Introduction

Quantifying abundance is fundamental for wildlife conservation and management. Indeed, reliable information on the number of individuals or wildlife population densities is key to support management decisions, harvest regulations, define conservation status, and further our understanding of ecosystems functioning. Large carnivores, and especially wolves, are highly controversial, and the assessment of their protection status or decisions about their management are often accompanied by intense political and public debate. This intense public scrutiny makes reliable and up-to-date population size estimates highly sought after by various government offices. Population size continues to be challenging to estimate in nature, especially when species are elusive, vagile, and distributed over large areas. This is because population sampling is never complete, making it necessary to use appropriate analytical methods that account for imperfect detection when estimating population size.

Important advances have been made over the years in understanding imperfect detection in part due to the wealth of capture-mark-recapture data generated by non-invasive genetic sampling methods, whereby individual genotypes are identified from DNA extracted from samples deposited in their natural environment. Here, the strict connection with recent developments in genetic techniques. Despite the recent multidisciplinary approaches and methodological advances mentioned above, estimating the size of wolf populations is currently even more challenging due to the recent spatial and numeric expansion of the species at the European scale. Monitoring large scale wolf populations has become more costly and logistically demanding, as it requires investigations over large areas with multipolitical jurisdictions, and the processing of numerous samples with harmonized protocols.

Hence, this conference will focus on recent progress in estimating large-scale wolf populations in Europe and discuss challenges, highlighting experiences where large wolf populations have been estimated in Europe and future perspective.

Spatial capture-recapture and the Scandinavian wolf: mapping population density and beyond

Pierre Dupont, Norwegian University of Life Sciences

The Scandinavian wolf population has been monitored using non-invasive genetic sampling for over 10 years in both Norway and Sweden. Using this unique dataset, we developed and applied large scale spatial capture-recapture models to map wolf population density and dynamics throughout both countries. In this talk, I outline the different developments we proposed over the years and how they allowed us to go beyond population size and towards a fully spatio-temporal model of the Scandinavian wolf population. Most importantly, this enabled us to provide crucial information to managers, such as estimates of population growth and recruitment rates or spatially explicit mortality probabilities.

For more information on the research activity:

• Bischof, R., Milleret, C., Dupont, P., Chipperfield, J., Tourani, M., Ordiz, A., ... & Kindberg, J. (2020). Estimating and forecasting spatial population dynamics of apex predators using transnational genetic monitoring. Proceedings of the National Academy of Sciences,117(48), 30531-30538.

For more information on the method developed by the team to estimate and map variation in survival throughout the wolf population in Scandinavia:

• Milleret, C., Dey, S., Dupont, P., Brøseth, H., Turek, D., de Valpine, P., & Bischof, R. (2023). Estimating spatially variable and density-dependent survival using open-population spatial capture-recapture models. Ecology,104(2), e3934.

For more information on the status of the wolf population in Sweden and Norway:

• Milleret, C., Dupont, P., Åkesson, M., Brøseth, H., Kindberg, J., & Bischof, R. (2021). Estimates of wolf density, abundance, and population dynamics in Scandinavia, 2012-2021.

A multidisciplinary approach to estimating wolf population size for longterm conservation in the Italian alpine regions with sampling optimization

Francesca Marucco 1 and Virginia Boiani 1,2

1 DBIOS-University of Turin, 2 University of Chester

As wolf population size estimate are increasingly demanded by administrations and wildlife management entities, we developed a system for comprehensive population estimation across the Italian alpine region (100,000 km2), involving 1513 trained operators representing 160 institutions. This extensive network allowed for coordinated genetic sample collection and landscape-level spatial capture-recapture analyses that transcended administrative boundaries to produce the first estimates of key parameters for wolf population status assessment. Wolf abundance was estimated at 952 individuals (95% credible interval 816-1120) and 135 reproductive units (i.e., packs) (95% credible interval 112-165) (Marucco et al. 2023). We also estimated that mature individuals accounted for 33-45% of the entire population. The monitoring effort was spatially estimated thereby overcoming an important limitation of citizen science data. We took advantage of the overall data collected to quantify how reducing the sampling and subsequent genotyping effort influence the estimate of wolf population size across the Italian Alps using spatial capture-recapture (SCR) methods. We investigated the effect of (i) decreasing the number of genotyped samples, (ii) decreasing the spatial extent of search transects and thus the number, and (iii) decreasing the number of search-transect repetitions on the SCR abundance estimates at the Alpine scale. By reducing the number of transects to 75% and fixing the number of search-transect repetitions to a maximum of 6, we obtained density estimates comparable to using the whole data set. While, neglecting the search effort and reducing only the genotyped samples resulted in higher estimates and had a great influence on the precision. This is an important approach for promoting wolf-human coexistence based on long term wolf abundance monitoring and an endorsement of large-scale harmonized conservation practices.

Marucco F., M. V. In 2020, a national wolf population estimation project was launched in Italy, as a result of a simultaneous and standardized demographic sampling of the two portions of the population. In central-southern Italy, given the vast area supposedly occupied by the species (about 150,000 km 2 ), the lack of previous knowledge in vast portions of the wolf distribution, and the limitations in the maximum achievable field effort, we adopted a design based on the demographic sampling of a portion of the whole population, identified through a stratified random sampling, and aimed at collecting both presence signs and noninvasive genetic data. The data collection strategy was articulated into: (i) an extensive survey in 25% of the study area, aimed at collecting wolf presence data; (ii) an intensive survey carried out in 13 sampling areas, defined by a 3x3 arrangement of 9 cells, and aimed at collecting non-invasive genetic sampling. Both the intensive and extensive data collection took place from 1st October 2020 to 30th April 2021. A network of field staff from several institutions concerned with wildlife management (N = 344), volunteers (N = 431) and forestry service staff (N = 725) systematically collected field data on a network of transects, selected along roads and trails, and through photo-traps placed across the study area.

For the estimation of wolf distribution, we built and analyzed a single-season, multiple data-source, multi-event occupancy model, which accounted for the different types of sampling processes involved in data collection (transects and photo-traps) and for the possibility of species misidentification when collecting wolf scats in areas where also dogs where present. For the estimation of wolf abundance, we integrated the occupancy model with a Spatially Explicit Capture-Recapture model (SECR). The occupancy model estimated that 74.2% of the study area was occupied by the species, for a total extent of the wolf distribution of 108,534 km 2 . The integrated model produced an estimate of total population size for the Apennine wolf population of 2,557 individuals (SD = 171.5; 95% CIs = 2,127 -2,844), with an associated CV = 6.7%. By merging our estimate with the one produced for the Alpine regions, we estimated a total population size of 3,501 wolves (SD = 249.5; 95% CIs = 2,949 -3,945) in the whole Italian peninsula.

Simulations showed that the integrated model was associated to an average tendency to slightly underestimate population size, with the average relative bias in the estimates being -12.9%. In future at least 30% of the study area should be sampled for presence signs and 30% for non-invasive genetic data, to provide a high probability of reducing bias below 10%.

Monitoring large carnivores over large scale: what's next for the successful recovery of wolves in France?

Christophe Wolves naturally recolonized France in the early 1990s from the Italian population, now covering all the alpine range. The highly human-dominated landscape of the Rhone valley, west of the French Alps, has slowed down the recovery of the species beyond the alpine chain. However, this barrier is still permeable to some dispersers as new packs have indeed appeared west of the river since 2019.

About 3500 field experts are monitoring the wolf presence in France. This extensive network collects around 1800 biological samples per year which are then analyzed with microsatellite genotyping method. Using multi-event open capture-recapture models, population size is estimated at the end of each winter along with confidence intervals accounting for detection heterogeneity. Population size estimates are requested by authorities early spring to define the maximum number of wolves that can be culled to reduce depredation. Wolf population size in 2023 in France is expected to range between 1000 and 1210 individuals.

A national action plan aims to reconcile outdoor livestock rearing and wolves, while maintaining the wolf population in a favorable conservation status. Governmental authority promotes and funds prevention measures against wolf depredation, including the use of culling. Wolf survival decreased from 0.76 to 0.68 as culling increased from 10% to 20% of the estimated population size. While Mortality remained stable for scent marking individuals (residents) the survival dropped down from 0.63 to 0.28 for transient individuals, thus being probably less impactful for the overall population dynamics. Transition probabilities from resident-to-transient showed significant higher turnover rates as culling increased, suggesting the destabilizing influence of wolf removals on pack dynamics, both producing more transients as well as a higher propensity to set up new packs.

This high turnover associated with increasing pack density makes it very challenging to accurately monitor the population in space and time at the national scale, leading to greater uncertainty in population size estimates. Among the alternatives, patch occupancy models appear as a relatively costefficient metric, useful in adaptive management in order to detect changes in the population while also accounting for imperfect detection. We optimized these models to document changes in wolf pack dynamics across space and time in France. This metric appears reliable to identify pack occurrences and disappearance also avoiding false positives, but is less performant to identify pack disappearance in areas of high wolf density. Stakeholders and the general public often focus on the number of wolves in the country, the hardest number to document accurately at large scale. The change in wolf occupancy as materialized through dynamic maps across years combined with studies on the underlying mechanisms driving population trends and livestock damages may become robust and cost-effective indicators to monitor the conservation status of the wolf population at the national scale as well as useful for local management issues.

Wolf monitoring in Germany -Keeping track of a fast-growing wolf population

Ilka Reinhardt, LUPUS -German Institute for Wolf Monitoring and Research

After 150 years of absence the first wolf pack was confirmed in Germany in 2000. From the Lausitz area in the German-Polish border region the wolf showed a rapid population growth and subsequent spread across western central Europe. In 2021 more than 200 wolf territories were confirmed in Germany, most of them in the North-Eastern part of the country. As Germany is a federal state wolf management including monitoring is under the jurisdiction of the 16 federal states. Wolf surveillance is conducted annually with the goal of obtaining reliable estimates for area of occurrence and population size at the national level. Population size survey is conducted as a minimum count of the number of packs and pairs. While the continuous generation of wolf monitoring data is funded, organized, and conducted on the federal state level, uniform data evaluation and interpretation across administrative borders is ensured by common national monitoring standards and yearly national monitoring meetings.

