η * ∂q 2 = (- 1 4 q -3 2 - 4λ α 2 (e λt + 1) (e λt -1) q 1 2 + 4λ 2 α 4
(e λt + 1) 2 (e λt -1) 2 q z 2 0 + e λt q 2 e λt -1 , and V (t, q) η * (t, q) = ( A q 2 + Bq 2 ) η * (t, q) = ( α 4 2 7 (δ -1)(δ -3) q -2 + λ 2 2 3 q 2 ) η * (t, q) = λ α 2 2 5 (δ -1)(δ -3) q -3 2 + λ 3 2α 2 q 5 2 e λt 2 z ν 0 (e λt -1)

I ν ( 4λz 0 e λt 2 q α 2 (e λt -1) )

× exp( λq 2 α 2 -2λ α 2 ( z 2 0 + e λt q 2 (e λt -1) )) .

Taking into account Equation (1.2) for the Bessel function I ν , it is then straightforward to see that for all (t, q) ∈ R *

+ × R * + , -θ 2 ∂η * ∂t = - θ 4 2 ∂ 2 η * ∂q 2 + V η * (C (V )
2 ) .

  Definition 1.1. Let θ > 0 denote a parameter. A Bernstein process z is a process satisfying a stochastic differential equation of the form

	dz(t) = θdw(t) + B(t, z(t))dt
	relatively to the canonical filtration of the Brownian motion w and the dual stochas-
	tic differential equation
	d * z(t) = θdw * (t) + B * (t, z(t))dt
	relatively to the canonical decreasing filtration of another Brownian motion w *
	which is independent of w, where
		∂η
	B: ≡ θ 2	∂q η	,
			∂η *
	B * : ≡ -θ 2	∂q η *	,


ON AN ARITHMETICAL QUESTION RELATED TO PERFECT NUMBERS

Introduction and statement of the problem

For n ≥ 1 an integer, we shall denote by σ(n) the sum of the (positive) divisors of n.

The line of thought leading to this note started with [3, Exemple III, p. 380], where two integers k such that σ(k 3 ) is a perfect square are given.

This suggests that we might look for numbers n such that σ(n) be a square. One rapidly notices that σ(66) = 144 = 12 2 ; furthermore σ(66) -2.66 = 144 -132 = 12, whence σ(66) = (σ(66) -2.66) [START_REF] Itard | Arithmétique et Théorie des Nombres[END_REF] .

We were thus led to the following definition :

Definition. An integer n is termed quadratically perfect if

σ(n) = (σ(n) -2n) 2 .
As we have just established, 66 is such a number. A quick search yields three others : 1, 3 and 491536.

As a matter of fact, one has Lemma 1. The only quadratically perfect primary integers are 1 and 3.

Proof. Let n = p k (p prime) be quadratically perfect ; if k = 0, n = 1. Let us then assume k ≥ 1 ; we have

σ(n) = 1 + p + ... + p k , whence, setting S := k-1 l=0 p l , we find σ(n) -2n = S -p k and (S -p k ) 2 = (σ(n) -2n) 2 = σ(n) = S + p k . ( * )
If k ≥ 2, ( * ) gives us, by working modulo p 2 , that S 2 ≡ S (mod p 2 ); as S ≡ 1 + p (mod p 2 ), it follows that

(1 + p) 2 ≡ 1 + p (mod p 2 ),
i.e. p 2 divides p, a contradiction. Therefore k = 1, S = 1 and ( * ) becomes Proof. Seeing that m and 2m -1 are coprime, one has

(1 -p) 2 = 1 + p, that is p 2 = 3p, p = 3 and n = p k = 3 1 = 3.
σ(n) = σ(m(2m -1)) = σ(m)σ(2m -1) = 2m(1 + (2m -1)) [as m is perfect and 2m -1 is prime] = 4m 2 . Therefore σ(n) -2n = 4m 2 -2m(2m -1) = 2m, and
(σ(n) -2n) 2 = (2m) 2 = 4m 2 = σ(n) ; n is quadratically perfect.
A partial converse holds : Theorem 1. Let n be an even quadratically perfect number ; then there exists an even perfect number m such that n = m(2m -1).

Remark. It is well-known (see e.g. [1,Theorem 277], or [2,pp.33-34]) that m can then be written as m = 2 p-1 (2 p -1) with 2 p -1 (hence also p) prime ; m = 6 corresponds to p = 2, and m = 496 to p = 5.

The proof of Theorem 1

For convenience, this will be broken up into four lemmas. Let us write n = 2 a x (a ≥ 1, x odd) ; then

σ(n) = σ(2 a )σ(x) = (2 a+1 -1)σ(x).
Lemma 2. There are integers b, e and f such that σ(x) = be 2 and bce 2 -(bc 2 + 1)f -e = 0.

