Charlotte Baron

Sabine De Bosscher

First part of a french validation of an Ambivalent Sexism toward Men Inventory

Introduction

We often forget sexism toward men and its deleterious effects. So, this study aimed to translate the AMI A,B in order to couple it with other scales in French and to show possible correlations (e.g., with adherence to rape myths C ). Also, this french version could complete the Ambivalent Sexism Inventory (ASI D ) which measures the sexism against women. The ASI was also validated in a french version E .

֍ The Ambivalent Sexism Toward Men can be divided into 6 sub-

Results

֍H1 : The 2-Factor structure appears to be the best to explain the variances in the data. The results showed satisfactory reliability (Table 1) and predictive validity (Figure 1). However, although Benevolence (B) is clearly divided into 3 subdimensions, it seems less appropriate to repeat this same division for Hostility (H). Indeed, the items (H) are significantly correlated with each other but don't show a distinction between H(G), H(P) and H(S) as clear as in (B).

֍ Item 20 B(M) : This item « Women ought to take care of their men at home, because men would fall apart if they had to fend for themselves » supposed to measure B(M), showed higher correlations with H(G) and H(P).

֍H2 : The gender effect was significant for items (H): women (♀) were more hostile than men (♂) compared to men. However, no significant effect of gender could be observed for the items (B). (Table 2)

Discussion

֍ The sexism toward ♂ is a set of ambivalent attitudes between (B) and (H). Maternalism also seems to be in this ambivalence (item 20). Indeed, the idea that ♀ have to protect and help ♂ ((B)) comes perhaps from the other idea that they are like children: immature, incapable ((H)).

֍ The difficulty of sub-categorization in (H) could come from the low number of items.

֍ Future Studies : First, we must continue and finish this scale validation by the method proposed by Vallerand F (2nd step in progress via the internet). Studies should also look more deeply at the link between B(M) and (H).

֍ Limits : a) Sample: small, students, women → Difficult generalization of the results b) Statistics: we have not studied equivalence and stability over time via a "test-retest" (it's currently in progress). We must not forgot that a factor-analysis remains only an approximation of the observed / observable effects.

֍ Implications : Prevention seems important. Indeed, knowing and better understanding sexism (towards ♀ and ♂) would certainly make it possible to reduce inter-gender behavioral differences. We can think of people who receive the speech from victims of sexual assault (magistrates, police officers, psychologists, etc.). We can also include education professionals : this prevention can be done from an early age ( sexist stereotypes aren't yet fully integrated into memory)

  groups of stereotypes between Benevolance (B) & Hostility (H) : -H(P) Resentment of parternalism : a rejection of the patriarchal ideology. (e.g.,« Most men pay lip service to equality for women, but can't handle having a woman as an equal ») -B(M) Maternalism : presupposes an obvious fragility in men. (e.g.,« Even if both members of a couple work, the woman ought to be more attentitve to taking care of her man at home ») -H(G) Compensory Gender Differentiation : Men are like children, they are incapable of anything without women. (e.g.,« Men would be lost in this world if women weren't there to guide them ») -B(G) Complementory Gender Differentiation : men are there to protect women who are fragile. (e.g.,« Men are more willing to take risks than women ») -H(S) Heterosexual Hostility : men are ready to do anything to show their domination (at work, in sex ...). (e.g.,«Men usually try to dominate conversations when talking to women ») -B(S) Heterosexual Intimacy : heterosexual romantic relationships are essential for personnal development. (e.g.,« Every woman needs a mal partner who will cherish her ») ֍ The aim is to carry out a first part of the translation of the AMI in a French version according to the method proposed by Vallerand F . ֍ Hypotheses : H1 : A 2-factor structure (a) completed by 6 sub-dimensions (b) makes it possible to explain a greater part of inter-subject variance. H2 : Women are more sexist (a) and more hostile (b) than men. Method ֍ The AMI includes 20 items divided in : 10 for (B) & 10 for (H) as follows : 3 H(P), 3 B(M), 3 H(G), 3 B(G), 4 H(S) and 4 B(S). Adherence to theses items is measured by a Likert-type scale ranging from : 0 (Disagree Strongly) → 5 (Agree Strongly) ֍Participants : Students in 1 rst Year of Psychology, n = 375 (324 ♀, 51 ♂) ; age between 18 and 38 (M = 19.05 ; sd = 1.90) ֍Time 1 : Translation. Two French versions were worked on simultaneously and then compared (with each other and with the ASI D ) in order to arrive at a third version.

֍Time 2 :

 2 Scale Infills. It took place at the beginning of lectures, without remuneration, and with debrifieng at the end.

  References A. Glick, P., & Fiske, S. T. (1999). The ambivalence toward men inventory: Differentiating hostile and benevolent beliefs about men. Psychology of Women Quarterly, 23, 519-536. https://psycnet.apa.org/doi/10.1111/j.1471-6402.1999.tb00379.x B. Glick, P., & Fiske, S. T. (2001). Ambivalent stereotypes as legitimizing ideologies: Differentiating paternalistic and envious prejudice. In J. T. Jost and B. Major (Eds.), The psychology of legitimacy. Emerging perspectives on ideology, justice, and intergroup relations. Cambridge (UK): Cambridge University Press. C. Lonsway, K. A., & Fitzgerald, L. F. (1994). Rape Myths : In Review. Psychology of Women Quarterly, 18(2), 133-164. https://doi.org/10.1111/j.1471-6402.1994.tb00448.x D. Glick, P., & Fiske, S. T. (1996). The Ambivalent Sexism Inventory: Differentiating hostile and benevolent sexism. Journal of Personality and Social Psychology, 70(3), 491-512. https://doi.org/10.1037/0022-3514.70.3.491 E. Dardenne, B., Delacollette, N., Grégoire, C., & Lecocq, D. (2006). Structure latente et validation de la version française de l' Ambivalent Sexism Inventory : l'échelle de sexisme ambivalent. L'année psychologique, 106(2), 235-263. http://doi.org/10.4074/S0003503306002041 F. Vallerand, R. J. (1989). Vers une méthodologie de validation trans-culturelle de questionnaires psychologiques: Implications pour la recherche en langue française [Toward a methodology for the transcultural validation of psychological questionnaires: Implications for research in the French language]. Canadian Psychology/Psychologie canadienne, 30(4), 662-680. https://doi.org/10.1037/h0079856

Figure 1 :

 1 Figure 1 : Inter-items Correlations in the 2 factors of sexim. (All the p<.001)

Table 2 :

 2

	Table 1
	Statistics
	tests

Gender differences in (H) and (B) between ♀ and ♂ (T-tests) * p<.05; ** p<.01; *** p<.001


Élisabeth Anstett
email: elisabeth.anstett@univ-amu.fr

Annual Burials and Reburials of Victims of Mass Violence in Present-Day Bosnia and Herzegovina

Keywords: delayed mourning, rituals, innovation, exhumation, body parts

The international armed conflict that took place in Bosnia and Herzegovina between 1992 and 1995 caused the death of around 100,000 people, the majority of whom were civilians. More than 25 years later, the process of search, identification, and burial of these victims is still not over, as a significant number of them were initially buried clandestinely or had their body moved and hidden in secondary, and sometimes tertiary, mass graves. These cover-ups bear a long-lasting legacy, as they have directly impacted funerary practices. The concealment and fragmentation of dead bodies have indeed delayed funerals and led them to be repeated numerous times. Focusing on the example of the annual collective burials, exhumations, and reburials occurring over the last two decades in Potočari cemetery, this article aims to show how the delay and repetition of burials may prevent families and society from finding closure but also contribute to the consolidation of a collective identity based on the shared experience of lasting mourning and resilience building.

Incipit: Potočari, Monday, July 12, 2021 The previous day, the funerals of nineteen victims drew a dense crowd of several thousand people. It is the eighteenth consecutive year that collective funerals have been held here on the anniversary of the Srebrenica massacre. Despite restrictions on international travel due to the pandemic, families, friends, neighbors, campaigners, and peace activists came in large numbers from all over the country and from abroad. Starting in mid-morning, various official speeches were made in the neighboring warehouse to an audience of members of the diplomatic corps and government officials. In the cemetery, where the coffins were all lined up on the ground, the funeral prayer Salāt al-Dženaza was said 1 at midday by the Reis-ul-Ulema in the presence of numerous imams. Each of the nineteen bereaved families then came forward to find their relative's coffin and lay it in the ground at last, twenty-six years after the death of their loved one. The men took turns carrying the coffin to the prepared grave, placing it inside, and shoveling earth on top to cover it, while the women prayed beside them. After a final moment of family remembrance, everyone left. After the funerals, dozens of volunteers worked to tidy the area up. By nightfall, it looked no different from usual except for the little mounds of earth and green-painted wooden planks marking the location of the fresh graves. In a few months, once the ground has settled, these planks will be replaced by white marble headstones bearing the name, birth and death years, and hometown of the deceased. On this warm, sunny July morning, the huge Potočari site is practically deserted, with almost no sign of yesterday's funerals. Here and there the grass has been trampled down where the families were standing, in some cases since daybreak to make sure of getting a shady spot. At the foot of the low wall on which the 8,372 names of the victims initially buried at the site are engraved, a mass of floral wreaths testifies to the number of foreign delegations who attended the funerals. Calm reigns. Nevertheless, there is still some activity. Around twenty gravestones scattered throughout the cemetery have been marked with a piece of barricade tape tied to the top. Beside some of them are people waiting in small groups of three or four, rarely more. Other graves are marked but have no one near them. A half dozen workers are busy in front of one of the tombs. Watched by five members of the family, they are trying to open the grave using a small backhoe loader. Precision, economy of movement: the gravediggers clearly know what they are doing. The grass and a top layer of earth are carefully lifted using the bucket. The final extraction is completed using shovels. The two workers who have gone down into the grave remove the decomposed remains of the green cloth and wooden planks that formed the coffin and place them beside the grave. A white plastic body bag is taken out of the narrow pit, and a few remaining clods of earth are brushed off. Written on it in permanent marker are a name and a reference code consisting of numbers and letters. Two other workers take over and carry the bag a few meters away to a road that can be accessed by vehicle. The exhumation takes about twenty minutes. A white pickup truck arrives, and the bag is loaded into the back; it leaves in the direction of the memorial and museum complex on the other side of the road running alongside the cemetery, where it disappears into a large warehouse. Inside the vast empty building, illuminated only by the sunlight falling through the high skylights, around ten people are working in silence: a team of forensic anthropologists from the International Commission on Missing Persons (ICMP) working in Bosnia and Herzegovina, their boss, who has come from the organization's headquarters in The Hague to help, four representatives of the Missing Persons Institute (MPI) acting on behalf of the state of Bosnia and Herzegovina, including a medical examiner, and a police officer representing the Bosnian judicial system. Next to the work area, several new, preprepared body bags are lying on the floor. They are marked with the numbers of all the "cases" that will be reviewed todayeleven in total, out of the 92 reassociations scheduled for this week. One of them corresponds to the bag that has just been brought in and placed, open, at the foot of the stainless-steel table. After the names and case numbers have been checked to make sure they match, the body bag brought by the gravediggers' vehicle is placed inside the new bag and opened. The bones it contains are removed, placed on a stainless-steel examination table, and arranged in anatomical position. The contents of the bag are meticulously inventoried and compared to the official report prepared during the previous burial. Other bone fragments, which are sitting at the side in a clear plastic bag with the same reference code, are added to the skeleton in their anatomical position.