The backbone of the current monitoring system is the genetic identification of wolf family groups, thus allowing genetic differentiation between neighbouring territories. In 2021 in about 75 % of the wolf territories at least one of the breeding individuals was genetically identified with large differences between the federal states. As the population is expected to farther increase alternative monitoring approaches are explored currently that are feasible for the German situation and meet the legal requirements as well as the public demand for information. It is obvious that there are no simple and cheap monitoring approaches that provide both reliable population estimates and detailed information on the current status of the wolf population for the public and the authorities, while at the same time providing the information necessary to carry out management removals in a meaningful way and to monitor the effects of such removals on the wolf population in a reliable way. As the country is for reasons of competence divided into 16 monitoring units (federal states) the current monitoring approach seems feasible also with a farther growing population, however ways for conducting monitoring and data evaluation in a more efficient way should be explored.

In order to support the wolf management and monitoring within the federal states several tools and resources are provided by the national authorities. In frame of the scientific consortium "Federal documentation and consultation centre on wolves" (DBBW) a monitoring data base was developed and is available for all states allowing for detailed data storage, exchange and analyses. The DBBW advises federal and state conservation agencies on all wolf-related issues upon request and compiles the wolf monitoring data annually on the national level. The results are published on the DBBW website https://www.dbb-wolf.de, providing timely and detailed information of wolf distribution and other aspects concerning the current recolonization process.

Large scale wolf monitoring without snow and genetics, and in a transboundary context. Some insights and challenges from Spain

José Vicente López-Bao 1 and Juan Carlos Blanco 2

1 Biodiversity Research Institute (University of Oviedo -CSIC -Principado de Asturias), Oviedo University, Mieres, Spain. 2 Consultores en Biología de la Conservación, Madrid, Spain

Estimating the size of wolf populations is a contentious issue, particularly when it comes to inform decision-making processes and to evaluate legal commitments. The number of wolf (breeding) packs is a reasonable target for wolf monitoring at large spatial scales, such as the case of Spain. This approach may make wolf population size estimates more homogeneous in a transboundary and regional context, and over time, as soon as the criteria used to confirm, or to estimate, a reproduction event are comparable. In Spain, the number of (breeding) packs as a monitoring goal is included in the two existing guidelines at the national level (from 2005 and 2022, non-legally binding), and is adopted at the regional level. But, the recommendation for the frequency of monitoring has varied between guidelines (2005: every 10 years, 2022: every 6 years in line with the Habitats Directive reporting period, following article 17 of the directive). However, wildlife conservation is under the jurisdiction of autonomous regions in Spain and, although a basic agreement on a common survey approach was reached in 2012 in the wolf working group, autonomous regions decide if to survey the wolf population, how, frequency of surveys, effort to invest, technical or scientific direction, and the collection and interpretation of field data. So far, the national wolf survey has been the outcome from the addition of regional surveys, coordinated by the national government, and avoiding double counting of transboundary (regional) packs. Political tensions at different administrative levels can make coordination difficult. Wolf monitoring in Spain is constrained by the lack of snow and, therefore, the monitoring of breeding packs is carried out in summer and early autumn. Different efforts have been carried out during the last decades to improve and to optimize the estimate of breeding events at that time of the year. Wolf monitoring based on the number of breeding packs is usually focused in confirming reproductions throughout the detection of pups in summer (e.g., using howling sessions, observations, or camera trapping). Additionally, different approaches to estimate reproductive events have been developed in recent years, such as linking the intensity of territorial marking (faeces and ground scratching) and the probability that a given pack has reproduced in a given area, or the implementation of multi-method and multi-state approaches (combining different sources of field information) to estimate a number of sites with wolf reproduction (within a credible interval). Additionally, efforts have been made to improve the use of sampling methods, like howling. Thus, since pup vocalizations are higher in pitch and shorter, compared to adults, and the acoustic energy is concentrated at lower frequencies in wolf choruses without pups, an increased effort have been made on the quantitative analyses of wolf choruses to estimate the probability of pup presence. Still, there is a need to improve the coordination in different aspects of wolf monitoring across autonomous regions, such as in the interpretation criteria for minimum number of confirmed (breeding) packs, how to estimate reproduction events, or the use of different spatiotemporal criteria to avoid double-counting of packs (e.g., application of distance criteria among den or rendezvous sites, or temporal criteria such as simultaneous howling sessions in two potential neighboring rendezvous sites or to concentrate in time the use of howling sessions in summer).

Opportunities and implications of HTS STR genotyping for standardized genetic surveys and monitoring

Marta De Barba -Biotechnical Faculty, University of Ljubljana, Slovenia.

I presented about a new approach for genotyping microsatellite markers based on high throughput sequencing (HTS) technologies. Microsatellites, often abbreviated as STRs -short tandem repeats, are among the most widely used molecular markers in wildlife studies. The new approach entails sequencing of PCR amplified STR on an HTS platform, and bioinformatic treatment of the sequence data to obtain multilocus genotypes based on both length and sequence polymorphism. Compared to the current genotyping system, this method enables standardization and automation of the genotyping process, provides greater accuracy and power to discriminate individuals, and allows direct transfer of genotype data between laboratories. In addition, the massive parallel sequencing capabilities of the new technologies and the scalable workflow makes it highly cost-effective for large genotyping projects. The utility of the approach for genetic monitoring of large carnivores was illustrated with a non-invasive genetic study for brown bears in Slovenia-Croatia, and with a novel application to individual genotyping from snow tracks eDNA of wolf, lynx and brown bears.

Litigation on wolf policy in Europe

Guillaume Chapron, Swedish University of Agricultural Sciences Guillaume.Chapron@slu.se

The return of large carnivores in Europe's human dominated landscapes is an unexpected conservation success. The wolf, in particular, has experienced a remarkable recovery with the species now being present and breeding in most European countries where it had previously become extinct. The return of the wolf, however, creates conflicts with several human activities which feed demands to know how large wolf populations are. Many countries allocate substantial efforts to provide reliable estimates of their wolf populations, as the talks during the symposium have illustrated. Wolf population estimates are political numbers in the sense that they may trigger political decisions which themselves may have consequences on wolf populations. Specifically, if it is considered that the number of wolves exceed what is considered desirable, for whatever reasons, authorities may decide to enact wolf culling or hunting. However, the wolf is by law a protected species and killing wolves can only happen under particular conditions. In many EU countries, wolves are listed in Annex IV of the Habitats Directive which grants the strictest level of protection. In some other countries, wolves appear in Annex V of that Directive, which is more flexible. In any case, European legal protection must always be reflected into the national laws of the Member States.

When national or regional authorities allow wolf killings, interested parties such as environmental nongovernmental organizations (NGOs) may challenge these decisions on legal grounds, arguing that these are not in line with the legal protection conferred to wolves and ask courts to carry a legal review. This is called public interest environmental litigation and it has become an integral part of environmental policy making in many countries. Importantly, litigation may also happen to oppose wolf conservation. Administrative litigation ("contentieux administratif" "contenzioso amministrativo" "Verwaltungsstreitverfahren") can take place in national courts (i.e. administrative courts, administrative courts of appeal, supreme administrative courts or constitutional courts) or at the European Court of Justice (through referral by the European Commission following an infringement or through a request for a preliminary ruling by a national court).

Litigation is a policy instrument that remains understudied. It may be seen by some in conservation academia as better avoided since it may be perceived to exacerbate conflicts. This is misleading as litigation may instead turn conflicts into disputes in a forum (the court) where parties agree on accepting democratic rules and the authority of the law. Litigation performs a diversity of additional social functions. It is a democratic practice where members of the public have the possibility to hold governing authorities accountable. Litigation clarifies what the law really means as laws are often purposely written in broad terms, with the courts being tasked to explain what the law implies in practice for the cases they handle. For authorities, litigation allows through trial and error to develop lawful policy: decisions rebutted by courts may be avoided in the future, while decisions validated by courts may be explored further. Finally, litigation may have a direct or indirect impact on wolf populations.

I present a research project funded by the Swedish research council FORMAS and aimed at developing a systematic mapping of litigation for species conservation in the European Union. We map the extent, scope, nature, and outcome of public interest litigation for species protection in the EU. More specifically, we map administrative, civil and constitutional court cases about the direct protection of wild animal species in the EU. The relevant laws are the Birds Directive and the Habitats Directive. We focus on litigation where members of the typically NGOs, take legal actions against local or governmental authorities for decisions, actions, or failures to act regarding species. I present preliminary results based on the wolf related court cases we have already collected and provide a deeper analysis of such court cases in France.

We collected a total of 275 French wolf related court rulings during the past 20 years. Unsurprisingly, we found that wolf litigation occurred more often in administrative courts located in regions where wolves first returned (i.e. South-East of France). Animal welfare or protection associations were the most active and successful plaintiffs. The State administration represented by its Préfets was also a plaintiff in lawsuits against illegal culling decisions made by mayors. The Préfet des Alpes Maritimes and the Minister of the Environment were regular defendants for decisions to cull wolves that were litigated by nature protection associations. These associations overall had a case winning rate higher than 50%. There were no immediately obvious inter-annual trends in wolf litigation. Our analysis did not allow us to quantify the total number of wolves that were effectively protected from culling decisions because court rulings made after the implementation of culling decisions did not specify whether the animals were already killed or not. Be it as it may, NGOs appear to conduct legally relevant litigation in view of the high success rate they achieve and conservation lawsuits belong to the portfolio of available conservation instruments.

Our project takes place while the legal landscape regarding wolves has become more dynamic, with legal changes to their protection being implemented or discussed. For example, in Italy, the province of Trentino has issued a culling permit for wolves for the first time. Austria has changed its legislation to fast-track wolf culling while limiting possibilities of appeal. The European Commission has recently launched a process to revise the wolf protection status. We may have now reached "peak wolf" in Europe and the coming years may see a reversal of the species recovery, with local re-extinctions. Legal aspects, including litigation, will be central.

