Mohammed Hichame Benbitour
email: mohammed-hichame.benbitour@ecp.fr

Evren Sahin
email: evren.sahin@ecp.fr

Evaluation of the Impact of Uncertain Advance Demand Information on Production/Inventory Systems

Keywords: supply chain, inventory management, Imperfect advance demand information, single stage production/inventory system, modified base-stock policy

In this paper, we conduct a simulation study to evaluate the impact of using imperfect advance demand information (ADI) in a single stage production/inventory system. We mean by ADI the fact that the production system receives customer orders in advance of their due-dates. We consider four types of ADI: 1) Perfect ADI, 2) ADI with imperfect due-dates, 3) ADI with imperfect demand quantities and 4) ADI with updates. The production system is controlled by a modified base-stock policy which has two parameters that are S, the base-stock level, and L, the production release lead time. We intend to shed light on the impact of imperfect ADI (ADI types 2, 3, 4) on the performance of production/inventory systems. The aim of the simulation study is twofold. Firstly, the objective is to find the optimal parameters of the modified base-stock policy and secondly, to evaluate the benefits of imperfect ADI in a production/inventory system.



1.INTRODUCTION

Developments in supply chain concepts and information technology have improved the flow of information between companies. Hence, various information can be shared in a supply chain: sales, inventory, order status, production schedules and demand forecasts (see [START_REF] Lee | Information sharing in a supply chain[END_REF]). Such information enables to improve the performance of existing production and inventory control policies. For example, MRP controlled systems use demand forecasts to better control production orders.

The aim of our work is to evaluate the benefits of demand information sharing. Precisely, we consider a context where the customer shares demand information, i.e. orders, in advance of the order due-dates. We call this type of information, advance demand information (ADI). ADI models can be divided in two classes: Perfect and Imperfect ADI. Under perfect ADI, the supplier receives reliable information about customer demand, so orders received do not change over time, in terms of both quantity demanded and due date. Under imperfect ADI, the order information is uncertain and can change before the due-date of the order. The objectives of this paper are: (1) to find optimal ways to use imperfect ADI and (2) to evaluate the impact of imperfect ADI in production/inventory systems.

Different examples of production/inventory systems with ADI have been found in the literature. [START_REF] Benjaafar | Productioninventory systems with imperfect advance demand information and updating[END_REF] cite the example of a supplier who provides components to an aircraft manufacturer. The production process of an aircraft is very complex and passes by many stages. If we assume that the supplier provides a component which should be assembled at the last stage of production, then, the manufacturer can send ADI by indicating the status of production process and expected time before using the needed component. Another interesting example of ADI is the automotive industry: Faurecia is a supplier of automotive parts which has as customers the original equipment manufacturers (OEM). Generally, manufacturing a car passes through four workshops consecutively: Stamping, Welding, Painting and Assembly. Parts supplied by Faurecia are involved at the end of the assembly workshop. In a just in time fashion, Faurecia receives a signal of demand when a car enters the assembly stage of the customer. This signal is considered to be perfect (the customer does not change the due-date of that signal) and represents an advance order information. The order is supposed to be delivered after a fixed amount of time called "the Demand lead time". Faurecia can also receive advance demand information from the earlier manufacturing workshops (Stamping, Welding and Painting), but these signals are considered to be imperfect because the lead time variability is high at these stages.

In this study, we consider a single stage single item production/inventory system controlled by a modified basestock policy that incorporates ADI. The system is assumed to use four types of ADI: i) Perfect ADI, ii) ADI with imperfect due-dates, iii) ADI with imperfect demand quantities and iv) ADI with updates. In fact, demand information has two components: due-date and quantity demanded. So we can have diverse types of ADI, for example, when information about customer order due-dates and quantity is exact, we say that we have a system with "perfect ADI". For a system where information regarding future quantities demanded by customer is exact but there is uncertainty associate with order due-dates, we say that we have ADI with imperfect due-dates. We show by means of a simulation model that the value of ADI decreases when information becomes imperfect. We provide interesting insights on the impact of imperfect ADI on the production system's performance.

The remainder of this article is organized as follows. In section 2, a literature review is presented. Section 3 describes the characteristics of our model and the related assumptions. Section 4 reports the simulation results and discussions. Finally, conclusions and some avenues for future research are presented in Section 5.

2.LITERATURE REVIEW

Our work is within the stream of papers that study the benefits of demand information sharing. We are essentially inspired by models introduced by [START_REF] Buzacott | Safety Stock versus Safety Time in MRP Controlled Production Systems[END_REF], [START_REF] Karaesmen | Integrating advance order information in make-to-stock production systems[END_REF] and [START_REF] Karaesmen | The Value of Advance Demand Information in Production/Inventory Systems[END_REF]. One possible classification of research related to ADI can be based on the type of information available. Systems with perfect ADI are studied differently than systems with imperfect ADI.

For systems with perfect ADI, [START_REF] Milgrom | Communication and inventory as substitutes in organizing production[END_REF] are among the first researchers that discuss the impact of ADI in production/inventory systems. They studied the issue of shifting between make-to-stock (MTS) to make-to-order (MTO) mode of production where information regarding inventories and demand are given by customers. They considered a single stage, multi-product system where demand follows a normal distribution. They found that for high level of price and large demand information, MTO strategy that uses ADI is optimal. [START_REF] Hariharan | Customer-Order Information, Leadtimes, and Inventories[END_REF] extended some existing inventory models by incorporating perfect ADI. They examined the effects of ADI on the performance of inventory systems. They concluded that the effect of using advance demand information is equivalent to reducing the supply lead time. [START_REF] Buzacott | Safety Stock versus Safety Time in MRP Controlled Production Systems[END_REF] studied the tradeoff between safety stock and safety time in MRP Controlled Systems. [START_REF] Toktay | Analysis of a Forecasting-Production-Inventory System with Stationary Demand[END_REF] studied a production/inventory system with stationary demand and dynamic forecast updates. They considered a single stage, single item, discrete-time, make-to-stock capacitated system managed by a base-stock policy. They found that a good forecast quality improves the performance of the system by reducing the expected costs and decreasing the base-stock level. They concluded also that the value of forecast information decreases when the system is saturated (high traffic). [START_REF] Karaesmen | Integrating advance order information in make-to-stock production systems[END_REF] studied a discrete time make to stock queue with advance demand information. They investigated two production control policies: i) An optimal policy found using dynamic programming and ii) A heuristic policy extended from the classic base-stock policy by integrating a release lead time parameter. [START_REF] Karaesmen | The Value of Advance Demand Information in Production/Inventory Systems[END_REF] studied an extension of the model proposed in [START_REF] Karaesmen | Integrating advance order information in make-to-stock production systems[END_REF]. They considered a continue M/M/1 single stage MTS queue with ADI. They found that the system can shift from MTS strategy to MTO for sufficient horizon of visibility. [START_REF] Altendorfer | A comparison of make-to-stock and make-to-order in multi-product manufacturing systems with variable due dates[END_REF] developed an optimization model that minimizes Finished Goods Inventory and backorder costs for a multi-product production system with perfect variable due-dates.

