Antoine Crinière

Jean Dumoulin

Laetitia Perez

Frédéric Bourquin

Study of an optimal heating duration indicator for square pulsed thermography applied to CFRP gluing quality control

Study of an optimal heating duration indicator for square pulsed thermography applied to CFRP gluing quality control

A. Crinière 1,2 , J. Dumoulin 1,2 , L. Perez 3 , F. Bourquin 2 1 INRIA/IRISA, I4S Team, Campus de Beaulieu, 35042 Rennes, France 2 Université Paris-Est, IFSTTAR, COSYS, F-77447, Marne-La-Vallée, France LUNAM Université, LTN, UMR-CNRS 6607, Rue Christian Pauc, 44306 Nantes France During previous works, square pulsed thermography was used to carry out non destructive testing of bonding quality of CFRP glued on civil engineering structures during reinforcement operations [START_REF] Crinière | Inverse model for defect characterization of externally glued CFRP on reinforced concrete structures: Comparative study of square pulsed and pulsed thermography[END_REF][START_REF] Théroux | Square heating applied to shearography and active infrared thermography measurements coupling: from feasibility test in laboratory to numerical study of pultruded CFRP plates glued on concrete specimen[END_REF]. The use of such wave form excitation was motivated by "on-site" requirements, but also by measurements duration, number of composite layers to test, depth of possible faulting areas versus temperature elevation allowed at composite level according to inner heat diffusion. Nevertheless, square pulsed excitation implies to choose an adapted heat duration. This duration is directly linked to the reliability of the parameter estimator [START_REF] Feuillet | Defect detection and characterization in composite materials using Square Pulse Thermography coupled with Singular Value Decomposition analysis and thermal quadrupole modeling[END_REF]. In fact, after a certain duration the standard deviation of the estimation procedure stagnates. According to these observations, an indicator able to predict the sufficient heating time when the reliability of the parameter estimator reached an asymptotic evolution behavior was studied. Based on the absolute thermal contrast, the proposed indicator Iph is defined with the maximum thermal contrast ∆Tmax and the time delay (tph) between the heating time tc and the appearance of the maximum contrast, as shown in figure 1 (left). A typical evolution of the Iph indicator is proposed in figure 1 (right). This indicator allows to take into acount the detectability as well as the induced flaw temporal effect on the thermal contrast shape evolution. It has been observed that the maximum of Iph is connected with the sufficient heating time when the standard deviation of the estimation procedure tends to be minimized. This paper will present the establishment of this indicator for optimal square heating time and present an analysis of results obtained with numerical simulations and laboratory experiments.

Figure 1

 1 Figure 1 Absolute thermal contrast characteristics (left) and indicator Iph evolution with heating time duration Keywords: NDT, Square pulse heating, Optimal Design, Civil engineering, Parameter estimation


O Briante

V Loscri

P Pace

G Ruggeri

N R Zema

COMVIVOR: an evolutionary communication framework based on survivors' devices reuse

Keywords: Intelligent Positioning, Disaster Recovery, Autonomous Networks

Mobile devices currently available on the market have a plethora of features and enough computing power to make them, at the same time, information consumers, forwarders and producers. Since they are also provided with a set of sensors and usually battery operating, they are perfect candidates to devise a network infrastructure tailored to function during disruptive events. When everything else fails, they could autonomously reorganize and provide to the civilians and rescue teams valuable services and information. In this paper we adapt and enhance a previous designed framework, capable to epidemically diffuse the proper software updates to its nodes, in order to deploy any kind of service as a prompt response to the needs raised in emergency situations. We further propose and integrate a new smart positioning strategy, to speed up the diffusion of software updates by also keeping under control the overall network overhead. The achieved results show the feasibility of our proposal and how the dynamics of diffusion are enhanced by the smart positioning algorithm.

Introduction

Large-scale environmental disasters such as earthquakes, floods or terrorist attacks could have a severe impact on the fixed communication infrastructure. Most of the services implemented in it, relying on standard wireless/wired networks, suddenly become non-functional in emergency situations due to both system damage and overuse [START_REF] Legendre | 30 Years of Wireless Ad Hoc Networking Research: What about Humanitarian and Disaster Relief Solutions? What are we still missing?[END_REF]. On the other side it is very likely that batterypowered wireless personal mobile communication devices will still be up and running after the disruptive event; therefore in this paper we design a framework which allows the reuse of survivors' devices to provide a basic networking communication facility in a post-disaster scenario.

Address(es) of author(s) should be given

Glancing at the plethora of personal mobile devices currently available on the market, we can see that most of them have the potentialities to behave as forwarding nodes [START_REF] Iera | Making a mesh router/gateway from a smartphone: Is that a practical solution? Ad Hoc Networks[END_REF]; moreover those devices, such as smart-phones, tablets, portable consoles and so on, are widespread, battery powered and, usually equipped with an IEEE 802.11 compliant wireless interface. They could hence be able to communicate even during disruptive events, characterized by power supplying interruptions, to support and restore a possibly damaged communication infrastructure. Furthermore, since those devices are commonly equipped with features as GPSs, microphones, cameras, accelerometers and other sensors, they can, if properly managed, collect information and provide useful services both to survivors and to rescue teams during civil distress periods.

Unfortunately, these devices are often programmed to only behave as end devices and, in spite of all their potentialities, they cannot actually communicate in front of a large fault of the traditional communication infrastructures by using their embedded software without making any explicit changes [START_REF] Aloi | Spontaneous Smartphone Networks As User Centric Solution for the Future Internet[END_REF]. In particular, the following ugly truths represent the main factors limiting the potential of these technologies [START_REF] Aloi | Spontaneous Smartphone Networks As User Centric Solution for the Future Internet[END_REF]: i) Although these commonly used devices can be easily reprogrammed to provide the required services, this operation usually requires the intervention of a skilled operator while the common user is usually technologically agnostic. ii) To protect the user's privacy, they do not provide any information about the sensed environment to any external entity if not specifically allowed by the user. iii) In a post-disaster scenario, they could be located in places not easily reachable by rescue teams, for example they can be left in an unsafe building or in a contaminated area.

For all these reasons, the devices are left unable to be exploited to the full spectrum of their capabilities. To further complicate the described context, it is worth to consider that, in a post disaster scenario, the services and the information required by rescue infrastructure can rapidly change in accordance with the evolution of the specific situation. Consequentially, as it could be necessary to continuously adapt the behavior of the network elements, the effective reuse of survivors' devices relies on an effective adaptation and autonomous evolution of such objects to support new services and protocols. These ones can also evolve and autonomously combine with each other to satisfy the needs coming out from the context. In our previous work reported in [START_REF] Aloi | Stem-net: an evolutionary network architecture for smart and sustainable cities[END_REF] we proposed a framework, called STEM-Net, which by leveraging an epidemic diffusion model and implementing network virtualization, is capable to epidemically diffuse entire protocol stacks to a set of nodes. As a natural consequence, in this paper we propose to apply STEM-Net to reprogram survivors' devices by making them able to autonomously organize and form or restore a communication infrastructure to help people during emergency situations or environmental distress.

According to our proposal, when a disaster happens, and all traditional communication services are jeopardized, an user can just enable the STEM-Net framework on his/her phone so that this latter can reconfigure itself by learning or acquiring new software updates which implement the new applications and/or communication protocols required by the situation. These software updates will be referred in the following as evolution modules.

In our view evolution modules are at first provided by an autonomous flying drone which is delivered on the disaster area by the rescue teams. In a second stage evolution modules are passed from device to device according to an epidemic paradigm. By adapting to the changing needs of situations and rescue teams, the evolution modules implement the needed new services and protocols and, by leveraging on single nodes' self-evolution, the performances of the network as a whole are also improved.

A second contribution of this paper comes from the consideration that the diffusion rate of an epidemic (i.e., the evolution modules in our case) strongly depends on the place the first infection take place [START_REF] Moreno | Epidemic outbreaks in complex heterogeneous networks[END_REF]. Hence, we also propose a new protocol, based on a smart positioning strategy, to effectively drive the flying drone deployed by the rescue teams with the aim of improving the STEM-Net performance, in terms of both increase in the diffusion speed of the evolution modules and reduction of the overall network overhead.

This enhanced communication framework, obtained through the combination of an improved STEM-Net framework with a wise drone driving protocol, has been named COMVIVOR (Communication for Survivors) in order to highlight the specific aim of such new approach.

We simulated the behavior of COMVIVOR throughout the Network Simulator 2 (ns-2) [START_REF]Network Simulator-ns (version 2[END_REF] by evaluating its performance in a realistic wireless simulation environment. The obtained results testify the effectiveness of our framework in performing the self-evolving process of network nodes; moreover, we also show how the performance of the whole system can be greatly improved by introducing a smart positioning technique originated by teorethical considerations of the epidemic model.

The rest of the paper is structured as follows. Section 2 summarises recent works in which autonomic devices reconfiguration and virtual network deployments have been used in emergency and disaster scenarios. Section 3 describes the epidemic spreading process within the COMVIVOR protocol whilst the new autonomic positioning strategy, based on a smart positioning strategy, is described in Section 4. The simulation campaigns and the obtained results are presented and discussed in Section 5; finally, the Section 6 concludes the paper presenting few future research directions and open issues.

Related Work

In the last years, ICT-related disaster monitoring and management have attracted many researches because of the huge impact on the modern society of a disastrous event that affects large numbers of people and cause loss of lives and infrastructures. In this section we briefly survey the literature for i) disaster monitoring and management frameworks and, ii) autonomic devices reconfiguration.

Disaster Monitoring and Management Approaches

Many frameworks and network strategies have thus been proposed and deeply investigated to address the issues related to a disaster scenario, from different points of view [START_REF] Bakht | Searchlight: Won't you be my neighbor?[END_REF][START_REF] Kashyab | A hierarchy based routing protocol for disaster recovery system[END_REF][START_REF] Martín-Campillo | Evaluating opportunistic networks in disaster scenarios[END_REF][START_REF] Minh | Tree-based disaster recovery multihop access network[END_REF][START_REF] Nelson | Event-driven, role-based mobility in disaster recovery networks[END_REF][START_REF] Nishiyama | Relay-by-smartphone: realizing multihop device-todevice communications[END_REF][START_REF] Ray | A framework for disaster management using wireless ad hoc networks[END_REF][START_REF] Sakano | Disaster-resilient networking: a new vision based on movable and deployable resource units[END_REF][START_REF] Uddin | A post-disaster mobility model for delay tolerant networking[END_REF]. Specifically, the contribution of [START_REF] Bakht | Searchlight: Won't you be my neighbor?[END_REF] consists in a general discovery protocol that is asynchronous. The authors verified its effectiveness both analytically and through real-world experiments on smartphones. In [START_REF] Minh | Tree-based disaster recovery multihop access network[END_REF], authors envisage the network auto-configuration software (NAS), as one of the fundamental requirements in the context of Discovery Access Network (DRAN) and they propose a Tree-based multihop Disaster Recovery Access Network (TDRAN). In [START_REF] Nelson | Event-driven, role-based mobility in disaster recovery networks[END_REF], authors propose a different mobility model to characterize and capture specific features of the topology after a disaster. This re-definition of the topology will be helpful for the development of specific routing approaches in the context of Disaster Recovery system [START_REF] Kashyab | A hierarchy based routing protocol for disaster recovery system[END_REF].

In [START_REF] Sakano | Disaster-resilient networking: a new vision based on movable and deployable resource units[END_REF] authors present concepts really close to our vision of the ICT role in the disaster recovery treatment. They propose a resilient network architecture based on resource units that are movable and deployable. Though their approach embraces, in some way, our phylosophy, we do not introduce specific units, but we leverage on user'mobile devices, by re-defining their role. Moreover, we consider and describe a better positioning of the plague-spreader node. In [START_REF] Nishiyama | Relay-by-smartphone: realizing multihop device-todevice communications[END_REF], authors show the feasibility of using a smartphone as relay for communication purpose and the advancements of user-driven networking powered by communication devices independent of operator networks.

In [START_REF] Ray | A framework for disaster management using wireless ad hoc networks[END_REF], authors propose a framework for disaster management by implementing ad hoc communication in three different scenarios, adaptively adjusting the transmission power to reduce the overall network energy consumption. However, they did not consider any mobility model to validate their analysis and they also did not address the issues related to the communication between devices supporting different interfaces and standards.

