Sophie Carles

Ghassen Saidani

Jean-Luc Le Garrec

Ngary Guen

James B A Mitchell

Albert A Viggiano

Nicholas S Shuman

Demonstration of the branching ratio inversion for the electron attachment to phosphoryl chloride POCl 3 in the gas phase between 300 and 200K

Keywords:

Electron attachment to phosphoryl chloride (phosphorus oxychloride) POCl 3 has been studied in the gas phase by mass spectrometry at several low temperatures (47.7, 74.5, 169.7 and 199.5K) with the CRESU method. By measuring over this temperature range and data from (11), we have demonstrated the inversion of the branching ratio between the exothermic non-dissociative exit channel POCl - 3 and the thermo-neutral dissociative exit channel POCl - 2 + Cl. A kinetic model in terms of statistical theory is used to fit the experimental data.

Introduction

Phosphorus oxychloride (POCl 3 ) is an organophorous compound widely used in several industrial processes including the synthesis of flame retardant and plasticizers for PVC (5), the chemical dehydratation process to obtain nitrile from amide (1), the gas diffusion for the solar cells N-type silicon solar cells [START_REF] Neuhaus | Industrial silicon wafer solar cells[END_REF]. However, from a more fundamental point of view, POCl 3 is a molecule that has proved to have a very interesting behaviour with respect to electron attachment. The reaction has been found (7) to have a large rate coefficient (0.6 -1.8 x 10 -7 cm 3 .s -1 in the range from 296-552 K), but what is particularly noteworthy is that the attachment products depend upon both pressure and temperature (18; 17).

The attachment process can be represented by the following dissociative processes: Email address: sophie.carles@univ-rennes1.fr (Sophie Carles*) all of which pass through an excited intermediate state POCl - * 3 that subsequently dissociates. ∆E are the reaction enthalpies at 298 K from G3 model chemistry calculations (4). An additional non-dissociative exit channel is also possible: e -+ POCl 3 → POCl - 3 (5) where the excited intermediate state POCl - * 3 is stabilized. This has been modeled and discussed in several experimental studies using a Flowing Afterglow Langmuir Probe (FALP), an ion mobility mass spectrometer apparatus, and also an electron beam experiment (18; 11; 12; 17). In swarm experiments, under low pressure conditions at temperature of between 300 and 600K, the dissociative attachment leading to a POCl - 2 product dominates representing more than 90% of the reaction. The non-dissociative attachment yielding POCl - 3 becomes more and more apparent as the ambient pressure is increased. At higher temperatures between 600 and 1200K, the other dissociative pathways open up yielding Cl -which dominates above 1000K with a small contribution from Cl - 2 . POCl -has been detected as a fragment product but only at higher electron energies in the beam experiment.

In this paper, we explore the electron attachment on POCl 3 in the gas phase, for a temperature range between 200 and 47.7K and for an electron energy near 0eV using the CRESU (Cinétique des Ractions en Ecoulement A c c e p t e d M a n u s c r i p t Supersonique Uniforme) apparatus. This experimental setup has previously been used in this laboratory for the study of electron attachment to CF 3 Br, CF 2 Cl 2 , SF 6 (6), CH 3 I (15), HBr, HCl (14) and HI, DI (2). In the latter publication, evidence was presented that in the supersonic flow of the CRESU apparatus, the electron temperature is not in equilibrium with the nitrogen buffer gas temperature due to the presence of vibrationally excited N 2 (v=1) that can transfer energy to the electrons. Thus, the present experiments are not under true thermal equilibrium conditions, with the internal energy of the attaching molecules at the buffer gas temperature, but the electrons initially at an elevated temperature and undergoing thermalization on the time scale of the experiment. A detailed modeling of the electron temperatures in the experiment has been presented elsewhere (2). At pressures approaching 1 atm (18), stabilization (i.e. reaction ( 5)) dominates and the branching ratios for attachment to POCl 3 appear insensitive to electron energy over the range of 0.04 -0.33 eV. The present experiments are at much lower densities, where more interesting effects can be explored.

Eperimental Method

The CRESU experiment has been described in detail elsewhere (9), only a brief description of this method illustrated by Figure 1 and details useful for understanding the present measurments are given here. The CRESU method consists of generating a supersonic uniform flow by the expansion of a buffer gas, here N 2 , through a Laval nozzle. The design of the Laval nozzle and its upstream and downstream pressures, maintained by a powerful pumping system (24,000 m 3 /hour), determine the temperature, the density, and the speed of the flow. The present design of the apparatus allows for a temperature range of 20 -300K. An important characteristic of the flow is its high density (10 16 -10 17 molecule cm -3 depending on the Laval nozzle design and the pressure conditions), which ensures local thermodynamic equilibrium of the gas. This expansion is uniform in gas temperature, speed, and density over tens of centimeters which corresponds to a hydrodynamic time from about 100 to 1000µs. This supersonic uniform flow can be regarded as a low temperature chemical reactor without walls, thus precluding condensation of the neutral species. The production of charged species is achieved with an electron beam crossing the neutral supersonic jet, a few millimeters downstream from the nozzle generating a weakly charged plasma (10 7 -10 9 e -cm -3 ). The electrons and ions in that plasma react with a neutral species, continuously introduced as a minority species into the buffer gas, to form anions and/or cations via collisional processes (charge transfer, dissociative electron attachment, etc.). The ions and electrons are monitored along the flow axis using a movable quadrupole mass spectrometer coupled with an electrostatic Langmuir probe constituted by a thin thoriated tungsten wire (25µm diameter and 1.5 cm length).

The neutral reactant phosphoryl chloride POCl 3 was obtained from the Aldrich R (98% purity). The vapour pressure of POCl 3 , which is solid at room temperature, is sufficient to obtain a He -POCl 3 mixture with a dilution factor f = 1.2 × 10 -4 . This mixture is continuously injected into the reservoir via a flow controller with a typical flow rate of Q m = 0.1-0.2 standard liters per minute. The density of this neutral reactant is obtained using the following formulae:

[POCl 3 ] = n tot Q m Q tot f (molecule cm -3 ) (6)
where [POCl 3 ] is the injected POCl 3 density, n tot is the total molecular beam density, and Q tot is the total flow rate of all the injected gases in the experiment. The [POCl 3 ] density is known with an accuracy of 5%.

Results and Discussion

Figure 2 shows normalized mass spectra for anions produced by non-dissociative attachment and dissociative attachment to POCl 3 . Measurements were made at three different temperatures 47.7, 74.5 and 199.5K at corresponding flow densities of 2.74, 1.67 and 5.29 ×10 16 molecules/cm 3 respectively and a neutral POCl 3 density of 2.3×10 11 cm -3 . Two sets of peaks have been observed between 0 and 200 Daltons and are primarily associated with the POCl - 3 and POCl - 2 exit channels (1) and ( 5) taking into account the major natural isotopes 1 for details) and the distance between the mass spectrometer and the electron beam is 9 cm. of the chlorine atom 35 Cl and 37 Cl. We did not observe POCl -, Cl -, or Cl - 2 anions, which correspond to the exit channels (2) (3) and ( 4) products, consistent with the endothermicity ∆E of these reactions.