For systems with imperfect ADI, [START_REF] Tan | Modelling imperfect advance demand information and analysis of optimal inventory policies[END_REF] evaluated the impact of using imperfect ADI on inventory policies. They used a general probability framework model that integrates imperfect ADI with ordering decisions. [START_REF] Tan | Using imperfect advance demand information in ordering and rationing decisions[END_REF] studied a rationing decision and inventory modeling problem using imperfect ADI. They showed that Imperfect ADI improves the performance of the system through reducing the inventory and the losing orders costs. [START_REF] Benjaafar | Productioninventory systems with imperfect advance demand information and updating[END_REF] analyzed a single stage production-inventory system where orders may become due prior to or later than the announced expected due-date or they can be canceled altogether. [START_REF] Gayon | Using Imperfect Advance Demand Information in Production-Inventory Systems with Multiple Customer Classes[END_REF] considered a MTS capacitated system with multiple customer classes where 1) order due-dates are not exactly known in advance, (2) orders can be cancelled by the customers, and (3) ADI is available only from a subset of the customers. [START_REF] Liberopoulos | On the effect of variability and uncertainty in advance demand information on the performance of a make-to-stock supplier[END_REF] evaluated the impact of variability and uncertainty of ADI on the performance of single stage production/inventory systems. By numerical experiments, they showed that in the case of unreliable ADI with capacitated system, safety stock and safety time are interchangeable. [START_REF] Bernstein | Advance Demand Information in a Multi-Product System[END_REF] investigated the impact of ADI on the performance of a multiproduct system. They considered both perfect and imperfect mix of information.

Through this study, we propose new models to evaluate the impact of imperfect ADI on inventory and backorder costs. To our knowledge, we are among the first authors that consider imperfect ADI in a single stage system where the customer order quantity is not a unit demand but a random variable following a Normal distribution. This work gives interesting results about the relation between base-stock levels and imperfect information.

3.MODEL DESCRIPTION AND ASSUMPTIONS

We consider a single stage, single product manufacturing system that operates in a make-to-stock environment with ADI. The system consists of one manufacturer and multiple customers (no priority rules) so we can represent them by a unique customer. Customer orders arrive in advance of their due-dates (required delivery date) and they are satisfied from the Finished Goods Inventory (FGI). We assume that there is an infinite supply of raw materials for the manufacturing stage and that the FGI is controlled by a modified base-stock policy. Additionally, we suppose that the manufacturing capacity is deterministic and that setup costs and setup times are negligible for simplification reasons.

In presence of advance demand information, the base-stock policy can be modified to include that information. In the classical base-stock policy (without ADI), the initial stock is equal to S (the base-stock level) and replenishment orders, i.e. production orders, are triggered at each demand arrival. If we assume that customer orders are placed in advance of their due-dates by a demand lead time H, then replenishment orders can be triggered H units of time before the customer order duedates. This approach would reduce stock outs, but it would generate high FGI levels which consequently increases the holding cost. Therefore, a control parameter L was proposed by [START_REF] Karaesmen | Integrating advance order information in make-to-stock production systems[END_REF] to regulate the timing of replenishment orders, it is called "the release lead time parameter". Instead of triggering replenishment orders H units of time before the customer order due-dates, replenishment orders are triggered L units of time before (where L≤H), so the release is delayed by H-L units of time. This modified basestock policy is called the (S,L) policy. In this model, the basestock level S can be viewed as a safety stock for a MTO system.

Perfect ADI

We consider a discrete simulation model. We assume that a customer order quantity denoted by O(t)arrives at the end of each period t. This order should be fulfilled after H H  units of time, i.e. at the end of period (t+H). The customer order quantity O(t) is assumed to follow a normal distribution with parameters ) , (

 

. Let us denote by X(t) the FGI level at the end of period t (X(t) is allowed to be negative under backordering assumption). P(t) is the quantity of products produced during period t and D(t) is the customer demand at the end of period t, we have

) ( ) ( H t D t O   . ( 1
)
In other words, customer demand observed at period t corresponds to the order received at period ) ( H t  .

The evolution of X(t) can be expressed by:

) ( ) ( ) 1 ( ) ( t D t P t X t X     . (2)
The total cost in period t, C(t), is expressed in (3), where h and b are the linear FGI holding and backorder costs per item per period, respectively.

    b*X(t) h*X(t) t C ) ( . ( 3
) Where   ) ( , 0 t X Max X(t)   and   ) ( , 0 t X Max X(t)    .
At the beginning of each period, the requirement R(t) is calculated following the logic of a modified base-stock policy. In a classical base-stock policy, the goal of the release mechanism is to maintain inventory position at level S, so we have

) 1 ( ) (    t X S t R . ( 4
)
In a (S,L) policy, the goal is to keep the base-stock level S and take into account the requirements for future (known) demands between periods t and t+L. We thus have

). 1 ( ) 1 ( ) 1 ( ) 1 ( ) ( 1 1                  t X i t D S t X H i t O S t R L i L i (5)
Being restricted by the limited manufacturing capacity Cap, the quantity produced during period t, P(t), is equal to

) ), ( ( ) ( Cap t R Min t P  . ( 6
)
We develop a simulation model to find a release policy (S,L) that minimizes the total average FGI holding and backorder costs. The optimization problem is

Min ] [ * ] [ * ) , (     X E b X E h L S C . (7)

ADI with Imperfect due-dates

In the previous section, we considered a model where information about future orders is perfect. In this case, we assume that due-dates of future orders are not known exactly. At the end of each period t, there is a customer order arrival O(t) with an estimated due-date for period (t+H). However, the customer who submits the due-date information can ask for his order before or after the estimated due-date. Hence, we assume that the demand lead time H is a random variable which follows a normal distribution with parameters ) , (

H H  .
It is important to note that the variability is on the timing of demand (i.e. H is a random variable) and not on the quantity demanded (i.e. O(t) does not change over time).