In [START_REF] Uddin | A post-disaster mobility model for delay tolerant networking[END_REF], authors propose a mobility model that includes the impact of the disaster on the transportation network by modeling the population and relief vehicle movement. The mobility model exploits a general communication scenario based on the DTN (Delay Tolerant Network) paradigm. They consider network devices that exchange buffered messages when they come into communication range. In addition, an opportunistic forwarding paradigm is used in [START_REF] Martín-Campillo | Evaluating opportunistic networks in disaster scenarios[END_REF] to investigate the potentials of novel routing methods applied to emergency scenarios. The authors evaluate the overall performance by taking into account parameters like the number of people involved or the number of victims, in order to measure their impact on the performance of routing methods.

Autonomic Devices Reconfiguration

Recently, the idea of using smart-phones to support the network infrastructure damaged by the disaster began to take hold as shown in [START_REF] Deb | Coordinating disaster relief operations using smart phone / pda based peer-to-peer communication[END_REF]. In this work the authors propose a smart-phone/PDA based disaster management system working on peer to peer communication only and supporting disconnected operation. By assuming the absence of a network infrastructure, the proposed approach tried to incorporate the notion of opportunistic network by using peer-to-peer communication between relief workers.

In contrast to these proposals, our approach tries to be more proactive because it supports the discovery of survivor devices and makes possible the implementation of behavioral changes by allowing the evolution of network devices through the reception of the evolving modules. Furthermore, we consider and evaluate the possibility to opportunistically move/position a mobile device which is in charge to provide the evolution modules required by the situation.

3 COMVIVOR: The epidemic spreading process Evolution modules are provided to mobile devices in the disaster area by means of an epidemic process. The epidemic process is triggered and sustained by special purpose nodes which: i) advertise evolution modules, and ii) provide long range connectivity, for example through a satellite link. In our vision those special nodes can also autonomously roam through the disaster area and in the following they will be referred as plague-spreaders. In a real case scenario, the plague-spreaders can be flying drones which go ahead of the rescue teams to collect the first information about the disaster area, create basic networking and monitoring facilities and pave the way for the later human intervention. The possibility of using such flying devices as an effective communication station is displayed in [START_REF] Giorgetti | Throughput per pass for data aggregation from a wireless sensor network via a uav[END_REF] [START_REF] Natalizio | Two families of algorithms to film sport events with flying robots[END_REF] [27] and [START_REF] Zema | Healing wireless sensor networks from malicious epidemic diffusion[END_REF].

We further assume that a consistent part of the end devices implements a COMVIVOR demon, this demon remains silent during all the normal life of a device but wakes up only when the user explicitly activate it in front of the disaster. Let us note that we expect that the effort to activate COMVIVOR demon will be comparable to the the effort required to run any other application on the device, hence we guess that it will be affordable by common users with no specific skills.

When the COMVIVOR demon is activated it looks on all the short range interfaces, like IEEE 802.11 or Bluetooth, for any incoming signal from a plague-spreader. In order to prevent any malicious use of COMVIVOR all the messages from a plague-spreader are signed with a secret key owned only by national security. A node accept an evolution module only after it has verified its signature by using a public key which is periodically published by government agencies.

Once a unit has received a message from a plague-spreader it goes through the infection process where the evolving module is passed from node to node like a virus going from sick beings to healthy ones.

Our current work takes its basic functionalities from our previous research on epidemic diffusion [START_REF] Aloi | Stem-net: an evolutionary network architecture for smart and sustainable cities[END_REF] where a fixed mobile station diffuses evolution modules. Since our present research focuses on the effectiveness of a better positioning of the plague-spreader node, herein we briefly remind the basic working principles of the original epidemic spreading process.

In the following, the dissemination process of a single evolving module is presented. A more general case, in which different modules are simultaneously distributed, can be accomplished by following the same procedures.

Basics

As assessed in [START_REF] Aloi | Stem-net: an evolutionary network architecture for smart and sustainable cities[END_REF], in order to disseminate an evolving module in the network, three concurrent processes are carried out: (i) the dissemination process, in which the spreading of information about new evolving modules takes place; (ii) the individual decision process, in which each node individually decides whether to accept or not an evolving module; (iii) the infection process, in which the executable implementation of an evolving module is transmitted to the requesting nodes.

Three types of node are defined in our framework: plague-spreader nodes, susceptible nodes and, infected nodes.

Mobile Plague-spreader nodes, now introduced, initially advertise the availability of an evolution module through the broadcasting of so-called DNApackets and provide, on demand, an executable implementation of it.

Susceptible nodes are potential recipient of an evolving module, which can be accepted with a given degree of susceptibility. In our framework the set of nodes susceptible to an evolution module is represented by all those nodes where: (i) the COMVIVOR demon has been activated and (ii) the evolution module has not been acquired yet.

Infected nodes are the ones that have acquired the evolving module and will contribute to advertise and spread it.

Dissemination process

The dissemination process is accomplished through the periodic broadcasting of DNA-packets that are generated by plague-spreader and infected nodes and processed by susceptible ones. To keep constant the signaling overhead, the emission rate of DNA-packets is dynamically adjusted so to remain constant in the network. Specifically, if a node senses that other nodes are broadcasting the same DNA-packet, then it reduces its transmission rate accordingly. A DNApacket carries the description of the advertised evolving module that contains the identifier (ID) of the plague-spreader, the ID of the module, the type (i.e., Protocol Stack or Service), and a description of the resources required to run the module, in terms of memory and computational load.

By analyzing the received DNA-packets, a node can infer other information as well as an estimation of its capability to run it. For instance, a node may estimate how popular is an evolving module by considering how many neighbors (plague-spreaders and infected nodes) are advertising it. Similarly, a node may estimate the distance from the source of a DNA-packet by considering the received signal strength.

Individual decision process

Although many evolution modules may be advertised in the network, not all of them could be suitable to any node. Thus, when a node receives a DNApacket goes through a decision process. This process considers several input parameters related to the node and the network conditions (e.g., hardware characteristics of the node, energy constraints, number of neighbors). As a result of the decision process, an evolving module can be stored, executed, or ignored. Analogously, a module already active may be switched off and discarded due to a change in the available node resources or in the network conditions. The devising of a suitable decision process, as a stimulus-response mechanism, has been addressed in our previous works reported in [START_REF] Aloi | Stem-net: an evolutionary network architecture for smart and sustainable cities[END_REF][START_REF] Di Felice | Smartphones like stem cells: cooperation and evolution for emergency communication in post-disaster scenarios[END_REF]. We refer the readers to those works for further detail on the decision process. For the purpose of this work we model the decision process by giving: (i) the susceptibility F of the node toward a specific module, and (ii) the possible switch off and discarding event G.

In more detail, F represents the probability for a given node to accept an evolution module that has been advertised in a DNA-packet, while G represents the average time after which a node is forced to discard an evolution module. It is worth to be noted that in a post-disaster scenario low values for G are expected due to: i) the high level of network disruption and ii) the impossibility to recharge the nodes which implies a short lifetime of latters.

Infection process

Through the infection process, the evolving module propagates from the plaguespreader to its one-hop neighbors, and then from the latter (if infected) to their one-hop neighbors thus following an epidemic diffusion. Therefore, hop-by-hop, the evolving module can reach all the nodes belonging to connected subsets in the network.

Whenever a susceptible node accepts a new evolving module, it starts the infection process; in particular, each implementation of an evolving module consists of an executable code divided into a number of elementary units called chunks that can be transmitted on the broadcast medium without any further fragmentation. COMVIVOR assumes that the number of chunks composing the executable code can vary from module to module and it is advertised by the DNA-packet. The download of the entire code is performed through the broadcasting of two packet types, the Interest and the Data. An Interest is used to request a specific set of chunks and includes the identifiers of both evolving module and the chunk set. When the plague-spreader (or any infected node) receives an Interest, it broadcasts the corresponding set of chunks within data packets. The Interest-Data exchange continues until the entire code is downloaded. In case of a packet loss, the Interest is transmitted again but the requested set of chunks is modified according to new searched data. In order to reduce the collision probability arising from multiple simultaneous transmissions (e.g., in the presence of two or more infected nodes that can reply to the same Interest), each node i waits a random defer time T D before transmitting the Data. During T D , i monitors the channel to detect the same transmission performed by other nodes. Finally, if anyone has issued the Data, it sends the packet.

COMVIVOR: The Spreader Autonomic Positioning

The application of a smart positioning technique, can be really advantageous in several and different network scenarios [START_REF] Akkaya | Distributed recovery from network partitioning in movable sensor/actor networks via controlled mobility[END_REF][START_REF] Akyildiz | A survey of mobility management in next-generation all-ip-based wireless systems[END_REF][START_REF] Aziz | A survey on distributed topology control techniques for extending the lifetime of battery powered wireless sensor networks[END_REF][START_REF] Mourad | Controlled mobility sensor networks for target tracking using ant colony optimization[END_REF][START_REF] Natalizio | Controlled mobility in mobile sensor networks: advantages, issues and challenges[END_REF]. In the COMVIVOR context, it is intended as the capability of the plague-spreader node to move to a different position according to a specific behavior or goal to be reached like, in our case, an improvement on the diffusion performances of the modules. To accomplish this we have devised a mechanism based on selective broadcast of node density information in probe packets in such a way that, after a probe phase that can be reiterated in case of topology changes, every node can know the value of the highest node density point of its network cluster area. By the knowledge of this point a spreader node can directly move to this position prior to any other action. Starting the infection from a denser area highly enhances the speed of infection as our results show. In our work the density perceived by a node is intended as the number of reachable neighbors versus the node's transmission range. If we consider that all the nodes are of the same kind, we can just use the total number of neighbors as an indication of the density of a circular area of which the sampling node is the center.

A different version of the algorithm has been devised in order to handle extreme cases in which the density value is not unique but it is uniform along the simulated area such as in an equispaced grid topology. In this case the spreader node is moved by considering a directional criterion: in particular, the final position is the one that maximize the directional dispersion of an epidemic (i.e., the barycenter of an equispaced lattice of points). According to the relation between highest density value and the estimated general density of the whole network area, the algorithm itself can choose the more convenient method to apply in order to speed up the infection process comparing the highest density value with the general area density and using the barycenter method only in the equispaced cases.

Topological values diffusion

In our work we suppose that a mobile flying device (i.e., Drone, UAV, etc...) is already present on the field as there are no long-range communication for the reasons illustrated beforehand. In an initial phase, every node can broadcast a probe packet containing the sender ID and ten fields as shown in table 1: the sender ID coordinates, its perceived density, the coordinates of the node that perceived the highest density the broadcaster node ever had information about and the actual value of perceived density associated with the latter; to those are added the coordinates of top and bottom points of the disaster area in order to calculate the barycenter as discussed in the previous section. We have used 8-bytes floating point numbers for the coordinate implementation and 2-bytes integers for the density values and IDs. In this way we model the possibility to use real GPS coordinates and a relatively large number of nodes [START_REF]GPS 16x Technical Specifications[END_REF]. In the very first broadcast phase, the only nonzero fields of the packet a node sent are its own coordinates and ID; once another node receive the probe packet, it can acquire the information about the actual presence of a neighbor in his receiving range. At the end of the first broadcast phase (after a time probe = T p), every node can have a first overview of the nodes in its surrounding (i.e., Perceived Nodes -PNs) and it can calculate its own initial perceived density to be broadcasted by using new probe packets. At the second broadcast run, every node will compare the received highest density value with the local and if found higher they will start to broadcast that as their highest density heard. As the broadcast process goes on, when a node receive a foreign density value that is higher than the local, the new value is stored and broadcasted in the new probe packet. At the end of the broadcast runs, every node connected to any subset of the total nodes in the network, will know the highest density value of that subset coupled with the coordinates of the related node.

This broadcasting information scheme needs a maximum time T M AX in order to guarantee the perfect knowledge of the denser area to all the nodes participating to the broadcasting process. This maximum time has been estimated by considering the worst case analysis applied to a standard IEEE 802.11g multi-hop wireless network as described in appendix 7.

According to this analysis, when a spreader node appears in the network it can know, after a quite reasonable time, the position where it should move to maximize the number of nodes to infect.

In the barycenter version, the same principles are applied for the propagation of the top and bottom values for every cluster involved. In this version only the highest or lowest values of the coordinates are propagated to any reachable node.