Proof. Let us define b as the square-free part of 2 a+1 -1 ; then

2 a+1 -1 = bc 2 , for some c ∈ N, c ≥ 1. As 2 a+1 -1 ≡ -1 (mod 4)), 2 a+1 -1 is not a square, therefore b ≥ 2 ; in fact, b ≥ 3 as b is odd. As (σ(n) -2n) 2 = σ(n), we have bc 2 σ(x) = (2 a+1 -1)σ(x) = σ(n) = (σ(n) -2n) 2 whence c divides σ(n) -2n : σ(n) -2n = cd, with ∈ {-1, 1} and d ≥ 1. It follows that bc 2 σ(x) = c 2 d 2 , i.e. bσ(x) = d 2 ,
≥ bc 2 x + bc 2 f ≥ bc 2 x + bcf = bc 2 x + x, thus = 1. We now get b 2 c 2 e 2 -(bc 2 + 1)bcf = cbe, that is bce 2 -(bc 2 + 1)f -e = 0.
Lemma 3. c = 1, i.e. 2 a+1 -1 is square-free.

Proof. We know that b > 1; then be 2 = σ(x) = σ(bcf ) ≥ bcf + cf, and, using Lemma 3.1 :

0 = bce 2 -(bc 2 + 1)f -e ≥ c(bcf + cf ) -(bc 2 + 1)f -e = (c 2 -1)f -e.
Assuming that c > 1, it would follow that f ≤ e c 2 -1 , whence that bce 2 = (bc 2 + 1)f + e

≤ (bc 2 + 1) e c 2 -1 + e = e c 2 -1 (bc 2 + c 2 ) = e(b + 1) c 2 c 2 -1 .
It would now appear that

e ≤ b + 1 b c c 2 -1 ≤ 4 3 3 8 = 1 2 < 1, a contradiction. Therefore c = 1.
Lemma 4. b is prime.

Proof. From Lemmas 3.1 and 3.2 follows that be 2 -e -(b + 1)f = 0.

As x = bcf = bf , one has σ(bf ) = σ(x) = be 2 .
We know that b ≥ 3 ; let us assume that b is not prime, and let p denote the smallest prime factor of b. Then We conclude that b does not divide f .

σ(b) ≥ b + p + 1 ≥ b + 4 and f (b + 4) ≤ f σ(b) ≤ σ(bf ) = be

Conclusion

We may now write and f = g(b(b + 1)g -1). In particular, g divides f , and g = f . As

(b + 1)σ(f ) = bf + (b + 1)g + f, we obtain σ(f ) = f + g ; therefore g = 1 and f is prime. Furthermore, f = b(b + 1) -1.

Final comments

It is reasonable to expect the converse of Proposition 2.1 to still hold for all odd m ≥ 5 ; combined with the proof of the long-standing conjecture that there is no odd perfect number, this would imply the nonexistence of odd quadratically perfect numbers other than 1 and 3.

I wish to express my gratitude to Dr. Numa Lescot for checking with the help of a computer that there is at least no such number between 5 and 10 6 .

2 .

 2 The main theorem On the positive side, one has Proposition 1. If m is a perfect number such that 2m -1 is prime, then n := m(2m -1) is quadratically perfect. Remark. m = 6 (2m -1 = 11) yields n = 66, and m = 496 (2m -1 = 991) yields n = 491536.

and b divides d 2 ;

 2 b being square-free, b divides d : d = be(e ≥ 1). Therefore bσ(x) = b 2 e 2 , i.e. σ(x) = be 2 . It now appears that b 2 c 2 e 2 -(bc 2 + 1)x = bc 2 .be 2 -2 a+1 x = (2 a+1 -1)σ(x) -2n = σ(n) -2n = cd = cbe; therefore, bc divides x. Let us then set x = bcf (f ≥ 1); as b > 1, bc > 1, be 2 = σ(x) ≥ x + f and b 2 c 2 e 2 = bc 2 σ(x)

, whence f b ≤ be 2 -

 2 2 , whence f (b + 4) ≤ be 2 = e + (b + 1)f, and 3f ≤ e. It follows that 3be 2 = 3e + (3b + 3)f ≤ 3e + (b + 1)e = e(b + 4), hence 3b ≤ 3be ≤ b + 4, 2b ≤ 4 and b ≤ 2, a contradiction. Therefore b is prime. Lemma 5. b does not divide f . Proof. Let us assume for a moment that b divides f ; then, as be 2 = e + bf + f, b divides e. It follows that be 2 = σ(bf ) ≥ bf + f + f b bf -f = e, thus f ≤ be. Now be 2 = e + bf + f ≤ e + b 2 e + be and be ≤ b 2 + b + 1 = b(b + 1) + 1 < b(b + 2). It appears that e < b + 2, hence e ≤ b + 1 < 2b ; as b divides e, e = b and (b + 1)f = be 2 -e = b 3 -b, thus f = b(b -1). Now we have σ(b 2 (b -1)) = σ(bf ) = be 2 = b 3 = b 2 (b -1) + b 2 , an obvious contradiction as σ(b 2 (b -1)) ≥ b 2 (b -1) + b 2 + b + 1.