  Duchamp 1 , S. Bauduin 1 , N. Jean 2 , O. Gimenez 3 & R. N. Simon 1 1 Office Français de la Biodiversité -Dpt. of Applied Research and Expertise -France 2 Office Français de la Biodiversité -Dpt. of Large Carnivore Management -France 3 Centre d'Ecologie Fonctionnelle et Evolutive -France

  https://www.clawsandlaws.org/courts-conservation/ The LIFE WOLFALPS EU project works to improve coexistence between the wolf and the people who live and work in the Alps and the Ligurian-Piedmontese Apennines by building and implementing shared solutions together with stakeholders to ensure the long-term conservation of the wolf in the Alps and along the Apennine corridor. LIFE WOLFALPS EU operates throughout the Alps and the Ligurian-Piedmontese Apennines, involving twenty Italian, Slovenian, French and Austrian partners and dozens of Institutions and associations that support the project.This conference was realized with the financial contribution of the LIFE Programme. Financial instrument of the European Union.

Estimating wolf distribution and abundance with integrated population modelling: the first wolf assessment in south-central Italy

	Vincenzo Gervasi, ISPRA-Italian Institute for Environmental Protection and Research
	Boiani, P. Dupont, C. Milleret, E. Avanzinelli, K. Pilgrim, A M. K. Schwartz, von Hardenberg, S.
	Perrone, O. Friard, A. Menzano, F. Bisi, U. Fattori, M. Tomasella, S. Calderola, S. Carolfi, P. Ferrari, C. Chioso, F.
	Truc, G. Bombieri, L. Pedrotti, D. Righetti, P. L. Acutis, F. Guglielmo, H. C. Hauffe, C. Rossi, R. Caniglia, P. Aragno,
	V. La Morgia, P. Genovesi, and R. Bischof. (2023). A multidisciplinary approach to estimate wolf population size
	for long-term conservation. Conservation Biology. DOI: 10.1111/cobi.14132

For more information on tracking techniques applied to cats and foxes: • Bischof R, Hansen NR, Nyheim ØS, Kisen A, Prestmoen L, Haugaasen T. Mapping the "catscape" formed by a population of pet cats with outdoor access. Sci Rep . 2022;12(1):5964. Available from: https://doi.org/10.1038/s41598-022-09694-9 • Bischof R, Gjevestad JGO, Ordiz A, Eldegard K, Milleret C. High frequency GPS bursts and path-level analysis reveal linear feature tracking by red foxes. Sci Rep . 2019;9(1):8849. Available from: https://doi.org/10.1038/s41598-019-45150-x

Scaling up wildlife monitoring and estimation: reconciling the individual with the population

Richard Bischof, Norwegian University of Life Sciences Applied wildlife ecology is often aimed at the management and conservation of animal populations. Yet, when we study wild animals, individuals tend to move into the foreground. This is not surprising, as populations are made up of individuals, and individual actions and experiences ultimately add up to the dynamics and fates of populations. On the other hand, a strong focus on individuals, especially small and nonrepresentative samples, can lead to skewed inferences about real systems, with potentially detrimental consequences for conservation policy and action. Using our group's research on carnivores as an example, I outline how modern wildlife monitoring methods and specialized models can connect information about individuals with studies of populations. The approach offers a comprehensive and mechanistic understanding of how individual life histories and movements propagate to the dynamics of populations. This, in turn, can facilitate scale-transcending population management and conservation.

For more information on wolves and spatial capture-recapture: Dupont, P., Brøseth, H., Flagstad, Ø., Kindberg, J., Svensson, L., and Bischof, R., 2023. Estimates of wolf density, abundance, and population dynamics in Scandinavia, 2013-2023-MINA technical report 85. 34 pp.


Philippe Fillion
email: philippe.fillion@cea.fr

DEVELOPMENT OF THE CATHARE 3 THREE-FIELD MODEL FOR SIMULATIONS IN LARGE DIAMETER HORIZONTAL PIPES

Keywords: LOCA, hot leg, droplet entrainment, three-field model

During hypothetical Loss of coolant accident in a PWR caused by a large break in the primary circuit, the reactor core is uncovered. Emergency core cooling is then activated and the accident enters the reflooding phase. Complex two phase flows and heat transfer phenomena occur during this phase and their analysis request detailed models, included in system codes such as CATHARE.

A three-field model has been previously implemented in CATHARE 3 and applied to reflooding calculations in the core. This paper aims to describe the extension of this three-field model to the primary circuit, especially for hot legs, in order to improve the simulation of liquid droplet transport from the core to the steam generators. For vertical flows, such as in the core, a comparison of correlations for the entrainment and deposition terms against steam-water data series allowed to select a model for the film-droplets mass exchange. These correlations established for vertical flows are not applicable in horizontal geometries with large diameters such as the hot legs where, during the reflooding phase, the flow regime is dispersed or stratified with entrained droplets. In this case, a specific deposition term based on a model developed from air-water experiments, accounting gravity and turbulent diffusion mechanisms, is used. Dedicated correlations for the entrainment rate and the droplet diameter complete the package of closure laws. These models are assessed against experimental data obtained in horizontal pipes with various diameters and with various inlet gas and liquid superficial velocities.

INTRODUCTION

During Loss of Coolant Accidents (LOCA) in Pressurized Water Reactors (PWR) featuring a large break in the primary circuit, after the first blowdown phase, emergency systems inject large amounts of liquid water into the circuit in order to refill the pressure vessel which has been previously partly emptied. The core is then quite hot and the reflooding phase may take some minutes or more. The boiling two phase flow at low pressure along the rod bundle is quite complex, resulting in small droplet production usually at the quench front. These droplets are carried out by a strong ascending steam flow towards the upper plenum and beyond, to the steam generators, where they vaporize, creating the "steam binding" effect. The droplet behavior out of the core is of great importance for the temperature history of the fuel and cladding and must be predicted as accurately as possible. Moreover, at the beginning of reflooding, strong flow rate oscillations may occur. In horizontal and sloping parts of hot legs, complex flows including stratified water and mist droplet flow have been predicted, sometimes in countercurrent conditions. Usual six-equation models used in system thermohydraulic codes may encounter difficulties in such situations. More advanced three-field models, sharing the liquid flow in a droplet part and a continuous part, improve the analysis capabilities [START_REF] Bestion | On the modelling of phase-flow in horizontal legs of a PWR[END_REF], as far as they can be validated against relevant experiments.

The three-field model of the CATHARE 3 system code [START_REF] Emonot | CATHARE-3: A new system code for thermalhydraulics in the context of the NEPTUNE project[END_REF] features a nine-equation system, three mass balance equations, three momentum equations and three energy balance equations for each of the 3 fields: continuous liquid, liquid droplets and gas phase. This model has been previously developed and assessed for vertical two-phase flows in tubes and rod bundles, especially for dryout prediction [START_REF] Jayanti | Prediction of dryout and post-dryout heat transfer at high pressures using a one-dimensional three-fluid model[END_REF][START_REF] Jayanti | Calculation of dry out and post-dry out heat transfer in rod bundles using a three field model[END_REF][START_REF] Spirzewski | Prediction of annular flows in vertical pipes with new correlations for the CATHARE 3 three-field model[END_REF][START_REF] Spirzewski | An improved phenomenological model of annular two-phase flow with high accuracy dryout prediction capability[END_REF]. Assessment of a first set of closure laws for the three-field model was done against reflooding separate effect tests (PERICLES and RBHT) and against the reflooding phase of a LBLOCA (Large Break Loss of Coolant Accident) simulated on BETHSY facility [START_REF] Valette | Revisiting large break LOCA with the CATHARE-3 three-field model[END_REF]. Simulation of the integral test BETHSY 6.7c showed the ability to predict the quench front motion and the clad temperatures in the core. The flow behavior at the outlet of the core is well predicted. However, the authors of the simulation concluded that the behavior of the droplet downstream of the core was not well captured since no adequate closure laws were available for the upper plenum and the hot leg. The conclusion in this previous paper mentioned some lacks in the validation matrix, especially for the upper plenum and hot leg geometries. Separate Effect Tests (SET) experiments, SEROPS-2 and REGARD, will be used respectively for these parts of the primary circuit.

The present paper aims at filling the gap and features new closure laws and validation in a hot leg geometry. Experimental databases in large diameter horizontal pipes, including REGARD, used as Separate Effect Tests for the development of new models and their validation. New closure laws for the entrainment and deposition terms and the droplet diameter dedicated to these conditions are suggested and analyzed.

EXPERIMENTAL DATA IN LARGE DIAMETER HORIZONTAL PIPES

Numerous experimental facilities for measurements of entrainment or deposition of droplets in horizontal pipes exist or have been built in the past, e.g. [START_REF] Dallman | Entrainment for horizontal annular gas-liquid flow[END_REF][START_REF] Williams | Droplet flux distributions and entrainment in horizontal gas-liquid flows[END_REF][START_REF] Mantilla | Mechanistic modeling of liquid entrainment in gas in horizontal pipes[END_REF]. The test section usually consists in a long pipe, of several meters or a few tens of meters, with an inner diameter comprise from a few centimeters to 10 cm, exception for a 15.3 cm diameter but inclined pipe reported by Mantilla [START_REF] Mantilla | Mechanistic modeling of liquid entrainment in gas in horizontal pipes[END_REF]. The length-to-diameter ratio L/D is roughly comprise between 250 and 500. In most cases, the flow is thus fully developed and the location of the measurements corresponds to equilibrium conditions in term of entrainment and deposition of droplets (i.e. the entrainment rate is equal to the deposition rate). When the diameter of the pipe increases, the vertical stratification of the droplets flow due to the gravity becomes more and more significant. Hot legs of a French PWR have a diameter about 0.7 m and a straight line about 6-7 m length. The L/D ratio is in this case close to 10. Because of the low L/D ratio, the expected flow is not fully developed, and probably has not reached the entrainment/deposition equilibrium and presents a strong stratification of the droplet field.

