Sébastien Konieczny
email: .konieczny@cril.fr

Stefano Moretti
email: stefano.moretti@lamsade.dauphine.fr

Ariane Ravier
email: ariane.ravier@lamsade.dauphine.fr

Paolo Viappiani
email: paolo.viappiani@lamsade.dauphine.fr

Selecting the most relevant elements from a ranking over sets

Keywords: social ranking, power relation, coalitions, social choice function

In this paper we study the problem of selecting the most relevant elements of a nite set N of elements from a ranking of non-empty subsets of N , that represents the performance of dierent coalitions. To solve this problem, we rst introduce the notion of coalitional social choice function (i.e., a map that associates each total preorder of the non-empty subsets of N with a subset of N ). Then, we provide four basic properties that a coalitional social choice function should satisfy to select the most relevant elements. Finally, we prove that the unique coalitional social choice function that satises such properties is the one selecting the elements ranked in the highest position of the lexicographic excellence ranking, which is computed according to a social ranking solution from the literature.

Introduction

The problem of dening a ranking of individual elements based on their contribution in establishing the position of groups or coalitions within a society has been recently introduced in the literature related to the notion of social ranking [START_REF] Bernardi | Ranking objects from a preference relation over their subsets[END_REF][START_REF] Haret | Ceteris paribus majority for social ranking[END_REF]. Taking as input a ranking of sets of objects like, for instance, a ranking over all possible research groups of a department, or over alternative combinations of attackers in a football team, or even a dichotomous order of winning or losing coalitions within a voting body, a social ranking generates a ranking of the individuals (researchers, football players, voters...) reecting the overall relevance of individuals within the rank of coalitions [START_REF] Bernardi | Ranking objects from a preference relation over their subsets[END_REF][START_REF] Algaba | Lexicographic solutions for coalitional rankings[END_REF][START_REF] Aleandri | Minimal winning coalitions and orders of criticality[END_REF].

In [START_REF] Bernardi | Ranking objects from a preference relation over their subsets[END_REF], for example, the individuals are ranked according to their frequency in the highest positions in the ranking of sets and a special social ranking function, called lexicographic excellence (lex-cel ), has been identied as the unique one satisfying a set of appealing properties. A generalization of the lex-cel has been introduced in papers [START_REF] Algaba | Lexicographic solutions for coalitional rankings[END_REF][START_REF] Béal | Lexicographic solutions for coalitional rankings based on individual and collective performances[END_REF] considering the size of coalitions, in addition to their positions in the ranking of sets. For other notions of social ranking solutions see also the papers [START_REF] Béal | Lexicographic solutions for coalitional rankings based on individual and collective performances[END_REF][START_REF] Allouche | Social ranking manipulability for the cp-majority, banzhaf and lexicographic excellence solutions[END_REF][START_REF] Khani | An ordinal banzhaf index for social ranking[END_REF] (for a software implementation of social ranking solutions from the literature, the interested reader is referred to the R package [START_REF] Fritz | socialranking : Social Ranking Solutions for Power Relations on Coalitions[END_REF]).

Sometimes, however, we are not interested in generating an entire ranking of the individual elements, but instead just want to select the most important ones.

Consider for instance the problem of identifying the most inuential scientists within an Academic Association (AA) based on the number and the quality of their publications. One of the major diculties in comparing scientists is taking into account their contributions to multi-authored publications [START_REF] Ausloos | A scientometrics law about co-authors and their ranking: the coauthor core[END_REF][START_REF] Vavry£uk | Fair ranking of researchers and research teams[END_REF].

Several bibliometric indices exist to compare the impact of research activity of individual scientists and of research groups, and the choice of the appropriate index goes beyond the scope of this paper (see, for instance, the paper [START_REF] Rubem | Comparative analysis of some individual bibliometric indices when applied to groups of researchers[END_REF] for an in-depth analysis of the problem). Nevertheless, each group of scientists can be characterized by a record of jointly published papers (possibly with other authors who do not belong to the AA). So, in principle, a ranking of groups of scientists can be established according to a predened bibliometric criterion for the comparison of the overall inuence of research groups (see Example 1 for a toy situation illustrating some naive approaches aimed at ranking groups starting from records of papers published in journals). Starting from such a ranking of groups, which scientists can be identied as the most inuential in a way that the scientists' ranking positions over dierent groups are considered?

In this paper we provide an answer to this question following a propertydriven approach. We rst single out a set of reasonable properties that a method selecting the set of most relevant elements (namely, a coalitional social choice function) from a ranking over groups should satisfy. Then we prove that the unique method satisfying those properties is the one which selects the best elements according to the ranking produced by the lex-cel.

As a rst property, we consider the All-Indierent-All-Winners axiom, which indicates that all the elements should be treated in the same way; so, if all groups are equally powerful and there is no reason to distinguish among the roles played by the elements, then all the elements should be selected. The second axiom we introduce is a Monotonicity for Winners property stating that if some elements are selected as the most relevant in a ranking of groups, and a new ranking over groups is produced improving the position of groups in the last equivalence class (but without aecting the comparison with groups in the other equivalence classes), then the most frequent elements over the improved groups should be considered as the most relevant ones also in the new ranking of groups. The third property, called the Dominance axiom, deals with the same kind of improvements described for the monotonicity axiom, but in this case it rules out the elements that are less represented among the improved groups. Finally, the Independence for Losers from the Worst Set axiom, prevents the elements that are excluded from being the most relevant ones in an original ranking of groups to become the most relevant in any other ranking of groups obtained from the original one by partitioning the last equivalence class. In other words, once the decision to exclude an element from the most relevant set is taken, the decision shall not be aected by changes involving the worst groups.

The roadmap of the paper is as follows. In the next section we introduce some preliminary denitions and a motivational example for the computation of the lex-cel. In Section 3, we formally introduce the four axioms and we illustrate them along the lines of the example introduced in Section 2. Section 4 is devoted to the axiomatic characterization of the lex-cel coalitional social choice function using the axioms introduced in Section 3, and to establishing their logical independence. Section 5 concludes with some comments and directions for future research.

Preliminaries

Let N = {1, . . . , n} be a nite set of elements (individuals, items, etc.) and let P(N ) be the set of the non-empty subsets of N (also called coalitions or groups).

A binary relation R ⊆ N × N is said to be: reexive, if for each i ∈ N , iRi; transitive, if for each i, j, k ∈ N , iRj and jRk ⇒ iRk; total, if for each i, j ∈ N , iRj or jRi; antisymmetric, if for each i, j ∈ N , iRj and jRi ⇒ i = j. A total preorder (also called a ranking) is a reexive, transitive and total binary relation. A total order is a reexive, transitive, total and antisymmetric binary relation.

R(N )

denotes the set of rankings (or total preorders) on a given set N .

A total preorder ∈ R(P(N )) is called a power relation. Given S, T ∈ P(N ),

S

T means that S is at least as powerful as T with respect to the power relation . We denote by ∼ the symmetric part of (i.e. S ∼ T if S T and T S) and by its asymmetric part (i.e. S T and not T S). So, for each pair of subsets S, T ∈ P(N ), S T means that S is strictly more powerful than T , whereas S ∼ T means that S and T are are equally powerful. The quotient order of is denoted as Σ 1 Σ 2 • • • Σ l in which the subsets S j are grouped in the equivalence classes Σ k generated by the symmetric part of . This means that all the sets in Σ 1 are equally powerful to S 1 and are strictly better than the sets in Σ 2 and so on.

Given a power relation and its associated quotient order Σ 1 Σ 2 • • • Σ l , we denote by i k = |{S ∈ Σ k : i ∈ S}| the number of sets in Σ k containing i for k = 1, . . . , l. Now, let θ (i) be the l-dimensional vector θ (i) = (i 1 , . . . , i l ) associated to . Consider the lexicographic order ≥ L among vectors i and j: i ≥ L j if either i = j or there exists t such that i t > j t and i r = j r for all r ∈ {1, . . . , t -1}.

