Ali Boudani
email: ali.boudani@irisa.fr

Alexandre Guitton
email: alexandre.guitton@irisa.fr

Bernard Cousin
email: bernard.cousin¡@irisa.fr

GXcast: Generalized Explicit Multicast Routing Protocol

Recently several multicast mechanisms were proposed that scale better with the number of multicast groups than traditional multicast does. These proposals are known as small group multicast (SGM) or explicit multicast (Xcast). Explicit multicast protocols, such as the Xcast protocol, encode the list of group members in the Xcast header of every packet. If the number of members in a group increases, routers may need to fragment an Xcast packet. Fragmented packets may not be identified as Xcast packets by routers. In this paper, we show that the Xcast protocol does not support the IP fragmentation. We show also that avoiding fragmentation limits the group size that can be handled by the Xcast protocol. First, we describe the Xcast protocol, the Xcast+ protocol (which is an extension of Xcast) and we compare these two protocols with traditional multicast protocols. We propose then a generalized version of the Xcast protocol, called GXcast, intended to permit the Xcast packets fragmentation and to support the increasing number of members in a multicast group. The behavior of the GXcast protocol is analyzed according to several criteria. Finally, we present and evaluate with simulations an improvement to GXcast and we conclude that GXcast is a feasible and promising protocol.

I. INTRODUCTION

Multicast, the ability to efficiently send data to a group of destinations, has become increasingly important with the emergence of network-based applications like IP telephony, video-conferencing, distributed interactive simulation and software upgrading. A multicast routing protocol should be simple to implement, scalable, robust, use minimal network overhead, consume minimal memory resources, and inter-operate with other multicast routing protocols.

Most of proposed multicast protocols like DVMRP [START_REF] Waitzman | Distance Vector Multicast Routing Protocol[END_REF] and MOSPF ( [START_REF] Moy | Multicast Extensions to OSPF[END_REF], [START_REF] Moy | MOSPF: Analysis and Experience[END_REF]) perform well if group members are densely packed. However, the fact that DVMRP periodically floods the network and the fact that MOSPF sends group membership information over the links, make these protocols not efficient in cases where group members are sparsely distributed among regions and the bandwidth is not plentiful.

To address these issues, the Protocol Independent Multicast (PIM) routing protocols are being developed by the Inter-Domain Multicast Routing (IDMR), working group of the IETF. PIM contains two protocols: PIM-Dense Mode (PIM-DM) [START_REF] Nicholas | Protocol Independent Multicast-Dense Mode (PIM-DM): Protocol Specification (Revised)[END_REF] which is adapted to groups where members are densely distributed, and PIM-Sparse Mode (PIM-SM) [START_REF] Estrin | Protocol Independent Multicast-Sparse Mode (PIM-SM): Protocol Specification[END_REF] which is adapted to group where members are sparsely distributed. Although these two protocols share similar control messages, they are essentially proposed for two different kinds of applications.

Traditional multicast protocols [START_REF] Ramalho | Intra-and Inter-domain multicast routing protocols: A survey and taxonomy[END_REF] can be used to minimize bandwidth consumption by using a delivery tree. However, a router has to keep a forwarding state for every multicast tree passing through it. Thus, traditional multicast protocols suffer from a scalability problem with the number of concurrently active multicast groups. Indeed, the number of forwarding states grows with the number of groups.

There seem to be two kinds of multicast that are important: a broadcast-like multicast that sends data to a large number of destinations and a narrow-cast multicast that sends data to a fairly small group. An example of the first kind of multicast is the audio and video multicasting of a presentation to all employees in a corporate intranet. An example of the second kind of multicast is a videoconference involving three or four parties [START_REF] Ramalho | Intra-and Inter-domain multicast routing protocols: A survey and taxonomy[END_REF]. Thus, a one-size-fits-all protocol will be unable to meet the requirements of all applications [START_REF]Reliable Multicast Transport IETF Working Group Web Site[END_REF]. Providing for many groups of small conferences (a small number of widely dispersed people) with global topological scope scales badly given the current multicast model [START_REF] Deering | Overview of the 1998 IAB Routing Workshop[END_REF].

Recently several multicast mechanisms were proposed that scale better with the number of multicast groups than traditional multicast does. These proposals are known as small group multicast (SGM) [START_REF] Ooms | Taxonomy of Xcast/SGM proposals[END_REF] or explicit multicast (Xcast) [START_REF] Boivie | Explicit multicast (Xcast) basic specification[END_REF]. Explicit multicast protocols, such as the Xcast protocol, encode the list of group members in the Xcast header of every packet. Xcast assumes that there is no packet fragmentation. However, if fragmentation occurs (e.g. if the group size or the data is too large) the fragmented packets will not be identified as Xcast packets by routers. In this paper we propose a generalized Xcast protocol to support the group size increasing and to overcome the fragmentation problem.

In Section II, we describe the Xcast [START_REF] Boivie | Explicit multicast (Xcast) basic specification[END_REF] protocol, the Xcast+ [START_REF] Yong-Jin | Explicit multicast extension (Xcast+) for efficient multicast packet delivery[END_REF] protocol (which is an extension of Xcast) and we present the Xcast packets fragmentation problem. In Section III, we describe the GXcast protocol and we study the effect of its parameter on several criteria. In Section IV, we propose an improvement of the GXcast protocol and we evaluate the gain in performance. Finally, we conclude in Section V that the GXcast protocol is feasible and promising.

II. THE XCAST AND THE XCAST+ PROTOCOLS

To solve the problems of traditional multicast protocols, Boivie et al. propose the Explicit Multicast protocol (Xcast). In this section, we describe the Xcast protocol [START_REF] Boivie | Explicit multicast (Xcast) basic specification[END_REF], the Xcast+ protocol [START_REF] Yong-Jin | Explicit multicast extension (Xcast+) for efficient multicast packet delivery[END_REF] (which is an extension of Xcast) and we compare them with traditional multicast protocols.

A. The Xcast protocol

The Xcast protocol [START_REF] Boivie | Explicit multicast (Xcast) basic specification[END_REF] is a newly proposed multicast protocol to support a very large number of small multicast groups. To send data to a given group, the source first explicitly encodes the list of destinations in the Xcast header of the packet. Then, the source parses the header, partitions the destinations based on each next unicast hop and forwards a packet with an appropriate header to each of the next hops. Each router along the path to destinations repeats the same processing on receiving an Xcast packet. If a router detects that there is only one destination in the destination list of a packet, the packet is converted to unicast. The algorithm realizing the conversion of an Xcast packet to a unicast packet is called Xcast-to-Unicast (X2U). This packet is then forwarded in unicast along the remainder of the route. Example: consider the network represented by Figure 1 and the group formed from the source ¡ and the six destinations ¢ ¤£ , ¢ ¦¥ , ¢ ¨ § , ¢ ¦© , ¢ ¦ and ¢ ¨ . The source ¡ generates an Xcast packet with the destination list ¢ £ ¢ ¨¥ ¢ ¨ § ¢ ¦© ¢ ¨ ¢ ¦ "! . ¡ proceeds the packet and remarks that # $£ is the next unicast hop for all the destinations. Consequently, ¡ sends the Xcast packet to # $£ . # $£ receives the packet and proceeds it similarly. It forwards the packet to the # ¥ router which also forwards it to # § . While proceeding the packet, the # § router remarks that # © is the next unicast hop for the two destinations ¢ £ and ¢ ¥ and that # is the next unicast hop for the remaining destinations ¢ § , ¢ © , ¢ and ¢ . # § sends to # © an Xcast packet containing the destination list ¢ %£ & ¢ ¨¥ '! and to # ( an Xcast packet containing the destination list )¢ § 0 ¢ ¦© 0 ¢ ¦ 0 ¢ ¨ '! . Upon receiving the Xcast packet, the # (© router detects that ¢ ¤£ and ¢ ¨¥ are two separated stations. # 1© then generates two unicast packets using the X2U algorithm and sends them to ¢ %£ and ¢ ¨¥ . Upon receiving the packet, ¢ 2£ extracts from the packet the data. The process is similar for routers # ( to # (3 and for the five remaining destinations.

B. The Xcast+ protocol

Xcast+ is an extension of Xcast for a more efficient delivery of multicast packets [START_REF] Yong-Jin | Explicit multicast extension (Xcast+) for efficient multicast packet delivery[END_REF]. Every source or destination is associated to a Designated Router (DR). Instead of encoding in the Xcast packet header the set of group members, Xcast+ encodes the set of their DRs. When a new member wants to join the group of source ¡ , it sends an IGMP-join message [START_REF] Cain | Internet Group Management Protocol, version 3[END_REF] to its DR. The DR will send a join-request message to the source ¡ . The DR of the source intercepts this message and analyzes it in order to keep track of all concerned DR addresses. When the source ¡ wants to send a message to the group , it sends a multicast packet. This packet is received by its DR and converted to an Xcast packet using the Multicast-to-Xcast algorithm (M2X).

The packet is then forwarded as in Xcast to the DRs, since the destination list in the Xcast header contains the DR addresses instead of the member addresses. Then, each DR converts the Xcast packet to a multicast packet using the Xcast-to-Multicast protocol (X2M) and sends it in its subnetworks.

Example: consider the same network represented by sends back to # 3 a registration reply message and does not forward the registration request message to ¡ . Thus, # $£ is able to know dynamically the set of DRs of the receivers and can fill the destination list of Xcast packets on receiving multicast packet from ¡ .

C. The IP fragmentation mechanism

Due to physical reasons, every link can transfer only a limited volume of information in each packet. The Internet protocol (IP) [START_REF] Postel | Internet Protocol[END_REF] contains a mechanism called fragmentation which makes this limitation transparent for end stations.

The fragmentation mechanism allows a packet to be cut into fragments in order to be suitably transferred on a link. Suppose that a router receives a packet. After having decided on which link this packet should be forwarded, the router checks the maximum capacity of this link which is the Maximum Transmission Unit (MTU). If the packet is too large and unless it is explicitly forbidden, the router cuts out it in order to respect the following constraints: each resulting fragment is an autonomous IP packet, with a valid IP header, each resulting fragment has a size less than or equal to the MTU, and the data is distributed between the fragments.

The algorithm used to fragment IPv4 packets is explained in [START_REF] Postel | Internet Protocol[END_REF]. The IPv6 protocol also have a fragmentation mechanism, described in [START_REF] Deering | Internet Protocol, version 6 (IPv6) Specification[END_REF]. Note that one goal of IPv6 is to avoid fragmentation. This will be discussed later.

D. Xcast packet fragmentation

Let us consider the Xcast packet fragmentation in a router. Since the Xcast packet header may be large, two cases can be considered: either the whole Xcast packet header is short enough to be kept in the first fragment, or the Xcast header has to be cut out. In both cases, the second fragment is not a valid Xcast packet since it has no Xcast header. Thus, these packets cannot be forwarded to receivers and the data they contain is lost. Moreover, in the second case the first fragment contains only a subset of receivers and no data. The first fragment may however be forwarded up to the mentioned receivers, inducing meaningful traffic.

These problems show that the fragmentation of an Xcast packet should be forbidden. This can be done in IPv4 by setting the appropriate flag (Don't Fragment, DF) in the IP header. In order to reach the receivers, the source has to limit the size of its packets to ¡ £¢ ¥¤ bytes which is the minimum MTU guaranteed by IPv4 on any link. This size limits the number of receivers in an Xcast group to ¦ ¨ § © (see Section III-B.1). However, the number of members of the group in Xcast is stored using a ¢ bits field, which leads to group containing no more than ¦ ¢ members. We consider that these two limits are too restrictive. What we propose is a simple mechanism to cancel these limitations in the size of Xcast groups. The performance and the scalability of our proposition will be analyzed.