Once the anthropological examination has been completed, all the bones are replaced and the reassociation procedure is written up in a new Official Report, duly dated and signed. Three sets of photographs are taken, respectively by the judicial police officer, the medical examiner, and one of the anthropologists from the ICMP team, who each in turn climb a stepladder in order to photograph the skeleton in its entirety. The Srebrenica massacre has been designated as a genocide by the International Criminal Tribunal for the Former Yugoslavia (ICTY). Because genocide is an imprescriptible crime, all evidence, including human remains, must remain at the disposition of the justice system for an unlimited time.

The various experts are focused on their task. The reassociation is carried out with no discussion other than that concerning their work, which began early in the morning and will continue for several days. It is occasionally punctuated by visits from some of the families who want to witness the reassociation, and to whom the medical examiner or one of the members of the forensic team explains the procedure being carried out. The white pickup truck arrives. A freshly unearthed bag is unloaded and placed on the floor, and another immaculate bag whose occupant has already undergone reassociation is loaded into the back. The truck leaves, taking the judicial police officer, who will witness the reburial and make sure that the exhumed individual is placed back into the correct grave. Inside the cemetery, the reburial proceeds quickly. Once the bag has been put back into the grave, the latter is filled in by shovel, with the youngest family members helping the gravediggers. An elderly woman crouches down for a last silent prayer, while her neighbor wipes his tears; their grief is still tangible. Not far away, the backhoe loader is in the process of opening another grave in the presence of three young men. In another part of the cemetery, two families are waiting patiently for their turn, while those whose relative's reassociation is complete leave the site. Until the next time. Accompanied by his mother and aunt, today Ahmet has buried his father's remains for the third time.

Introduction: Exploring the effects of repeated collective funerals

The international armed conflict waged in the small territory of Bosnia and Herzegovina between 1992 and 1995, involving the Bosnian (majority Muslim), Serb (majority Orthodox Christian), and Croatian (majority Roman Catholic) communities, triggered a migration crisis generating 2 million refugees and left almost 100,000 dead, more than a third of whom were non-combatants. Many of the dead were victims of bombs and snipers, but there were also numerous victims of atrocities specifically targeted at civilians. In Bosnia, civilian populations were subjected to raids, detention and torture, arbitrary execution, and more broadly activities that representatives of the international community referred to euphemistically at the time as "ethnic cleansing" or "purification" [START_REF] Naimark | Fires of hatred, ethnic cleansing in twentieth-century Europe[END_REF]. Among the numerous episodes of extreme violence, some were subsequently qualified under criminal law as war crimes, crimes against humanity, or genocide, as was the case for the massacre in the Srebrenica enclave, 2 where troops of the Bosnian Serb Army (Vojska Republike Srpske) under the command of General Ratko Mladić murdered more than 8,000 men and boys in mid-July 1995. At Srebrenica, the roundup, sorting, detention, and execution of victims were also followed by the singular practice of concealing the remains. During the Bosnian War, the bodies of executed civilians were often buried in mass graves, but also sometimes thrown into waterways or natural wells, or off cliffs in particularly inaccessible mountainous areas, with the result that several thousand remain unfound to this day. The concealment of bodies is thus one of the hallmarks of the violence committed in Bosnia and Herzegovina [START_REF] Komar | Patterns of mortuary practice associated with genocide: Implications for archaeological research[END_REF]. Among these concealment practices, the use of secondary and tertiary graves (whereby remains are first buried in an initial series of mass graves, before being exhumed and reconcealed in a second series of graves, from where they may even be moved again and reburied in a third series of secret graves) is one of the characteristic features of the violence perpetrated at Srebrenica. These repeated exhumations and concealments had the twofold aim of making it more difficult to discover and document the crimes by increasing the number of secret burial sites, and of lending credit to a narrative used by the perpetrators in their defense: the secondary and tertiary graves were located along the front line, supporting the claim that those buried there were combatants rather than civilians. As well as complicating the documentation of the crimes, these concealment practices had numerous other consequences, particularly concerning the reburial of victims. The care taken to conceal the remains made it difficult to find them, delayed their discovery, and thus also delayed the moment when they would be returned to their families and funerals could be held [START_REF] Claverie | Réapparaître. Retrouver les corps des personnes disparues pendant la guerre en Bosnie[END_REF]. The human remains that took the longest to be located were especially difficult to identify because they had been fragmented by the mechanical diggers used to move them from grave to grave: the remains of a single body might be scattered between several secondary and tertiary graves. Only the systematic use of DNA identification, which started to be used in the early 2000s,3 allowed forensic anthropologists to understand the perpetrators' modus operandi, to identify this use of secondary and tertiary graves, and to begin reconstituting the scattered remains [START_REF] Jugo | Ghosts of the past: The competing agendas of forensic work in identifying the missing across Bosnia and Herzegovina[END_REF]. We will return in more detail later on to the issues addressed by the funerary and commemorative site dedicated to those who died in the enclave of Srebrenica. The site is located in the village of Potočari4 right next to the place (a former industrial complex) where the victims initially took refuge under the protection of UN soldiers and were then detained by Serb troops before being taken to be executed in the surrounding area. At this point, we simply want to point out that the creation of this memorial was conceived as one element in a wider plan for exiting violence that was initiated by the Dayton Accords and was intended to facilitate the reconstruction of Bosnian and Bosniak society by enabling the discovery, identification, and burial of victims. Nevertheless, things are not so simple. The burials held at Potočari are unique in three ways: the time lag between the massacre and the burials (the first burials took place in 2003, almost eight years after the killings, and in 2022, more than 25 years later, they have still not been completed); their repetitive nature (they are held every year on July 11, the anniversary of the massacre, with human remains identified during the previous year buried at each annual ceremony); and their group nature (the burials at Potočari have comprised as many as 775 burials at one time, as on July 11, 2010).

Although researchers have already discussed the time lag of the funerals, their collective and repetitive nature, involving the simultaneous and recurring burial of dozens-or even hundreds-of individuals, has received little attention so far, even though it poses a very particular challenge to the nomenclature established by funerary archaeologists to distinguish between "collective" burials (the successive deposit of bodies) and "multiple" burials (the simultaneous deposit of several different bodies) [START_REF] Schmitt | Qu'est-ce qu'une sépulture collective? Vers un changement de paradigme[END_REF] Based on observation of the practices deployed at the Potočari memorial and drawing on a series of field studies carried out between 2013 and 20225 in Bosnia and Herzegovina in which we spoke to the principal actors involved in the search, identification, and burial of victims of the Srebrenica genocide, this article has a twofold objective. It aims first of all to investigate the effects of the simultaneous reburial of civilian victims of armed conflicts. Funerals involving the joint burial of several individuals, whether in separate graves or not, have already raised questions, such as those posed regarding Bosnia by the social anthropologist Sarah [START_REF] Wagner | To know where he lies[END_REF]. And these simultaneous burials are not unique to Bosnia. They have also been observed elsewhere in Europe, as in Cyprus following the war with Turkey [START_REF] Sant Cassia | Bodies of evidence: Burial, memory and the recovery of missing persons in Cyprus[END_REF] or in Spain for those who died during the civil war and the Francoist period [START_REF] Renshaw | Missing bodies near-at-hand: The dissonant memory and dormant graves of the Spanish Civil War[END_REF]. The use of simultaneous burial within a single sepulchral space is also attested, although still little documented, in the case of Holocaust victims, particularly in the collective reburial in 2011 of Jewish victims of the pogrom in Iaşi, Romania. Nevertheless, the questions raised by these group funerals have so far tended to focus on their temporal deferment, without fully exploring the implications of their simultaneity. Continuing the research into ritual innovation begun by the social anthropologists of religion Emma Gobin and Maxime Vanhoenacker (2017), these group funerals in contexts of mass violence raise the central question of the efficacy and potential plasticity of funeral rites. In the vast majority of cultures, funerary rituals are designed for a single individual, so what happens when it becomes necessary to bury many people at the same time? Burials related to disasters have been studied by archaeologists, especially in the context of epidemics, but the question of whether funerary rituals are slightly altered or radically transformed by the shift to the "collective," and so the extent to which a crisis of mortality is accompanied by a crisis of funerary practices, remains little discussed in social anthropology. 6Moreover, besides their collective and simultaneous dimension, which deserves investigation in its own right, the funerals of the Srebrenica victims are also unusual because of their repetitive nature. With that in mind, this article also aims to study the social effects of repeated burials: the Potočari memorial complex is characterized not just by having hosted large-scale collective funerals on July 11, the anniversary of the Srebrenica massacre, every year since 2003, but also by the fact that the remains of a single individual may have been buried there on several occasions, in several successive sequences. One of the direct consequences of the bodily fragmentation caused by the use of secondary and tertiary graves is that the remains of a single victim may be discovered and identified in different places and at different times. The remains of a single individual are rarely all dealt with simultaneously, and they are not always buried at the same time. Continuing the analysis developed by Valérie Robin Azevedo in Peru (Robin Azevedo, 2015, and in this volume) and by Clara Duterme in Guatemala (Duterme, 2021, and in this volume), these multiple partial burials raise the question of how, given the impact of the psychological fatigue that has been identified and discussed in the context of other mortality crises [START_REF] Haram | We are tired of mourning": The economy of death and bereavement in a time of AIDS[END_REF], the moral and political economies of lengthy and regularly reactivated mourning are collectively constructed, where mourning is understood in this case as a sociocultural process that takes place over several years or even decades. What meaning should we ascribe to the act of burial-a stage of funerary rituals that is normally intended to be conclusive-when it is repeated?