In this model, the requirement R(t) is calculated based on the expected demand lead time associated with each future order, i.e. H . Hence, R(t) can be expressed by:

) 1 ( ) 1 ( ) ( 1          t X H i t O S t R L i . (8)
Note that in this model, we can have the phenomenon of "Cross demands" where more than two different customer orders end with the same due-date. The FGI level X(t) evolves according to (2). Equation ( 1) is no longer verified, but (3), ( 6) and ( 7) hold for this model. Equation ( 5) is replaced by (8).

ADI with imperfect demand quantities

In this model, we assume that the quantities associated with future orders are not known exactly, i.e. order quantities are uncertain. It is important to note that the variability is on the quantity demanded (i.e. O(t) is a random variable) and not on the quantity demanded (i.e. H is fixed and equal to H ).

Hence, at the end of each period t, there is a customer order arrival with a due-date for period H t  . The customer who submits demand information can ask more or less than the quantity initially requested O(t). Thus, we have

q t t O H t D     ) ( ) ( , (9)
where q t  is the variability associated with the quantity of products demanded at the end of period t. We assume that q t  follows a normal distribution with parameters ) , 0 ( O  .

ADI with updates

This model is an extension of the models with imperfect duedates and imperfect quantities. We consider a system in which a manufacturer may receive several signals for the same order in different time horizons, where only the last signal is perfect.

To start with, we assume that there are two signals. The last signal is perfect and the former one is imperfect Figure 1 illustrates the updating quantities mechanism: Fig. 1. The updating quantities mechanism

The updating mechanism can concern quantities or due-dates: updating quantities consists in receiving a first advance signal at the end of period t with an exact demand lead time H1, i.e. The order has to be satisfied at the end of period t+H1. The quantity requested by the customer for period t+H1 is O(t). Then, after (H1-H2) units of time (i.e. at period t+H1-H2), the manufacturer receives a second signal for the same order where the quantity to be delivered at t+H1 is updated and is no more equal to O(t).

Updating due-dates consists in receiving a first advance signal at the end of period t with an estimated demand lead time H1, ie. The order is estimated to be satisfied at the end of period t+H1. The quantity requested by the customer for period t+H1 is O(t). That signal has an exact information about the quantity demanded. After (H1-H2) units of time (i.e. at period t+H1-H2), the manufacturer receives a perfect information about the same customer order with a new exact demand lead time H2 where H1> H2. In this section, we present results of the simulation study. The purpose is to find the optimal parameters (S,L) of the base stock-policy, and to examine the benefits of using ADI. Matlab R2012b is used to execute the simulation experiments. We run the simulation for 20000 periods and we calculate the average total cost for a list of S and L values to find which combination (S,L) minimizes the total cost. We assume that the capacity Cap is constant and takes values from 50.1 to 100. The demand quantity D(t) follows a normal distribution N (50, σ) where σ can take values: 0.05, 1, 5 and 10. As for cost parameters, we take the linear holding cost h=1, and the linear backorder cost b=10, 100 and 1000. We assume that the demand lead time 24  H .

Case of ADI with imperfect due-dates

Our goal is to evaluate the impact of ADI with imperfect duedates in single stage systems. We make a comparison between the performances of a system with perfect ADI and another system with different degrees of imperfect due-dates (H = 0, 0.5, 1, 2, 5). The case where H = 0, is equivalent to the model with perfect ADI.

It is intuitive that increasing the demand lead time uncertainty would deteriorate the performance of ADI. Figure 3 shows the impact of the error variability on the performance of a capacitated single stage system. From Figures 3 and4, it can be seen that for a sufficiently large demand lead time H , the average cost is decreasing linearly in L until a threshold level L*, and is increasing in L when * L L  . So, the cost is minimized at * L L  . As the release lead time 𝐿 increases from zero to L*, the optimal base stock level S* appears to decrease linearly with L. The insight behind these results is that the optimal parameter L* defines the desired demand lead time H, because any further increase in L beyond L* is useless and does not improve performance in terms of average total cost reduction. Another important conclusion is that the value S*(L*) sometimes equals to zero which means that it is optimal to operate the system in a make-to-order environment. (In a continuous review system, the value S*(L*) is always equal to zero). We observe that increasing the error variability increases the optimal average cost, the optimal base-stock level and the optimal release lead time. Table 1 shows the impact of capacity increment on system performance when due-dates are imperfect. S*(L=0) represents the optimal base-stock level for a system operating in a pure make-to-stock environment without ADI. C*(L=0) represents the average total cost for the release policy (S*(L=0), L=0). C* is the cost of operating the system in a pure make-to-order environment. It can be seen that the optimal release lead time L* decreases with capacity. It means that a smaller demand lead time H is required to shift from a make-to-stock to a make-to-order system. We also observe that the optimal base-stock level decreases with capacity increment. In a system with perfect ADI, both relative and absolute values of gain is increasing in capacity and take their maximum values for higher amount of capacity. However, this is not the case when ADI is imperfect.

Case of ADI with imperfect quantities

Figure 5 shows the impact of forecast quantity on the performance of ADI. We consider five values of error variability. The results indicate that the error variability increment increases the optimal average total cost, the optimal base-stock level and the optimal release lead time.

Additionally, we observe that the effect of imperfect quantities on system performance is less significant than imperfect duedates.

Fig. 5. The effect of imperfect quantities on the average cost.

ADI with updates

We show results for a system which receives one update on quantities. We assume that the first signal of customer order arrives in advance with H1=24 units of time and contains imperfect information about demand quantity. The second signal is perfect and arrives in advance with H2=18 units of time. We make a comparison between the costs of managing the system with the two signals. It can be seen that the first signal corresponds to the case of ADI with imperfect quantities, while using the second signal corresponds to a model with perfect ADI. In fact, it is difficult to decide which signal to choose, especially, when the demand lead time of the second signal H2 is sufficiently long to attain the value of the optimal release lead time L*. In such a situation, we recommend to run simulations for the two signals and decide whether to choose the first or the second signal.