III. THE GXCAST PROTOCOL

As explained in the previous section, the Xcast protocol can not support large groups due to its incompatibility with the IP fragmentation mechanism. In this section, we propose a generalized Xcast routing protocol, the GXcast protocol, which is designed basically to avoid the fragmentation. Moreover, the GXcast protocol can be parameterized in order to improve the Xcast behavior.

A. The GXcast protocol

The GXcast protocol is a simple generalized version of the Xcast protocol: instead of sending a message to the destinations, the source limits the number of destinations in a packet to . Thus, the list of destinations is cut into sub-lists of at most destinations. Each sub-list corresponds to a destination list for an Xcast packet. Several packets may have to be sent in order to deliver data to all the destinations.

is the parameter of the GXcast protocol and it impacts the protocol performance in terms of several criteria. The choice of is justified in section III-B. GXcast packets are similar to Xcast packets: they have the same header and are treated in the same way by intermediate routers, DR destinations and destinations. The only difference between the Xcast protocol and the GXcast protocol is the packet process at the source or at the DR of the source. The Xcast protocol and the GXcast protocol can therefore inter-operate easily.

Example: consider the same network represented by Figure 2 and the group formed from the source ¡ and the six members ¢ %£ , ¢ ¦¥ , ¢ ¦ § , ¢ © , ¢ ¦ and ¢ ¨ . As in the Xcast+ protocol, the DR of the source keeps track of only the three DRs representing the subnetworks that contain all the destinations: # 1© , # ¡ and # (3 . For this example, is fixed to1 . The source sends a multicast packet to its DR, # $£ . # £ translates it from a multicast packet to an Xcast packet using the M2X algorithm. # £ notices that there are three destinations in the list for the next hop. Since equals to , this list is cut into two sub- lists: one contains the first two destinations # © and # and the second contains the last destination, # 32 . Each generated packet is treated as a normal Xcast packet as shown on Figure 2.

B. Study of the GXcast parameter

The behavior of the GXcast protocol greatly depends on the value of the parameter. Indeed, as we will see in this subsection, there is a number of criteria that are directly influenced by the chosen value. In the following, we denote by ¢ ¤£ ¦¥ the value of the minimum guaranteed MTU which depends on the IP version used, by § the size of the IP header plus the size of the Xcast header and by © the size of an IP address. will represent the number of destinations in the group and the volume in bytes of data to transfer.

1) Simple behavior:

As we have seen in subsection II-C, since a packet has to contain at least one byte of data, the maximum number of destinations allowed in an Xcast packet is defined as: ! ¢ ¤£ ¦¥ #" § " ¦ © $ %

The values & ¦ § © and ' ¢ ¤ correspond respectively to the IPv4 and to the IPv6 specifications. The simplest behavior GXcast can have is to fix the value to the value. However, this is not efficient for groups having a lot of members (typically more than ). For example, suppose that IPv6 is used and suppose that ( ¢ 0)

members have joined the group. Each message can contain only ¦ ) © bytes of information 3 . In order to send a volume of ¦ ) 1) ) 1) bytes, 2 ¢ packets are needed. However, less packets would be the result of a better choice of . Choosing equals to § 13 allows ¤ ¦ ¤ bytes per packet, which results into the emission of only § © packets to reach the destinations. This is approximately three times less.

2) The number of members influenced by a fault: If a drop occurred on a GXcast packet, every member having its address in the member list will be concerned by the drop. To reduce the number of destinations concerned by such errors, small values of should be chosen.

3) Number of generated packets: Considering a group of destinations and a volume of bytes to transmit to these members, the number of packets e f ! sent by the GXcast protocol with a parameter of is defined as: e g ! h pi q i ¢ ¤£ r¥ s" § " ẗ© % Iu wv yx ! q %

Recall that in the GXcast protocol, the list of destina- tions is cut into sub-lists of size at most . The left part of the expression of e represents the number of sub- lists that will be generated by the GXcast protocol. The right part of the expression of e represents the number of packets needed to transmit bytes of data. In order to study the behavior of e in terms of , we will consider two cases: and ( . In the first case, we have: e g ! i ¢ ¤£ ¦¥ #" § " % ẗ© q %

This expression of e does not depend on . The GXcast protocol behaves in this case in the same way than the Xcast protocol. In the second case where , we assume the following approximation:

e g ! ¢ ¤£ ¦¥ s" § " ẗ© % e g ! %

The e function admits a minimum value for: ¢ ¤£ ¦¥ #" § % © C % Since this optimal value does not depend neither on nor on , it is very simple to calculate and provides good results in terms of the number of generated packets, we propose it as a default value for the GXcast protocol.

4) Global processing time:

To send a fixed amount of data, several packets are generated. The global processing time 2! of the GXcast protocol is the sum of the header processing times of these packets. The global processing time for a GXcast packet having destinations is approximately I "£ ¥ ¨ , where "£ is the processing time of the IP and the GXcast header and ¥ is the processing time for an entry in the list of destinations (lookup in routing table, generation of packets per outgoing interface, etc.). We have then:

! h pi q 1
"£ ¥ % The ! function is strictly decreasing and admits a minimum for p . Meanwhile, choosing is not realistic as shown in section III-B.1. On the other side, choosing a small value for greatly increases the global processing time. The default value of , 1d fe Ig ¥ , leads to a global processing time which is close to the optimal and is therefore a good compromise.

5) Total delay added by the GXcast protocol:

Packets generated at the source or at branching routers may be delayed. At the source, packets destinated to at most members are sent successively. Therefore, the last generated packet experiences a larger delay than all previous packets. At a branching router, generated packets per outgoing interface are different and cannot be sent simultaneously. In the same way, latest packets are also delayed.

¡ ¡ ¡ ¡ ¢ ¡¢ ¢ ¡¢ ¢ ¡¢ ¢ ¡¢ £ ¡£ £ ¡£ £ ¡£ £ ¡£ ¤ ¡¤ ¤ ¡¤ ¤ ¡¤ ¤ ¡¤ ¥ ¡¥ ¥ ¡¥ ¥ ¡¥ ¥ ¡¥ ¦ ¡¦ ¦ ¡¦ ¦ ¡¦ ¦ ¡¦ § ¡ § § ¡ § § ¡ § § ¡ § ¡¡ ¡ ¡ ¨© ¡© © ¡© © ¡© © ¡© ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ! ¡! ! ¡! ! ¡! ! ¡! " ¡" " ¡" " ¡" " ¡" # ¡# # ¡# # ¡# # ¡# $ ¡$ $ ¡$ $ ¡$ $ ¡$ frag replacements % % % & (' & (' & (' & 0) & 0) & 0) 1 2' 1 2' 1 2' 1 3) 1 3) 1 3) 1 54 1 54 1 54
1 76 1 76 1 76 8 @9 6 8 @9 ) 8 @9 ' Let us consider the network represented by Figure 3, where the source is called ¡ and the A © members are respectively ¢ ¦ , ¢ , ¢ § and ¢ © . Let A ¢ ¤! be the added delay experienced by a destination ¢ . In a first time, we consider that © . ¡ sends a message to )¢ ¦ ¢ ¢ § ¢ © ! . The branching router # ¦ sends a first message to ¢ ¦ "! and a second message (slightly different) to ¢ ¢ § ¢ © ! . Thus, A ¢ ¦ "! ) and the three destinations ¢ , ¢ § and ¢ © will experience a small delay. Since the branching router # sends a first message to ¢ 0! , a second to )¢ § ! and a third to )¢ © ! . The added delay for the destination ¢ © will be the time for the router # ¦ to do a copy plus the time for the router # to do two copies, i.e., three units of time ( A )¢ © ! § ). The total added delay for © is ) ¦ ¦ § ¤ . For information, when ,

A )¢ ¦ "! ) , A )¢ 0! w ¦ , A )¢ § ! ¦ and A )¢ © ! ,
which yields a total added delay of © . When ( ¦ ,

A )¢ ¦ "! ) , A )¢ 0! w ¦ , A )¢ § ! and A )¢ © ! § ,
which yields a total added delay of ¤ . The choice of has an impact on the total delay added by the protocol. We have seen that A )¢ CB ! is due to the time for the source to send several packets and to the time for branching node routers to send several packets. Destination ¢ CB is in the DB " ¦ "! FE ! ¦ $ -th packet sent by the source. The time for the source to send the previous packets is GB " ¦ &! HE ¥ $ units of time. The number of branching nodes is harder to compute since this number depends on the topology, which is unknown by the protocol. However, in the worst-case, the packet to ¢ IB may be delayed DB " ¦ "! mod times. Thus, we have:

A )¢ CB ! QP GR £ 1 $ GB " ¦ &! mod .
Let S be such as TE US . For every member ¢ CB , we have: GB " ¦ "! FE $ S " ¦ and GB " ¦ "! mod TE VS " ¦ . Finally, we obtain A ¢ CB ! S r TE US " . This function admits a minimum value for S XW , thus, for TE VS & YW , the total added delay is limited to a minimum value interval.

We propose to choose W for applications that are delay-sensitive. For IPv4, the value for is 11 and for IPv6, the value for is 8.

C. Using Path MTU instead of minimum MTU

In Section III-B, we defined ¢ ¤£ r¥ as the minimum MTU guaranteed by IP. However, the value of the Path MTU (PMTU) can also be used since we do not make any assumptions on the stability of the ¢ ¤£ ¦¥ value in our study. The PMTU is the minimum value of the MTU on the links of a path. It can be noticed that the PMTU value is easy to obtain in GXcast, since unicast paths are used. Moreover, the IPv6 protocol stores the PMTU for unicast paths to every destinations.

IV. PROPOSED IMPROVEMENT However, a certain locality can be deduced from the longest common prefix of two IP addresses [START_REF] Postel | Internet Protocol[END_REF]. What we propose is to sort the destination list in the source in order to increase the chance of having a good regrouping. In order to analyze the computation time induced by this sort, three operations have to be considered: the join of a new member to the group, the leave of a member of the group, and the send operation, i.e., the cutting operation.

To reduce the complexity of the proposed sort, we choose to store the set of members as a red-black tree [START_REF] Cormen | Introduction to Algorithms[END_REF]. Inserting, searching or deleting a node in a redblack tree can be done in logarithmic time. In such a tree, the cutting operation can be realized in linear time with two steps: first, an infix walk of the tree produces a sorted list, and then, this sorted list is cut into sublists of elements. Thus, managing a sorted list of members can be done in an efficient way.

To evaluate the impact of the members list sort, we simulate the GXcast protocol on the Abilene topology [START_REF]Abilene Network[END_REF]. The Abilene topology is an experimental Internet2 backbone for educational and research purposes. It consists of ¦ ¦ nodes and ¦ © edges.

Figure 4 shows the average cost of the trees for # ¦ ) destinations, divided in packets of ¢ ) destinations, on ¦ ) 1) 1) simulations. The number of LAN per node varies from ¦ to ) . Class C IP subnetworks were randomly attributed to each LAN, which ensures that two IPs in the same LAN are close. There is no relation between two IPs of different LANs, even if these two LANs are connected to the same node. Since the IP are randomly chosen and uniformly distributed, the cost of the trees is § ) (since TE ¥ § , three trees are generated. Without sorting, each tree spans all the ¦ ¦ nodes of the Abilene topology, covering ¦ ) edges; the total cost of the three trees is therefore § ) ) for every simulation. Indeed, for each of the three trees, every node is represented at least once in the destination list and therefore, the eleven edges of the Abilene topology are used by each tree. When there are few LANs per node, sorting the destination list greatly reduces the cost of the trees. Indeed, some locality can be deduced: for example, a node that contains LAN having only high IP adresses is not represented in the destination list for the first tree. Figure 5 shows the average cost of the trees on the same topology when the number of packets increases and when there are ¡ LANs per node, for ¦ ) ) 1) simulations. The number of packets varies from ¦ to ) . As the number of packets increases, the cost of the trees without sorting grows linearly (each tree covers all nodes). The more packets are generated, the better the sorting performs. For ¦ ) generated packets, the gain of sorting is close to 50%. Therefore, sorting the destination list is very important in the GXcast protocol. The more packets are generated, the better the sorting performs.