Let ∈ R(P(N )). The lexicographic excellence (lex-cel) [START_REF] Bernardi | Ranking objects from a preference relation over their subsets[END_REF] is the binary relation R le such that for all ∈ R(P(N )) and all i, j ∈ N

: i R le j ⇐⇒ θ (i) ≥ L θ (j).
(in the remaining I le and P le stand for the symmetric part and the asymmetric part of R le ( ), respectively.)

We illustrate the computation of the lex-cel via the following toy example.

Example 1. Consider an AA formed by three scientists N = {1, 2, 3} and the problem of identifying the most inuential scientist(s) within the AA based on the number of papers published in journals ranked according to the SCImago Journal Rank (SJR) and awarding the publication in the highest position of the SJR. Suppose that the three scientists never worked all together on a research project, so there is no paper co-authored by the three scientists together. Instead, some published papers exist which were co-authored by pairs of authors. Moreover, while scientists 1 and 2 are more experienced and they published some papers without the help of the others, 3 is a young researcher and he never published a paper alone. Journals related to the AA discipline are grouped into four quartiles Q 1 , Q 2 , Q 3 and Q 4 , where Q 1 is the rst quartile formed by the top 25% of journals in the SJR list, Q 2 is the second quartile (from 25% to 50%), Q 3 the third one (from 50% to 75%) and Q 4 the last one (from 75% to 100%).

The number of papers co-authored by each group of scientists is summarized in Table 1.

Table 1. Number of published papers co-authored by groups of scientists in each SJR quartile.

Groups Q1 Q2 Q3 Q4 {1, 2, 3} 0 0 0 0 {1, 2} 5 0 0 15 {1, 3} 0 10 5 0 {2, 3} 5 0 5 0 {1} 5 0 0 15 {2} 0 0 10 5 {3} 0 0 0 0
To reward the publications in the highest-ranked journals in the SJR, one may argue that the performance of the above groups may be ranked according to the lexicographic comparison (from the rst quartile to the fourth one) of the rows of Table 1. As a result, we obtain the following power relation (in the following, to avoid cumbersome notation, commas and parentheses for sets are omitted, e.g., we use 123 instead of {1, 2, 3}):

23 1 ∼ 12 13 2 3 ∼ 123.
So, the quotient order of is

Σ 1 = {23} Σ 2 = {1, 12} Σ 3 = {13} Σ 4 = {2} Σ 5 = {3, 123}.
Assume that the lex-cel is adopted to rank the scientists according to their inuence over the ranking of groups. To compute the lex-cel on , we need to lexicographically compare the vectors θ (i), i ∈ N , dened earlier in this section.

So,

θ (2) = (1, 1, 0, 1, 1) > L θ (3) = (1, 0, 1, 0, 2) > L θ (1) = (0, 2, 1, 0, 1)
and therefore the lex-cel R le gives a total order over N such that 2P le 3P le 1.

Notice that alternative approaches, less oriented to rewarding the highest positions in the SJR, might be adopted. Therefore, a dierent power relation could be generated from Table 1. For instance, the use of the total number of publications as bibliometric index would produce the power relation

1 ∼ 12 2 ∼ 13 23 3 ∼ 123.
It is easy to verify that the lex-cel in this case gives the ranking 1P le 2P le 3.

Properties for coalitional social choice functions

We dene the notion of coalitional social choice function (cscf ) as a map B : R(P(N )) → P(N ) that associates to each power relation ∈ R(P(N )) a non-empty subset B( ) ∈ P(N ) which is interpreted as the set of most relevant elements or winners in . We now introduce some properties for a cscf.

The rst axiom states a principle of neutrality for elements: if all coalitions are indierent then there is no way to distinguish a major contribution of any element and all the elements should be considered winners.

Axiom 1 (All-Indierent-All-Winners (AIAW)) Consider a power relation ∈ R(P(N )) such that for all S, T ∈ P(N ):

S ∼ T. (1)
Then a cscf B satises the property All-Indierent-All-Winners if it holds that B( ) = N.

Example 2. Consider again an AA with N = {1, 2, 3}, as in Example 1, but suppose now that all non-empty subsets of the set N of scientists have published precisely the same number of papers in each SJR quartile. Based on such information, all coalitions may be considered equally powerful. So, we have a power relation such that.

1 ∼ 2 ∼ 3 ∼ 12 ∼ 13 ∼ 23 ∼ 123.
A cscf B satisfying Axiom 1 selects as winners B( ) = {1, 2, 3}.

The second axiom is a particular monotonicity condition for winners: improving the position of some coalitions in the worst equivalence class, but keeping the same relation among coalitions in the other equivalence classes, should not aect the status of most represented winners (over the improved coalitions). In the following, given a family of coalitions Σ ⊆ P(N ) and any element i ∈ N , we use the notation i Σ to denote the number of coalitions in Σ to which i belongs,

i.e. i Σ = |{S ∈ Σ : i ∈ S}|.

Axiom 2 (Monotonicity for Winners (MW)) Consider two power relations , ∈ R(P(N )) and their respective quotient orders and such that:

Σ 1 Σ 2 • • • Σ l , Σ 1 Σ 2 • • • Σ l-1 Σ Σ l \ Σ, with Σ ⊆ Σ l .
Take a cscf B and let T ⊆ B( ) be the set of most represented winners over Σ,

i.e.

T = {i ∈ B( ) : i Σ ≥ j Σ ∀j ∈ B( )}.
(

) 2
We

23 1 ∼ 2 ∼ 3 ∼ 12 ∼ 13 ∼ 123.
According to relation [START_REF] Algaba | Lexicographic solutions for coalitional rankings[END_REF], the set of most represented winners is T = {2, 3}. So, a cscf B satisfying Axiom 2 is such that 2, 3 ∈ B( ). Notice that, based on Axiom 2, we cannot arm whether element 1 belongs to B( ) or not.

The third axiom states that after making an improvement to coalitions in the worst equivalence class (as in the previous axiom), the winners that are now strictly less represented than other winners over the improved coalitions (i.e., they are dominated by other winners) become losers (no longer winners).

Axiom 3 (Dominance (D)) Consider two power relations , ∈ R(P(N ))

and their respective quotient orders and such that:

Σ 1 Σ 2 • • • Σ l , Σ 1 Σ 2 • • • Σ l-1 Σ Σ l \ Σ, with Σ ⊆ Σ l .
Take a cscf B and let L ⊆ B( ) be the set of winners that are strictly less represented than other winners over Σ, i.e.

L = {j ∈ B( ) : ∃i ∈ B( ) with i Σ > j Σ }. (3)
We say that B satises the Dominance property if it holds that B( ) ⊆ N \ L. Finally, the last axiom states that once an element becomes a loser it remains a loser over all possible power relation obtained fractioning the worst equivalence class of the quotient order. Axiom 4 (Independence for Losers from the Worst Set (ILWS))

We say that a cscf B satises the property of Independence for Losers from the Worst Set if for any power relation ∈ R(P(N )) with the associated quotient order such that

Σ 1 Σ 2 • • • Σ l and i ∈ N such that i / ∈ B( ),
then for any partition T 1 , . . . , T m of Σ l and for any power relation ∈ R(P(N ))

with the associated quotient order such that

Σ 1 Σ 2 • • • Σ l-1 T 1 • • • T m , it holds that i / ∈ B( ).
Example 5.

The lex-cel coalitional social choice function A particular cscf denoted as B le is based on the lex-cel and it associates to a each power relation ∈ R(P(N )) the set of elements in the highest positions in the ranking R le provided by the lex-cel.

Denition 1 (Lex-cel cscf). Let ∈ R(P(N )). The lex-cel cscf is the map B le : R(P(N )) → P(N ) such that for all ∈ R(P(N )): To axiomatically characterize the lex-cel cscf we rst need to introduce the following lemma. Lemma 1. Consider two power relations , ∈ R(P(N )) and their respective quotient orders and such that:

B le ( ) = {i ∈ N : i R le j ∀j ∈ N }.
Σ 1 Σ 2 • • • Σ l , Σ 1 Σ 2 • • • Σ l-1 Σ Σ l \ Σ, with Σ ⊆ Σ l . Then, B le ( ) = T = {i ∈ B le ( ) : i Σ ≥ j Σ ∀j ∈ B le ( )}.
Proof. First, notice that for each i ∈ N

θ (i) = (i 1 , . . . , i l-1 , i Σ , i l -i Σ ). ( 4
)
where i k = |{S ∈ Σ k : i ∈ S}| is the number of sets in Σ k containing i for k = 1, . . . , l. We prove that, for each i ∈ T , we have i ∈ B le ( ) or, equivalently by the denitions of lex-cel, that θ (i) ≥ L θ (j) ∀j ∈ N.