Spreader Node Movement

The density information can be updated periodically by taking into account the lower bound due to the natural delay of the propagation scheme (i.e., see the computation of the T M AX value) and also differentiating the nodes that are already mutated from the ones that still need to be reached. In this fashion, once the area with maximum density has been discovered, the spreader node can directly move to the densest area, starting there the mutation process. As our result will show, the speed of mutation is dramatically increased just from moving the node only one time.

Of course after the individual mutation process, each node can be automatically excluded by the density calculation algorithm; in this way, when the density discovery algorithm starts again, the spreader node will be eventually reached by the updated information about the densest area of un-mutated nodes. Since, the density information exchanges are faster than the mutation process of the nodes, when they are activated both, the former can feed up information to the spreader while the infection is ongoing. This feature shows its importance mostly when the number of nodes in a network is not fixed. In the disaster recovery scenarios it is quite likely that large subsets of nodes deactivate temporarely while another subset joins the network. In this case it is necessary to swiftly update the information about the nodes distribution in reaction to these events.

In algorithm 1 is shown a pseudo-code description of the proposed topological diffusion strategy based on a selective broadcast information mechanism. The pseudo-code has been differentiated to describe both the spreader and the generic node behaviors.

After a certain number of broadcast we have observed that the values converge within the cluster the nodes belong to.

Dedicated simulation shows that the algorithm can deliver information about the presence of nodes agglomeration point in a bounded number of iterations (see fig. 1). In these preliminary simulations we used a grid scenario of 100 equispaced nodes with the distance among them roughly equal to their transmission range, their characteristics equal to the ones we used in next simulation by running COMVIVOR meta-protocol. We put a small aggregation point consisting of a set of nodes placed in a corner, effectively doubling the local density, and a plague spreader node in different positions: in the opposite corner (16 hops far), in the center of the lattice (8 hops far) and in an adjacent corner (9 hops far). From the figure we can observe that the needed number of iteration required to compute an estimation of the center of the agglomeration point to the monitoring node, is roughly dependent on the distance in hops from them. Specifically the same number of iterations as hop count is needed for the first information to arrive (un-tuned to the actual point of maximum

Positioning algorithm: simulations and results

To evaluate the performance of COMVIVOR we implemented the proposed framework in ns-2 [START_REF]Network Simulator-ns (version 2[END_REF] and considered the scenario in Fig. 2, which represents an urban area as a 500m-wide square grid. For sake of simplicity, at this stage we considered all nodes equipped with a single IEEE 802.11 physical interface. A Ricean fading model accounts for multi-path effects due to various obstacles, buildings and trees. We further assumed that the executable code of the evolving module is 1M B long. Regarding the fragmentation, COMVIVOR assumes each chunk is 1024 bytes long. For the simulation purposes, we translated the parameters F and G into floating point numbers representing the probabilities of accepting an evolving module and the averaged time to trigger a discarding event.

In our previous work [START_REF] Aloi | Stem-net: an evolutionary network architecture for smart and sustainable cities[END_REF] we already evaluated the effectiveness of the diffusion mechanism and we proceed to assess its capabilities in presence of an opportunistic smart positioning of the plague-spreader node.

As shown in Table 2, Physical and MAC parameters are based on IEEE 802.11g and transmission power and receiver sensitivity figures are taken from data-sheets of devices available on the market.

In our set of simulation we tested the system by placing the wireless nodes within the simulation area according to the specific scenarios shown in Fig. 4: i) equispaced grid, ii) Gaussian distribution, iii) realistic distribution. In particular, the Gaussian scenario has been generated by following a Normal distribution criterion with the aim of creating denser areas, even if the full connectivity between all the nodes of the networks is not guaranteed. On the contrary, the realistic topology has been generated by using a specific framework called NPART [START_REF] Milic | NPART -node placement algorithm for realistic topologies in wireless multihop network simulation[END_REF], capable of designing network topologies whose statistical characteristics are similar to the ones measured in real networks. All the simulated scenarios have been designed to support a fixed density value of 6.5 * 10 -3 nodes/m 2 , an expected E{G} ∈ [150, +∞[ seconds and we fixed the value of susceptibility F = 0.5. The value E{G} → +∞ corresponds to the case when infected nodes never discard the evolving module while the lower values try to reproduce the high network disruptive environment where nodes could suddenly cease to function and others may replace them in the geographical vicinity. We analyzed the time evolution of the infection by evaluating the number of infected nodes over the total of nodes. Each value is the result of a 20 simulative runs which outputs were processed with 95% confidence intervals and shown where applicable. Initially the plague-spreader node is located at the center of the network area for the Gaussian topology and in a randomly selected corner for the others that vary within runs. (d) E{G} = ∞ Fig. 3: Fraction of infected nodes varying time and topology. 6.05 • 10 -3 N odes/m 2 ; F = 0.5.

Our main objective is to evaluate if COMVIVOR allows to obtain a steady state in the epidemic diffusion process, and how fast it is reached. It is worth to be noted that the target of the spreading algorithm is to reach a given fraction of the nodes, which is not necessary the totality of them. This could be the case of an applicative service which requires that only a small fraction of nodes execute the proper evolution module. In Figure 3 it is shown the cumulative curve of infected nodes versus time with different E{G} and grouped by topology. Regardless the considered scenario, the proposed mobility framework achieves better performances in terms of mutation dynamics by allowing to reach the steady state in much less time. From figures 3c and 3d it can be shown that, by using the mobility framework the system always reaches the steady state in less time rather than when it uses the fixed positioning system. However when E{G} is low, as shown in 3a and 3b, the whole system has difficulties in reaching a fairly reasonable number of mutated nodes regardless of a better positioning of the plague-spreader node. This means that the system has difficulties in the evolving module delivery when the average lifetime of a node is comparable to the time needed to a module transfer. However, this possibility for the E{G} value is quite uncommon. For example in Figure 3a the network does not even reach 20% of mutated nodes due to an E{G} value of 90s. The scenario in which we noticed the lowest system performance is the grid topology one, where the overhearing of data packets is limited by the equispaced dispersion of the nodes by making the mutation process more time consuming and prone to disruption in the cases of low average lifetime. However, the smart positioning of the plague-spreader node allows to improve the system performance also in this unfavourable scenario. To better clarify the obtained results, we also shown the cumulative curve of infected nodes, grouped by the E{G} values, in order to make similar considerations. Figure 6 shows that, starting the mutation process from the previously computed network barycenter, it is possible to greatly improve the speed of mutation. Figure 5 shows instead the results varying E{G} with the Gaussian topology. The curves crossings are due to the fact that the biggest percentage of the whole network nodes lies in the region where the mobile node moves to and, even if this percentage is infected first, there is still a sensible number of isolated peripheral nodes. Those last, are reached first if the node remains nearer to the network periphery rather than moving to the point of maximum agglomeration. Finally figure 7, related to a Realistic NPART-generated topology, shows a general improvement using the mobility framework in all of the previously introduced aspects.

Furthermore, in order to quantify the impact of the mobility framework on the system overhead, we also measured the so called mutation overhead representing the number of bytes transmitted by the nodes with the specific purpose of epidemic diffusion, averaged by the number of mutations. The obtained results are shown in figure 8 in which the mutation overhead is consistent with the topology and mobility scenarios while it varies with the average node lifetime. In fact, the increase of this parameter causes less mutations, less congestion and less bytes sent as a consequence. Finally, the figure 9 shows the peak value of mutated nodes by varying: i) network scenario, ii) presence of mobility framework and iii) discarding time E{G}. As expected, the peak value of mutated nodes is exactly equal to the amount of nodes within the network when the lifetime if infinite and the network connectivity is always guaranteed (i.e., Grid and Realistic scenarios); however, as the node lifetime decreases, the peak value of mutated nodes will decrease concordantly. Starting from the grid scenario, where by definition there are no high concentration points, the lower lifetime greatly decreases the peak number of mutated nodes without the diffusion enhancement due to the positioning system. In the other case, by using the mobility framework, the spreader node is capable to place itself where it can maximize an omnidirectional diffusion and compensate for the network disruption. The same concept can be applied to the other scenarios where the presence of mobility framework enhance the infection diffusion as it keeps the most dense area of nodes with readily available software modules and constantly spreads the infection to the nodes that have recently discarded the evolving module.

Conclusion

In this paper we designed and tested COMVIVOR, a communication framework suited for managing, in an effective way, emergency and disaster scenarios.

Leveraging on concepts like virtual networking, smart positioning strategy and epidemic networking, the proposed framework takes advantages from the reuse of the users devices in situations of public distress. In particular, it enables the devices scattered in the field to autonomously cooperate during large scale environmental disaster and ease the operations of the rescue teams by providing a solution to restore connectivity and load on-demand services and protocols. We have shown the improvements that the an opportunistic and smart positioning can add to the whole framework by the proposal of an algorithm for autonomous network analysis linked to a smart mobility scheme applied to flying mobile devices. We have demonstrated that, providing the values of local network density to a mobile entity in order to make it able to start the diffusion of evolving modules from the most dense area, would greatly improve the speed and maintenance of the infection even in highly disruptive conditions.

As future works, we plan to continue our research paths in multiple ways.

For instance, even if the system is already capable to handle multiple plaguespreaders, it should be devised an active coordination system between them coupled with a careful tuning of the network parameters. It should also be analyzed the case of multiple concurring infection and a decision engine that is capable to discern between multiple module options in a comprehensive manner. Moreover, it would be very interesting the evaluation of the controlled mobility paradigm on the effectiveness of the epidemic propagation speed. In practice, the controlled mobility approach would allow the plague-spreader to move again by following the same logic, once the region where it previously moved has been infected.

APPENDIX A: Density discovery delay

In this section we provide an analytical estimation of the maximum delay time in order to acquire the perfect knowledge of the densest area to all the participating nodes. The proposed worst case analysis has the aim of demonstrating that, even in presence of a high amount of collisions and quite old and slow transmission technology such as IEEE 802.11b (Table 3), the density discovery mechanism proposed in this paper and implemented by using a selective broadcasting mechanism, can be executed in a fair amount of time. Let T header be the time spent in all the protocol headers on each PROBE packet including the PLCP and 802.11 headers. We consider that PLCP header is transmitted at 1 Mbps while the others are transmitted at R Mbps (e.g. 11 Mbps). The time spent by each probe packet T P ROBE for one single hop is the sum of DIFS period, the station's back-off computed for the worst case, the headers transmission, the link-layer ACK and the SIFS period. In particular the T Bof f value, representing the most dominant terms, is computed according to a worst case analysis in which we suppose there are six consecutive collisions at MAC layer before to gain a valid transmission and the congestion window dimension CW min , is doubled at each collision as shown in the following formula: Due to the back-off timer mechanism, the slot duration is not always a fixed value but it differs depending by the specific status (i.e., idle, occupied, successful transmission, unsuccessful transmission due to frame error or collisions). By following the worst case analysis, we always considered the greatest value for the slot duration T SW represented by the average time the channel is sensed busy by each station because of a successful transmission:

T Bof f = [
T SW = T header + T L-P ROBE + SIF S + T L-ACK + DIF S = 213µs + 1091µs + 10µs + 11µs + 50µs = 1375µs Thus the maximum T Bof f value can be computed as follow: furthermore, we can argue that the time for sending each probe packet towards a single hop transmission, is slightly bigger than the back-off collision time:

T Bof f = [
T P ROBE = T Bof f + ε
In conclusion, to compute the ∆T M AX value to propagate the broadcasting information in a multi-hop path of N hops and to receive back the related ack message, we can use the following relation:

∆T M AX = 2 * N hops * T P ROBE
In particular, by considering a network 7 hops long, it is possible to complete the broadcast process, to be aware of the most dense location area, in less than one minute (i.e. about 58.52s).

