Biru Zhou
email: biru.zhou@umontreal.ca

Ph.D Stéphane Guay
email: stephane.guay@umontreal.ca

Alain Marchand

Running head: COPING WITH WORKPLACE VIOLENCE 1 I See so I Feel: Coping with Workplace Violence Among Victims and Witnesses

Keywords: Workplace aggression, Psychological consequences, Work functioning, Vicarious experiences, Paraformal helping

BACKGROUND: Workplace violence is a serious concern for workers' mental health and well-being in high risk work sectors. OBJECTIVE: This study examined victims' and witnesses' experiences after exposure to workplace violence, and the types of help they used to cope with the violent event. METHODS: Workers (n = 211) from five different work sectors participated in our study. Multiple mediation analysis was used to investigate the indirect effects through psychological and work consequences on victims' versus witnesses' differential likelihood of using formal, paraformal and informal helping. RESULTS: Results showed that workplace violence has detrimental effects on both victims and witnesses. Direct victims were more negatively affected psychologically and at work than witnesses. The indirect effect through psychological difficulty after experiencing workplace violence was significant in predicting formal helping. The indirect effect through reduced work functioning in predicting paraformal helping was also significant. No significant indirect effect was found in predicting informal helping. CONCLUSIONS: Both victims and witnesses used multiple types of helping to cope with the violent event. This study has practical implications on management and clinical practices for better organizations of resources in helping victims and witnesses to cope with workplace violence.

I See so I Feel: Coping with Workplace Violence Among Victims and Witnesses Workplace violence is often portrayed misleadingly by the media as angry employees (or formal employees) lashing out and shooting supervisors and co-workers [START_REF] Denenberg | Workplace violence and the media: The myth of the disgruntled employee[END_REF]. However, most incidents of workplace violence are not "inside jobs" [START_REF] Neuman | Workplace violence and aggression: when you do not want your company on the news[END_REF]. Rather, the most prevalent perpetrators of workplace violence are from outside of the organization, such as clients/visitors, patients or any other persons for whom an organization provides services [START_REF] Barling | Predicting workplace aggression and violence[END_REF][START_REF]Nonfatal occupational injuries and illnesses requiring days away from work 2014[END_REF][START_REF] Hills | A review of research on the prevalence, antecedents, consequences and prevention of workplace aggression in clinical medical practice[END_REF]. Previous research shows that more than half of violence directed at employees at work are committed by outsiders [START_REF] Harrell | Workplace violence 1993-2009[END_REF][START_REF] Piquero | Assessing research on workplace violence, 2000-2012[END_REF]. The number of witnesses to workplace violence may far outnumber direct victims. From a clinical perspective, witnessing a traumatic event is considered a potential trigger for developing symptoms of post-traumatic-stress-disorder (PTSD) [START_REF]Diagnostic and statistical manual of mental disorders[END_REF]. Despite the alarming psychological effects of workplace violence [START_REF] Lanctôt | The aftermath of workplace violence among healthcare workers: a systematic literature review of the consequences[END_REF], little research has examined witnesses' experiences after exposure to workplace violence with a few exceptions [START_REF] Dupré | Harm to those who serve: effects of direct and vicarious customer -initiated workplace aggression[END_REF][START_REF] Eriksson | Trauma exposure and PTSD symptoms in international relief and development personnel[END_REF][START_REF] Vartia | Consequences of workplace bullying with respect to the well-being of its targets and the observers of bullying[END_REF]. The goal of this study was to investigate outsider-initiated workplace violence and its psychological and work consequences among not only victims but also witnesses. We will also examine different types of help used by direct victims and witnesses after the violent event.

Workplace aggression and workplace violence are often interchangeable in the literature.

In order to fully inform our research on the current topic, we will draw on empirical evidence and conceptual models from both workplace aggression and workplace violence research to facilitate our understanding of this phenomena. For the purpose of this study, we defined workplace violence "on the basis of type of offence (assaults or threats); what the victim was doing at the time of the incident (at work or working); and the relationship between victim and offender (domestic violence is excluded)" to guide our research [13, p. 3]. The type of offence in this study included physical violence and death threats.

Consequences of Workplace Violence

Workplace violence has multiple consequences on both organizations and individuals [START_REF] Lanctôt | The aftermath of workplace violence among healthcare workers: a systematic literature review of the consequences[END_REF]. Barling et al. argue that workplace aggression is a stressor at the workplace that relates to a range of physical, psychological and behavioural strains [START_REF] Barling | Handbook of work stress[END_REF]. For instance, incidents of nonfatal assaults at the workplace can result in various types of physical injuries, ranging from bruising to concussions. [START_REF] Lanctôt | The aftermath of workplace violence among healthcare workers: a systematic literature review of the consequences[END_REF]13]. Psychological consequences include anger, fear, anxiety, stress, frustration [START_REF] Gerberich | An epidemiological study of the magnitude and consequences of work related violence: the Minnesota Nurses' Study[END_REF][START_REF] Gillespie | Violence against healthcare workers in a pediatric emergency department[END_REF][START_REF] Wilson | Violence against teachers: prevalence and consequences[END_REF], and symptoms of post-traumatic-stress-disorder [START_REF] Gates | Violence against nurses and its impact on stress and productivity[END_REF][START_REF] Lilly | Gender and PTSD: what can we learn from female police officers?[END_REF][START_REF] Rojas-Flores | Exposure to violence, posttraumatic stress, and burnout among teachers in El Salvador: testing a mediational model[END_REF].

Moreover, reduced work functioning, as a behavioural consequence related to workplace violence [START_REF] Gates | Violence against nurses and its impact on stress and productivity[END_REF][START_REF] Banyard | The impact of interpersonal violence in adulthood on women's job satisfaction and productivity: the mediating roles of mental and physical health[END_REF], is viewed as employees' productivity and performance at work given a certain state of health [START_REF] Lagerveld | Factors associated with work participation and work functioning in depressed workers: a systematic review[END_REF]. It can be quantified by self-reported loss of productivity and experiences of limitations at work [START_REF] Lagerveld | Factors associated with work participation and work functioning in depressed workers: a systematic review[END_REF]. Both quantitative [START_REF] Gates | Violence against nurses and its impact on stress and productivity[END_REF] and qualitative [START_REF] Gillespie | Violence against healthcare workers in a pediatric emergency department[END_REF] studies show that workplace violence initiated by patients/visitors is related to increased psychological difficulty and decreased work productivity.

Witnessing Workplace Violence

Workers not only experience workplace violence directly as victims, but they may also experience it indirectly by witnessing or hearing about incidents of workplace violence.

Witnesses of violent acts at work might experience similar psychological and behavioural outcomes as direct victims [START_REF] Ferguson | I know what you did: the effects of interpersonal deviance on bystanders[END_REF][START_REF] Hockley | The impact of workplace violence on third party victims: a mental health perspective[END_REF][START_REF] Namie | Active and passive accomplices: the communal character of workplace bullying[END_REF]. Our research on witnesses' experiences after exposure to workplace violence is guided by Figley's [START_REF] Figley | Compassion fatigue: coping with secondary traumatic stress disorder in those who treat the traumatized[END_REF][START_REF] Figley | Burnout as systemic traumatic stress: a model for helping traumatized family members[END_REF] Trauma Transmission Model. This theoretical model was initially developed for secondary traumatization for caregivers, such as psychotherapists and family members of trauma victims. In this study, we apply this model to examine witnesses' experiences after being exposed to workplace violence. According to the model, witnesses may be "swept up" by the emotions of the victims, which then lead to developing similar psychological difficulty as the victims. This process is operated partially through the witness's identification with the victim in terms of interpersonal relationships, as well as through witness's empathetic ability to notice and feel the suffering of others via emotional contagion [START_REF] Figley | Burnout as systemic traumatic stress: a model for helping traumatized family members[END_REF][START_REF] Figley | Compassion fatigue: psychotherapists' chronic lack of self care[END_REF]. Figley argued that people around the victim, such as witnesses, might indeed experience similar emotional responses as the victim due to their effort in generating an understanding of the victim and the traumatic event.

A study by Eriksson et al. examined 195 returned humanitarian staff workers who had been directly and/or indirectly exposed to life-threatening events during their deployment [START_REF] Eriksson | Trauma exposure and PTSD symptoms in international relief and development personnel[END_REF]. Results showed that both personal exposure and vicarious exposure to life-threatening events positively predicted PTSD severity six months after their return. In fact, about 10% of the participants met the full diagnostic criteria for PTSD, 19% reported experiencing partial PTSD, and more than half experienced moderate problems for at least one PTSD symptom cluster. Similarly, Dupré et al. used structural equation modeling to examine the negative effects of direct vs. vicarious experiences of workplace aggression [START_REF] Dupré | Harm to those who serve: effects of direct and vicarious customer -initiated workplace aggression[END_REF]. Both victims and witnesses experienced mental and physical health issues, as well as turnover intentions after exposure, with stronger effects for victims than for witnesses. In other words, both direct and vicarious experiences of workplace violence exert similar patterns of negative effects on workers' mental health and work outcomes, with direct victims experiencing stronger effects than witnesses [START_REF] Vartia | Consequences of workplace bullying with respect to the well-being of its targets and the observers of bullying[END_REF].