At lower temperatures, 74.5 and 47.7K, neutral clusters may be formed, however, these species lie above the observable 0 -200 Dalton range of the mass spectrometer. A process which could contribute to the POCl - 3 signal is the dissociative electron attachment to (POCl 3 ) n≥2 clusters: (POCl 3 ) n + e -→ POCl - 3 + (POCl 3 ) n-1 . Dissociative electron attachment to clusters is expected to be energetically more favourable than the studied nondissociative attachment to POCl 3 (3) and could correspond to an additional source term for the evolution of the anion signal and a loss term for the electron density. To verify the absence of this process for both the 74.5K and 47.7K temperatures, we have monitored the electron density as a function of the Laval nozzle distance from the electron beam (see Figure 1) while injecting a large concentration 1×10 11 cm -3 of POCl 3 . By moving the nozzle backwards, clusters would have more time to form in the supersonic beam before intersecting the electron beam. This would lead to a change in the electron density if they were present. The measured electron density was found to be constant for a Laval nozzle displacement of ∆z = -30 cm, and therefore, cluster formation can be neglected.

Table 1 shows the proportions of anion signals at 47.7, 74.5, and 199.5K. Contrary to the situation at 199.5K, where competition between the two exit channels (1) and ( 5) is evident, at the lower temperatures there is a clear predominance of the non-dissociative exit channel (5). We have explored the density effect with an additional measurement which has been performed with a supersonic gas flow at 169.7K and a density lower by one order of magnitude, i.e. 0.572×10 16 molecule cm -3 . The corresponding normalized mass spectrum for anions (Figure 3) shows that the fraction of the dissociative exit channel ( 1) is increased compared to the higher density data at the slightly higher temperature of 199.5K. This is qualitatively consistent with the need of a third body stabilization process for the non-dissociative channel (17). Note that the decrease in temperature from 200 to 170K will promote the non-dissociative exit channel over the dissociative one, all else being equal. That the dissociative channel is increased in the 170K data indicates that this effect is much smaller than the decrease in the non-dissociative channel due to the lowered density.

Figure 4 provides a comparison between the POCl the 47.7, 74.5, 169.7 and 199.5K temperatures and the branching ratios of the exit channel ( 5) and ( 1) from (11).

Assuming that the quadrupole mass filter transmission is at least greater for the lighter anions and is independent of the temperature and considering that all the measurements have been performed under the same experimental detection conditions (same biasing on the electrostatic lens and same resolution), the signal proportion of POCl - 3 is probably underestimated compared to that for POCl - 2 . We deduced from these observations that the branching ratio inversion between the exit channels ( 5) and (1) occurs between 300 and 200K.

A temperature dependent branching ratio inversion has also been observed for the electron attachment reaction to CF 3 between 300 and 600K (13). The competition between the dissociative and a non-dissociative exit channels as a function of temperature and pressure were successfully analysed using a kinetic model. Details of the modeling have been presented elsewhere (11; 10; 16). A similar kinetic modeling approach has been previously applied to the system of interest here, electron attachment to POCl 3 , at temperatures between 300 -1200K. We have extended the kinetic modeling to the temperature and pressure conditions here, using the system-dependent parameters previously determined from the higher temperature data and treating the electron temperature as a variable parameter. The resulting branching ratios calculated using the model are presented in Figure 5 as a function of electron temperature. At 170 and 200K, the product branching fraction is a strong function of electron temperature, and the data is only explained by assuming electron temperatures of 450K. This value is in good agreement with prior results (2) indicating that the CRESU apparatus produces an initial electron temperature distribution of 500 -1000K, which thermalizes on the timescale of the experiment. Note that the kinetic modeling was fairly constrained, and could only have reasonably explained a product branching fraction of 0.5-0.7 at 200K and of 0.35-0.65 at 170K. At the lower temperatures of 50 and 75K, the product branching fraction is dominated by the non-dissociative channel and is a weaker function of the electron temperature. The kinetic modeling can explain the observed data by assuming electron temperatures of 100 and 150K, respectively.

Conclusions

In this work we have demonstrated that a temperature dependent branching ratio inversion for the electron attachment to phosphoryl chloride POCl 3 in the gas phase between occurs 300 and 200K. This reaction leads to four dissociative exit channels including the thermoneutral dissociative POCl - 2 + Cl channel and one exother-4

A c c e p t e d M a n u s c r i p t mic non-dissociative POCl - 3 exit channel. In this latter process, the POCl - 3 anion formation requires a third body for its stabilization thus the pressure condition is fundamental. In this work, the POCl - 2 and POCl - 3 ion signals have been observed by mass spectrometry for several temperatures 47.7, 74.5 and 199.5K and a high density about few 10 16 molecule/cm 3 . A kinetic model has been used to obtained the branching fraction of attachment to POCl 3 as a function of the electron temperature. It has been demonstrated that at very low temperature (47.7K), the non-dissociative exit channel predominates while as the temperature is increased, competition begins between the two exit channels. The data confirm that electron temperatures in the CRESU experiment are not thermalized, and is consistent with the previously modeled behavior of the electron temperature (2). The success of the kinetic modeling in explaining the present data suggests that its extrapolation is robust for temperatures down to 50K.

p t e d M a

Highlights

• Charged products from electron attachment reaction on POCl 3 in gas phase are observed by mass spectrometry;

• Temperature dependance of the corresponding exit channels is studied down to 47.7K;

• A kinetic model is used to fit the experimental data.

e→ POCl 2 +→ POCl + Cl - 2 ∆E = 0. 4 eV ( 3 )→

 2243 -+ POCl 3 → POCl - 2 + Cl ∆E = 0.0 eV (1) Cl -∆E = 0.11 eV (2) POCl -+ Cl 2 ∆E = 1.53 eV (4)

Figure 1 :

 1 Figure 1: Supersonic cold uniform beam in the CRESU apparatus at Rennes.

2

 2

  A c c e p t e d M a n u s c r i p t

Figure 2 :

 2 Figure 2: Normalized negative mass spectra for three different temperatures: 47.7, 74.5 and 199.5K. The typical density is about n tot = 10 16 molecules cm -3 (see Table1for details) and the distance between the mass spectrometer and the electron beam is 9 cm.

Figure 3 :

 3 Figure 3: Normalized negative mass spectra for a molecular beam corresponding to a temperature T=169.7K and a density n tot = 0.572× 10 16 molecules/cm 3 .

  -

2 and

 2 POCl - 3 anion signal proportions (%) obtained at 3 A c c e p t e d M a n u s c r i p t

Figure 4 :

 4 Figure 4: ( , ) Black full circles and triangles correspond to the observed POCl - 3 and POCl - 2 signal proportions (%) for the different temperatures 47.7, 74.5 and 199.5K. ( , ) Gray circles and triangles correspond to the POCl - 3 and POCl - 2 signal proportions respectively at 169.7K for which the density is lower, i.e. 0.572 × 10 16 molecule.cm -3 . ( , ) Open circles and triangles are the POCl - 3 and POCl - 2 branching ratios from (11).

Figure 5 :

 5 Figure 5: Non-dissociative branching fraction of attachment to POCl 3 (solid lines) from the kinetic model at the indicated densities and gas temperatures as a function of electron temperature. Full circles (•) are CRESU experimentally measured branching fractions at ill-defined electron temperatures and are plotted at the T e where agreement with the modeling is best.

[ 1 ]

 1 Shuman, N. S., Miller, T. M., Viggiano, A. A., Troe, J., Sep. 2011. International Journal of Mass Spectrometry 306 (2-3), 123-128.