(b + 1 2 =

 12 )σ(f ) = σ(b)σ(f ) [by Lemma 3.3] = σ(bf ) [as b and f are coprime, by Lemmas 3.3 and 3.4] = be bf + e + f ; thus b + 1 divides e : e = (b + 1)g, hence (b + 1)f = e(be -1) = (b + 1)g(b(b + 1)g -1)

Now let m := 2 a

 2 (2 a+1 -1) = 2 a b ; then m is perfect([1,Theorem 276], or [2,p.33]) and even, 2m -1 = 2 a+1 b -1 = (b + 1)b -1 = f is prime, and m(2m -1) = 2 a bf = 2 a x = n.

Acknowledgement

I wish to thank the (anonymous) referee, whose comments helped me improve the style and readability of the article.


Xiwen Dai
email: xiwen.dai@sjtu.edu.cn.

Yves Aurégan
email: yves.auregan@univ-lemans.fr.

Hydrodynamic instability and sound amplification over a perforated plate backed by a cavity

The long-wavelength hydrodynamic behaviour over a cavity-backed perforated plate, in a duct with a mean shear flow, is studied numerically using the multimodal method, where the acoustic and hydrodynamic disturbances are calculated from the linearized Euler equations. The flow-acoustic coupling near the perforated plate is first solved hole by hole, and results indicate a well-defined large-scale hydrodynamic wave over the plate, with a wavelength close to the plate length at the peak sound amplification frequency when a plane acoustic wave is introduced from the upstream duct. Since the hydrodynamic wavelength is one order larger than the period of the perforation, the effect of the perforated plate is then described by a homogeneous plate impedance. It is shown that the homogenized approach approximately represents the discrete approach in this problem.

I. Introduction

L are effective in mitigating noise emissions from aero-engines. The interaction between flow and sound in the vicinity of the lined walls of aero-engine nacelles is complex and, therefore, a precise description of the sound attenuation in a lined duct with flow still attracts lots of attention [START_REF] Tam | Experimental validation of numerical simulations for an acoustic liner in grazing flow: self-noise and added drag[END_REF][START_REF] Zhang | Numerical investigation of a honeycomb liner grazed by laminar and turbulent boundary layers[END_REF][START_REF] Khamis | Acoustic boundary conditions at an impedance lining in inviscid shear flow[END_REF][START_REF] Khamis | Viscous effects on the acoustics and stability of a shear layer over an impedance wall[END_REF]. On the other hand, it is surprising to find that, under certain circumstances, a liner can also act as a sound amplifier [START_REF] Brandes | Sound amplification in flow ducts lined with a periodic sequence of resonators[END_REF][START_REF] Ronneberger | Sound absorption, sound amplification, and flow control in ducts with compliant walls[END_REF][START_REF] Aurégan | Experimental evidence of an instability along an impedance wall with flow[END_REF][START_REF] Marx | PIV and LDV evidence of hydrodynamic instability over a liner in a duct with flow[END_REF][START_REF] Marx | Effect of turbulent eddy viscosity on the unstable surface mode above an acoustic liner[END_REF][START_REF] Aurégan | Experimental observation of a hydrodynamic mode in a flow duct with a porous material[END_REF][START_REF] Alomar | Particle image velocimetry measurement of an instability wave over a porous wall in a duct with flow[END_REF][START_REF] Dai | A cavity-by-cavity description of the aeroacoustic instability over a liner with a grazing flow[END_REF] or a "singer" [START_REF] Tam | Experimental validation of numerical simulations for an acoustic liner in grazing flow: self-noise and added drag[END_REF][START_REF] Tsui | Self-induced sound generation by flow over perforated duct liners[END_REF]. For those phenomena, a type of instability is usually involved in the flow-acoustic coupling.

Flow passing the holes of a perforated plate can give rise to shear layer instability, which causes small-scale instability waves or vortex in each of the holes. A feedback loop consisting of the Kelvin-Helmholtz instability wave and an upstream propagating pulse within each individual small hole can lead to a self-noise tone at a very high frequency [START_REF] Tam | Experimental validation of numerical simulations for an acoustic liner in grazing flow: self-noise and added drag[END_REF]. When such instability couples with the acoustic resonant modes of the flow ducts or cavities, strong pure-tone whistling occurs near the frequency of the acoustic resonance [START_REF] Tsui | Self-induced sound generation by flow over perforated duct liners[END_REF]. In those cases, the instability waves have wavelengths of the order of the size of the small holes. The same coupling between the shear layer instability in a small opening and acoustic resonant modes inside a cavity can also be found in the self-sustained oscillations in a single cavity [START_REF] East | Aerodynamically induced resonance in rectangular cavities[END_REF][START_REF] Yamouni | Interaction between feedback aeroacoustic and acoustic resonance mechanisms in a cavity flow: a global stability analysis[END_REF], a side-branch [START_REF] Bruggeman | Self-sustained aero-acoustic pulsations in gas transport systems: Experimental study of the influence of closed side branches[END_REF][START_REF] Ziada | Strouhal numbers of flow-excited acoustic resonance of closed side branches[END_REF], and a Helmholtz resonator [START_REF] Ma | Fluid mechanics of the flow-excited Helmholtz resonator[END_REF][START_REF] Dai | Flow-excited acoustic resonance of a Helmholtz resonator: Discrete vortex model compared to experiments[END_REF].