Coping With Workplace Violence

When facing stressful or traumatic events, humans are motivated to use different strategies to cope with the threatening situations that are impinged upon them. Coping is generally referred to behaviours or responses that protect individuals from being harmed by problematic life events [START_REF] Pearlin | The structure of coping[END_REF]. It can also serve to prevent, avoid and/or control emotional distress during times of duress. Barker et al. categorize three types of help -formal, informal and paraformal -people use to cope with difficult situations [START_REF] Barker | Psychotherapy and social support[END_REF]. Formal helping refers to the use of professional help from psychiatric services, psychological therapy, counseling and so on; informal helping on the other hand refers to help and support from ordinary people in everyday settings, such as support from friends, families and colleagues; paraformal helping refers to the use of help and support from individuals who have some specialized training or experiences in psychological helping, such as clergymen and family doctors [START_REF] Barker | Psychotherapy and social support[END_REF]. Paraformal helping is a type of helping that lies between the formal and informal helping continuum. Barker et al. posit that even though formal and informal helping are the two extremes on the helping continuum with paraformal helping at the middle, these three types of helping are not mutually exclusive [START_REF] Barker | Psychotherapy and social support[END_REF]. Workers exposed to workplace violence are likely to use multiple types of helping to deal with their trauma [START_REF] Aquino | Workplace victimization: aggression from the target's perspective[END_REF][START_REF] Rospenda | Patterns of workplace harassment, gender, and use of services: an update[END_REF]. Historically, most research on coping had focused on formal and informal helping [START_REF] Ang | Effects of gender and sex role orientation on help-seeking attitudes[END_REF][START_REF] Barbee | Effects of gender role expectations on the social support process[END_REF][START_REF] Day | Gender differences in perceptions of stressors and utilization of social support among university students[END_REF][START_REF] Galdas | Men and health help-seeking behaviour: literature review[END_REF], and neglected other types of helping, such as paraformal helping [START_REF] Wills | Social support and interpersonal relationships[END_REF][START_REF] Wills | Interpersonal analysis of the help-seeking process[END_REF]. The oversight of paraformal helping in the literature may indeed fail to account for personal resources and meaningful ways of coping used by victims and witnesses. To fill in this gap, the current study examine the three types of helping used by victims and witnesses after encountering workplace violence.

Model and Hypothesis

According to previous research [START_REF] Dupré | Harm to those who serve: effects of direct and vicarious customer -initiated workplace aggression[END_REF][START_REF] Eriksson | Trauma exposure and PTSD symptoms in international relief and development personnel[END_REF], it was expected that both victims and witnesses would experience psychological difficulty and reduced work functioning after exposure, with victims showing more psychological difficulty and higher levels of reduced work functioning than witnesses (path a 1 and a 2 in Figure 1(a)). Both psychological difficulty and reduced work functioning were then hypothesized to positively predict the use of formal, informal and paraformal helping (path b 1 and b 2 in Figure 1(a)) [START_REF] Barker | Psychotherapy and social support[END_REF][START_REF] Aquino | Workplace victimization: aggression from the target's perspective[END_REF]. The total effect of exposure status (victims versus witnesses) on the usage of different types of helping is denoted as path c in Figure 1(a). The direct effect between exposure status and the usage of different types of helping after controlling for all other effects in the model is denoted as c .

Furthermore, the mean differences on psychological difficulty and reduced work functioning experienced by victims versus witnesses were expected to be positively related to the odds of using formal, informal and paraformal helping. In other words, the indirect effects through psychological difficulty and reduced work functioning would be significant to explain the differential likelihood of using different types of helping between victims and witnesses.

Insert [Figure 1]

Materials and Method

Data

This cross-sectional self-repot questionnaire based study was part of a larger project which examined workplace aggression and violence among 2889 French-speaking workers from five different work sectors (i.e., police officers, administrative workers in civil services, bus drivers, healthcare staffs and social workers) in the province of Québec in Canada.

These sectors are within the justice, healthcare and social services industries which are prone to workplace violence, with police officers having the highest risk [START_REF] Piquero | Assessing research on workplace violence, 2000-2012[END_REF]. Participants were contacted by e-mail or on-site to complete an online questionnaire between January 2011 and October 2012 and their participation was completely voluntary. Participants were informed that they could withdraw from the study at any time without any penalty or harm. The survey was anonymous and the information they provided was confidential. This study was approved by the ethics committee of the Institut universitaire en santé mentale de Montréal.

Since the primary interest of this paper was on outsider-initiated workplace violence, only a subset of participants who indicated they had encountered outsider-initiated workplace violence in the past 12 months were included in our dataset for analysis (N = 326). About 14.9% responses were missing in our dataset. Little's missing completely at random test1 , using age, sex and exposure status (victims vs. witnesses) as covariates, was not significant, which indicated that the data were missing completely at random, χ 2 (152) = 162.86, p = .26. After deleting cases with missing values, the available worker sample size was n = 211 (female = 118).

Materials

The online questionnaire was organized in three parts. The first part included socio-demographic questions regarding participants' age, sex ("male" was coded as 0, "female" was coded as 1), marital status ("single" was coded as 0, "not single" was coded as 1), annual personal income, and work shifts ("night shift or over night shift" were coded as 0, "day shift" was coded as 1). The risk of workplace violence according to work sector was also dichotomized as police officers (coded as 1) and the other four sectors (coded as 0). The second part of the questionnaire contained questions regarding victimization and witnessing experiences of six forms of serious workplace violence -physical violence, robbery, armed robbery, sexual touching, sexual assaults and death threats. Participants were asked to indicate the frequency of each of the six violent acts they encountered as a victim or a witness in the past 12 months. They were then asked to indicate the one most disturbing workplace violence they have encountered either as a victim or as a witness in the past 12 months. The third part of the survey included questions regarding the consequences (i.e., psychological and work) of being exposed to the one most disturbing workplace violent event and what kind of strategies (subdivided as formal, informal and paraformal) participants used to regain their normal levels of functioning after exposure.

Exposure to workplace violence. Exposure to workplace violence was assessed in three folds: general exposure, one most disturbing event, and the elapsed time from the most disturbing event. According to previous research, general exposure to workplace violence on a daily basis could have a negative effect on individual workers' psychological well-being [START_REF] Brewin | Meta-analysis of risk factors for posttraumatic stress disorder in trauma-exposed adults[END_REF].

In order to avoid recall bias, we have asked participants to report one most disturbing workplace violent event they have encountered in the past 12 months. General exposure to workplace violence was measured by summing the frequencies of the six forms of workplace violence as a victim or a witness in the past 12 months. The exposure status for the one most disturbing event had two categories -victim (coded as 1) and witness (coded as 0). The elapsed time of the most disturbing workplace violent event was also recorded by months in the second part of the survey.

Consequences of workplace violence. Psychological difficulty was measured by summing nine symptoms ("presence of symptom" was coded as 1, "absence of symptom" was coded as 0): flashbacks, nightmares, avoidance, loss of interest in important or interesting activities, sleeping problems, hypervigilance, concentration problems, irritability and guilt. More symptoms being present indicated higher levels of psychological difficulty after exposure to workplace violence. Crobach's α was .88 for the full sample for this scale (Crobach's α was .74 for the victims and .73 for the witnesses).

Work related consequence (i.e.,reduced work functioning) was measured by asking participants to indicate their percentage of reduction in work functioning from 0% to 100%, after taking into account of their usual levels of work functioning prior to the occurrence of the most disturbing workplace violent event. Higher values indicated more reductions in work functioning. "Presence of physical injuries" was coded as 1 and "absence of physical injuries" was coded as 0.

Types of Helping. Three types of helping were measured according to the categories identified in Barker et al.'s paper [START_REF] Barker | Psychotherapy and social support[END_REF]. The use of formal helping was quantified as a dichotomous (Yes/No) variable by participants' indications of consulting a psychologist or psychiatrist, and/or using services from employee assistance programs (EPA). The use of informal helping was quantified as a dichotomous (Yes/No) variable by participants' indications of talking about the most violent event to family, friends and/or colleagues.

Similarly, the use of paraformal helping was also quantified as a dichotomous (Yes/No) variable by participants' indications of consulting a general physician and/or seeking services from the union. "Yes" was coded as 1 and "No" was coded as 0 for using different types of helping.