Funerary and mortuary practices

First, we will turn to the Potočari funeral complex in order to describe the practices that take place there in more detail and explore their implications.

1.a. The cemetery and memorial

Begun at the start of the 2000s, the construction of the complex was supported by international financing,7 enabling its rapid completion in a social, economic, political, and topographical8 environment that was still heavily marked by the ordeal of the armed conflict. The official name of the memorial, "Srebrenica-Potočari Memorial and Cemetery for the Victims of the 1995 Genocide,"9 emphasizes its twofold nature as a memorial (commemorative space) and a cemetery (funerary space), and the fact that it was designed to host both burials and commemorations. The social anthropologist Sarah Wagner analyzed the issues surrounding the creation of the site very early on [START_REF] Wagner | Tabulating loss, entombing memory: The Srebrenica-Potočari memorial center[END_REF]. For the families of the victims, who insisted on the choice of location, it was about reinstalling order-the order of funeral arrangements-in areas devastated by war and extreme violence, but also about rendering the extent and materiality of the disaster intelligible by displaying the accumulation of graves in one place. The principal challenge was to demonstrate the scale and the unique nature of the violence perpetrated against the Bosniak civilian population in a local geopolitical context dominated (then and still now, in many respects) by denial of the crimes. The choice of the site, which is located in the territory of Republika Srpska-the Serbian part of the state of Bosnia and Herzegovina 10 -itself gives the ceremonies, which bring Bosniak families together in Serbian territory, an undeniable political overtone. With nearly two decades having passed since its opening in 2003, the memorial now has a history that can be read in the evolution of its layout. The muddy slopes of the early days have been succeeded by an enclosed, wooded, and landscaped site laid out in the shape of a daisy, which has become the symbol of the Srebrenica massacre, with carefully maintained lawns and paved pathways. The memorial is arranged around a sheltered central space where the imams gather to conduct funerals, with fountains for the ablutions that precede Muslim rituals, as well as raised cement stands on one side, from where important guests (most often foreign) can watch the funerals at a distance from the crowd. The greenpainted wooden steles of the early years have been gradually replaced by white marble tombstones in the shape of posts, and the thousands of graves now fill the site with their immaculate mineral verticality. Gone are the tense funerals of the early days; in their place is, if not a routine, then at least a logistically well-oiled process (and one that is also tightly controlled by the police forces of Republika Srpska and Bosnia and Herzegovina). It is now possible for several thousand people to arrive via a single narrow road, gather in a small area, and depart via the same road in the course of a few hours.

1.b. Funerary and post-funerary

This complex thus raises the question of the nature of the practices conducted there: Are they funerary, post-funerary-i.e., commemorative activities after burial-or simply mortuary 11 practices with no funerary element? As well as the funerals organized every year on the anniversary of July 11, when any victims identified during the previous year are buried, there are also regular exhumations and reburials related to the reassociation of human remains identified at a later date. The simultaneous burials are public events of an undeniable funerary and religious character. They draw large crowds, ranging from a few thousand in ordinary years to tens of thousands on symbolic anniversaries (10 years, 15 years, 20 years, etc.). 12 Although more than 6,500 victims have already been buried in the complex, numerous others are still awaiting burial, including those that have not yet been found. The ICMP and MPI teams estimate that there are around a thousand victims of the Srebrenica genocide whose remains have still not been found; they believe there are probably one or two large graves containing hundreds of bodies that are still to be discovered, with no guarantee that they ever will be. In addition to the fact that the number of individuals simultaneously buried has gradually declined (only 19 in 2021 and 71 in 2017, compared to 409 in 2013 and 580 in 2005, the tenth anniversary of the massacre), this raises the question of when the funerals will end. The site is inevitably destined to see its funerary role gradually sidelined by its commemorative one. Nevertheless, it is funerary practices that have dominated there until now. By contrast, the reassociation of bones belonging to an individual who has already been buried, which involves the exhumation of the remains and their reburial after the addition of the new bone fragment(s), is a private, non-religious event. As the funeral prayer has already been said, these later stages in the treatment of the remains are in effect excluded from the order of Muslim funerary ritual. They therefore fall outside the religious register and belong instead to the simple register of burial management, which we refer to as mortuary following the distinctions established by archaeologists. Although they tend to be carried out in summer (when families, some members of which may live abroad, come together for their vacation), reburials of reassociated remains do not take place at the same time as the July 11 burials. The distinction between a funerary and religious burial and a non-funerary and non-religious burial is carefully maintained. These secondary burials, and the more modest family gatherings they occasion, are likely to continue because there is a high chance that any human remains still to be discovered13 or in the morgue 14 but not yet identified will belong to individuals who have already been buried. In Potočari, mortuary and post-funerary (i.e., commemorative) practices are thus destined to gradually gain ground on funerary practices; the time of collective mourning will come to an end even as an abundance of individual mourning processes, which may entail very different experiences, will continue to be annually reactivated.

The collective in death: mortality crisis, funerals in crisis?

Like any crisis, mortality crises give rise to contradictory processes and practices that divide and fragment social groups, as well as practices that bring them together and contribute to their cohesion. The mortality crisis brought about by the war in Bosnia and Herzegovina between 1992 and 1995 caused long-lasting political and social fragmentation [START_REF] Džihić | Failing promises of democracy: Structural preconditions, political crisis and socioeconomic instability in Bosnia and Herzegovina[END_REF], but did it affect funerary practices? The anthropologists Admir Jugo and Sari Wastell have already shown the extent to which the establishment of a systematic DNA-based procedure for identifying human remains found in secondary and tertiary graves contributedat least at the forensic examination stage-to reassembling Bosniak society [START_REF] Jugo | Disassembling the pieces, reassembling the social: The forensic and political lives of secondary mass graves in Bosnia and Herzegovina[END_REF]. To what extent, then, do the simultaneous burials and collective funerals held at Potočari also help to rebuild or strengthen a socius? Can burial rituals maintain their role as drivers of social cohesion in these contexts?

2.a. A succession of individuation effects

It seems important, first of all, to emphasize that in the case of the Srebrenica genocide, although the victims were all murdered in the same way (executed by gunshot), at the same time (in the days following July 11, 1995), and in the same area, the unfolding of administrative, judicial, and forensic procedures imposes very disparate timelines on the families of the victims. Differences in expert assessments (based on DNA samples taken from several of each victim's family members) and administrative or judicial decisions (particularly concerning the recognition of death or of the status of victim of a crime) affect the date on which the remains of the deceased are actually returned to their families, and so when the funerals can be organized. This succession of stages, which in some cases can be drawn out over several decades, gives rise to individuation effects that the collective burial event works to overcome. In effect, the funerals held at Potočari propose a single, almost identical funerary treatment for several individuals, who are placed in coffins at the same moment, in the same way (with the bodies enclosed in a white body bag and placed in a wooden coffin covered with a green cloth), and in the same place (in the vast majority of cases, the morgue in Visoko, around 20 kilometers from Sarajevo), before being transported together in trucks from Sarajevo to Potočari a few days before the ceremony, when they are buried simultaneously. Nevertheless, at Potočari as at other Bosnian sites of the same nature that attempt to make a tragic common fate intelligible, the place of the individual is maintained. Each victim is buried in a separate, named tomb, and some victims may even be buried outside these memorial spaces if their families so desire. Thus, although 6,609 of the 7,012 victims of the Srebrenica massacre identified to date have already been buried, not all are buried at Potočari. 15 This choice to maintain the individual's visibility within the collective is not always adopted in collective funerals. In some cases, such funerals may involve burial in a single tomb, with (as in Spain for some victims of the civil war16 ) or without (as in Rwanda for the victims of the genocide against the Tutsi17 ) individuation of the remains. The decision to maintain the principle of individual tombs within a collective space intended to symbolize a common fate is, however, shared by all the cemeterymemorials for civilians in Bosnia and Herzegovina, and indicates a concern to maintain the balance between the individual and the collective.

2.b. The collective and cohesion

Despite this succession of individuation effects and despite their different experiences of mourning, some of which can be extremely complex,18 the families of the Bosniak victims experience a sense of cohesion at these collective funerals. Because of the conditions under which their loved ones died, the family members are already part of a collective. According to the definition in the 1948 Convention on the Prevention and Punishment of the Crime of Genocide,19 genocide is characterized by the intention to "destroy, in whole or in part, a national, ethnical, racial or religious group, as such." And it was the Bosniak group, to which the victims' relatives also belong, that was targeted by the violence, rather than the individuals themselves. These multiple deaths-especially when several members of a family were murdered at the same time-triggered a rearrangement of family ties around the missing links, the manifold consequences of which have already been studied in the context of large-scale genocides like the Holocaust or the Armenian genocide,20 and which we know is reflected in the importance ascribed to commitments to the collective. At the family level, the funeral of a victim of the Srebrenica genocide is an event that concerns and mobilizes all genealogical ties of consanguinity and kinship without distinction, and one that brings together extended family groups that may be scattered by migration across several different countries or even continents. 21At the vast memorial site, the design of the funerary area itself encourages gatherings of relatives: when the layout of the thousands of tombs was planned, the funeral home in charge of the arrangements 22 made sure that all members of a single family (father, sons, brothers) could be buried side by side, deliberately leaving empty spaces where necessary for remains that had still not been discovered. To this mobilization of kinship ties, the simultaneous and public funerals add that of neighborly ties, whether those established during the lifetime of the deceased, those formed by survivors following the end of the war, or those generated through the proximity of tombs. The notion of neighborhood (komşiluk) has a specific role in collective ideas about community life in the Balkans, and the part it has played in the production and suppression of collective violence has already been amply discussed [START_REF] Sorabji | Bosnian neighborhoods revisited: Tolerance, commitment and komšiluk in Sarajevo[END_REF][START_REF] Odak | Concluding remarks on identity and Komšiluk[END_REF]. But komşiluk also refers to the experience of long-term cohabitation in the funerary space [START_REF] Baskar | Komšiluk and taking care of the neighbor's shrine in Bosnia-Herzegovina[END_REF], both in Potočari and in other similar funerary sites. The size of the event and the fact that the funerals attract several thousand people each year extends the reunion experience to further types of social bond, whether friendly, generational, or professional. To avoid traffic, many families arrive at Potočari very early in the morning, or even the day before if they can find accommodation nearby. The long hours leading up to the burial offer an opportunity for family members to walk around the site, greet each other and hug, and exchange news or a few words with all those recognized as relatives, friends, neighbors, or simply acquaintances. In this respect, the bereaved family members who come to Potočari undergo a simultaneous experience of collective mourning and extended social cohesion. For the youngest family members, the funerals are also an occasion for learning (particularly about religious rituals and the performance of prayer) and for entering into adulthood. There are few young children at Potočari, but many adolescents and young adults (boys and girls). The gathering also enables reunions with those who migrated during or after the war and whose children have grown up far away, without much knowledge of or familiarity with their parents' society of origin. In this respect, the funerals and large gatherings occasioned by these burials have the twofold effect of increasing the cohesion of family groups reunited around their relatives' graves, but also of allowing the youngest to experience, through their participation in the funerals, a sense of belonging to a family collective, but also to a broader cultural, national, and religious collective. The impact of simultaneous funerals of Srebrenica victims on Bosniak society should not, therefore, be underestimated. These events, which receive extensive media coverage, trigger the mobilization of various national and international groupings (based around sport, culture, or religion) that converge on Potočari on the date of the funeral. The simultaneous funerals thus create cohesion through metonymy: each individual laid to rest is recognized as a representative of Bosniak society as a whole, in both its political (as a national collective) and religious (as a Muslim community) dimensions. These identities are not exempt from political connotations, in that the religious identity of the victims is sometimes exploited by certain states (such as Turkey, Egypt, Malaysia, or Saudi Arabia), whose official delegations and representatives, equipped with flags, attend the funerals at Potočari. In a recent study of cemeteries in Europe and the Middle East (in Bosnia and Herzegovina, Cyprus, Greece, and Libya), Thorsen Kruse's research team has shown the extent to which funerary spaces may be used as tools in political or 22 The company in charge of the burials is also responsible for developing the Potočari site and performs the exhumations carried out there; interviews were conducted with its director in 2013, 2017, and 2021. geopolitical power games in multicultural and multifaith conflict contexts (Kruse, Faustmann, & Rogge, 2020). Potočari is no exception.