Table 1 :

 1 Proportions of POCl - 2 and POCl - 3 anion signals in this work. The gray lines are branching ratios from 4

	T	n tot	POCl -2	POCl -3	Cl -2	Cl -
	(K)	(×10 16 molecules.cm -3 )	(%)	(%)	(%) (%)
	550	3.2	94	2	4	0.5
	500	3.2	93	3.5	3	0.3
	400	3.2	89	10	2	-
	300	3.2	74	25	1.2	-
	199.5	5.29	41	59	-	-
	169.7	0.572	56	44	-	-
	74.5	1.67	5	95	-	-
	47.7	2.74	2	98	-	-


Pierre-Yves Piriou

Jean-Marc Faure

Jean-Jacques Lesage

A formal definition of Minimal Cut Sequences for dynamic, repairable and reconfigurable systems

published or not. The documents may come

INTRODUCTION

Modern systems are expected to be more and more flexible and dependable. To deal with these needs, system designers have defined reconfiguration strategies, i.e. switching mechanisms that change on-line the system structure and/or behavior. Such reconfigurations can be motivated by functional requirements (e.g. to change the phase of the mission), fault tolerance objectives (management of redundant resources), predictive maintenance policies, production needs etc... The complexity and diversity of reconfiguration strategies are even more important when the system is repairable. These reconfiguration strategies have a significant impact on system safety. In particular, we have shown in [Piriou2014] that the failure of the control part that manages the reconfiguration strategies may have a huge impact on the failure of the whole system.

Qualitative safety analysis aim to link the failure of the system to the failure of its components, what provides important expert knowledge in the design process. For dynamic systems, this knowledge has been formalized through the notion of cut sequence (CS): a scenario (sequence of event occurrences) that leads the system from its initial state to a failure state (without passing through another failure state). For dynamic systems, since the set of CS is very huge, Minimal Cut Sequences (MCS -that can be informally defined as the subset of CS sufficient to the knowledge of all CS) have been defined.

Several definitions of MCS coexist in the literature ( [Rauzy2011], [Walker2007], [Tang2004], [Chaux2013]). Only the two latter ones will be considered in this paper because the other ones do not consider repairable systems. It will be shown first, by using counter-examples, that the definitions of MCS which are proposed in these references are not suitable for reconfigurable systems. A new formal definition will be afterwards proposed, based on the postulate that a dysfunctional sequence is characterized by the order of the events occurrences it includes and by the set of faulty components at the end of the sequence.

In order to illustrate the benefit of this new definition for repairable and reconfigurable systems, the paper proposes next an algorithm for computing MCS from dysfunctional models built by using Generalized Boolean logic Driven Markov Processes (GBDMP). A case study is presented for illustrating both the GBDMP modeling power and the MCS computing principle. The set of MCS obtained is afterwards compared to the one obtained by using the previous definitions of MCS and the differences are analyzed in terms of implication for safety.

The outline of the paper is the following. In first section, a selection of existing definitions of MCS are recalled. Section 2 proposes a new definition allowing to deal with dynamic repairable and reconfigurable systems. Section 3 shows how the MCS set can be computed from a GBDMP model. A case study is addressed in section 4 to illustrate the approach. Finally, concluding remarks and perspectives are drawn up in section 5.

BACKGROUND

[Chaux13] defines the MCS set as the minimal set of sequences of minimal length that are necessary and sufficient to represent the whole set of cut sequences. In this work, the main problem consists in A formal definition of Minimal Cut Sequences for dynamic, repairable and reconfigurable systems P.Y Piriou1 , J.M. Faure 2 & J.J. Lesage 3 1 -Electricité de France, R&D, 78400 Chatou, France 2 -LURPA, ENS Cachan, Univ. Paris-Sud, Supmeca, Univ. Paris-Saclay, 94235 Cachan, France 3 -LURPA, ENS Cachan, Univ. Paris-Sud, Univ. Paris-Saclay, 94235 Cachan, France ABSTRACT: MCS (Minimal Cut Sequences) computation is the main objective of qualitative safety analysis of dynamic systems. This paper shows first that the existing definitions of MCS are not suitable when systems are repairable and reconfigurable, then proposes a new definition for this class of systems. This proposal is illustrated on a case study from power industry. Comparison of the obtained MCS to those which are yielded by algorithms based on previous definitions permits to highlight the relevance of the approach. formally defining the order relation "to represent" between two cut sequences.

The usual notations and vocabulary of the language and automata theory ( [Meduna2012]) are used in what follows.

A language-theory-based problem statement

Let us note:

  the alphabet of events that are to be considered for safety analysis (failures, repairs, etc.).  LE   * the evolution language of the system, i.e. the set of all scenarios built on  that may happen within the system.  LF  LE the failure language of the system i.e. the set of all evolution scenarios that lead the system from its initial state to a failure one.  LCS  LF the set of cut sequences:

LCS = {   LF |      ' , ' ) ( Pref '     

LF}.

 LMCS(R )  LCS the set of Minimal Cut Sequences according to the order relation R,:

LMCS(R )={   LCS|   '  LCS, '  R      ' }
In this definition, the problem is to define the order relation R that relevantly translates the informal relation "to represent".

To illustrate these definitions, let us consider a toy example: two components A and B in standby redundancy driven by a third control component C. When C fails, components A and B remain stuck in their current configuration until C be repaired. The system is faulty when the currently active component (A or B) is faulty. The automaton A that models the dysfunctional behavior of this system is depicted on Figure 1 1 . For this automaton, we have:   {fA, rA, fB, rB, fC, rC}, LE= L(A), LF= Lm(A) and LCS= Lm(A'), where A' is the automaton A for which all transitions leaving a marked state have been removed.

First basic definition of MCS

The first proposition for R is the sequence inclusion, denoted "  " (see [Tang2004] and [Chaux2012]): a sequence  is included into another sequence '  if and only if all events of  are in '  in the same order. This definition is convenient only for non-repairable systems. Indeed, for the example given at Figure 1, we have: LMCS(  )={fAfB, fCfA}. In particular, fAfCrAfB LMCS(  ) because fAfB  fAfCrAfB. Nevertheless, the longer sequence expresses supplementary information on the system dysfunctional behavior: the effect of the failure of C (the resumption of A after its repair cannot be processed). Then fAfB does not represents fAfCrAfB according to the safety point of view, and  is therefore not a suitable order relation for repairable systems.

A definition of MCS based on coherence rules

A promising definition of MCS for dynamic and repairable systems has been proposed in [Chaux2013]. It is based on a definition of the coherence: a system is said coherent if and only if, starting from any failure sequence (   LF) by adding events according to the two following applications a new failure sequence is obtained:

1. insertion of a single failure event.

2. ordered distribution of a set of events that let unchanged the covering cut.

Let us call respectively f1( )  LE and f2( )  LE the sets of all possible sequences that can be obtained by adding events   to  by using applications 1 and 2 mentioned above. f1 and f2 can easily be extended to languages (  L  LE, f1(L)=∪ 𝜎∊L f1( ) and f2(L)=∪ 𝜎∊L f2( )). A coherent system can then be defined as a system verifying the following property:

 L  LF,  nℕ, f2(f1 n (L))  LF (1)
According to [Chaux2013], a sequence  "represents" another sequence '  if and only if '  can be coherently built from  (i.e. by using f1 and f2). Hence a new proposition of R definition arises: the coherence relation ⊨.

   ' ,  LE 2 ,  ⊨   '  nℕ|  '  f2(f1 n ( )) (2)
For the example depicted in Figure 1, with definition (2), LMCS(⊨)={fAfB, fCfA, fAfCrAfB} what is satisfactory. Nevertheless the next section shows that this definition is not suitable for reconfigurable systems.