Analysis

In order to test whether the mean differences between victims and witnesses on psychological difficulty and reduced work functioning were related to the likelihood of using different types of helping, three sets of multiple mediation models were used following the procedures recommended by Hayes et al. [START_REF] Hayes | Statistical mediation analysis with a multicategorical independent variable[END_REF]. Each type of helping -formal, informal and paraformal -was analyzed separately. The indirect effect in a multiple mediation model with a dichotomous independent variable is interpreted as the amount by which two groups that differ by one unit on the dichotomous independent variable are estimated to differ on the outcome variable as a result of the effect of the independent variable on the mediator which in turn affects the outcome variable. To put it differently, an indirect effect represents the mean differences between the two groups on the outcome variable resulting from the indirect pathway [START_REF] Hayes | Statistical mediation analysis with a multicategorical independent variable[END_REF]. In our case, we set out to test the effect of workplace violence exposure status (victim versus witness) on the likelihood of using three different types of helping through the influences of psychological difficulty and reduced work functioning, controlling for socio-demographic variables (sex, age, marital status, income, working as day shift or not, time elapsed since the event, and work sector), physical injuries and general exposure to workplace violence in the past 12 months. Specifically, workplace violence exposure status (victim versus witness) was the dichotomous independent variable and the outcome variables were formal, informal and paraformal helping. The two mediators -psychological difficulty and reduced work functioning -were entered at the same time into the multiple mediation analysis for each type of helping (see Figure 1(a)). Since the two mediators were continuous variables and the three outcome variables were dichotomous variables, the estimates for a paths in each multiple mediation model were OLS regression based coefficients whereas the b paths were logistic regression based coefficients. The p-value was set at .01 in this paper to avoid the inflation of Type I error due to repeated testing of multiple mediation models. All analyses were conducted using SPSS v20 unless otherwise specified.

The multiple mediation analyses were conducted using the open source macro PROCESS in SPSS2 [START_REF] Hayes | Introduction to mediation, moderation, and conditional process analysis: a regression-based approach[END_REF]. The indirect effects were calculated as the product of a * b in this macro and the confidence intervals were based on 10,000 bootstrap samples [START_REF] Preacher | Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models[END_REF]. In order to adjust for the bias that may arise in the bootstrap distributions, bias-corrected bootstrap confidence intervals were used in all multiple mediation analyses.

Results

Table 1 presents descriptive statistics between victims and witnesses, and their differences on all measures in this study. Table 2 presents the bivariate correlations among all variables for victims and witnesses separately. Multicollinearity tests were conducted for victims and witnesses separately for all independent variables using the "collin" package in Stata v13. The variance inflation factor (VIF) ranged between 1.07 and 1.47 with an average of 1.27 for victims; and the VIF ranged between 1.08 and 2.02 with an average of 1.37 for witnesses. These values are much lower than the threshold of 10, indicating our data do not have multicollinearity problems [START_REF] Chen | Regression diagnostics. In: Regression with Stata[END_REF]. In addition, we can not rule out the possibility of common method variance bias in this study because all measures were from the same source.

Harman's single factor test [START_REF] Podsakoff | Common method biases in behavioral research: a critical review of the literature and recommended remedies[END_REF] indicated that the common factor did not account for the majority of the variance (only 19.2%), suggesting common method variance bias in this study to be small.

Insert [Table 1 and Table 2]

Formal Helping

The multiple mediation analysis for formal helping indicated that the direct effect of exposure status (i.e., victim vs. witness) was marginally significant for the usage of formal helping, c = 1.59, SE = .63, p = .01, bias-corrected 99% CI[-.02, 3.21], with victims reporting higher percentage in using formal helping. Victims and witnesses significantly differed on their usage of formal helping (total effect path c) as a result of the indirect effect through psychological difficulty (point estimate: .73, SE = .38, bias-corrected 99% CI[.06, 1.71]; Figure 1(b)). Victims reported higher levels of psychological difficulty than witnesses after being exposed to workplace violence (b = 1.23, p < .001) and the higher levels of psychological difficulty were then related to a higher likelihood of using formal helping (b = .59, p < .001). In other words, there were significant mean differences between victims and witnesses on their usage of formal helping and these group differences were transmitted through psychological difficulty. Similarly, victims reported greater reduction in work functioning than witnesses (b = 1.59, p = .002), and the reduced work functioning was positively related to using formal helping (b = .20, p = .006). However, the mediation analysis showed that the indirect effect through reduced work functioning was not significant in this model (point estimate: .31, SE = .21, bias-corrected 99% CI[-.11, .97]). Among the covariates, only general exposure to workplace violence in the past 12 months was a significant predictor for psychological difficulty (b = .03, p = .001). All other coefficients for each type of helping are presented in Table 3.

Informal Helping

The same multiple mediation analysis was conducted for predicting informal helping, but a different pattern of results emerged comparing to formal helping. The direct effect of exposure status was not significant for predicting the usage of informal helping, c = .39, SE = .53, p = . [START_REF] Walling | Community violence exposure, adverse childhood experiences, and posttraumatic distress among urban development workers[END_REF] 1(c)). There was no significant mean differences on the usage of informal helping between victims and witnesses. Neither psychological difficulty nor reduced work functioning was related to informal helping (see Table 3). No covariate was significant in predicting informal helping.

Paraformal Helping

The last set of multiple mediation analysis was conducted for paraformal helping. The results of the analysis showed that there was no significant direct effect on paraformal helping between victims and witnesses, c = .70, SE = .43, p = .10, bias-corrected 99% CI[-. 40, 1.80]. The indirect effect of exposure status through psychological difficulty was not significant (point estimate: .31, SE = .18, bias-corrected 99% CI[-.02, .91]).

Nonetheless, psychological difficulty was significantly related to a higher likelihood of using paraformal helping (b = .25, p = .005). The indirect effect for reduced work functioning was significant in predicting paraformal helping (point estimate: .33, SE = .18, bias-corrected 99% CI[.03, .94]; Figure 1(d)). Victims reported more reduction in work functioning than witnesses (b = 1.59, p = .002), which was then associated with higher likelihood of using paraformal helping (b = .21, p = .002). Among all the covariates, having physical injuries was the only significant covariate predicting paraformal helping (b = 1.10, p = .007). No other covariate was significant in this model.

Insert [Table 3]

Discussion

This study examined victims' versus witnesses' experiences and their usage of different types of helping after encountering violent events at work. Our overall results suggest that workplace violence has detrimental effects on both victims and witnesses. Direct victims are more negatively affected psychologically and at work after exposure, compared to witnesses.

As predicted, these mean differences on psychological difficulty and work functioning reduction between victims and witnesses significantly predicted the usage of different types of helping.

Consistent with previous research [START_REF] Dupré | Harm to those who serve: effects of direct and vicarious customer -initiated workplace aggression[END_REF][START_REF] Eriksson | Trauma exposure and PTSD symptoms in international relief and development personnel[END_REF][START_REF] Vartia | Consequences of workplace bullying with respect to the well-being of its targets and the observers of bullying[END_REF][START_REF] Lerias | Vicarious traumatization: symptoms and predictors[END_REF] and our hypothesis, victims experienced greater psychological and work consequences after exposure to workplace violence than witnesses. Victims have direct contact with the perpetrator while witnesses experience the violent event indirectly and vicariously by identifying with the victim and by repeated exposure to the environment where the violence takes place [START_REF] Figley | Burnout as systemic traumatic stress: a model for helping traumatized family members[END_REF][START_REF] Figley | Compassion fatigue: psychotherapists' chronic lack of self care[END_REF][START_REF] Lerias | Vicarious traumatization: symptoms and predictors[END_REF]. This is particularly pertinent in the case of workplace violence. Victims might be the co-workers or supervisors of the witnesses. According to Figley's Trauma Transmission Model [START_REF] Figley | Burnout as systemic traumatic stress: a model for helping traumatized family members[END_REF][START_REF] Figley | Compassion fatigue: psychotherapists' chronic lack of self care[END_REF], both the relationship with the victim and the workplace serve as vivid reminders of the violent event on a daily basis for the witness. Even though direct exposure of workplace violence could create greater personal meaning for victims than indirect exposure of the same event for witnesses, one does not need to be directly victimized to experience the negative impacts of violence [START_REF] Dupré | Harm to those who serve: effects of direct and vicarious customer -initiated workplace aggression[END_REF][START_REF] Walling | Community violence exposure, adverse childhood experiences, and posttraumatic distress among urban development workers[END_REF]. Therefore, both victims and witnesses could experience similar patterns of negative psychological and work outcomes, with stronger effects for victims [START_REF] Neuman | Workplace violence and aggression: when you do not want your company on the news[END_REF][START_REF] Vartia | Consequences of workplace bullying with respect to the well-being of its targets and the observers of bullying[END_REF]. Furthermore, our results on the usage of formal, paraformal and informal helping partially supported our predictions. When facing stressful and traumatic life situations, such as serious workplace violence, individuals might use different types of strategies and resources to maintain psychosocial adaptation [START_REF] Aquino | Workplace victimization: aggression from the target's perspective[END_REF]. Since formal helping is closely related to the traditional sense of psychological helping by psychiatrists and/or psychotherapists [START_REF] Barker | Psychotherapy and social support[END_REF], it is not surprising that victims who experience greater psychological difficulty are more likely than witnesses to use formal helping in order to try to regain psychological well-being.