Fig. 1 :

 1 Fig. 1: Error in computed Epidemic Position vs. Number of Iterations.

Fig. 2 :

 2 Fig. 2: General simulation scenario.

Fig. 4 :

 4 Fig. 4: Simulation scenarios 6.05 • 10 -3 N odes/m 2 : a) Grid scenario, b) Gaussian Scenario, c) Realistic Scenario

Fig. 5 :

 5 Fig.5: Fraction of infected nodes varying time and E{G}. 6.05 • 10 -3 N odes/m 2 ; F = 0.5. Gaussian Topology.

Fig. 6 :

 6 Fig.6: Fraction of infected nodes varying time and E{G}. 6.05 • 10 -3 N odes/m 2 ; F = 0.5. Grid Topology.

Fig. 7 :

 7 Fig. 7: Fraction of infected nodes varying time and E{G}. 6.05 • 10 -3 N odes/m 2 ; F = 0.5. Realistic Topology.

Fig. 8 :

 8 Fig. 8: Mutation Overhead.

Fig. 9 :

 9 Fig. 9: Peak value of mutated nodes.

T

  header = T P LCP + T 802.11 = 192 + 28 * 8/R = 192 + 224/R By considering a data rate of 11 Mbps we have T header ∼ = 213µs

5 i=0 2 i

 52 * CW min * T slot ] + [CW max * T slot ]

5 i=0 2 i

 52 * CW min * T SW ] + [CW max * T SW ] = 4.18s

Table 1 :

 1 Probe Packet composition.

	Local Values	Bytes	Best Values	Bytes	Top Values	Bytes	Bottom Values	Bytes
	Position X	8	Position X	8	Position X	8	Position X	8
	Position Y	8	Position Y	8	Position Y	8	Position Y	8
	Perceived Density	2	Highest Density	2

  Algorithm 1 Density Discovery and Diffusion: Node and Spreader behaviors ⇒ i th Node Behavior: while T <= endtime do while K * T p ≤ T ≤ (K + 1) * T p do Collect P robes f rom P erceived N odes P N ;

	if T = (K + 1) * T p then
		for j = 1 to P N do
		Calculate distance (i, j);
		evaluate M axDistance() = M d;
		perceivedDensity = P N/(M d * π 2 )
		if perceivedDensity > HighestDensity then
		HighestDensity = localData;
		Send P robe();
		K + +;
	⇒Spreader Node Behavior:
	Every	T M AX ;
	if M aximum Density area detected then
	Move to denser area
	else
	Move to Barycenter

density) while it is necessary to add a number of iterations roughly equal to twice the diameter of the epidemic in order to deliver a good estimation of the epidemic center (i.e., the point of highest node density).

Table 2 :

 2 802.11g Simulation Parameters.

	PHY Parameter	Value	MAC Parameter	Value
	Frequency	2.4 GHz	SlotTime	9 µs
	Receive Sensitivity	-86 dBm	SIFS	16 µs
	Transmission Power	18 dBm	Preamble Length	96 bit

Table 3 :

 3 IEEE 802.11b relevant parameters

	Parameter	Value
	Channel rate	11Mbps
	SIFS	10 µs
	DIFS	50 µs
	Slot Time T slot	20 µs
	P LCP Header	192 µs
	[CW min , CWmax]	[32,1024]
	Probe packet	1500 byte
	Ack packet	14 byte

T P ROBE = DIF S + T Bof f + T header + T L-P ROBE + +SIF S + T P LCP + T L-ACK = 50µs + T Bof f + 213µs + (1500 * 8/R) + +10µs + 192µs + (14 * 8/R)

Acknowledgements This work has been carried out under the framework of STEM-Net, PRIN-National Italian Project # H21J11000050001, financed by the Italian Ministry of University and Research. The research of Nicola Roberto Zema is partially supported by European Union (EU), European Social Fund (ESF) and Calabria Local Goverment. This paper reflects the views only of the authors, and the EU, the ESF and Calabria Local Government cannot be held responsible for any use which may be made of the information contained therein.


É Delaygue

ARITHMETIC PROPERTIES OF APÉRY-LIKE NUMBERS

Keywords: 2010 Mathematics Subject Classification. Primary 11B50; Secondary 11B65 05A10 Apéry numbers, Constant terms of powers of Laurent polynomials, p-Lucas property, Congruences

We provide lower bounds for p-adic valuations of multisums of factorial ratios which satisfy an Apéry-like recurrence relation: these include Apéry, Domb, Franel numbers, the numbers of abelian squares over a finite alphabet, and constant terms of powers of certain Laurent polynomials. In particular, we prove Beukers' conjectures on the p-adic valuation of Apéry numbers. Furthermore, we give an effective criterion for a sequence of factorial ratios to satisfy the p-Lucas property for almost all primes p.

1. Introduction 1.1. Classical results of Lucas and Kummer. It is a well-known result of Lucas [START_REF] Lucas | Sur les congruences des nombres eulériens et les coefficients différentiels des fonctions trigonométriques suivant un module premier[END_REF] that, for all nonnegative integers m, n and all primes p, we have

m n ≡ k i=0 m i n i mod p, (1.1)
where m = m 0 + m 1 p + • • • + m k p k and n = n 0 + n 1 p + • • • + n k p k are the base p expansions of m and n.

In particular, a prime p divides the binomial m n if, and only if there is 0 ≤ i ≤ k such that m i < n i . Precisely, Kummer proved in [START_REF] Kummer | Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen[END_REF] that, for all natural integers m ≥ n, the p-adic valuation ( 1 ) of the binomial m n is the number of carries which occur when n is added to m -n in base p. As a consequence, we have m n ∈ p α Z, where α = # 0 ≤ i ≤ k :

m i n i = 0 . (1.2)
In this article, we show that many sequences of Apéry-like numbers satisfy congruences similar to (1.1), that is, for all nonnegative integers n and all primes p, we have

A(n) ≡ k i=0 A(n i ) mod p,
where n = n 0 + n 1 p + • • • + n k p k is the base p expansion of n. Furthermore, we prove that an analogue of (1.2) holds for those numbers, that is A(n) ∈ p α Z, where α = # 0 ≤ i ≤ k : A(n i ) ≡ 0 mod p , which proves Beukers' conjectures on the p-adic valuation of Apéry numbers. Those sequences were used in 1979 by Apéry in his proofs of the irrationality of ζ(3) and ζ(2) (see [START_REF] Apéry | Irrationalité de ζ(2) et ζ(3)[END_REF]). In the 1980's, several congruences satisfied by those sequences were demonstrated (see for example [START_REF] Beukers | Some congruences for the Apéry numbers[END_REF], [START_REF] Beukers | Another congruence for the Apéry numbers[END_REF], [START_REF] Chowla | Congruences properties of Apéry numbers[END_REF], [START_REF] Gessel | Some congruences for Apéry numbers[END_REF], [START_REF] Mimura | Congruence properties of Apéry numbers[END_REF]). In particular, Gessel proved in [START_REF] Gessel | Some congruences for Apéry numbers[END_REF] that A 1 satisfies the p-Lucas property for all prime numbers p, that is, for any prime p, all v in {0, . . . , p -1} and all natural integers n, we have

A 1 (v + np) ≡ A 1 (v)A 1 (n) mod p.
Thereby, if n = n 0 + n 1 p + • • • + n N p N is the base p expansion of n, then we obtain

A 1 (n) ≡ A 1 (n 0 ) • • • A 1 (n N ) mod p.
(1.3)

In particular, p divides A 1 (n) if and only if there exists k in {0, . . . , N} such that p divides A 1 (n k ). Beukers stated in [START_REF] Beukers | Congruence properties of coefficients of solutions of Picard-Fuchs equations[END_REF] two conjectures, when p = 5 or 11, which generalize this property (2 ). Before stating these conjectures, we observe that the set of all v in {0, . . . , 4} (respectively v in {0, . . . , 10}) satisfying A 1 (v) ≡ 0 mod 5 (respectively A 1 (v) ≡ 0 mod 11) is {1, 3} (respectively {5}).

Conjecture A (Beukers, [START_REF] Beukers | Congruence properties of coefficients of solutions of Picard-Fuchs equations[END_REF]). Let n be a natural integer whose base 5 expansion is n = n 0 + n 1 5 + • • • + n N 5 N . Let α be the number of k in {0, . . . , N} such that n k = 1 or 3. Then 5 α divides A 1 (n).

Conjecture B (Beukers, [START_REF] Beukers | Congruence properties of coefficients of solutions of Picard-Fuchs equations[END_REF]). Let n be a natural integer whose base 11 expansion is n = n 0 + n 1 11 + • • • + n N 11 N . Let α be the number of k in {0, . . . , N} such that n k = 5. Then

11 α divides A 1 (n).
Similarly, Sequence A 2 satisfies the p-Lucas property for all primes p. Furthermore, Beukers and Stienstra proved in [START_REF] Beukers | On the Picard-Fuchs equation and the formal Brauer group of certain elliptic K3-surfaces[END_REF] that, if p ≡ 3 mod 4, then A 2 p-1 2 ≡ 0 mod p, and Beukers stated in [START_REF] Beukers | Congruence properties of coefficients of solutions of Picard-Fuchs equations[END_REF] the following conjecture.

Conjecture C. Let p be a prime number satisfying p ≡ 3 mod 4. Let n be a natural integer whose base p expansion is n = n 0 + n 1 p + • • • + n N p N . Let α be the number of k in {0, . . . , N} such that n k = p-1 2 . Then p α divides A 2 (n).

Conjectures A-C have been extended to generalized Apéry numbers and any prime p by Deutsch and Sagan in [START_REF] Deutsch | Congruences for Catalan and Motzkin numbers and related sequences[END_REF]Conjecture 5.13] but this conjecture is false for at least one generalization of Apéry numbers. Indeed, a counterexample is given by

A(n) = n k=0 n k 2 n + k k 3 , since A(1) = 9 ≡ 0 mod 3 but A(4) = A(1 + 3) = 1152501 is not divisible by 3 2 .
The main aim of this article is to prove Theorem 1, stated in Section 1.4, which demonstrates and generalizes Conjectures A-C. First, we introduce some notations which we use throughout this article.

• m + n = (m 1 + n 1 , . . . , m d + n d ); • m • n = m 1 n 1 + • • • + m d n d ; • mλ = (m 1 λ, . . . , m d λ); • |m| = m 1 + • • • + m d ; • m (k) = m k ;
• m ≥ n if, and only if, for all i in {1, . . . , d}, we have m i ≥ n i . Furthermore, we set 0 = (0, . . . , 0) ∈ N d , 1 = (1, . . . , 1) ∈ N d and we write 1 k for the vector in N d , all of whose coordinates equal zero except the k-th which is 1. If p is a prime number and n is nonzero, then we say that n = n 0 + n 1 p + • • • + n N p N is the base p expansion of n if, for all i in {0, . . . , N}, we have n i ∈ {0, . . . , p -1} d , and n N = 0. For all primes p, we write Z p for the ring of p-adic integers. If A = A(n) n∈N d is a Z p -valued family, then we say that A satisfies the p-Lucas property if and only if, for all vectors v in {0, . . . , p -1} d and n in N d , we have

A(v + np) ≡ A(v)A(n) mod pZ p .
(1.4)

We write f A for the generating series of A defined by f A (z) :

= n∈N d A(n)z n , where, if z = (z 1 , . . . , z d ) is a vector of variables and n = (n 1 , . . . , n d ) ∈ N d , z n denotes z n 1 1 • • • z n d d .
In addition, we write Z p (A) for the set of all vectors v in {0, . . . , p -1} d such that A(v) ∈ pZ p . For every nonzero vector n in N d whose base p expansion is n = n 0 + n 1 p + • • • + n N p N , we write α p (A, n) for the number of k in {0, . . . , N} such that n k ∈ Z p (A), and we set α p (A, 0) = 0. Thereby, to prove Conjectures A-C, it is enough to show that

A i (n) ∈ p αp(A i ,n) Z with i = 1, p = 5 or 11 and i = 2, p ≡ 3 mod 4.
Given tuples of vectors in N d , e = (e 1 , . . . , e u ) and f = (f 1 , . . . , f v ), we write |e| = u i=1 e i and, for all vectors n in N d and all natural integers m, we set

Q e,f (n) := u i=1 (e i • n)! v i=1 (f i • n)! and S e,f (m) := n∈N d ,|n|=m Q e,f (n).
Let S := {1 ≤ i ≤ u : e i ≥ 1}. For every positive integer r, we say that e is r-admissible

if #S + min 1≤k≤d #{1 ≤ i ≤ u : i / ∈ S, e i ≥ d1 k } ≥ r.
For all primes p, we write F d p for the set of all functions g : N d → Z p such that, for all natural integers K, there exists a sequence (P K,k ) k≥0 of polynomial functions with coefficients in Z p which converges pointwise to g on {0, . . . , K} d . For all tuples e and f of vectors in N d , all g ∈ F d p and all natural integers m, we set

S g e,f (m) := n∈N d ,|n|=m Q e,f (n)g(n).
Finally, we set θ := z d dz and we say that a differential operator L in Z p [z, θ] is of type I if there is a natural integer q such that:

• L = P 0 (θ) + zP 1 (θ) + • • • + z q P q (θ) with P k (X) ∈ Z p [X] for 0 ≤ k ≤ q; • P 0 (Z × p ) ⊂ Z × p ; • for all k in {2, . . . , q}, we have P k (X) ∈ k-1 i=1 (X + i) 2 Z p [X]. We say that a differential operator L in Z p [z, θ] is of type II if • L = P 0 (θ) + zP 1 (θ) + z 2 P 2 (θ) with P k (X) ∈ Z p [X] for 0 ≤ k ≤ 2; • P 0 (Z × p ) ⊂ Z × p ; • P 2 (X) ∈ (X + 1)Z p [X].
1.4. Main results. The main result of this article is the following. Theorem 1. Let e and f = (1 k 1 , . . . , 1 kv ) be two disjoint tuples of vectors in N d such that |e| = |f |, for all i in {1, . . . , v}, k i is in {1, . . . , d}, and e is 2-admissible. Let p be a fixed prime. Assume that f S e,f is annihilated by a differential operator L ∈ Z p [z, θ] such that at least one of the following conditions holds:

• L is of type I.