We distinguish three cases: i) j ∈ T : since i, j ∈ T , we have that i Σ = j Σ ; moreover, i, j ∈ B le ( ) implies that θ (i) = θ (j)). Then, by relation (4), θ (i) = θ (j); ii) j ∈ B le ( ) \ T : since i, j ∈ B le ( ), we have that i k = j k for all k = 1, . . . , l; moreover, j ∈ B le ( ) \ T implies that i Σ > j Σ . Then, by relation (4),

θ (i) > L θ (j) iii) j ∈ N \B le ( ): since i ∈ B le ( ) but j / ∈ B le ( ), it must exist k = 1, . . . , l -1 such that i k > j k . Then, by relation (4), θ (i) > L θ (j).
So, T ⊆ B le ( ). Moreover, by the the above points ii) and iii) for each j ∈ N \T we have that θ (i) > L θ (j) for all i ∈ T . So B le ( ) ⊆ T , which concludes the proof.

We are now ready to introduce the main result of this section.

Theorem 5. The cscf B le is the unique cscf fullling Axioms 1, 2, 3 and 4.

Proof. We rst prove that the cscf B le satises the four axioms.

To see that B le satises Axiom 1, consider a power relation ∈ R(P(N )) such that S ∼ T for all S, T ∈ P(N ). We have that θ (i) = (2 n-1 ) for any i ∈ N since any element i belongs to precisely 2 n-1 coalitions in the same equivalence class Σ 1 = P(N ) of the quotient order .

The fact that B le satises Axiom 2 immediately follows from Lemma 1, for B( ) = T for any , ∈ R(P(N )) with the properties considered in Axiom 2.

The fact that B le satises Axiom 3 immediately follows from Lemma 1 and Remark 1, for B( ) = T = B( ) \ L ⊆ N \ L for any , ∈ R(P(N )) with the properties considered in Axiom 3.

Finally, to see that B le satises Axiom 4, consider any power relation ∈ R(P(N )) with quotient order such that

Σ 1 Σ 2 • • • Σ l and i ∈ N such that i / ∈ B( ).
For any partition T 1 , . . . , T m of Σ l and for any power relation ∈ R(P(N )) with quotient order such that

Σ 1 Σ 2 • • • Σ l-1 T 1 • • • T m , we have that θ (i) = (i 1 , . . . , i l-1 , i T1 , . . . , i Tm ), (5)
where i Tp = |{S ∈ T p : i ∈ S}| is the number of sets in T p containing i for p = 1, . . . , m.

Since i / ∈ B( ), by Denition 1 there must exist j = i such that θ (j) > L θ (i). So, for the sum l k=1 i k = 2 n-1 for all i ∈ N , there must exist s ∈ {1, . . . , l -1} such that i k = j k for all k ∈ {1, . . . , s -1} and i s < j s . Consequently, by relation [START_REF] Béal | Lexicographic solutions for coalitional rankings based on individual and collective performances[END_REF], θ (j) > L θ (i); so i / ∈ B le ( ).

We now prove that if a cscf B satises the four axioms then B = B le . Consider a cscf B that satises Axioms 1, 2, 3 and 4. We want to prove that for any power relation ∈ R(P(N )) (and its quotient order such that

Σ 1 Σ 2 • • • Σ l ) it holds that B( ) = B le ( ).
The proof is by induction to the number l of equivalence classes in the quotient order . If l = 1 (therefore the power relation ∈ R(P(N )) is such that S ∼ T for all S, T ∈ P(N )), by Axiom 1 and the rst part of this proof, we have that B( ) = N = B le ( ). Now, let l ≥ 1 and suppose that the assertion B( ) = B le ( ) has been proven for any power relation such that the quotient order contains precisely

l equivalence classes. Let ∈ R(P(N )) be such that the quotient order is Σ 1 • • • Σ l+1 . Consider a new power relation ∈ R(P(N )) such that the quotient order is Σ 1 • • • Σ l ∪ Σ l+1
, containing precisely l equivalence classes. By Lemma 1 with Σ l in the role of Σ, we have that

B le ( ) = T = {i ∈ B le ( ) : i l ≥ j l ∀j ∈ B le ( )}. (6)
Application of the induction hypothesis on yields B( ) = B le ( ). Now, by Axioms 2 and 3 on B, with Σ l in the role of Σ, it follows that

T ⊆ B( ) ⊆ N \ L, (7)
where L = {j ∈ B( ) : ∃i ∈ B( ) with i l ≥ j l }. Then, by Axiom 4, with partition T

1 = Σ l , T 2 = Σ l+1 of the last equivalence class Σ l ∪ Σ l+1 in , we have B( ) ⊆ B le ( ) = L ∪ T. (8)
Finally, by relations [START_REF] Fritz | socialranking : Social Ranking Solutions for Power Relations on Coalitions[END_REF] and [START_REF] Haret | Ceteris paribus majority for social ranking[END_REF] we have that T ⊆ B( ) ⊆ T . So, by relation [START_REF] Bernardi | Ranking objects from a preference relation over their subsets[END_REF], we have proven that B( ) = T = B le ( ).

We now show that Axioms 1, 2, 3 and 4 are logically independent, which means that they are necessary for the axiomatic characterization of the lex-cel cscf provided in Theorem 5 .

Proposition 1. Axioms 1, 2, 3 and 4 are logically independent. Proof. We want to prove that axioms 1, 2, 3 and 4 are necessary in order to uniquely characterize the cscf B le . Therefore, we show that for any combination of three out of the four axioms, a cscf B : R(P(N )) → P(N ), with N = {1, . . . , n}, that satises such three axioms does not necessarily satises the fourth one. Axiom 1 is not satised: Axiom 2 is not satised:

Let i ∈ N and consider a cscf B 1 : R(P(N )) → P(N ) such that B 1 ( ) = {i} for all ∈ R(P(N )). Clearly B 1 does not satisfy Axiom 1 if |N | ≥ 2. B 1 trivially satises Axiom 2, since T = {i} = B 1 ( ) for all power relations , ∈ R(P(N )) as in Axiom 2.
For any S ∈ P(N ), let b(S) = min{i ∈ S} and consider a cscf B 2 : R(P(N )) → P(N ) such that for all ∈ R(P(N ))

B 2 ( ) = N, if = 0 , {b(B le ( ))}, otherwise. (9)
where 0 ∈ R(P(N )) is a power relation with quotient order 0 having a unique equivalence class Σ 1 = P(N ) (all coalitions are indierent in 0 ). B 2 does not satisfy Axiom 2: take for instance the power relations , ∈ R(P(N )) such that the quotient order is

Σ 1 = P(N )
and the quotient order is {12} P(N ) \ {12}.

Taking Σ = {12}, with the notations of Axiom 2, we have T = {1, 2}, but B 2 le ( ) = {1}, which violates the axiom.

Notice that the conditions of Axiom 4 for and apply with the partition of the worst equivalence class in into the three disjoint sets T 1 = {2, 23}, T 2 = {13} and T 3 = P(N ) \ (T 1 ∪ T 2 ). However, it easy to check that 1P d ( )i for all i ∈ N \ {1}, so, 2 / ∈ B 4 ( ), whereas 2P d ( )j for all j ∈ N \ {2} and so, 2 ∈ B 4 ( ).

We leave to the reader to verify that B 4 satises Axioms 1, 2 and 3 by following the same steps as in the proof of Theorem 5 where we proved that B le satises those axioms.

We have therefore proven that the cscf B le is the unique solution satisfying the four proposed axioms and that these four axioms are logically independent.

So we obtained a full characterization of this method.