Similarly, victims who experience greater work functioning reduction would also be more likely than witnesses to use paraprofessionals' services to file complaints, process sick leave, and obtain referrals for psychiatric and counselling services, etc. In fact, victims, who had direct contact with the perpetrators resulting in physical injuries, would be treated by physicians and therefore might have greater access to obtaining referrals to receive psychological services through physicians. However, this may not be the case for witnesses due to lack of physical injuries. Different from our hypothesis, the indirect effects through psychological difficulty and reduced work functioning did not explain the group differences between victims and witnesses on their likelihood of using informal helping. The majority of our victims (90.7%) and witnesses (86.5%) indicated they had used informal helping to cope after the violent event. In this case, we might be encountering a ceiling effect where the outcome measure demonstrated almost no variation at the upper end of its range due to a large number of participants using informal helping as a way to cope. This result does not imply that victims and witnesses do not use informal helping, but instead it shows victims and witnesses both use informal helping to a similar extent. In fact, informal helping was used the most comparing to other types of helping (see Table 1). Specifically, victims used informal helping three times more than formal helping, and about two times more than paraformal helping.

Witnesses used informal helping 14 times more than formal helping, and about 6 times more than paraformal helping. This is consistent with previous research that informal helping from family and friends is the primary source of support individuals use to cope with stress-related issues, whereas formal helping is used only after informal helping is consulted [START_REF] Wills | Social support and interpersonal relationships[END_REF][START_REF] Wills | Interpersonal analysis of the help-seeking process[END_REF]. According to Table 1, informal helping plays a very important role for workers to cope with outsider-initiated workplace violence, particularly for witnesses when other types of helping may not be accessible. In conclusion, supporting the Trauma Transmission Model [START_REF] Figley | Compassion fatigue: coping with secondary traumatic stress disorder in those who treat the traumatized[END_REF][START_REF] Figley | Burnout as systemic traumatic stress: a model for helping traumatized family members[END_REF], both victims and witnesses were negatively impacted by workplace violence, and victims reported higher levels of psychological and work difficulties than witnesses [START_REF] Dupré | Harm to those who serve: effects of direct and vicarious customer -initiated workplace aggression[END_REF][START_REF] Barling | Handbook of work stress[END_REF].

These psychological and work difficulties also explained victims' versus witnesses' differential use of formal, informal and paraformal helping.

Limitations and Future Directions

The present study has some limitations that should be taken into consideration while interpreting the results. The analysis in this study cannot imply causality due to its cross-sectional design. However, it is unlikely that psychological difficulty, reduced work functioning and the three types of helping could cause participants' exposure status (i.e., being a victim versus a witness). In the questionnaire, participants were first asked about the most disturbing workplace violence incident they have encountered in the past 12 months. They were then asked to report their psychological and work difficulty, and to indicate what types of help they used to regain normal levels of functioning following the incident. Although the structure of the questionnaire followed a logical flow that was congruent with our proposed model (Figure 1 Secondly, the measure for psychological difficulty in this study resembles the measure for symptoms of PTSD, but it is not a standardized measure for PTSD. Previous studies showed that workplace violence could lead to the development of PTSD symptoms for both victims and witnesses [START_REF] Eriksson | Trauma exposure and PTSD symptoms in international relief and development personnel[END_REF][START_REF] Gates | Violence against nurses and its impact on stress and productivity[END_REF][START_REF] Lilly | Gender and PTSD: what can we learn from female police officers?[END_REF][START_REF] Laposa | Posttraumatic stress disorder in the emergency room: exploration of a cognitive model[END_REF]. This is an important mental health issue in the workforce that warrants further investigation using well validated and standardized measures, such as the Penn Inventory for posttraumatic stress disorder [START_REF] Hammarberg | Psychometric review of the Penn Inventory for Post Traumatic Stress Disorder[END_REF], to capture the severity and duration of the psychological consequences of workplace violence.

The participants in our study were primarily White/Caucasian Francophone speakers in Québec. Our sample may not represent the diverse cultural/ethnic populations in Canada.

Due to this limitation, the results of our study might not be generalizable to other populations. Future studies should extend this line of research to other cultural/ethnic populations in order to examine the universality (or lack thereof) of using different types of helping after encountering workplace violence.

This study focused on the differential effects of workplace violence according to exposure status, but neglected potential sex differences on the consequences of workplace violence. Some studies observed female gender as a risk factor for experiencing various psychological difficulties after a traumatic event [START_REF] Lilly | Gender and PTSD: what can we learn from female police officers?[END_REF][START_REF] Brewin | Meta-analysis of risk factors for posttraumatic stress disorder in trauma-exposed adults[END_REF][START_REF] Olff | Gender differences in posttraumatic stress disorder[END_REF], but other studies had found little or no evidence of gender differences [START_REF] Elwood | Cognitive vulnerabilities to the development of PTSD: a review of four vulnerabilities and the proposal of an integrative vulnerability model[END_REF][START_REF] Santos | Gender differences in exposure to co-worker and public-initiated violence: assessing the impact of work-related violence and aggression in police work[END_REF]. Our results, using sex as a covariate, supported Hyde's [START_REF] Hyde | The gender similarities hypothesis[END_REF] hypothesis that males and females react very similarly in most psychological measures. Specifically, there was no significant sex differences on psychological difficulty and reduced work functioning, nor were there differences on formal, paraformal or informal helping usage in our study. Hyde argues that most of the sex differences observed in psychological measures are very sensitive to contextual factors. Thus, it might prove fruitful for future studies to take into account working conditions, such as human resources practices for zero-tolerance of workplace violence, as well as organizational and occupational culture [START_REF] Neuman | Workplace violence and aggression: when you do not want your company on the news[END_REF][START_REF] Lilly | Gender and PTSD: what can we learn from female police officers?[END_REF], in order to better unpack potential sex differences on the negative consequences of workplace violence.

Conclusions and Implications

Undeterred by the limitations, the present study provided empirical evidence which suggests the detrimental outcomes of indirect exposure to workplace violence and has great implications on management and clinical practices. Our study shows that violent incidents at work initiated by outsiders have extremely adverse effects on both victims and witnesses.

With informal helping being used the most to cope with the aftermath of workplace violence, clinicians should not overlook the importance of social support from patients'/clients' family, friends and colleagues. Family therapy or incident debriefing for family members and friends will help to create better social support after the violent event.

While measuring the types of helping used by victims and witnesses, it was unclear whether the type of helping was offered mandatorily at the workplace, or victims and witnesses sought out different types of help voluntarily. It is also unclear whether victims and witnesses have prior knowledge of the services that are available to them after encountering workplace violence. Therefore, it is important for organizations to analyze and survey their employees' knowledge regrading workplace violence policies [START_REF] Dillon | Workplace violence: Impact, causes, and prevention[END_REF] in order to create a safe and open environment for witnesses to use the same services as victims to overcome the psychological and work difficulties witnesses may experience. Students from high risk professions, such as police officers or healthcare workers, should learn about workplace violence before they enter the profession and during orientation training. A social-ecological model may also be applied in this context to facilitate the implementation of effective workplace violence policies in organizations [START_REF] Gillespie | Individual, relationship, workplace, and societal recommendations for addressing healthcare workplace violence[END_REF].

In addition, even though there is a general consensus in the literature that workers use multiple ways to cope with workplace violence, empirical evidence for victims and witnesses using paraformal helping is lacking. This study suggests that paraformal helping is indeed used by both victims and witnesses, and it may have a bridging function to direct workers to use other services, such as formal helping. Hence, it is imperative for human resources departments, union workers or general physicians to understand workplace violence affects everybody, not just direct victims, so that they may provide or direct appropriate services without delay. Last but not least, the categorization of formal, informal and paraformal helping could serve as a basic framework for organizations and healthcare institutes to better adjust and integrate different resources to facilitate victims' and witnesses' paths to recovery.

  (a)), it is necessary for future studies to use longitudinal designs to examine the causal relation between psychological and work consequences of workplace violence on different types of helping.

  Indirect effect was significant for psychological difficulty only. Indirect effects were significant for both psychological difficulty and reduced work functioning.

Figure 1 .

 1 Figure 1 . Multiple Mediation Models. Exposure status coded as victim = 1 and witness = 0. Types of Help include formal, informal or paraformal helping.

Table 1

 1 Descriptive Statistics Note. *p < 0.01. † T-tests were used for continuous variables and Z-tests were used for categorical variables to compare group differences. Only the frequencies for categorical variables were included. -.04 .03 .28 -.14 -.28 -.04 -.11 -.22 -.26 .04 .27 -Note. Upper diagonal coefficients are for witnesses and lower diagonal coefficients are for victims. r < -.20 or > .20 are significant at p < .05.

				Victims			Witnesses
				(n = 107)			(n = 104)
	Variables		Mean Frequency SD	Mean Frequency SD Differences †
	Consequences of workplace
	violence
	1 Psychological difficulty		3.98	-	2.59	2.68	-	2.30	-1.30*
	2 Work functioning		4.42	-	4.04	2.86	-	2.79	-1.57*
	Types of Helping
	3 Formal helping (%)		29.9	32	-	5.8	6	-	24.1 *
	4 Informal helping (%)		90.7	97	-	86.5	90	-	4.2
	5 Paraformal helping (%)	39.3	42	-	14.4	15	-	24.9 *
	Covariates
	6 Sex (%)		51.4	55	-	60.6	63	-	9.2
	7 Age		38.49	-	11.10	39.87	-	11.48	1.38
	8 Marital status (%)		66.4	71	-	73.1	76	-	6.7
	9 Income	48 925.23	-13 229.26 53 365.38	-12 855.48 -4440.15*
	10 Day shift (%)		41.1	44	-	51.9	54	-	10.8
	11 Time elapse		5.90	-	3.92	5.11	-	3.57	-0.79
	12 High risk sector (%)		58.9	63	-	69.2	72	-	10.3
	13 Physical injuries (%)		50.5	54	-	26.0	27	-	24.5 *
	14 General exposure		40.04	-	26.01	33.59	-	24.15	-6.45

This test was conducted using the "mcartest" package in Stata v13.