Hydrodynamic instability with wavelength much larger than the size of the perforations and an associated sound amplification, near the resonance frequency of the liner, have been observed when the liner has a low resistance and the flow velocity is relatively high [START_REF] Brandes | Sound amplification in flow ducts lined with a periodic sequence of resonators[END_REF][START_REF] Ronneberger | Sound absorption, sound amplification, and flow control in ducts with compliant walls[END_REF][START_REF] Aurégan | Experimental evidence of an instability along an impedance wall with flow[END_REF][START_REF] Marx | PIV and LDV evidence of hydrodynamic instability over a liner in a duct with flow[END_REF]. The instability is due to the coupling of a hydrodynamic mode in the shear flow with the cavity resonance, and such an instability over the liner can still exist even when the Kelvin-Helmholtz instability of the shear flow over the cavities or holes does not occur. It has also been found that this type instability wave does not scale on any streamwise geometrical dimensions of the liner, i.e. neither the size of the holes nor the total length of the lined wall [START_REF] Dai | A cavity-by-cavity description of the aeroacoustic instability over a liner with a grazing flow[END_REF]. A second type large-scale instability over a perforated plate backed by a single cavity, where the instability wavelength is of the order of the total plate length, has also been reported. It was first observed in experiments in water channels that long-wavelength instability along perforated and slotted plates can occur without acoustic or gravity wave resonance [START_REF] Celik | Shear layer oscillation along a perforated surface: a self-excited large-scale instability[END_REF][START_REF] Celik | Coupled oscillations of flow along a perforated plate[END_REF][START_REF] Sever | Oscillations of shear flow along a slotted plate: small-and large-scale structures[END_REF]. The coupling of this large-scale instability with the acoustic [START_REF] Zoccola | Effect of opening obstructions on the flow-excited response of a Helmholtz resonator[END_REF] and gravity wave [START_REF] Ekmekci | Oscillation of shallow flow past a cavity: Resonant coupling with a gravity wave[END_REF] resonance of the bounding cavities was also investigated.

The large-scale hydrodynamic instability over a non-local liner, which consists of a cavity covered by a same-length perforated plate, with a wavelength close to the plate length is considered in this paper. The problem is sketched in Fig. 1, where the non-local liner is attached to a duct containing a mean shear flow. Both discrete approach where the flow-acoustic coupling at the perforated plate is described hole-by-hole and homogenized approach in which the perforated plate is represented by a homogeneous impedance are used to model this large-scale hydrodynamic behaviour.

II. Model

Calculation of the modal scattering of a non-local liner attached to a flow duct containing a mean shear flow is sketched in Fig. 2. To solve this problem of linear propagation in a shear flow, the multimodal method is used [START_REF] Kooijman | Multimodal method for scattering of sound at a sudden area expansion in a duct with subsonic flow[END_REF][START_REF] Kooijman | Influence of mean flow profile and geometrical ratios on scattering of sound at a sudden area expansion in a duct[END_REF], where the disturbances in the ducts are expressed as a linear superposition of acoustic and hydrodynamic tranverse modes. For details of the modal scattering calculation in a duct-cavity system with a shear flow, the reader is also referred to [START_REF] Dai | Acoustic of a perforated liner with grazing flow: Floquet-Bloch periodical approach versus impedance continuous approach[END_REF][START_REF] Dai | A cavity-by-cavity description of the aeroacoustic instability over a liner with a grazing flow[END_REF].

In the discrete approach, as sketched in Fig. 2(a), the configuration is divided into zones of three types denoted by I, II, and III. In each zone, the modal profiles and wavenumbers of transverse modes are solved numerically. In zones I and II, duct modes include acoustic modes propagating or decaying in the ±x directions and hydrodynamic modes travelling in the +x direction with the mean flow. Zone III does not have mean flow, so only acoustic modes are solved. A resistance R denoted by the horizontal dashed lines in Fig. 2(a) is introduced at the entrenance of each hole in the plate to mimic the resistance without flow due to thermo-viscous effects. The transverse modes in each zone are then matched using the continuity of pressure p, velocity v, and ∂p/∂ x at the interfaces between zones, and ∂p/∂ x = 0 on the vertical walls inside the small holes and the backing cavity. The continuity and wall conditions can be put in the form of a large matrix that links incoming waves to out-going waves and to all the internal variables in the non-local liner. From this large matrix, the scattering matrix of the liner is obtained.

In a homogenized approach, the perforated plate is described by its acoustic impedance Z p denoted by the red dashed lines in Fig. 2(b), which leads to a pressure jump at y = 1 for 0 < x < L. Duct modes are solved in each zone, and the modal matching at the interfaces between the zones gives the scattering matrix.