As already noticed in [START_REF] Bernardi | Ranking objects from a preference relation over their subsets[END_REF], the dual lex-cel R d (cf. to its denition in the proof of Proposition 1) penalizes elements appearing many times in the worst coalitions of a power relation. As a consequence, the dual lex-cel cscf B 4 shows a similar behaviour for the selection of the most relevant elements, as illustrated in the following example.

Example 7. Consider again the power relation of Example 1. It is easy to verify that the dual lex-cel ranking R d is such that 1P d 2P d 3. So, the corresponding dual lex-cel cscf B 4 yields the most relevant element B 4 ( ) = {1}. In contrast with the lex-cel cscf, which rewards the excellence of element 2 as a member of the best coalitions (specically, coalitions 23 and 12) according to the power relation , the dual lex-cel csfc punishes the presence of element 2 in the worst coalitions (specically, coalitions 123 and 2) according to .

Conclusion and Future Work

There are many situations where sets of elements are compared; for instance, in sports, in job performance evaluations, etc. A common task in these situations is to reason about the elements in order to assess the performance; typically one might want to compare elements based on their performance when part of different teams. An approach that has been recently proposed is the lexicographic excellence (lex-cel) method [START_REF] Bernardi | Ranking objects from a preference relation over their subsets[END_REF], that provides a full ranking of the elements given the full ranking of all possible coalitions (called power relation). The peculiar characteristic of the lex-cel is its qualitative nature and the fact that is a relatively simple rule to use.

In this work we addressed the problem of determining the most relevant element(s) given in input the ranking over coalitions. We dene a coalitional social choice function (cscf ) as a mapping from a power relation to a (typically small) subset of winners. We adopted an axiomatic approach and stated four axioms that we believe constitute reasonable behaviors of the desired cscf. We then showed that the cscf which returns the rst element(s) in the ranking obtained by lex-cel is the only cscf compatible with the four axioms. So we proposed the rst, as far as we know, method for determining the most relevant elements given a ranking of coalitions, as well as its corresponding characterization in terms of four intuitive axioms. Although there are similarities between some of the axioms introduced in this paper and those for social rankings introduced in [6], we would like to point out some important dierences suggesting a novel interpretation of the lex-cel ranking when it is applied to select the most relevant elements from a power relation. First, our axiomatic characterization of the lex-cel cscf does not use any property directly related to the Coalitional Anonymity axiom proposed in [START_REF] Bernardi | Ranking objects from a preference relation over their subsets[END_REF], which is a strong property requiring that the relative ranking of two individual elements should exclusively depend on the relative positions of groups containing just one of them and disregarding the structure or the size of those groups. Second, our axioms highlight the fundamental mechanism driving the selection of the most relevant elements, which is essentially based on changes (i.e., improvements or partitioning) in the structure of the worst equivalence class of a power relation (in [START_REF] Bernardi | Ranking objects from a preference relation over their subsets[END_REF], only the Independence from the Worst Set axiom concerned the worst equivalence class). In conclusion, we believe that the set of (easy to understand) axioms used in this paper reveals in an explicit way the crucial role of the worst equivalence class in determining the most relevant elements according to the lex-cel method.

Concerning future works, we believe that an important direction is to deal with situations where there is uncertainty about the preference order over the coalitions. Preference uncertainty can arise because of a lack of information on how certain coalitions compare, or because of the user's reluctance to rank an exponential number of coalitions (due to cognitive cost or time constraints).

Indeed, the main obstacle to the adoption of lex-cel lies in the exponential size of its input (the exponential number of elements in a power relation). Therefore, we think it is worth studying the application of the lex-cel (and as well of similar methods) when a partial order of the coalitions is given as an input. This partial order can be interpreted in the sense of strict uncertainty [START_REF] Konczak | Voting procedures with incomplete preferences[END_REF]: we can assume that there exists a true complete ranking * ∈ R(P(N )), but that it is unknown to us and only a reduction of the ranking is given. We can then design methods to reason about all possible completion of the partial ranking given as input, and view these methods as coalitional social choice functions taking partial orders as input. Considering the running example dealing with the determination of the most inuential scientists with an AA, this could lead to simplication of the process of comparison of collaborations: for instance, it could be possible to have m sub-committees tasked with ranking certain subsets of P(N ), i.e. the work of comparing all potential collaborations is divided between these m committees.

From their conclusions, the application of a cscf equipped to work with missing information will be used to determine the ranking over the researchers' inuence.

In this way, it will be possible to deal with real-life situations, where a complete order is seldom given.

Let∈

  R(P(N )) be a power relation of the form S 1 S 2 • • • S |P(N )| .

Remark 1 .

 1 Notice that the set L in Axiom 3 is the complement in the set of winners of the set T in Axiom 2, so T ∪ L = B( ) and T ∩ L = ∅. Example 4. Consider again the power relation of Example 2 and the power relation of Example 3. According to Remark 1, we have L = N \ {2, 3} = {1}. So, a cscf B satisfying Axiom 3 is such that scientist 1 does not belong to the set B( ) of most important elements.

Example 6 .

 6 Consider the power relation of Example 1 and the lex-cel ranking computed over in the same example. Then, B le ( ) = {2}.

B 1

 1 trivially satises Axiom 3, since L = ∅ and B 1 ( ) = {i} ⊆ N for all power relations , ∈ R(P(N )) as in Axiom 3. B 1 also satises Axiom 4, for all j ∈ N \ {i}, j / ∈ B 1 ( ) and j / ∈ B 1 ( ) for all power relations , ∈ R(P(N )) as in Axiom 4.

  say that B satises the Monotonicity for Winners property if it holds that T ⊆ B( ). Example 3. Consider again the power relation of Example 2. Now, suppose that, after an update of the publication records, it turns out the number of jointpapers published by scientists 2 and 3 together (and without author 1) improves with respect to SJR. As a consequence, a new power relation is considered where coalition 23 is the most powerful coalition:

  ) (like inExample 4). A direct consequence of Axiom 4 is that 1 does not even belong to B( ) .

	obtained from	via a partition of Σ 2 and such that
		23	1 ∼ 12	13	2	3 ∼ 123.
	Notice that	coincides with	of Example 1. Consider a cscf B satisfying
	Axiom 4 and suppose 1 / ∈ B(

Consider of Example 3 and its quotient order such that Σ 1 = {23} and Σ 2 = R(P(N )) \ {23}. Now, suppose that after an update in the publications records of groups of scientists, we obtain a new power relation

Acknowledgements We acknowledge a nancial support from the project THEMIS ANR-20-CE23-0018 of the French National Research Agency (ANR).

Clearly B 2 satises Axiom 1. Since B le satises Axiom 3, the same holds for B 2 , as the set L in Axiom 3 remains precisely the same for both cscfs.

Finally, since all elements that do not belong to B le ( ), ∈ R(P(N )), do not neither belong to B 2 ( ), it follows that B 2 also satises Axiom 4.

Axiom 3 is not satised

Consider a cscf B 3 : R(P(N )) → P(N ) such that B 3 ( ) = N for all ∈ R(P(N )). B 3 does not satisfy Axiom 3: take for instance two power relations , ∈ R(P(N )) such that the quotient order is

and the quotient order is {1, 12} P(N ) \ {12, 1}.

Taking Σ = {1, 12}, with the notation of Axiom 3 we have L = {2}, but B 3 le ( ) = N N \ {2}, which violates the axiom.

On the other hand, B 3 clearly satises Axiom 1. B 3 trivially satises Axiom 2, as we have T ⊆ N . B 3 also satises Axiom 4, as the set of elements not in B 3 ( ) is empty, for all ∈ R(P(N )).

Axiom 4 is not satised:

To dene the last solution B 4 : R(P(N )) → P(N ), we rst need to introduce the notion of dual lex-cel (see [START_REF] Bernardi | Ranking objects from a preference relation over their subsets[END_REF]). Consider the lexicographic * order ≥ L * among vectors i and j: i ≥ L * j if either i = j or there exists t such that i t < j t and i r = j r for all r ∈ {t + 1, . . . , l}. The dual lex-cel is the binary relation R d such that for all ∈ R(P(N )) and all i, j ∈ N :

i R d j ⇐⇒ θ (i) ≥ L * θ (j).