Dedicated experiments providing data in a scaled-down hot leg geometry exist. This is the case of the MHYRESA facility at CEA-Grenoble, aimed at studying the entrainment of liquid from the upper plenum to the steam generator (SG) inlet chamber, which can occur in case of LOCA or Loss of Heat Removal System (LORHR) during mid-loop operation [START_REF] Geffraye | Analysis of the MHYRESA hot leg entrainment tests[END_REF]. Several runs were performed with different air-water flow configuration and geometries with a pipe diameter of 0.118 m and 0.351 m. More recently, effect on the droplet entrainment on the CCFL phenomenon during the reflux condensation mode of a SBLOCA were studied in the COLLIDER test facility [START_REF] Issa | Droplets entrainment ratio in a PWR hot-leg pipe geometry[END_REF], consisting of a scaled-down (1/3.9) PWR hot-leg pipe, with a diameter of 190 mm. However, such of experiments have not been investigated in the present paper, focused on the analysis of flows within straight horizontal pipes, leaving aside for the moment the consideration of the bend in the leg.

REGARD experiments

REGARD experimental program has been designed to overcome some of the lack of data in horizontal pipes, which could be extrapolated in PWR Hot Leg conditions, allowing construction for dedicated models of entrainment or deposition rates [START_REF] Valette | Droplet entrainment over a stratified flow in a PWR hot leg: results of the REGARD experiment[END_REF][START_REF] Henry | A model for droplet entrainment rate in horizontal stratified flow[END_REF][START_REF] Neiss | Modélisation et simulation de la dispersion turbulente et du dépôt de gouttes dans un canal horizontal[END_REF]. This air-water experiment features a 24 cm diameter, 4 m long horizontal Plexiglas pipe (see Figure 1).

Figure 1. REGARD test section (from [13]).

The entrainment campaign was focused on measurements in wavy-stratified flow conditions. Air and water at near atmospheric pressure and at a temperature equal to 30°C were injected at the inlet of the test section at several volumetric flowrates: four air flowrates of 3100, 3900, 5400 and 6000 Nm 3 /h (Normalized flowrate corresponding to a volume reduced to 0°C and 1013 hPa), corresponding to superficial velocities Jg of 19.8, 24.8, 38.2 and 38.2 m/s respectively, and two different volumetric flowrates for the water, 3 m 3 /h and 6 m 3 /h, corresponding to superficial velocities Jl equal to 0.018 m/s and 0.036 m/s respectively. Several instrumentation techniques were used for the entrainment campaign. Local measurements of droplet flow rates, diameter and velocities, as well as liquid film thickness and surface wave amplitude and celerity have been performed along the test section. Three laser beams were used to quantify the droplet velocity and droplet diameter (laser granulometry). The uncertainty of the droplet diameter has not been calculated. An isokinetic probe was located just downstream the laser beams and placed at different locations of the cross section for the measurement of the local mass flowrate of droplets. The estimated uncertainty of the droplet mass flowrate is 0.003 g/s. A resistive wire network, called BANJO, was devoted for the measurement of the liquid film (interfacial wave frequency, wave velocity, height ...). The total uncertainties on the liquid level reaches up to ± 7.5 %. The measurements by BANJO and by the laser and the isokinetic probe are available at different locations along the test section. These devices were supplemented by direct visualization of the flow, using a high speed camera.

The analysis of several series of runs allowed to build entrainment rate values in steady flows. The large diameter is closer to the reactor conditions than any other known experiment with detailed measurement.

Williams experiments

These air-water experiments at near atmospheric pressure, analyzed by Williams et al. [START_REF] Williams | Droplet flux distributions and entrainment in horizontal gas-liquid flows[END_REF], were performed in a long horizontal test section (26 m) including a pipe with an inner diameter of 0.0953 m. Local droplet fluxes were measured using a sampling tube. Total entrainment flux was provided in steady state conditions with a large range of water and gas flow rates. Different procedures, discussed in [START_REF] Williams | Droplet flux distributions and entrainment in horizontal gas-liquid flows[END_REF], were used to calculate the total entrainment flux. For this quantity, the calculation from the vertical profile of the entrainment flux allows an accuracy within 5%. Local film heights were also measured using a conductance technique, with an accurary of about 5%. This test series has been retained in CATHARE validation matrix in order to evaluate the diameter effect for the closure laws. For future simulations of transient tests involving droplet entrainment performed in BETHSY facility, the Williams test section also takes the advantage to have a diameter close to but smaller than the diameter of the horizontal legs of the BETHSY facility, allowing, with REGARD experiment, to frame the BETHSY geometry for this point.

Runs were carried out with different conditions in term of superficial velocities. Air superficial velocities vary from 26 to 88 m/s whereas water superficial velocities are comprise in the range 0.02-0.12 m/s. In these conditions, the flow is stratified for the lowest values of the gas velocities. According to Williams et al. [START_REF] Williams | Droplet flux distributions and entrainment in horizontal gas-liquid flows[END_REF], the flow becomes stratified-annular for air superficial velocities and water superficial velocities around 40-50 m/s and 0.03 m/s. For the highest air superficial velocities, 68 m/s and more, asymmetric annular flow is obtained in the pipe. Local film heights and local droplet flux were measured at different location of the pipe, allowing to obtain vertical profiles of the droplet flux and the droplet concentration.

CATHARE MODEL

The three-field model of the CATHARE 3 system code is based on a nine-equation system, including mass balance equation, momentum equation and energy balance equation for each of the 3 fields: continuous liquid, liquid droplets and gas phase. Specific correlations dedicated to vertical geometries exist for the entrainment and deposition rates, and for the estimation for the droplet diameter. In this section, we describe the new correlations developed for large diameter horizontal pipes.

Entrainment model

Pan and Hanratty [START_REF] Pan | Correlation of entrainment for annular flow in horizontal pipes[END_REF] based their entrainment model in horizontal pipes on the correlation for the entrainment rate developed by several authors in tubes (Dallman et al. [START_REF] Dallman | Entrainment for horizontal annular gas-liquid flow[END_REF], Lopez de Bertodano [START_REF] Lopez De Bertodano | Entrainment rate of droplets in the ripple annular regime for small vertical ducts[END_REF]). The correlation reads:

𝑚 𝐸 = 𝑘 𝐴 ′ 𝑉 𝑔 2 (𝜌 𝑔 𝜌 𝑙 ) 1 2 ⁄ 𝜎 ( 𝑄 𝑙 𝑃 𝑤 -Γ 𝑙oE ) (1)
where 𝜌 𝑙 and 𝜌 𝑔 are the liquid and gas density, 𝜎 the surface tension, 𝑃 𝑤 the perimeter of the pipe and 𝑄 𝑙 the film flowrate. The dimensionless atomization constant 𝑘 𝐴 ′ is determined from experiments. The critical film flow Γ 𝑙oE corresponds to the liquid film flowrate per unit length at which the entrainment occurs (oE: onset of entrainment). In the Pan and Hanratty approach, the critical film flow depends on the film velocity around the pipe, which is not constant due to the gravity. This method requires thus the knowledge of the local film around the pipe and has not been retained in the one-dimensional approach of CATHARE, where an averaged critical film flow on the cross section of the pipe is used, simply given by Γ 𝑙oE = 𝑄 𝑙oE /𝑃 𝑤 , equivalent to the formulation used in vertical flows. Pan The proposed entrainment model overestimates the experimental entrainment rate for the lowest superficial velocities whereas the entrainment rate tends to be underestimated for the highest gas velocities. For Jl = 0.036 m/s series, the discrepancy between the model the experiment are more important for the lowest gas superficial velocities.

Deposition model

Neiss [START_REF] Neiss | Modélisation et simulation de la dispersion turbulente et du dépôt de gouttes dans un canal horizontal[END_REF] developed a one-dimensional deposition model dedicated to large diameter horizontal pipes. In such of geometries, two mechanisms acts on the droplet deposition: gravity and turbulent diffusion. The model suggested by Neiss takes into account these two phenomena, following the works of several authors (e.g. Williams et al. [START_REF] Williams | Droplet flux distributions and entrainment in horizontal gas-liquid flows[END_REF]), leading to a deposition rate written as:

𝑚 𝐷 = 𝑘𝐶 = (𝑘 D,grav + 𝑘 D,diff )𝐶 (3)
In this equation, 𝐶 is the concentration of the within the gas core and 𝑘 D,grav and 𝑘 D,diff are the deposition velocities relative to the turbulent diffusion and relative to the gravity, respectively.

Following approaches of several authors [START_REF] Williams | Droplet flux distributions and entrainment in horizontal gas-liquid flows[END_REF][START_REF] Pan | Correlation of entrainment for annular flow in horizontal pipes[END_REF], Neiss related the deposition constant to the terminal falling velocity of the droplet:

𝑘 D,grav = 𝑔𝜏 𝑝 𝑆 grav 𝑆 tot (4)
where 𝜏 𝑝 is the relaxation time of the droplet. Neiss suggests a value of 𝜏 𝑝 close to 0.1 s, deduced from REGARD experiments, and assumes the deposition due to the gravity only occurs on the lower part of the pipe, so 𝑆 grav 𝑆 tot = 1 2 ⁄ ⁄ . In our model, the variation of the droplet diameter 𝛿 is taking account in the calculation of the relaxation time,

𝜏 𝑝 = 1 18 𝛿 2 𝜌 𝑑 𝜇 𝑔 (5)
corresponding to a particle following the Stokes regime (𝐶 𝐷 = 24/Re). In this approach, the relaxation time is calculated from the force balance between the drag and buoyancy forces, and the droplet diameter corresponds to the Sauter mean diameter.

Using the heat transfer / mass transfer analogy, the diffusion turbulent part is modeled by Neiss as

𝑘 D,diff = 0.023 𝑉 𝑔 𝑅𝑒 𝑔 -0.2 𝑆𝑐 𝑔 -2/3 1 1 + 2.5𝛼 𝑑 𝜌 𝑑 𝜌 𝑔 𝑆 𝑑iff 𝑆 tot (6)
where 𝑅𝑒 𝑔 is the gas Reynolds number, 𝑆𝑐 𝑔 the Schmidt number, 𝛼 𝑑 the volumetric droplet fraction, 𝜌 𝑑 the droplet density and 𝑆 diff 𝑆 tot ⁄ the ratio of the surface where the diffusion turbulent mechanism induced a deposition flux over the total surface of the duct. In the current analysis, the deposition flux is assumed to be identical on the walls of the pipe, so 𝑆 diff 𝑆 tot ⁄ = 1.