• L is of type II and p -1 ∈ Z p (S e,f ).

Then, for all natural integers n and all functions g in F d p , we have S e,f (n) ∈ p αp(S e,f ,n) Z and S g e,f ∈ p αp(S e,f ,n)-1 Z p . In Section 1.6, we show that Theorem 1 applies to many classical sequences. In particular, Theorem 1 implies Conjectures A-C. Indeed, we have A 1 = S e 1 ,f 1 and A 2 = S e 2 ,f 2 with d = 2, e 1 = (2, 1), (2, 1) and f 1 = (1, 0), (1, 0), (1, 0), (1, 0), (0, 1), (0, 1) , and e 2 = (2, 1), (1, 1) and f 2 = (1, 0), (1, 0), (1, 0), (0, 1), (0, 1) .

Furthermore, it is well known that f A 1 , respectively f A 2 , is annihilated by the differential operator L 1 , respectively L 2 , defined by

L 1 = θ 3 -z(34θ 3 + 51θ 2 + 27θ + 5) + z 2 (θ + 1) 3 and L 2 = θ 2 -z(11θ 2 + 11θ + 3) -z 2 (θ + 1) 2 .
Since L 1 and L 2 are of type I for all primes p, the conditions of Theorem 1 are satisfied by A 1 and A 2 , and Conjectures A-C hold. In addition, for all primes p and all natural integers n and α, we obtain that

n k=0 k α n k 2 n + k k 2 ∈ p αp(A 1 ,n)-1 Z and n k=0 k α n k 2 n + k k ∈ p αp(A 2 ,n)-1 Z.
We provide a similar result which applies to the constant terms of powers of certain Laurent polynomials. Consider a Laurent polynomial

Λ(x) = k i=1 α i x a i ∈ Z p [x ± 1 , . . . , x ± d ],
where a i ∈ Z d and α i = 0 for i in {1, . . . , k}. Recall that the Newton polyhedron of Λ is the convex hull of {a 1 , . . . , a k } in R d . Hence we have the following result.

Theorem 2. Let p be a fixed prime. Let Λ(x) ∈ Z p [x ± 1 , . . . , x ± d ] be a Laurent polynomial, and consider the sequence of the constant terms of powers of Λ defined, for all natural integers n, by A(n) := Λ(x) n 0 . Assume that the Newton polyhedron of Λ contains the origin as its only interior integral point, and that f A is annihilated by a differential operator L in Z p [z, θ] such that at least one of the following conditions holds:

• L is of type I.

• L is of type II and p -1 ∈ Z p (A). Then, for all natural integers n, we have

A(n) ∈ p αp(A,n) Z p .
For example, Theorem 2 applies to Apéry numbers A 1 thanks to the following formula of Lairez [18]:

A 1 (n) = (1 + z)(yz + z + 1)(1 + x)(xy + x + y) xyz n (0,0,0)
.

By a result of Samol and van Straten

[23], if Λ(x) ∈ Z p [x ± 1 , . . . , x ± d ]
contains the origin as its only interior integral point, then [Λ(x) n ] 0 n≥0 satisfies the p-Lucas property, which is essential for the proof of Theorem 2. Likewise, the proof of Theorem 1 rests on the fact that S e,f satisfies the p-Lucas property when |e| = |f |, e is 2-admissible and f = (1 k 1 , . . . , 1 kv ).

Since those results deal with multisums of factorial ratios, it seems natural to study similar arithmetic properties for simpler numbers such as families of factorial ratios. To that purpose, we prove Theorem 3 below which gives an effective criterion for Q e,f to satisfy the p-Lucas property for almost all primes p ( 3 ). Furthermore, Theorem 3 shows that if A := Q e,f satisfies the p-Lucas property for almost all primes p, then, for all natural integers n and all primes p, we have A(n) ∈ p αp(A,n) Z.

To state this result, we introduce some additional notations. For all tuples e and f of vectors in N d , we write ∆ e,f for Landau's function defined, for all x in R d , by

∆ e,f (x) := u i=1 ⌊e i • x⌋ - v i=1 ⌊f i • x⌋ ∈ Z,
where ⌊•⌋ denotes the floor function. Therefore, according to Landau's criterion [START_REF] Landau | Sur les conditions de divisibilité d'un produit de factorielles par un autre, collected works[END_REF] and a precision of the author [START_REF] Delaygue | A criterion for the integrality of the Taylor coefficients mirror maps in several variables[END_REF], we have the following dichotomy.

• If, for all x in [0, 1] d , we have ∆ e,f (x) ≥ 0, then Q e,f is a family of integers;

• if there exists x in [0, 1] d such that ∆ e,f (x) ≤ -1, then there are only finitely many primes p such that Q e,f is a family of p-adic integers. In the rest of the article, we write D e,f for the semi-algebraic set of all x in [0, 1) d such that there exists a component d of e or f satisfying d • x ≥ 1. Observe that ∆ e,f vanishes on the nonempty set [0, 1) d \ D e,f . Theorem 3. Let e and f be disjoint tuples of vectors in N d such that Q e,f is a family of integers. Then we have the following dichotomy.

(1) If |e| = |f | and if, for all x in D e,f , we have ∆ e,f (x) ≥ 1, then for all primes p, Q e,f satisfies the p-Lucas property; (2) if |e| = |f | or if there exists x in D e,f such that ∆ e,f (x) = 0, then there are only finitely many primes p such that Q e,f satisfies the p-Lucas property. Furthermore, if Q e,f satisfies the p-Lucas property for all primes p, then, for all n in N d and every prime p, we have

Q e,f (n) ∈ p αp(Q e,f ,n) Z.
Remark. Theorem 3 implies that Q e,f satisfies the p-Lucas property for all primes p if and only if all Taylor coefficients at the origin of the associated mirror maps z e,f,k , 1 ≤ k ≤ d, are integers (see Theorems 1 and 3 in [START_REF] Delaygue | A criterion for the integrality of the Taylor coefficients mirror maps in several variables[END_REF]). Indeed, if ∆ e,f is nonnegative on

[0, 1] d and if |e| = |f |, then there exists k in {1, . . . , d} such that |e| (k) > |f | (k) .
Coster proved in [START_REF] Coster | Congruence properties of coefficients of certain algebraic power series[END_REF] similar results to Theorems 1-3 for the coefficients of certain algebraic power series. Namely, given a prime p ≥ 3, integers a 1 , . . . , a p-1 , and a sequence A such that

f A (z) = (1 + a 1 z + • • • + a p-1 z p-1 ) 1 1-p ,
3 Throughout this article, we say that an assertion A p is true for almost all primes p if there exists a constant C ∈ N such that A p holds for all primes p ≥ C.

Coster proved that, for all natural integers n, we have

v p A(n) ≥ α p (A, n) + 1 2 .
1.5. Auxiliary results. The proof of Theorem 1 rests on three results which may be useful to study other sequences.

Proposition 1. Let p be a fixed prime and A a Z p -valued sequence satisfying the p-Lucas property with A(0) in Z × p . Let A be the Z p -module spanned by A. Assume that (a) there exists a set B of Z p -valued sequences with A ⊂ B such that, for all B in B, all v in {0, . . . , p -1} and all positive integers n, there exist A ′ in A and a sequence

(B k ) k≥0 , B k in B, such that B(v + np) = A ′ (n) + ∞ k=0 p k+1 B k (n -k); (b) f A (z) is annihilated by a differential operator L in Z p [z, θ]
such that at least one of the following conditions holds:

-L is of type I. -L is of type II and p -1 ∈ Z p (A).
Then, for all B in B and all natural integers n, we have

A(n) ∈ p αp(A,n) Z p and B(n) ∈ p αp(A,n)-1 Z p .
In Proposition 1 and throughout this article, if (A(n)) n≥0 is a sequence taking its values in Z or Z p , then, for all negative integers n, we set A(n) := 0. Therefore, to prove Theorem 1, it suffices to show that S e,f satisfies the p-Lucas property and Condition (a) of Proposition 1 with B = {S g e,f : g ∈ F d p }. To that purpose, we shall prove the following results.

Proposition 2. Let e and f be disjoint tuples of vectors in N d such that |e| = |f | and, for all x in D e,f , ∆ e,f (x) ≥ 1. Assume that e is 1-admissible. Then, S e,f is integer-valued and satisfies the p-Lucas property for all primes p. Proposition 3. Let p be a fixed prime. We write Γ p for the p-adic Gamma function. Then, there exists a function g in F 2 p such that, for all natural integers n and m, we have

Γ p (m + n)p Γ p (mp)Γ p (np) = 1 + g(m, n)p.
1.6. Application of Theorem 1. By applying Theorem 1, we obtain similar results to Conjectures A-C for numbers satisfying Apéry-like recurrence relations which we list below. Characters in brackets in the last column of the following table form the sequence number in the Online Encyclopedia of Integer Sequences [START_REF]The On-Line Encyclopedia of Integer Sequences[END_REF].

Sequence Q e,f (n 1 , n 2 ) L Reference n k=0 n k 2 n + k k 2 (2n 1 + n 2 )! 2 n 1 ! 4 n 2 ! 2 [1, (γ)] Apéry numbers (A005259) n k=0 n k 2 n + k k (2n 1 + n 2 )!(n 1 + n 2 )! n 1 ! 3 n 2 ! 2 [26, D] Apéry numbers (A005258) 2n n = n k=0 n k 2 (n 1 + n 2 )! 2 n 1 ! 2 n 2 ! 2 type I Central binomial coefficients (A000984) n k=0 n k 3 (n 1 + n 2 )! 3 n 1 ! 3 n 2 ! 3 [26, A] Franel numbers (A000172) n k=0 n k 4 (n 1 + n 2 )! 4 n 1 ! 4 n 2 ! 4 [14],[15] (A005260) n k=0 n k 2k k 2(n -k) n -k (n 1 + n 2 )!(2n 1 )!(2n 2 )! n 1 ! 3 n 2 ! 3 [1, (d)] (A081085) n k=0 n k 2 2k k (n 1 + n 2 )! 2 (2n 1 )! n 1 ! 4 n 2 ! 2 [26, C]
Number of abelian squares of length 2n over an alphabet with 3 letters (A002893)

n k=0 n k 2 2k k 2(n -k) n -k (n 1 + n 2 )! 2 (2n 1 )!(2n 2 )! n 1 ! 4 n 2 ! 4 [1, (α)] Domb numbers (A002895) n k=0 2k k 2 2(n -k) n -k 2 (2n 1 )! 2 (2n 2 )! 2 n 1 ! 4 n 2 ! 4 [1, (β)] (A036917)
All differential operators listed in the above table are of type I for all primes p, except the one associated with A 5 (n) := n k=0 n k 4 which reads

L 5 = θ 3 -z2(2θ + 1)(3θ 2 + 3θ + 1) -z 2 4(θ + 1)(4θ + 5)(4θ + 3).
Hence L 5 is of type II for all primes p. By a result of Calkin [8, Proposition 3], for all primes p, we have A 5 (p -1) ≡ 0 mod p, i. e. p -1 is in Z p (A 5 ). Thus we can apply Theorem 1 to A 5 .

Observe that the generating function of the central binomial coefficients is annihilated by the differential operator L = θ -z(4θ + 2) which is of type I for all primes p.