A NEW DEFINITION OF MCS

Limitation of the coherence-based definition of MCS

The dysfunctional behavior of reconfigurable systems cannot be always described with only failure and repair events. For example, Figure 2 shows the behavior of a system including two components (A and B) that perform a mission in two phases. During the first phase A and B are in standby redundancy, whereas during the second one both are required. On Figure 2, 𝜑 represents the event of phase switching. This system is not coherent according to (1). Indeed, the inserting of event    at the end of the failure sequence 𝜑fA does not result in a failure sequence although the covering cut remains unchanged. Therefore the notion of coherence cannot be directly extended to all reconfigurable systems.

A new minimality criterion for CS

For dynamic systems the relative order of event occurrences matters to determine the system failure. Therefore the set of MCS should be defined using a relation that preserves this order (like the sequence inclusion). Nevertheless, as for static systems, the dysfunctional behavior of a dynamic system is linked to the failure of its components. The set of MCS should consequently be defined using a relation that preserves the set of faulty components (like the set inclusion).

This observation allows us to postulate that a cut sequence is characterized not only by the order of events but also its covering cut. Let us remark that this postulate complies with the nature of applications f1 and f2 defined in subsection 1.3.

Based on this postulate we can propose a new order relation to define the MCS set: the cut sequence inclusion ⋐.

   ' ,  LE 2 ,  ⋐         ' ' '          (3)
This relation is less restrictive than the coherence relation. Indeed, we can easily check (4) but the inverse is not true.

   ' ,  LE 2 ,  ⊨  '   ⋐ '  (4)
In particular, the relation ⋐ allows to determine satisfying sets of MCS for the first example (cf. Figure 1 The role of a switch is to set/reset the requirement statuses of the nodes that are connected to its outputs according to the values of its inputs and the reconfiguration strategy which is described by the associated Moore machine. In the Moore machine M1 at Figure 3 c, let q0 be the current state. In this state, C1 is activated in operation mode 1, C2 is activated in operation mode 1 and C3 is deactivated. The transition between state q0 and state q1 is fired if the associated condition "(W,True)" is true, i.e. if the SMP of C4 (input #0 of S1) is in state W and if the output of Gate G2 (input #1 of S1) is True. The firing of this transition implies the change of active state of M1 (which becomes q1) and the change of outputs values: the requirement of G2 (output #0 of S1) is reset and the requirement of G3 (output #1 of S1) is set. As a consequence, C1 is deactivated, C2 is activated in operation mode 2 and C3 is activated in operation mode 1.

Translation of a GBDMP into a finite state automaton.

For qualitative analysis, assuming that the initial state of a GBDMP is deterministic, its semantics can be seen as a finite deterministic state automaton. This subsection describes how a well-formed2 BDMP <V, E, K, υ, str, smp> can be translated into an automaton < 𝛴, Q, q0, QM, 𝛿 >.

The GBDMP evolution is driven by the occurrence of two types of events. On one hand the spontaneous events (solid arrows in Figure 3 b) correspond to the perturbations of the system (mainly failures and repairs of a leaf). On the other hand the provoked events (dashed arrows in Figure 3 b) correspond to the reconfigurations of the system (operation mode change of a leaf). Obviously all spontaneous events are important for safety, whereas only the provoked events that change the failure status of the leaf are. Assuming that components cannot be repaired during an operation mode change, only the failure provoked events (e.g. on-demand failures) have to be considered for safety analysis. Hence, alphabet 𝛴 is constituted of the set of all spontaneous events (solid arrows of the SMP of all leaves) and failure provoked events (dashed arrows from a faultless state to a faulty state in the SMP of every leaf).

The global state of a GBDMP model is fully determined from the local states of every SMP and Moore machine. According to the initial state of Moore machines, it is possible to determine the initial state of SMP, what gives the initial global state q0 of the GBDMP model. Hence the set of states Q is the set of combinations of these local states reachable from q0.

The definition of the transition function 𝛿 can easily be deduced from the evolution rules of a GBDMP, which are formally described in [Piri-ou2016]. Q can be computed recursively by applying 𝛿 from q0.

Finally, for a given analysis, one must select a gate of the GBDMP whose failure is the undesirable event. Then the set of marked states QM is simply the subset of states where the selected gate is faulty.

An algorithm to calculate MCS from a GBDMP model.

Algorithm 1 allows computing the set of MCS on the fly by performing a breadth-first exploration of the state space Q. The breadth-first exploration ensures that for each new cut sequence found, it is possible to determine if it is minimal by comparing it with the already found MCS (lines 14-16). The exploration from a sequence is stopped if one of the two following conditions is met (lines 12-13):

1 which gives the set of events that may occur in a given state of the GBDMP.

Let us remark that this algorithm can easily be applied with other definitions of MCS, by replacing the relation ⋐ at line 15 by another one (assuming that this relation is also based on the growing of sequences).

The complexity of this algorithm is 𝓞(Card(Q))!²). Despite this complexity, the algorithm is applicable in practice because the set of MCS is computed on the fly by increasing order of length. For qualitative analysis, the longer a MCS is, the less relevant it is. The set of MCS computed by the algorithm in a reasonable time constitutes therefore generally a relevant enough qualitative result (even if incomplete).

APPLICATION

Controlled coolant feeding system

In order to illustrate the presented approach, the system depicted in Figure 4 is used. It is a very simplified version of a part of the cooling system of nuclear power plants proposed by the company Electricité de France. Its main function is to feed a downstream system with a cooling fluid by using two groups of pumps. The first group of three pumps is powered by a heavily redundant electric power supply whose components are repairable. The system fails when fluid can no more be provided. Three stages of electric energy supply and two pumping stages can be identified:  The fluid is extracted by the group of pumps Ex1, Ex2 and Ex3 (E). Only two pumps are used during operation; if one pump fails the third one is activated (standby redundancy 2 out of 3).


 Pumps Pr1 and Pr2 (F) pressurize the fluid; only one pump is used during operation. If the main pump Pr1 fails, the spare pump Pr2 is activated (standby redundancy 1 out of 2).

The subsystem {B, C, D} is duplicated in order to provide a standby redundancy for powering the ex-traction pumps. We consider that diesel generators and pumps may fail when they are active or in standby, whereas the other components may only fail when they are active.

Thanks to the multiple redundancies, lots of reconfigurations are possible in this system for maintaining the production, even in case of multiple failures of components.

GBDMP model

The GBDMP of Figure 5 models the possible dysfunctional behaviors of the coolant feeding system depicted at Figure 4. The structure of the system that makes that certain ordered combinations of failure and repair events of components lead to the global failure te (top event) is given by the fault tree Figure 5a.

Each one of the 14 components (including the grid) is associated to a leaf of the GBDMP. Components that may fail in working mode and in standby mode are associated to a leaf of type SF ("Standby Failure"); components that may fail only in working mode are associated to a leaf of type F ("simple Failure"). The corresponding SMP are given in Fig- ure 5b.

Each time a redundancy between several components has to be managed, a switch is introduced. The reconfiguration strategy which is chosen for each switch is modeled by a Moore machine, associated to this switch. Two kinds of switches are used in this study case. Switches of type M3 express a strategy "1 out of 2, latest replacement, latest resumption" (replacement occurs when the main component fails if the spare component is available, and resumption occurs when the spare fails if the main is available); the switch of type M2 expresses a strategy "2 out of 3, latest replacement, latest resumption". The corresponding Moore machines are given in Figure 5c. Finally, a control equipment in charge of executing the reconfiguration is associated to each switch (leaves RA to RE depicted by grey boxes at Figure 3a). A failure of the switch controller implies the loss of reconfiguration ability.