III. Results

The geometrical parameters are: H * = 15 mm, T * = 1 mm, L * o = 1 mm, and L * s = 0.2 mm. The perforated plates contains 10 holes, thus the length of the plate and the backing cavity is L * = 12 mm. The Mach number averaged over the cross-section is M 0 = 0.1 and the velocity profile is given by a simple polynomial law with a unity average value, f = (1y m ) (m + 1)/m, where the parameter m = 10 is used. At the entrance of each hole a thin sheet with a normalized resistance R = 0.016 has been added. The number of the discrete points in the duct is 300, so 900 transverse modes are solved from in the duct. Transmission coefficient for a plane acoustic wave passing through the cavity segment in the flow direction is presented in figure 3 for four different cavity depths: D * = 24 mm, 34 mm, 44 mm and 54 mm. It is observed that sound can be amplified by the liner at certain frequencies.

The fields of the pressure p and transverse velocity v are shown in Fig. 4(a,b) at the peak sound amplification frequency. A convectively growing wave appears over the perforated plate. The amplitude and phase of v approximately remain the same across the plate, as shown in Fig. 4(c,d), which are in agreement with experimental results of [START_REF] Celik | Coupled oscillations of flow along a perforated plate[END_REF]. It can also be found in Fig. 4(c) that the growth always happens in the small holes, which can be understood as a result of the Kelvin-Helmholtz instability of the shear flow [START_REF] Schmid | Stability and Transition in Shear Flows[END_REF] in the holes. However, the observed wavelength is close to the length of the perforated plate L rather than the individual holes L o , as shown by the fields and also by the phase angle in Fig. 4(d). In other words, the Strouhal number of the sound amplification scales on the length of the perforated plate rather than the holes (Sr = ωL/U 0 ). This means that it is difficult to explain the large-scale hydrodynamic wave and the related sound amplification by only examining the instabilities in the individual holes. Moreover, the long wavelength, compared to the period of the perforated plate (for the present case λ hy /(L o + L s ) ≈ 10), suggests that a homogenized description of the perforated plate can be used in this problem.

To understand the large-scale convectively growing wave over the perforated plate, the simplified model based on the plate impedance will be utilized in the following analysis. Also, since the goal is not to describe every detail of the flow and acoustics, but rather to obtain a better understanding of the phenomenon, the classic and simple plate impedance model of [START_REF] Guess | Calculation of perforated plate liner parameters from specified acoustic resistance and reactance[END_REF] is used to describe the perforated plate with grazing flow, Z p = (R + R f )/σ + iω(T + δ)/σ where, for a low sound pressure level:

R f = 0.3(1 -σ 2 )M 0 and δ = 0.85L o (1 -0.7 √ σ)(1 + 305M 3 0 ) -1
, and σ is the open area ratio of the perforated plate σ = L o /(L o + L s ). The results of the transmission coefficient for a plane acoustic wave passing through the liner in the flow direction are shown in Fig. 5. Compared to results in Fig. 3, the amplitudes are over predicted, which could be due to that the effect of the flow on the plate resistance is underestimated. However, the amplification frequencies and the trend of the peak amplification frequency and amplitude with the increasing cavity depth are well captured by the simplified model.

Iso-colour plots of the real part of p and v at the peak amplification frequency shown in Fig. 5 for D = 2.933 are presented in Fig. 6 (a,b). A large-scale convectively growing hydrodynamic wave with wavelength close to the length of the liner can be seen. Comparison of the fields in Figs. 4 and6, respectively from the discrete and homogenized models for the perforated plate, shows a qualitative agreement. The variation of the transverse velocity along the perforated plate are given in Fig. 6 (c,d). The exponential growth along the plate in the amplitude of the hydrodynamic part of the velocity indicates that the unstable hydrodynamic mode of the shear flow is excited at the upstream end of the plate. The slope of the phase angle of the hydrodynamic velocity is constant, which means that such unstable mode is convected downstream at a constant velocity. The acoustic velocity also has a growing amplitude with increasing x along the plate, especially near the downstream end, and the phase angle only changes where x > 0.5 but almost keeps unvarying in the range 0 < x < 0.5. These suggest that the acoustic waves are excited near the downstream edge, and they are mainly the high-order modes. Thus, sound amplification comes from the excited unstable hydrodynamic mode that scatters sound waves near the downstream edge.

IV. Conclusion

The long-wavelength hydrodynamic instability over a non-local liner, which consists of a cavity covered by a same-length perforated plate, in a flow duct is studied by a two-dimensional (2D) linear analysis. The mean shear flow in the duct is assumed unchanged in the streamwise direction and the acoustic and hydrodynamic disturbances are described by the linearized Euler equations with an artificial damping at the entrance of each hole in the plate. The results of the hole-by-hole approach show that the present simple model can describe the appearance of the large-scale instability over the non-local liner. The homogenized approach approximately represents the hole-by-hole approach in describing the large-scale instability. However, the plate impedance model used can lead to inaccuracy in describing wave propagation, especially when the flow velocity is high [START_REF] Dai | Acoustic of a perforated liner with grazing flow: Floquet-Bloch periodical approach versus impedance continuous approach[END_REF]. For this reason, only a qualitative comparison can be made between the discrete and homogenized models, and a quantitative agreement requires a more precise homogenization of a perforated plate with flow.