Consider the dual lex-cel cscf B 4 : R(P(N )) → P(N ) such that for all ∈ R(P(N )):

B 4 does not satisfy Axiom 4: take for instance two power relations , ∈ R(P(N )) such that the quotient order is {1, 12} P(N ) \ {1, 12}

and the quotient order is {1, 12} {2, 23} 13 P(N ) \ {1, 12, 2, 23, 13}.


Chaimae Bariki

Fabien Halter

Raik Hesse

Christian Chauveau

Heinz Pitsch

Joachim Beeckmann

Experimental measurements of laminar flame speeds for highly N2-diluted ethanol flames under microgravity conditions

Keywords: Buoyancy, Microgravity, Laminar Flame Speed, Spherical Flame, Radiation

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Chemically inert diluents, such as nitrogen, have proven to be an important class of fire extinguishing agents in confined environments such as spacecraft [START_REF] Qiao | A study of the effects of diluents on near-limit h2-air flames in microgravity at normal and reduced pressures, Combust[END_REF][START_REF] Qiao | Extinction of premixed methane/air flames in microgravity by diluents: Effects of radiation and lewis number[END_REF][START_REF] Zhang | Effects of diluents on the ignition of premixed h2/air mixtures[END_REF]. Nitrogen is safe for use in occupied spaces and poses no threat to the environment. Several studies focused on assessing flame extinction limits caused by N2 diluent [START_REF] Liao | Flammability limits of combustible gases and vapors measured by a tubular flame method[END_REF] and its effects on ignition processes [START_REF] Zhang | Effects of diluents on the ignition of premixed h2/air mixtures[END_REF]. Flame propagation is also an aspect investigated by several groups since it also occurs in the fire suppression process [START_REF] Qiao | A study of the effects of diluents on near-limit h2-air flames in microgravity at normal and reduced pressures, Combust[END_REF][START_REF] Qiao | Extinction of premixed methane/air flames in microgravity by diluents: Effects of radiation and lewis number[END_REF][START_REF] Hu | Experimental and numerical study on the effect of composition on laminar burning velocities of h2/co/n2/co2/air mixtures[END_REF]. Highly N2-diluted flames are slow-burning near the flammability limits. They are affected both by radiation heat losses and buoyancy. As a result, these flames are heavily distorted due to the gravitational force and are prone to experimental errors. It is worth stressing that the available experimental data measured for slow propagating flames are sparse in the literature. In addition, modeling real fuels is difficult because these are mixtures of hundreds of components, making it challenging to unravel their combustion chemistry. For this reason, fuel surrogates containing simpler mixtures of well-selected components are used to model the combustion behavior. This study is essential for a fundamental understanding of (1) how fast flames are when approaching the flammability limit and (2) how radiation heat losses affect flame propagation. Experiments and simulations are used to highlight the complexity of extracting accurate flame speeds of such slowly propagating flames, even for well-known fuels.

The laminar flame speed is a fundamental physicochemical property of a combustible mixture because it notably influences the mixture's burning rate. Measuring the laminar flame speed is essential for validating kinetic mechanisms [START_REF] Holley | Sensitivity of propagation and extinction of large hydrocarbon flames to fuel diffusion[END_REF] and serves as an input parameter in turbulent combustion models [START_REF] Poinsot | Prediction and control of combustion instabilities in real engines[END_REF]. An accurate measurement of this property for slowly propagating flames is difficult because the effects of gravity and radiation cannot be neglected anymore. Owing to these difficulties, measuring the laminar flame speeds of highly diluted premixed mixtures remains a formidable challenge.

The closed vessel method with spherically propagating flames has been widely used to measure the laminar flame speed due to its large pressure-range applicability and well-defined stretch rate. For slowly burning flames, the gravitational force can lead to a strong deformation of the initially spherical shape, making the extraction of this parameter extremely difficult. The methods found in the literature to assess buoyant flames differ quite significantly between studies. For example, Pfahl et al. [START_REF] Pfahl | Flammability limits, ignition energy, and flame speeds in H2-CH4-NH3-N2O-O2-N2 mixtures[END_REF] suggested a method evaluating only the horizontal radii temporal evolution of optically investigated flames in a spherical vessel. Accordingly, Sun et al. [START_REF] Sun | Buoyant unstable behavior of initially spherical lean hydrogen-air premixed flames[END_REF] proposed two parameters, a buoyant flow rate, and a buoyant critical flame radius, to investigate gravity effects quantitatively from the flame front's projection area determined using a Schlieren technique. Recently, Berger et al. [START_REF] Berger | A DNS study of the impact of gravity on spherically expanding laminar premixed flames[END_REF] analyzed the effect of buoyancy through direct numerical simulations (DNS) of spherically expanding laminar premixed flames whose motion is affected by buoyancy. They proposed a new technique to extract the laminar flame speed. However, it cannot be directly determined from experimental images recorded during the flame propagation. The robustness and validity of existing post-processing techniques for slowly burning flames are still unclear and the only way to suppress buoyancy effects is to conduct experiments under microgravity conditions, for example in a drop tower facility or on a parabolic flight.

Radiative heat losses inherently affect laminar flame speeds measured in a spherical chamber and can no longer be neglected for slowly propagating flames, as it was recently demonstrated by Hegetschweiler et al. [START_REF] Hegetschweiler | Data Reduction Considerations for Spherical R-32(CH2F2)-Air Flame Experiments[END_REF] and Hesse et al. [START_REF] Hesse | Low globalwarming-potential refrigerant CH2F2 (R-32): Integration of a radiation heat loss correction method to accurately determine experimental flame speed metrics[END_REF]. As propagation time increases, radiation plays an increasingly important role due to the cooling of burned gases. Several correction schemes relying on empirical corrections or thermodynamics were proposed for spherically expanding flames measured under quasiisobaric conditions. These models were almost exclusively applied to standard hydrocarbons or hydrogen, which have significantly faster flame speeds than highly N2-diluted flames. Yet, the nitrogen chemistry has not been taken into account in the radiation correction models. For instance, Santner et al. [START_REF] Santner | Uncertainties in interpretation of high pressure spherical flame propagation rates due to thermal radiation[END_REF] proposed a correction based on a thermodynamic calculation to correct for radiation. This approach estimates the thermal and flow effects caused by radiation separately. Yu et al. [START_REF] Yu | Radiation-induced uncertainty in lam-inar flame speed measured from propagating spherical flames[END_REF] proposed an empirical correlation approach, which has been validated for flame speeds of 10 cm/s and faster. However, there is no proof that it can be applied to mixtures resulting in flame speeds below 10 cm/s, such as conditions encountered during the combustion of highly N2-diluted mixtures or fuels such as ammonia and refrigerants.

Given these considerations, the main goal of the present paper is to investigate the combined effects of stretch and radiation on N2-diluted premixed spherically expanding flames. First, measurements were conducted under terrestrial gravity to elucidate the flame distortion issue due to the gravitational force and the difficulty in extracting the laminar flame speed based on the existing models. Second, experiments were carried out in a pressure-release-type dual chamber, adequate for aircraft-related safety requirements, during parabolic flights onboard the Airbus A310 Zero-G to suppress the buoyancy effect under microgravity conditions. Since only a few selected liquid fuels are allowed onboard the aircraft, ethanol was chosen for this work.

Experimental framework

Experimental setups

Experiments were conducted under microgravity conditions to suppress the effect of buoyancy for slowly propagating flames in a pressure-release-type dual chamber, which is adequate for aircraft-related safety requirements [START_REF] Renoux | Experimental study of the morphology of two-phase flame instabilities in microgravity[END_REF][START_REF] Thimothée | Experimental investigation of the passage of fuel droplets through a spherical two-phase flame[END_REF]. These conditions were obtained using the parabolic flights of the Airbus A310 ZERO-G, during which a 10 -2 g gravity level is achieved. A schematic of the experimental facility is given in Fig. 1. It consists of a 1 L spherical chamber inserted in an 11 L high-pressure chamber. The spherical combustion chamber is equipped with eight symmetrically distributed evacuation valves to release pressure during flame propagation once 0.025 MPa increase is reached. Note that in all cases, pressure release occurs largely after the flame exceeded the window limit. The pressure variation is very low although the flame has traveled half the chamber radius. The inner chamber is filled with the flammable mixture, while the high-pressure chamber is filled at the same time with nitrogen.