V. CONCLUSIONS The Xcast and Xcast+ protocols permit to manage efficiently a large number of small multicast groups.

A major drawback of these protocols is that they are incapable to manage packet fragmentation. In addition, there is a limit for the multicast group size. In this paper, we proposed an extension to these protocols, named GXcast. GXcast solves the fragmentation problem of the Xcast protocol. We studied optimization criteria like sending less packets or minimizing the header processing time in routers. We showed that sorting the destination list using the IP adresses enhances the performance of the GXcast protocol. Finally, we deduced that the GXcast protocol could manage a large number of small size groups, with members in different sub-networks.

Fig. 1 .

 1 Fig. 1. The forwarding of data in the Xcast protocol.

Figure 2 Fig. 2 .

 22 Figure 2 and the group formed from the source ¡ and the five destinations ¢ ¤£ , ¢ ¨¥ , ¢ ¨ § , ¢ ¦© and ¢ ¨ . The figure shows where the M2X and X2M algorithms are used. Between the DR of the source and the DRs of the destinations, the packets are forwarded as normal Xcastpackets. Suppose that ¢ is a new member which wants to join the group . ¢ initiates the join of the group by sending an IGMP message for the group 4¡ 5 6! . The DR of ¢ , # 3 , receives the join request and sends a registration request message toward ¡ . When the DR of ¡ , # $£ , receives the registration request message, it

Fig. 3 .

 3 Fig. 3. Delay added by the GXcast protocol.

Fig. 4 .

 4 Fig. 4. The sorting performs better when there are few LANs per node.

  Fig. 5.The more packets are generated, the better the sorting performs.

This assumption is done to make the example easier. In real cases, 4 65 will usually have larger values.

An improvement to better choose the sub-lists will be described in section IV.

We consider that 7 98 A@ CB ED GF IH for IPv6, P 'Q SR A8 9F IT U B and V CW 8 XF IH . For IPv4, we would have taken 7 Y8 `T B 0D aF IH , P 'Q SR &8 'b Cc dH and V CW 8 &@ .


A Bruand

P Pe

Ârez Ferna Ândez

& O Duval

Use of class pedotransfer functions based on texture and bulk density of clods to generate water retention curves

Keywords: Soil horizon classi®cation, water retention, porosity, structure

Water retention properties of 219 horizons were measured in Cambisols, Luvisols and Fluvisols, mainly from the Paris basin. We derived class pedotransfer functions (class PTFs) based on texture alone and in a second stage class PTFs based on classes combining texture and clod bulk density. The performance of these two types of PTFs were discussed at ±330 and ±15000 hPa water potential on an independent set of 221 horizons. Results showed that PTFs based on sets grouped by texture and clod bulk density provide estimates with an accuracy that is (i) greater than with class PTFs based on texture alone, and (ii) similar to the estimation accuracy recorded with continuous PTFs. As a consequence, the lack of interest in class PTFs should be reconsidered to bridge the gap between the available basic soil data and hydraulic properties which are generally missing, particularly when pertinent soil characteristics can be derived from the data available in soil databases. The two types of class PTFs providing gravimetric water contents at seven water potentials ranging from ±10 to ±15 000 hPa were converted to volumetric water content using the soil bulk density. Finally, the parameters of van Genuchten's water retention curve model were computed for every class PTF.

I N T R O D U C T I O N

W ater retention properties of representative soil hor- izons of France have been studied for several years in order to test the validity of pedotransfer functions (PTFs) that can be found in the literature. Bastet et al. (1999) evaluated PTFs that predict points of the water retention curve and others that predict the parameters of a mathematical model ®tting the water retention curve. Their results showed that performance of the tested PTFs varied according to the pedological origin of the soils. Generally, the best performances were recorded with PTFs established on soils showing many similarities in parent material and pedogenesis. This is in agreement with Wo Èsten et al. (2001) who concluded that PTFs should not be used to make predictions for soils that are outside the range of soils used to derive the PTFs. Bastet et al. (1999) showed also that the performance of tested PTFs was not related to the number or type of basic soil characteristics that were used as predictors. Simple PTFs like those of Bruand et al. (1996) performed as well as the more complex PTF of Vereecken et al. (1989).

Since the water retained at a particular water potential is related to both the nature and arrangement of the elementary soil particles, most PTFs use particle size distribution, organic carbon content and bulk density as predictors in the regression equations (Wo Èsten et al. 2001). However, these estimators are not independent, bulk density being related to particle size distribution and organic carbon content. On the other hand, many PTFs use soil characteristics that are expressed both on a mass basis (particle size distribution, organic carbon content) and on a volume basis (bulk density, volumetric water content at a particular water potential). It is more consistent to use the same reference system for the estimators and predicted characteristics.

Besides continuous pedotransfer functions that enable the estimation of the parameters of models of the water retention curve (Rawls et al. 1992;Minasmy et al. 1999;Wo Èsten et al. 2001), there are class pedotransfer functions (class PTFs) that receive little attention because their accuracy is considered to be limited (Wo Èsten et al. 1995). Class pedotransfer functions provide average water contents at particular water potentials or one average water retention curve for each texture class. However, due to the range in particle size distribution, clay mineralogy (Leij et al. 1999 Wo Èsten et al., 1999), organic matter content and structural development within each texture class, water retention properties for individual soils can vary considerably. Despite their possible inaccuracies, class PTFs are easy to use because they require little soil information and are well adapted to prediction of water retention properties over large areas (Wo Èsten et al. 1995;Lilly et al. 1999;Wo Èsten et al. 1999).

In this article, we derived class PTFs based on texture alone; then, in a second stage, we derived class PTFs based on classes combining texture and clod bulk density. Clod bulk density was preferred to horizon bulk density because the latter includes macropores that do not intervene in water retention and vary in tilled topsoils with time and management. Accuracies of the two types of class PTFs were compared. We showed that taking both texture and clod bulk density into account leads to an increase in prediction accuracy of the gravimetric water content. Applications in agronomy and hydrology require the volumetric water content, so the gravimetric water content was converted into volumetric water content using the horizon bulk density. Finally, the parameters of the van Genuchten's water retention curve model (van Genuchten 1980) were computed for each class PTF.

M A T E R I A L S

The soils studied were Cambisols, Luvisols and Fluvisols (ISSS Working Group RB 1998), mainly from the Paris basin with some from the western coastal marshlands and from the Pyrenean piedmont plain. The class PTFs were derived from a set of 219 horizons comprising 58 A horizons (54 ploughed), 16 E horizons, 98 B horizons and 47 C horizons. Another set of 221 horizons was used to test the proposed class PTFs. These horizons comprised 29 A horizons (23 ploughed), 15 E horizons, 119 B horizons and 58 C horizons. The soils developed on a large range of parent materials (Table 1). For the set of horizons used to establish the PTFs and according to the FAO triangle of textures, texture was very ®ne (VF) for 22 horizons, ®ne (F) for 42 horizons, medium ®ne (MF) for 62 horizons, medium (M) for 69 horizons and coarse (C) for 24 horizons (Figure 1a, Table 2). For the set of horizons used to test the PTFs, texture was VF for 25 horizons, F for 105 horizons, MF for 39 horizons, M for 35 horizons and C for 17 horizons (Figure 1b).

M E T H O D S

The horizons were sampled in winter when the soil was near to ®eld capacity. Undisturbed samples 100±1000 cm 3 in volume were collected and stored at 5°C in sealed plastic containers to reduce biological activity and to avoid water loss. The dry bulk densities of 114 horizons (D h b ) were measured using 1236 cm 3 cylinders. (These horizons are representative of the whole set of horizons used to derive the class PTFs.) Particle size distribution was measured using the pipette method after pre-treatment of samples with hydrogen peroxide and sodium hexametaphosphate (Robert & Tessier 1974). The cation exchange capacity (CEC, in cmol c kg ±1 of oven-dried soil) was measured using the cobalt-hexamine trichloride method (Ciesielski & Sterckeman 1997) and organic carbon by oxidation using excess potassium bichromate in sulphuric acid at 135°C (Baize 2000). Results were expressed by mass after drying at 105°C. Clods 5±10 cm 3 in volume were separated by hand from the stored samples. We measured the dry bulk density of the clods at ®eld conditions (D c b , in cm 3 g ±1 of oven-dried soil) by using the kerosene method (Monnier et al. 1973). Gravimetric water contents (W, grams of water per gram of oven-dried soil) at the seven water potentials, h, viz. ±10 (W 10 ), ±33 (W 33 ), ±100 (W 100 ), ±330 (W 330 ), ±1000 (W 1000 ), ±3300 (W 3300 ), and ±15 000 hPa (W 15 000 ) were measured using pressure membrane or pressure plate apparatus. Clods were placed on a paste made of <2 mm particles of kaolinite to establish continuity of water between the clods and the membrane or the porous plate of the apparatus (Bruand et al. 1996). Speci®c water content and volume were expressed with respect to the dry mass of the sample after oven-drying at 105°C for 24 hours. Twelve to ®fteen clods were used for each sample to determine the mean values of W at the different values of water potential. For horizons used to test the class PTFs we measured W 330 and W 15 000 only, using similar methodology.

R E S U L T S A N D D I S C U S S I O N

The soils studied The mean composition of the horizons used to test the class PTFs was somewhat ®ner, since the clay content was 8.4% greater and the sand content was 6.5% smaller (CEC 2.6 cmol c kg ±1 greater) than in the set of horizons used to derive the class PTFs (Table 2). The mean organic carbon content Table 2. Characteristics of the horizons used to derive the class of pedotransfer functions and of those used to test their validity.

Particle size distribution (%)

Organic CaCO and the mean D c b were similar for the two sets of horizons. The mean water content of the horizons used to test the class PTFs was 0.031 g g ±1 greater at ±330, and was 0.035 g g ±1 greater at ±15 000 hPa because of the greater mean clay content (Table 2). Figure 2

3 CEC D c b Gravimetric water content W log(-h) (g g ±1 ) carbon (g kg ±1 ) (cmol c (g cm 3 ) < 2 mm 2±50 mm 0.05±2 mm (g kg ±1 ) k g ±1 ) W 1.0 W 1.5 W 2.0 W 2.5 W 3.0 W 3.5 W 4

Derivation of the class pedotransfer functions

For horizons belonging to a class of texture, we computed the mean W at the seven water potentials studied that de®ne the class PTF for each texture class (Table 2). Then within each texture class we used D c b to derive a new set of class PTFs. Unlike D h b , D c b excludes the large pores that do not contribute to water retention and D c b can be considered as an estimator of the effective volume of pores involved in the water retention properties. Thus for each texture class, we distributed the horizons according to D c b and, for each of these classes combining texture and D c b , we computed the mean W at the seven selected water potentials that de®ned the class PTF for the texture class and clod density considered (Table 3).