In the horizontal pipe with a large diameter, such as in the REGARD experiments, it exists a stratification of the flow. It results a strong concentration of the droplets in the lowest quarter of the pipe. This phenomenon is taken into account by Neiss considering an effective deposition velocity:

𝑘 D,eff = 𝑓 𝐷 . 𝑘 D [START_REF] Valette | Revisiting large break LOCA with the CATHARE-3 three-field model[END_REF] where 𝑓 𝐷 = 4.

For REGARD tests, the gravity term is preponderant and contributes roughly to 90-95% to the deposition process. In the case of Williams runs, the contribution diminishes, due to the lowest pipe diameter and higher gas superficial velocities and higher droplet volume fractions.

Droplet diameter model

The standard correlation of the droplet diameter in CATHARE 3 used in the three-field calculations was obtained by Jayanti and Valette [START_REF] Jayanti | Prediction of dryout and post-dryout heat transfer at high pressures using a one-dimensional three-fluid model[END_REF] from analysis of high-pressure post-dryout experiments in vertical tubes. For the 6-equation model, the estimation of the maximum droplet diameter takes into account a critical Weber criterion. These correlations significantly overestimate the droplet size obtained during the REGARD entrainment campaign and tests performed in a pipe with the same diameter as for Williams experiments. A short literature review has been achieved in order to find and evaluate correlations for the droplet size in the studied conditions. Schimpf et al. [START_REF] Schimpf | Development of droplet entrainment and deposition models for horizontal flow[END_REF] considered, from an analysis conducted by McCoy and Hanratty, that the volume median diameter 𝑑 50 best describes the rate at which droplets deposit into the liquid film at the bottom of a pipe. Schimpf et al. developed of a new correlation of the droplet diameter, including REGARD data, on the work of Al-Sarkhi and Hanratty [START_REF] Al-Sarkhi | Effect of pipe diameter on the drop size in a horizontal annular gas-liquid flow[END_REF]. These last authors developed correlations of 𝑑 50 and the Sauter mean diameter 𝑑 32 from their own data in a horizontal 0.0254 m diameter pipe and from the Simmons and Hanratty data [START_REF] Simmons | Droplet size measurements in horizontal annular gas-liquid flow[END_REF] (diameter 0.0953 m). They provide a simplified expression of 𝑑 32 without dependence against the liquid properties

( 𝑑 32 𝐽 𝑔 2 𝜌 𝑔 𝜎 ) 0.55 ( 𝑑 32 𝐷 ) 0.36 = 0.154 (8)
The expression of the relaxation time, related to the droplet terminal velocity, appearing in the deposition model following the Neiss' approach was obtained from the balance of the buoyancy force and the drag force. The Sauter mean diameter is used as the droplet size in this approach. This diameter is also used for in the expression of the gas-droplet interfacial friction used in the code.

In a review of droplet entrainment in annular flow, Berna et al. [START_REF] Berna | Review of droplet entrainment in annular flow: Interfacial waves and onset of entrainment[END_REF] propose a correlation of the volume mean droplet diameter 𝑑 𝑣𝑚 in horizontal pipes, from the analysis of three databases with various pipe diameters but smaller than REGARD one, from 0.0254 m to 0.0953 m. The database with the largest pipe diameter is that published by Simmons and Hanratty [START_REF] Simmons | Droplet size measurements in horizontal annular gas-liquid flow[END_REF], and was obtained on the same test section used

To avoid the dependence against the film properties, and because this dependence is weak, the volume mean diameter 𝑑 𝑣𝑚 has been expressed in CATHARE from the gas properties only. The experimental droplet diameters are reasonably predicted in case of the Al-Sarkhi and Hanratty, and Simmons and Hanratty data (Figure 3a) by the proposed model (equations ( 10)-( 11)) and the simplified Al-Sharki model (equation ( 8)). Both model have the same trends, but the error between the correlation and the experiment is slightly reduced using the proposed model. However, both models do not fit the REGARD experimental values (Figure 3b). One of this reason of these discrepancy is due to the fact that in REGARD, the measured droplet distributions according to their diameter is completely different to those provided by Simmons and Hanratty (see Figure 4).

In Valette and Henry [START_REF] Valette | Droplet entrainment over a stratified flow in a PWR hot leg: results of the REGARD experiment[END_REF], the analysis of the droplet diameter spectrum points out, apart of the pipe diameter effect, that the droplet diameter data produced by Simmons and Hanratty, and Al-Sharki and Hanratty are obtained at the centre line (or at 3/4" above and below the centre line) of these experiments whereas the REGARD data are obtained using the laser measurement system in the lower quadrant of the pipe cross-section. Literature correlations for volume median diameter, such as Tatterson or Kataoka ones, do not predict the REGARD experimental results. From this assessment, the analysis also suggests that "to the fact that in REGARD a considerable number of small droplets disappeared by vaporization in contact to the air. As the number of vaporized droplet is less sensitive to the air velocity than the increase of created droplet, the volume median diameter is more affected by the loss of its smaller droplets at low air velocity than at high air velocity.". Schimpf et al. observed that the volume median diameters obtained in REGARD show an inverse trend with increasing gas velocity, comparing to those of Simmons and Hanratty, and Al-Sharki and Hanratty. We also remark that the REGARD tests were performed with low droplet volume fraction (< 0.6×10 -4 for the considered droplet diameter data) instead of a range 2×10 -4 -11×10 -4 for the other databases. For a dilute droplet population, the coalescence or breakup of the droplet phenomema affecting the droplet distribution are less sensitive. According to these remarks, it seems to be difficult to correlate the droplet diameter from these three experiments without at least to reanalyze the REGARD databank. Nevertheless, REGARD experimental data for the Sauter mean diameter provided in [START_REF] Valette | Droplet entrainment over a stratified flow in a PWR hot leg: results of the REGARD experiment[END_REF] have the same trend (decrease of the diameter with increasing gas superficial velocity) to those obtained on the two other considered experiments (see Figures 3a and3b). So, our current study, we first preferred to use a specific correlation fitted on REGARD data for the space-average Sauter mean diameter in the CATHARE simulations of this experiment: 𝑑 32 = 3.67 × 10 -3 𝑊𝑒 𝑔 -0.157 𝐷 (12) This specific correlation can only be used for the assessment for the entrainment and deposition models against REGARD data, whereas the correlation ( 10)-( 11) can be used in a larger domain, including Williams experiment.

CATHARE RESULTS

REGARD experiments

Figures 5 and6 show a comparison between the calculated droplet flowrate function of the gas superficial velocity using the Neiss model, the Pan-Hanratty model, the droplet diameter calculated from [START_REF] Issa | Droplets entrainment ratio in a PWR hot-leg pipe geometry[END_REF], implemented in CATHARE, and the REGARD experiments with an inlet water volumetric flowrate Ql = 3 m 3 /h and Ql = 6 m 3 /h. The droplet flowrate has been normalized by the maximum experimental flowrate obtained for each series. CATHARE well reproduces the increasing of the droplet flowrate at the beginning of the test section. In general, a good prediction of the droplet flowrate profiles along the test section is found, except for the couple (Ql = 6 m3/h, Jg = 34.6 m/s), and the code tends to underestimate the experimental values for the highest gas superficial velocities for Ql = 3 m 3 /h. The value of the atomization constant k'A=4×10 -7 is obtained from the analysis of the REGARD tests. However, the experimental entrainment rates used to determine this value do not correspond to equilibrium conditions and are underestimated. Consequently, the atomization constant for the Williams experiments, in fully developed conditions, is probably underestimated. Calculations were thus performed with a higher value of the atomization constant k'A=4.5×10 -7 , which gives better results (Figure 8) with a relative mean absolute error of 17.7% on the entrainment fraction. If the results for the REGARD tests are slightly degraded consequently, the global error on the entire database diminishes.

CONCLUSIONS

This work aimed at addressing the assessment of the three-field model of the system code CATHARE 3 against large diameter horizontal pipes, in wavy-stratified and asymmetric annular flow conditions with entrained droplets. This first step of assessment gave encouraging results. Correlations due to Pan-Hanratty and Neiss, with proposed corrections or modifications for the modeling of the entrainment and deposition of the droplets were assessed against REGARD entrainment and Williams databases.

CATHARE well reproduces the increasing of the droplet flowrate at the beginning of the REGARD test section. Pan-Hanratty model with an atomization constant k'A=4.5×10 -7 can be used both in REGARD and Williams conditions, and allows, with the suggested models for the deposition term and the droplet diameter, to reproduce experimental data of Williams with a relative mean absolute error of 17.7% on the entrainment fraction.

This new set of correlations included into CATHARE 3 could be improved by the deposition term due the gravitational settling, by a better modeling of the relaxation time of the droplet. In particular, the current model requires a correlation of the droplet size covering various large pipe diameters and flow conditions. Further analysis of the raw data of REGARD for the droplet size distributions will probably give information for the improvement of this correlation. The effective factor introduced by Neiss, allowing to take into account the vertical stratification of the droplets within the pipe, could also be improved from REGARD data or other experiments, considering the experimental vertical profiles of the droplet concentration at different locations and different gas superficial velocities.

Figure 2

 2 Figure 2 shows the model prediction (without CATHARE simulations) against the experimental REGARD data obtained at L/D=3.8. A value of 𝑘 𝐴 ′ = 4 × 10 -7 is used, close to that proposed by Schimpf et al. (𝑘 𝐴 ′ = 3.8 × 10 -7) used in SPACE code[START_REF] Schimpf | Development of droplet entrainment and deposition models for horizontal flow[END_REF].

Figure 2 .

 2 Figure 2. Comparison of the entrainment model with the REGARD data at L/D=3.8.

Figure 3 .