Fig. 1 Fig. 2

 12 Fig. 1 Sketch of the flow-acoustic coupling over a perforated plate baked by a cavity

Fig. 3

 3 Fig. 3 Transmission coefficients for a plane acoustic wave passing through the liner in the flow direction, calculated with the hole-by-hole description of the perforated plate.

Fig. 4 Fig. 5 Fig. 6

 456 Fig. 4 Iso-colour plots of the real part of p (a) and v (b) for a plane sound wave incidence at the peak amplification shown in Fig. 3. In the calculation, D = 2.9333 (44 mm) and ω = 0.2356 (860 Hz). Note that the fields are only plotted for the duct, the perforated plate, and part of the backing cavity. (c) and (d) are the amplitude and phase angle of v along the perforated plate at y positions (blue thick lines: just below the plate; black segmented thick lines: in the middle of the holes of the plate; red thin lines: just above the plate), denoted by the dashed-lines in (b).

Acknowledgments

X.D. gratefully acknowledges support from the National Natural Science Foundation of China no. 51876120; Y.A. gratefully acknowledges support from the ANR international project FlowMatAc no. ANR-15-CE22-0016-01.


Antonietta Specogna

The meetings, of which the 13 th will take place in November as traditionally, are called "Meetings on the new philosophical practices".

To develop language, initiate the students' reflection or usher them in the phrasing of their thoughts are fundamentals objectives in primary education that are also pushed forward by philosophically oriented discussions.

The concept of philosophically oriented discussions (POD) seems relevant for it allows to put students in a situation in which their thoughts are valued. It is hence possible to disagree with a myth read by the professor or even discuss a quote written on the board; space normally dedicated to the correct and right. The playing PODs in this learning of doubt and questioning are a major element allowing the student to develop their critical thinking. This doubt also allows them to, notably in the current conditions of communication, to question the fake or misdirected information, personal beliefs being collective… The objective of this article is to study the progression of thought between oral and written based on the contribution of philosophically oriented discussions. We will analyse the philosophically oriented discussion as a developing medium of critical thinking and written linguistic competences.

Prior to the analysis of the discussions, notions of philosophically oriented discussions and critical thinking should be defined.

State of the art

What are PODs?

Philosophically oriented discussions or "Philosophy for Children" or "Community of philosophical Inquiry" (CPI) (Miller & Tapper, 2012) emerged in the United States in the 80s with the initiative of the philosopher Matthew Lipman [START_REF] Lipman | Philosophy in the classroom[END_REF][START_REF] Lipman | Renforcer le raisonnement et le jugement par la philosophie [Strengthening reasoning and judgement through philosophy[END_REF]. Supporting the practices since 1995, the UNESCO 1 every year since 2000, on the occasion of the World Philosophy Day, links these academic practices with others (all innovations regarding philosophical practices) on an international level as it considers the development of this practice as an indicator of the country's democratic degree. "Philosophy is a discipline encouraging critical and independent thinking, able to operate for a better understanding of the world and to promote tolerance and peace" (general conference of the Organisation, UNESCO2 , 2005). Tozzi, trained philosopher, explains that the construction of thinking mechanisms starts very early, and that pushing children to question since kindergarten, contributes to arouse their judgment. Philosophise hence amount to resort to "three processes of thinking", conceptualise, argue and problematize. [START_REF] Tozzi | Faire philosopher les enfants constats, questions vives, enjeux et propositions[END_REF] explains that DOPs contribute to the "acquiring of a bigger self-esteem" since the child is less considered as a child than as an individual expressing thought. New philosophical practices insist on the fact of philosophising rather than philosophical knowledge [START_REF] Tozzi | Faire philosopher les enfants constats, questions vives, enjeux et propositions[END_REF].

Furthermore, the POD tends towards the confrontation of ideas without passion. It aims at making students understand that it is possible to confront, oppose ideas without conflicting with the individual. The culmination being to be convinced by others' ideas to the point of being able to abandon one's own opinions and build a thought collectively [START_REF] Specogna | Enonciations d'élèves et tentative de construction collective de l'enseignante : regard de la pragmatique[END_REF].

What is critical thinking?

The concept of "critical thinking" appears in the United States at the beginning of the 60s. It is inspired by the works of Dewey and his notion of "reflective thoughts", that considers that "a universal knowledge does not exist or, in other terms, that the "true" idea is the one victorious from verification trial" [START_REF] Daniel | La nécessaire praxis de la pensée et du dialogue critiques en classe[END_REF]. Critical thinking also allows a "reflective feedback on oneself" allowing in particular a reconsideration, an improvement [START_REF] Lipman | Renforcer le raisonnement et le jugement par la philosophie [Strengthening reasoning and judgement through philosophy[END_REF]. [START_REF] Daniel | La nécessaire praxis de la pensée et du dialogue critiques en classe[END_REF] compares the development of critical thinking to a scaffolding, or to a "proximal zone of development that extends and retracts". Indeed, the author explains that students tend to an intersubjective thinking, meaning they become more and more capable to take into consideration others' thoughts to build their own judgement. However, she underlines the nonlinear character of this construction of thinking and the necessity to take into consideration the context, of the knowledge about the studied concept, of the position of the teacher and the quality of interaction between members. On this matter, Saint Dizier, [START_REF] Saint-Dizier De Almeida | L'activité communicationnelle enseignante lors des discussions à visée philosophique[END_REF] show the effect of the teacher's style in the construction and development of POD's sessions.