2.c. Funerary practices, arrangements, and innovations

As we know, mortuary rituals materialize and render intelligible the inscription of an individual into a set of social and symbolic orders beyond the moment of death. Funerary customs in Bosnia exclude cremation 23 and dictate the use of earth burials in local cemeteries, whether single or multifaith. In rural areas, different communities have maintained the widespread use of religious rituals, characterized for Muslims by the washing of the body (gasuljenie), the performance by the men of the community of the Dženaza, the funeral prayer recommending the soul of the deceased to God, and rapid burial-ideally the day after death-in a white cotton shroud (ćefin); the use of a wooden coffin or casket has gradually increased in response to administrative guidelines. These funerary practices were significantly impacted by the war. From the beginning of the armed conflict, families appealed to the Islamic religious authorities to resolve the problems raised by the funerary treatment of civilian victims: as well as leaving behind bodies that were fragmented, severely decomposed, or simply absent, making the ritual washing of the body extremely difficult, the Srebrenica genocide specifically targeted men, creating a lack of men to lead the funeral rites. 24 Having traveled to Cairo to study the various existing jurisprudences, the Bosnian Islamic religious authorities issued a number of rulings with the force of law (fatwa) that introduced innovations into the funerary customs of the school of jurisprudence to which Bosnian Islam belongs. 25The first decision taken by the Islamic authorities was to declare the civilian victims of the war as shahid (saints or martyrs), thus absolving their families of the need to wash their bodies before burial. The decisions to allow funerals in absentia using the Salat al-Gha'ib prayer 26 and to allow partial burials without waiting for the recovery of the entire body also represent attempts to respond to the difficulties posed by the concealment and fragmentation of bodies. These innovations in funerary customs mean that the "nestaly" (literally, "those who are no longer there," a term used to refer to those who disappear in war) could receive funerary treatment as quickly as possible. Finally, permission for women to participate in funerals represented an alternative solution given the disappearance of entire age groups of men. In Potočari, the presence of women and their participation in the funeral prayer at the cemetery, standing alongside the men in no particular spatial order, as well as their role in organizing the funerals through the Association of the Mothers of Srebrenica, 27 are particularly noticeable. These innovations in the funerary treatment of the bodies of civilian victims in Bosnia did not fundamentally alter the principles of the preexisting funerary ritual, and they even made it possible to maintain a formal link with it through the use of ordinary coffins in which the bodies or human remains were placed, with the white cotton shroud replaced by a white plastic body bag for victims who are likely to be reexhumed at some point. The transformation of funerary practices can also be seen in the systematic use of standardized stone monuments. The fact that the many civilians killed during the war have been declared as shahid, a status that calls for the use of a stone stele, as well as the standardized character of the stones used (similar in color, size, shape, and texture), has profoundly transformed the landscape of Bosnian cemeteries, as well as their management: the principle of reoccupationcustomary for ordinary deceased persons-is now difficult to envisage in such spaces. The management of funerary spaces in a small, mountainous country where there is a shortage of suitable burial grounds, almost all of which are now occupied by shahid cemeteries, is one of the challenges that Bosnia must now confront in this context. The creation of a crematorium in Sarajevo28 may provide a partial solution, although it is not possible at this stage to judge whether the introduction of cremation should be interpreted as a final impact of the war, or more generally as an effect of the secularization of funerary practices in Europe.

Repetition

We have seen the extent to which the collective nature of the funerals held at Potočari has helped to rebuild and strengthen Bosniak society after the war by increasing its resilience. In a context marked by violence and a mortality crisis that specifically affected men, burial rituals have been able to maintain their function of promoting social cohesion thanks to the transformation of some of their details. But what about the effects of repetition? In Potočari, as in other funerary complexes in Bosnia associated with wartime violence, such as in Kamičani, burials are not just collective, but repeated.

3.a. Annual funerals and the effect of ritualization

The first repetition that we ought to discuss is that of the simultaneous funerals themselves. To begin with, it should be emphasized that the choice to hold the funerals at the same time and to do so annually is not unique to Bosnia. After the genocide against the Tutsi in 1994, which left more than 800,000 dead in just three months, the Rwandan state, faced with the huge number of exhumations to be carried out and human remains to be brought to national memorials, decided to hold all collective funerals and genocide commemorations solely in the month of April (the month in which the massacre began29 ) in order to limit the impact on the general population [START_REF] Korman | Mobilizing the dead? The place of bones and corpses in the commemoration of the Tutsi genocide in Rwanda. Human remains and violence[END_REF]. In Bosnia and Herzegovina, the choice to hold the simultaneous burials at Potočari on the anniversary of the Srebrenica genocide, July 11, was taken very soon after the first burials in March 2003 and the official opening of the site in September 2003. The choice of a single date allowed the families of the Bosniak victims to limit the number of times they had to travel to Republika Srpska at a time when access to the site was still not entirely safe. The collective funerals have thus been repeated annually at Potočari since the site was opened. They are always alike (held on the same day, in the same place, following the same religious funerary ritual) yet always different, both because they concern distinct individuals and so different family groups and because they are sometimes distinguished by unique events.

The commemorations of the twentieth anniversary of the genocide in 2015 were thus unusual in more than one way. They were partly affected by the exceptional measures 30 imposed to ensure the security of the international figures present (including former US president Bill Clinton, Princess Anne of England, the Turkish prime minister Ahmet Davutoğlu, and Queen Noor of Jordan), and partly by the scandal caused by the presence of the Serbian prime minister, Aleksandar Vučić, who was reproached by the Bosniak population for his words inciting hatred a few days before the genocide 31 and who was booed out of the site. The growing importance of the geopolitical issues at stake in the funerals has led representatives of the families, and more specifically the Association of the Mothers of Srebrenica, 32 to request that strictly funerary activities and spaces be distinguished from commemorative events with political overtones. Speeches by government officials and foreign delegations have thus been moved outside the cemetery and are now given in the warehouse located on the other side of the road, 33 where a cultural complex has been established. Even if some national flags or banners still indicate the presence of foreign delegations at the memorial, the general spatial and symbolic division between the speeches and political events taking place outside the site and the religious and private practices inside the memorial remains in place. This distinction shows that attention and reflection have already been devoted to ensuring that the moral aspects of the funerals have priority over the political aspects. Our observations of successive collective funerals between 2013 and 2022 have, nevertheless, revealed a slow transformation, and in particular a progressively reaffirmed Bosniak national or Bosnian civic dimension in a local context where relations with Republika Srpska are still tense. The increased visibility of signs of national identity, like the flag of the Republic of Bosnia and Herzegovina that was used by the Bosniaks during the war 34 and is now placed on the coffins of all the victims buried at Potočari and on the truck that transports them to the cemetery, suggests that the tension between the religious and political dimensions of the funerals has not yet stabilized. The repetition of the collective funerals has the effect of producing a shared contemporary national history, in a very similar way to the processes analyzed in Spain by Zahira Aragüete-Toribio in the context of the reburial of victims of Francoism (Aragüete-Toribio, 2017). The annual repetition of the burials thus produces a form of ritualization, which obliges us to consider these ceremonies no longer as events but as stages or sequences in a larger diachronic process. The ritualization of the funerals allows the establishment of a twofold capitalization mechanism (of the number of victims buried, and of the experience of social cohesion) that counters the potentially destructive effects of the repetition of the ordeal of loss.