 3 Figure 3. Comparison of the predicted mean Sauter diameter using the simplified Al-Sharki model and the proposed model against experimental data.

Figure 4 .

 4 Figure 4. Comparison of the droplet diameter distributions from Simmons and Hanratty and REGARD (figure extracted from [13]).

Figure 5 .

 5 Figure 5. Comparisons of the REGARD experimental droplet flowrates profiles along the test section and the CATHARE simulations for Ql = 3 m 3 /h.

Figure 6 .Figure 7

 67 Figure 6. Comparisons of the REGARD experimental droplet flowrates profiles along the test section and the CATHARE simulations for Ql = 6 m 3 /h.

Figure 7 .Figure 8 .

 78 Figure 7. Comparisons of the Williams experimental entrainment fraction E and the CATHARE simulations using an atomization constant k'A=4×10 -7 .

  and Hanratty calculated 𝑄 𝑙oE from an onset of entrainment Reynolds number: 𝑅𝑒 𝑙𝑜𝐸 = 7.3(log 10 𝑤) 3 + 44.2 (log 10 𝑤) 2 -263 log 10 𝑤 + 439(2)Parameter 𝑤 = 𝜇 𝑙 /𝜇 𝑔 √(𝜌 𝑔 /𝜌 𝑙 ), depending on the liquid and gas viscosities 𝜇 𝑙 and 𝜇 𝑔 , is valid in the range[1.8;28].

  by Williams et al.[START_REF] Williams | Droplet flux distributions and entrainment in horizontal gas-liquid flows[END_REF], The correlation obtained by Berna et al. for the mean volume diameter 𝑑 𝑣𝑚 depends on the pipe diameter 𝐷 and is expressed as a function of the gas and film Reynolds numbers and the gas Weber number based on the pipe diameter as the scale length:

	𝑑 𝑣𝑚 𝐷	= 2.634 𝑊𝑒 𝑔 -0.23 𝑅𝑒 𝑔 -0.54 𝑅𝑒 𝑙	0.13

ACKNOWLEDGMENTS

CATHARE-3 is developed in the framework of the NEPTUNE project, financially supported by CEA, EDF, Framatome and IRSN.


The long shadow of the COVID-19 pandemic on children and adolescents' mental health.

In May 2023, the World Health organisation announced the end of the COVID-19 pandemic [START_REF]Statement on the fifteenth meeting of the IHR (2005) Emergency Committee on the COVID-19 pandemic[END_REF], which killed nearly 7 million persons worldwide since 2020 [START_REF]WHO Coronavirus (COVID-19) Dashboard[END_REF]. One of the well-described consequences of the pandemic and of the sanitary crisis which resulted from it was a rapid and durable deterioration in the mental health of children and adolescents: in particular, a notable increase in levels of anxiety and depression [START_REF] Miao | Impact of the COVID-19 pandemic on the mental health of children and adolescents: A systematic review and meta-analysis of longitudinal studies[END_REF]. While this increase was highest during the course of 2020, it appears that contrary to what has been observed in adults (4), there has been no major improvement in the psychological health of young people since 2022. What is more, there is evidence of increases in self-harm and suicidal attempts [START_REF] Madigan | Comparison of paediatric emergency department visits for attempted suicide, self-harm, and suicidal ideation before and during the COVID-19 pandemic: a systematic review and meta-analysis[END_REF], that is clinically-significant mental health problems that require medical attention. While the deterioration in youths' mental health during the pandemic, when fear of illness and death due to COVID-19 made the headlines, schools were closed, sports and leisure activities were curtailed and social relations were limited, is understandable [START_REF] Viner | School closures during social lockdown and mental health, health behaviors, and well-being among children and adolescents during the first COVID-19 wave: a systematic review[END_REF], what could explain the persistence of what appears to be a mental health crisis among young people? Several articles in this issue of European Child and Adolescent Psychiatry shed light upon some of the mechanisms that may be at play.

First, the study conducted by Calvano et al [START_REF] Calvano | Almost 2 years into the COVID-19 pandemic: an update on parental stress, parental mental health, and the occurrence of child maltreatment[END_REF], shows an increase in levels of verbal, physical and sexual abuse towards children during the course of the COVID-19 pandemic, as well as a worsening of parental mental health. Child maltreatment and abuse are among the best described risk factors of children's poor mental and physical health over the long-term, and the increase in the prevalence of violence among parents and toward children could have lasting effects [START_REF] Li | Maltreatment in childhood substantially increases the risk of adult depression and anxiety in prospective cohort studies: systematic review, meta-analysis, and proportional attributable fractions[END_REF]. Second, the study conducted by Rapport et al. shows that children's mental health additionally fluctuated with the prevalence of COVID-19 infection rates [START_REF] Rappaport | Monthly correlates of longitudinal child mental health during the COVID-19 pandemic according to children and caregivers[END_REF]. Clearly, COVID-19 infection rates were a source of stress in and of themselves and were additionally related to the stringency of protective measures implemented, as well as decreased parental mental health in part due to a worsening in employment and income [START_REF] Hajek | Depression and anxiety in later COVID-19 waves across Europe: New evidence from the European COvid Survey (ECOS)[END_REF][START_REF] Hecker | Impact of work arrangements during the COVID-19 pandemic on mental health in France[END_REF]. Several reports highlighted the relationship between parental levels of stress, anxiety and depression as well as their fluctuations and children's mental health [START_REF] Moulin | Risk and protective factors related to children's symptoms of emotional difficulties and hyperactivity/inattention during the COVID-19-related lockdown in France: results from a community sample[END_REF][START_REF] Monnier | Children's mental and behavioral health, schooling, and socioeconomic characteristics during school closure in France due to COVID-19: the SAPRIS project[END_REF][START_REF] Wolf | Scoping review: longitudinal effects of the COVID-19 pandemic on child and adolescent mental health[END_REF][START_REF] Fischer | Internalizing problems before and during the COVID-19 pandemic in independent samples of Dutch children and adolescents with and without pre-existing mental health problems[END_REF]. It may be that in some families the COVID-19 pandemic precipitated the occurrence of parental symptoms of anxiety and depression, which in turn had a negative impact on parent-child interactions -possibly during critical periods of development -and therefore exert long-term consequences on youth wellbeing [START_REF] Thomson | Parent psychological distress and parent-child relationships two years into the COVID-19 pandemic: Results from a Canadian cross-sectional study[END_REF]. Third, the COVID-19 appears to have had an impact on social behaviours that may have consequences on children's and adolescents' psychological difficulties -in particular an increase in the levels of screen use [START_REF] Shoshani | The longitudinal impact of the COVID-19 pandemic on adolescents' internalizing symptoms, substance use, and digital media use[END_REF] and a decrease in social leisure activities which, as shown by the study conducted by Timonen et al in this month's issue of ECAP are an important sources of positive mental health and appear to mediate the relationship between early signs of psychological distress and later psychiatric morbidity [START_REF] Timonen | Social leisure time activities as a mediating link between self-reported psychological symptoms in adolescence and psychiatric morbidity by young adulthood: the Northern Finland 1986 Birth Cohort study[END_REF].

Overall, these studies highlight negative changes in multiple risk factors of children's and adolescents' psychological well-being in the context of the COVID-19 pandemic, some of which persist over time which could explain relatively stable rates of young people's levels of depression, anxiety and self-harm. One of the issues that seems unresolved at this time is the extent to which the deterioration in youth mental health is a lasting phenomenon and whether it will predict a rise in the number of cases of psychiatric disorders in the future. Since the COVID-19 pandemic, the world has provided children and adolescents which much reason to worry -the Russia-Ukraine war, an economic shock and massive increases in economic and energy prices exerting pressure on family budgets, unresolved climate change, and now the war between Israel and the Hamas. In parallel, exposure to known risk factors of stress, anxiety and depression such as violence in the context of family or school, as well as school pressure, do not appear to have receded [START_REF] Boer | National-level schoolwork pressure, family Structure, Internet use, and obesity as drivers of time trends in adolescent psychological complaints between 2002 and 2018[END_REF]. Finally, child and adolescent mental health care systems are experiencing a serious shortage of professionals and underfunding across several European countries, making access to care more difficult for a substantial fraction of children in need of psychological and psychiatric treatment [START_REF] Ball | Inequalities in children's mental health care: analysis of routinely collected data on prescribing and referrals to secondary care[END_REF]. Almost two years after the end of the COVID-19 pandemic, the global sanitary crisis casts a long shadow on children's and adolescents' mental health and all efforts should be made to implement actions that may help address the needs of those who require psychological attention, from broad-ranging, population-wide, psychosocial support, through psychological care all the way to psychiatric treatment whenever necessary.


Keywords: Classical Core Logic, Relevance, and Substructurality

come

Classical Core Logic, Relevance, and Substructurality

Classical Core Logic (CCL) is a system put forward by [START_REF] Tennant | Core Logic[END_REF] for relevant deductive reasoning. Like many relevant logics, CCL achieves relevance by restricting the structural rule of Weakening. Unlike relevant logics in the R family, it blocks the Lewis argument for explosion not by rejecting Disjunctive Syllogism, but by rejecting unrestricted Cut:

A ⊢ A ∨ B A ∨ B, ¬A ⊢ B (Cut) A, ¬A ⊢ B
Like R, and unlike Meyer's LR [START_REF] Meyer | Topics in Modal and Many-Valued Logics[END_REF] (often presented as the most natural relevant substructural logic), it validates the distribution of conjunction over disjunction.

Although CCL deserves the title of substructural logic, if only because it restricts Weakening and Cut (as pointed out recently by Tennant himself [START_REF] Tennant | On the Adequacy of a Substructural Logic for Mathematics and Science[END_REF]), this arguably remains a somewhat superficial characterization. It is often pointed out that interesting substructural logics need to preserve the operational rules of the logic the structural rules of which are being restricted. From this point of view, CCL behaves in a rather unusual way. The goal of this paper is to bring out the unusual features of Classical Core Logic qua substructural logic, and provide a principled explanation for them, based on the specific understanding of the notion of relevance that Classical Core Logic explicates. The upshot is that Classical Core Logic is interesting qua Substructural Logic, although not for the usual reasons.