According to the recurrence relation found by Almkvist and

Zudilin (see Case (d) in [1]), A 6 (n) := n k=0 n k 2k k 2(n-k) n-k
is also Sequence E in Zagier's list [START_REF] Zagier | Integral solutions of Apéry-like reccurence equations, Groups and symmetries[END_REF], that is

A 6 (n) = ⌊n/2⌋ k=0 4 n-2k n 2k 2k k 2 .
Furthermore, according to [START_REF] Richmond | Counting abelian squares[END_REF], Domb numbers

A 8 (n) = n k=0 n k 2 2k k 2(n-k) n-k
are also the numbers of abelian squares of length 2n over an alphabet with 4 letters. Now we consider the numbers C i (n) of abelian squares of length 2n over an alphabet with i letters which, for all positive integers i ≥ 2, satisfy (see [START_REF] Richmond | Counting abelian squares[END_REF])

C i (n) = k 1 +•••+k i =n k 1 ,...,k i ∈N n! k 1 ! • • • k i ! 2 .
According to [START_REF] Borwein | Some arithmetic properties of short random walk integrals[END_REF], C i (n) is also the 2n-th moment of the distance to the origin after i steps traveled by a walk in the plane with unit steps in random directions.

To apply Theorem 1 to C i , it suffices to show that f C i is annihilated by a differential operator of type I for all primes p. Indeed, by Proposition 1 and Theorem 2 in [START_REF] Borwein | Some arithmetic properties of short random walk integrals[END_REF], for all j ≥ 2, C j (n) satisfies the recurrence relation of order ⌈j/2⌉ with polynomial coefficients of degree j -1:

n j-1 C j (n) + i≥1 n j-1 α 1 ,...,α i i k=1 (-α k )(j + 1 -α k ) n -k n -k + 1 α k -1 C j (n -i) = 0, (1.5)
where the sum is over all sequences of positive integers α 1 , . . . , α i satisfying α k ≤ j and α k+1 ≤ α k -2. We consider i ≥ 2 and i positive integers α 1 , . . . , α i ≤ j satisfying α k+1 ≤ α k -2. We have

n j-1 i k=1 n -k n -k + 1 α k -1 = n j-1 n α 1 -1 i-1 k=1 (n -k) α k -α k+1 (n -i) α i -1 , with j -α 1 ≥ 0, α k -α k+1 ≥ 2 and α i -1 ≥ 0. Then, f C j (z)
is annihilated by a differential operator L = P 0 (θ) + zP 1 (θ) + • • • + z q P q (θ) with P 0 (θ) = θ j-1 and, for all i ≥ 2,

P i (θ) ∈ i-1 k=1 (θ + i -k) 2 Z[θ] ⊂ i-1 k=1 (θ + k) 2 Z[θ],
so that L is of type I for all primes p, as expected.

1.7. Structure of the article. In Section 2, we use several results of [START_REF] Delaygue | A criterion for the integrality of the Taylor coefficients mirror maps in several variables[END_REF] to prove Theorem 3. Section 3 is devoted to the proofs of Theorem 2 and Proposition 1. In particular, we prove Lemma 1 which points out the role played by differential operators in our proofs. In Section 4, we prove Theorem 1 by applying Proposition 1 to S e,f . It is the most technical part of this article.

Proof of Theorem 3

First, we prove that if |e| = |f |, then, for all primes p, all a in {0, . . . , p -1} d and all n in N d , we have

Q e,f (a + np) Q e,f (a)Q e,f (n) ∈ u i=1 ⌊e i •a/p⌋ j=1 1 + e i •n j v i=1 ⌊f i •a/p⌋ j=1 1 + f i •n j (1 + pZ p ). (2.1)
Indeed, we have

Q e,f (a + np) Q e,f (a)Q e,f (n) = Q e,f (a + np) Q e,f (a)Q e,f (np) • Q e,f (np) Q e,f (n)
.

Since |e| = |f |, we can apply [11, Lemma 7] (4 ) with c = 0, m = n and s = 0 which yields

Q e,f (np) Q e,f (n) ∈ 1 + pZ p .
Furthermore, we have

Q e,f (a + np) Q e,f (a)Q e,f (np) = 1 Q e,f (a) u i=1 e i •a j=1 (j + e i • np) v i=1 f i •a j=1 (j + f i • np) = u i=1 e i •a j=1 1 + e i •np j v i=1 f i •a j=1 1 + f i •np j ∈ u i=1 ⌊e i •a/p⌋ j=1 1 + e i •n j v i=1 ⌊f i •a/p⌋ j=1 1 + f i •n j (1 + pZ p ),
because, if p does not divide j, then 1 + (e i • np)/j belongs to 1 + pZ p . This finishes the proof of (2.1). Now we prove Assertion (1) in Theorem 3. Let p be a fixed prime number. It is well known that, for all natural integers n, we have

v p (n!) = ∞ ℓ=1 n p ℓ .
Thus, for all vectors n in N d , we have

v p Q e,f (n) = ∞ ℓ=1 ∆ e,f n p ℓ .
Let fix n in N d and a in {0, . . . , p-1} d . Let {•} denote the fractional part function. For any vector of real numbers x = (x 1 , . . . , x d ), we set {x} := ({x 1 }, . . . , {x d }). Since |e| = |f |, we have

v p Q e,f (a + np) = ∞ ℓ=1 ∆ e,f a + np p ℓ ≥ ∆ e,f a p ,
because ∆ e,f is nonnegative on [0, 1] d . On the one hand, if a/p is in D e,f , then both Q e,f (a + np) and Q e,f (a)Q e,f (n) are congruent to 0 modulo p. On the other hand, if a/p is not in D e,f , then, for all d in e or f , we have ⌊d • a/p⌋ = 0 so that (2.1) yields

Q e,f (a + np) ≡ Q e,f (a)Q e,f (n) mod pZ p ,
as expected. This proves Assertion (1) in Theorem 3.

Now we prove Assertion

(2) in Theorem 3. If |e| = |f | then, since ∆ e,f is nonnegative on [0, 1] d , there exists k in {1, . . . , d} such that |e| (k) -|f | (k) = ∆ e,f (1 k ) ≥ 1.
Thereby, for almost all primes p, we have

v p Q e,f (1 k + 1 k p) = ∞ ℓ=1 ∆ e,f 1 k + 1 k p p ℓ ≥ ∆ e,f 1 k p + 1 k ≥ 1, but v p Q e,f (1 k ) = 0 so that Q e,f
does not satisfy the p-Lucas property.

Throughout the rest of this proof, we assume that |e| = |f |. According to Section 7.3.2 in [START_REF] Delaygue | A criterion for the integrality of the Taylor coefficients mirror maps in several variables[END_REF], there exist k in {1, . . . , d} and a rational fraction R(X) in Q(X), R(X) = 1, such that, for all large enough prime numbers p, we can choose a p in {0, . . . , p -1} d satisfying Q e,f (a p ) ∈ Z × p , and such that, for all natural integers n, we have (see [11, (7.10)

]) Q e,f (a p + 1 k np) ∈ R(n)Q e,f (a p )Q e,f (1 k n)(1 + pZ p ).
We fix a natural integer n satisfying R(n) = 1. For almost all primes p, the numbers R(n), Q e,f (1 k n) and Q e,f (a p ) are invertible in Z p , and R(n) ≡ 1 mod pZ p . Thus, we obtain

Q e,f (a p + 1 k np) ≡ Q e,f (a p )Q e,f (1 k n) mod pZ p ,
which finishes the proof of Assertion (2) in Theorem 3.

Now we assume that |e| = |f | and that, for all x in D e,f , we have ∆ e,f (x) ≥ 1. Hence, for every prime p, we have

Z p (Q e,f ) = v ∈ {0, . . . , p -1} d : v/p ∈ D e,f .
Furthermore, if v/p belongs to D e,f , then, for all positive integers N and all vectors a 0 , . . . , a N -1 in {0, . . . , p -1} d , we have

v p ≤ a 0 + a 1 p + • • • + a N -1 p N -1 + vp N p N +1 ∈ D e,f , so that, for every n in N d , n = ∞ k=0 n k p k with n k ∈ {0, . . . , p -1} d , we have v p Q e,f (n) = ∞ ℓ=1 ∆ e,f ℓ-1 k=0 n k p k p ℓ ≥ α p (Q e,f , n),
and Theorem 3 is proved.

Proofs of Theorem 2 and Proposition 1

3.1. Induction via Apéry-like recurrence relations. In this section, we fix a prime p. If A is a Z p -valued sequence, then, for all natural integers r, we write U A (r) for the assertion "For all n, i ∈ N, i ≤ r, if α p (A, n) ≥ i, then A(n) ∈ p i Z p ". As a first step, we shall prove the following result.

Lemma 1. Let A be a Z p -valued sequence satisfying the p-Lucas property with A(0) in Z × p . Assume that f A is annihilated by a differential operator L ∈ Z p [z, θ] such that at least one of the following conditions holds:

• L is of type I.

• L is of type II and p -1 ∈ Z p (A). Let r be a natural integer such that U A (r) holds. Then, for all n 0 in Z p (A) and all natural integers m satisfying α p (A, m) ≥ r, we have

A(n 0 + mp) ∈ p r+1 Z p .
Proof. Since A satisfies the p-Lucas property, we can assume that r is nonzero. The series f A (z) is annihilated by a differential operator L = P 0 (θ) + zP 1 (θ) + • • • + z q P q (θ) with P k (X) in Z p [X] and P 0 (Z × p ) ⊂ Z × p . Thus, for every natural integer n, we have

q k=0 P k (n -k)A(n -k) = 0. (3.1)
We fix a natural integer m satisfying α p (A, m) ≥ r. In particular, since r is nonzero and A(0) is invertible in Z p , we have m ≥ 1. Furthermore, for all v in {0, . . . , p -1}, we also have α p (A, v + mp) ≥ r. According to U A (r), we obtain that, for all v in {0, . . . , p -1}, A(v + mp) belongs to p r Z p so that A(v + mp) =: β(v, m)p r , with β(v, m) ∈ Z p . By (3.1), for all v in {q, . . . , p -1}, we have

0 = q k=0 P k (v -k + mp)A(v -k + mp) = p r q k=0 P k (v -k + mp)β(v -k, m) ≡ p r q k=0 P k (v -k)β(v -k, m) mod p r+1 Z p ,
because, for all polynomials P in Z p [X] and all integers a and c, we have P (a + cp) ≡ P (a) mod pZ p . Thus, for all v in {q, . . . , p -1}, we obtain

q k=0 P k (v -k)β(v -k, m) ≡ 0 mod pZ p . (3.2)
We claim that if v is in {1, . . . , q -1}, then, for all k in {v + 1, . . . , q}, we have

P k (v + mp -k)A(v + mp -k) ∈ p r+1 Z p . (3.3)
Indeed, on the one hand, if L is of type II, then we have q = 2 and P 2 (X) belongs to (X + 1)Z p [X] which yields

P 2 (-1 + mp)A(-1 + mp) ∈ pA p -1 + (m -1)p Z p .
Since 0 is not in Z p (A), we have α p (A, m -1) ≥ r -1 which, together with p -1 ∈ Z p (A), leads to

α p A, p -1 + (m -1)p ≥ r.
According to U A (r), we obtain that pA p -1 + (m -1)p is in p r+1 Z p , as expected. On the other hand, if L is of type I, then for all v in {1, . . . , q -1} and all k in {v + 1, . . . , q}, we have

v p P k (v + mp -k) ≥ v p k-1 i=1 (v + mp -k + i) 2 .
Writing k -v = a + bp with a in {0, . . . , p -1} and b in N, we obtain k -1 ≥ a + bp so that

v p k-1 i=1 (mp + i -a -bp) ≥ b if a = 0; b + 1 if a ≥ 1.
.

Thus, it is enough to prove that

A(v + mp -k) ∈ p r+1-2b Z p if a = 0; p r-1-2b Z p if a ≥ 1. . ( 3
α p (A, v + mp -k) = α p A, p -a + (m -b -1)p ≥ r -b -1.
Hence Assertion U A (r) yields

A(v + mp -k) ∈ p r-b Z p if a = 0; p r-1-b Z p if a ≥ 1.
.

If a = 0, then b ≥ 1 so that (3.4) holds and (3.3) is proved.