3.b. Burials, reburials: Never-ending funerals

The funerals are not the only burials to be repeated. Every year there are exhumations and reburials carried out at Potočari following the reassociation of belatedly identified body parts with the bodies they belong to. A victim of the Srebrenica genocide may, therefore, have his remains laid to earth on several different occasions as new fragments are found and identified. Nearly one hundred of these reburials took place between July 12 and 18, 2021, and another hundred between July 12 and 20, 2022, for example. This second series of repetitive practices, which themselves form part of a repeated mortuary process, represent a fairly rare and in many respects transgressive configuration. In the Bosniak case, the fragmentation of bodies is the result of an entirely intentional criminal act, but other accidental or criminal scenarios (for example, explosions or air crashes) have given rise to similar situations. As we conclude our study, these configurations allow us to raise the question of the border between the funerary and non-funerary management of human remains, and more broadly the question of the plasticity of symbolic or ritual arrangements for dealing with fragmented remains. In Bosnia, victims may be buried on multiple successive occasions, sometimes many years apart, depending on the family's wishes. 35 Some families choose to wait for the body to be complete, or almost complete, before holding a funeral, while others prefer to proceed to burial as soon as a single bone fragment has been identified, agreeing to be informed in the event that further fragments are identified and if necessary to perform one or several reburials. Other families, meanwhile, choose to bury their relative (immediately or after a period of gathering the remains) and indicate that they do not wish to be notified if further fragments are identified. Each option is legitimate from the point of view of the administrative and religious authorities, who leave the decision regarding the type of burial up to the families; none of the options is recommended or forbidden. In that respect, the treatment of fragmented remains in Bosnia is the object of neither prohibition nor prescription. Nevertheless, only the first burial is considered as a funeral from a religious point of view and receives ritual treatment. There is no ritual procedure stipulated for reburials following reassociations. And although the reassociations of human remains are carried out with good intentions and are described as being intended to enable families to move forward by burying their relatives in their entirety, the effect of repeated burials conducted outside any ritual framework may turn out to be less positive than hoped. As the son of one of the victims buried at Potočari explained on social media a few days before the reassociation of his father's remains, repeated burials can also constitute an emotional ordeal:

On Monday, July 12, 2021, the belatedly identified remains of my father A* will be reassociated. Srebrenica teems with martyrs who we see, and with martyrs that many people who live here do not want to see. 37 The repetition of the burial process undoubtedly has an impact on the mental health of the mourners. The deleterious effects of repetition of the separation stage are reflected particularly in a psychological fatigue that has been documented in the context of other mortality crises such as epidemics, like the AIDS epidemic in Tanzania studied by Liv [START_REF] Haram | We are tired of mourning": The economy of death and bereavement in a time of AIDS[END_REF]. In Bosnia, these effects are explicitly mentioned by certain victims' families who do not travel to attend each reburial, and who rather feel a need to distance themselves from this endless mourning. 38 In contrast to the collective funerals whose annual ritualization helps to unite the Bosniak national collective, extended families, and neighborhoods around a shared experience, the repeated ordeal of exhumation and reburial is experienced by each family in a unique, isolated way, especially because these steps in the treatment of the body are not accompanied by any particular ceremony. Individuals and families are left to cope alone with the necessary logical and symbolic arrangements involved in dealing with a fragmented body in several stages. The final question raised by the fragmentation of bodies, therefore, concerns the status given to human body parts and the preservation measures or treatment they should receive. In this field, studies carried out on the World Trade Center attacks all emphasize how families' expectations can conflict with the logic of scientific expertise [START_REF] Aronson | Who owns the dead? The science and politics of death at ground zero[END_REF][START_REF] Toom | Finding closure, continuing bonds, and codentification after the 9/11 attacks[END_REF]. In the Bosniak case, the cumulative social effects of concealment and commingling reveal even more clearly the tension between the political concerns of administrative management by the state and the moral concerns of ritual management by the religious institution when it comes to dealing with large numbers of fragmented human remains [START_REF] Wagner | The social complexities of commingled remains[END_REF]: the moral and religious order ultimately yields to the administrative and political order in the case of procedures of reassociation and reburial. In this context, the fragmentation of bodies serves to intensify that of a community that does not have any kind of system in place for mitigating the social effects of such fragmentation. The funerary practices at the site of Potočari thus reveal singular modalities of mourning that take place repeatedly over a long period of time (several decades) and simultaneously mobilize and construct the Bosniak collective. Ethnographic observation of these practices shows that their strictly moral and social stakes do not necessarily diminish with time. The materiality of the bodies and skeletonized remains, which are buried but can be unburied again to be reassembled, in some cases on several occasions, reveals a "stubborn will to mean, to signify" (Mbembe, 2003, p. 35). This capacity of human remains to serve as a focal point for necropolitical issues over the long term calls for a large-scale comparative study of societies that hold deferred funerals, sometimes after a very long time, in order to explore how various aspects of the available ritual systems are deactivated, activated, or modified in each case, and what these different arrangements reveal about these societies' relationship to death and the dead.

  Almost half of his bones will be buried again. I don't know if anyone else in history, apart from us Bosnians, has buried their relatives several times? […] Only we Bosnians, natives of this land of suffering, know what it is to bury one's father, grandfather, or uncle severaltimes. A third of the body the first time, another third the second time, and a final third that may never be found. How could I leave Bosnia and leave behind the dismembered bodies of my loved ones? How could I live somewhere where I would not exist? Without a soul, without a body, without being myself. In Bosnia lies our pain and our remedy. And when a new A* is born, God willing, 36 he will also stay in this country, and he will bear his grandfather's burden, as do Azem Mustafa, Assim Mehan, Semso, Perad, Niazia, Ramo… and so many others.

Five prayers punctuate the day in Islam; at Potočari, the funeral prayer is said after the midday prayer (Zuhr).

The Podrinje Identification Project was established for that purpose under the aegis of the ICMP in 1999. See the ICMP website: https://www.icmp.int/press-releases/podrinje-identification-project-to-mpi/

A small village in the municipality of Srebrenica; before the war it had around 1,000 inhabitants, and it now has around 800.

Seven field studies were carried out inJuly 2013July , 2015July , 2017July , 2018[START_REF]Translated from Bosnian to French with the help of Sabina Subasič. 38 Interview with the mother of a victim buried at Potočari[END_REF]July , 2021July , and 2022. They included a series of observations at places where bodies and human remains are dealt with (forensic anthropology laboratories, morgues, funeral homes, cemeteries) and a series of non-directive and unrecorded interviews with representatives of the administrative, judicial, and religious authorities (local, national, and international), victims' associations, and funeral parlors in Bosnia and Herzegovina.

The COVID-19 pandemic has opened up a burgeoning field of inquiry into these questions. See in particular the double issue of Human Remains and Violence published in fall 2021 and spring 2022.

On the way these funds were raised, and more broadly on the memory policies implemented at the beginning of the 2000s, see[START_REF] Duijzings | Commemorating Srebrenica: Histories of violence and the politics of memory in Eastern Bosnia[END_REF]. I would like to thank Galia Valtchinova for having drawn my attention to this text.

It took several years to clear the country of mines, and certain areas are still considered dangerous.

See the text of the law of 2007 altering the initial decision to create the site on October 25, 2000, in order to include a reference to the genocide committed at Srebrenica in accordance with the ICTY's designation: https://web.archive.org/web/20110606075959/http://www.ohr.int/print/?content_id=40028.

The Dayton Accords that put an end to the war in 1995 also ratified the composition of the state of Bosnia and Herzegovina out of two distinct components: the Federation of Bosnia and Herzegovina (Croat-Bosniak) and Republika Srpska (Serb).

Archaeothanatology has studied the complexities of funerary behavior for almost twenty years. See in particular[START_REF] Duday | Archaeothanatology and funeral archaeology: Application to the study of primary single burials[END_REF] and[START_REF] Schmitt | Defining collective burials: Three case studies[END_REF].

The last large-scale funeral ceremony was held in 2015 on the twentieth anniversary of the massacre. The twenty-fifth anniversary, which should have been commemorated in July 2020, was impacted by the restrictions on international travel and large gatherings during the COVID-19 pandemic.

Another grave, probably secondary or tertiary, containing the remains of around ten previously identified individuals associated with the Srebrenica genocide was found in June 2021. See the ICMP website: https://www.icmp.int/news/victims-found-in-mass-grave-in-kalinovik-linked-to-july-1995-srebrenica-genocide/.

The Bosniaks use the term "morgue," although it is actually a storage space, more like an "ossuary." In 2012, the chief prosecutor of Bosnia and Herzegovina began an inventory of the country's 12 mortuaries, which was completed in 2016. This inventory enabled the identification of more than one hundred additional victims, as well as around a thousand bone fragments belonging to previously buried victims. It also revealed that some identifications carried out prior to the use of DNA techniques were probably incorrect, initiating a process of exhumations for identity verification purposes that is still ongoing.

See the figures of the ICMP, updated in 2022: https://www.icmp.int/?resources=srebrenica-infographic-2017.

See Francisco[START_REF] Ferrándiz | Death on the move: Pantheons and reburials in Spanish Civil War exhumations[END_REF] analysis of the issues surrounding the reburial of victims of Francoism, particularly in cases where several bodies were found in a single vault.

In a context where victims were not individually identified.

Some women simultaneously lost their husband, one or more sons, some of whom were still teenagers, and one or more brothers; some men simultaneously lost their father, brothers, sons, and nephews.

See the full text here: https://www.ohchr.org/EN/ProfessionalInterest/Pages/CrimeOfGenocide.aspx.

See the works of psychiatrists and psychoanalysts who have worked with the descendants of victims of the Holocaust (Katz-Gilbert, 2020) and the Armenian genocide(Altounian & Altounian, 2009) for discussion of the mechanisms by which family memories are constructed.

Population displacement caused by the armed conflict was followed by the migration of many Bosniaks to countries in Europe and beyond, as shown by the license plates of the private cars in the Potočari parking lots.

This was the case even during the socialist period, when secular funerals were preferred. In 2021, there were still no crematoria in Bosnia and Herzegovina. Interview with the director of the Pokop funeral parlor, Sarajevo, in July 2017.

Interview with Husein Kavazović (current Reis-ul-Ulema of Bosnia), in July 2013.

The dominant school of jurisprudence in Bosnia is the Sunni Hanafi school.

Several other branches of Islam also use the Salat al-Gha'ib, a prayer explicitly intended for performing a funerary rite in the absence of the deceased, particularly for those who died far from home or in circumstances where it would not have been possible to perform an Islamic rite.

Interview with Munira Subašić (Association of the Mothers of Srebrenica), in July 2013 and 2018.

The construction of which remained incomplete in 2022.

As the genocide was carried out between April 7 and July 17, 1994, the date of April 7 was chosen to commemorate it.

Interview with the teams of the Podrinje Identification Project in July 2013.

In Bosnia, it is customary for boys, especially firstborn sons, to be given their paternal grandfather's name.


A Busic
email: ana.busic@inria.fr

A Cadas

J Doncel
email: josu.doncel@ehu.eus

J M Fourneau

Product form solution for the steady-state distribution of a Markov chain associated with a general matching model with self-loops

We extend the general matching graph model to deal with matching graph where every node has a self loop. Thus the states on the Markov chain are associated with the independent sets of the matching graph. We prove that under i.i.d. arrivals assumptions the steady-state distribution of the Markov chain has a product form solution.