First, we provide an mset:mset sequent calculus for CCL, which avoids a non-standard reading of the comma in the published set:set and set:fmla 0 and presentations of Tennant's [START_REF] Tennant | Perfect Validity, Entailment and Paraconsistency[END_REF][START_REF] Tennant | Core Logic[END_REF], and facilitates the comparison with familiar mset:mset sequent calculi for Classical Logic (CL). One striking feature of this mset:mset calculus is that the rules for conjunction and disjunction are unassorted in the following sense: disjunction is multiplicative on the left and additive on the right, while conjunction is additive on the left and multiplicative on the right. This unassortedness is in fact crucial to the validation of disjunctive syllogism and the failure of Cut. Another striking feature is that the conditional validates only one half of the residuation property, which suggests that in CCL a conditional formula A → B is nothing more than an abbreviation for ¬A ∨ B.

(R⇒) A, B ⊢ CCL C only if A ⊢ CCL B → C
Of course, it is always possible to add a new conditional to CCL for which the residuation property holds in both directions, but then one loses the basic property of CCL that whenever Γ ⊢ CL ∆, then Γ ′ ⊢ CCL ∆ ′ for some Γ ′ ⊆ Γ and some ∆ ′ ⊆ ∆. (Counterexample: ⊢ CL p → (q → p).)

Second, building on this insight about the unassortedness of the rules for conjunction and disjunction, we provide a closure frame semantics for CCL, which is essentially a special case of a semantics described in [1] for cognate systems of non-transitive relevant entailment. What is unusual here is that a closure frame semantics is adequate for a distributive logic. Once again, the assortedness of the rules for conjunction and disjunction is responsible for this, for what is expressed by the sequent

A ∧ (B ∨ C) ⊢ (A ∧ B) ∨ (A ∧ C) may be understood as A ⊓ (B ⊕ C) ⊢ (A ⊗ B) ⊔ (A ⊗ C)
which is valid in LR N D with additive and multiplicative conjunctions (⊓, ⊗) and disjunctions (⊔, ⊕).

Third, the general picture that emerges from this proof-theoretic and semantic analysis of CCL is a commitment to premise/conclusion relativism, i.e., the view according to which connectives may receive different interpretations depending on whether they occur as premises or conclusions in a sequent. If one takes a classically valid sequent to be relevant whenever all formulas make a contribution to the validity of the sequent, conjunction will have to behave extensionally on the left, but intensionally on the right: from the fact all formulas make a contribution in Γ ⊢ A ∧ B, ∆, it does not follow that all make a contribution in Γ ⊢ A, ∆ (compare A, B ⊢ A ∧ B and A, B ⊢ A). Dually, disjunction has to behave extensionally on the right, but intensionally on the left: from the fact that all formulas make a contribution in Γ, A ∨ B ⊢ ∆, it does not follow that all make a contribution in Γ, A ⊢ ∆ (compare A ∨ B ⊢ A and A ⊢ A, B). This accounts for the peculiar features of CCL when seen as a substructural logic, which makes CCL an interestingly peculiar substructural logic, if not an standardly interesting one.


Rozenn Texier-Picard

Ghislaine Gueudet

Gender equality and students' group work in mathematics at tertiary level

Keywords: Gender equality, students' group work, topos, topogenetic position

In this paper we study gender equality in the context of students' group work in mathematics at tertiary level. We use a specific framework, combining the Anthropological Theory of the Didactics and the Joint Action Theory of the Didactics, to examine students' interactions and their roles in a group in relation with the knowledge at stake. Analysing the case of three students in the context of a project-based course, we investigate epistemic symmetry (the equivalence of the roles) according to the students' gender at the macro-level of this course, the meso-level of a session and the microlevel of a given episode. We observe differences at each of these levels in terms of tasks and of gestures performed by each student.

Background

The theme of gender and mathematics has been extensively investigated (see [START_REF] Leyva | Unpacking the Male Superiority Myth and Masculinization of Mathematics at the Intersections: A Review of Research on Gender in Mathematics Education[END_REF] for a recent survey), and the complexity of the gender concept is now acknowledged by research. In this study, we refer to gender as a social construct, and more precisely "a dynamic social construct performed differently across contexts and individuals". (Leyva, 2017, p. 398). We use the expression "gender equality", referring to Collet who considers that there is equality in a teachinglearning situation if the pedagogical environment makes it possible "to establish for everyone a relationship to knowledge that is as independent as possible of social relationships". (Collet, 2021, p. 5). We claim that specific theoretical and methodological tools are required for analysing the gender equality in a given situation.

Our study belongs to "equity research", defined by [START_REF] Adiredja | Taking the sociopolitical turn in postsecondary mathematics education research[END_REF] as: "research that explicitly focuses on efforts to understand and mitigate systemic differences in ways that people experience and are afforded educational opportunities". (p. 446). This research orientation is developing at university level. In particular, studies about Inquiry-Based Learning or Inquiry-Oriented Instruction (IBL or IOI) approaches investigate a possible gendered impact of student-centred teaching approaches. [START_REF] Laursen | Benefits for Women and Men of Inquiry-Based Learning in College Mathematics: A Multi-Institution Study[END_REF] collected data in 100 courses sections, including 42 IBL courses sections. They observed that in the IBL courses, the women's confidence in their ability to do mathematics increased more than the men's. The opposite was observed in the non-IBL courses: women's confidence decreased substantially, compared to a minor decrease for their male classmates. [START_REF] Reinholz | When Active Learning Is Inequitable: Women's Participation Predicts Gender Inequities in Mathematical Performance[END_REF] made a statement that can seem to contradict the previous one: comparing IOI classes with non-IOI classes, they found a greater gender inequity-in terms of performance-in favour of men in the IOI classes. Assuming that this may be a consequence of participation patterns in the IOI classes, the authors studied the rate of participation of women and men in these classes, and the nature of this participation. They evidenced that the rate of participation was a predictor of performance; moreover, in some of the classes inequitable opportunities for participation were offered to women and men.

This study evidences the importance for gender equality of the classroom interactions. [START_REF] Jungwirth | Interaction and gender? Findings of a microethnographical approach to classroom discourse[END_REF], using a symbolic interactionism perspective, analysed the teacher-students interactions in secondary school mathematics classes. Her focus was not on the quantity of interactions, but on their nature. She identified gender-specific practices in these interactions, which made it more likely for boys to participate successfully. These practices directly impacted the mathematical activity of the students, and their relation with the mathematics content at stake. [START_REF] Barnes | Collaborative group work in mathematics: Power relationships and student roles[END_REF] studied the interactions within groups of students working in an inquiry approach at secondary school. She used a post-structuralist perspective focusing on the roles played by the students in the discourse and on the power relations it produces. Through precise analyses of the discourse within groups of students, she evidenced that the collective inquiry offered opportunities for female students, who were enabled to "position themselves in ways which are empowering for them" [START_REF] Barnes | Collaborative group work in mathematics: Power relationships and student roles[END_REF].

In our study, like [START_REF] Barnes | Collaborative group work in mathematics: Power relationships and student roles[END_REF], we focus on the work of groups of students; the teacher does not intervene. We consider how the work is shared and also analyse the interactions concerning the mathematical knowledge at stake. Considering that the group work takes place within a given institution, and that this institution shapes the mathematical knowledge, we developed a specific framework that we introduce in the next section.

Proposition of a theoretical construct

By networking theoretical elements coming from two germane theories: the Anthropological Theory of the Didactics (ATD, [START_REF] Pensée Sauvage | L'analyse des pratiques enseignantes en théorie anthropologique du didactique [Analysis of teaching practices in anthropological theory of the didactics[END_REF] and the Joint Action Theory of the Didactics (JATD, Sensevy & Mercier, 2007), we propose a theoretical construct providing tools for identifying and understanding gendered mathematical practices in the context of students' collective work.

ATD considers that any human being is a subject of multiple institutions. These institutions shape the relations of their subjects to knowledge by proposing praxeologies. A praxeology comprises a type of tasks (a set of precise tasks) T, a technique τ to accomplish this type of tasks, a technology θ which is a discourse explaining or justifying the technique and a theory Θ. [T, τ] constitute the praxis part of the praxeology and [θ, Θ] the logos part of it. In the context of mathematics at tertiary education, the subjects are students or teachers (our focus being here on the students); the types of tasks can be mathematical, like "proving that a matrix is invertible", but also organisational, like "coordinating the work of a team". In ATD, the set of the tasks that a student performs by themselves is called the topos of this student. The concept of topos describes the responsibility of a student as a dynamic area (within the different praxeologies present in the classroom). The topoi of the students and of the teacher are interdependent, they develop in the classroom along a process named topogenesis [START_REF] Chevallard | La Transposition didactique[END_REF]. ATD also considers cooperative tasks, involving several subjects of the institution. For a cooperative task a subject performs gestures, which refer to the set of means that this subject uses to accomplish the task [START_REF] Pensée Sauvage | L'analyse des pratiques enseignantes en théorie anthropologique du didactique [Analysis of teaching practices in anthropological theory of the didactics[END_REF].

The topogenesis concept has been further developed in the Joint Action Theory of the Didactics (JATD, Sensevy & Mercier, 2007). JATD is a socio-cultural theory devoting a central place to the joint construction of knowledge by the teacher and the students in class, interacting within a milieu, encompassing material objects, signs, and knowledge. Sensevy and Mercier introduced in particular the concept of topogenetic position, described as the symbolic place occupied in the classroom by a student or a teacher, in relation with the knowledge at stake. These authors distinguish between high and low topogenetic positions, according to what they call the epistemic density of the task performed by a subject, i.e., the potential of the task in terms of modification of the milieu. In our study concerning cooperative tasks, in order to characterise the students' topogenetic positions, we consider the gestures performed by each student to accomplish the task.