Performance analysis of the class pedotransfer functions

In order to discuss the validity of the class PTFs proposed, we computed the mean error of prediction (MEP w ) and the standard deviation (SDP w ) of prediction as follows:

MEP w 1 n n W p À W m Y SDP w 1 n n W p À W m À MEP 2 & ' 1a2
where W p is the predicted gravimetric water content, W m is the measured gravimetric water content and n is the number of horizons. The sign of MEP w indicates whether the PTF overestimated (positive) or underestimated (negative) the water content, whereas SDP w measures the precision of the predictions. To compare our results with the literature we computed also the root mean square error (RMSE q ) as follows:

RMSE 1 n n p À m 2 ! 1a2
where q p and q m are the predicted and measured volumetric water content, respectively. For horizons lacking a measured D h b , q p and q m were computed as follows:

q p = W p Q D c b q m = W m Q D c b
Thus the RMSE q that was computed using D c b as an estimator of D h b can be considered as an overestimation of the RMSE q of the measured D h b because:

D h b `Dc b
Validity of the class pedotransfer functions based on texture alone Overall, the class PTFs derived using texture alone underestimated W 330 and W 15 000 by 0.014 and 0.013 g g ±1 , respectively, and the precision was 0.038 and 0.031 g g ±1 , respectively (Table 4). The class PTFs underestimated W 330 and W 15 000 for all texture classes except for Coarse at ±330 hPa and ±15 000 hPa water potential and for Medium Fine at ±15 000 hPa water potential. The greatest underestimation was recorded for Fine textures (0.028 g g ±1 at ±330 hPa and 0.023 g g ±1 at ±15 000 hPa) and the greatest overestimation was recorded with Coarse (0.038 g g ±1 at ±330 hPa and 0.019 g g ±1 at ±15 000 hPa) (Table 4). The least precise was the Fine class irrespective of the water potential, and the most precise at ±330 hPa for the Medium Fine class and at ±15 000 hPa for the Coarse class.

Validity of the class pedotransfer functions based on texture and bulk density of clods

When the set of horizons was taken as a whole, results showed a smaller underestimation (0.009 g g ±1 at ±330 hPa and 0.011 g g ±1 at ±15 000 hPa) and a greater precision (0.021 g g ±1 at ±330 hPa and 0.022 g g ±1 at ±15 000 hPa) than with class PTFs based on texture alone (Table 4). Thus the estimation bias was reduced by 36 and 18% and the precision increased by 45 and 29% at ±330 hPa and ±15 000 hPa water potential, respectively. When the bias was greater than 0.010 g g ±1 , analysis of the results for each texture class showed a decrease in the estimation bias, with the class PTFs based on texture and clod density. Precision for all textures increased at ±330 and ±15 000 hPa water potential with the greatest increase in the Fine class. Results showed that RMSE q 's recorded at ±330 and ±15 000 hPa water potential were smaller than with class PTFs based on texture alone (Table 4) and similar to those recorded with continuous PTFs (Wo Èsten et al. 2001).

Proposed use of the class pedotransfer functions

Estimation of the water retention properties of a soil pro®le requires the component horizons to be identi®ed. Then, from the texture class in the FAO triangle for each horizon and using either the PTFs based on texture or texture + D c b , the water retention properties can be estimated (Figure 3). Our results showed that PTFs based on texture and clod bulk density provide estimates of gravimetric water content of soil horizons at seven water potentials ranging from ±10 to ±15 000 hPa with an accuracy that is (i) greater than with class PTFs based on texture alone, and (ii) similar to the estimation accuracy recorded with continuous PTFs. As a consequence, the lack of interest in class PTFs should be reconsidered to bridge the gap between basic soil data which is available and missing hydraulic properties (Wo Èsten et al. 2001), particularly when pertinent soil characteristics can be derived from the data available. For example, the clod bulk density, which is more pertinent than the horizon bulk Appendix II. Volumetric water contents at the different water potentials using the class pedotransfer functions based on texture and bulk density (see Table 2) and parameters of the van Genuchten's (1980) Volumetric water content, q log(-h) (cm 3 cm ±3 ) Parameters of van Genuchten's model class q 1.0 q 1.5 q 2.0 q 2.5 q 3.0 q 3.5 q 4.2 q s q r n a R 2

Very

Figure 1 .

 1 Figure 1. Texture of the horizons used to derive the class pedotransfer functions (a) and of those used to test their validity (b). For texture codings, see Materials section.

  shows that D c b À D h b ranged from ±0.05 to +0.35 g cm ±3 for A horizons with a maximum for +0.05 < D c b À D h b `+0.15 g cm ±3 , from ±0.05 to +0.25 g cm ±3 for E and B horizons with a maximum for ±0.05 < D c b À D h b `+0.05 g cm ±3 and from ±0.05 to +0.15 g cm ±3 for C horizons with a maximum for ±0.05 < D c b À D h b `+0.05 g cm ±3 . These differences between D c b and D h b are consistent with a decrease in the macroporosity with depth.

Figure 2 .

 2 Figure 2. Distribution of horizons according to the difference between the bulk density measured on clods (D c b ) and the bulk density measured with large cylinders (D h b ). The classes showing D c b ±D h b are centred on ±0.1, 0.0, +0.1, +0.2, +0.3 and correspond to the intervals ]±0.05, +0.05], ]+0.05, +0.15], ]+0.15, +0.25] and ]+0.25, +0.35].

  1. When D c b is unknown (i.e. not measured and cannot be inferred from D h b or from any other horizon characteristics) and if the horizon characteristics known are within the range of horizons used to derive the class PTFs (Table 2), we can use the class PTFs based on texture alone. Then, by selecting the appropriate D h b proposed in Appendix I, values of q are available. If D h b were not measured, it might be inferred from other horizon characteristics or from other soil data. Finally, parameters of van Genuchten's model (1980) that were computed for each class PTF using the RETC code (van Genuchten et al. 1991) are shown in Appendix I and the corresponding curves in Figure 4. 2. When D c b is known (i.e. measured or inferred from D h b or from other horizon characteristics) and if the horizon characteristics known are within the range of horizons used to derive the class PTFs (Table 2), we can use the class PTFs based on texture and bulk density of clods that are shown in Appendix II to provide values of q.

Finally

  , parameters of the van Genuchten's model (1980) that were computed for each class PTF as in point 1 are shown in Appendix II. C O N C L U S I O N

Figure 4 .

 4 Figure 4. Water retention curves for different values of horizon bulk density (D h b ) corresponding to the class PTFs based on texture alone and computed using van Genuchten's model (1980) (see Appendix I). For texture codings, see Materials section.

Fine

  model adjusted on the volumetric water contents.

  ISTO, UMR CNRS-UO 6113, Universite  d'Orle Âans, Ge Âosciences, BP 6759, 45067 Orle Âans cedex 2, France. 2 INRA, Unite  de Science du Sol, Centre de Recherche d'Orle Âans, BP 20619, 45166 Olivet cedex, France.

; 1

Table 1 .

 1 Number of soil horizons per type of parent material.

	Parent material	Number
		of
	horizons
	Horizons used to derive the class pedotransfer functions (n = 219)
	Marls and limestones	46
	Clayey materials resulting from weathering	2
	Old alluvium (river terraces)	28
	Recent alluviums	24
	Colluviums	13
	Aeolian materials	106
	Horizons used to test the class pedotransfer functions (n = 221)
	Marls and limestones	97
	Clayey materials resulting from weathering	37
	Old alluvium (river terraces)	6
	Recent alluviums	23
	Colluviums	14
	Aeolian materials	44

Table 3 .

 3 Water retained at different water potentials (h) for each class of texture and bulk density of clods.

	Texture class	Class of D c b		D c b (g cm ±3 )	W 1.0	W 1.5	Gravimetric water content W log(-h) (g g ±1 ) W 2.0 W 2.5 W 3.0	W 3.5	W 4.2
	Very ®ne	[1.2±1.3]	mean	1.25	0.421	0.411	0.392	0.372	0.342	0.334	0.263
			SD	0.02	0.039	0.036	0.034	0.036	0.027	0.033	0.022
			min	1.22	0.341	0.346	0.337	0.327	0.301	0.280	0.235
			max	1.28	0.465	0.459	0.447	0.435	0.380	0.367	0.292
		]1.3±1.4]	mean	1.36	0.365	0.360	0.346	0.331	0.308	0.297	0.238
			SD	0.03	0.040	0.032	0.034	0.036	0.036	0.036	0.018
			min	1.32	0.293	0.310	0.281	0.260	0.240	0.228	0.199
			max	1.40	0.432	0.404	0.388	0.371	0.360	0.359	0.257
		]1.4±1.5]	mean	1.44	0.337	0.329	0.320	0.307	0.291	0.266	0.219
			SD	0.02	0.030	0.030	0.028	0.033	0.013	0.024	0.010
			min	1.42	0.316	0.308	0.300	0.284	0.282	0.249	0.212
			max	1.45	0.358	0.350	0.340	0.330	0.300	0.283	0.226
	Fine	[1.3±1.4]	mean	1.36	0.340	0.318	0.310	0.289	0.273	0.246	0.200
			SD	0.03	0.033	0.051	0.039	0.057	0.045	0.041	0.032
			min	1.33	0.301	0.276	0.280	0.236	0.234	0.217	0.167
			max	1.39	0.381	0.390	0.364	0.368	0.335	0.305	0.240
		]1.4±1.5]	mean	1.46	0.304	0.291	0.276	0.263	0.240	0.223	0.189
			SD	0.02	0.037	0.040	0.044	0.046	0.044	0.055	0.040
			min	1.43	0.268	0.249	0.227	0.212	0.182	0.153	0.139
			max	1.48	0.360	0.340	0.333	0.326	0.291	0.289	0.233
		]1.5±1.6]	mean	1.55	0.247	0.244	0.236	0.226	0.210	0.190	0.167
			SD	0.03	0.035	0.028	0.030	0.031	0.028	0.030	0.028
			min	1.51	0.154	0.164	0.152	0.129	0.129	0.117	0.111
			max	1.60	0.294	0.290	0.281	0.266	0.249	0.251	0.215
		]1.6±1.7]	mean	1.65	0.231	0.220	0.214	0.202	0.189	0.183	0.160
			SD	0.03	0.019	0.016	0.013	0.014	0.018	0.017	0.014
			min	1.62	0.202	0.199	0.197	0.185	0.172	0.164	0.134
			max	1.69	0.264	0.245	0.235	0.226	0.227	0.207	0.179
		]1.7±1.8]	mean	1.72	0.209	0.208	0.195	0.180	0.177	0.167	0.145
			SD	0.03	0.017	0.011	0.008	0.008	0.006	0.018	0.012
			min	1.71	0.181	0.197	0.182	0.172	0.171	0.160	0.132
			max	1.77	0.223	0.223	0.202	0.192	0.186	0.207	0.164
	D c

b : bulk density of clods; W log(-h) : gravimetric water content at water potential, h, in hPa.

Table 3 .