Comparative study

The application of Algorithm 1 to the GBDMP of Figure 5 gives the result L1 = LMCS(⋐). A selection of elements of this language is reported on Table 1 3 . This result has been produced in 45 minutes on a standard laptop (2,9 GHz). The first 200 sequences were found in less than one minute. Moreover, all cut sequences of length inferior or equal to 5 (resp. 6) were found in less than 5 minutes (resp. 55 minutes).

This case study has been addressed differently in [Chaux2012]. Firstly the system was modeled using BDMP and not GBDMP (the modifications correspond to the grey elements on Figure 5). Secondly, the definition of MCS was based on ⊆ instead of ⋐.

Let us compare L1 with L2 = LMCS(⊆), computed from the BDMP model. Bold sequences in Table 1 do not belong to L2; moreover some sequences of length 5 and 6 belong to L2 but not to L1 (of course they are not displayed in this table).

One of the most important improvement from BDMP to GBDMP is the replacement of the trigger primitive by the switch one. Both translate a reconfiguration mechanism in the model, but a trigger refers to a unique strategy of switching that may not fail [Piriou2016]. On the other hand, in GBDMP, the Moore machines associated to the switches allow to describe multiple reconfiguration strategies that may fail. Hence the sequences that are not in the intersection of L1 and L2 are explained by at least one of the three following reasons:

1. ⋐ is less restrictive than ⊆ for the MCS calculus. fa Grid fa Di1 rp Grid fa TEa2 fa TEb1 . It is not a MCS according to the relation ⊆ because it includes the MCS fa Grid fa Di1 fa TEa2 . However, it is a MCS according to the relation ⋐ because: [fa Grid fa Di1 fa TEa2 ] = {Grid, Di1, TEa2}, whereas [fa Grid fa Di1 rp Grid fa TEa2 fa TEb1 ] = {Di1, TEa2, TEb1}.

The differences coming from the modeling of reconfigurations failures can be illustrated by the sequences fa RB fa Pr1 and fa Pr1 fa RB rp Pr1 fa Pr2 that are respectively symptomatic of the failure of the replacement and the resumption of the pump Pr1.

Let us now exemplify the differences explained by the third reason using two sequences:  fa Ex1 rp Ex1 fa Ex3 fa Ex2 ∈ L1 \ L2. Indeed, according to the GBDMP model, the resumption of Ex1 does not occur after its own repairs. Then fa Ex3 fa Ex2 can occur after that. On the contrary, with the BDMP model, Ex1 is resumed just after it is repaired and Ex3 is put in standby mode, the event fa Ex3 is consequently not possible. Then the considered sequence is not even in the evolution language LE of the model.

 fa TEa1 fa TEb1 rp TEa1 fp Di2 fa Tea1 ∈ L2 \ L1. Indeed, according to the GBDMP model, the resumption of power supply on the first line following to the repairs of TEa1 is not immediate (contrarily to what describes the BDMP model). Then Di2 may fail after that only while it is active, what would result in the sequence fa TEa1 fa TEb1 rp TEa1 fa Di2 fa Tea1 which is not minimal because: fa TEa1 fa TEb1 fa Di2 ⋐ fa TEa1 fa TEb1 rp TEa1 fa Di2 fa Tea1 . Finally, some differences between L1 and L2 are explained by a combination of the three abovementioned reasons. For example, the sequence fa Pr1 rp Pr1 fa RB fa Pr2 ∈ L1 \ L2 because of the three reasons.

CONCLUSIONS

In this paper a new definition of MCS has been proposed to perform a more accurate qualitative safety analysis for dynamic repairable and reconfigurable systems. This definition is suitable for any dynamic system. Moreover an algorithm to compute the MCS set of a system modeled in GBDMP has been proposed in order to take advantages of the new definition. The comparative study performed shows the benefits of this approach: the model is more precise and the analysis results are more relevant.

Nevertheless, even if the qualitative analysis supplies relevant information on the system dysfunctional behavior, it is not enough to validate that a design meets its safety requirement. Indeed a short MCS can be highly improbable. To combine the MCS calculus with the probability assessment of the system failure is a promising idea to improve the relevance of safety analysis. [Brameret2015] defined a factor to order the states of a Markov chain according to their probabilistic relevance. Given that a GBDMP model describes a Markov chain by intention, the Algorithm 1 can be improved by introducing a heuristic based on the probabilistic relevance factor to drive the state space exploration.

Figure 1 .

 1 Figure 1. Automaton modeling two components A and B in standby redundancy driven by a third control component C.

Figure 2 .

 2 Figure 2. Automaton modeling two components A and B that perform a phased mission.

  ): LMCS(⋐) = LMCS(⊨); and for the second (cf. Figure 2): LMCS(⋐) = {fAfB, fA𝜑, 𝜑fA, 𝜑fB}. The next section shows how this new definition can be used to compute the set of MCS from a GBDMP model.4 COMPUTING MCS FROM A GBDMP MODEL4.1 Recall on GBDMP formalismGeneralized Boolean logic Driven Markov Processes (GBDMP) is an extension for reconfigurable systems of the BDMP formalism defined in [Bouissou03] for safety analysis of dynamic and repairable systems. Basically, a GBDMP integrates in the same model a representation of the system structure (Fault Tree), the dysfunctional behaviors of the components of the system (Switched Markov Processes) and the reconfiguration mechanisms (Moore Machines).The bases of GBDMP syntax and semantics are now briefly given and exemplified. They have been formally and broadly presented in[Piriou2016].Definition 1. A Generalized Boolean logic Driven Markov Process is a 6-tuple <V, E, K, υ, str, smp> where: V = N  S = G  L  S isa set of vertices partitioned into the nodes N (i.e. the gates G and the leaves L) and the switches S;  E = EF  ES is a set of oriented edges, such that EF  G X N and ES  (N X S)  (S X N);  K: G  ℕ* is a function that determines the gates kind. This function is the same as the one used in BDMP [2];  υ : E  ℕ is a function that associates an integer label to each edge;  str: S  𝕄 is a function that associates a Moore machine (which represents a reconfiguration strategy) to each switch. 𝕄 designates the set of Moore machines;  smp: C  ℙ is a function that associates aSMP to each component (a k-SMP for a component with k operation modes). ℙ designates the set of Switched Markov Processes.A simple GBDMP is shown at Figure3. The structure of the system is represented by a fault tree (part a of Figure3). It is composed of 3 gates (G1 is an AND gate, G2 and G3 are OR gates), 3 basic components (leaves C1, C2, C3 and C4) and a switch (S1 depicted with a dashed rectangle). The solid (resp. dashed) arrows are the edges of EF (resp. ES), which connect the gates to the nodes (resp. the switches to the nodes and the nodes to the switches). The dysfunctional behavior of the leaves C1, C2 and C3 is depicted by the SMP "Pu" at part b of Figure3. The component C4 is in charge of the control of the switch, its dysfunctional behavior is depicted by the SMP "Co" at part b of Figure3. The reconfiguration strategy implemented in switch S1 is modelled by the Moore machine at part c of Figure3. The label, given by function υ, of an edge of EF and ES permits to associate respectively a branch to an operation mode of a leaf and a number of input or output of the Moore machine to a node.