Liquid fuel injection is performed using a multiinjection system with a switching valve which delivers an accurate volume of ethanol to obtain the targeted equivalence ratio. The fresh gases are heated by an electric heating cable that encircles the chamber. Additionally, a pre-heater is used to pre-vaporize the air-fuel mixture before it arrives in the combustion chamber. A K-type thermocouple of 13 µm diameter and a pressure sensor monitor the temperature and the pressure during the filling process.

Both chambers are equipped with aligned transparent windows to allow optical access. A conventional shadowgraphy system is used to visualize the flame morphology. An Energetiq LDLS (Laser-Driven-Light-Source) is used as light source, and a highspeed camera (Phantom v1210) continuously records the flame propagation at 1000 fps, with a field of view of 768 × 768 pixels 2 . The resulting magnification ratio is 16.4 px/mm. The combustible mixtures were ignited at the chamber's center using a pair of tungsten electrodes of 0.3 mm diameter, arranged face to face. The electrode distance was set to 1.5 mm. The spark was generated using a newly developed two-step highvoltage system: the first output stage feeds an ignition coil, which is responsible for ionizing the mixture between the electrodes and generating the primary ignition spark. If the mixture is conductive, the current can flow through the gap between the electrodes allowing the second stage of the ignition system to feed more energy into the spark. A variable voltage of 500 to 1800 V DC, fed by the power capacitors, can be generated. More details are given in the Supplementary Material for completeness. The applied voltage and its corresponding delivered energy are reported in Table 1. Experiments under terrestrial gravity conditions were conducted in the RWTH high-pressure, hightemperature spherical chamber [START_REF] Bariki | Combined isochoric and isobaric acquisition methodology for accurate flame speed measurements from ambient to high pressures and temperatures[END_REF][START_REF] Beeckmann | Experimental investigation of the laminar burning velocities of methanol, ethanol, n-propanol, and n-butanol at high pressure[END_REF][START_REF] Wu | Understanding the antagonistic effect of methanol as a component in surrogate fuel models: A case study of methanol/n-heptane mixtures[END_REF].

The schematic of the setup and the respective characteristic parameters are given in Fig. S1 and Table S1 of the Supplementary Material, and only a brief description is given here. Flame propagation is imaged with a Schlieren technique. A pulsed high-power LED emitting green light is used with a high-speed CMOS camera (LaVision HighSpeedStar 6). The acquisition rate was set to 5 kHz with a field of view of 448 × 448 pixels 2 resulting in a magnification ratio of 11.41 px/mm. All fuel/air mixtures were prepared in a separate premixing vessel using the partial pressure method. Two temperature-compensated pressure transducers were used during the mixture preparation with different ranges of 3 and 10 bar to accurately perform the filling process. The combustible mixtures were ignited at the chamber's center using a pair of tungsten electrodes of 0.3 mm diameter, arranged face to face, with an electrode distance of 1.5 mm. The spark was generated using a similar two-step highvoltage system as the one used for parabolic flight experiments.

Selected conditions

For aircraft-related safety reasons, gaseous fuels, such as ammonia and refrigerants, are prohibited, and only a few selected liquid fuels are allowed onboard the aircraft. Radiation is caused mainly by the combustion products H2O and CO2 and by soot. Here, a fuel with high sooting propensity was not chosen because the windows would become dirty after only a few measurements under microgravity conditions. This is a critical point to be considered, given the short time between parabolas. Therefore, ethanol was selected to avoid this problem and still be able to evaluate radiation effects. Moreover, previous campaigns widely investigated this fuel in the parabolic flight setup at conditions facilitating significantly faster flame speeds. One way to obtain highly-buoyant flames is to decrease the content of oxygen. A preliminary parametric study was conducted to define the content of oxygen to be used, and it was found that a synthetic mixture of 12% O2 and 88% N2 in volume is sufficient to obtain mediumto highly-buoyant flames under normal-gravity conditions. The severity of the buoyancy impact was varied by approaching lean and rich flammability limits. Note that it is unnecessary to modify the O2 content to obtain buoyant flames near the fuel-rich flammability limit, which corresponds to 9.8% for ethanol [START_REF] Brooks | Flammability envelopes for methanol, ethanol, acetonitrile and toluene[END_REF].

Decreasing the O2 content reduces not only the flame temperature but also the flame speed and the propensity for cellular structure formation, thus, enabling the analysis of radiation and buoyancy effects independently. Note that the effect of buoyancy is governed by the Richardson number, which is defined as Ri = (ρu -ρ b )/ρu • gR f / Ṙ2 f . Here, ρu and ρ b represent the density in the unburned and burned gas, g is the gravitational acceleration, R f is the flame radius, and Ṙf refers to the propagation speed.

An initial temperature was fixed at 373 K for all conditions. It is important to highlight that igniting the selected mixtures would not be possible without the newly developed high-voltage ignition system and measurements under these conditions were extremely challenging. Table 2 summarizes the experimental conditions investigated. The repeatability of the results was ensured by conducting two to three experiments for each condition.

Simulation framework

Flame characteristics were calculated for a 1D unstretched laminar flame configuration, using the opensource FlameMaster code [START_REF] Pitsch | Flamemaster: A C++ computer program for 0d combustion and 1d laminar flame calculations[END_REF]. The ITV chemical mechanism developed by Cai et al. [START_REF] Cai | Exploring the combustion chemistry of a novel lignocellulose-derived biofuel: cyclopentanol. part ii: experiment, model validation, and functional group analysis[END_REF], which was validated extensively for a large number of fuels including ethanol, was chosen to compare experimental and predicted results.

In order to ensure that the mechanism by Cai et al. [START_REF] Cai | Exploring the combustion chemistry of a novel lignocellulose-derived biofuel: cyclopentanol. part ii: experiment, model validation, and functional group analysis[END_REF] is valid for the investigated conditions, an indicative sensitivity analysis has been performed for different ethanol mixtures to identify the key reactions influencing laminar flame speeds. Sensitivity coeffi-cients S α S L on elementary reaction rate constants for an initial pressure of 3 bar were calculated as

S α S L = kα SL dSL dkα . (1)
Here, kα is the rate constant of reaction α of the chemical mechanism. Three dilution rates of 21/79, 16/84, and 12/88 O2/N2 were considered.

The sensitivity coefficients are shown in Fig. 2 for stoichiometric mixture condition. As expected, small-species reactions such as H + O2 ↔ O + OH, CO + OH ↔ CO2 + H and HCO (+ M) ↔ H + CO (+ M) are important reactions that enhance flame reactivity, while the reactions CH3 + H (+ M) ↔ CH4 (+ M) and H + O2 (+ M) ↔ HO2 (+ M) reduce flame speed predictions for all mixtures. The fuel-specific reactions with considerable sensitivities are the H-abstraction reactions by hydrogen radical, which have a mild negative sensitivity, and those by hydroxyl radical, which have a small positive sensitivity. Similar behaviors are observed for the three dilution rates in Fig. 2. One can deduce that the reaction pathways are not strongly influenced when decreasing the content of oxygen.