Introduction

Intuitively, a Matching model describes the waiting times suffered by items before they match and disappear immediately once they are matched. It is an easy representation of multiple types Rendez-Vous between items. Following [START_REF] Mairesse | Stability of the stochastic matching model[END_REF] a Matching model is a triple (G, Φ, µ) formed by 1. a matching graph G = (V, E) which is an undirected graph whose vertices in V are classes of items and whose edges in E models the allowed matching of items. G is called the compatibility graph or the Matching graph. 2. Φ is a matching policy. It states the couple of items which is chosen upon arrival when an arriving item matches one or more types of item already waiting. 3. a distribution of probability µ to model the arrivals of items. Alternatively one can consider a collection of Poisson processes for a continuous-time model.

The Matching graph represents the classes of items and the compatibility among classes of items. Upon arrival, an item is queued if there are not compatible items present in the system. A matching occurs when two (or more) compatible items are present and it is performed according to the matching discipline. Typical matching disciplines are First Come First Match (an analog of First Come First Served in this approach) or Match the Longest Queue. Once they are matched, both items leave the system immediately (no need for service). Note that even if a Matching model has a queueing theory flavor, the items play the roles of both customers and servers. In some sense it has also some links with two well known stochastic models: networks with positive and negative customers and stochastic Petri nets. In a network with positive and negative customer proposed by Gelenbe in [START_REF] Gelenbe | Product-form queuing networks with negative and positive customers[END_REF] a negative customer can provoke the instantaneous deletion of a positive customer but it is never queued in the network. In [START_REF] Fourneau | G-networks with multiple classes of positive and negative customers[END_REF] the deletion depends on the type as in a matching. In a stochastic Petri net, tokens wait until they match but the places where the tokens match are usually associated with non negative delays [START_REF] Marco Ajmone Marsan | On petri nets with stochastic timing[END_REF]. Under some structural conditions, stochastic Petri nets may have a product form steady-state solution (see for instance [START_REF] Lazar | Markovian Petri net protocols with product form solution[END_REF][START_REF] Haddad | Product-form and stochastic petri nets: a structural approach[END_REF]). Even if it possible to associate a Petri net with a matching graph model, our results differ from [START_REF] Lazar | Markovian Petri net protocols with product form solution[END_REF][START_REF] Haddad | Product-form and stochastic petri nets: a structural approach[END_REF].

Despite its simple formulation, Matching models were not so simple to analyze. Assuming independent Poisson arrivals of items, and FCFM discipline the model is associated with an infinite Markov chain. Under these assumptions, a necessary condition of stability and a product form solution were proved in [START_REF] Moyal | A product form for the general stochastic matching model[END_REF] and [START_REF] Mairesse | Stability of the stochastic matching model[END_REF]. Moreover we recently established that there exists some performance paradox for FCFM matching models [START_REF] Cadas | Flexibility can hurt dynamic matching system performance[END_REF]. When one add new edges in the compatibility graph, one may expect that the expectation of the total number of customers decreases. In [START_REF] Cadas | Flexibility can hurt dynamic matching system performance[END_REF] we have given some examples which show that it is not always the case and we prove a sufficient condition for such a performance paradox to exist. Thus adding flexibility on the matching does not always result in a performance improvement.

The general matching model proposed in [START_REF] Moyal | A product form for the general stochastic matching model[END_REF] and [START_REF] Mairesse | Stability of the stochastic matching model[END_REF] was considering a general undirected matching graph G and it is assumed that the arrivals of items occur one at a time. It is important to avoid the confusion with Bipartite Matching Model (see for instance [START_REF] Bušić | Stability of the bipartite matching model[END_REF] and references therein) where the matching graph is bipartite and two items of distinct classes arrive at the same time. Bipartite Matching Models were motivated by analysis of the public housing [START_REF] Caldentey | FCFS infinite bipartite matching of servers and customers[END_REF]. In this model, households which apply for public housing and housings which become available both arrive over time. Once the matching is done, the housing is occupied for a long time period. Thus it is more convenient to represent them as an arrival streams of items rather than traditional servers in a queue. Another application studied in the literature was the kidney exchanges [11,[START_REF] Unver | Dynamic kidney exchange[END_REF]. The kidney exchange arises when a healthy person who wishes to donate a kidney is not compatible (blood types or tissue types) with the receiver. Two incompatible pairs (or maybe more) can form a cyclic exchange, so that each patient can receive a kidney from a compatible donor (see [START_REF] Ashlagi | Kidney exchange in dynamic sparse heterogenous pools[END_REF] and reference therein for a presentation of the problem and the modeling and algorithmic issues).

Here, we further extend the type of graph to represent the general matching. We assume that all the nodes in the matching graph have a self loop (see Fig. 1 for an example of such a graph) and this was clearly forbidden by the previous assumptions in [START_REF] Mairesse | Stability of the stochastic matching model[END_REF]. Note that the results obtained in [START_REF] Moyal | A product form for the general stochastic matching model[END_REF] and [START_REF] Mairesse | Stability of the stochastic matching model[END_REF] are not valid anymore because of some technical details in the proofs of product form. It is required in these papers that the matching graph does not contain self loops.

Without self loops the Markov chain is infinite and the system stability has to be studied taking into account the matching discipline.

For our new model, the stability problem is not an issue. As all the nodes in the compatibility graph have a self loop, the system can contain at most one item of each type. Therefore the Markov chain associated with the population is finite. If the chain is irreducible, it is therefore ergodic (see more details in section 2). Such a model may represent how we can organize fair competition between players with roughly the same ranking (for instance ELO points for chess). Such an application was not possible for the model described in [START_REF] Mairesse | Stability of the stochastic matching model[END_REF].

A more general result for matching with multigraph has recently been presented in [START_REF] Begeot | A general stochastic matching model on multigraphs[END_REF]. In that paper a multigraph is defined as a graph where the self loops are allowed. Therefore the proof they proposed is more general than our result because the self loops are not mandatory leading to potential infinite Markov chains. However our proof is based on simpler arguments (balance equations rather than reversibility of an extended process followed by an aggregation) and we hope it has its own value.

The technical part of the paper is as follows. We begin in the next section with the notations. Section 3 is devoted to a simple example built for a matching graph with 4 nodes. We obtained the steady-state distribution and checked that a multiplicative solution holds. In Section 4, we prove that this product form solution holds for every matching graph with self loops for all the nodes.

Notation and Assumptions

Let G = (V, F) be the matching graph. Nodes in V are also denoted as letters. An ordered list of letters is called a word. Assume that x is a letter from V, Γ (x) is the set of neighbors of x in G. As all nodes in V carry a loop, we have x ∈ Γ (x) for all node x.

Let m be an arbitrary word.

-(m|x) is the word obtained by appending letter x at the end of word m while (m + x) represents the set of words obtained by adding an x anywhere inside word m (even before word m). -|m| will denote the size of word m (i.e. the number of letters).

m(ψ) will be the letter of m at position ψ.

-Γ (m) = ∪ x∈m Γ (x).

-P re(m, ψ) is the prefix of m with size ψ. The prefix of size 0 is the empty word denoted as E or ∅. -Similarly Suf (m, ψ) is the suffix of m with size ψ.

-Finally, In(m, x, ψ) is the word of size |m|+1 built from m after the insertion of x at position ψ.

We consider a discrete time model. We assume i.i.d. arrivals of a letter at every time slot. We consider the First Come First Match policy or FCFM (sometimes denoted as the First Come First Served discipline in the literature). An arriving letter will be added at the end of the word if it does not match any letter in the current word. If the arriving letter matches one or several letters of the current word, both the oldest matching letter and the arriving letter vanish immediately.

α i will denote the probability of arrival of letter i while α 0 is the probability that there is no arrival. The state of the system at time t is a word. Under these assumptions, the process {m t }, with t ∈ N is a Discrete Time Markov chain. If α 0 > 0, this Markov chain is aperiodic. Furthermore it is clear that all the states of the Markov chain are independent sets of the matching graph. Indeed, if two letters are neighbors in the matching graph, they cannot be in a state of the Markov chain. Furthermore, as all the nodes of the matching graph have a self loop, it is not possible to have several occurrences of the same letter in a node of the chain. This last property does not hold for matching models without loops (see for instance [START_REF] Moyal | A product form for the general stochastic matching model[END_REF]). Therefore the Markov chain is finite (again this is not true in the model studied in [START_REF] Moyal | A product form for the general stochastic matching model[END_REF] and [START_REF] Mairesse | Stability of the stochastic matching model[END_REF]). Clearly if α i > 0 for all i, the chain is irreducible. Therefore we do not have to study the stability problem as in [START_REF] Mairesse | Stability of the stochastic matching model[END_REF]. The chain is always ergodic.

For all subset S of nodes of G, α S will denote the probability of arrival of letters in S:

α S = x∈S α x
To simplify the formulation of the steady-state distribution, we make the following remark:

Remark 1 Let m be an arbitrary state and x be an arbitrary letter, such that (m|x) is a valid state of the chain, we have

Γ (m + x) = Γ (m|x) = Γ (x|m).
Remember that Γ (x) is the set of neighbors of state x.

Path of length 4

We begin with a simple example. We consider as a matching graph, a path of length 4 (usually denoted as P 4) with loops on every node (see Fig. 1). We first build the Markov chain associated with this matching graph. The states are based on the independent sets of this graph. The states take into account the order of arrivals (remember that we consider FCFM discipline).

These states (and sets) contain up to two letters. A state is labelled by the letters which are included while E will denote the empty state. (x|y) represents the state containing letter x followed by letter y. The associated independent set is {x, y}. Therefore set {x, y} is associated with states (x|y) and (y|x).

This Markov chain has 11 states: E, 1, 2, 3, 4, (1|4), (4|1), (1|3), (3|1), (2|4), (4|2). The graph of the Markov chain obtained by the XBorne [START_REF] Fourneau | Xborne 2016: A brief introduction[END_REF] tool is depicted in Fig. 2. We do not add the transition probabilities to make the figure more understandable. For the same reason, we do not draw either the loop on every state associated with the null arrival event (with probability α 0 ). We will write the global balance equations for an arbitrary node of the Markov chain, taking into account the graph properties of the nodes. We consider the following partition of the states of the chain based on their properties.

1. State E. 2. States associated with maximal independent sets. For this matching graph, a maximal independent set contains two letters and they are denoted (x|y). 3. States which are neither the empty state nor a state associated with a maximal independent set. Here, these states are words with a single letter.

A similar partition will be used in the next section to make the proof in the general case.

To write the transition probability matrix, we consider that the states are in the following order E, 1, 2, 3, 4, (1|4), (4|1), (1|3), (3|1), (2|4), (4|2). The partition is added into the matrix presentation to make the block structure more visible.