Florin, Professor in psychology, (cited by Péroz, 2010, p16), has leaded works showing that one third of students remain silent during dialogues with the teacher, and this, no matter the taught subject. [START_REF] Péroz | Apprentissage du langage oral à l'école maternelle[END_REF] enlightens the importance to include all students in the dialogues, given that few children at the end of kindergarten have the capacity to put in place an "internal dialogue", which is to wait in silence, to listen to what is being said and then synthetize or note an incoherence or omission. And this observation unfortunately does not end at the beginning of elementary school, since a child who has not learned to converse, orally, with his peers or with the adult, will still not be able to do so in elementary school. Péroz indicated that, indeed, the child can only do alone what he previously learned with others.

To what extent does the regular practice of a philosophically oriented discussion contribute to the development of critical thinking and the language skills to write it?

Methodology

Context of the discussion

In order to be able to answer the aforementioned problem, we will rely on six one-hour philosophical discussions with CE2 (3rd year of elementary school in France) students.

In order to provide each student with sufficient space to express themselves, the philosophically oriented discussions took place in groups of 12 students. The debating group discusses while the other half class observes.

At the end of the discussions, a personal time for reflection on the debate that has just taken place is set up, which can allow each student to express their thoughts more accurately. That is why a personal written synthesis is produced. This trace thus allows to observe the students' progress throughout the six philosophically oriented discussions, in written form and following a personal phase of reflection. The group that debated and the one that observed prepare their written records. This choice is made in order to involve all students in the discussed theme, and also to study if observers are capable of "internal dialogue", as defined by [START_REF] Péroz | Apprentissage du langage oral à l'école maternelle[END_REF] in the ability to remain be silent, to listen to what is said, and to synthesize it in order to produce a new statement.

Research methodology

We will study the comparison between the students' oral productions during the discussions and their written syntheses.

After the discussions, students are invited to an individual written reflective phase, so that each student takes the time, after hearing their peers' thoughts, to express their own as accurately as possible. The construction of the oral concept will be compared to the students' written syntheses. The focus is put on three students (Aïda, Elias and Lilou) and will be an opportunity to see what they remember once the discussion is over. These are three students with very different profiles, Aïda is a studious student, very active orally during all lessons and diligent in writing. Lilou is a studious and discreet student, very involved in her written work but in retreat orally, she hardly intervenes spontaneously verbally even though she has the "right answer" when she is asked. Elias is a student with academic difficulties, he speaks frequently but his reasoning skills are to be built.

Two analysis times observed:

-when students actively participated in the POD either verbally: a comparison was made between the opening and closing remarks of the discussions for the writing of the synthesis that they retained as a record of this elaboration. -when students have not actively participated in the POD neither verbally, the skills to practice or not "internal dialogue", as expressed by [START_REF] Péroz | Apprentissage du langage oral à l'école maternelle[END_REF], are analysed.

Results

POD 1 SHOULD WE ALWAYS TELL THE TRUTH? STUDENT 1 Aïda

POD Synthesis

We should tell the truth, except in two cases, for surprises and for our protection (terrorism)

Say the truth yes, but not if we need to protection as in the case of terrorism

STUDENT 2 Elias

We should always tell the truth, even if "the person is very annoying". After six interventions from his peers tempering this obligation to truth, Elias also points out the importance to denounce oneself in the case of "misbehaving" as anyways "if everyone lies, we will know, if everyone says it's not me".

He explains, in his synthesis, that if we do not say the truth we will "get scolded".

STUDENT 3 Lilou

Did not participate actively.

It is necessary to always say the truth, including in the case of " misbehaving " such as the broken glass event that occurred at school. She only accepts lies for surprises.

Aïda's thinking does not vary between her oral interventions and her written synthesis, she seems to have built her thought alone and does not make it evolve during its drafting. On the other hand, it appears that either Elias was convinced by the ideas of his peers or that he adjusted his remarks according to his audience since his oral remarks evolve during the discussion. It is also possible that he may have felt authorized, in response to the interventions of his lie-tolerant classmates, to admit certain cases of lies. He seems to be engaged in internal dialogue since he synthesizes his written thinking around the need to tell the truth in order to avoid scolding.

Lilou incorporates in her synthesis the idea of lying to make a surprise, this example has been repeated several times and accepted by all. It therefore seems impossible to know whether this is a reflection of Lilou's own or whether the POD has changed her point of view. On the other hand, she talks about misbehaving and seems to synthesize the exchange on the "broken glass" that just took place. She seems to be demonstrating internal dialogue, as developed by [START_REF] Péroz | Apprentissage du langage oral à l'école maternelle[END_REF], since she silently synthesizes, by expressing her point of view, on what she just heard.

It is also necessary to point out that not participating actively does not mean being unable to conceive and embrace a collective concept.