Figure 3 :

 3 Figure 3: Example of GBDMP. a) Structure modelling; b) SMP Pu (associated to C1, C2 and C3) and Co (associated to C4); c) Moore machine M1 (associated to S1) The behavior of a leaf is modeled by a k-SMP which is composed of k Markov chains. Each Markov chain corresponds to an operation mode and comprises faultless and faulty states; the transitions between these states are stochastic and they model mainly failures and repairs. In the example of Figure 3 b, the 3-SMP associated to the leaves C1, C2 and C3 comprises three Markov chains (one for each line) to represent a component with two working modes and one standby mode; in this model, it is assumed that no failure occurs in the standby mode and that the failure rate in the second working mode is greater than the corresponding rate in the first working mode. k(k-1) probabilistic transfer functions between the chains of a k-SMP must be defined. The value of the transfer function between two states of two different chains (in dashed arrows) is equal to 1 if no failure on-demand is considered (case of Figure 1 b) when the operation mode is changed and belongs to [0,1] otherwise.



  An electric transformer Tr1 (A) connected to the grid, is used to provide low voltage electricity; if it fails, a second transformer Tr2 is available thanks to a standby redundancy.  A distribution board TEb1 (B) powering a second distribution board TEa1 (D) using one of the two transformers Tr1 or Tr2. A diesel generator Di1 (C) is in standby redundancy with subsystem TEb1. The lower level distribution board TEa1 (D) powers the group of extraction pumps Ex1, Ex2 and Ex3 (E), using one of the two possible sources (TEb1 or Di1).

Figure 4 .

 4 Figure 4. Physical architecture of the coolant feeding system.

  TEb1 fa Di1 fp Di2 fa Tr2 fa Grid rp Grid fa RD fp Di2 fa Di1 fa Tr1 fa RE fa RD rp Tr1 fa Tr2 fa Grid fa Di1 rp Grid fa TEa2 fa TEb1 Length 6: 1560 fa RE fa TEb1 fa Di1 rp TEb1 fp Di2 fa Tr1 fa Grid fa Di1 rp Grid fa TEa2 fa RE fa Tr1 fa Tr1 fa Tr2 fa RD fa Di1 rp Tr1 fa Di2 fa TEa1 fa Tr1 rp TEa1 fa Tr2 fp Di1 fa TEa2

  Figure5. GBDMP of the controlled coolant feeding system.

RB fa Pr1 fa RD fa Grid

	Total number of MCS	Selection of MCS
		fa Ex1 fa Ex2
	Length 2:	fa TEa1 fa TEa2
	14 fa Length 3: Fp Di1 fa Grid fa Di2 fa TEa1 fa TEb2 fa Di2
	37

fa RE fa Grid fa Di1 fa RC fa RD fa TEb1

	fa Tr1 fp Di1 fa Tr2 fa TEa2
	Length 4:
	93

fa Ex1 rp Ex1 fa Ex3 fa Ex2 fa Pr1 fa RB rp Pr1 fa Pr2 fa Pr1 rp Pr1 fa RB fa Pr2

Table 1 .

 1 Extract of LMCS(⋐) for the coolant feeding water 3 f and r mean failure and repairs; a and p mean active and passive. Then fp Tr1 represents the event: failure of the leaf Tr1 while it is passive (mode 0).

fX: failure of X ; rX: repair of X ; white states: A is active and B is inactive ; grey states: A is inactive and B is active. The marked states are double-circled.

In[Piriou2016] 5 syntactic properties guaranteeing that a GBDMP is well-formed are given.


Erwann Delay
email: erwann.delay@univ-avignon.fr

CONFORMALLY COVARIANT PARAMETERIZATIONS FOR RELATIVISTIC INITIAL DATA

Keywords: Conformal riemannian geometry, non linear elliptic systems, vectorial laplacian, general relativity, constraint equations 53C21, 53A45, 53A30, 58J05, 35J61

We revisit the Lichnerowicz-York method, and an alternative method of York, in order to obtain some conformally covariant systems. This type of parameterization is certainly more natural for non constant mean curvature initial data.

Introduction

On a smooth manifold M of dimension n, given a riemannian metric g, we denote by R g its scalar curvature and ∇ its Levi-Civita connection. If h is a symmetric covariant two tensor field, we define its divergence, as the 1-form given by (div g h) i = -∇ k h ki .

The vacuum initial data for the relativistic Einstein equations are given by a riemannian metric g and a symmetric two tensor field K, satisfying the constraint equations (C) R g -| K| 2 g + (Tr g K) 2 = 0 div g K + d(Tr g K) = 0 .

This system is highly under-determined because it contains (n + 1) equations for n(n + 1) unknowns . It is natural to fix some of the unknowns, and to look for the (n + 1) remaining ones.

In the usual conformal parameterization, appears the 1944 Lichnerowicz equation [START_REF] Lichnerowicz | L'intégration des équations de la gravitation relativiste et le problème des n corps[END_REF] together with the 1973 York decomposition [START_REF] York | Conformally invariant orthogonal decomposition of symmetric tensors on riemannian manifolds and the initial value problem of general relativity[END_REF]. It was actively studied by many authors, on compact or non compact manifolds (asymptotic to some models : euclidean, hyperbolic, cylindrical,...).

For a glimpse at the subject, we just mention the recent paper [START_REF]The conformal method and the conformal thin-sandwich method are the same[END_REF] and its references. Otherwise, too many articles should be quoted.

We only recall that after the approaches realized for constant mean curvature τ , like [START_REF] Isenberg | Constant mean curvature solutions of the Einstein constraint equations on closed manifolds[END_REF] for instance, several attempts were made in order to allow for variable mean curvature.

The classical method (historically method A, see [START_REF] Bartnik | The constraint equations, The Einstein equations and the large scale behavior of gravitational fields[END_REF] section 4.1 for instance, also called conformal TT method [START_REF] Cook | Comparing initial-data sets for binary black holes[END_REF]) starts with a given metric g, together with a trace free and divergence free symmetric two tensor σ (a TT-tensor), and a function τ . One looks for the solutions of (C) of the form

g = φ N -2 g , K = τ n g + φ -2 (σ + Lg W ), (P )
where N = 2n n-2 , and the unknowns are a function φ > 0 and a one form W and where

( Lg W ) ij = ∇ i W j + ∇ j W i - 2 n ∇ k W k g ij .
Setting w = |σ + Lg W | g , we infer from (C) and (P ) the coupled system (S)

P g,w φ := 4(n-1) n-2 ∇ * ∇φ + R g φ -w 2 φ -N -1 = 1-n n τ 2 φ N -1 (L) div g Lg W = 1-n n φ N dτ (V )
The Lichnerowicz equation (L) has a covariant conformal property (see section 7.2 for a precise definition). Indeed, if φ is a solution of (L), and ϕ is any positive function, we may define

g = ϕ N -2 g , w = ϕ -N w , φ = ϕ -1 φ.
Doing so, we find

P g, w φ = ϕ 1-N P g,w φ = 1 -n n τ 2 φN-1 .
In contrast to (L), the vectorial equation (V ) does not possess such a property. Moreover, the conformal transformation of w used here for (L) does not correspond to a natural transformation of σ and W .

However we note that the operators appearing in (V ) are separately conformally covariant. Specifically

div g h = ϕ -N div g (ϕ 2 h) , Lg X = ϕ N -2 Lg (ϕ 2-N X). (C)
But the vector Laplacian div • L is not so.

Finally, we recall the York decomposition [START_REF] York | Conformally invariant orthogonal decomposition of symmetric tensors on riemannian manifolds and the initial value problem of general relativity[END_REF] valid for instance if M is compact and g has no conformal Killing fields (i.e. ker Lg is trivial). Any covariant symmetric trace free two tensor field h splits in a unique way as h = σ + Lg W, (Y ) where σ is a TT-tensor and W a 1-form.

We define in the section 2 a parameterization (method B of York) derived from the York decomposition relative to g instead of g. This parameterization gives rise to a conformally covariant system.