This macro is available at http://www.processmacro.org/.


PHD, Gerald Billard
email: gerald.billard@univ-rouen.fr

FRENCH PROFESSIONAL ICE HOCKEY IS IT REALLY FRENCH? A GEOGRAPHY OF FOREIGN INFLUENCE ON FRENCH ICE HOCKEY

Number one sport in Canada, ice hockey gained its independence in France only on April 29th, 2006. Previously affiliated with the Federation of the sports of ice which counted in its rows the figure and speed skating or the curling, French hockey thus hopes to obtain an additional visibility. Of course, this emancipation appears still too recent to see emerging a competitive and economically completely viable professional hockey within a French sporting landscape largely dominated by soccer, rugby or cycling. Today, whereas the Olympique Lyonnais, (currently the best soccer French team) announces a budget of 200 millions euros for the season 2007/2008 1 , the budget of the best teams of hockey reaches a maximum with approximately 450.000 euros; the average wages of a professional soccer player in France rise with 40.000 euros per month 2 against almost 1500/2000 euros for the hockey players ; whereas the League One (soccer) records more than 8 million people in the stadiums into 2004/2005, hockey brings together around the skating rinks 195.000 spectators for the season 2006/2007 but with a average rate of filling of 75%, the same one as for soccer. However, the hockey experiences real progress with the creation of complementary competitions (French Cup, League Cup) to the League Magnus, main professional championship, the modification of the rules of the game (more penalties are called) or the broadcasting of French hockey games on a private channel.

The organization of the French Cup final game in the Palais Ominisports of Paris Bercy, which brought together 12 5000 fans, also shows the ambition of the new federation whereas the best average of spectators is held by the Grenoble Team with 3 400 persons per match! Beyond its institutionalization, one of the great specificities of professional hockey in France remains the high number of foreign players in the League Magnus championship. In the Eighties, the contribution of Canadian and Finnish players raised the level of French hockey and was "assets" with the development of this sport in France. In 1995, the Bosman ruling (liberalizing the movement of the

European players within Europe) transformed the "French-Canadian" championship into a more European championship with the massive arrival of European players from the Eastern European countries (Slovakia, Czech Republic…). In the same way, the performances of the National French are the fruit of foreign trainers. The eighth place at the Olympic Games of Albertville in 1992 is the result of a Swedish trainer Kjell Larsson whereas today the French national team is ranked only at the nineteenth world row (IIHF, 2006) and always pains to find technicians to take its lead.

Through this contribution, we will try to evaluate the level of penetration of the French hockey by foreign players. And then, map a geography of the hearths of players emission since 17 years (1990)(1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007). In a more analytical part, we will see the factors which make evolve the origin of the foreign players in our championship elite (League Magnus) and discus, in the final part, the future of ice hockey in France with or without foreign players Then in the Twenties, hockey was diffused by contagion in the alpine areas and not by hierarchy, bus much of small cities like Chamonix and Gap adopted the innovation before the great agglomerations like Grenoble. These cities were favorable to the hockey, because of the natural presence of the cold and the ice. From 1923, Chamonix becomes the French reference as regards hockey, with 30 titles of champion of France. In 1930, the city organizes the championships of the world of hockey. From the Sixties, one attends the constitution of teams in north and south-west, thanks to the notoriety of the Olympic Games of Grenoble in 1968 but also following the construction of skating rinks covered and permanent in many towns of plains: Amiens, Rouen, Bordeaux, Angers (Billiards, Lemoine, 2005)…

The Seventies and Eighties are remembered by the creation and the expansion of clubs of hockey on all the French territory. However, it is necessary to underline the role essential by the local communities with the level of the infrastructures (inauguration of new skating rinks) but especially of the direct or indirect subsidies; sometimes this contributes artificially to maintain in life certain professional and semi-professional teams.

At the time of the Olympic Games of Albertville in 1992, the mediatization of the good service of the team of France makes pass the number of bachelors of 29.000 to more than 32.000 people but the high professional level of hockey underwent a major crisis in 1991-92, been dependant on the mechanism of a missed professionalism. Indeed, the formulas of the championship of France were dubious. The championship 1991-1992 presented a new final phase which included/understood the national league with 6 clubs and the first division with 10 clubs. Championship 92-93 gathered the 16 clubs, including 4 professional teams survivors of the crisis of the previous year, which had resulted in the disappearance of professional teams in Briançon, Bordeaux, Epinal or Clermont-Ferrand with sometimes due the withdrawal to important and often single sponsor (example of Bayer in Caen) and the limitation of the subsidies of the local communities on the level of the professional sport.

The French hockey launched out in an inflationary professionalism to forced marches, in a French space with weak culture hockey, following the growth of the licenses and seeking to exploit international successes (eighth place with the Olympic Games of Albertville). But the media interest remains fragile, the passion having held with the Olympic test with a vast public not really won over to this discipline. Since 2004, the situation of the French hockey seems to improve with creation of an Executive Authority of the French Hockey (A.E.H.F the hockey body within the French Ice Sports Federation) under the direction of a former player of hockey Mr Luc Tardif. This authority wants to be the starter of an autonomy of the hockey and she wants to give again with the French hockey an image of cohesion, organization and the serious one. In 2003, the limitation of the wage bill of the clubs of the Elite with 450.000 of euros per annum, then the adoption in June 2004, rule of a maximum of four foreign players (except Europe) in the teams disputing the League Magnus go in the direction of a healthy financial management of French professional hockey3 while limiting the recourse to the Canadian players, thus supporting the formative clubs.

A GEOGRAPHY OF THE PROFESSIONAL PLAYERS OF HOCHEY ON ICE IN FRANCE

In this context of revival of the hockey in France, the question of the presence of the foreign players is important because it returns us at the same time to questions of quality of play and thus of sporting interest of the Magnus League but also to the problems of integration on the most level of the young French players. Many federations wonder today about becoming to it certain national championships in which the best teams present themselves sometimes on the ground with a very vast majority from abroad. In France, in 2006, in the various championships elites men, the share from abroad is 43.21 % for the basketball (Pro A), 41,31 % for volley ball (Pro A), 39,58 % for soccer (League 1), and 33,20 % for Rugby (Signal 14) 8 . According to our calculations, the rate of penetration of the foreign players in the Magnus League for season 2006/2007 is at 36.88 %. If this rate is less low than for other sports, it has been however in progression for one year (32.39 % in 2005/2006). All in all, whereas the number from abroad was 12.93% in 1990, it passes to more than 20 % starting from the middle of the Nineties (25.93% into 1995Nineties (25.93% into /1996) and more than 30 % in 2002 (cf. Fig. 2). Figures balanced with the Of course, there are however important differences between the clubs on the level of the presence of foreign players in their row (cf 4). Thus the team of Amiens with 12,5% foreign players or Villard de Lans (12,5 %) posts rates definitely lower than the French average contrary in Briançon (57,14 %), Epinal (56,52%), or Rouen (52,17 %). These high figures are explained is by the incapacity of the clubs to form young loan for the high level or of a point seen budgetary impossibility of engaging the best young people French players of other clubs. It should be specified that contrary to the North-American system, the young French players are not trained by the universities but by the clubs directly. Their first integration in the Magnus League does not depend on a draft open to all the clubs but on a relatively traditional market answering supply and. In lack of financial means, certain teams are turned over thus towards the market in particular is European with "cheap" players. The case of Rouen is undoubtedly atypical because which forms a lot of young players seeks to up skills them by good foreign players.

Towards a new geography of the professional foreign players in France?

When we speak today about globalization of the economy, there is not certainly a sphere of activities other than the professional sport where the international mobility of employment is also intense9 . In Europe in particular, the deregulation of the labour market for the sportsmen in the European Union countries since 1995, liberalization extended in 2002, to the other European countries out-union, the Maghreb and several tens of country of the Africa-Caribbean-Pacific zone. This extension of the European chart in the field of the professional sport obviously did not save the French hockey; it is a new geography of the French professional hockey which is redrawn since 1990.