We contend that investigating the topoi and the topogenetic positions of female and male students working together can contribute to an analysis of gender equality. We complement these theoretical elements by the concept of epistemic symmetry introduced by Gerin (2021). In her investigation of teaching strategies promoting gender equality (in the context of writing at primary school), Gerin, referring to JATD, considers that epistemic symmetry is achieved when the female students and the male students working together have the same responsibility towards knowledge. Thus the research question we study here is: In the context of a project-based course, does the students' group work have the characteristics of epistemic symmetry, i.e., are the topoi and topogenetic positions the same for male and female students?

Case study

Context and methods

Our study is situated in France, in the context of the first year of scientific preparatory classes (preparing students for highly selective engineering schools). In the context of a more general project concerning gender equality in such classes, we considered in particular students' cooperation in a project-based course named « supervised personal interest work » (TIPE in French). Teams of students perform during the second semester a collective project involving one or several scientific disciplines (mathematics, physics, engineering, computer science). The teachers do not propose projects, the students are free to choose any topic related to the subjects they are studying. They must then define a problematic and study it both theoretically and experimentally. When the topic is related to mathematics, the experiments usually consist of numerical simulations. For 16 weeks, two hours per week are devoted to this course, the physics and mathematics teachers are present and can be asked to answer a question or explain a new knowledge. At the end of the semester, all groups present their work orally in front of the other students and the teachers, using a slide show.

We presented our research project in two classes (we did not mention our focus on gender equality to avoid influencing the students' behaviours). Four teams of students volunteered for participating in the project; we selected the two teams comprising male and female students, and followed them along the semester. Each team recorded weekly in a logbook shared with the researcher what each of them had done during the session, which resources they had used and possibly what they planned for the next session. Moreover, two sessions were video-recorded and transcribed, the first time after two months of collective work, and the second time one month later. For the sake of brevity, we focus here on one group of three students: two female students, Alice and Clara, and a male student, Thomas (pseudonyms) who chose to work on neural networks and classification. The group was first formed on the basis of a personal affinity between Clara and Alice, then Thomas joined out of interest in the chosen theme. We selected this group for the richness of their notes in the logbook. The first session recorded took place before they reached a clear delimitation of their problematic; for the second session recorded, the problematic was quite clear and the group was working on the numerical experiment.

We distinguish between three levels in our analysis: the macro-level of the whole project-based course; the meso-level of one session; and the micro-level of episodes where the students interact around a type of tasks. At the macro-level, we focus on the students' topoi, and triangulate data from the logbook and the video-recording of the sessions. At the meso-level, we use the videorecording to investigate both the topoi and the topogenetical positions by characterising each student's gestures and classifying them for a given session. We distinguish between high-position gestures (explain, present, fix, solve, articulate with old knowledge), neutral gestures (nod, approve), low-position gestures (ask a question, expose some difficulty). This classification is so far exploratory, it will be refined in further studies. At the micro-level, we analyse the students' discourse and refine the analysis of the topogenetical positions. For each level, we compare the topoi and topogenetical positions according to the gender of the students.

Results

We firstly present our analyses at the macro-level, drawing mainly on the logbook. During the beginning of the course, the students search for resources, explore different directions before choosing a problematic. From that moment on, the types of tasks to be undertaken by the group have a different nature, digital experimentation takes an important part, and the search for resources becomes secondary. Nevertheless we can observe a difference between male and female students' topoi, as illustrated by the following extract of the logbook for May 12, 2022 (our translation).

Thomas:

Creation and implementation of neurons and of the genetic algorithm. Clara:

I have continued searching how to create generations. The PhD Theses and scientific papers are a bit lacking in concreteness so for the next sessions I will code neural networks with Thomas. However these researches will allow me to mathematically justify the result of the algorithm. Alice:

research of the optimal number of neurons for the network, result: there is no predefined recipe for the number of neurons, so we try with one hidden layer. + Watching a 'machine learnia' video about programming a two-layer neural network.

Thomas focuses on the praxis part, mainly the implementation of the algorithms, whereas Alice and Clara continue to look for mathematical justifications and optimisation for these algorithms, taking the responsibility for the logos part. This observation also holds for other parts of the logbook, and is confirmed by the video-recording of the second session. A possible explanation is that Thomas is considered an expert in coding, which is close to gender stereotypes. This choice leads Alice and Clara to confront the widely open question of justification and optimisation of these algorithms, which explains the difficulties they encounter. Eventually they give up on justifying the algorithms and choose to help implement them.

We now turn to the meso-level and focus on the first video recorded session. It can be divided into several parts: during the first part, Thomas briefly exposes his difficulties with the notion of gradient, then Alice tells her classmates about a video she watched during the previous session, and presents the numerical tests she implemented. The video and the tests regard the k-means method for unsupervised classification. The second part of the session consists of a collective reflection on the directions to give to the work, and the tasks and objectives to be distributed within the group. In the third part, the three students work individually on their own objective, mainly watching or reading resources they found on the internet. At the very end of the session, the students interact with the teacher, explain what they have done and their blocking points.

At the meso-level of the whole session, we listed all the students' gestures and classified them in terms of high-position, neutral or low position. The result is presented in Figure 1.

This figure shows the epistemic asymmetries between male and female students. Although the major episode of part 1 is devoted to Alice explaining the k-means method and presenting her numerical experiments to her classmates, paradoxically her topogenetic position according to her gestures is globally lower than Thomas'. For both female students, 40% of their gestures are associated with a low position, whereas only 10% of Thomas' gestures are low-position gestures. In order to refine these observations, and link them with the mathematical content, we now focus at a micro-level on the first part of the session and compare Alice's and Thomas' topogenetic positions when they report about what they did in the previous session.

Thomas exposes his difficulties in terms that seem to discourage his classmates to help.

Thomas: I had begun to compute the gradients, but uh… it… it doesn't lead anywhere in the sense that… it leads to things that are not homogeneous, I mean I multiply matrices but with the wrong sizes and all. Clara (seems impressed): Oh, it's matrices? Thomas:

Yes indeed, because I compute gradients of matrices with respect to matrices.

[Clara nods.] You know, since we vectorialised the equations at the very beginning. You know, you put you know the gradients of capital X, capital W, and all.

The concept of gradient has not been introduced yet by the mathematics teacher, and Thomas presents it as a very complicated concept, whereas in reality he has to compute gradients of a scalar function of two variables. At this stage, Alice gets out of the discussion, she no longer looks at her classmates and tries to focus on her computer's screen. When these difficulties occur, Thomas gives instructions to help Alice fix the problem.

Thomas: Suppress the one you have just had. Close it.

His instructions happen to be inefficient and finally Alice modifies her code and finds by herself the solution of the problem.

In this episode we see that Thomas adopts a high topogenetic position, by linking new knowledge with old knowledge, and by trying to fix Alice's difficulties. Thus at this micro-level we also conclude that there is no epistemic symmetry between female and male students for this group.

Discussion and further directions

In the analyses presented above, we evidenced different kinds of epistemic asymmetries in the group of students followed. At the macro-level of the whole course, the male student was in charge of the programming, praxis aspect, while the female students started by working on the justification of the algorithm, the logos aspect, with an a priori high epistemic density. Nevertheless it turned out to be so difficult that eventually their actual task became to assist with programming. At the meso-level of the session, we observed the asymmetry in terms of percentages of low-position and high-position gestures. At the micro-level of the episode studied, we identified asymmetries between Alice and Thomas during their discussions about the work each of them has conducted. Thomas adopts from the start a high topogenetic position. When presenting difficulties he met, he emphasises the complexity of the concepts he used: gradient, matrices. Although this might not be conscious, we interpret this emphasis on complexity as a way to discourage the female students to offer their support. When Alice is in a high position, presenting the k-means method, Thomas tries to appear equally high, by foregrounding his understanding of her discourse. He links this method with previous knowledge and offers his help. We do not claim that these observations can be generalised to all the groups in this class or in other classes proposing project-based courses. We do not even claim that gender was the factor explaining the differences observed. Other factors like the perceived ability of the student in the class can impact the interactions during the work in groups.

Our aim here was to investigate the epistemic symmetry between male and female students in the case studied. Identifying the causes of the differences observed requires a further study.

Like [START_REF] Jungwirth | Interaction and gender? Findings of a microethnographical approach to classroom discourse[END_REF], we observe that the interactions-between students in our case-are central in the learning of mathematics. We also align with [START_REF] Barnes | Collaborative group work in mathematics: Power relationships and student roles[END_REF] about the power relation produced by discourses. The combined use of ATD and JATD provides us with conceptual tools for studying these interactions and power relations, and for linking them with the mathematical content. At the macro-level, a power-relation can result from the fact that some students have a praxis-centred topos, while others are logos-centred, thus with more difficult tasks. This leads to unequal opportunities to develop useful skills. At the micro-level, we noted that a male student can exaggerate the complexity of the mathematical content in order to hinder a cooperation in which his topogenetical position might be low.

In our further work, we intend to investigate further the theoretical construct introduced here, to complement and refine it. One important direction concerns the determination of the topogenetical positions. Focusing on cooperative tasks, we looked at gestures and introduced an exploratory classification, considering for example that "explain" was a high-position gesture and "expose a difficulty" a low-position gesture. Nevertheless a systematic classification is missing, and the link of this classification with the mathematical content at stake also requires a further investigation.

Figure 1 :

 1 Figure 1: Analysis of gestures with respect to knowledge

  Clara tries to concentrate and understand Thomas' indications but it seems she can't figure out how to help him.A few minutes later, Alice presents what she has understood of the k-means method. Thomas and Clara listen to her and nod regularly, showing their interest and understanding. Thomas tries to link this method to an algorithm called "nearest-neighbour algorithm", previously encountered in an assignment, Alice confesses she did not envisage this link. She goes on presenting her numerical tests, but her computer is very slow and after a while the results of the test are not displayed, leading Alice to express doubts.

	Alice:	Normally, normally it displays uh… that's strange, why doesn't it want to display? Ah. It doesn't want to. Because normally it displays the graph with the evolution of inertia with respect to the number of groups you make. […] OK, that's it. Wait, because there… Oh yes. Wait. It mixes both… Uh… OK. Wait, that's strange, it seems it still displays the other one.