The latter one leads to a new vectorial laplacian, which is self adjoint for a weighted measure. This laplacian will be studied in section 3.

In section 4 we further discuss some aspects of the method B. Amazingly, this method seems widely ignored in the literature neither pure nor numerical (see however [START_REF] Cook | Comparing initial-data sets for binary black holes[END_REF]).

In section 5 we propose further parameterizations (including conformally covariant ones) and compare them.

Finally, in section 6, we give some comments related to the continuation of this work.

Acknowledgements: I am grateful to Ph. Delanoë for his multiples comments and to J. Isenberg for pointing out to me that J. York [START_REF] York | Conformally invariant orthogonal decomposition of symmetric tensors on riemannian manifolds and the initial value problem of general relativity[END_REF] had already discovered the method B, as independently proposed in the earlier version of this paper. I also thank P. Chruściel and R. Gicquaud for their comments.

A first conformally covariant parameterization

We recall here the method B of York (see section 4.1 of [START_REF] Bartnik | The constraint equations, The Einstein equations and the large scale behavior of gravitational fields[END_REF] or the original paper [START_REF] York | Conformally invariant orthogonal decomposition of symmetric tensors on riemannian manifolds and the initial value problem of general relativity[END_REF] page 461), also called physical TT method (see [START_REF] Cook | Comparing initial-data sets for binary black holes[END_REF] for instance). Getting back to (C) and now using the York decomposition relative to g, namely

K - τ n g = σ + L g W ,
we find that (C) is equivalent to

(C ′ )        R g -| σ + L g W | 2 g = 1 -n n τ 2 div g L g W = 1 -n n dτ div g σ = 0
If we are looking for a solution in a conformal class g = φ N -2 g, from (C), it is natural to introduce σ = φ -2 σ, where σ is a TT-tensor. The third equation is then automatically satisfied. Still using (C), and expressing its second equation in terms of g = φ 2-N g, we are prompted to set W := φ N -2 W .

Sticking to the same fixed g, σ, τ , we can thus parametrize the solutions of the constraint (C) by

(P ′ ) g = φ N -2 g , K = τ n g + φ -2 (σ + φ N Lg W )
With this parameterization in (C), setting

ω = ω(σ, φ, W, g) := |σ + φ N Lg W | g ,
we obtain the new system

(S ′ ) P g,ω φ = 1-n n τ 2 φ N -1 (L ′ ) ∆ g,φ W := φ -N div g (φ N Lg W ) = 1-n n dτ. (V ′ ) We now make the conformal changes: g = ϕ N -2 g , φ = ϕ -1 φ , σ = ϕ -2 σ , W = ϕ N -2 W.
Note that by (C), the tensor σ is still TT for g. Now, we further have :

ω := ω(σ, φ, W , g) = ϕ -N ω
so the scalar equation (L ′ ) behaves like (L) as explained above. As for the corresponding vectorial equation (V ′ ), we now have : ∆ g, φ W = ∆ g,φ W, due to (C). In other words, this equation is now conformally covariant.

Remarks :

• When ϕ = φ we find ∆ g,φ W = ∆ g,1 W .

• Since ω depends on φ, let us give the more explicit form of (L ′ ), namely :

4(n -1) n -2 ∇ * ∇φ + R g φ -|σ| 2 φ -N -1 -2 σ, Lg W φ -1 -| Lg W | 2 g φ N -1 = 1 -n n τ 2 φ N -1 . (L ′ )
• Let us consider the operator given by

P g   φ W σ   :=   φ 1-N P g,ω φ ∆ g,φ W φ -N div g σ   .
This operator is conformally covariant :

P g   φW σ   = P g   φ W σ   .
The system (S ′ ) (modulo the third equation added) simply reads :

P g   φ W σ   = 1 -n n   τ 2 dτ 0   .
• There has been other attempts to modify the Lichnerowicz-York method, like the "Conformal Thin-Sandwich"(see for instance [START_REF]The conformal method and the conformal thin-sandwich method are the same[END_REF] for a comparison, see also [START_REF]Initial data in general relativity described by expansion, conformal deformation and drift[END_REF]). In particular, there exists already parameterizations where

K = τ n g + φ -2 (σ + f Lg W )
but so far the function f would usually be given and would not depend on φ, whereas f = φ N is the simplest one to make the system conformally covariant. We will see in section 5 further possibilities, including conformally covariant ones.

• More generally, for each conformally covariant differential linear operator of order k, we would interpose a function between the operator and its formal adjoint, in order to produce an operator of order 2k, conformally covariant positive and self adjoint for a weighted measure.

The vectorial equation

Let us consider the vectorial laplacian, of Witten type, obtained previously :

∆ g,φ W = φ -N div g φ N Lg W = div g Lg W -N Lg W (∇ ln φ, .).
We choose to work here on a compact manifold. Since 2 div g is the formal L 2 (dµ g ) adjoint of Lg , we have

M ∆ g,φ W, V g φ N dµ g = 1 2 M Lg W, Lg V g φ N dµ g
Our Laplacian is then L 2 (φ N dµ g ) self adjoint and its kernel is reduced to conformal killing 1-forms.

We want to solve for W the equation

∆ g,φ W = 1 -n n dτ , (V ′ )
with g, φ, τ smooth for simplicity (we could also choose φ ∈ L ∞ ). By the Fredholm alternative, a necessary and sufficient condition for solving (V ′ ) is orthogonality of the right-hand side dτ to the kernel:

M X, dτ g φ N dµ g = 0, (⊥)
for all X in ker Lg . Uniqueness of W occurs up to the addition of an element of ker Lg .

Note that if X = ϕ N -2 X , X, Y g φN dµ g = X, Y g φ N dµ g .
Hence the condition (⊥) is conformally covariant due to (C). It can be written

M τ d * g (φ N X)dµ g = 0. (⊥ * )
Of course, if the metric g does not possess some conformal Killing field, the equation (V ′ ) always has a unique solution whatever the (positive) function φ is.

If K = ker L is not trivial, we may define Π g,φ the orthogonal projection on K for the weighted scalar product :

(X, Y ) g,φ = M X, Y g φ N dµ g .
We can then solve

∆ g,φ W = 1 -n n (Id -Π g,φ )dτ, (V ′ Π )
W being unique if it is chosen orthogonal to K (for the weighted scalar product).

Remark : By analogy with the Witten laplacian, we could consider the similar laplacian φ -N 2 div(φ

N 2 φ N 2 Lφ -N 2 ).

Further versions of the method B

We develop some variants of the method B.

A linear version.

Let us linearise operator P g introduced in section 2. We denote by (ψ, V, h) the variations of (φ, W, σ), υ the variation of τ , we keep here the metric g fixed. Because of the conformally covariant properties of P g , the computation of the linearisation can be done at φ = 1. We find

DP (1,W,σ) g   ψ V h   =   P g,σ,W -2 K, L. , -2 K, . -N Lg W (∇ g ., .) div g Lg 0 0 0 div g     ψ V h   where K = σ + Lg W, and
P g,σ,W = 4 n -2 (n -1)∇ * ∇ -R g + n|σ| 2 + n σ, Lg W = 4 n -2 (n -1)∇ * ∇ -R g + n 2 |σ| 2 + | K| 2 -| Lg W | 2 .
The latter operator, evaluated at a solution of (S ′ ), can be written using the Yamabe laplacian, because

R g = | K| 2 + 1 -n n τ 2 .
The linear version of the system becomes

DP (1,W,σ) g   ψ V h   = 1 -n n   2τ υ dυ 0   .
If σ is fixed, it forces h = 0, if not, the global form of the system suggests to look at other possibilities.