Close cultural links with Canada

Without context, the Canadian players and in particular Québécois, represent the most important quota for a long time foreign players in France. The enormous number of young Canadian players and the cultural proximity with France, in particular for the French-speaking players, provide the foundation of privileged bonds between the two countries. Moreover much from Canadian will obtain French nationality and will be able to thus take part in the World Championship or the Olympic Games within The next map (Fig. 4) shows perfectly this natural opening towards Canada. For 17 years, 213

Canadian players have arrived to France, far in front of Swedish and the Slovak ones. However, if over the duration this distribution appears relatively clear, since 1995, the things evolved/moved. On 6, Canadian with nearly 30 % of the foreign players, are largely most present within the French clubs in 1990 (Fig. 5). 10 We can speak in particular about the case of naturalized French-Canadian B. Laporte, S. Woodburne and S. Poudrier who played for the French national Team which was quarter of finalist to the JO of Albertville in 1992. It is undoubtedly necessary here to moderate the speech of a massive arrival of Eastern countries players in our championship 12 even if the players of ex-Czechoslovakia represents in 2006, 42 % from abroad evolving in Magnus League (Fig. 7). If they generally better are gone than their North-American or Finnish counterparts, the recruitment carried out by the clubs is also based on a sporting logic. For a long time, the Canadian players certainly more physical than Europeans had no problem to familiarize themselves with the French play but today the context have changed. The defensive play is 11 Supplemented by the Simutenkov ruling in 2005 12 The Polish plumber symptom more severely sanctioned and the referees called more easily penalties, which disadvantages the "violent" play more of the North-Americans to the profit of more technical players ready to shine in more during the phases of power-plays (Tardif, 2007, Op. Cit.). Moreover, the larger size of the ice ring playgound in France compared to that of the skating rinks in the NHL is sometimes a handicap for the American players: "some North-American players even with large resume could not adapt to the European standards" (Tardif, 2007, Op. Cit.).

THE FUTURE OF FRENCH HOCKEY

Which will be the weight of the foreign players in French professional hockey ? Questioned on the subject, Luc Tardif, current president of the French Federation of Ice Hockey, affirms first of all that the contribution of the foreign players is essential to maintain overall a good level of plays within the French championship. Firstly, this "foreign legion" allows the club to supplement their manpower in the absence of a production of young French players sometimes insufficient. Moreover, for the French players, the young people in particular, the daily contact with the drive or confrontation in match with players Canadian, Finnish, Czech, Swedish or Slovak represent certainly a factor of sporting progression. But a balance must be found between foreign influence and training of the French players. This question is important so that French hockey can exist on the international scene bus without French hockey players, not of team of powerful France to the Olympic Games or in Championship of the World (group A if possible!). This element is far from being anecdotic when the impact is known, on the young practice and the sponsors, of the incredible epopee of the French national team to the Olympic Games of Albertville of 1992 (Quarterfinal).

If the NHL is recognized the best league of the world, it mainly owes it with the contribution of the foreign players. A simple assumption is thus posed: does the arrival of the best European players raise the quality of the North-American championship while weakening the level of the American and Canadian players because the young North-American have of weaker advisability of playing in a team and less time of play ? Since 1990, the European national teams reached 11 times the Gold medal of IIHF World Championship against 4 times for Canada (1994, 1997, 2003 and 2004) A study published in 2006 by the International Ice Hockey Federation (IIHF) worried about the important presence of European players often too young and badly prepared for the NHL 16 . The American League plans to make pass the ratio to 70 % of North-Americans in the clubs to 80 % to thus make fall the weight of Europeans to 20 % instead of 30 % today. This measurement would thus make it possible to better integrate the young players of the two countries into the world which has the most bachelors (1 million for the USA and 550.000 for Canada against 90.000 for the Czech Republic). Of course, the League Magnus is not the NHL but the problems of the formation and the integration of the young national players and thus in the long term, the performances of the national teams also pass by an interrogation on the weight of the foreign players in the championship. Maintain artificially an elite team during a few years with "mercenaries" is undoubtedly not the best way for ensuring the perennial of a club and of allowing the engagement of private partners on the long term. One of the proposals of the new federation is for example to maintain the principle of the levelling off of the wage bill (454 000 € is US$ 581.000/year) but with possible removals of the upper limit for the

  SPECIFIC ELEMENTS OF THE RISE OF THE FRENCH HOCKEYInFrance, as of 1891 of the players gathered by Pierre De Coubertin play for the very first time on the channel of Versailles (Paris). The first club (the club of the skaters of Paris) was besides Hockey club of Paris, founded in 1901 and which with the great surprise défet "Princess Ice Hockey Club of London" which had been champion of the world for 7 years. In 1905, this sport was in Net progress, the Parisian clubs started to multiply and dominated the French championship until 1922, only the club of Lyon creates in 1905 and first champion of France of hockey, represented the remainder of France.

For

  the French hockey, season 2005/2006 was very paradoxical. It however begin with exclusion from the club from Mulhouse the 2005 League Magnus champion in prey with financial difficulties 4 : Mulhouse was unable to engage a team in 2006. Worse, the finalist of the League Magnus (Tours) also finds itself excluded from the competition for the same reasons 5 and relegated in Division 2. The results of the French national team are also disappointing. In the city of Amiens (France), in April 2006, French finishes second of the world championship (2006 IIHF -World Championship -Division 1 Groupe A) and leaves Germany goes to World Elite Championship. However in April 2006, of Hockey (FFHG is born the French Federation) and the professional championship Elite (Magnus chain + and the play off give place to meetings with suspense (final victory of Grenoble). One of the events of the season in February 2007, the final of the cut of France, brings together 12.250 people in Bercy (Paris), and at the same time popular, sporting and media success for French hockey. In 2007, the FFHG counts 17.025 permit-holders 7 and 117 clubs located in 108 cities but the way is still long, in spite of many assets, to settle definitively in the French sporting landscape.

Fig. 1

 1 Fig. 1 Location of the League Magnus Teams in 2006/2007

FigFig. 3

 3 Fig.3shows it.

French

  national Team, an opportunity undoubtedly impossible in their country of birth 10 . Late Luc, Canadian of origin, chair current French Federation of Hockey (FFHG) and former professional player inChamonix (1975) andRouen (1984) wire-drawer also that at the exit of the University, the arrival in France represents for certain Canadian students an almost initiatory voyage."For a long time, recruitment foreigners was focused mainly on Canadian and in particular the Québécois. The bringing together was facilitated, because Québécois and French have the same language. The hockey is born in France with the arrival of the Canadian players. First Canadians came to France were former students, curious to know Europe. Their notion of ice hockey was more cultural than professional."(Luc Tardif, interview of March 27th, 2007).

Fig. 4

 4 Fig. 4 Mother country and total number of foreign players in the Elite Ice Hockey Championship in France between 1990 and 2006

Fig. 6

 6 Fig. 6 Rate of foreign players by mother country in the League Magnus (France) in 2006/2007

Fig. 7

 7 Fig. 7 Nationality of foreign players in the League Magnus (France) in 2006/2007

  and zero times for the United States 13 ! At the time of the last eight Olympic Games, the United States obtained the gold medal once (1980 -university players) and Canada once (2002) against six times for the European teams. In 2006, in Turin, Sweden (Gold), Finland (Silver) and the Czech Republic (Bronze) obtained amedal. This situation is all the more paradoxical as since 1990, the team of less than twenty years of Canada gained 10 times the world title and finished 4 times second 14 . This as well confirmed as Canada remains certainly the country which produces the best young players but the passage at the professional level perhaps appears delicate because of the overseas recruitment carried out in NHL 15 .

  formative clubs 17 . Rise of budget are granted height of 5 % into 2006/2007 for the integration of 4 young French players, even of 10 % into 2007/2008 per 5 young players. An effort is also carried on the training of the young players (TEAM 2010) on the most level with for ambition a title of champion of the world (-of 18 or -20 years) from here 2010. The question of the identity (if it really exists) French hockey but more still of its future than it is as a professional sport or amateur, does not rest only on the problems of the presence of the foreign players on our national ice floes. The young federation of hockey born in June 2006 must be structured today in-house but more still, to reorganize at the same time the professionalization of the discipline while making the promotion of hockey amateur, including female. E 2006, Luc Late, president of the federation, presented before the general assembly the main trends of its plan of durable development of the hockey in France. The federation is given thus 4 years to ensure sporting and financial perenniality discipline while supporting, for example, on the following points: -referee and trainer, training of young people, -a better framing and means increased for the teams of France (Senior, Hopes, Women…), -the search for deprived financial partners and the renegotiation of the rights TV, -the maintenance of financial control on the clubs, -the development of the communication around the hockey, -the promotion of female hockey, leisure and for the people handicapped… The question of the infrastructures is also essential today bus in France, few skating rinks are entirely dedicated to hockey; some are also today unsuited to the high level sport because in particular of their small size. In 2006/2007, only 3 clubs (Amiens, Grenoble and Rouen) have a skating rink of more than 2.500 places. Is a plan of development of the skating rinks (if possible in double track with one reserved for hockey) thus carried by the federation to modernize the infrastructures, to fix a minimum number of seats (1 500?) and if possible to allow the emergence of clubs elites in the large French cities (Marseilles, Strasbourg, Nantes, Bordeaux, Lyon and of course Paris).

In a press release 2005, march, the 8th, the FFSG announced that four clubs from League Magnus were seen meaning their exclusion of championship 2005/2006 if the state of their finances did not improve.

Deficit of 290.000 euros($371,000)

Deficit of 250,000 Euros ($319,700)

In March 2007, the Management Committee of the FFHG made the decision to maintain 14 teams in the Magnus League and that for 3 years.

By comparison, soccer counts 2 283.268.

Milanovic B, 2006, "Le foot, industrie sans frontière", Le Monde, 7 juillet 2006.