 3 Continued. , bulk density of centimetric clods; W log(-h) , gravimetric water content at water potential, h, in hPa.

	Texture class	Class of D c b		D c b (g cm ±3 )	W 1.0	W 1.5	Gravimetric water content W log(-h) (g g ±1 ) W 2.0 W 2.5 W 3.0	W 3.5	W 4.2
	Medium	[1.4±1.5]	mean	1.45	0.263	0.252	0.240	0.216	0.182	0.152	0.133
	®ne		SD	0.03	0.009	0.007	0.008	0.009	0.029	0.033	0.032
			min	1.41	0.247	0.239	0.229	0.204	0.126	0.090	0.079
			max	1.49	0.275	0.264	0.251	0.234	0.219	0.183	0.163
		]1.5±1.6]	mean	1.56	0.240	0.230	0.219	0.198	0.169	0.137	0.117
			SD	0.03	0.013	0.011	0.009	0.015	0.019	0.023	0.024
			min	1.51	0.219	0.213	0.205	0.170	0.132	0.096	0.068
			max	1.60	0.280	0.260	0.240	0.234	0.210	0.172	0.158
		]1.6±1.7]	mean	1.64	0.224	0.217	0.208	0.196	0.170	0.143	0.119
			SD	0.02	0.012	0.012	0.011	0.010	0.019	0.024	0.023
			min	1.60	0.201	0.195	0.190	0.178	0.125	0.084	0.074
			max	1.69	0.247	0.240	0.230	0.211	0.199	0.169	0.146
	Medium	[1.5±1.6]	mean	1.57	0.230	0.219	0.201	0.177	0.149	0.133	0.113
			SD	0.02	0.009	0.014	0.013	0.022	0.026	0.027	0.020
			min	1.52	0.213	0.190	0.176	0.146	0.111	0.083	0.078
			max	1.60	0.246	0.240	0.220	0.213	0.188	0.179	0.142
		]1.6±1.7]	mean	1.65	0.212	0.205	0.193	0.173	0.146	0.117	0.092
			SD	0.03	0.009	0.019	0.019	0.023	0.026	0.024	0.019
			min	1.60	0.173	0.152	0.137	0.110	0.099	0.089	0.055
			max	1.70	0.241	0.232	0.221	0.215	0.210	0.189	0.128
		]1.7±1.8]	mean	1.75	0.184	0.177	0.171	0.161	0.149	0.129	0.105
			SD	0.03	0.015	0.014	0.013	0.011	0.013	0.016	0.022
			min	1.70	0.165	0.156	0.149	0.140	0.130	0.093	0.063
			max	1.79	0.209	0.201	0.190	0.181	0.168	0.155	0.133
		]1.8±1.9]	mean	1.84	0.168	0.162	0.155	0.147	0.143	0.129	0.098
			SD	0.03	0.010	0.008	0.007	0.007	0.014	0.024	0.031
			min	1.80	0.156	0.148	0.145	0.136	0.124	0.091	0.061
			max	1.87	0.187	0.172	0.163	0.155	0.161	0.154	0.141
	Coarse	[1.6±1.7]	mean	1.65	0.191	0.168	0.127	0.110	0.086	0.069	0.054
			SD	0.02	0.032	0.022	0.019	0.019	0.018	0.020	0.017
			min	1.62	0.147	0.142	0.105	0.087	0.060	0.048	0.034
			max.	1.68	0.233	0.196	0.158	0.140	0.112	0.094	0.076
		]1.7±1.8]	mean	1.74	0.160	0.144	0.110	0.088	0.069	0.057	0.049
			SD	0.03	0.048	0.050	0.052	0.043	0.036	0.030	0.006
			min	1.70	0.083	0.072	0.047	0.033	0.029	0.021	0.018
			max	1.78	0.201	0.196	0.179	0.158	0.137	0.107	0.078
		]1.8±1.9]	mean	1.85	0.164	0.152	0.139	0.122	0.099	0.089	0.069
			SD	0.03	0.015	0.017	0.019	0.014	0.019	0.010	0.006
			min	1.81	0.145	0.133	0.119	0.107	0.080	0.077	0.065
			max	1.88	0.178	0.172	0.156	0.143	0.128	0.099	0.078
	D c b

Table 4 .

 4 Validity of the class pedotransfer functions derived after strati®cation by texture alone and after strati®cation by texture and clod bulk density. Flowchart illustrating the estimation method of the water retention properties for a soil horizon.

		n	Mean error of prediction (MEP W ) (g g ±1 )	Standard deviation of prediction (SDP W )	Root mean squared error (RMSE q ) (cm 3 cm ±3 )
			±330 hPa	±15 000 hPa	±330 hPa	±15 000 hPa	±330 hPa	±15 000 hPa
	All textures together without any strati®cation
		219	±0.030	±0.034	0.076	0.097	0.121	0.110
	After strati®cation by texture alone
	Very ®ne	22	±0.003	±0.015	0.030	0.025	0.040	0.037
	Fine	42	±0.028	±0.023	0.048	0.036	0.079	0.061
	Medium ®ne	62	±0.001	0.003	0.020	0.023	0.032	0.037
	Medium	69	±0.006	±0.012	0.027	0.028	0.045	0.049
	Coarse	24	0.038	0.019	0.027	0.021	0.078	0.046
	All textures together after textural strati®cation	219	±0.014	±0.013	0.038	0.031	0.064	0.052
	After strati®cation by texture and bulk density of clods
	Very ®ne	25	0.010	±0.013	0.020	0.020	0.029	0.032
	Fine	105	±0.021	±0.020	0.022	0.022	0.047	0.047
	Medium ®ne	39	±0.002	±0.002	0.017	0.022	0.028	0.035
	Medium	35	±0.009	±0.011	0.019	0.025	0.039	0.047
	Coarse	17	0.032	0.018	0.020	0.016	0.067	0.045
	All textures together after strati®cation	221	±0.009	±0.011	0.021	0.022	0.044	0.045
	by texture and bulk density of clods

Universidade de Santiago de Compostela, Escuela Politecnica Superior, Campus Universitario, 27002 Lugo, Spain. *Corresponding author. Fax: 33 (0) 23841 7308. E-mail: Ary.Bruand@ univ-orleans.fr

density but not usually available in most databases, can be derived by combining data such as bulk density of the soil, structure and macroporosity development. Volumetric water content, which is required for applications in agronomy and hydrology, was derived from the estimated gravimetric water content using the horizon bulk density and parameters of the van Genuchten water retention curve model that we computed for each class PTF. Finally, because of the signi®cance of clay mineralogy and pedogenesis (Bruand & Tessier 2000), further development of class PTFs should take into account other criteria such as clay mineralogy, type of parent material or type of pedogenesis.

R E F E R E N C E S

Baize D 2000. Guide des analyses en pe Âdologie. INRA Paris. Bastet G Bruand A Voltz M Bornand M & Que Âtin P 1999. Performance of available pedotransfer functions for predicting the water retention properties of French soils. In: MTh. van Genuchten & F.J. Leij, eds, Characterization and measurement of the hydraulic properties of unsaturated porous media. Proceedings of the International Workshop, Riverside, California, October 22±24, 1997 pp 981±992. Bruand A Duval O Gaillard H Darthout R & Jamagne M 1996. Variabilite  des proprie Âte Âs de re Âtention en eau des sols : importance de la densite  apparente. Etude et Gestion des Sols 3, 27±40. Bruand A & Tessier D 2000. Water retention properties of the clay in soils developed on clayey sediments: Signi®cance of parent material and soil history. European Journal of Soil Science 51, 679± 688. Ciesielski H & Sterckeman T 1997. Determination of cation exchange capacity and exchangeable cations in soils by means of cobalt hexamine trichloride. Effects of experimental conditions. Agronomie 17, 1±7. Leij FJ Alves WJ van Genuchten MTh Williams JR 1999. The UNSODA unsaturated soil hydraulic database. In: MTh van Genuchten & FJ Leij, eds, Characterization and measurement of the hydraulic properties of unsaturated porous media. Proceedings of the International Workshop, Riverside, California, October 22±24, 1997 pp 1269±1281. Lilly A Wo Èsten JHM Nemes A Le Bas C 1999. The development and use of the HYPRES database in Europe. In: MTh van Genuchten & FJ Leij, eds, Characterization and measurement of the hydraulic properties of unsaturated porous media. Proceedings of the International Workshop, Riverside, California, October 22±24, 1997 pp 1283±1204. Van Genuchten MTh 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soil. Soil Science Society of America Journal 44, 892±898. Van Genuchten MTh Leij FJ & Yates SR 1991. The RETC code for quantifying the hydraulic functions of unsaturated soil. USDA, US Salinity Laboratory, Riverside CA. United States Environmental Protection Agency, document EPA/600/2±91/065. ISSS Working Group RB 1998. World reference base for soil resources: introduction, eds, J.A. Deckers, F.O. Nachtergaele & O.C. Spaargaren, 1st edition. International Society of Soil Science. ISRIC-FAO-ISSS-Acco Leuven Belgium. Misnamy B McBratney AB & Bristow KL 1999. Comparison of different approaches to the development of pedotransfer functions for waterretention curves. Geoderma 93, 225±253. Monnier G Stengel P & Fie Ás JC 1973. Une me Âthode de mesure de la densite  apparente de petits agglome Ârats terreux. Application a Á l'analyse des syste Ámes de porosite  du sol. Annales Agronomiques 24, 533±545. Rawls WJ Ahuja LR Brakensiek DL 1992. Estimating soil hydraulic properties from soils data. In: MTh van Genuchten & FJ Leij, eds, Indirect methods for Estimating the Hydraulic Properties of Unsaturated Soils. Proceedings of the International Workshop, Riverside, California, October 11±13, 1989, 329±340. Robert M & Tessier D 1974. Me Âthode de pre Âparation des argiles des sols pour les e Âtudes mine Âralogiques. Annales Agronomiques 25, 859±882. Vereecken H Maes J Feyen J & Darius P 1989. Estimating the soil moisture retention characteristics from texture, bulk density and carbon content. Soil Science 148, 389±403. Wo Èsten JHM Finke PA & Jansen MJW 1995. Comparison of class and continuous pedotransfer functions to generate soil hydraulic characteristics. Geoderma 66, 227±237. Wo Èsten JHM Lilly A Nemes A & Le Bas C 1999. Development and use of a database of hydraulic properties of European soils. Geoderma 90, 169±185. Wo Èsten JHM Pachepsky YA & Rawls WJ 2001. Pedotransfer functions: bridging the gap between available basic soil data and missing soil hydraulic characteristics. Journal of Hydrology 251, 123±150. Received August 2002, accepted after revision April 2003.

Appendix I. Volumetric water contents at the different water potentials using the class pedotransfer functions based on texture alone (see Table 2) and parameters of the van Genuchten's (1980) Volumetric water content, q log(-h) (cm 3 cm ±3 ) Parameters of van Genuchten's model q 1.0 q 1.5 q 2.0 q 2.5 q 3.0 q 3.5 q 4.2 q s q r n a R


Rapport

Arnaud Crolais

Monsieur Félix

Von Pechmann

Le plan Bus 2025 : la longue marche d'une innovation La rupture dans la continuité

de thèse professionnelle pour le master PAPDD, année universitaire 2015-2016. Pour le compte du département Matériel roulant bus (MRB) de la RATP.

e transport par autobus est une activité historique de la RATP depuis sa création en 1948. A côté de l'essor formidable des infrastructures ferroviaires, le mode bus a pu sembler un peu oublié et dénigré alors qu'il bénéficie de formidables atouts : sa proximité et la finesse de son maillage sur le territoire, sa flexibilité, son coût, son intégration dans la ville. Pour la RATP, le bus représente 3 millions de voyages par jour pour un chiffre d'affaires de 1,36Md€ en 2013, avec L

Le plan Bus 2025 : une innovation radicale

En mars 2014, la RATP annonce son intention de convertir son parc de bus diesel en un parc « zéro émission » en 2025 qui sera composé de 80% de bus électriques et 20% de bus biogaz (GNV). Le plan Bus 2025 vient alors modifier en profondeur l'outil de production de l'entreprise publique et redéfinir son modèle d'exploitation du réseau de surface. Cette transformation de son parc d'autobus, en éradiquant le diesel, a pour objectif de répondre aux enjeux de transition énergétique et de qualité de l'air, bien que le bus ait une empreinte très limitée à l'échelle de la métropole. En inscrivant le bus dans le développement de la ville durable, le plan est porteur d'une forte charge symbolique. La RATP choisit alors de développer, d'une part, la traction 100 % électrique en retenant la solution de charge lente en centre bus de nuit, et d'autre part la traction biogaz.