Results and discussion

Terrestrial gravity measurements

Experiments were conducted under terrestrial gravity conditions, and the temporal evolution of the typical Schlieren images recorded are shown in Fig. 3. During the first phase of the flame at an initial pressure of 3 bar (cf. Fig. 3 (first row)), it propagates spherically. As the flame continues to grow, the gravitational force leads to a strong deformation of the initially spherical shape. Due to the substantial density difference between the burned and unburned gas, the flame develops an upward motion, leading to a flame deformation and attaining a mushroom-like shape. After reaching the top of the chamber, the flame spreads laterally and downward to consume the remaining mixture in the combustion chamber. By further increasing the initial pressure to 10 bar (cf. Fig. 3 (second row)), the flame motion is significantly affected by buoyancy. It can be seen that the upward flame motion induces entrainment of unburned gas beneath the flame, resulting in a complex flow. Consequently, different points of the flame front propagate with different absolute speeds, making the extraction of laminar flame speed extremely difficult. Similar behavior recently reported by Berger al. [START_REF] Berger | A DNS study of the impact of gravity on spherically expanding laminar premixed flames[END_REF] where DNS spherically expanding premixed flames were performed for flames featuring very low flame speeds. Existing post-processing techniques, such as the horizontal radius or the equivalent radius based on the flame's cross-section, are not feasible for these highly buoyant flames. Extracting the correct flame-front contour is extremely challenging, especially beneath the flame. An advanced optical technique, such as particle image velocimetry (PIV), would need to be used to overcome this problem. This is not discussed further because it is out of the scope of the present paper. To extract the laminar flame speed accurately for these highly-buoyant flames based on the recorded images, experiments un-der microgravity conditions are necessary to suppress the formation of these non-spherical shapes.

Microgravity measurements

A series of parabolic flights onboard the Airbus A310 Zero-G was conducted under microgravity conditions to suppress the effect of buoyancy for slowly propagating ethanol/O2/N2 flames. An example is given as in Fig. 4 for a fuel-rich ethanol flame obtained at two initial pressures. For both conditions (i.e., P0 = 3 and 5 bar), the flame expands in an almost perfectly spherical manner until it reaches the limit of the window. Only images corresponding to a flame front radius greater than 9 mm were used to overcome ignition effects. It is worth stressing that the usable data range is larger than for terrestrial gravity experiments where flames quickly reach the upper windows' limit. (3) the radiation-induced regime. These three regimes are described in the following and can be clearly observed in Fig. 5 for the conditions at P0 = 5 bar & φ = 1.06, and P0 = 3 bar & φ = 1.23. In the first regime, the flame kernel propagation is mainly driven by ignition energy deposition. Both flame propagation and stretch decrease as the ignition kernel propagates outwardly until it reaches a critical radius, which is in those cases at R f = 0.9 cm. Note that a successful ignition is only achieved when the ignition kernel can propagate beyond the critical flame initiation radius. Subsequently, the flame propagation speed is only affected by stretch rate, which represents the second regime. One can see that S L,b changes almost linearly with K. Hence, this regime is used to deduce the unstretched laminar flame speed and the burned-gas Markstein length L b , which represents the sensitivity of flame speed to stretch. In the third regime and for small stretch rates, S L,b does not linearly evolve with K, but rather reaches a maximum and then decreases. There are two major reasons for this decrease: the first one is the slight pressure increase in the spherical chamber because the valves open once 0.025 MPa increase is reached, and the second is radiation effects. Chen [START_REF] Chen | Effects of radiation and compression on propagating spherical flames of methane/air mixtures near the lean flammability limit[END_REF] performed detailed numerical simulations of radiation in spherical flames and demonstrated that radiation heat loss decreases the flame speed via two mechanisms:

(1) a thermal-flow effect caused by the burned gas cooling, which generates an inward flow towards the cooling core of the burned gas region and a reduction of the flame's expansion velocity, and (2) a thermalchemical effect caused by the decreased flame temperature, which slows the actual flame propagation because of a decrease in the overall reaction rate. Three ways to correct for radiation heat losses can be found in the literature. The first one is to perform DNS together with employing a statistical narrow band model (SNB). The SNB considers reabsorption by unburned gas and therefore gives quite accurate predictions. However, combining a DNS with an SNB model is computationally expensive, especially when the reaction mechanism used contains a large amount of species and reactions. The second way is to use an empirical formula proposed by Yu et al. [START_REF] Yu | Radiation-induced uncertainty in lam-inar flame speed measured from propagating spherical flames[END_REF], which was developed by employing DNS and the SNB model and considers corrections both of the thermal-flow and the thermal-chemical effects (cf. Eq. ( 2) in Section 4.2.2). The third method is to apply the analytical model by Santner et al. [START_REF] Santner | Uncertainties in interpretation of high pressure spherical flame propagation rates due to thermal radiation[END_REF], which estimates the thermal and flow effects caused by radiation separately.

In our recent work [START_REF] Hesse | Low globalwarming-potential refrigerant CH2F2 (R-32): Integration of a radiation heat loss correction method to accurately determine experimental flame speed metrics[END_REF], we employed the Santner et al. [START_REF] Santner | Uncertainties in interpretation of high pressure spherical flame propagation rates due to thermal radiation[END_REF] model. We showed that without using the radiation correction for refrigerant flames, the obtained flame speeds would be 15% to 20% lower than using the correction. However, the Santner et al. [START_REF] Santner | Uncertainties in interpretation of high pressure spherical flame propagation rates due to thermal radiation[END_REF] analytical model is not applicable in the present work because it does not capture the high N2-dilution levels involved in the mixtures investigated. This can be seen in Fig. S3 of the Supplementary Material, where the temperature decay due to radiation heat losses is found to be marginal.

For the larger equivalence ratios shown in Fig 5, the propagation speed S L,b decreases with the decreasing stretch rate K as the flame propagates outwardly resulting in negative burned-gas Markstein length, L b . The effect of stretch on the flame speed changes with equivalence ratio. Moreover, the change of S L,b with the stretch rate is reasonably linear over the measured range of flame radius. It is worth noting that the decay in S L,b observed previously vanishes when φ becomes even richer. Consequently, a linear extrapolation of S L,b to zero stretch can be applied to obtain the unstretched propagation speed. A similar behavior was obtained by Wang et al. [START_REF] Wang | Laminar burning velocities and markstein lengths of premixed methane/air flames near the lean flammability limit in microgravity[END_REF] investigating the flammability limits of methane flames under microgravity conditions.

The effect of pressure is also investigated at a fixed equivalence ratio φ = 1.23 and the results are shown in Fig. 6. As expected, the propagation speed decreases with increasing pressure. Negative burnedgas Markstein lengths are obtained with higher magnitude for higher pressures. The flame-radii range to be used to extrapolate back to zero-stretch is of paramount importance and should be carefully selected. The lower limit is usually chosen to reduce the effects of initial spark ignition, and the upper bound is often selected to ensure that the pressure change does not exceed 1%. Several extrapolation techniques exist, for instance, the linear [START_REF] Matalon | Flames as gas dynamic discontinuities[END_REF] and non-linear methods [START_REF] Halter | Nonlinear effects of stretch on the flame front propagation[END_REF][START_REF] Kelley | Premixed flame propagation in a confining vessel with weak pressure rise[END_REF]. Assessments of these techniques have been presented in the literature [START_REF] Wu | Uncertainty in stretch extrapolation of laminar flame speed from expanding spherical flames[END_REF][START_REF] Beeckmann | Flame propagation speed and markstein length of spherically expanding flames: Assessment of extrapolation and measurement techniques[END_REF].

Laminar flame speed

Due to the non-linearity of radiation for cases at P0 = 5 bar & φ = 1.06 and P0 = 3 bar & φ = 1.23, extrapolation can involve large errors. For this reason, the flame radius-range used for those specific conditions was limited to flames where no decay in the flame propagation speed S L,b was observed, which is linked to a nonzero burned gas velocity resulting in slowing down the flame propagation. This means that the radius range considered is 0.9 cm ≤ R f ≤ 1.5 cm. For the remaining conditions, the extrapolation was straightforward, and a linear method is applied for a radius ranging between 0.9 cm ≤ R f ≤ 1.8 cm.