Proposal made par the League Magnus Commission and ratified by the Management Committee of March 12, 2007.

Reuters, 3/04/2007. The average budget of aclub 2 F. Barthez (former French National Team goal keeper) gained nearly

http://www.linternaute.com/sport/


Andrei Halanay
email: halanay@mathem.pub.ro

Marius Cornel
email: cornel.murea@uha.fr

Murea

Uniform Estimation of a Constant Issued from a Fluid-Structure Interaction Problem

Keywords: star-shaped domains, fluid-structure interaction, fictitious domain

We prove that the domain obtained by small perturbation of a Lipschtz domain is the union of a star-shaped domains with respect to every point of balls, such that the radius of the balls is independent of the perturbation. This result is useful in order to get uniform estimation for a fluid-structure interaction problem.

Introduction

In [START_REF] Halanay | Existence and approximation for a steady fluid-structure interaction problem using fictitious domain approach with penalization[END_REF] and [START_REF] Halanay | Existence of a steady flow of Stokes fluid past a linear elastic structure using fictitious domain[END_REF] the existence of a solution is studied for an elastic structure immersed in an incompressible fluid.

Let D ⊂ R 2 be a bounded open domain. We denote by Ω S 0 the undeformed structure domain and by u = (u 1 , u 2 ) : Ω S 0 → R 2 its displacement. A particle of the structure with initial position at the point X will occupy the position x = Φ (X) = X + u (X) in the deformed domain Ω S u = Φ Ω S 0 . In [START_REF] Halanay | Existence of a steady flow of Stokes fluid past a linear elastic structure using fictitious domain[END_REF], we have assumed that ∂Ω S u has the uniform cone property and the geometry of the cone is independent of u. The fluid occupies the domain Ω F u = D \ Ω S u , see Figure 1. It is possible to construct an uniform extension operator E from {v ∈

H 1 Ω S u 2 ; ∇ • v = 0 in Ω S u } to H 1 0 (D) 2 such that ∇ • E(v) = 0, in D (1)
E(v) = v, in Ω S u (2)
E(v) 1,D ≤ K v 1,Ω S u (3)
where the constant K > 0 is independent of Ω S u , but it depends on the geometry of the cone. This construction is obtained by solving the Bogowskii problem in Ω F u , see for example [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF], Lemma 3.1, page 121. There exists w ∈ H

1 0 Ω F u 2 such that ∇ • w = f , in Ω F u (4) w = 0, on ∂D (5) w = 0, on ∂Ω S u (6) |w| 1,Ω F u ≤ K 1 f 0,Ω F u (7)
where

Ω F u f dx = 0.
If Ω F u is star-shaped with respect to any point of a ball of radius R u , we have the following estimation of K 1 > 0:

K 1 ≤ c 0 diam(Ω F u ) R u 2 1 + diam(Ω F u ) R u (8)
where diam(Ω F u ) is the diameter of Ω F u . A similar result holds if the domain Ω F u is union of star-shaped domains with respect to any point of a ball, see Theorem 3.1, p. 129, [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF]. We have diam(Ω F u ) ≤ diam(D), for all u such that Ω F u ⊂ D. The aim of this paper is to prove that, under some geometrical assumptions,one can choose R u = R = constant, for small u.

We define the domain, see Figure 2

Ω ϕ = (x 1 , x 2 ) ∈ R 2 ; -r < x 1 < r, -r < x 2 < a + ϕ(x 1 ) . ( 12
)
Let R ∈ (0, r). If L < a-2r 2r , then the domain Ω ϕ is star-shaped with respect to every point of the ball B R (0).

Proof. Since R < r, then B R (0) ⊂ Ω ϕ . Let x ∈ B R (0)
. We suppose that a ray starting from this point cuts ∂Ω ϕ in two points y and z and y ∈ (x, z), i.e. y is on the segment with ends x and z.

Let us denote the top boundary of Ω ϕ by

Γ ϕ = (x 1 , a + ϕ(x 1 )) ∈ R 2 ; -r < x 1 < r .
If y, z ∈ ∂Ω ϕ \ Γ ϕ it follows that the ray cuts the boundary of the strip

C = (x 1 , x 2 ) ∈ R 2 ; -r < x 1 < r, -r < x 2 ,
twice, which is not true because the C is a convex set and a convex set is starshaped with respect to every interior point.

If we have only y ∈ ∂Ω ϕ \ Γ ϕ , then z belongs to the exterior of the strip C, consequently z / ∈ Γ ϕ . So, we will study only two cases: Let ξ = (ξ 1 , ξ 2 ) be the point in the plane, such that the line passing throw ξ and x is parallel to the axis Ox 1 and the line passing throw ξ and z is parallel to the axis Ox 2 .

i) y, z ∈ Γ ϕ ; ii) y ∈ Γ ϕ and z ∈ ∂Ω ϕ \ Γ ϕ . z x y z x 1 1 1 y ξ Γ ϕ x y z x 1 1 y ξ 1 =r z q Γ ϕ
We denote by α the angle xzξ. Since x ∈ B R (0) using (10), we get |z -ξ| ≥ a -2r and |x -ξ| ≤ 2r, then

tan α = |x -ξ| |z -ξ| ≤ 2r a -2r
.

By assumption we have y, z ∈ Γ ϕ , then y = (y 1 , a + ϕ(y 1 )) and z = (z 1 , a + ϕ(z 1 )) with -r < y 1 , z 1 < r. We have also, tan α = This implies that 2r a -2r

≥ 1 L
which is in contradiction with the hypothesis L < a-2r 2r .

Case ii): y ∈ Γ ϕ and z ∈ ∂Ω ϕ \ Γ ϕ . As in the Case i), we construct ξ, we denote by α the angle xzξ and we get

tan α = |x -ξ| |z -ξ| ≤ 2r a -2r
.

Since z cannot belong to the bottom boundary of C, we assume that z belongs to the right boundary of C, so z = (r, z 2 ). We denote by q the point at the right top corner of Ω ϕ , so q = (r, a + ϕ(r)). We have W 1,∞ (-r, r) ⊂ C 0 ([-r, r]) (see [START_REF] Brezis | Analyse fonctionnelle : théorie et applications[END_REF], Theorem VIII.7, page 129), so ϕ(r) is well defined.

Let α be the angle yqξ. Then

tan α = |y 1 -r| |ϕ(y 1 ) -ϕ(r)| ≥ 1 L .
If z 2 > a + ϕ(r) it follows that z belongs to the exterior of Ω ϕ not to ∂Ω ϕ , so z 2 ≤ a + ϕ(r). In this case α ≥ α and we get

1 L = tan α < tan α ≤ 2r a -2r which is in contradiction with the hypothesis L < a-2r 2r . Proposition 2. Let ζ : [-r, r] → R be a Lipschitz function of constant L, i.e. ∀x, y ∈ [-r, r], |ζ(x) -ζ(y)| ≤ L|x -y|. Denote by Γ ζ = {(x, ζ(x)) ∈ R 2 , x ∈ [-r, r]}, the graph of ζ. Let u = (u 1 , u 2 ) : Γ ζ → R 2 be such that ∃η 1 ∈ 0, r 2 , sup x∈Γ ζ |u(x)| ≤ η 1 ( 13
)
∃η 2 ∈ 0, 1 √ 1 + L 2 , ∀x, y ∈ Γ ζ , |u(x) -u(y)| ≤ η 2 |x -y|. ( 14
)
Then there exits a Lipschitz function ϕ : -r 2 , r 2 → R of constant less than

L+η2 √ 1+L 2 1-η2 √ 1+L 2 , such that its graph is included in (Id + u) (Γ ζ ). If ζ(0) = 0, then ∀z ∈ - r 2 , r 2 , |ϕ(z)| ≤ r 1 + L 2 + 1 2 .
Proof. Let us introduce the function

φ : [-r, r] → R, φ(x) = x + u 1 (x, ζ(x)).
We will prove that φ is strictly increasing, so it is injective. To simplify, we assume that ζ and u are of class C 1 . We get

φ (x) = 1 + ∂u 1 ∂x 1 (x, ζ(x)) + ∂u 1 ∂x 2 (x, ζ(x))ζ (x). Since ζ is Lipschitz function of constant L, |ζ (x)| ≤ L. Similary, we have |∇u(x)| ≤ η 2 .
Using Cauchy-Schwarz we get

∂u 1 ∂x 1 (x, ζ(x)) + ∂u 1 ∂x 2 (x, ζ(x))ζ (x) ≤ ∂u 1 ∂x 1 2 + ∂u 1 ∂x 2 2 1 + L 2 ≤ η 2 1 + L 2 .
It follows that

φ (x) ≥ 1 -η 2 1 + L 2 > 0,
and then φ is strictly increasing, so φ is a bijection from [-r, r] to [φ(-r), φ(r)].

From (13), we obtain φ(-r) < -r 2 and φ(r) > r 2 .

Let us introduce the function

ϕ : -r 2 , r 2 → R, ϕ(z) = ζ(x) + u 2 (x, ζ(x)),
where x = φ -1 (z). The function is well defined because if z is in -r 2 , r 2 , then x = φ -1 (z) is in (-r, r).