P =                   α 0 α 1 α 2 α 3 α 4 0 0 0 0 0 α 1 + α 2 α 0 0 0 0 α 4 0 α 3 0 0 α 1 + α 2 + α 3 0 α 0 0 0 0 0 0 0 α 4 α 2 + α 3 + α 4 0 0 α 0 0 0 0 0 α 1 0 α 3 + α 4 0 0 0 α 0 0 α 1 0 0 0 α 2 0 α 3 + α 4 0 0 α 1 + α 2 α 0 0 0 0 0 0 α 3 + α 4 0 0 α 1 + α 2 0 α 0 0 0 0 0 α 3 + α 4 0 α 1 + α 2 0 0 0 α 0 0 0 0 α 2 + α 3 + α 4 0 α 1 0 0 0 0 α 0 0 0 0 α 4 0 α 1 + α 2 + α 3 0 0 0 0 α 0 0 0 α 3 + α 4 0 α 1 + α 2 0 0 0 0 0 α 0                  
Consider now the global balance equations. Let us begin with a maximal independent set. Writing a balance equation for a state (x|y) we get:

π(x|y) = π(x|y)α 0 + π(x)α y ,
from which we easily obtain for state (x|y):

π(x|y) = π(x) α y 1 -α 0 .
Now consider a state with one letter (x) . For instance consider State (3).

π(3)(1 -α 0 ) = π(1|3)(α 1 + α 2 ) + π(E)α 3 + π(3|1)α 1
Using the relations we already obtained, we substitute π(1|3) and π(3|1):

π(3)(1 -α 0 ) = π(1)(α 1 + α 2 ) α 3 1 -α 0 + π(E)α 3 + π(3)α 1 α 1 1 -α 0
One can check with some simple algebraic manipulations of these equations, that the solution we propose in Eq. 1 is the solution of the balance equations.

π(x|y) = π(x)α y /(1 -α 0 ) π(1) = π(E)α 1 /(α 1 + α 2 ) π(2) = π(E)α 2 /(α 1 + α 2 + α 3 ) π(3) = π(E)α 3 /(α 2 + α 3 + α 4 ) π(4) = π(E)α 4 /(α 3 + α 4 ) (1)
and π(E) is obtained by normalization.

Note that this solution is the solution proved by Moyal et al. in [START_REF] Moyal | A product form for the general stochastic matching model[END_REF] for a general matching model without loops on the matching graph:

π(w 1 |..|w k ) = C k i=1 α(w(i)) α(Γ (w(1), ..., w(i))) (2)
Thus one may expect that the multiplicative solution for the steady-state still holds under our assumptions.

Steady-State distribution

We now prove that the Markov chain associated with any matching graph which has a loop on all nodes has a steady state distribution which has a multiplicative form. To prove the theorem, we use the following characterization of the steadystate distribution.

Property 1 Consider a Markov chain with state space E and transition matrix P . Let π be a finite measure (i.e. ||π|| < ∞). If matrix Q defined by for all i and

j in E π(i)P [i, j] = π(j)Q[j, i]
is a stochastic matrix, then the steady state distribution of the Markov chain associated with P is obtained through normalization of π (i.e. π/||π||).

Remark 2 If P = Q, the chain is reversible.

Assuming that the multiplicative solution already known for matching graphs without loop still holds, one can formally obtain matrix Q and check if this matrix is stochastic. This is the key idea for the proof.

Remark 3 As for all x we have P [x, x] = α 0 , then, by construction, Q[x, x] = α 0 .

P 4 revisited

Consider again the example of the Markov chain associated with a P4 matching graph. We compute Q t (instead of Q to simplify the presentation) by computing the product with the probabilities π() given in Eq. 1. For instance Q[i, i] = α 0 for all i, and:

Q[(1|4), (1)] = π((1)) π((1|4)) P [(1), (1|4)] = 1 -α 0 α 4 α 4 = 1 -α 0
We denote β 0 = 1 -α 0 to simplify the matrix formulation.

Q t =                      α 0 α 1 + α 2 α 1 + α 2 + α 3 α 2 + α 3 + α 4 α 3 + α 4 0 0 0 0 0 0 α 1 α 0 0 0 0 β 0 0 β 0 0 0 0 α 2 0 α 0 0 0 0 0 0 0 β 0 0 α 3 0 0 α 0 0 0 0 0 β 0 0 0 α 4 0 0 0 α 0 0 β 0 0 0 0 β 0 0 α3(α3+α4) β0 0 0 α1(α3+α4) β0 α 0 0 0 0 0 0 0 α4(α1+α2) β0 0 0 α1(α1+α2) β0 0 α 0 0 0 0 0 0 α3(α3+α4) β0 0 α1(α2+α3+α4) β0 0 0 0 α 0 0 0 0 0 α3(α1+α2) β0 0 α 2 1 β0 0 0 0 0 α 0 0 0 0 0 α 2 4 β0 0 α2(α3+α4) β0 0 0 0 0 α 0 0 0 0 α4(α1+α2+α3) β0 0 α2(α1+α2) β0 0 0 0 0 0 α 0                     
Clearly Q t is column stochastic, thus Q is a stochastic matrix. And the result (i.e. Eq. 1) holds.

Main result

Let us now proceed with our main result on the steady-state distribution of the Markov chain associated with the matching graph. We first state the result. Before proceeding with the proof, we give some technical lemmas for the 3 types of nodes, as mentioned during the analysis of the P 4 example.

Theorem 1 Let G be a graph with a loop on each vertex. Let n be the number of nodes of G. Let α 0 , α 1 , ..., α n be a proper distribution of probability. The steadystate distribution of the Markov chain associated with matching graph G has a multiplicative form:

π(m) = C n ψ=1 α m(ψ) n ψ=1 α(Γ (P re(m, ψ))) .
where C is a normalization constant equal to π(E).

Corollary 1 Assume that state (m|x) exists, then we have:

π(m|x) = π(m) α x α(Γ (P re(m|x)))
.

The proof of the theorem is based on the analysis of matrix Q for the three types of node. First we need to study the graph properties of the Markov chain. We begin with the description of the edges. Proof: It is a clear consequence of the description of the effect of an arrival. Lemma 2. A state m of the Markov chain with a positive number of letters (i.e. |m| > 0) has only one transition to a state with size |m| -1 and the loop transition with probability α 0 . The other transitions lead to states with size |m|+1 Furthermore, if m is not a maximal independent set, for all the letters x such that (m|x) exists, we have P (m, m|x) = α x .

Proof: First, let m be a state with a positive number of letter. We can write m = (j|x), where j is a word (one can have j = E)). There exists a unique transition from j to (j|x) with rate α x due to the arrival of a letter x and the FCFM matching discipline. Indeed, the only possibility to increase the size of the state is the arrival of a letter which must be the last one due to the FCFM discipline.

Finally according to the previous lemma, all the remaining transitions leads to states with one more letter. And if (m|x) exists, the only transition from m to (m|x) is the arrival of letter x and it has probability α x . Lemma 3. [Global balance equation for a maximal independent set] Assume that the size of the maximal independent sets is at least 2. A state is a maximal independent set if it is a word (m|x) where m is also an independent set which does not contain an x. The only transition (except the loop) entering such a state comes from state m and has probability α x . There exist at least two outgoing transitions: the loop with rate α 0 and all the transitions provoked by arrivals which delete one letter in (m|x). Thus the global balance equation for such a state is:

(

1 -α 0 )π(m|x) = π(m)α x
Proof: Clearly, it exits a transition with probability α x going from m to (m|x) if x does not match with a letter of m. According to Lemma 1, the states which precede (m|x) have size |m|, |m| + 1 (due to the loop) or |m| + 2. Clearly, the only transition entering (m|x) from a state with a smaller number of letters is the transition going from m due to the FCFM discipline. Let us prove now by contradiction that there does not exist any transition from a state (m |x) to a state (m|x) with |m | = |m| + 2. Assume that it is possible, then state (m |x) has a size larger than the size of (m|x) and as an arrival provokes a transition from (m |x) to (m|x), all the letters of (m|x) are also in (m |x). Thus (m|x) cannot be a maximal independent set as it is contained in a larger independent set.

Finally, state (m|x) has an output degree which is at least 3. Indeed there exists a transition from (m|x) to m since the arrival of x deletes the last letter x in the word (this is the effect of the loop on x in the matching graph). Furthermore, the arrivals of all the letters in m delete a letter of m and to not delete x. Therefore they provoke the transition to (m |x) with |m | = |m| -1. Finally we add the loop and the output degree is at least 3. The global balance equation is now trivial.

Property 2 Let m be a state of the chain and x an arbitrary letter which is in V \ Γ (m). We define the following subset of states:

Γ -x (m) = {p |p| = 1 + |m|, and p = m + x}
Intuitively, the arrival of letter x in state p provokes a transition from p to m because letter x is deleted. Such a subset is only defined when x is in V \ Γ (m). Indeed, if the word has size |m| + 1, it means that letter x does not interact with word m. Furthermore the cardinal of Γ -x (m) is |m| + 1. Indeed, one can add letter x anywhere inside word m. Lemma 4. Let j be a state (i.e. a word) of size n. For all letter x in V \ Γ (j), we have:

l∈Γ -x (j) Q(j, l) = α x .
As the proof of this lemma is technical, it is postponed after the proof of the main theorem.

Proof of the Theorem Let us now proceed with the proof the main theorem. Remember we partition the states of the Markov chain into three subsets:

1. Empty state E 2. Maximal Independent states 3. Other states We make the proof for the three types of states:

-Empty State E: for all letter x we have by construction:

Q(E, x) = P (x, E) π(x) π(E)
From the definition of π in Theorem 1, π(x) = π(E)αx α(Γ (x)) and P (x, E) = α(Γ (x)). After simplification,

Q(E, x) = α x Thus x∈V Q(E, x) = x∈V α x = 1 -α 0 and Q(j, j) = α 0 .
Therefore the row sum is equal to 1 for state E.

-Maximal Independent set: Let p be a maximal independent set. According to Lemma 3, there exists now only one entry in the column associated with p in P . Thus there exists only one entry in the row associated with p in Q.