4.2.

Another conformally covariant parameterization. In the parameterization of the section 2, we fixed a T T -tensor σ and defined σ = φ -2 σ in order to verify immediately the last equation of the system (C ′ ). Without this particular choice , (C ′ ) has 2n + 1 equations. Alternatively, we can fix instead a trace free tensor σ and look for the unique TT-tensor σ for g, such that

φ -2 σ = σ -L g Y .
Doing so we should define

σ = φ -2 (σ + φ N Lg Y ), ̟ = |σ + φ N Lg Y + φ N Lg W |.
It yields the conformally covariant system

(S ′′ )        φ 1-N P g,̟ φ = 1-n n τ 2 (L ′′ ) ∆ g,φ W = 1-n n dτ (V ′′ 1 ) ∆ g,φ Y = -φ -N div g σ. (V ′′
2 ) The matching linear operator has the form :

  φ 1-N P g,̟ φ ∆ g,φ ∆ g,φ   ′ =   P g,σ,W,Y -2 K, L. , -2 K, L. -N Lg W (∇ g ., .) ∆ g,1 0 -N Lg Y (∇ g ., .) 0 ∆ g,1   where K = σ + Lg Y + Lg W, and
P g,σ,W,Y = 4 n -2 (n -1)∇ * ∇ -R g + n|σ| 2 + n σ, Lg (W + Y ) = 4 n -2 (n -1)∇ * ∇ -R g + n 2 |σ| 2 + | K| 2 -| Lg (W + Y )| 2 .
This operator evaluated at a solution of (S ′′ ), can again be written with the Yamabe laplacian, because

R g = | K| 2 + 1 -n n τ 2 .
Getting back to (S ′′ ), we realize that X = Y + W is the only important variable and a natural system to solve is

(S ′′′ ) φ 1-N P g,̟ φ = 1-n n τ 2 (L ′′′ ) ∆ g,φ X = 1-n n dτ -φ -N div g σ (V ′′′ )
where the function τ and the trace free symmetric two tensor σ are given. The parameterization for the solutions of (C) will then be

g = φ N -2 g , K = φ -2 (σ + φ N Lg X) . (P ′′′ )
Remark : We recover the system (S ′ ) if we choose σ divergence free.

Further parameterizations and matching

Let us propose a general parameterization, from which the conformal methods A and B, and the conformal thin sandwich method are particular cases.

Let g = ϕ N -2 g be another conformal metric. The York decomposition (Y ), related to g, of the tensor

φ 2 ϕ -2 ( K - τ n g),
leads to the parameterization

(P ϕ ) g = φ N -2 g , K = τ n g + φ -2 (σ + ϕ N Lg W ).
Using this parameterization in (C), and setting

ω ϕ = ω(σ, ϕ, W, g) := |σ + ϕ N Lg W | g ,
we obtain the new system

(S ϕ ) P g,ωϕ φ = 1-n n τ 2 φ N -1 (L ϕ ) φ -N div g (ϕ N Lg W ) = 1-n n dτ (V ϕ ).
If g has no conformal Killing field, we infer from (Y) that, for each one form W and each positive function ϕ, there exists a unique 1-form V and a TT-tensor σ ′ such that

σ ′ + Lg V = ϕ N Lg W.
It follows that the solutions of the systems (S ϕ ) match, but for different σ.

Explicitly, (L ϕ ) reads in the following way

4(n -1) n -2 ∇ * ∇φ + R g φ -|σ| 2 φ -N -1 -2 σ, Lg W ϕ N φ -N -1 -| Lg W | 2 g ϕ 2N φ -N -1 = 1 -n n τ 2 φ N -1 (L ϕ ).
Here, it is important to note that we could let ϕ depend on φ and possibly on some other parameters in the (then abusively denoted) system (P ϕ ).

The conformal method A consists in choosing ϕ = 1, the conformal method B arises when ϕ = φ, and for ϕ a fixed positive function, we obtain the conformal thin sandwich method.

But many other choices can be made. For instance ϕ N = φ N F 2 (| LW | g ), for a given function F , will provide again a conformally covariant system of order of derivation 2. When possible, the parameterizations

ϕ N = c|τ |φ N or ϕ N = c |τ |
| LW |g φ N seem also to yield interesting systems.

Comments and prospects

We give here some comments and hints at a future study of conformally covariant systems, starting with the case of compact manifolds.

• Other York decompositions : Conformally covariant or not, in section 5, some other choices of ϕ = ϕ(φ, W, σ, τ, t, f, ...) in (S ϕ ) may be judicious.

• Constraints with right-hand side : The parameterizations works for some other stress energy tensors like a scalar field for instance.

• Matching of parameterizations : Due to the matching between the differents parameterizations, already existing solutions of (S) provide solutions of (S ′ ), using a change of parameters. If we view the solutions set of (C) as a manifold, we expect the conformal method to give a local chart, whereas a conformally covariant method would give a larger chart and allow for a larger choice of (τ, σ) (or (τ, σ) for (S ′′′ )).

• Variational study : The systems we obtained are not triangular like for the classical conformal method. However, a variational study seems well suited.

• fixed point : As in many previous studies of (S), a classical method can be used to solve a system like (S ′ ), namely by considering a map T : E → E, with an appropriate function space E, defined as follows. If there is no conformal Killing fields, given ψ in E, we consider the solution W of (V') with φ replaced by ψ. We then define ω := ω(σ, ψ, W, g) and solve (L ′ ) (see for instance [START_REF] Maxwell | A class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature[END_REF]). The solution defines the map T (ψ) := φ. Now, we look for a fixed point φ > 0 of T . If necessary, here, one could insert some φ fitting some ψ in the choice of ω (and/or in (L ′ )) then solve another scalar equation, similar to (L ′ ) on φ, for example linear with respect to φ. If there exists a conformal Killing field, we can replace in the above process, the solution of (V ′ ) by that of (V ′ Π ). The projection disappears in the limit under appropriate conditions.

• The limit equation : In [START_REF] Gicquaud | A limit equation associated to the solvability of the vacuum einstein constraint equations by using the conformal method[END_REF] a (family of) limit equation(s) is proposed for (S). If the limit equation admits only the zero solution, it has the striking property to guarantee the existence of solutions of (S). We will have to check if the limit equation measures the asymmetry of the usual parameterization (method A) or if a similar equation exists for the system (S ′′′ ) (method B). Depending of the choice of ϕ in section 5, an associated limit equation may also appear. A differential system will be said conformally covariant if it can be written in the form P g (e) = f , for a conformally covariant operator P g .

7 . Appendix 7 . 1 . 7 . 2 .

 77172 Vector laplacian.div g Lg = ∇ * g ∇ g -Ric g + n-2 2 dd * g = ∆ Hodge -2 Ric g + n -Conformal covariance.Let us consider three products of tensor bundles over M,E = E 1 × ... × E k , F = F 1 × ... × F l , G = G 1 × ... × G m ,and a differential operator acting on the sections :P g : Γ(E) -→ Γ(F ),with coefficients determined by g = (g 1 , ..., g m ) ∈ G. We will say that P g is conformally covariant if there exist a = (a 1 , ..,a k ) ∈ R k , b = (b 1 , .., b l ) ∈ R l and c = (c 1 , .., c m ) ∈ R m suchthat for each smooth section e of E, and every smooth function ϕ on M, we haveϕ b ⊙ P ϕ c ⊙g (ϕ a ⊙ e) = P g (e),where ϕ a ⊙ e = (ϕ a 1 e 1 , ..., ϕ a k e k ).