We will prove that ϕ is a Lipschitz function. Let z, w ∈ -r 2 , r 2 and x = φ -1 (z), y = φ -1 (w).

|ϕ(z) -ϕ(w)| = |ζ(x) -ζ(y) + u 2 (x, ζ(x)) -u 2 (y, ζ(y))| ≤ |ζ(x) -ζ(y)| + |u 2 (x, ζ(x)) -u 2 (y, ζ(y))| ≤ L|x -y| + η 2 |(x, ζ(x)) -(y, ζ(y))| = L|x -y| + η 2 (x -y) 2 + (ζ(x) -ζ(y)) 2 ≤ L + η 2 1 + L 2 |x -y|.
We have

|z -w| = |x -y + u 1 (x, ζ(x)) -u 1 (y, ζ(y))| ≥ |x -y| -|u 1 (x, ζ(x)) -u 1 (y, ζ(y))| ≥ |x -y| -η 2 |(x, ζ(x)) -(y, ζ(y))| ≥ |x -y| -η 2 (x -y) 2 + (ζ(x) -ζ(y)) 2 ≥ 1 -η 2 1 + L 2 |x -y|.
We deduce that for all z, w ∈ -r 2 , r

2 |ϕ(z) -ϕ(w)| |z -w| ≤ L + η 2 √ 1 + L 2 1 -η 2 √ 1 + L 2 .
Let (z, ϕ(z)) be a point on the graph of ϕ, where z ∈ -r 2 , r 2 . Then there exits x = φ -1 (z) ∈ (-r, r). From the definition of φ and ϕ, we have

(z, ϕ(z)) = (x + u 1 (x, ζ(x)), ζ(x) + u 2 (x, ζ(x))) = (x, ζ(x)) + (u 1 (x, ζ(x)), u 2 (x, ζ(x))) = (Id + u) (x, ζ(x)) which proves that the graph of ϕ is included in (Id + u) (Γ ζ ). Also |ϕ(z)| ≤ |(z, ϕ(z))| = |(x + u 1 (x, ζ(x)), ζ(x) + u 2 (x, ζ(x)))| ≤ |(x, ζ(x))| + |(u 1 (x, ζ(x)), u 2 (x, ζ(x)))| ≤ x 2 + (ζ(x)) 2 + η 1 ≤ r 2 + (ζ(x)) 2 + r 2 If ζ(0) = 0, since ζ is a Lipschitz function of constant L, we have |ζ(x)| = |ζ(x) -ζ(0)| ≤ L|x -0| < Lr. Then |ϕ(z)| ≤ r √ 1 + L 2 + 1 2 .
3 Uniform Estimation of the Radius of the Ball for Small Perturbation of Lipschitz Domain Definition 1 (see [START_REF] Chenais | On the existence of a solution in a domain identification problem[END_REF]). Let r, a, L be three positive numbers and Ω an open bounded set of R 2 . We say that the boundary ∂Ω is uniform Lipschitz if, for every x 0 ∈ ∂Ω, there exits a Cartezian coordinates system {x 1 , x 2 } of origin x 0 and a Lipschitz function

ζ : (-r, r) → (-a, a) of constant L, such that ζ(0) = 0, ∂Ω ∩ P (x 0 ) = {(x 1 , ζ(x 1 )), x 1 ∈ (-r, r)}, Ω ∩ P (x 0 ) = {(x 1 , x 2 ), x 1 ∈ (-r, r), ζ(x 1 ) < x 2 < a},
where P (x 0 ) = (-r, r) × (-a, a).

We will use the same notation as in the first section. We will prove a result similar to the Lemma 3.2, p. 40, from [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF], but the radius of the balls is the same for all admissible displacements. Theorem 1. Let ∂Ω S 0 be an uniform Lipschitz boundary of parameters r, a, L. We set R ∈ 0, r 2 . There exists two constants

η 1 ∈ 0, r 2 , η 2 ∈ 0, 1 √ 1+L 2
and two natural numbers m, n ∈ N * , such that for all u : ∂Ω S 0 → R 2 , such that

sup x∈∂Ω S 0 |u(x)| ≤ η 1 (15) ∀x, y ∈ ∂Ω S 0 , |u(x) -u(y)| ≤ η 2 |x -y| (16)
we have the decomposition

Ω F u = m i=1 Ω i   m+n j=m+1 B η1 (x j )  
where

Ω i = Ω F u ∩ Q(x i ) for 1 ≤ i ≤ m are star-shaped domains with respect to every point of a ball B i of radius R, B i ⊂ Ω i , Q(x i ) are rectangles of center x i ∈ ∂Ω S 0 congruent to -r 2 , r 2 × (-a, a) and B η1 (x j ) for m + 1 ≤ j ≤ m + n are balls of center x j ∈ Ω F u and radius η 1 , B η1 (x j ) ⊂ Ω F u .
Proof. Let us remark that if Ω S 0 is Lipschitz of parameters r, a, L, then it is also for the parameters r , a, L, for r < r. Consequently, we can choose r small enough such that 2L + √ 1 + L 2 + 5 2 < a r . Let P (x) be the rectangle of center x ∈ ∂Ω S 0 given by the Definition 1. We denote by Q(x) the rectangle -r 2 , r 2 ×(-a, a) using the same local coordinates. We have ∂Ω S 0 ⊂ ∪ x∈∂Ω S 0 Q(x) and since ∂Ω S 0 is compact, then there exits m ∈ N * and x i ∈ ∂Ω S 0 , 1 ≤ i ≤ m, such that

∂Ω S 0 ⊂ m i=1 Q(x i ).
We will use the notation

O η = {x ∈ R 2 ; d(x, ∂Ω S 0 ) ≤ η}, Ω F 0,η = {x ∈ Ω F 0 ; d(x, ∂Ω S 0 ) > η} where d(x, A) = inf y∈A |x -y| is the distance function. There exists η 0 > 0, such that O η0 ⊂ m i=1 Q(x i ). ( 17
) Set η 1 = min( η0 2 , r 2 ).
Let u : ∂Ω S 0 → R 2 be such that (15) holds. Then (Id+u)(∂Ω S 0 ) ⊂ O η1 ⊂ O η0 , for all admissible u and it follows that

Ω F u ⊂ O η0 ∪ Ω F 0,η0 . (18)
Since Ω F 0,η0 is bounded, then there exists n ∈ N * and x j ∈ Ω F 0,η0 for m + 1 ≤ j ≤ m + n, such that 17), ( 18), (19), we get

Ω F 0,η0 ⊂ m+n j=m+1 B η1 (x j ). ( 19
) But η 1 ≤ η0 2 , so B η1 (x j ) ⊂ Ω F u . From (
Ω F u = m i=1 Ω F u ∩ Q(x i )   m+n j=m+1
B η1 (x j )   . Using the Proposition 2, (Id + u)(∂Ω S 0 ) ∩ Q(x i ) is the graph of a Lipschitz function ϕ : -r 2 , r 2 → (-a, a) of constant We have 2L + 1 < a-r( √ 1+L 2 + 3 2 ) r by the choice of r at the begining of the proof. Using a similar argument as in the Proposition 1, we get that Ω i = Ω F u ∩ Q(x i ) is a star-shaped domain with respect to every point of the ball of radius R and center (0, -a + r 2 ).

L = L + η 2 √ 1 + L 2 1 -η 2 √ 1 + L 2 and |ϕ| ≤ r √ 1 + L 2 + 1 2 . If η 2 ∈ [0,
Corollary 1. Using the Theorem 3.1, p. 129, from [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF], we get that the constant K 1 from the inequality (7) depends on min(η 1 , R) and diam(D), but it is independent on the displacement u.

Fig. 1 .

 1 Fig. 1. Fluid and structure domains.

Fig. 2 .Proposition 1 .

 21 Fig.2. A star-shaped domain with respect to every point of a ball.

Fig. 3 .

 3 Fig. 3. Case i) at the left and case ii) at the right

  |y1-z1| |ϕ(y1)-ϕ(z1)| and using (11) we get tan α ≥ 1 L .

Fig. 4 .

 4 Fig. 4. The graphs of ζ and ϕ.

Fig. 5 .

 5 Fig. 5. The rectangle -r 2 , r 2 × (-a, a) is of center xi ∈ ∂Ω S 0 , the ball of radius η1 is of center xj ∈ Ω F 0,η 0 .

1 2 √

 12 1+L 2 ), then L ∈ [L, 2L + 1).

2

  Small Perturbation of the Boundary of a Star-Shaped Domain with Respect to any Point of a Ball We denote by B R (x 0 ) the open ball of radius R centered at x 0, i.e. B R (x 0 ) = {x ∈ R 2 ; |x 0 -x| < R}, where | • | is the euclidean norm. A domain Ω is starshaped with respect to every point of a ball B R (x 0 ) such that B R (x 0 ) ⊂ Ω, ifand only if, for every x ∈ B R (x 0 ) and y ∈ Ω, the segment with ends x, y is included in Ω. A characterization of such a domain is that every ray starting from a point x ∈ B R (x 0 ) intersects the boundary of the domain at only one point.