5.CONCLUSION AND FUTURE WORK

In this paper, we propose simple control policies that incorporate perfect and imperfect advance demand information. We consider a periodic review capacitated single stage system with normally distributed demand and a fixed processing capacity. We evaluate the impact of four types of ADI: i) perfect ADI, ii) ADI with imperfect due-dates, iii) ADI with imperfect demand quantities and iv) ADI with updates in quantity. We found that the imperfectness of demand information reduces the benefits of ADI. We also found that imperfect due-dates deteriorate the system's performance more than imperfect demand quantities. This result is quite intuitive because of the phenomenon of "Cross demands". Simulation results show that in system with updates, we can use only imperfect signals if the demand lead time related to the perfect signal is not large enough to achieve the optimal release lead time L*. Simulation results confirm earlier insights of [START_REF] Milgrom | Communication and inventory as substitutes in organizing production[END_REF] and [START_REF] Karaesmen | Integrating advance order information in make-to-stock production systems[END_REF]: advance demand information and inventory are substitutes. For sufficiently large demand lead times, information can replace the base-stock which allows the system to shift from a make-to-stock to a make-to-order setting. In terms of relative gains, imperfect demand information has not a significant impact on cost reductions, especially, for systems with small capacities. In fact, systems with limited capacities cannot respond to information about future customer orders.

There are several avenues for future research in ADI.

According to the literature review, it would be of interest to study systems with multiple products and imperfect ADI. For example, we can consider systems with multiple customer classes having different priorities. In each period, every class of customers send advance orders for multiple products. The objective is to find a release mechanism that minimizes the average holding and backorder costs. Almost all research papers used the base-stock policy to evaluate the benefits of ADI (the base-stock policy was proved to be optimal) but industrials use another practical policies as MRP and Kanban. We propose to find new methods to incorporate ADI in those most used policies. We can consider hybrid basestock/Kanban policies with imperfect ADI in multi-stage production/inventory systems.

Figure 2

 2 Figure 2 depicts the updating due-dates mechanism:

Fig. 3 .

 3 Fig. 3. The effect of imperfect due-dates on the average cost.

Fig. 4 .

 4 Fig. 4. The effect of the error variability on the optimal basestock level.

Fig. 6 .

 6 Fig. 6. Variations of average cost in system with updates.

Fig. shows

  Fig. shows the variations of a system which uses both perfect and imperfect signals. It can be observed that there is a slight difference between the average total costs of the two signals.In fact, it is difficult to decide which signal to choose, especially, when the demand lead time of the second signal H2 is sufficiently long to attain the value of the optimal release

Table 1 . The impact of capacity, for h=1, b=10, Demand~N(50,10), imperfect due-dates (0,2)

 1

	Cap	C*(L=0)	S*(L=0)	C*	L*
	50,1	382,89	621	374,61	12
	50,2	321,83	475	308,68	9
	50,5	186,01	333	177,00	7
	51	138,78	265	133,98	5
	52	125,18	229	119,66	4
	55	116,89	193	113,14	4
	60	112,69	170	109,59	3
	65	109,38	156	106,82	3
	70	106,94	148	104,88	3
	80	103,67	137	102,66	2
	90	101,47	128	101,05	2


M Xu

D Masson

D Ryckelynck

A Chesnaud

A Thorel

Modelling and Prediction of Deformation During Sintering of a Metal Foam Based SOFC (EVOLVE)

Stacking of cells in a SOFC stack requires that each element be perfectly flat and deprived, as much as possible, of internal stresses while maintaining their electrochemical capabilities. The EVOLVE concept introduces a metal foam based anode in which the foam plays the role of current collector, gas diffuser and thermo-mechanical deformation buffer. Owing to the different mechanical behaviour of the anode components, the deformation during sintering cannot be intuitively anticipated. Therefore, the global deformation of the cell was modelled and simulated by Finite Element considering a phenomenological approach of the anisotropic sintering. The thermo-mechanical parameters of each component were determined experimentally by dilatometry and three-point bending tests operated under conditions identical to those of the sintering. Results provide relevant indications on components composition and morphology, and on the sintering conditions for producing flat and stackable cells.

Introduction

Solid Oxide Fuel Cell (SOFC) is one of the most attractive energy conversion system for stationary applications owing to its efficiency (80% net in cogeneration) and fuel flexibility (hydrocarbons, CO, bio-fuels,…). Although the feasibility and potentiality of SOFCs have already been demonstrated for ages, their repeated delaying large-scale commercialization still proves the existence of scientific and technological challenges which are imperative to solve. The main limitation of SOFCs that restrict their bringing to market remains the fast-acting deterioration of materials through redox cycling at the operating temperature, particularly on the anode side. In this day and age, three main configurations of planar SOFC have been experimented either at the laboratory or industrial scale. The electrolyte-supported cell (ESC) presents the advantage of being solid and adaptable despite its very low level of performances. Nowadays, the high power output of the anode-supported cell (ASC) makes this configuration the most mature design [START_REF] Han | the 10 th European Fuel Cell Forum[END_REF][START_REF] Lang | the 10 th European Fuel Cell Forum[END_REF], nevertheless its life span remains strongly limited owing to mechanical and chemical degradation mechanisms promoted at high temperature by the presence of nickel particles [START_REF] Lohsoontorn | [END_REF](4)(5)(6)(7). Even if the operating temperature is reduced, coking and sulphur poisoning become the main source of performance degradation through redox cycles. The metal-supported cell (MSC) brings significant advantages in terms of mechanical stability and robustness under transient conditions (8), however its power density remains low compared to the ASC (9).

The originality of the EVOLVE concept resides in the implementation of an innovative architectured current collector, on the anode side, based on materials that can withstand mechanical and chemical stresses. The current collector is constituted of a NiCrAl foam impregnated with La 0.1 Sr 0.9 TiO 3+δ (LST), that also plays the role of gas diffuser as well as mechanical support (Figure 1) for the following sequence: the anode (LST or LST + GDC10), the electrolyte (YSZ), the diffusion barrier (GDC10) and the cathode (LSCF48). This configuration combines all attractive features of an ESC, i.e. the robustness and flexibility, with the advantage of an ASC, the electrical power supplied, along with the electrochemical stability under redox cycle of a MSC. Therefore, such architecture represents a strategy to improve performances, reliability and durability of SOFCs, along with their tolerance regarding sulphur. A predictive thermo-mechanical model that is able to describe the global deformation of this cell during sintering is proposed. The numerical simulation was carried out considering a phenomenological approach of the anisotropic sintering. This theoretical study aimed at reducing the "trials-error" experimental approach for optimizing the sintering cycle, guide the choice of materials and cell dimensions for producing flat and stackable cells.