By (3.3), for all natural integers v satisfying 1 ≤ v ≤ min(q -1, p -1), we have

0 = q k=0 P k (v -k + mp)A(v -k + mp) ≡ v k=0 P k (v -k + mp)A(v -k + mp) mod p r+1 Z p ≡ p r v k=0 P k (v -k + mp)β(v -k, m) mod p r+1 Z p ≡ p r v k=0 P k (v -k)β(v -k, m) mod p r+1 Z p .
Thus, for all natural integers v satisfying 1 ≤ v ≤ min(q -1, p -1), we have

v k=0 P k (v -k)β(v -k, m) ≡ 0 mod pZ p . (3.5)
Both sequences (β(v, m)) 0≤v≤p-1 and (A(v)) 0≤v≤p-1 satisfy Equations (3.2) and (3.5). Furthermore, for all v in {1, . . . , p -1}, P 0 (v) and A(0) are invertible in Z p . Hence there exists γ(m) in {0, . . . , p -1} such that, for all v in {0, . . . , p -1}, we have β(v, m) ≡ A(v)γ(m) mod pZ p so that

A(v + mp) ≡ A(v)γ(m)p r mod p r+1 Z p .
Since n 0 is in Z p (A), we obtain that A(n 0 + mp) belongs to p r+1 Z p and Lemma 1 is proved.

Proof of Theorem 2.

Let p be a fixed prime number. For every positive integer n, we set ℓ(n) := ⌊log p (n)⌋ + 1 the length of the expansion of n to the base p, and ℓ(0) := 1. For all natural integers n 1 , . . . , n r , we set

n 1 * • • • * n r := n 1 + n 2 p ℓ(n 1 ) + • • • + n r p ℓ(n 1 )+•••+ℓ(n r-1 )
, so that the expansion of n 1 * • • • * n r to the base p is the concatenation of the respective expansions of n 1 , . . . , n r . Then, by a result of Mellit and Vlasenko [21, Lemma 1], there exists a Z p -valued sequence (c n ) n≥0 such that, for all positive integers n, we have

A(n) = n 1 * ••• * nr=n 1≤r≤ℓ(n), nr>0 c n 1 • • • c nr and c n ≡ 0 mod p ℓ(n)-1 Z p . (3.6)
For every natural integer r, we write U(r) for the assertion: "For all n, i ∈ N, i ≤ r, if α p (A, n) ≥ i, then A(n), c n ∈ p i Z p ". To prove Theorem 2, it suffices to show that, for all natural integers r, Assertion U(r) holds.

First we prove U(1). By Theorem 1 in [START_REF] Mellit | Dwork's congruences for the constant terms of powers of a Laurent polynomial[END_REF], A satisfies the p-Lucas property. In addition, if v is in Z p (A), then v is nonzero because A(0) = 1, and by (3.6) we have c v = A(v) ∈ pZ p . Now, if a natural integer n satisfies ℓ(n) = 2 and α p (A, n) ≥ 1, then Equation (3.6) yields A(n) ≡ c n mod pZ p , so that c n is in pZ p . Hence, by induction on ℓ(n), we obtain that, for all natural integers n satisfying α p (A, n) ≥ 1, c n belongs to pZ p , so that U(1) holds.

Let r be a positive integer such that U(r) holds. We shall prove that U(r + 1) is true. For all positive integers M, we write U M (r + 1) for the assertion:

"For all n, i ∈ N, n ≤ M, i ≤ r + 1, if α p (A, n) ≥ i, then A(n), c n ∈ p i Z p ". Hence U M (r + 1) is true if ℓ(M) ≤ r.
Let M be a positive integer such that U M (r + 1) holds. We shall prove U M +1 (r + 1). By Assertions U(r) and U M (r + 1), it suffices to prove that if α p (A, M + 1) is greater than r, then A(M + 1) and c M +1 belong to p r+1 Z p . In the rest of the proof, we assume that α p (A, M + 1) is greater than r.

If u and n 1 , . . . , n u are natural integers satisfying 2 ≤ u ≤ ℓ(M + 1) and n 1 * • • • * n u = M + 1 with n u > 0, then, for all i in {1, . . . , u}, we have n i ≤ M and

α p (A, n 1 ) + • • • + α p (A, n u ) = α p (A, M + 1) ≥ r + 1. Then there exist 1 ≤ a 1 < • • • < a k ≤ u and 1 ≤ i 1 , . . . , i k ≤ r + 1 such that α p (A, n a j ) ≥ i j and i 1 + • • • + i k ≥ r + 1. Thereby, Assertion U M (r + 1) yields c n 1 • • • c nu ∈ p r+1 Z p , so that n 1 * ••• * nu=M +1 2≤u≤ℓ(M +1), nu>0 c n 1 • • • c nu ∈ p r+1 Z p .

By (3.6), we obtain

A(M + 1) ≡ c M +1 mod p r+1 Z p and c M +1 ≡ 0 mod p ℓ(M +1)-1 Z p .
Hence it suffices to consider the case ℓ(M +1) = r+1. In particular, we have M +1 = v+mp where v is in Z p (A) and m is a natural integer satisfying α p (A, m) = r. Since U(r) holds, Lemma 1 yields A(M + 1) ∈ p r+1 Z p . Thus we also have c M +1 ∈ p r+1 Z p and Assertion U M +1 (r + 1) holds. This finishes the proof of U(r + 1) so that of Theorem 2.

Proof of Proposition 1.

Let p be a prime and A a Z p -valued sequence satisfying hypothesis of Proposition 1. For every natural integer n, we write α(n), respectively Z, as a shorthand for α p (A, n), respectively for Z p (A). For every natural integer r, we define Assertions

U(r) : "For all n, i ∈ N, i ≤ r, if α(n) ≥ i, then A(n) ∈ p i Z p .", and
V(r) : "For all n, i ∈ N, i ≤ r, and all B ∈ B, if α(n) ≥ i, then B(n) ∈ p i-1 Z p ".
To prove Proposition 1, we have to show that, for all natural integers r, Assertions U(r) and V(r) are true. We shall prove those assertions by induction on r.

Observe that Assertions U(0), V(0) and V(1) are trivial. Furthermore, since A satisfies the p-Lucas property, Assertion U(1) holds. Let r 0 be a fixed positive integer, r 0 ≥ 2, such that Assertions U(r 0 -1) and V(r 0 -1) are true. First, we prove Assertion V(r 0 ).

Let B in B and m in N be such that α(m) ≥ r 0 . We write m = v + np with v in {0, . . . , p -1}. Since r 0 ≥ 2 and 0 does not belong to Z, we have n ≥ 1 and, by Assertion (a) in Proposition 1, there exist A ′ in A and a sequence (B k ) k≥0 , with B k in B, such that

B(v + np) = A ′ (n) + ∞ k=0 p k+1 B k (n -k). (3.7)
In addition, we have α(n) ≥ r 0 -1 and, since 0 is not in Z, we have α(n -1) ≥ r 0 -2. By induction, for all natural integers k satisfying k ≤ n, we have α(n -k) ≥ r 0 -1 -k. Thus, by (3.7) in combination with U(r 0 -1) and V(r 0 -1), we obtain

A ′ (n) ∈ p r 0 -1 Z and p k+1 B k (n -k) ∈ p k+1+r 0 -2-k Z p ⊂ p r 0 -1 Z p ,
so that B(v + np) belongs to p r 0 -1 Z p and V(r 0 ) is true.

Now we prove Assertion U(r 0 ). We write U N (r 0 ) for the assertion:

"For all n, i ∈ N, n ≤ N, i ≤ r 0 , if α(n) ≥ i, then A(n) ∈ p i Z p ".
We shall prove U N (r 0 ) by induction on N. Assertion U 1 (r 0 ) holds. Let N be a positive integer such that U N (r 0 ) is true. Let n := n 0 + mp ≤ N + 1 with n 0 in {0, . . . , p -1} and m in N. We can assume that α(n) ≥ r 0 .

If n 0 is in Z, then we have α(m) ≥ r 0 -1 and, by Lemma 1, we obtain that A(n) belongs to p r 0 Z p as expected. If n 0 is not in Z, then we have α(m) ≥ r 0 . By Assertion (a) in Proposition 1, there exist A ′ in A and a sequence (B k ) k≥0 with B k in B such that

A(n) = A ′ (m) + ∞ k=0 p k+1 B k (m -k).
We have m ≤ N, α(m) ≥ r 0 and α(m -k) ≥ r 0 -k, hence, by Assertions U N (r 0 ) and V(r 0 ), we obtain that A(n) belongs to p r 0 Z p . This finishes the induction on N and proves U(r 0 ). Therefore, by induction on r 0 , Proposition 1 is proved.

Proof of Theorem 1

To prove Theorem 1, we shall apply Proposition 1 to S e,f . As a first step, we prove that this sequence satisfies the p-Lucas property.

Proof of Proposition 2. For all x in [0, 1] d , we have ∆ e,f (x) = ∆ e,f ({x}) ≥ 0 so that, by Landau's criterion, Q e,f is integer-valued. Let p be a fixed prime, v in {0, . . . , p -1} and n a natural integer. We have This finishes the proof of Proposition 2.

S e,f (v + np) = k 1 +•••+k d =v+np k i ∈N Q e,f (k 1 , . . . , k d ). Write k i = a i + m i p with a i in {0, . . . , p -1} and m i in N. If a 1 + • • • + a d = v,
If e is 2-admissible then e is also 1-admissible. Furthermore, if f = (1 k 1 , . . . , 1 kv ), then, for all x in D e,f , we have

∆ e,f (x) = u i=1 ⌊e i • x⌋ ≥ 1.
Hence, if e and f satisfy the conditions of Theorem 1, then Proposition 2 implies that, for all primes p, S e,f has the p-Lucas property and S e,f (0) = 1 is invertible in Z p . Thereby, to prove Theorem 1, it remains to prove that S e,f satisfies Condition (a) in Proposition 1 with B = S g e,f : g ∈ F d p . First we prove that some special functions belong to F 1 p .

4.1. Special functions in F 1 p . For all primes p, we write | • | p for the ultrametric norm on Q p (the field of p-adic numbers) defined by |a| p := p -vp (a) . Note that In the rest of the article, for all primes p and all positive integers k, we set Ψ p,k,0 (0) = 1, Ψ p,k,i (0) = 0 for i ≥ 1 and, for all natural integers i and m, m ≥ 1, we set

(Z p , | • | p ) is a compact space. Furthermore, if (c n ) n≥0 is a Z p -valued sequence, then ∞ n=0 c n is convergent in (Z p , | • | p ) if
Ψ p,k,i (m) := (-1) i σ m,i 1 k , 1 k + p , . . . , 1 k + (m -1)p ,
where σ m,i is the i-th elementary symmetric polynomial of m variables. Let us remind to the reader that, for all natural integers m and i satisfying i > m ≥ 1, we have σ m,i = 0. The aim of this section is to prove that, for all primes p, all k in {1, . . . , p -1} and all natural integers i, we have

i!Ψ p,k,i ∈ F 1 p . (4.1)
Proof of (4.1). Throughout this proof, we fix a prime number p and an integer k in {1, . . . , p -1}. Furthermore, for all nonnegative integers i, we use Ψ i as a shorthand for Ψ p,k,i and N ≥i as a shorthand for the set of integers larger than or equal to i. We shall prove (4.1) by induction on i. To that end, for all natural integers i, we write A i for the following assertion: "There exists a sequence (T i,r ) r≥0 of polynomial functions with coefficients in Z p which converges uniformly to i!Ψ i on N". First, observe that, for all natural integers m, we have Ψ 0 (m) = 1, so that Assertion A 0 is true. Let i be a fixed positive integer such that assertions A 0 , . . . , A i-1 are true. According to the Newton-Girard formulas, for all integers m ≥ i, we have

i(-1) i σ m,i (X 1 , . . . , X m ) = - i t=1 (-1) i-t σ m,i-t (X 1 , . . . , X m )Λ t (X 1 , . . . , X m ), where Λ t (X 1 , . . . , X m ) := X t 1 + • • • + X t m .
Thereby, for all integers m ≥ i, we have

iΨ i (m) = - i t=1 Ψ i-t (m)Λ t 1 k , . . . , 1 k + (m -1)p . ( 4.2)
For all natural integers j and t, we have

1 (k + jp) t = 1 k t 1 (1 + j k p) t = 1 k t + ∞ s=1 (-1) s k t t -1 + s s j k s p s , (4.3)
where the right hand side of (4.3) is a convergent series in

(Z p , | • | p ) because k is invertible in Z p . Therefore, we obtain that Λ t 1 k , . . . , 1 k + (m -1)p = m k t + m-1 j=0 ∞ s=1 (-1) s k t t -1 + s s j k s p s = m k t + ∞ s=1 (-1) s k t+s t -1 + s s p s m-1 j=0 j s . (4.4)
According to Faulhaber's formula, for all positive integers s, we have

p s m-1 j=0 j s = s+1 c=1 (-1) s+1-c s + 1 c p s B s+1-c s + 1 (m -1) c ,
where B k is the k-th first Bernoulli number. For all positive integers s and t, we set R 0,t (X) := X/k t and R s,t (X) := 1

k t+s t -1 + s s s+1 c=1 (-1) 1-c s + 1 c p s B s+1-c s + 1 (X -1) c , so that Λ t 1 k , . . . , 1 k + (m -1)p = ∞ s=0 R s,t (m).
In the rest of this article, for all polynomials P (X) = N n=0 a n X n in Z p [X], we set P p := max |a n | p : 0 ≤ n ≤ N .