Cette décision radicale est prise alors qu'aucun bus disponible sur le marché n'est en mesure de répondre aux contraintes d'exploitation de la RATP et que l'entreprise défend depuis des an-nées la motorisation diesel norme Euro 6 qui revendique de bonnes performances environnementales. Il est également intéressant de relever que la RATP conduit depuis des années des expérimentations sur des alternatives au diesel (Aquazole, diester, GPL, gaz, hybride) sans les inscrire dans un plan global et aller jusqu'à l'industrialisation. La question qui est abordée ici est de comprendre comment la RATP décide de bousculer son modèle et de partir sur un développement de bus électriques qui n'est à ce stade pas industrialisé.

Pour aborder cette question, l'étude s'est focalisée sur la phase d'initialisation de l'innovation sans développer le processus de diffusion des savoir et d'exploration. Cela permettait d'identifier la manière dont a été prise la décision et d'en dégager des propriétés fondamentales qui pourrait être ap-pliquées pour conduire des changements ou des innovations dans d'autres contextes et domaines.

Méthodologie de l'étude de cas

La présente étude s'est appuyée dans un premier temps sur une analyse de la littérature en sciences humaines et social sur l'innovation et la décision en choisissant de s'intéresser au temps de la décision. A partir de références théoriques, une analyse du déroulement de la construction et de la décision du plan Bus 2025 a pu être réalisée.

Puis, après avoir réalisé un travail d'analyse des données produites par la RATP et plus généralement les publications sur le plan Bus 2025, plusieurs entretiens ont été réalisés avec des personnes qui sont intervenues dans l'émergence du plan Bus 2025 aussi bien à la RATP qu'au niveau politique avec le STIF.

Une décision dans un espace d'opportunité

A partir d'un travail d'entretiens et d'analyse des publications sur le plan Bus 2025, il est apparu que la décision de faire évoluer rapidement le parc d'autobus de la RATP vers l'électrique et le biogaz repose sur la convergence d'éléments qui permettent de créer une configuration favorable, appelée aussi « fenêtre d'opportunité » reprenant la théorie de la mise sur agenda développée en sciences politiques [Kingdon J., 1984]. Ainsi, la décision d'innovation est éloignée des idées reçues qui voudraient qu'une décision s'appuie sur une étude de faisabilité, sur la maturité technique d'une solution et soit portée par un décideur de génie. En réalité, la décision est plus complexe et elle se construit comme une négociation, et partant, peut être vue comme le résultat d'un jeu de pouvoir, dans le cas de l'étude entre la RATP, le STIF, les élus et les industriels. Ainsi, sans la convergence des dimensions politiques exacerbées par la campagne des municipale à Paris et la mise en avant des questions de pollution au particules, des dimensions techniques avec la perception de « signaux » rendant la solution électrique réaliste à moyen terme, d'une dimension individuelle avec le renouvellement du Président directeur général de la RATP, et d'un contexte industriel particulier, la décision du plan Bus 2025 n'aurait vraisemblablement pas été prise. Le plan Bus 2025 est finalement apparu comme la solution permettant d'apporter une réponse satisfaisante aux différentes dimensions tout en offrant à la RATP la possibilité de conserver le pouvoir sur son avenir technologique. Ce n'est peut-être pas la solution optimale, mais c'est la solution qui suscite l'accord du plus grand nombre.

Cette analyse de la décision montre ainsi que pour être en capacité de prendre une décision de changement majeur, un décideur doit l'inscrire dans une fenêtre d'opportunité qui façonnera nécessairement la décision. Ainsi, les décideurs doivent être attentifs aux éléments conjoncturels qui pourraient rendre possible un changement. Il paraît en outre important que toute décision de changement soit vue, à certains égards, comme opportuniste et devant s'adapter par rapport au contexte dans lequel elle s'inscrit. Enfin, il ressort qu'en matière de décision, il est bon de ne pas se contenter des modèles classiques de rationalité. Au sein des organisations, le processus de prise de décision doit être appréhendé à travers les négociations entre les acteurs, et au cours desquelles la première solution qui apparaît répondre au problème est finalement retenue [Allison G., 1971].

La longue marche de la solution électrique

Une fois la décision de changement prise par la présidence de la RATP, il s'agit de construire ce changement.

L'étude de cas a montré comment des chemins technologiques préexistants permettaient d'alimenter et de guider la construction du changement, même si celui-ci est radical. En effet, il ressort que l'innovation du plan Bus 2025 s'inscrit sur une trajectoire technologique préexistante, l'électrique, qui était jugée possible, bien que non majoritaire à ce moment là. Les expérimentations sur les bus hybrides, l'exploitation de quelques minibus électriques ainsi que les partenariats avec des industriels peuvent, à ce titre, être considérés comme la base de la trajectoire technologique électrique. L'ensemble des documents produits par la RATP et les personnes rencontrées s'accordent sur le faut que « l'hybride a toujours été vue comme un premier pas vers la traction électrique. » Dès lors, il a semblé que le changement attaché au plan Bus 2025 n'est pas tant une rupture qu'une accélération et une redirection sur une trajectoire préexistante. Cela invite à considérer l'innovation non pas comme une rupture mais plutôt comme une évolution dans une direction cadrée par l'histoire et les routines de l'orga-nisation, confirmant les théories évolutionnistes en innovation [Nelson & Winter, 1982]. Dès lors, les routines et connaissances développées conditionneraient l'évolution organisationnelle de l'entreprise et pourraient se recombiner pour cadrer le changement et rétrécir les possibilités. Il apparaît donc important pour conduire une innovation d'avoir conscience de ses éléments et de l'influence de ces routines.

L'étude de cas montre enfin qu'il est possible de sortir d'une trajectoire technologique préétablie comme cela est le cas pour le diesel. Ainsi, même si le coût d'adaptation reste élevé, en particulier pour les infrastructures et les personnels, la RATP décide de modifier profondément sont outil productif en passant du diesel à l'électrique et au biogaz. Dès lors, ce cas montre que les phénomènes de type lock-in peuvent être dépassés si cela s'inscrit dans un contexte favorable (fenêtre d'opportunité) [David P, 1985]. Dans le cas du plan Bus 2025, les gains espérés aussi bien politiques, que personnels, mais aussi industriellement dans la perspective de l'ouverture à la concurrence rendent le changement intéressant même avec des coûts d'adaptation importants.

Une adhésion construite autour de l'imaginaire et de la force symbolique du plan Bus 2025

Le plan Bus 2025, même s'il est porteur d'un changement important, réussit à créer l'adhésion et à être accepté par l'ensemble des acteurs. Un des éléments essentiels fondant cette adhésion réside dans la charge symbolique et l'imaginaire portés par le plan Bus 2025. En donnant une nouvelle image au bus, en l'inscrivant dans l'imaginaire écologique actuellement à l'oeuvre dans la société, le plan bus construit le fondement de sa légitimité. En ce sens, le cas du plan Bus 2025 semble particulier et cette légitimité a priori ne pourrait être appliqué à toute sorte de change-ments. Néanmoins, cela montre que la communication, l'imaginaire attaché à un changement, permettent de créer l'adhésion aussi bien en interne qu'en externe.

En outre, à partir de la théorie de l'intéressement développée par Michel Callon [Callon M., 1986], il a été montré qu'au delà de la qualité intrinsèque du projet, l'enrôlement est possible si le porteur de l'innovation démontre que sa solution est incontournable pour l'ensemble des acteurs, répond à leurs attentes et qu'une négociation per-met d'adapter la solution. La mise en place d'alliés institutionnels et avec les industriels (Renault, Edf, Engie) ressort également comme un facteur de légitimation du plan tout en permettant à la RATP de disposer d'expertises nécessaires à la conduite de l'innovation.

Pour finir, un changement dans une entreprise industrielle telle que la RATP reste très attaché aux enjeux techniques. La qualité technique du projet, qui s'illustre au travers de l'expérimentation sur une ligne complète de bus électriques produits par le groupe Bolloré, sert à créer l'adhésion autour du projet. Ainsi, la technique est au centre du dispositif de conduite du changement au sein de la RATP. A ce titre, les expérimentations dans un projet d'innovation ne doivent pas uniquement être vues comme un moyen de tester techniquement une solution, mais comme un outil de communication et de conviction par rapport aux acteurs internes. Le rôle d'une expérimentation ou d'un prototype est donc bien d'alimenter la conduite du changement et son adhésion.

Conclusion

L'étude de cas du plan Bus 2025 avait pour objectif de proposer une analyse de la manière dont la décision d'un changement majeur, qui peut être vu comme une rupture, est prise au sein d'une grande entreprise publique comme la RATP. Il ressort que le changement se construit au sein de l'entreprise sur la base des expériences et des compétences développées par le passé qui permettent la mise en place de trajectoires technologiques mineures à côté du modèle socio-technique dominant. L'analyse de la décision d'engager ce changement met en lumière la limite d'une vision trop rationnelle qui doit intégrer les phénomènes de négociations de pouvoir entre les acteurs (approche politique), et l'influence de l'organisation, conformément à la dynamique de prise de décision développée par Graham Allison [Allison G., 1971]. Dans ces conditions, il n'est pas étonnant qu'une fenêtre d'opportunité apparaisse strictement nécessaire à la prise de décision. Ces contraintes semblent devoir être intégrées par les décideurs dans les phases de décision pour développer des innovation et des changements dans leurs organisations.

Bibliographie

AKRICH Madeleine, CALLON Michel, LATOUR Bruno (1988), A quoi tient le succès des innovations ? 1 : L'art de l'intéressement ; 2 : Le choix des porte-parole, Les Annales des Mines, Gérer et comprendre, n°11 et n°12, pp.4-17 & 14-29. ALLISON Graham (1971), The essence of Decision : Explaining the Cuba Missile Crisis, Harper Collons Publishers CALLON Michel (1986), Eléments pour une sociologie de la traduction. La domestication des coquilles Saint-Jacques et des marins pêcheurs dans la baie de Saint-Brieuc, L'année sociologique 36, pp 169-208 KINGDON John (1984), Agendas, alernatives andpublic policies, Boston, Little Brown andCompany. MARCH James, COHEN Michael D., OLSEN Johan P. (1972), A Garbage Can Model of Organizational Choice, Administrative Science Quarterly, Vol 17, n°1, Mar., pp1-25. NELSON Richard et WINTER Sidney Graham (1982), An evolutionary theory of economic change, Cambridge (Mass.), Belknap Press/Harvard University Press.

500 bus et un tiers des effectifs de l'entreprise publique. Aujourd'hui, les impératifs de transition énergétique, les enjeux de santé et d'environnement, l'évolution de l'offre de transport en Ilede-France avec la mise en oeuvre du réseau ferré du Grand Paris Express ont conduit la RATP à réinterroger son système de transport par bus pour qu'il réponde aux enjeux de mobilité d'une métropole du 21 ème siècle.


Frédéric Chapoton

Baptiste Rognerud

On the wildness of cambrian lattices

In this note, we investigate the representation type of the cambrian lattices and some other related lattices. The result is expressed as a very simple trichotomy. When the rank of the underlined Coxeter group is at most 2, the lattices are of finite representation type. When the Coxeter group is a reducible group of type A 3 1 , the lattices are of tame representation type. In all the other cases they are of wild representation type.