Experimental section

Fabrication of the anode-architectured cell

A sequence of processes is implemented in order to fabricate a complete anodearchitectured cell. The current collector is made from a NiCrAl foam impregnated with a LST powder that is further cold rolled to decrease its thickness and porosity down to 500-600 µm and 15% to 20%, respectively. The 50 µm LST (or GDC10-LST) functional anode is then deposited by screen printing or bar coating. Initial porosities of 45% are usually measured from SEM images recorded on microstructures of layers fabricated with such processes. The smooth surface of the anode is afterwards covered with a 15-20 µm thick YSZ electrolyte layer using an electron-beam physical vapor deposition technique achieved at 600 °C. This latter is repeated for the coating of the 2-5 µm thick GDC10 diffusion barrier. The advantage of this technic is that fully dense materials (relative density in the range 95-99%) can be directly deposited at 600 °C without any additional post thermal treatment. The final LSCF48 cathode layer with a thickness varying from 30 to 50 µm is then coated by screen printing; the initial layer porosity is about 45%. At last, the whole multilayered structure is sintered at 1100 °C for 5 hours in air.

Dilatometry tests

The anisotropic shrinkage curve for each component of the cell was deduced from dilatometry data of corresponding materials. The unidirectional variation of the dimension of samples, i.e. expansion and shrinkage, were successively recorded as a function of the temperature using a differential horizontal dilatometer DIL 402 CD from NETZSCH. This apparatus is equipped with two high resolution inductive sensors (digital resolution of 1.25 nm) connected with an alumina pushrod that enables to apply continuously a pressure on the sample during measurement. Data collection was carried out in static air, between 35°C and 1570°C with a heating rate of 5 °C min -1 while a static load of 1.59×10 -2 MPa (45 cN applied on a surface of 28.3 mm 2 ) was continuously applied. The thermal expansion of the sample holder was systematically subtracted from the raw data applying the DIN 51045-1 calibration correction and using alumina calibration standards. The data were recorded and analysed using the Netzsch-Proteus® software (v 6.1.0).

Three-point bending tests

The three-point bending tests were carried out in static air using a vertical multi-units SETARAM TMA-92 thermo-mechanical analyser equipped with a dedicated sample holder constituted of a flange supporting two alumina triangle-shaped edges spaced 8 mm. A constant load of 45 cN is continuously applied during measurement by means of an ended-edge supporting shaft having a knife-shaped section. The alumina shaft is connected to a linear variable differential transducer (LVDT) which records the vertical deflection of the middle line of the specimen. The displacement range was fixed to ± 2.0 mm with a linearity of 0.3% and the vertical deflexion was measured with a resolution of 0.1 µm. Beam-shaped samples of 10 mm long by 3 mm wide by 500 µm thick were used to neglect shear stresses. Vertical deflexion data were collected between 35°C and 1570°C with a heating rate of 5°C min -1 and the maximal temperature was maintained for 5 hours. The data were collected and processed using the CALISTO v 1.076 software.

Model and Numerical Simulation

Modelling of the deformation

In order to predict the deformation of the cell during heating, a phenomenological approach of the sintering was considered. The procedure consists in following the total deformation of the cell over time during the heat treatment. This deformation occurs according to four successive processes: two reversible phenomena, i.e. (i) the elastic deformation (ε e ) and (ii) the thermal dilatation (ε th ), followed by two simultaneous phenomena, i.e. (iii) a viscoplastic deformation (ε vp ) related to the creep and (iv) a shrinkage (ε s ) associated to sintering. The global behaviour law of the deformation kinetic can be expressed by:

s vp th e total ε ε ε ε ε + + + = [1]
The viscoplastic deformation is driven by a thermally activated creep deformation Norton's law; the sintering shrinkage, also a thermally activated process, expresses the densification of the material. Elastic and thermal deformations were not taken into accounts as they contribute by an insignificant margin compared with irreversible (ε vp and ε s ) processes. The model integrates morphological (initial f o and final f ∞ porosities) and mechanical characteristics of each constituent layer, as well as the cell dimensions (Figure 2a). In the present case, the improvement of the model application was achieved by considering the real mechanical of each material that was deduced from dilatometric and creep tests. These latter were operated in the same heat and atmosphere conditions as the real sintering. In addition, the anisotropic sintering was considered and detailed hereinbelow. For the modelling, the thickness of each layer of the cell was fixed (Table I), while cell diameters of 10 mm, 20 mm, and 40 mm were implemented. The numerical simulation was carried out integrating the functional anode material either as single phase or composite (Table I). For symmetry reasons, only a section of the cell was meshed ; a quadratic element (c2d8) without model reduction was used and applied to the cell element of diameter Ø cell /2 (Figure 2a andb). The meshing was refined close to the layer interfaces. The numerical simulations were performed in 2D using the Z-set finite element code, implemented in the "Zebulon" software and developed by MINES-ParisTech (10). For the calculations, only the horizontal displacement (U 1 ) of the nodes on the left side of the cell element was blocked as if the cell was subjected to "free sintering". The initial porosity of each constituting layer of the cell (Table I) was set so that it is similar to that obtained by the corresponding shaping process. However, the "realistic condition" of sintering should be that one blocking the creep of the cell put on a substrate.

Modelling of the anisotropic shrinkage during sintering

Experimentally, the samples for dilatometry tests are cylindrical with a diameter of 6 mm. Given that pressure and temperature fields imposed to the samples are uniformly distributed on their surface, the mesh geometry and dimension will not impact results obtained from the simulation. As a matter of fact, the numerical geometry of the samples can be simplified using a 3D meshing of a linear cubic element (Figure 3a). For the calculation, the three displacements U 1 (along x), U 2 (along y) and U 3 (along z) of the (0,0,0) origin point were blocked. In addition, the displacement U 3 of the (0,0,1) point was blocked in order to avoid rotation of the element. An anisotropy model of the sintering was considered to simulate the material behaviour (11). The deformation kinetic of sintering depends on temperature and porosity of the sample. It can be expressed by a two-order tensor:

( ) ( ) ( ) ( )             = T f, ε 0 0 0 T f, ε 0 0 0 T f, ε T f, ε s z s y s x s [2]
The sintering being thermally activated, the kinetic deformation of sintering in each direction can be written:

( ) ( )       - × - × - = ∞ RT Q exp f f A T f, ε s β o s z y, x, [3]
A o is the pre-exponential factor, f and f ∞ the time-dependent and final porosities, respectively, β a constant that depends on the material, Q s the sintering activation energy, and R the ideal gas constant. By introducing anisotropy coefficients for the sintering to describe the component of the deformation in the direction perpendicular to the cylinder axis, i.e. y, the kinetic of deformation along x and z can be expressed as a function of ( ) T f, ε s y and K 1 and K 2 respectively:

( ) ( ) T f, ε K T f, ε s y 1 s x × = [4] ( ) ( ) T f, ε K T f, ε s y 2 s z × = [5]
with K 1 = K 2 in the case of a cylinder-shaped sample.