Let m be this predecessor of p and let x the letter appended to m to obtain p (i.e. p = (m|x)). By construction:

Q(p, m) = P (m, p)π(m)/π(p).
From Lemma 3, P (m, p) = α x , and (1-α 0 )π(p) = α x π(m). Thus, Q(p, m) = 1 -α 0 . Furthermore Q(p, p) = α 0 . And all other entries of matrix Q for row p are 0. Therefore the row sum is equal to 1 for a state which is a maximal Independent set. -Other states: we know due to Lemma 2 that, if it is not a maximal independent set, a state j of size n has one predecessor with size n -1,several predecessors of size n + 1 and itself (with probability α 0 ). We will consider these three sets of predecessors in a separate way.

• Let m the predecessor of size n -1. Let x the letter which has been appended (i.e. j = m|x). From Lemma 2 we have: P (m, j) = α x . From Corollary 1, we have

π(j) = π(m|x) = π(m)α x α(Γ (m|x)) Thus Q(j, m) = α(Γ (m|x)) = α(Γ (j))
• We now consider the predecessors of j which have size n + 1. Let H(j) be this set. We will partition this set of states according to the letter x which provokes the transition. Thus,

H(j) = ∪ x∈V \Γ (j) Γ -x (j).
As subsets Γ -x (j) do not intersect, this is a true partition. Thus,

l ∈H(j) Q(j, l) = x∈V \Γ (j) l∈Γ -x (j) Q(j, l)
Technical Lemma 4 states that: l∈Γ -x (j) Q(j, l) = α x . Thus,

l ∈H(j) Q(j, l) = x∈V \Γ (j) α x .
Combining both results, we get:

l Q(j, l) = α 0 +α(Γ (j))+ l∈H(j) Q(j, l) = α 0 +α(Γ (j))+ x∈V \Γ (j) α x = x∈V α x = 1
And the proof is complete.

Proof of the technical lemma

We want to prove that for an arbitrary word j of size n and for all letter x in V \ Γ (j), we have:

l∈Γ -x (j) Q(j, l) = α x .
Let us first explain the terms involved in the summation and the way we combine them. By definition we have: Q(j, l) = P (l, j)π(l)/π(j). and by assumptions, the solution is:

π(l) = π(E) n+1 ψ=1 α l(ψ) n+1 ψ=1 α(Γ (P re(l, ψ)))
.

As l ∈ Γ -x (j), one can write l = j + x and Let us now study the denominator. Consider an arbitrary word (j + x). There exists an index ψ (which can be 0) such that this word is (P re(j, ψ)|x|Suf (j, n-ψ)). This formulation allows to obtain the transition probability and the steadystate distribution for all the values of ψ and obtain an induction on partial sums on ψ

Let us begin the induction with ψ = 0 associated with Q(j, (x|j)). Clearly,

P ((x|j), j) = α(Γ (x)).
Now the denominator of π(x|j) is according to the assumptions equal to We now have to compute the term associated with ψ = 1 (remember that we want to make an induction on partial sums). The state of the chain is (P re(j, 1)|x|Suf (j, n -1)). The transition from this state to j is provoked by the arrival of letters which match with x but which are not matched with the first letter of this word due to the FCFM matching discipline. More formally: P ((P re(j, 1)|x|Suf (j, n -1)), j) = α(Γ (x) \ Γ (P re(j, 1))), and π((P re(j, 1)|x|Suf (j, n-1))) = π(E)α x n ψ=1 α j(ψ) α(Γ (P re(j, 1))) n ψ=1 α(Γ ((x|P re(j, ψ)))) , and finally after substation and cancellation, we get:

Q(j, (P re(j, 1)|x| Suf (j, n -1))) = α x α(Γ (x) \ Γ (P re(j, 1))) n ψ=1 α(Γ (P re(j,ψ))) α(Γ (P re(j,1))) n ψ=1 α(Γ ((x|P re(j,ψ)))) .
We now compute the sum of these first two elements. After factorization:

Q(j, (x|j)) + Q(j, ( P re(j, 1)|x|Suf (j, n -1))) = α x n ψ=1 α(Γ (P re(l,ψ))) n ψ=1 α(Γ ((x|P re(j,ψ))) α(Γ (x))
α(Γ ((x|P re(j,0)))) + α(Γ (x)\Γ (P re(j,1))) α(Γ (P re(j,1)))

.

As (P re(j, 0)) is the empty word, we have Γ ((x|P re(j, 0)))) = Γ (x), and the first part of the summation simplifies. Q(j, (x|j)) + Q(j, (P re(j, 1)|x| Suf (j, n -1))) = α x n ψ=1 α(Γ (P re(j,ψ))) n ψ=1 α(Γ ((x|P re(j,ψ))) 1 + α(Γ (x)\Γ (P re(j,1))) α(Γ (P re(j,1)))

.

Thus:

Q(j, (x|j)) + Q(j, (P re(j, 1)|x| Suf (j, n -1))) = α x n ψ=1 α(Γ (P re(l,ψ))) n ψ=1 α(Γ ((x|P re(j,ψ)))
α(Γ (P re(j,1)))+α(Γ (x)\Γ (P re(j,1)))

α(Γ (P re(j,1)))

.

For all words a and b, we have:

α(Γ (a) \ Γ (b)) + α(Γ (b)) = α(Γ (a|b)).
Using a = x and b = P re(j, 1), after substitution we get:

Q(j, (x|j)) + Q(j, (P re(j, 1)|x| Suf (j, n -1))) = α x n ψ=1 α(Γ (P re(j,ψ))) n ψ=1 α(Γ ((x|P re(j,ψ))) α(Γ ((x|P re(j,1))))
α(Γ (P re(j,1))) .

α(Γ ((x|P r(j, 1)))) is in the numerator and the denominator. We cancel this term and we get: Let us now consider the induction on the number of terms we add. Assume that we consider them and accumulate them according to the index of x in the word. Assume that we have proved that for all ψ between 0 and ν -1, we have stated that the summation of the first ν terms is equal to: .

Q(j, (
We now consider term with index ν. We have to compute Q(j, (P re(j, ν)|x|Suf (j, nν))) and add it to the previous partial sum. As usual due to the FCFM matching discipline P ((P re(j, ν)|x|Suf (j, n -ν)), j) = α(Γ (x) \ Γ (P re(j, ν))),

By assumption on the multiplicative solution for the steady-state: Thus, Q(j, (P re(j, ν)|x| Suf (j, n -ν))) = α(Γ (x) \ Γ (P r(j, ν)))

αx n ψ=1 α(Γ (P re(j,ψ))) ν ψ=1 α(Γ (P re(j,ψ))) n ψ=ν α(Γ ((x|P re(j,ψ)))) .

After summation with the previous partial sum (given by the induction assumption) and factorization, we get that the new partial summation is equal to: 1 + α(Γ (x) \ Γ (P re(j, ν))) α(Γ (P re(j, ν))) .

We use a similar argument to simplify:

1 + α(Γ (x) -Γ (P re(j, ν))) α(Γ (P re(j, ν))) = α(Γ ((x|P re(j, ν)))) α(Γ (P re(j, ν))) .

After substitution the sum is equal to: Thus the induction holds. Now let us compute the sum of all the elements for ψ between 0 and n. According to the induction assumptions which is now proved, this sum is equal to α x n ψ=1 α(Γ (P re(j, ψ))) n ψ=1 α(Γ (P re(j, ψ)))

= α x , and the proof of the Lemma is complete.

Conclusions and Remarks

This paper is a sequel of [START_REF] Cadas | Flexibility can hurt dynamic matching system performance[END_REF] where we proved that adding a new edge in a matching graph without loops may lead to a performance paradox: the expectation of the total number of customers increase after the addition of the edge. See also [START_REF] Cadas | Flexibility can hurt dynamic matching system performance, extended version on arxiv[END_REF] for an extended version of this paper.

Our aim was to prove or disprove the existence of the same paradox for matching graph with loops. The first step was to prove that the steady state solution has a multiplicative form. However even with this result, the existence of a paradox similar to the one shown in [START_REF] Cadas | Flexibility can hurt dynamic matching system performance[END_REF] is still an open problem as all the examples studied so far do not exhibit the same paradox we found in [START_REF] Cadas | Flexibility can hurt dynamic matching system performance[END_REF]. Note however that the chains we obtain with this new model are all finite while the chains studied in [START_REF] Cadas | Flexibility can hurt dynamic matching system performance, extended version on arxiv[END_REF] are infinite.

Fig. 1 .

 1 Fig. 1. Matching graph: P 4.

Fig. 2 .

 2 Fig. 2. Graph of the Markov chain associated with Matching graph P 4.

Lemma 1 .

 1 A state m of the Markov chain a positive number of letters (i.e. |m| > 0) only has transitions to itself (because of null arrival event) and to states with size |m| + 1, or |m| -1.

  n+1 ψ=1 α(Γ (P re((x|j), ψ))). It is easy to remark that:n+1 ψ=1 α(Γ (P re((x|j), ψ))) = n ψ=0 α(Γ (x|P re(j, ψ)))This remark is used to simplify the factorization.π(x|j) = π(E)α x n ψ=1 α j(ψ) n ψ=0 α(Γ ((x|P re(j, ψ)))) and, π(j) = π(E) n ψ=1 α j(ψ) n ψ=1 α(Γ (P re(j, ψ)))and finally after cancellation of terms:Q(j, (x|j)) = α x α(Γ (x))n ψ=1 α(Γ (P re(j, ψ))) n ψ=0 α(Γ ((x|P re(j, ψ))))

  x|j)) + Q(j, (P re(j, 1)|x| Suf (j, n -1))) = α x n ψ=1 α(Γ (P re(j,ψ))) α(Γ (P re(j,1))) n ψ=2 α(Γ ((x|P re(j,ψ))) .

  Γ (P re(j, ψ))) ν-1 ψ=1 α(Γ (P re(j, ψ))) n ψ=ν α(Γ ((x|P re(j, ψ)))

  π((P re(j, ν)|x| Suf (j, n -ν))) = π(E) αx n ψ=1 α j(ψ) ν ψ=1 α(Γ (P re(j,ψ))) n ψ=ν α(Γ ((x|P re(j,ψ)))) .

  Γ (P re(j, ψ))) n ψ=ν α(Γ ((x|P re(j, ψ)))

  αx n ψ=1 α(Γ (P re(j,ψ)))ν-1 ψ=1 α(Γ (P re(j,ψ))) n ψ=ν α(Γ ((x|P re(j,ψ)))) α(Γ ((x|P re(j,ν)))) α(Γ (P re(j,ν))) = αx n ψ=1 α(Γ (P re(j,ψ)))ν ψ=1 α(Γ (P re(j,ψ))) n ψ=ν+1 α(Γ ((x|P re(j,ψ)))) .