Introduction

By a famous result of Y. Drozd [START_REF] Drozd | Tame and wild matrix problems[END_REF], associative algebras can be classified according to their representation type, which can be finite, tame or wild. Algebras of finite representation type have a finite number of isomorphism classes of indecomposable modules. They are quite rare in some sense, and a random algebra, meaning roughly any sufficiently complicated algebra, should be expected a priori to be wild, unless there is a specific reason for the contrary. Algebras of tame type are living on the frontier between finite type and wild type, and should not be expected to appear frequently either.

Given any finite poset P and a base field k, one can define the incidence algebra of P over k, which is a finite dimensional associative algebra of finite global dimension. It is a natural question to ask what is the representation type of incidence algebras. A general description of the representation-finite case has been made in [START_REF] Loupias | Représentations indécomposables des ensembles ordonnés finis[END_REF][START_REF] Loupias | Indecomposable representations of finite ordered sets[END_REF]. The tame case has been considered in [START_REF] Leszczyński | Representation-tame incidence algebras of finite posets[END_REF][START_REF] Leszczyński | The completely separating incidence algebras of tame representation type[END_REF].

In this short note, an answer is given for some families of posets, namely the cambrian lattices and some other related lattices. Cambrian lattices have been introduced by N. Reading [START_REF] Reading | Cambrian lattices[END_REF][START_REF] Reading | Sortable elements and Cambrian lattices[END_REF] to provide a description of the most combinatorial aspects of the theory of cluster algebras of S. Fomin and A. Zelevinsky. More than that, these posets allow to extend this combinatorial description from finite root systems to finite Coxeter groups, therefore encompassing the non-crystallographic finite Coxeter groups of type I and H.

Our main aim in this note is to tell which of the cambrian lattices have finite representation type, and which have tame or wild representation type. It turns out that for this family of posets the trichotomy is particularly simple: when the rank of W is 1 or 2, then the cambrian lattices are of finite representation type. Otherwise they are of wild representation unless they are isomorphic to the cube which is tame. This last case arises exactly when

W = A 1 × A 1 × A 1 .
In the remaining part of the note, the same result is obtained for two related families, namely lattices of order ideals of root posets and Stokes lattices.

Representation theory of posets

The incidence algebra of a poset P over a field k has a basis I a,b indexed by pairs of comparable elements a ≤ b in P and its associative product is defined on this basis by

I a,b I c,d = 0 if b = c, I a,d if b = c.
(1)

Every poset can be seen as a small category, where there is a (unique) map from a to b if and only if a ≤ b in P . The category of finite-dimensional modules over the incidence algebra of P over k is equivalent to the category of functors from the poset P to the category of finite-dimensional vector fields over k, denoted by F P,k .

This can also be restated, using the Hasse diagram of the poset P , as the category of representations of a quiver with relations. The arrows a → b of the quiver are the cover relations a ⊳ b in the poset P , and the relations are commuting relations, saying that any two paths sharing their ends are equal.

By a slight abuse of notation, we will say that two posets P and Q are derived equivalent when the derived categories of modules over their incidence algebras are triangle-equivalent.

Similarly, we will say that P has finite, tame or wild representation type if the incidence algebra of P has this property.

In the rest of this note we assume that k is an algebraically closed field. We will systematically use the following well-known Lemma. Recall that a subposet of a poset Y is the induced partial order on a subset of Y .

Lemma 2.1. Let X be a subposet of a finite poset Y .

1. If X is of wild representation type, then Y is of wild representation type.

2. If X is of tame representation type, then Y is either of tame or of wild representation type.

Proof. We view X and Y as finite categories. Then, the embedding i of X in Y is a fully-faithful functor. The precomposition by this functor induces a functor i -1 from F Y,k to F X,k . By usual arguments, it has a left adjoint i ! given by the left Kan extension along i. If F ∈ F X,k , then by Proposition 3.7.3 of [START_REF] Borceux | Handbook of categorical algebra. 1[END_REF], we have (i -1 • i ! )(F ) ∼ = F . It follows that the functor i ! sends an indecomposable object of F X,k to an indecomposable object of F Y,k . Moreover, for F 1 and F 2 in F X,k , we have i ! (F 1 ) ∼ = i ! (F 2 ) if and only if

F 1 ∼ = F 2 .
We will also need the following well-known result.

Proposition 2.2. Let X and Y be two finite posets and f : X → Y be a surjective morphism of posets such that f -1 (y) is connected for every y ∈ Y . Then, if Y is wild representation type, so is X.

Proof. This is Proposition 1.3 of [START_REF] Loupias | Indecomposable representations of finite ordered sets[END_REF]. This follows from the fact that the functor f -1 : F Y,k → F X,k is a fully-faithful embedding.

Derived equivalences provide powerful tools for the study of finite dimensional algebras and finite posets. When two algebras share the same derived categories, they share a number of invariants such as the number of isomorphism classes of simple modules, or the center. However, the representation type is not preserved by derived equivalences. Still, when a finite dimensional algebra is derived equivalent to a hereditary algebra, its representation type is dominated 1 by the representation type of the hereditary algebra. Proof. Let F : D b (A) → D b (H) be an equivalence of triangulated categories. As usual, we view the indecomposable A-modules as complexes concentrated in degree zero. Then, the functor F sends an indecomposable A-module to an indecomposable object of D b (H). Moreover, there exist two integers m and n, independent of X (but which depend on the choice of the equivalence F ) such that the homology of F (X) is concentrated in degrees in [m, n]. Since the category H is hereditary, the indecomposable objects of D b (H) are just shifts of indecomposable objects of H. So the images of the indecomposable A-modules under F lie in a finite number of copies of H. For more details, see for example Section 2 of [START_REF] Happel | Homological properties of piecewise hereditary algebras[END_REF].

Proposition 2.4 (Ladkani). Let X be a finite poset with a unique maximal element 1. Let Y = X\{ 1} ⊔ { 0} where 0 is the unique minimal element of Y . Then, the posets X and Y are derived equivalent.

Proof. This is a special case of the much more general flip-flop of Ladkani (see Corollary 1.3 of [START_REF] Ladkani | Universal derived equivalences of posets[END_REF]). In this simple case, the derived equivalence can be realized by the following tilting complex. If x ∈ X, we denote by P x the projective indecomposable module corresponding to the element x. If x = 1, we consider the complex T x = P 1 f → P x , where f is the canonical embedding of P 1 into P x and P 1 is in degree zero of the complex. Then, T = P 1 1 =x∈X T x is a tilting complex and its endomorphism algebra is the incidence algebra of Y . n . Let X = Y ⊔ {ω} be a poset whose Hasse diagram contains exactly two more edges, in such a way that ω is the maximal element or the minimal element of a commutative square. Then, X is of wild representation type.

Proof. Let us remark that it is always possible to produce such a poset X. The orientation of the diagram

A (1)
n is non circular, so there are subposets of A

(1)

n of the form a • → • ← b • and of the form c • ← • → d •.
If we set ω > a and ω > b we have a poset X such that ω is the maximal element of a commutative square. If we set ω < c and ω < d, then ω is the minimal element of a commutative square. By duality, it is enough to prove the result when ω is the minimal element of a commutative square. We exhibit a two-parameter family of pairwise non-isomorphic indecomposable representations for the poset X inspired by a similar construction for the preprojective algebra of type A 6 (see [START_REF] Rickard | Why is the A 6 preprojective algebra of wild representation type? mathoverflow.net[END_REF]). This implies that X is of wild representation type. Let i : A

(1) n → X be the canonical embedding. By precomposition, we have a functor i -1 :

F X,k → F A (1) n ,k
which is nothing but the obvious restriction to the subposet A

(λ) in F A (1) n ,k . For λ, µ ∈ k, it is easy to see that Hom F A (1) n ,k M (λ), M (µ) ∼ = k if λ = µ, 0 otherwise.
Here, the isomorphism k ∼ = End(M (λ)) is given by the application that sends t ∈ k to the natural transfor-

mation (t x ) x∈A (1) n ∈ End(M (λ)), where t x : M (λ)(x) = k → M (λ)(x) = k is the multiplication by t.
Since the functor M (λ) has a local endomorphism algebra, it is indecomposable.

Let λ, µ ∈ k such that λ = µ. Then, we consider the functor M (λ, µ) in F X,k defined as follows:

i -1 M (λ, µ) = M (λ) ⊕ M (µ), and M (λ, µ)(ω) = k.
For the two arrows g 1 and g 2 starting at ω, we let M (λ, µ)(g i ) be the diagonal embedding △ of k into k ⊕ k.

In other terms, the representation M (λ, µ) has the following shape:

k △ % % J J J J J J J J △ y y t t t t t t t t k ⊕ k • • • k ⊕ k λ 0 0 µ α k ⊕ k • • • k ⊕ k % % J J J J J J J k ⊕ k y y t t t t t t t • • • k ⊕ k k ⊕ k k ⊕ k
where all the non-labelled edges correspond to the identity matrix.

Let f be an endomorphism of M (λ, µ).

Then, i -1 (f ) is an endomorphism of M (λ) ⊕ M (µ).
In particular, because one assumes that λ = µ, it is of the form f 1 ⊕ f 2 where f 1 is an endomorphism of M (λ) and f 2 is an endomorphism of M (µ). Moreover, we know that f 1 (resp. f 2 ) is the natural transformation given by the multiplication by an element t 1 ∈ k (resp. t 2 ∈ k). It follows that for every x ∈ A

(1)

n , we have f x = t1 0 0 t2 . If we look at the situation at the vertex ω, we have the following commutative square:

k fω / / △ k △ k ⊕ k f1⊕f2 / / k ⊕ k.
So, we have

t 1 = t 2 = f ω (1). It is now clear that End M (λ, µ) ∼ = k. In particular, the functor M (λ, µ) is indecomposable. Let λ ′ = µ ′ ∈ k. If there is an isomorphism f : M (λ, µ) → M (λ ′ , µ ′ ), then by restriction, there is an isomorphism from M (λ)⊕M (µ) to M (λ ′ )⊕M (µ ′ ). This implies that {λ, µ} = {λ ′ , µ ′ }.
Since the roles of λ and µ are symmetric, there is an isomorphism M (λ, µ) ∼ = M (µ, λ). In other words, there is an isomorphism between the indecomposable representations M (λ, µ) and M (λ ′ , µ ′ ) if and only if {λ, µ} = {λ ′ , µ ′ }.

Cambrian lattices

Let W be a finite Coxeter group. Let n denote the rank of W , which is the number of simple generators {s 1 , . . . , s n }.

When the Coxeter graph of W is connected, so that W is not a product of smaller Coxeter groups, we will say that W is irreducible, and reducible otherwise.

Recall that a Coxeter element c in W is the product of the simple generators in some order. Once a Coxeter element c is chosen, one can define the cambrian lattice Cambrian(W, c) using restriction of the weak order on W to the set of c-sortable elements. The reader is referred to the original articles [START_REF] Reading | Cambrian lattices[END_REF][START_REF] Reading | Sortable elements and Cambrian lattices[END_REF] for the precise definition.

For a given W , the cambrian lattices for various choices of c share the same number of elements, which is called the Coxeter-Catalan number of type W . Much more is true, namely their Hasse diagrams are orientations of the same unoriented n-regular graph, which is the exchange graph of cluster theory when W is crystallographic.

Something deeper also holds, at least in the case of simply-laced crystallographic W . S. Ladkani has proved in [START_REF] Ladkani | Universal derived equivalences of posets of cluster tilting objects[END_REF], in the much more general case of quivers, the following statement.