Modelling of the 3-point bending tests

In order to simulate the vertical deflexion of the rod-shaped samples submitted to 3point bending tests that will lead to the identification of creep kinetic parameters, a 3D meshing as illustrated in Figure 3b was used. Since a force of 45 cN is uniformly applied on a rectangular surface (3 mm length and 2 mm width) of the top surface of the element, the charge can be converted to a uniform pressure fields. For the calculation, the displacements U 1 and U 2 of the bottom left segment as well as the displacement U 2 of the right bottom segment of the 3D element were blocked. Besides, the U 3 displacement of the (0,0,0) origin was blocked. The viscous flowing is supposed to be linearly dependent of the stress in accordance with a creep associated with a high temperature diffusion process. Nevertheless, the kinetic of deformation also depends on the material porosity. So the kinetic of creep deformation is written as:

( ) ( ) ( ) σ : f M T t, η 1 T f, ε s vp × = [6]
Where η(t,T) represents the viscosity of the material according to the Arrhenius law:

( ) ( )         + × × = RT Q exp K f - 1 1 T t, η vp o o [7]
Q c is the creep activation energy, R the ideal gas constant and T the temperature. The tensor ( )

f M
of order 4 was introduced to model the effects of the porosity on the kinetic of deformation. The creep is assumed to be isotropic, therefore ( )

f M
can be written:

( ) ( ) ( ) I f F J f C 2 3 f M × + × × = [8]
With C(f) and F(f) two coefficients that depend on the porosity defined as:

( ) c n o f C 1 f C × + = [9]
( )

F n o f F f F × = [10]
Otherwise, the tensors of order 4, i.e. I and J are expressed by:

( ) I tr : I × σ = σ [11] ( ) I tr 3 1 : J × σ - σ = σ [12]
For NiCrAl foam, only the creep phenomenon occurs so that its deformation during sintering can be described according to a "Norton-Hoff" viscoplastic model given by: ( ) ( )

n vp o n vp RT Q exp k T K T f, ε + + σ ×         - × = σ = [13]
The creep activation energy Q vp is the same as the sintering activation energy Q s assuming that both phenomena are controlled by self-diffusion.

Results and Discussion

Dilatometric measurements and modelling

The longitudinal shrinkage (ε l ) of all materials constituting the cell was measured on compacted powders, in the form of cylinder, as a function of the temperature. Tests were carried out in static air under a load of 45 cN. The result is displayed in Figure 4a for the YSZ electrolyte material. In the low temperature region (< 1000 °C), the compacted powder expands which results in an increase of the sample length. From 1000 °C, the sintering takes place and the sample drastically shrinks up to 1400 °C and then its dimensions remain unchanged. During cooling, the contraction makes possible the determination of the thermal expansion coefficient (TEC) of the sample.

The technical curve was built by extracting the reversible thermal deformation ε th (equation 14) from the dilatometric data ε dil . The resultant curve (Figure 4a) obtained is only comprised of the irreversible processes, i.e. the sintering and viscoplastic deformations that occur almost simultaneously. The sample length is reduced by 24% as illustrated by the evolution of the shrinkage curve recorded between room temperature (RT) and 1570 °C. The shrinkage curve shows that the sample length does not evolve from RT to 600 °C.

( ) ( )

) T (T 100 TEC T t, ε T t, ε i dil tech - × × - = [14]
T i (= 35 °C) is the temperature from which dilatometric data were recorded. ε tech and ε dil are the technical and dilatometric shrinkages, respectively. TEC is the thermal expansion coefficient deduced from the slope of the linear portion of the dilatometric curve (Figure 4a) recorded during the cooling step. The evolution of the material porosity as a function of the temperature, displayed in Figure 4b, was then calculated using the theoretical approach reported by K. Maca et al. (12). The mechanical parameters A o , β and Q s of the sintering behaviour law were refined for each material by fitting the experimental curve displaying the porosity as a function of the temperature. In order to do this, all parameters were adjusted until both the simulated and experimental curves match. Figure 4b illustrates an example of such a refinement achieved on the YSZ electrolyte material and leading to a perfect agreement between experimental and simulated curves. The morphological parameters f o and f ∞ are data measured on the sample before and after sintering, respectively, and anisotropy factors (K 1 = K 2 ) are deduced from the total shrinkage ε l (inset in Figure 4a) along the cylinder axis. The same procedure was followed for the other materials. The experimental and simulated curves are compared in Figure 5a and refined parameters are gathered in Table II. Apart from materials containing the LST phase, a very good agreement between experimental and simulated data is noticed. Indeed, L. Amaral et al. reported a two-step densification process in nonstoichiometric LST ceramics (13).

Three-point bending tests and modelling

The deflexion of each sample under constant load of 45 cN was measured as a function of the temperature (Figure 5b) As with dilatometric measurements, dilatation effects were first removed from the raw data prior to numerical simulation.

Modelling of deformation in complete cells

The influence of the sintering temperature was investigated on laboratory-scale cells with Ø cell = 10 mm. Layer thicknesses as indicated in Table I were implemented in the model. An initial porosity of 15% was set for the NiCrAl-LST current collector. Three calculations were carried out in order to simulate deformation in the cell after sintering at 1100 °C, 1200 °C and 1300 °C for 5 hours. For the simulations, a heating rate of 5 °C min -1 was implemented. As illustrated in Figure 6, a sintering at either 1100 °C and 1200 °C has a limited influence on the cell deformation. At higher temperature, e.g. 1300 °C (Figure 6c), the cell significantly bends as a direct consequence of the important shrinkage of the LSCF48 cathode material that reaches its optimal sintering temperature. In a second time, the influence of cell diameter on the total deformation was regarded. The calculations were carried out using initial thickness and porosities gathered in Table I and the sintering was operated at 1100 °C for 5 hours with a heating rate of 5 °C min -1 . At such a sintering temperature, the Figure 7 shows that the cell diameter has an unimportant influence on the total deformation of the cell. Strong deformation will take place at a temperature from which the creep of materials becomes significant, that is to say 1200 °C.

Figure 7. Scale effect on the curvature of the cell at 1100 °C for 5 hours.

Conclusion

A thermo-mechanical model was developed in order to predict deformation during sintering of a self-supported current collector SOFC. This model was fed with mechanical and morphological parameters deduced from dilatometric and creep tests. The numerical simulations were carried out assuming that materials are homogeneous media and implementing various cell diameters and sintering temperature as these two parameters are likely to influence deformation of the multilayered structure after thermal treatment. Finite element simulations demonstrated that plane and stackable cells can be fabricated after sintering at 1100 °C for 5 hours if and only if the starting powders are identical to those ones used for dilatometric and three-point bending tests. Indeed, sinterability of a pulverulent material depends on its granulometry while creep properties are linked to the microstructure of the related dense bodies.

Figure 1 .

 1 Figure 1. Schematic view of the layer configuration in the anode supported SOFC. The connected spheres represent the skeleton of the metal foam.

Figure 2 .

 2 Figure 2. (a) Longitudinal section of a 2D-asymmetrical element extracted from the cylindrical cell and (b) its corresponding meshing. One layer corresponds to one grey level.

Figure 3 .

 3 Figure 3. Geometrical elements used to model (a) the technical shrinkage and (b) the 3point bending tests.

Figure 4 .

 4 Figure 4. (a) Dilatometric (unbroken line) and technical shrinkage (dashed line) curves; the inset represents the measurement configuration in the sample-holder of the dilatometer; (b) evolution of the experimental (open squares) and simulated (light grey unbroken line) porosity of the YSZ material as a function of the temperature.

Figure 5 .

 5 Figure 5. Evolution of (a) the porosity and (b) the deflexion of the sample versus the temperature. Symbols and unbroken light grey lines are used for experimental data and simulation, respectively.

Figure 6 .

 6 Figure 6. Finite element modelling of the curvature of the complete cell during sintering at (a) 1100 °C, (b) 1200 °C and (c) 1300 °C. The grid arrangement (in red) symbolizes the initial meshing of the cell.

TABLE I .

 I Thickness of each layer implemented in the thermo-mechanical model.

	Designation	Composition	Initial thickness	Initial porosity
	Current collector	NiCrAl-LST	600 µm	15%
	Anode	LST (or LST-GDC10)	50 µm	45%
	Electrolyte	YSZ	15 µm	1%
	Diffusion barrier	GDC10	2 µm	1%
	Cathode	LSCF48	30 µm	45%
	Cell	-	t cell = 697 µm	-

TABLE II .

 II Refined mechanical parameters and final porosities for all materials of the anode-architectured cell.

	Parameters	GDC10	YSZ	LSCF48	LST	LSCF48 GDC10-	LST GDC10-	NiCrAl
	A o /s -1	8.25×10 5	1.43×10 9	5.12×10 4	4.50×10 3	5.32×10 5	1.29×10 3	-
	K o	12.1	2.11×10 3	208	54.1	2.08×10 5	81.5	-
	β	1.55	1.64	1.18	0.70	1.93	2.00	-
	f ∞ /%	1.20	1.60	10.4	9.50	15.0	4.40	-
	K 1 (= K 2 )	0.940	0.948	0.985	0.567	0.937	0.951	-
	α exp /×10 6 K -1	12.8	12.7	16.5	14.4	14.3	13.8	14.9
	η o /MPa.s	8.26×10 -2	4.74×10 -4	4.81×10 -3	1.85×10 -2	4.81×10 -6	1.23×10 -2	-
	Q/kJ mol -1	264	370	225	240	225	202	30
	C o	1.50	1.50	1.10	1.50	1.10	1.50	-
	n c	1.60	1.20	1.80	1.20	1.80	1.20	-
	F o	1.00	1.00	1.00	1.00	1.00	1.00	-

& & & & &

Acknowledgments

The research leading to these results has received funding from the European Union's Seventh Framework Programme (FP7/2007-2013) for the Fuel Cells and Hydrogen Joint Technology Initiative under grant agreement N° 303429. The author thanks the CerPoTech company for supplying the powders.


Zoé Chatzidakis

Milan Perera

A criterion for p-henselianity in characteristic p

come

A criterion for p-henselianity in characteristic p

Zoé Chatzidakis, Milan

Preliminaries

Throughout this paper, all fields have characteristic p > 0. First we recall some definitions and notations. Let O v := {x ∈ K | v(x) ≥ 0} be the valuation ring associated with v. It is a local ring, and

M v := {x ∈ K | v(x) > 0} is its maximal ideal. Let K v := O v /M v = {a = a + M v | a ∈ O v }
be the residue field (or K when there is no danger of confusion). We let K(p) denote the compositum of all finite Galois extensions of K of degree a power of p.

A valued field (K, v) is p-henselian if v extends uniquely to K(p). Equivalently (see [START_REF] Engler | Valued fields[END_REF], Thm 4.3.2), if it satisfies a restricted version of Hensel's lemma (which we call p-Hensel lemma) : K is p-henselian if and only if every polynomial P ∈ O v [X] which splits in K(p) and with residual image in K v [X] having a simple root α in K v , has a root a in O v with a = α. Furthermore, another result (see [START_REF] Engler | Valued fields[END_REF], Thm 4.2.2) shows that (K, v) is p-henselian if and only if v extends uniquely to every Galois extension of degree p.

The aim of this note is to give a complete proof of the following result:

Theorem. Let (K, v) be a valued field. (K, v) is p-henselian if and only if M v ⊆ {x p -x | x ∈ K}.
This result was announced in [START_REF] Koenigsmann | p-Henselian Fields[END_REF], Proposition 1.4, however the proof was not complete. The notion of p-henselianity is important in the study of fields with definable valuations, and in particular it is important to show that the property of p-henselianity is an elementary property of valued fields.

The proof we give is elementary, and uses extensively pseudo-convergent sequences and their properties. Recall that a sequence {a ρ } ρ<κ ∈ K κ indexed by an ordinal κ is said to be pseudo-convergent if for all α < β < γ < κ :

v(a β -a α ) < v(a γ -a β ). (1)
A pseudo-convergent sequence {a ρ } ρ<κ is called algebraic if there is a polynomial P in K[X] such that v P (a α ) < v P (a β ) ultimately for all α < β, i.e:

∃λ < κ∀α, β < κ (λ < α < β) ⇒ v P (a α ) < v P (a β ) . (2)
Otherwise, it is called transcendental.