We claim that, for all natural integers s and t, t ≥ 1, we have

R s,t (X) ∈ Z p [X],
R s,t p -→ s→∞ 0 and R s,t (0) = 0. (4.5)

Indeed, on the one hand, if p = 2 and s = 1, then we have

R 1,t (X) = -t k t+1 X -1 + (X -1) 2 ∈ XZ 2 [X].
On the other hand, if p ≥ 3 or s ≥ 2, then we have p s > s + 1 so that v p (s + 1) ≤ s -1.

Furthermore, according to the von Staudt-Clausen theorem, we have v p (B s+1-c ) ≥ -1.

Thus, the coefficients of R s,t (X) belong to Z p . To be more precise, we have v p (s + 1) ≤ log p (s + 1), so that R s,t p -→ s→∞ 0 as expected. In addition, we have

R s,t (0) = - p s (s + 1)k t+s t -1 + s s s+1 c=1 s + 1 c B s+1-c = - p s (s + 1)k t+s t -1 + s s s d=0 s + 1 d B d = 0,
where we used the well known relation satisfied by the Bernoulli numbers

s d=0 s + 1 d B d = 0, (s ≥ 1).
According to A 0 , . . . , A i-1 , for all j in {0, . . . , i -1}, there exists a sequence (T j,r ) r≥0 of polynomial functions with coefficients in Z p which converges uniformly to j!Ψ j on N. According to (4.2) and (4.5), for all natural integers N, there exists S N in N such that, for all r ≥ S N and all m ≥ i, we have

i!Ψ i (m) ≡ - i t=1 (i -1)! (i -t)! T i-t,r (m) r s=0 R s,t (m) mod p N Z p .
Thus, the sequence (T i,r ) r≥0 of polynomial functions with coefficients in Z p , defined by

T i,r (x) := - i t=1 (i -1)! (i -t)! T i-t,r (x) r s=0 R s,t (x), (x, r ∈ N), (4.6)
converges uniformly to i!Ψ i on N ≥i . To prove A i , it suffices to show that, for all m in {0, . . . , i -1}, we have

T i,r (m) -→ r→∞ 0. (4.7)
The aim of this section is to prove Proposition 3.

Proof of Proposition 3. Let p be a fixed prime number. For all natural integers n and m, we have Let k in {1, . . . , p -1} be fixed. By (4.1), for all positive integers i, there exists a sequence (P i,ℓ ) ℓ≥0 of polynomial functions with coefficients in Z p which converges pointwise to i!Ψ p,k,i .

Γ p (m + n)p Γ p (mp)Γ p (np) = (
We fix a natural integer K. For all positive integers N, we set f N (x, y) := 1 + K+1 i=1 (-1) i x i p i i! P i,N (y).

If n and m belong to {0, . . . , K}, then we have

R N := 1 + ∞ i=1 (-1) i n i p i Ψ p,k,i (m) -f N (n, m) = K+1 i=1
(-1) i n i p i i! i!Ψ p,k,i (m) -P i,N (m) -→ N →∞ 0.

Furthermore, we have f N (x, y) ∈ 1 + pZ p [x, y]. Indeed, if i = i 0 + i 1 p + • • • + i a p a with i j in {0, . . . , p -1}, then we set s p (i) := i 0 + • • • + i a so that, for all positive integers i, we have

i -v p (i!) = i - i -s p (i) p -1 = i(p -2) + s p (i) p -1 > 0.
Hence, by (4.10), we obtain that there exists a function g in F 2 p such that, for all natural integers n and m, we have In particular, there is τ ′ a in F d p such that τ a = p 2j τ ′ a . Hence, for all a in {0, . . . , p -1} d , we have This shows that S e,f and B satisfy Condition (a) in Proposition 1, so that Theorem 1 is proved.

1. 2 . 1 (

 21 Beukers' conjectures on Apéry numbers. For all natural integers n, we set A

1. 3 .

 3 Notations. Let d be a positive integer. If m = (m 1 , . . . , m d ) and n = (n 1 , . . . , n d ) belong to R d and if λ ∈ R and k ∈ {1, . . . , d}, then we write:

. 4 )

 4 We have v + mp -k = -a + (m -b)p. If -a + (m -b)p is negative, then A(v + mp -k) = 0 and (3.4) holds. If m -b is nonnegative, then we have α p (A, m -b) ≥ r -b. Thus, we have either a = 0 and α p (A, v + mp -k) ≥ r -b, or a, m -b ≥ 1 and

  and only if |c n | p tends to 0 as n tends to infinity. In addition, if ∞ n=0 c n converges, then (c n ) n∈N is a summable family in (Z p , | • | p ).

(- 1 )

 1 X, T 1 , . . . , T m be m + 1 variables. Then, we havem j=1 (X -T j ) = X m + ∞ i=1 i σ m,i (T 1 , . . . , T m )X m-i . i n i p i Ψ p,k,i (m) .(4.10)

4 . 3 .

 43 g(n, m)p, which, together with (4.9), finishes the proof of Proposition 3. Last step in the proof of Theorem 1. Let A be the Z p -module spanned by S e,f . We set B = {S g e,f , g ∈ F d p }. We shall prove that S e,f and B satisfy Condition (a) in Proposition 1.Obviously, B is constituted of Z p -valued sequences and A is a subset of B. For all a in {0, . . . , p -1} d and m in N d , we haveQ e,f (a + mp) = u i=1 (e i • mp)! e i •a k=1 (e i • mp + k) v i=1 (f i • mp)! f i •a k=1 (f i • mp + k).For every natural integer n, we have(np)! n! = p n (-1) np Γ p (np), so that we have u i=1 (e i • mp)! v i=1 (f i • mp)! = p (|e|-|f |)•m Q e,f (m) u i=1 (-1) e i •mp Γ p (e i • mp) v i=1 (-1) f i •mp Γ p (f i • mp).Furthermore, we haveu i=1 e i •a k=1 (e i • mp + k) v i=1 f i •a k=1 (f i • mp + k) = u i=1 e i •a k=1,p∤k (e i • mp + k) v i=1 f i •a k=1,p∤k (f i • mp + k) • p ∆ e,f (a/p) u i=1 ⌊e i •a/p⌋ k=1 (e i • m + k) v i=1 ⌊f i •a/p⌋ k=1 (f i • m + k) .Let g be a function in F d p . For all a in {0, . . . , p -1} d and m in N d , we setτ a (m) := g(a + mp)h a (m)p ∆ e,f (a/p) u i=1 ⌊e i •a/p⌋ k=1 (e i • m + k),so that τ a ∈ Z p + pF d p . Therefore, for all v in {0, . . . , p -1} and n in N, we haveS g e,f (v + np) = 0≤a≤1(p-1) m≥0 |a+mp|=v+np g(a + mp)Q e,f (a + mp) = 0≤a≤1(p-1) m≥0 |a+mp|=v+np Q e,f (m)τ a (m). If |a + mp| = v + np, then we have |a| = v + jp with 0 ≤ j ≤ min n, d(p -1) -v p =: M.Furthermore, we have ⌊|a|/p⌋ = j and there is k in {1, . . . , d} such that a (k) ≥ (v + jp)/d. Since e is 2-admissible and f is constituted by vectors 1 k , we obtain that ∆ e,f (a/p) = u i=1 e i • a p ≥ 2j.

Q

  e,f (m)τ ′ a (m).Therefore, there exist A ′ in A and a sequence (B k ) k≥0 , with B k in B, such thatS g e,f (v + np) = A ′ (n) + pB 0 (n) + ∞ k=1 p k+1 B k (n -k).

  then we have a 1 + • • • + a d ≥ p and there exists i in {1, . . . , d} such that a i ≥ p/d. Since e is 1-admissible, (a 1 , . . . , a d )/p belongs to D e,f so that ∆ e,f (a 1 , . . . , a p )/p ≥ 1 and Q e,f (k 1 , . . . , k d ) is in pZ p . In addition, by Theorem 3, Q e,f satisfies the p-Lucas property for all primes p. Hence we obtainS e,f (v + np) ≡ a 1 +•••+a d =v 0≤a i ≤p-1 m 1 +•••+m d =n m i ∈N Q e,f (a 1 + m 1 p, . . . , a d + m d p) mod pZ p (a 1 , . . . , a d )Q e,f (m 1 , . . . , m d ) mod pZ p ≡ S e,f (v)S e,f (n) mod pZ p .

	≡	Q e,f
	a 1 +•••+a d =v	m 1 +•••+m d =n
	0≤a i ≤p-1	m i ∈N

If p is 2,

or 7, then for all v in {0, . . . , p -1}, A 1 (v) is coprime to p so that, according to(1.3), for all natural integers n, A 1 (n) is coprime to p.

The proof of this lemma uses a lemma of Lang which contains an error. Fortunately, Lemma 7 remains true. Details of this correction are presented in[START_REF] Delaygue | On Dwork's p-adic formal congruences theorem and hypergeometric mirror maps[END_REF] Section 2.4].

Research supported by the project Holonomix (PEPS CNRS INS2I 2012).

1

Observe that Equations (4.6) and (4.5) lead to T i,r (0) = 0. In particular, if i = 1, then (4.7) holds. Now we assume that i ≥ 2. For all m ≥ 2, we have

Thereby, for all j in {1, . . . , m -1}, we obtain that

Thus, there exists a sequence (U r ) r≥0 of polynomials with coefficients in Z p such that, for all positive integers N, there exits a natural integer S N such that, for all r ≥ S N and all m ≥ i + 1, we have

But, if V 1 (X) and V 2 (X) are polynomials with coefficients in Z p and if there exists a natural integer a such that, for all m ≥ a, we have V 1 (m) ≡ V 2 (m) mod p N Z p , then, for all integers n, we have V 1 (n) ≡ V 2 (n) mod p N Z p . Indeed, let n be an integer, there exists a natural integer v such that n + vp N ≥ a. Thus, we obtain that

In particular, Equation (4.8) also holds for all positive integers m. Furthermore, according to A i-1 , for all m in {0, . . . , i -2}, T i-1,r (m) tends to zero as r tends to infinity. Thus, for all positive integers N, there exists a natural integer S N such that, for all r ≥ S N and all m in {1, . . . , i -1}, we have

Since T i,r (0) = 0, we obtain that T i,r (m) ≡ 0 mod p N Z p for all m in {0, . . . , i -1}, so that (4.7) holds. This finishes the induction on i and proves (4.1).

4.2.

On the p-adic Gamma function. For every prime p, we write Γ p for the p-adic Gamma function, so that, for all natural integers n, we have

Let α 1 , . . . , α d be natural integers with α i 0 ≥ 1 for some i 0 in {1, . . . , d}. By Proposition 3, there exists a function h in F d p such that, for all natural integers m 1 , . . . , m d , we have

Hence, there exists a function h ′ in F d p such that, for all natural integers m 1 , . . . , m d , we have

Since f is only constituted by vectors 1 k , there exists g ′ in F d p such that, for all m in N d , we have

Furthermore, if k is an integer coprime with p, and d a vector in N d , then for every m in N d , we have

so that there is a function g ′′ in F d p such that, for all m in N d , we have

Hence, for all a in {0, . . . , p -1} d , there exist a p-adic integer λ a and a function g a in F d p such that, for all m in N d , we have

Since f is only constituted by vectors 1 k , for all i in {1, . . . , v}, we have ⌊f i • a/p⌋ = 0.

Thereby, for all a in {0, . . . , p -1} d , there exists a function h a in Z p + pF d p , such that, for all m in N d , we have