Proposition 3.1. Let W be a simply-laced crystallographic root-system. Let c and c ′ be two Coxeter elements for W . Then the incidence algebras of Cambrian(W, c) and Cambrian(W, c ′ ) are derived equivalent.

This proposition is expected to hold in all cases, but has not been proved yet, to the best of our knowledge.

Easy types

There are two situations when one can easily determine the representation type of the poset Cambrian(W, c). Lemma 3.2. If the rank n of W is at most 2, every poset Cambrian(W, c) is of finite representation type.

Proof. In rank 1, this is trivial for the quiver with one vertex.

In rank 2, for the Coxeter groups of type I 2 (h), the Hasse diagram of Cambrian(W, c) is obtained by adding a minimal element and a maximal element to the disjoint union of the chain poset of size h -1 and the poset with one element.

By Proposition 2.4, the poset Cambrian(W, c) is derived equivalent to a poset which is just a quiver, and has type D h+2 . This quiver is therefore of finite representation type, and the conclusion follows by Proposition 2.3.

If the rank n is at least 4, then one can proceed as follows. Recall that the cambrian lattice of a Coxeter group W of rank n has a n-regular Hasse diagram. Otherwise, every element covering one of the b i covers at least another b j . Because P is a lattice, for every pair 1 ≤ i < j ≤ 4, there is at most one element c i,j covering both b i and b j . So in particular there are at most 6 elements covering one of the b i . Using the hypothesis that the Hasse diagram is 4-regular, one can deduce that, for every 1 ≤ i < j ≤ 4, there exists an element c i,j covering exactly both b i and b j . Note that there can be no arrows between the c i,j . Now consider the subquiver on the set {b 1 , b 2 , b 3 , c 1,2 , c 2,3 , c 1,3 , c 1,4 }. Without c 1,4 , this would be an affine quiver of type A

(1) 5 . This is therefore a quiver of wild representation type. In the case of rank n = 3, the situation is less clear. By picking appropriate subsets of vertices, we will now show that the posets Cambrian(W, c) are wild unless they are isomorphic to the cube.

Reducible types of rank 3

Lemma 3.5 (Lenzing). The cube poset is of tame representation type.

Proof. The subposet of the cube consisting of all the vertices except the minimal and the maximal elements of the cube is a quiver of type A

(1) 5 . By Lemma 2.1, the cube has an infinite representation type. Moreover, by Example 18.6.2 of [START_REF] Lenzing | Coxeter transformations associated with finite-dimensional algebras[END_REF] or by Example 2.2 of [START_REF] Ladkani | Which canonical algebras are derived equivalent to incidence algebras of posets?[END_REF], the incidence algebra of the cube is derived equivalent to the weighted projective line X(3, 3, 3). By Section 4 of [START_REF] Lenzing | Sheaves on a weighted projective line of genus one and representations of a tubular algebra[END_REF], the category coh(X(3, 3, 3)) is tame. Since it is a hereditary category, the result follows from Proposition 2.3. Proposition 3.6. Let h 3. Then, the poset Cambrian(W A1I2(h) , c) is wild.

Proof. The poset attached to the reducible type A 1 I 2 (h) is the cartesian product of a segment and a (h-1, 1)cycle: a poset obtained by adding a minimal element and a maximal element to the disjoint union of the chain poset of size h-1 and the poset with one element. We exhibit a subposet which has wild representation type by Proposition 2.5. The induced quiver-with-relations on the marked vertices is just a quiver, as one can check that no relation is implied by the commuting relations in the initial Hasse diagram. Removing the vertex 2 in this quiver gives a quiver of affine type A The induced quiver-with-relations on the marked vertices is just a quiver, as one can check that no relation is implied by the relations in the initial Hasse diagram. Removing the vertex 26 in this quiver gives a quiver of affine type A

Weak order on the Coxeter group

The cambrian lattice Cambrian(W, c) is known to be a quotient of the weak order on W such that the fibers of the quotient are intervals in the weak order. By Proposition 2.2, when the rank of W is at least 3 and W is not A 1 × A 1 × A 1 , the weak order poset on W has a wild representation type. When W = A 1 × A 1 × A 1 , then the weak order poset is isomorphic to the corresponding cambrian lattice, which is a cube. In particular, it has a tame representation type. Finally, it is easy to check that when the rank of W is at most 2, then the weak order poset has a finite representation type. Indeed, in these cases, the posets are obtained by adding a minimal and a maximal element to the disjoint union of two chains. These posets are known to be of finite representation type (see for instance [START_REF] Chaptal | Objets indécomposables dans certaines catégories de foncteurs[END_REF]).

Antichain posets

This is another family of posets, expected but not known to be derived equivalent to cambrian lattices. Let Φ be a finite root system. The set Φ + of positive roots in Φ is endowed with a partial order by the relation α ≤ β if and only if βα is a positive linear combination of simple positive roots. This poset is called the root poset of Φ. If n is the rank of the root system, this poset has n minimal elements, the simple positive roots.

Given a finite root system Φ, the poset NonNesting(Φ) is defined as the lattice of order ideals in the root poset of Φ. Recall that an order ideal in a poset P is a subset L of P such that x ∈ L and y ≤ x implies y ∈ L. The partial order on the set of order ideals is given by inclusion.

For S a subset of Φ + , let us denote by L(S) the order ideal generated by S. Every order ideal can be uniquely written as L(S) where S is an antichain in the root poset.

The poset NonNesting(Φ) has a unique minimal element, the empty order ideal. It also has a unique maximal element, the full root poset. Moreover the restricted poset on the set of order ideals contained in L(α 1 , . . . , α n ) is isomorphic to the n-cube poset.

Easy types

If the rank n ≤ 2, then every poset NonNesting(Φ) has a finite representation type. Indeed, in this case Hasse diagram of the poset is a commutative square with a (possibly empty) chain attached to the maximal element of the square. By Proposition 2.4, it is derived equivalent to a Dynkin diagram of type D. The result follows from Proposition 2.3.

If the rank n ≥ 4, then NonNesting(Φ) contains a 4-cube, hence one can apply Lemma 3.3 to get that representation type is wild.

There remains only to handle the cases of rank 3.

Types of rank 3

For the reducible type (A 1 ) 3 , the poset NonNesting(Φ) is isomorphic to the cube poset. By Lemma 3.5, it has a tame representation type. Assume now that Φ is not of type (A 1 ) 3 . Then there exist at least two simple roots (say α 1 and α 2 ) in the root poset that are covered by a common root β. Then β covers only α 1 and α 2 . This property also holds for the substitute of the root poset in type H 3 introduced by D. Armstrong in [Arm09, §5.4.1].

The Hasse diagram of the poset NonNesting(Φ) contains a cube, as its restriction to the order ideals contained in L(α 1 , α 2 , α 3 ). The order ideal L(β) covers L(α 1 , α 2 ) and no other vertex of the cube.

Consider the induced poset Q on the vertices L(α i ), L(α i , α j ) and L(β). This is just a quiver, with no commuting relation. Removing L(β) gives an affine quiver of type A

(1) 5 . Therefore Q has wild representation type, and so does NonNesting(Φ). Since the article [START_REF] Garver | Oriented Flip Graphs and Noncrossing Tree Partitions[END_REF], the Stokes lattices can even be considered as a special case of a more general class of lattices, attached to dissections of polygons. Our arguments below work just the same for this extended class, which has very similar properties.

Just as the cambrian lattices, the Stokes lattices also have n-regular Hasse diagrams, where n is a parameter called the rank of the Stokes lattice.

Easy types

There are essentially the same easy cases as in the Cambrian setting.

In rank n ≤ 2, the only possible Stokes posets are cambrian lattices of type A, already considered before. In rank n ≥ 4, one can also use Lemma 3.3 to get wild representation type. So once again, there remains to handle the case of rank 3.

Stokes lattices of rank 3

Excluding the intersection with the cambrian cases of type A (reducible or not), there remains only one case to consider, which is a poset with 12 vertices and the following shape.

Proposition 2. 3 .

 3 Let A be a finite dimensional k-algebra. Let H be an abelian hereditary category. If D b (A) ∼ = D b (H), then the representation type of A is dominated by the representation type of H.

  Proposition 2.5. Let k be an algebraically closed field. Let n 3. Let Y be a finite poset whose Hasse diagram is a non circular orientation of an affine diagram of type A (1)

  fix α an arrow of the Hasse diagram of X which is not in the commutative square. Let λ ∈ k. For x ∈ A (1) n , we set M (λ)(x) = k. We let M (λ)(α) : k → k be the multiplication by λ. And for every arrow α ′ in the Hasse diagram of A (1) n , we let M (λ)(α ′ ) = Id k . Extending M (λ) to any morphism of the finite category A (1) n , we have a functor M

Lemma 3. 3 .

 3 Let P be a lattice whose Hasse diagram is a 4-regular graph. Then P is of wild representation type. Proof. Let a be the unique minimal element of P and let b 1 , b 2 , b 3 , b 4 be the elements that cover a. Assume first that some b i is covered by an element c that does not cover any other b j . The induced quiver on the set {a, b 1 , b 2 , b 3 , b 4 } has affine type D (1) 4 . Then, the induced quiver on the set {a, b 1 , b 2 , b 3 , b 4 , c} has wild representation type.

Lemma 3. 4 .

 4 Let P be a poset that has a minimal element a covered by n ≥ 5 elements b 1 , . . . , b n . Then P is of wild representation type. Proof. The induced quiver on {a, b 1 , . . . , b n } has wild representation type because n ≥ 5.

  Let us show that the Tamari lattice with 14 vertices has a wild representation type by exhibiting a wild subquiver.

  is the similar wild sub-quiver for the other cambrian lattice of type A 3 .Let us now consider one of the cambrian lattices of type B 3 , with 20 vertices. Let us again show that this has wild representation type by exhibiting a wild subquiver. -with-relations on the marked vertices is just a quiver, as one can check that no relation is implied by the relations in the initial Hasse diagram. Removing the vertex 10 in this quiver gives a quiver of affine type A(1) 9 .Here is the similar wild sub-quiver for the other cambrian lattice of type B 3 .Let us now consider one of the cambrian lattices of type H 3 , with 32 vertices. Let us again show that this has wild representation type by exhibiting a wild subquiver.

  is the similar wild sub-quiver for the other cambrian lattice of type H 3 . Note the similarity in the shape of the wild subquivers for A 3 , B 3 and H 3 . They all consist of something like a middle belt between a left part and a right part, plus one extra vertex. They have been obtained by carefully removing well-chosen maximal or minimal elements, until reaching this kind of configuration.

  Let us turn to another family of posets, containing the cambrian lattices of type A. The Stokes lattices were introduced by the first author as posets in[START_REF] Chapoton | Stokes posets and serpent nests[END_REF] (inspired by previous work of Baryshnikov in[START_REF] Baryshnikov | On Stokes sets[END_REF]) and they were proved to be lattices byGarver and McConville in[START_REF] Garver | Oriented Flip Graphs and Noncrossing Tree Partitions[END_REF].

Here dominated means smaller in the order where (finite) ≤ (tame) ≤ (wild).

* Cet auteur a bénéficié d'une aide de référence ANR-12-BS01-0017). † Cet auteur a bénéficié d'un financement de l'IDEX BMM/PN/AM/N o 2016-096c.

1

By Proposition 2.5, the subquiver on vertices {0, 1, 2, 3, 5, 7, 8, 10, 11} and its commutative square (0, 2, 10, 7) is of wild representation type. So, the poset is of wild representation type.