We assume familiarity with the properties of pseudo-convergent sequences, see [START_REF] Kaplansky | Maximal Fields with Valuation[END_REF] for more details, and in particular Theorem 3, Lemmas 4 and 8.

Proof of the theorem

First, we prove a lemma in order to restrict our study to immediate extensions:

Observation. Let (K, v) be a valued field and (L, w) be a Galois extension of degree a prime ℓ. Then, if (L, w)/(K, v) is residual or ramified, w is the unique extension of v to L.

Proof. The fundamental equality of valuation theory (see [START_REF] Engler | Valued fields[END_REF], Thm 3.3.3) tells us that if L is a Galois extension of K, then

[L : K] = e(L/K)f (L/K)gd (3)
where e(L/K) is the ramification index, f (L/K) the residue index, g the number of extensions of v to L and d, the defect, is a power of p. Thus, as ℓ is a prime, if e(L/K)f (L/K) > 1, then necessarily g = d = 1, and in particular, v has a unique extension to L.

2

Now, let us prove the result announced in the preliminaries:

Theorem. Let (K, O v ) be a valued field of characteristic p. Then, (K, O v ) is p-henselian if and only if M v ⊆ K (p) := {x p -x | x ∈ K}.
Proof. The forward direction is an immediate application of the p-Hensel Lemma. Conversely, assume that M v ⊆ K (p) := {x p -x | x ∈ K}. Every Galois extension of K of degree p is an Artin-Schreier extension, i.e is generated over K by a root a of a polynomial X p -X -b = 0, with b ∈ K \ K (p) . The previous observation gives us the result when K(a)/K is not immediate. Let L be an immediate Galois extension of degree p and ṽ an extension of v to L (hence with the same value group Γ and residue field L = K as K). We can write L = K(a) where a p -a = b ∈ K \ K (p) .

Step 1: (Claim) The set Step 2: We extract a strictly well-ordered increasing and cofinal sequence from C. If we write P (X) := X p -X -b, we get a sequence {a ρ } ρ<κ in K such that the sequence {v P (a ρ ) } ρ<κ is stricly increasing and cofinal in C. Thus, the sequence {P (a ρ )} ρ<κ is pseudo-convergent (with 0 one of its limits). As v P (a α ) < 0, we have v(a β -a α ) = 1 p v P (a α ) = γ α for α < β < κ. Thus, the sequence {a ρ } ρ<κ is also pseudo-convergent. Furthermore, {a ρ } ρ<κ has no limit in K: if l ∈ K is a limit of {a ρ } ρ<κ then P (l) is a limit of {P (a ρ )} ρ<κ . As {v P (a ρ ) } ρ<κ is cofinal in C, v P (l) would be a maximal element of C: contradiction.

C = {v(x p -x -b) | x ∈ K} = v K (p) -b is contained in Γ <0 and has no last element. First observe that C ⊆ Γ ≤0 : if v(c p -c -b) > 0, then the equation X p -X + (c p -c -b) has a root in K, so that (a -c) ∈ K: contradiction. Let γ ∈ Γ, d ∈ K such that v(d p -d -b) = γ. As L/K is immediate there is c ∈ K such that ṽ a -(d + c) > ṽ(a -d). If ṽ(a -d) = 0 then ṽ a -(d + c) > 0 and (d + c) p -(d + c) -b = (d + c -a) p -(d + c -a) in M v ,
Step 3: (Claim) Let P 0 (X) ∈ K[X], and assume that v P 0 (a α ) is strictly increasing ultimately. Then deg P 0 (X) ≥ p. We take such a P 0 of minimal degree, assume this degree is n < p, and will derive a contradiction. One consequence of Lemma 8 in [START_REF] Kaplansky | Maximal Fields with Valuation[END_REF] is that:

v P 0 (a ρ ) = δ ′ + γ ρ ultimately for ρ < κ (4)
where δ ′ is the ultimate valuation of P ′ 0 (a ρ ) and γ ρ is the valuation of (a σ -a ρ ) for ρ < σ < κ (which does not depend on σ as {a ρ } ρ<κ is pseudo-convergent). We write P (X) = m i=0 h i (X)P 0 (X) i with deg(h i ) < n, ∀i ∈ {1, . . . , m}. Then, {h i (a ρ )} ρ<κ is ultimately of constant valuation, and we let λ i be this valuation. As {a ρ } ρ<κ has no limit in K, it is easy to see that n > 1, so that m < p. By Lemma 4 in [START_REF] Kaplansky | Maximal Fields with Valuation[END_REF], there is an integer i 0 ∈ {1, . . . , m} such that we have ultimately:

∀i = i 0 (λ i + iδ ′ ) + iγ ρ > (λ i 0 + i 0 δ ′ ) + i 0 γ ρ . (5)
Then, ultimately:

pγ ρ = v P (a ρ ) = v m i=0 h i (a ρ )P 0 (a ρ ) i = λ i 0 + i 0 (δ ′ + γ ρ ). (6)
Thus, we have ultimately (p -i 0 )γ ρ = λ i 0 + i 0 δ ′ . As p > m ≥ i 0 , the left hand side of the equation increases strictly monotonically with ρ. But the right hand side is constant: it has no dependence in ρ! We have a contradiction, thus n = p.

Step 4: Clearly, {a ρ } ρ<κ is of algebraic type. By Theorem 3 in [START_REF] Kaplansky | Maximal Fields with Valuation[END_REF], if a ∞ is a root of P , we get an immediate extension (L ′ , v ′ ) = (K(a ∞ ), v ′ ). Let a ∞ = a, we have (K(a), v ′ ) isomorphic to (K(a), ṽ). Thus:

∀Q ∈ K p [X] ṽ Q(a) = v ′ Q(a) = v Q(a ρ ) ultimately (7)
This shows the uniqueness of ṽ and concludes the proof of the theorem.

  which as above give a contradiction. Hence ṽ(a -d) < 0, and from d p -d -b = (d -a) p -(d -a), we deduce that γ = pṽ(a -d) < 0, and v (d + c) p -(d + c) -b = p ṽ a -(d + c) > γ. This shows the claim.

* Partially supported by ANR-13-BS01-0006

Adresses and contacts of the authors: DMA, ENS -45 rue d'Ulm, 75230 Paris cedex 05, FRANCE e-mail: zoe.chatzidakis@ens.fr, milan.perera@ens.fr