POD 2 THE MYTH OF PANDORA'S BOX STUDENT 1 Aïda

POD Synthesis

Did not partake actively retranscription of presumed reaction a strong interrogation: why did the gods give Pandora a box she should not open

STUDENT 2 Elias

Did not partake actively He recalls that Pandora is curious and that's the reason she opened the box.

STUDENT 3 Lilou

Speaks positively of curiosity Shifts away from the myth and its negative connotation.

è Curiosity may help us or

She declares not open it for security reasons, even when her classmates say they would have asked what was in the box before opening it. Lilou is cautious, and expresses the idea that the person to whom we communicate our fears can lie to us and harm us.

The risk of danger can be found written in her synthesis

Lilou develops her reasoning by relying on interactions with her classmates but always by nuancing the given comments. She demonstrates "decentration" (Daniel, quoted by Nonnon, 2015) by distancing herself from the myth to evoke more generally the question of curiosity. She perfectly synthesizes her thinking in writing by addressing again the link between danger and curiosity.

Aïda does not bring new elements in her synthesis compared to what was addressed in the discussion. She simply writes what was said by the group, explaining what she would have done had she been in Pandora's place. She therefore does not seem to have practiced internal dialogue, except to put herself in the place of Pandora and explain what her reaction would have been. She therefore still works on her written language skills by producing a personal synthesis. Elias remains factual, and does not distance himself from the myth. He does not shift his focus nor does he question himself about the other causes and consequences of the myth.

POD 3 What is the purpose of freedom? STUDENT 1 Aïda

POD Synthesis

Did not partake actively She synthesizes the exchanges by explaining that freedom is used to live better, but that there are limits

STUDENT 2 Elias

Did not partake actively Freedom is for pleasure

STUEDNT 3 Lilou

She synthesizes the interventions that legitimize limits to our freedom. In her second, she develops the intervention preceding her own, which explains that we are free in order to live better, by completing it with examples In writing she uses the same examples she mentioned orally Lilou constructs her reflection in mimicry in relation to the interventions immediately preceding her own. She does not refine her thinking in her synthesis and only translates it into writing based on the examples she proposed orally.

Aïda summarizes in her synthesis what was stated during the discussion. Elias speaks of the notion of pleasure induced by freedom. It thus appears that he has not evolved from his initial thinking, or that he was convinced by the comments explaining that there are no limits to freedom.

Discussion

This comparison between the construction of the concept in oral and written form, during the course of the discussions, appears as Daniel's "scaffolding" (2016), i.e. non-linear and strongly correlated to the theme of the discussion. Students build their critical thinking, but this slow edification depends on their comfort with the theme and their interactions with their classmates.

We notice that many students do not succeed in decentralizing themselves (Daniel, quoted by Nonnon, 2015) from the proposed theme and that they do not all succeed in practicing an internal dialogue [START_REF] Péroz | Apprentissage du langage oral à l'école maternelle[END_REF] and therefore in constructing a personal thought alone, in writing. However, it appears that even in cases where students use in their syntheses the heard arguments or examples, they develop their language skills, in writing, by producing their syntheses. Indeed, they choose the appropriate vocabulary, work on syntax to give meaning to their text and try to base their thoughts on examples.

It seems essential to practice philosophically oriented discussions, in order, first of all, to disrupt the classic pattern of seeking the "right answer". Students appeared destabilized by the plurality of opinions of their classmates, but more importantly, by the absence of right or wrong answers.

By studying the syntheses written at the end of the PODs, it appears that some students modify their thinking, disagree with the words they have used during the discussions or, on the contrary, that others do not vary and do not change their thought. In all cases, the philosophically oriented discussions, thanks to the posterior reflective phase, allow students to refine their language skills in written expression. Indeed, they develop the habit of reformulating, synthesizing, giving examples and putting their thoughts into words, in writing.

In order to allow students to see for themselves how their thinking progresses, not only through the different PODs but also within the discussions themselves, it would be appropriate to have them reflect individually and in writing on their representations of the theme, before discussing it. They could thus become aware of the disruption of their initial representations in relation to their final thoughts. The teacher would no longer be the only one aware of the socio-cognitive conflict encountered by the student. The student himself could observe it, accept it and thus understand that his thought has the right to change. According to Doise and Mugny, quoted by [START_REF] Buchs | Conflits et apprentissage. Régulation des conflits sociocognitifs et apprentissage[END_REF], socio-cognitive conflict is a collective construction mechanism. It allows the child to become aware of the existence of various responses and therefore of the possibility, through interaction with peers, of modifying their thinking.

Finally, it would be interesting to conclude with the work of [START_REF] Perrin | Ateliers de philosophie à l'école primaire[END_REF], Professor of Philosophy, who is interested in the place of students with difficulties in the philosophically oriented discussion system. In particular, she explains that they often find their place with more ease in the philosophical activities and "surprise by the richness of their ideas" [START_REF] Perrin | Ateliers de philosophie à l'école primaire[END_REF]. Also, it would be interesting to see how in these situations students engage in writing? If what is still a complex part of the academic exercise can be supported in a different way and allow challenged students to find other resources.

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