After extrapolating back to zero-stretch, i.e., S 0 L,b , the laminar flame speed is determined from mass continuity through a planar unstretched flame, S 0 L,u = ρ b /ρu S 0 L,b , where ρ b and ρu represent the burned-and unburned-gas densities. It is important to highlight that these results are radiation-affected and are shown in Fig. 7 over the equivalence ratio (filled symbols). The lines in Fig. 7 represent laminar flame speeds obtained from adiabatic unstretched premixed simulations using the kinetic mechanism by Cai et al. [START_REF] Cai | Exploring the combustion chemistry of a novel lignocellulose-derived biofuel: cyclopentanol. part ii: experiment, model validation, and functional group analysis[END_REF]. The total differences between radiation-induced flame speeds and simulations are in the order of 10% near stoichiometric conditions and become larger to approximately 45% at fuel-rich conditions. These results were expected since the experimental data obtained are not yet radiation corrected. The change in laminar flame speed is clearly nonnegligible for the conditions investigated and would be even more pronounced for fuels with a high sooting tendency. A radiation-correction is therefore necessary. A radiation correction is conducted as a final step. The generalized empirical correlation by Yu et al. [START_REF] Yu | Radiation-induced uncertainty in lam-inar flame speed measured from propagating spherical flames[END_REF] is used to obtain radiation-free laminar flame speeds. Recall that this model was developed by employing DNS and the SNB model. It is also based on several types of fuels, from C1 to C8 hydrocarbon fuels, syngas H2/CO at various ratios, and fuels containing oxygen functional groups such as dimethyl ether. This indicates that the model should be applicable for alcohols, such as ethanol. Moreover, a wide range of equivalence ratios and different initial pressures and temperatures were chosen to quantify the uncertainty associated with radiation. It is also important to highlight that the flame radius range, similar to our experiments, of 1.0 cm ≤ R f ≤ 2.0 cm was selected to deduce their correlation. Thus, the correlation considers also the contraction velocity effect appropriately. It is given as

S corr L,u SL,u = 1 + 0.82 SL,u S0 -1.14 T0 Ti P Pi -0.3 , (2)
where S0 = 1 cm/s, Ti = 298 K, and Pi = 1 atm. Radiation-corrected laminar flame speeds obtained using Eq. ( 2) are shown in Fig. 7 by open symbols. The corrected experimental results and the predicted data are now comparable and agree very well, especially on the rich side, where a wider flame-radius range was used. The choice of the radius range to extrapolate back to zero stretch can significantly affect the results when radiation due to flow effects is pronounced. Therefore, the obtained experimental data, which were extremely difficult to measure and postprocessed very carefully, can be used as a reference data set to validate kinetic schemes.

It is worth stressing that, because the flame thickness cannot be extracted experimentally, it was evaluated using the 1D unstretched laminar flame configuration of the open-source FlameMaster code [START_REF] Pitsch | Flamemaster: A C++ computer program for 0d combustion and 1d laminar flame calculations[END_REF],

where the mechanism by Cai et al. [START_REF] Cai | Exploring the combustion chemistry of a novel lignocellulose-derived biofuel: cyclopentanol. part ii: experiment, model validation, and functional group analysis[END_REF] was employed. The flame thickness increases as the mixture becomes richer for a defined initial pressure. From dimensional analysis (SL,u = D/l f , where D is the thermal diffusivity), the laminar flame speed is reciprocal to the flame thickness l f . The time scales become longer. Thus, the relative radiation heat loss effect on the flame speed becomes significant.

Concluding remarks

An experimental study was carried out to investigate laminar flame speeds of premixed ethanol/O2/N2 mixtures with very high N2-dilution. These flames exhibit very low burning velocities that are difficult to measure quantitatively with good accuracy. The severe flame-front distortion under terrestrial gravity is shown by conducting experiments using the highpressure, high-temperature RWTH spherical chamber employing a Schlieren optical technique to visualize the flame front propagation. Part of the difficulty stems from the fact that the flames become more sensitive to experimental errors when they burn with low propagation speeds. The effect of buoyancy was suppressed under microgravity conditions in a series of parabolic flights onboard the Airbus A310 Zero-G. The setup used at reduced gravity consists of a pressure-release-type dual chamber adequate for aircraft-related safety requirements, combined with a conventional shadowgraphy system to visualize the flame morphology. Unstretched 1D freely propagating flame simulations were performed using the Cai et al. [START_REF] Cai | Exploring the combustion chemistry of a novel lignocellulose-derived biofuel: cyclopentanol. part ii: experiment, model validation, and functional group analysis[END_REF] mechanism and were compared to the experimental data, both non-radiation-corrected or radiation-corrected, with the model given by Yu et al. [START_REF] Yu | Radiation-induced uncertainty in lam-inar flame speed measured from propagating spherical flames[END_REF]. The main outcomes of the present study are summarized as follows:

• Under reduced gravity, the flame expands in a spherical manner until it reaches the windows' limit. The propagation speeds were deduced based on the flame radius detected from shadowgraph images. A pronounced decrease in the burned-gas propagation speed owing to the nonzero burned-gas velocity is observed at large flame radii. • Flame speed reduction induced by radiation is found to be in the order of 10% to 45% for the near-stoichiometric to very rich conditions, respectively. Radiation-corrected laminar flame speeds were obtained by applying the empirical correlation presented by Yu et al. [START_REF] Yu | Radiation-induced uncertainty in lam-inar flame speed measured from propagating spherical flames[END_REF]. Excellent agreement between the corrected experimental results and the predicted ones using the Cai et al. [START_REF] Cai | Exploring the combustion chemistry of a novel lignocellulose-derived biofuel: cyclopentanol. part ii: experiment, model validation, and functional group analysis[END_REF] mechanism is observed, highlighting that the new provided data can be used as a reference to validate/improve kinetic schemes.

Fig. 1 :

 1 Fig. 1: Schematic of the parabolic flight setup.

Fig. 2 :

 2 Fig. 2: Laminar flame speed sensitivities on reaction rate coefficients of ethanol mixtures with various dilution rates using the Cai et al. model [22] (φ = 1.0, P 0 = 3 bar, T 0 = 373 K).

Fig. 3 :

 3 Fig. 3: Sequence of typical Schlieren images of highlybuoyant ethanol flames under terrestrial gravity obtained using the RWTH setup (φ = 1.4, T 0 = 373 K, P 0 = 3 and 10 bar, oxidizer: 12% O 2 -88% N 2 in vol.).

Fig. 4 :Figure 5

 45 Fig. 4: Sequence of typical shadowgraph images of ethanol flames under microgravity conditions during parabolic flights (T 0 = 373 K, oxidizer: 12% O 2 -88% N 2 in vol.). (a), (b) and (c) correspond to a flame radius of 0.8 cm, 1.4 cm, and 1.8 cm, respectively.

Fig. 5 :

 5 Fig. 5: Evolution of the propagation speed as a function of the flame stretch for ethanol mixtures at two initial pressures of 3 and 5 bar, and various equivalence ratios shown by different symbols. Lines represent the filtered data.

Fig. 6 :

 6 Fig. 6: Evolution of the propagation speed as a function of the flame stretch for ethanol-rich mixtures at different initial pressures shown by different symbols. Lines represent the filtered data.

Fig. 7 :

 7 Fig. 7: Laminar flame speed vs equivalence ratio. Filled and open symbols correspond to non-and radiation corrected flame speeds, respectively. Lines represent results obtained using the mechanism by Cai et al. [22].

Table 1 :

 1 Applied voltage and delivered energy output in air by the high-voltage ignition system using a 1.5 mm gap at the electrode tips at 1 atm.

	Voltage [kV] 0.50 0.80 1.00 1.30 1.50 1.80
	Energy [J]	0.24 0.92 1.62 3.05 4.25 6.43

Table 2 :

 2 Experimental investigated conditions under microgravity and earth gravity conditions of ethanol mixtures.

	Fuel	Oxidizer (% in vol.) T0 [K] P0 [bar]	Equivalence ratio φ [-]
	0 g Ethanol	12% O2 / 88% N2	373	1	1.23
	Ethanol	12% O2 / 88% N2	373	3	1.06, 1.23, 1.37, 1.52, 1.66
	Ethanol	12% O2 / 88% N2	373	5	0.81, 1.06, 1.23, 1.4, 1.58
	Ethanol	12% O2 / 88% N2	373	7	1.23
	1 g Ethanol	12% O2 / 88% N2	373	3	1.4
	Ethanol	12% O2 / 88% N2	373	10	1.4

Acknowledgments

The authors gratefully acknowledge the financial support provided by the Deutsches Zentrum fur Luft-und Raumfahrt (DLR, German Aerospace Center), Grant no. 50WM2073. We are also thankful to the ICARE laboratory.

Supplementary material

Supplementary material associated with this article can be found, in the online version, at doi: