William V Gehrlein

Vincent Merlin
email: vincent.merlin@unicaen.fr

On the Probability of the Ostrogorski Paradox

The notion of the existence of a Strict Ostrogorski Paradox presents an interesting phenomenon that could lead to some very unsettling outcomes in group decision-making situations. This phenomenon cannot be observed in two-issue situations, and when three-issue situations are considered, the probability that such an outcome will be observed never reaches a likelihood of as much as two percent for large electorates, regardless of the propensity of voters to align their preferences on issues with the standards of political parties. The probability of observing a Strict Ostrogorski Paradox in four-issue situations is nearly zero when using an assumption that exaggerates the likelihood that such a paradoxical outcome will be observed. We conclude that it is very unlikely that a Strict Ostrogorski Paradox would ever be observed in any real voting situation with a large electorate

Introduction

Suppose that there are two parties {R,L} that have opposing positions on each of m different issues that are being considered. Each of n voters has preferences on the individual issues that are in agreement with the position of either Party R or of Party L, but a given voter does not necessarily agree with the position of the same party on every issue. However, a given voter will be considered to have an overall alignment with a particular party whenever that voter is in agreement with that party's positions over a majority of the issues that are being considered. If a voter has preferences that agree with both Party R and Party L on m/2 issues each when m is even, then there is a tie and that voter is not considered to be aligned with either party. The majority party is the one with which the greater number of voters have an overall alignment, excluding voters that are not aligned with either party. Each issue is voted on individually, and the outcome of the vote will be in agreement with the position of either Party R or Party L, based on the outcome of majority rule voting on the issue. Voters are assumed to vote sincerely on each of the individual issues.

We are interested in developing representations for the probability, k m PA , that the voting results on the m different issues produce exactly k outcomes on issues that are in agreement with the positions of the majority party. m m PA is the probability that there is complete agreement between voting outcomes on issues and the majority party positions. 0 m PA is the probability that there is complete disagreement between voting outcomes on issues and the positions of the majority party. The situation of complete disagreement is referred to as an occurrence of a Strict Ostrogorski Paradox, loosely following some terminology that is introduced in Deb and [START_REF] Kelsey | On constructing a generalized Ostrogorski paradox: necessary and sufficient conditions[END_REF]. The general problem of such disagreement was originally presented in [START_REF] Ostrogorski | La démocratie et l'organisation des partis politiques[END_REF] and it was discussed in [START_REF] Daudt | The Ostrogorski Paradox: A peculiarity of compound majority decision[END_REF]. This phenomenon is often referred to in research that is related to paradoxical outcomes that can produce problems in group decision making. For example, see [START_REF] Brams | The paradox of multiple elections[END_REF], [START_REF] Nurmi | Voting Paradoxes and How to Deal with Them[END_REF], [START_REF] Saari | The sum of the parts can violate the whole[END_REF], [START_REF] List | The probability of inconsistencies in complex collective decisions[END_REF] and [START_REF] Gehrlein | Condorcet's Paradox[END_REF]. [START_REF] Laffond | Single-switch preferences and the Ostrogorski Paradox[END_REF] consider restrictions on voters' preferences that preclude the existence of an occurrence of Ostrogorski's Paradox. [START_REF] Mbih | La vulnérabilité de la règle de la majorité aux paradoxes d'Anscombe et d'Ostrogorski : une analyse comparative[END_REF] derive the likelihood of related paradoxes, Ostrogorski's Paradox and Anscombe's Paradox, under the Impartial Anonymous Culture assumption for m = 3 issues, as a function of n. They also provide estimations for these paradoxes under the Impartial Culture assumption via Monte-Carlo simulations for small values of n.

Values of

k m PA for 0 < k < m represent probabilities of increasingly less paradoxical outcomes as k increases, with partial agreement between voting outcomes and the positions of the majority party. Attention is focused on developing these representations for the limiting case in voters, with ,   n when voters form their preferences on each of the issues independently of the preferences of other voters. The study begins by considering specific cases with a small number of issues, and then considers general results.

The Case of Three Issues

There are 2 3 = 8 different possible combinations of voters' preference agreement with party positions on issues with m = 3. Let q i denote the probability that a randomly selected voter will have the associated i th such combination for 1 ≤ i ≤ 8, as shown in Table 1.

Table 1 Feasible voter preferences for sequential elections on three issues.

1 q 2 q 3 q 4 q 5 q 6 q 7 q 8 q Issue 1 R R R R L L L L Issue 2 R R L L R R L L Issue 3 R L R L R L R L Overall R R R L R L L L
The probability of complete voter agreement with the positions of Party R (L) for a randomly selected voter is q 1 (q 8 ) in Table 1. Let a = q 1 +q 8 measure the probability that a randomly selected voter shows complete agreement with the issue positions of one of the parties. Such voters can be viewed as staunch supporters of one party's positions. There is greater degree of complete voter agreement with party positions as a increases, but the preferences of the voters will also reflect an increasing degree of polarization as a increases, since we further assume that q 1 = q 8 = a/2 so that neither party has any advantage over the other party. The motivation behind this party parity assumption will be explained in detail later. The remaining six rankings represent more moderate voters who show less than complete agreement with any party, with position agreement with some party on two issues and disagreement on one issue. The parity concept is also used here, since it is assumed that q i = (1-a)/6 for 2 ≤ i ≤ 7.

We start the development of a representation for the limiting probability of complete agreement,

  a PA 3 3 , as   n
for a specified value of a by defining a binary variable j i Y for the j th voter, to denote the party association of that voter's preference on the i th issue. For Issue 1:

j Y 1 = +1: 4 3 2 1 q q q q    (1) -1: 8 7 6 5 q q q q    .
This definition of j Y 1 requires that Issue 1 must have an outcome that is in agreement with Party R by majority rule for n voters whenever

q q q q    -1: 8 7 4 3 q q q q    (2) j Y 3 = +1: 7 5 3 1 q q q q    -1: 8 6 4 2 q q q q    . As above,     0 3 2   j j Y E Y E and 1 2 3 2 2               j j Y E Y E . Variable j
Y 4 is then defined to account for the party alignment of voter j, with j Y 4 = +1:

5 3 2 1 q q q q    (3) -1: 8 7 6 4 q q q q    . Party R will be the majority party if   n j j Y 1 4 > 0, or 0 4  j Y
. We also note that

  0 4  j Y E and that       2 4 j Y E
= 1 with the adopted definition of the q i probabilities. Issues 1, 2 and 3 will then each have outcomes that are in majority rule agreement with Party R, while Party R is the majority party with the joint probability that 0  j i Y for each 1 ≤ i ≤ 4. The q i definitions make this equivalent to the joint probability that

n Y j i >       n Y E j i for 1 ≤ i ≤ 4 since   0         j i j i Y E Y E for 1 ≤ i ≤ 4. The Central Limit Theorem requires that the distribution of the n Y j i variables is multivariate normal as   n
, and the corresponding correlation terms for these variables are obtained directly from the  

j k j i Y Y E of the original variables, since   0  j i Y E and 1 2        j i Y E
for all 1 ≤ i ≤ 4, with:

  j j Y Y E 2 1 = 8 7 6 5 4 3 2 1 q q q q q q q q        =   3 1 4 / a  .   j j Y Y E 3 1 = 8 7 6 5 4 3 2 1 q q q q q q q q        =   3 1 4 / a    j j Y Y E 3 2 = 8 7 6 5 4 3 2 1 q q q q q q q q        =   3 1 4 / a  (4)   j j Y Y E 4 1 = 8 7 6 5 4 3 2 1 q q q q q q q q        =   3 1 2 / a    j j Y Y E 4 2 = 8 7 6 5 4 3 2 1 q q q q q q q q        =   3 1 2 / a    j j Y Y E 4 3 = 8 7 6 5 4 3 2 1 q q q q q q q q        =   3 1 2 / a  .
Since the probability of observing any specific value, including zero, in a continuous distribution is equal to zero, the limiting probability as

  n
that Issues 1, 2 and 3 have outcomes that are in majority rule agreement with Party R, while Party R is the majority party is equivalent to the four-variate normal positive orthant probability,

  1 4 R  , that n Y j i >       n Y E j i for 1 ≤ i ≤ 4 with correlation matrix 1 R :                            1 1 1 1 3 1 2 3 1 2 3 1 4 3 1 2 3 1 4 3 1 4 1 a a a a a a R .
(5)

The probability of having complete agreement for the case of three issues with two parties and a given a is therefore obtained from

    1 4 3 3 2 R   a PA
, and some results follow immediately. Since all correlation terms in 1 R increase as a increases, a result from [START_REF] Slepian | The one-sided barrier problem for Gaussian Noise[END_REF] can be used to show that   a PA positive orthant probabilities in [START_REF] Gehrlein | A representation for quadrivariate normal positive orthant probabilities[END_REF] and it leads to a representation for

  a PA 3 3 , with   . dz z a z a Sin z a Sin a Sin a PA a                                           3 1 2 2 2 1 2 1 1 3 3 0 6 2 4 3 1 4 1 1 2 2 3 3 1 4 3 1 2 4 3 8 1   (6)
For the specific case with a = 1, an exact result is obtained from direct integration of (6), with  

1 3 3 PA = 1. Table 2 lists values of   a PA 3 3
for each value of a = .00(.10)1.00 that were obtained by numerical integration from (6). The computed value for a = .25 is also included since this corresponds to the situation in which all voter preference combinations are equally likely to be observed, which is referred to as the Impartial Culture Condition (IC) in the general literature.

Table 2 Computed values of   a PA 3 3 ,   a PA 2 3 ,   a PA 1 3 and   a PA 0 3 .
The voting outcomes on all three issues agree with the position of Party R under simple majority rule if

n Y j i >       n Y E j i for all i = 1,2,3, and Party L is the majority party if n Y j 4 <   n Y E j 4 . A representation for this probability of A   a PA 3 3   a PA 2 3   a PA 1 3   a PA 0 3
.00

. , can be obtained quite easily by going through the analysis that was presented above while replacing variable

j Y 4 with j Y 4  , which leads to   dz z a z a Sin z a Sin a Sin a PA a                                           3 1 2 2 2 1 2 1 1 0 3 0 6 2 4 3 1 4 1 1 2 2 3 3 1 4 3 1 2 4 3 8 1   . (7)
An exact solution of ( 7) can be obtained for the special case of

1  a , with   1 0 3 P = 0. Computed values of   a P 0 3
are listed in Table 2 for each a = .00(.10)1.00 that were obtained from numerical integration from ( 7), along with the value for a = .25. These results indicate that the probability of observing a Strict Ostrogorski Paradox is quite small over the range of all possible values of a.

An alternative representation for   a P 0 3 is found by using a different approach to the problem that follows [START_REF] Tataru | On the relation of the Condorcet winner and positional voting rules[END_REF], [START_REF] Saari | The likelihood of dubious election outcomes[END_REF] and [START_REF] Merlin | On the probability that all decision rules select the same winner[END_REF][START_REF] Merlin | On the likelihood of Condorcet's profiles[END_REF] For any a, the sum

                                         0 2 1 2 1 4 2 2 1 8 1 2 2 2 2
    a PA a PA 0 3 3 3  is simply obtained as two times the joint probability that n Y j i >       n Y E j i
for all i = 1,2,3. This joint probability is the orthant probability of the multivariate normal distribution

  2 3 R  where all correlation terms in 2 R are obtainable from 1 R and are equal to   3 1 4 / a  .
Using the three-variate extension of Sheppard's Theorem of Median Dichotomy (see [START_REF] Johnson | Distributions in Statistics: Continuous Multivariate Distributions[END_REF], pg. 92), it is simple to show that  

a PA 3 3 +   a PA 0 3 = 2   2 3 R  =          3 1 4 2 3 4 1 1 a Sin  . ( 9
)
3 The Case of Partial Agreement on Three Issues Party R will be the majority party and Issue 1 will have the only election outcome that is in agreement with the position of Party L if

n Y j 1 <   n Y E j 1 and if n Y j i >   n Y E j i
for 2 ≤ i ≤ 4. This is the same as the joint probability that

n Y j i >   n Y E j i for 2 ≤ i ≤ 4 minus the probability that n Y j i >   n Y E j i for 2 ≤ i ≤ 4.
The first probability can be obtained directly from the three-variate extension of Sheppard's Theorem and the second probability is

  1 4 R 
. We must also account for the fact that there are three issues that could be the single issue that is not in agreement with the majority party and for the fact that there are two parties that could be the majority party. After algebraic reduction, we get a representation for

  a PA 2 3 as   . dz z a z a Sin z a Sin a Sin a PA a                                           3 1 2 2 2 1 2 1 1 2 3 0 6 2 4 3 1 4 1 1 2 2 9 3 1 4 3 1 2 4 3 8 3   (10)
An exact integral solution of (10) can be found for the special case of

1  a with   1 2 3 PA = 0. Computed values of   a PA 2 3
are listed in Table 2 for each a = .00(.10)1.00 that were obtained from numerical integration of (10), along with the value for a = .25. A representation for

  a PA 1 3 is obtained from the identity relation   a PA 1 3 = 1 -   a PA 3 3 -   a PA 2 3 -   a PA 0 3
. After substituting the representations from ( 9) and ( 10) into this identity relation, algebraic reduction leads to

  . dz z a z a Sin z a Sin a Sin a PA a                                           3 1 2 2 2 1 2 1 1 1 3 0 6 2 4 3 1 4 1 1 2 2 9 3 1 4 3 1 2 4 3 8 3   (11)
An exact integral solution of (11) can be found for the special case of a = 1,

with   0 1 1 3  PA . Computed values of   a PA 1 3
are listed in Table 2 for each a = .00(.10)1.00 that were obtained from numerical integration from (11), along with the value for a = .25.

The Impact of the Party Parity Assumption

Some discussion is in order regarding the impact of the party parity assumption that q 1q 8 = a/2 and q i = (1-a) /6 for 2 ≤ i ≤ 7. The possible voters' preferences on party positions on issues in Table 1 indicate that this assumption is equivalent to saying that the probability that any voter has a given set of preferences on issues is identical to the probability that the voter has preferences on issues with all of the R and L entries reversed. This leads to parity in voters' preferences for issues positions of Parties R and L, such that voters are equally likely to have an overall party alignment with either party and the majority party is equally likely to be either party. Situations of this nature with a complete balance of outcome possibilities will obviously tend to exaggerate the probability that paradoxical events are observed for large electorates, since the introduction of any consistent bias that favors the position of either party on the issues will typically lead to a very high probability of complete agreement with the majority party position on issues as

  n
. However, such a parity situation is not a completely implausible scenario, despite the fact that it does represent an extreme case.

Other more extreme theoretical models can be developed to obtain significantly greater probabilities that a Strict Ostrogorski's Paradox is observed. For example, consider a scenario in which the q i probabilities are obtained with the following process. Randomly generate two variables,  and  , from some probability distribution on the interval [0, 1/8] , and let     denote the propensity of voters to lean toward Party R (L) partisanship. Thus, voters are generally more disposed to favor the issue positions of Party R than Party L whenever    . The q i probabilities for a partisanship model can then defined on the basis of  and  , as shown in Table 3.

Table 3 Feasible voter preferences with a partisanship model.

 3 8 1    8 1   8 1   8 1   8 1   8 1   8 1  3 8 1  Issue 1 R R R R L L L L Issue 2 R R L L R R L L Issue 3 R L R L R L R L Overall R R R L R L L L The definitions from Table 2 lead to         2 j i Y E for i = 1,2,3 and   0 4  j Y E
. If we suppose without a loss of generality that

   as   n
, then Party R will be the majority rule winner on all issues with probability approaching one. However, Party R will only be the majority party with probability .5 with this

model since   0 4  j Y E
, so there is a very significant chance that a Strict Ostrogorski Paradox will be observed. However, the impact of this striking observation must be weighed against the relative degree of rationality that this partisanship model associates with the electorate.

Suppose that  is significantly greater than  , that we have a population that has a strong bias toward adopting the issue positions that are taken by Party R. A randomly selected voter is very predictably most likely to have preferences that are in complete agreement with Party R on all issues, which is quite a rational outcome for this model. Unfortunately, Table 3 then tells us that a randomly selected voter is least likely to have agreement with Party R on two out of three issues, suggesting an electorate that displays very odd behavior for a group that is supposedly predisposed to be highly favorable toward the issue positions of Party R. So, while it is possible to define such a theoretical model, it falls out of the realm of plausibility.

In the same vein, it is possible to develop other models that give a significantly large probability of observing other forms of an Ostrogorski Paradox by making assumptions about different intensities of importance that parties might place on the passage of the various issues that are being considered. While these models can indeed fall into the realm of plausibility, they typically rely on the assumption that the majority party has some subset of issues for which it takes a position, but where it has a low intensity of concern about the ultimate vote outcome. However, it would not be particularly paradoxical or disconcerting for the majority party if the minority party position won on such issues that are considered to be of little importance.

The party parity model that we use in the current study attempts to give an upper bound on the estimate of the paradox probabilities with a not implausible model that assumes that the parties take issue positions with a real concern about the voting outcome on the issues, without making any a priori assumptions that are intentionally creating a specific situation that is tailored to produce the paradoxical outcome that is being studied.

The Case of Two Issues

When there are only two issues that are being considered, there are four possible sets of preferences that each voter might have on issues, as listed in Table 4. The possibility of ties in determining the party alignment of voters exists with even m, and no designation for party alignment is made for the voter in such cases.

Table 4 Feasible voter preferences for sequential elections on two issues

1 t 2 t 3 t 4 t Issue 1 R R L L Issue 2 R L R L Overall R - - L
Let b denote the probability that there is complete agreement between a randomly selected voter's preference and the position of the same party on each issue. The party parity assumption is then applied with t 1 = t 4 = b/2 and t 2 = t 3 = (1-b)/2, and the special case with b =1/2 is equivalent to IC.

Following the same general logic that led to the development of the representation for

  a PA 3 3 , the probability   b PA 2 2 is given by   3 3 2 R  , where                1 1 1 2 1 3 b b b R . ( 12
)
As a result of [START_REF] Slepian | The one-sided barrier problem for Gaussian Noise[END_REF],

  b PA 2 2
does not decrease as b increases, and the three-variate extension of Sheppard's Theorem leads to

      b Sin b Sin b PA 1 1 2 2 1 1 2 2 1 4 1         . ( 13
)
We note from identities for special cases that

  0 0 2 2  PA ,   1 1 2 2  PA
, and 13) are listed in Table 5 for each value of b = .00(.10)1.00. , that the Strict Ostrogorski Paradox occurs with two items

  . / / 3 1 4 1 2 2  PA Computed values of   b PA 2 2 from (
      b Sin b Sin b PA 1 1 0 2 1 1 2 2 1 4 1         . ( 14
)
Direct integration of ( 14) shows that

  0 0 0 2  PA ,   0 1 0 2  PA
, and

  0 4 1 2 2  / PA
, which suggests that a Strict Ostrogorski Paradox cannot be observed when m = 2. This general result is proved rigorously in Corollary 1 of Deb and Kelsey (1980).

We can use this observation with ( 13) and ( 14) to obtain a simpler representation for

  b PA 2 2 as b   b PA 2 2   b PA 1 2 .00
.0000 1.0000 .10

.

    1 2 1 2 1 1 2 2     b Sin b PA  . ( 15
)
It then follows directly from earlier discussion that

    1 2 1 2 1 1 1 2     b Sin b PA  . ( 16
)
Then, ( 15) and ( 16) require that

    b PA b PA   1 1 2 2 2
, as shown in Table 5.

The General Case of m Issues

The development of these types of representations becomes significantly more complex as the number of issues increases, since the number of possible combinations of voter agreements with party positions increases as 2 m . Attention is therefore restricted to the IC assumption when we consider m ≥ 4. The initial analysis that was presented above for the two and three-issue cases is generalized by defining m binary variables to determine if the position of Party R is the winner by majority rule on each issue. Variable There are an equal number of possible voters' party position agreement combinations in both the +1 and -1 categories in the variable definitions in (17). Since each possible combination is equally likely with IC, it then follows directly that   0

 j i Z E and 1 2        j i Z E for all 1 ≤ i ≤ m.
In order to determine   Each of the possible combination of a voter's party agreement on issues can be paired with the equally likely combination in which the Party R and L positions are interchanged. Then, either both members of this pair do not have a party alignment or one is aligned with Party R while the other is aligned with Party L. It is therefore

j k j i Z Z E for 1 ≤ I < k ≤ m,
obvious that   0 1   j m Z E with IC.
If m is odd, ties for party alignment of a voter cannot exist, so it must be true that

1 2 1         j m Z E , for odd m. ( 19
)
When m is even, there are exactly issues in S(R) that will have exactly i issues in agreement with the position of Party L, and then Party R will then be the majority party if 0 ≤ i ≤ m/2. The total number of combinations in S(R) for which Party R is the majority party is therefore given by #S(R), with

         2 1 0 1 / m i m i C R S # , for odd m (21)          2 2 0 1 / m i m i C R S #
, for even m.

(

) 22
The remaining half of the possible combinations have agreement with Party L on Issue h, and we denote this subset as S(L). There are

1  m i C combinations that
have exactly i issues in agreement with the position of Party L in the remaining m-1 issues, and each such combination will have Party R as the majority party if 0 ≤ i ≤ (m-2)/2. The total number of combinations in S(L) for which Party R is the majority party is therefore given by #S(L), with

         2 3 0 1 / m i m i C L S # , for odd m (23)          2 4 0 1 / m i m i C L S #
, for even m.

(24)

The value of variable

j m j h Z Z E 1  with IC is then obtained from             m j m j h / L S # R S # Z Z E 2 1 1 1      . ( 25
)
The correlation between

j h Z and j m Z 1
 for all 1 ≤ h ≤ m follows from all of the above as

  1 1 2 1 1 2      m m / m m , h C  , for all m h   1 (odd) (26)     m / m m m m / m m , h C C 2 2 1 2 2 1 2 2       
, for all

m h   1 (even). ( 27
)
Let 1  m W denote the correlation matrix for the m+1 variables that are defined in ( 17) and ( 18), with components j , i  . The neutrality of IC toward the two parties gives the limiting probability of complete agreement with the majority party when

  n as     1 1 2     m m m m IC PA W .

The General Case of m Issues -Partial Agreement

Suppose that we are interested in the probability that there is nearly complete agreement, in the sense that only Issue 1 has a majority rule outcome for a party position that is in disagreement with the majority party. This would be determined by finding the resulting correlation matrix   for all 1< g ≤ m. The only difference between

1 1  m W and 1  m W is that 1 1 1 1 1     m , m ,  
. With the neutrality of IC toward the two parties and the symmetry of IC with respect to the m possible issues that could be the single issue in disagreement with the majority party position, it follows that

    1 1 1 1 2      m m m m m IC PA V .
This logic can easily be extended to the general case in which exactly k issues have majority rule agreement with party positions that are in disagreement with the majority party positions. The correlation matrix that is applicable to the associated probability is

    k m m m k k m m C IC PA 1 1 2      W .
This observation can be extended to produce some interesting results.

Theorem 1.

    IC PA IC PA k m k m m   , . / m k 2 0 for   Proof. Given the definition of k m 1  W , * k j , i k j , i    for all 1 ≤ i < j ≤ m+1 when k < k*. This observation is contingent upon the requirement that 0 1   m , i



, which is true from ( 26) and ( 27). It then follows from [START_REF] Slepian | The one-sided barrier problem for Gaussian Noise[END_REF] that

    * k m k m 1 1      W W . Given that m m-k m k C C  ,     k m m m k m k m m k C C        1 1 2 2 W W if k ≤ m/2. QED Theorem 2.       1 2 1     m m k k m k m m C IC PA IC PA
for m ≥ 2 with 0 ≤ k ≤ m. We then subtract the second probability that there are exactly k majority rule outcomes on issues that are in disagreement with the position of Party R, when Party L is the majority party. This second probability is obtained by using the assumptions that led to the development of

  k m m 1 1    W
, except that the signs on variable

  k m m m     1 1 W . As a result, we find that       k m m m m k m m          1 1 2 1 1 1 W W       k m m m m k m m k k m m m k C C C           1 1 1 2 1 1 1 2 2 W W       1 2 1 1 1 1 1 2 2            m m k k m m m m k m k m m m k C C C W W
.

QED

A related observation then follows directly from the proof of Theorem 2. Representations for multivariate normal positive orthant probabilities generally become extremely complex in cases with more than four variables, except for special cases in which very restrictive conditions can be placed on the associated correlation matrix for the distribution.

A reasonable representation is obtained for   5 5 W  by appealing to Boole's Equation (see [START_REF] Johnson | Distributions in Statistics: Continuous Multivariate Distributions[END_REF], pg. 52), which describes a procedure that can be used to express positive orthant probabilities with an odd number of dimensions in terms of a linear combination of positive orthant probabilities with fewer dimensions. With the correlation matrix

  5 5 W  =                   .                          4 4 4 4 3 3 3 3 2 2 2 2 4 6 4 4 6 2 1 5 1 2 1 U Z U Z U Z (28)
Here, j Z denotes a correlation matrix for a distribution on j variables with all correlation terms are equal to zero, as above. The correlation matrix j U is defined on j variables with terms 0  h , i u for all 1 ≤ i < h < j and . There are four issues that could be the single issue that is in disagreement with the issue position of the majority party, and there are two parties that could be the majority party. The symmetry of IC with respect to issues and parties leads to the conclusion that   IC PA

Conclusion

The possibility of the existence of a Strict Ostrogorski Paradox presents a very interesting phenomenon that could lead to very unsettling outcomes in group decision-making situations. However, this phenomenon cannot exist in two-issue voting situations for any n. When three-issue situations are considered, the results of Table 2 indicate that the probability of such an outcome never reaches as much as a two percent for large electorates, regardless of the propensity of voters to align their

  Table5clearly show that the probability of complete agreement increases dramatically as b increases for the case of two issues.Using the logic from earlier arguments, we obtain a representation for the probability,

  of +1 (-1) when the j th voter is in agreement with the position of Party R (L) on the i th issue. The m variables are formally defined as j i Z = +1 : For j th voter agreement with the position of Party R on Issue i (17) -1 : For j th voter agreement with the position of Party L on Issue i.

Z

  we partition the set of all possible combinations of voter agreements with party positions on issues into 2 m-2 subsets of cardinality four such that each of can have values of +1 or -1 in each subset, while the party agreements on issues on the remaining m-2 issues are identical within each of the subsets. Obviously, is then defined to denote the contribution that the party alignment of the j th voter makes toward Party R being the majority party, with j m Z 1  = 1  : If j th voter is aligned with Party R (18) 0: If j th has no party alignment -1: If j th voter is aligned with Party L.

  of the 2 m possible voter agreements on the m issues for which a voter has no party alignment, assigns an equally likely probability of 1/2 m to each possible combination, ' party agreement on Issue h in the 2 m different possible combinations of voter agreements to obtain   with Party R on Issue h, and this subset is denoted as S(Ragreements on the remaining m-1 party positions for

  +1 [-1] for each combination of possible voter agreements in S(R)[S(L)]. The expected value  

  1 ≤ i ≤ k. The same probability value will be obtained, regardless of which specific set of k issues are selected to have k issues and there are two parties that could be the majority party, so

  the probability that Party R is the majority party and that there are exactly k majority rule outcomes on issues that are in disagreement with the position of Party R. This orthant probability can alternatively be obtained as the difference in two probabilities. The first of these probabilities represents the situation in which there are exactly k majority rule outcomes on issues that are in disagreement with the position of Party R. This situation makes no determination of the majority party, and the associated correlation matrix

  associated positive orthant probability that is equivalent to

  Theorem can be used to obtain simple representations for to the situation in which there is only partial agreement with four issues, with the development of a representation for

Table 5

 5 Computed values of

3does not decrease as a increases. The correlation terms in 1 R fit the special case for four-variate normal

W , Boole's Equation results in

views with the standards of political parties. The results of (31) indicate that the probability of observing a Strict Ostrogorski Paradox in four-issue situations is nearly zero with IC for large electorates.

Given our discussion regarding the propensity of models with the assumption of part parity to exaggerate the likelihood that such paradoxical outcomes will be observed, we can conclude that it is very unlikely that a Strict Ostrogorski Paradox, or any other extreme form of Ostrogorski's Paradox, would ever be observed in any real situation with large electorates over the range of the number of issues that we have been considering.

An alternative approach to the problem of deriving these probability representations in the limit as   n that was mentioned previously has been used to obtain alternative forms to verify numerical results from all of the representations that are given above for m = 2, 3, 4. Details of these derivations are available from the authors upon request.

Aknowledgements

The authors are grateful for comments from two extremely helpful reviewers of an earlier version of this paper.


Chong Wang
email: chong.wang@insa-strasbourg.fr

Cyrille Chazallon
email: cyrille.chazallon@insa-strasbourg.fr

Pierre Hornych
email: pierre.hornych@univ-eiffel.fr

Peng Jing
email: peng.jing@bjut.edu.cn

Saida Mouhoubi
email: saida.mouhoubi@insa-strasbourg.fr

Wang C Writer

Chazallon C Author

Hornych P Author

Jing P Author

Mouhoubi S Author

Study of mechanical behavior and self-cementing properties of recycled concrete aggregates under repeated loading

Keywords: recycled concrete aggregates, permanent deformation, resilient modulus, self-cementing properties, repeated load triaxial test, unbound granular materials

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

INTRODUCTION

In Europe, over 800 million tonnes of construction and demolition wastes (CDW) are produced each year (Eurostat, 2018), with concrete wastes representing up to 75% of total CDW by weight Proceedings Paper Formatting Instructions -2 -Rev. 9/2020 [START_REF] Nwakaire | Utilisation of recycled concrete aggregates for sustainable highway pavement applications; a review[END_REF]. The traditional method to treat those wastes, disposed in landfills, is gradually being abandoned by more and more countries due to the environmental and economic concerns. To achieve a sustainable development of pavement construction, recycling and reuse of concrete wastes into pavement base and subbase layers has become a topic of global concerns in recent years [START_REF] Lu | Resilient and permanent deformation behaviors of construction and demolition wastes in unbound pavement base and subbase applications[END_REF][START_REF] Yaghoubi | Impact of compaction method on mechanical characteristics of unbound granular recycled materials[END_REF].

After pavement construction, a continuous increase of strength and stiffness of pavement base and subbase layers built with recycled concrete aggregates (RCA) can be observed [START_REF] Korkiala-Tanttu | Long-term behavior of crushed concrete in road structure[END_REF]Lancieri et al., 2006), due to the hydration of unhydrated cement inside RCA, known as self-cementing properties. Some researches have been conducted to study the selfcementing properties of RCA [START_REF] Arm | Self-cementing properties of crushed demolished concrete in unbound layers: results from triaxial tests and field tests[END_REF][START_REF] Jitsangiam | Recycled Concrete Aggregates in Roadways: Laboratory Examination of Self-Cementing Characteristics[END_REF][START_REF] Poon | The cause and influence of self-cementing properties of fine recycled concrete aggregates on the properties of unbound sub-base[END_REF]. [START_REF] Arm | Self-cementing properties of crushed demolished concrete in unbound layers: results from triaxial tests and field tests[END_REF] and [START_REF] Jitsangiam | Recycled Concrete Aggregates in Roadways: Laboratory Examination of Self-Cementing Characteristics[END_REF] investigated the effect of self-cementing properties on resilient modulus. It was found that resilient moduli show a clear increase with curing time. [START_REF] Poon | The cause and influence of self-cementing properties of fine recycled concrete aggregates on the properties of unbound sub-base[END_REF] reported that self-cementing properties are most likely governed by the fine fraction of RCA (< 0.15 mm and 0.3-0.6 mm), as the increase of compressive strength was only observed with these two fine fractions. However, the influence of self-cementing properties on the mechanical performance has been rarely studied.

In this work, the influence of self-cementing properties on the permanent and resilient deformation behavior of RCA was explored. Firstly, the self-cementing properties of two RCA (NRCA and ORCA) were evaluated by pH value measurements and calorimeter tests. Then, the permanent and resilient deformation behavior were investigated by conducting a series of repeated load triaxial tests on NRCA and ORCA specimens, after different curing times (1, 180, 360 and 720 days).

MATERIALS AND METHODS

The two different RCA materials, NRCA and ORCA, sourced from different demolished buildings and crushed in France with unknown compositions, were used in this study. NRCA was crushed at the beginning of the study in 2020 (new RCA), while ORCA was crushed in 2012 (old RCA). After crushing process, two RCA were reconstituted to a same grading curve close to the mean value of the upper and lower limits used for pavement base and subbase layer (NF EN 13285, 2018), as shown in Figure 1. Figure 2 shows the Modified Proctor curves of NRCA and ORCA. The optimum moisture contents (OMC) of NRCA and ORCA are 12.9% and 11.5% respectively, significantly higher than those of most natural aggregates (typically between 4 % and 6 %) due to the attached mortar. The maximum dry densities (MDD) are 1.98 Mg/m3 (NRCA) and 2.0 Mg/m3 (ORCA). The main properties of two RCA materials are summarized in Table 1. . Proceedings Paper Formatting Instructions -3 -Rev. 9/2020

Self-cementing properties

To assess self-cementing properties of NRCA andORCA, pH value (BS 1377-3, 2018) and heat of hydration (NF EN 196-9, 2010) were measured. For pH value measurement, 30 g of RCA fines aggregates (0-2 mm) and 75 g of distilled water were mixed and stirred for 2 minutes. Then, the pH value of each suspension was measured by a pH meter. The heat of hydration was measured Proceedings Paper Formatting Instructions -4 -Rev. 9/2020 by semi-adiabatic method. The previously weighted 360 g RCA and 180 g water were mixed quickly and placed in calorimeter device to measure the heat of hydration of RCA for 24 hours.

Repeated load triaxial tests

For repeated load triaxial tests (RLTT), the mixtures with targeted water content were compacted in a split mould to the targeted dry density. Same targeted water content (OMC-1%) and targeted dry density (maximum dry density at the targeted water content) were selected from the Modified Proctor curves (Figure 2) for two RCA mixtures. Finally, the specimen was demolded and wrapped with plastic bags and stored indoors for 1, 180, 360 and 720 days.

A procedure following the European Standard NF EN 13286-7 ( 2004) was employed to study the permanent and resilient deformation behavior of RCA under the traffic loadings, which consists of a conditioning phase and followed by a resilient phase. In the conditioning phase, the specimen was subjected to a constant confining pressure (70 kPa) and a cyclic axial stress (340 kPa) at the frequency of 1 Hz for 20000 loading cycles. After the condition phase, six different confining pressures, 20 kPa, 35 kPa, 50 kPa, 70 kPa, 100 kPa and 150 kPa, and two different cyclic deviator stress q (low stress level and high stress level) for each confining pressure were applied on the specimen at a frequency of 0.1 Hz for 100 cycles, to study resilient deformation behavior. The stress paths for the RLTT are shown in Figure 3. The deviator stress q and mean normal stress p are defined by: 13 q =  -

(1)

13 2 3 p  +  = ( 2
)
where σ1 is the vertical stress and σ3 is the confining pressure.

RESULTS AND DISCUSSION

Self-cementing properties

Proceedings Paper Formatting Instructions -5 -

Rev. 9/2020
The pH values of NRCA and ORCA are also shown in Table 1. The results show that the pH value of NRCA (13.07) is much higher than that of ORCA (10.53), indicate a much stronger self-cementing property of NRCA while a negligible self-cementing property of ORCA after longterm storage [START_REF] Bestgen | Mechanical and Environmental Suitability of Recycled Concrete Aggregate as a Highway Base Material[END_REF][START_REF] Poon | The cause and influence of self-cementing properties of fine recycled concrete aggregates on the properties of unbound sub-base[END_REF].

Figure 4 shows the total released heat of NRCA and ORCA, resulting from the hydration of unhydrated cement. It shows that the heat was released once the water was added to the two RCA. The released heat of NRCA is around 14.7 J, while only 2 J of hydration heat was released by ORCA due to the long-term storage. As a result, self-cementing properties of NRCA are much stronger than these of ORCA, in agreement with the pH value measurement.

Repeated load triaxial tests

Figure 5 (a) and (b) show the permanent axial deformation ε1 p and the water content of each NRCA and ORCA specimen, after different curing times. To better show the effect of self-cementing properties on permanent deformation, Figure 6 plot the final ε1 p of NRCA and ORCA after 20000 loading cycles versus curing times. For NRCA, crushed recently, showing strong self-cementing properties, the results show ε1 p decreases quickly after curing few months (180 days), then the influence of self-cementing properties on ε1 p is diminished as ε1 p reaches to a constant value. A power relationship can be obtained between final permanent axial deformation and curing time t, as can be seen in Figure 6. As for ORCA, stored for a long time, showing negligible self-cementing properties, ε1 p is influenced by water content as other unbound granular materials [START_REF] Jing | Permanent deformation behaviour of a granular material used in low-traffic pavements[END_REF][START_REF] Soliman | Permanent deformation behavior of unbound granular base materials with varying moisture and fines content[END_REF], instead of self-cementing properties, as can be seen in Figure 5 An increase of Mr during the first load cycles can be observed, and then gradually reaches to a constant value, for both NRCA and ORCA after different curing times except the NRCA specimens curing 360 and 720 days. This behavior is due to the particle rearrangement and consolidation during the conditioning phase, which is observed also for unbound granular materials (UGM) in a shakedown state by other researchers [START_REF] Jing | Unsaturated mechanical behaviour of a granular material[END_REF][START_REF] Yang | Correlation Between Resilient Modulus and Plastic Deformation for Cohesive Subgrade Soil Under Repeated Loading[END_REF]. Interestingly, for NRCA specimens after curing 360 and 720 days, a decrease of Mr during the first Proceedings Paper Formatting Instructions -7 -Rev. 9/2020 loading cycles can be observed, especially for NRCA curing 720 days. It indicates that some weak bonds between particles, due to the self-cementing properties, could break under the traffic loadings. Resilient modulus is a key input parameter in pavement design. The widely used k- model [START_REF] Seed | Prediction of flexible pavement deflections from laboratory repeated-load tests[END_REF] was used to analyze the test results. The resilient modulus Mr and k- model are defined as:

1 r r q M  = (3) 2 1 K r MK  = (4)
where q and 1 r is the deviator stress and resilient axial strain,  is the bulk stress (σ1+σ2+σ3), and K1 and K2 are model parameters determined by regression analyses. Figure 8 (a) and (b) show the experimental data and k- model predictions of resilient modulus Mr for NRCA and ORCA after different curing times. It can be observed that k- model can fit the experimental results well, as all the coefficients of determination (R 2 ) are higher than 0.90, except for the NRCA specimens curing for 720 days (R 2 =0.761). The scattered tested data of NRCA curing for 720 days are attributed to the low resilient deformations (high stiffness), under the tested stresses, as observed also by [START_REF] Jitsangiam | Recycled Concrete Aggregates in Roadways: Laboratory Examination of Self-Cementing Characteristics[END_REF]. Besides, a significant influence of self-cementing properties on the Mr of NRCA can be observed (Figure 8 (a)), as Mr increases with curing time, under a same stress state, in accordance with previous studies [START_REF] Arm | Self-cementing properties of crushed demolished concrete in unbound layers: results from triaxial tests and field tests[END_REF][START_REF] Jitsangiam | Recycled Concrete Aggregates in Roadways: Laboratory Examination of Self-Cementing Characteristics[END_REF]. The increasing rate of Mr increases with curing time, until the tested curing time (720 days). For example, under the highest stress level (σ3=150 kPa, high stress level), Mr increases 0.3 (180 days), 0.8 (360 days) and 2.2 (720 days) times than that curing 1 day. However, the curing time has no influence on the Mr of ORCA (Figure 8 Table 2 summarizes the k- model parameters for NRCA and ORCA, after different curing times. Figure 9 depicts the evolution of k- model parameters K1 and K2 of NRCA with curing time. Given that the self-cementing properties of ORCA are negligible, only the parameters of NRCA were presented. Parameter K1 is an indicator of the resilient modulus magnitude, while K2 represents the stress dependency of materials. It can be observed that K1 increases while K2 decreases with increasing curing time, indicating the improvement of stiffness and the reduce of stress dependency of NRCA as curing time increases, as observed also by other researchers [START_REF] Arm | Self-cementing properties of crushed demolished concrete in unbound layers: results from triaxial tests and field tests[END_REF][START_REF] Jitsangiam | Recycled Concrete Aggregates in Roadways: Laboratory Examination of Self-Cementing Characteristics[END_REF]. Furthermore, the variation of K1 and K2 is more pronounced with increasing curing time, which is in good agreement with the improvement of Mr.

CONCLUSION

In this study, the influence of self-cementing properties on the permanent and resilient deformation behavior of RCA was investigated, by conducting a series of repeated load triaxial tests.

The results of pH values and heat of hydration measurements showed that the selfcementing properties of NRCA are very strong, indicated by high pH value and more total released heat, while the self-cementing properties of ORCA, after long-term storage, are negligible.

The permanent deformation of NRCA, with strong self-cementing properties, decreases quickly after curing 180 days, then reaches to a constant low value. During permanent deformation test, the resilient modulus of NRCA curing 1 and 180 days increase with loading cycles, while these of NRCA curing 360 and 720 decrease, to a constant value.

The resilient modulus of NRCA increases with curing time till the studied curing time (720 days). Besides, the increasing rate of resilient modulus also increases with curing time. The modelling results with k- model reveal that the stress dependency of Mr decreases with curing time.

While for ORCA, after long-term storage, the curing effect has no influence on the permanent deformation and resilient deformation behavior due to the negligible self-cementing properties.

Finally, the study shows that RCA with good self-cementing properties (like NRCA) represent excellent materials for use as unbound pavement layers, because their properties increase significantly with time, and can reach, in the long term, much higher values than those of natural unbound granular materials. However, this evolution, evolved from unbound RCA layers, governed by rutting failure in the short term, to partially bound RCA layers, governed by fatigue and cracking in the long term, could lead to other deterioration mechanisms, which should be further investigated.

Figure 1 .

 1 Figure 1. Particle size distributions of NRCA and ORCA.

Figure 2 .

 2 Figure 2. Modified Proctor curves of NRCA and ORCA

Figure 3 .

 3 Figure 3. Stress paths for repeated load triaxial tests.

Figure 4 .

 4 Figure 4. Hydration heat of NRCA and ORCA.

  Figure5(a) and (b) show the permanent axial deformation ε1 p and the water content of each NRCA and ORCA specimen, after different curing times. To better show the effect of self-cementing properties on permanent deformation, Figure6plot the final ε1 p of NRCA and ORCA after 20000 loading cycles versus curing times. For NRCA, crushed recently, showing strong self-cementing properties, the results show ε1 p decreases quickly after curing few months (180 days), then the influence of self-cementing properties on ε1 p is diminished as ε1 p reaches to a constant value. A power relationship can be obtained between final permanent axial deformation and curing time t, as can be seen in Figure6. As for ORCA, stored for a long time, showing negligible self-cementing properties, ε1 p is influenced by water content as other unbound granular materials[START_REF] Jing | Permanent deformation behaviour of a granular material used in low-traffic pavements[END_REF][START_REF] Soliman | Permanent deformation behavior of unbound granular base materials with varying moisture and fines content[END_REF], instead of self-cementing properties, as can be seen in Figure5(b).

Figure 6 .

 6 Figure 6. Final permanent axial deformation of NRCA and ORCA.

Figure 7

 7 Figure 7 plots the resilient modulus Mr of NRCA and ORCA, during permanent deformation test, after different curing times. A clear increase of Mr for NRCA with curing time can be observed due to the self-cementing properties. By contrast, the final Mr of ORCA is constant, which is consistent with the evolution of permanent deformation.An increase of Mr during the first load cycles can be observed, and then gradually reaches to a constant value, for both NRCA and ORCA after different curing times except the NRCA specimens curing 360 and 720 days. This behavior is due to the particle rearrangement and consolidation during the conditioning phase, which is observed also for unbound granular materials (UGM) in a shakedown state by other researchers[START_REF] Jing | Unsaturated mechanical behaviour of a granular material[END_REF][START_REF] Yang | Correlation Between Resilient Modulus and Plastic Deformation for Cohesive Subgrade Soil Under Repeated Loading[END_REF]. Interestingly, for NRCA specimens after curing 360 and 720 days, a decrease of Mr during the first

Figure 7 .

 7 Figure 7. Resilient modulus of NRCA and ORCA during permanent deformation test: (a) NRCA; (b) ORCA.

  (b)), as observed also from permanent deformation.Figure 8. Test results and the k- model predictions of M r for NRCA and ORCA after varying curing times: (a) NRCA; (b) ORCA.

Figure 9 .

 9 Figure 9. Evolution of k- model parameters K 1 and K 2 of NRCA with curing time

Table 1 . Properties of RCA materials.

 1

	Materials	Particle density-	Water absorption-	Fine content	Grading	pH
		coarse (10-20mm)	coarse (10-20mm)	(63μm)	mm
		Mg/m 3	%	%
	ORCA	2.57	4.30	2.2	0/20	10.53
	NRCA	2.65	4.64	2.1	0/20	13.07

Table 2 . Summary of k- model parameters for NRCA and ORCA.

 2

	Parameters		NRCA			ORCA
		1 day 180 days 360 days 720 days	1 day	28 days 360 days
	K 1	1.91	2.40	5.39	31.69	2.69	2.49	1.90
	K 2	0.87	0.88	0.81	0.64	0.81	0.83	0.87
	R 2	0.903	0.963	0.958	0.761	0.959	0.965	0.984

ACKNOWLEDGMENTS

This work is supported by the China Scholarship Council.


Pierre Pébereau
email: pierre.pebereau@lip6.fr

Subfield attack: leveraging composite-degree extensions in the Quotient Ring transform

.

Notations

Let q = p e for p prime and let F q denote the finite field of characteristic p with q elements. Vectors are assumed to be column vectors and are denoted by bold letters: x, y, o, . . .. Matrices are denoted by capital letters, and transposition is written A T . Given a field F and an integer n, we denote F[x 1 , . . . , x n ] or F[x] the polynomial ring of F in n indeterminates. The restriction of a function f to a set E is denoted f |E .

Unbalanced oil and vinegar

A UOV key pair for parameters (n, m, q) is composed of a secret key (A, F) and a public key P, with:

-A ∈ GL n (F q ) -F = (F 1 , . . . , F m ) a quadratic map with F i (e j ) = 0 for all i and for j ≤ m -P = F • A a quadratic map If we represent the quadratic maps with matrices, we have:

∀1 ≤ i ≤ m, F i = 0 F (1) i F (2) i F (3) i
This formulation was introduced by Patarin in [START_REF] Patarin | The oil and vinegar signature scheme[END_REF] and the original motivation was that the system F(x) = t is linear in x 1 , . . . , x m . These variables are distinguished from the rest of variables and are named "oil variables". The remaining ones are "vinegar variables". The knowledge of A allows the signer to efficiently solve P(x) = t using this property. The set of accepted signatures for a message t ∈ F m q , noted V(t) = {x ∈ F n q , P(x) = t}, is an algebraic variety of dimension n -m generically.

VOX

VOX is a signature scheme submitted to the NIST alternative signature round [START_REF] Cogliati | Vox-sign[END_REF]. It relies on the same core principles as UOV, but adds "noise" to the public key to hide the structure of the UOV trapdoor. VOX also relies on additional structure, the QR-transform [START_REF] Hiroki Furue | A new variant of unbalanced oil and vinegar using quotient ring: Qr-uov[END_REF], which is akin to the construction of structured lattices. We view VOX as "QR-UOV +".

UOV

+

We start by defining UOV +. A UOV + key pair for parameters (o, v, t, q) is composed of a secret key (S, A, F) and a public key P, with:

-A ∈ GL o+v (F q ) -S = I t S ′ 0 I o-t , S ′ ∈ F (o-t)×t q , S ∈ GL o (F q )
-F = (F 1 , . . . , F o ) a quadratic map with F i (e j ) = 0 for i > t and j ≤ o.

-

P = S -1 • F • A a quadratic map
We will write n = o+v and let F = (F t+1 , . . . , F o ) be the "VOX oil polynomials". Fig. 1: VOX parameters sets submitted to NIST [START_REF] Cogliati | Vox-sign[END_REF] and alternative parameters [START_REF] Macario-Rat | Rectangular attack on vox[END_REF].

QR-UOV and security assumptions

In [START_REF] Hiroki Furue | A new variant of unbalanced oil and vinegar using quotient ring: Qr-uov[END_REF], the authors introduce a way to reduce key sizes for UOV by representing a UOV public key using structured matrices. Essentially, the scheme may be considered as a "block matrix" variant of UOV, where the matrices in the key pair are block matrices with elements belonging to a matrix ring F l×l q . To achieve a gain in performance, we consider the quotient ring R = F q [x]/(f ), where deg(f ) = l, and embed it in F l×l q using an injective ring homomorphism. We will call the integer l the "QR parameter". Not every choice of f will lead to a secure scheme, as observed by the authors of QR-UOV (a subset of the authors had previously attacked a similar construction called BAC-UOV, see [START_REF] Szepieniec | Block-anti-circulant unbalanced oil and vinegar[END_REF] and [START_REF] Hiroki Furue | A structural attack on block-anti-circulant uov at sac 2019[END_REF]). In short, f must be irreducible to ensure the security of the schemes, and in this case notice that

F q [x]/(f ) ∼ = F q l .
The security assumptions highlighted by the authors of [START_REF] Hiroki Furue | A new variant of unbalanced oil and vinegar using quotient ring: Qr-uov[END_REF] are the following: An instance of QR-UOV for parameters (q, v, m, l) is secure only if two (generalized) UOV instances are secure:

1. UOV(q l , v l , m l , m), a UOV(q l , v l , m l ) instance with m equations instead of m l 2.
UOV(q, v, m), a plain UOV instance.

Analysis of QR-UOV

+

The same reduction applies to VOX when interpreted as QR-UOV +. In [START_REF] Cogliati | Vox-sign[END_REF], the parameter l is noted c. We keep the original notation l in the following. An instance of QR-UOV + for parameters (q, v, o, l, t) is secure only if two (generalized) UOV + instances are secure:

1. UOV +(q l , v l , o l , o, t), a UOV +(q l , v l , o l , t) instance with o equations. 2. UOV +(q, v, o, t), a plain UOV + instance. Let N = v+o l , O = o l , V = v l .
Let (P 1 , . . . , P o ) be a UOV +(q l , V, O, o, t) instance. Let I = ⟨p 1 (x), . . . , p o (x)⟩ the ideal generated by the corresponding system of equations. Let ( Pt+1 , . . . , Po ) be the underlying UOV(q l , V, O, o) instance. Define the underlying UOV variety:

V( Î) = {x ∈ (F q l ) N , x T Pt+1 x = . . . = x T Po x = 0}
This variety may be defined by an overdetermined system of quadratic equations, but it is never empty as O ⊆ V( Î). When o > N , this inclusion is generically an equality. Define the UOV + variety:

V(I) = {x ∈ (F q l ) N , x T P 1 x = . . . = x T P o x = 0}
Notice that this variety is the intersection of V( Î) with t generic quadratic hypersurfaces defined by p 1 (x) = 0, . . . , p t (x) = 0. Therefore, if dim

V( Î) = d, then we expect, if d -t ≥ 0, that dim V(I) = d -t. If this quantity is negative, then V(I) is empty.
The interesting case is when o > N and O -t > 0. Then V(I) is not empty and V(I) ⊂ O. If I is radical, this implies in particular that the linear equations defining O belong to I. Therefore, they must belong to a Gröbner basis of the ideal for a graded monomial ordering. We summarize this in Proposition 1.

Proposition 1. Let O be a linear subspace of F n q of dimension o. Let I = ⟨p 1 , . . . , p m ⟩ be a radical ideal of F q [x] such that V(I) ̸ = ∅ and V(I) ⊂ O. Then, a Gröbner basis of I for any graded monomial ordering contains linear equations

H 1 (x), . . . , H n-o (x) such that O = ∩ n-o i=1 H i Proof.
By assumption, V (I) ⊂ O. Therefore, if H is a linear form such that O ⊂ ker(H), then H(x) ∈ I(V (I)). By the Nullstellensatz, this implies that H(x) ∈ √ I. Since I is radical, then √ I = I and H(x) ∈ I. Next, let ≤ be a graded monomial ordering. On the first hand, a graded monomial ordering is a monomial ordering which first compares the total degree before breaking ties. Therefore, if p is a polynomial in F q [x], then the leading term of p with respect to ≤ has degree equal to the total degree of p.

On the other hand, a Gröbner basis of

I for ≤ is a set G = {g 1 , . . . , g t } ⊂ I such that ⟨LT ≤ (g 1 ), . . . , LT ≤ (g t )⟩ = ⟨LT ≤ (I)⟩
This implies that a Gröbner basis of I must contain polynomials whose leading terms are of minimal degree, in our case 1. The collection of linear equations included in a Gröbner basis must have rank at least n -o, otherwise we could find a linear equation in I linearily independant from the ones in the basis. We add that in the reduced Gröbner basis for ≤, there are only independant linear equations.

⊓ ⊔

The ideal generated by a general collection of polynomials is radical as a consequence of the theorem of Bertini. Since UOV polynomials are certainly not generic, it is not obvious that the ideal generated by a UOV public key is radical. Experimentally, we observe that the systems we obtain define radical ideals, hence we use the previous result.

Using Proposition 1, we mount a key recovery attack on the scheme by computing a grevlex Gröbner basis of the ideal generated by the system P(x) = 0.

x =∈ F N q such that        P 1 (x) = 0 . . . P o (x) = 0 (1)
Assuming semi-regularity, the degree of regularity d reg is found as the first nonpositive coefficient in the Hilbert Series:

H R/I (t) = (1 -t 2 ) o (1 -t) N
Therefore, if O -t > 0 and N < o, using dense linear algebra with a matrix multiplication cost identified as O(n ω ) arithmetic operations, we upper bound the number of arithmetic operations required by our attack by

O N + d reg d reg ω (2)
Following NIST methodology, an arithmetic operation in F q is taken to be equal to 2 log 2 (q) 2 + log 2 (q) gates. Once a Gröbner basis has been computed, we retrieve the linear terms, and dismiss the remaining higher degree polynomials. This directly applies to the initial VOX parameter sets, and exploits the same fault as the rectangular Min-Rank attack of Furue and Ikematsu [START_REF] Furue | A new security analysis against mayo and qr-uov using rectangular minrank attack[END_REF], namely O > t.

We describe in the next section a trick to attack a subset of the new parameters proposed for VOX in [START_REF] Macario-Rat | Rectangular attack on vox[END_REF] when l is a composite number. We expect the Rectangular MinRank attack of Furue and Ikematsu [START_REF] Furue | A new security analysis against mayo and qr-uov using rectangular minrank attack[END_REF] to also apply in this case.

Factoring the QR parameter The previous paragraph showed that manipulating the equations of the QR-UOV + instance may be more tricky due to the fact that we keep the same number of vinegar equations but have a smaller subspace O when working in the extension F q l . Therefore, if l is large, then o l < t and there is no intersection between O and the (generic) vinegar variety, which means that we are unable to attack the UOV + instance (q l , V, O, t, m) without inverting S. This happens for the VOX parameters proposed in [START_REF] Macario-Rat | Rectangular attack on vox[END_REF] precisely because the inequality O > t enabled the rectangular MinRank attack on the scheme, therefore the opposite was enforced to defeat it.

We use the following trick to bypass this obstacle in some instances of VOX: if l ′ divides l, then F q l is a field extension of degree l l ′ of F q l ′ .

Remark 1. If l ′ divides l, we may interpret a QR-UOV + instance for parameters (q, v, m, l) as a QR-UOV + instance for parameters (q, v, m, l ′ ). This holds also for QR-UOV.

For VOX, it enables the simple attack described in Equation ( 1) on intermediate fields for QR-UOV +, while the attack would fail in the field q l because of the dimension considerations. In Figure 1, the attack applies to the parameter sets with c in bold. The best complexity will be obtained by minimizing the number of variables, therefore we choose the largest divisor l ′ of l such that o l ′ > t. Notice that this attack will only be able to target the parameters Ic, IIIa, Vb, since for all other parameter sets, l is prime. For parameter sets I,III,V, the attack applies in the large field F q l without using an intermediate field.

Performing the attack We evaluate the attack for the largest admissible factor of l. Figure 2 gives the estimation for the VOX parameters submitted to NIST, and for the parameters proposed in [START_REF] Macario-Rat | Rectangular attack on vox[END_REF]. We generate random instances of the corresponding UOV +(q l ′ , v l ′ , o l ′ , o, t) scheme, and solve them using msolve [START_REF] Berthomieu | msolve: A library for solving polynomial systems[END_REF] on a laptop with an i7-1165G7 CPU running at 2.80GHz with 8 cores and 8 GB of RAM. Parameter sets that are omitted are not concerned by the attack. Notice that the variety V(I) is zero-dimensional in case t = O. In this case, we believe that the attack does not applies in the large field F q l , because the only solution of the homogeneous system is (0, . . . , 0).

MinRank attack on VOX

In [START_REF] Guo | A practical minrank attack against vox[END_REF], Guo and Ding introduce a MinRank attack that also targets the new parameters proposed for VOX. The attack is practical and breaks all of the new parameters sets proposed for VOX under 2 58 arithmetic operations. The attack of Guo and Ding does not exploit the same vulnerability we exploit in Section 1.3. In particular, they target all parameter sets, while our attack only targets the parameter sets with composite c. Guo and Ding propose alternative parameters for VOX to improve the security of the scheme against the MinRank attack. We recall these possible parameters and their expected security in Figure 3. Fig. 3: Possible VOX parameters proposed in [START_REF] Guo | A practical minrank attack against vox[END_REF].

q,t
To illustrate the difference in our approach, notice that our attack applies to the subset of these new parameters where c is composite. We detail the cost in arithmetic operations to match the estimates in [START_REF] Guo | A practical minrank attack against vox[END_REF].

Parameter set l l ′ log 2 ops Previous [START_REF] Guo | A practical minrank attack against vox[END_REF] GD-Ic 9 3 99. Fig. 4: Binary operation counts for the intermediate field attack on some VOX parameters proposed in [START_REF] Guo | A practical minrank attack against vox[END_REF].

This shows that while our attack is not as general as the attack of Guo and Ding, it is significantly more efficient when it applies.

Patching the scheme

We identify the following countermeasure to our attack: use a QR parameter l that is prime.

Fig. 2 :

 2 Fig. 2: Gates counts for the intermediate field attack on QR-UOV + and practical results.

  Parameter set o/c, v/c, c Security (log 2 ops)

		GD-Ia	4, 7, 13	78.1
	251, 6	GD-Ib	5, 9, 11	99.8
		GD-Ic	6, 11, 9	134
		GD-IIIa	5, 9, 15	85
	1021, 7	GD-IIIb	6, 11, 13	101.5
		GD-IIIc	7, 13, 11	129.55
		GD-Va	6, 11, 17	90.6
	4093, 8	GD-Vb	7, 13, 14	113.7
		GD-Vc	8, 15, 13	130.6


Niéli Daffé

Dr Marcin Sikora

Mauro Rovezzi

Nadejda Bouldi

Dr Véronica Gavrilov

Dr Sophie Neveu

Dr Fadi Choueikani

Philippe Ohresser

Dr Vincent Dupuis

Dr Dario Taverna

Dr Alexandre Gloter

Dr Marie-Anne Arrio

Dr Philippe Sainctavit

Dr Amélie Juhin
email: amelie.juhin@impmc.upmc.fr

Nanoscale Distribution of Magnetic Anisotropies in Bimagnetic Soft Core-Hard Shell MnFe 2 O 4 @CoFe 2 O 4 Nanoparticles

Keywords: core-shell nanoparticles, magnetic anisotropy, spinel ferrites, XAS, RIXS, XMCD, RIXS-MCD

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

technique, emergent magnetic properties were revealed. The coercive field is not constant in a whole nanospinel. The very thin (0.5 nm) CoFe2O4 hard shell imposes a strong magnetic anisotropy to the otherwise very soft MnFe2O4 core: a large gradient in coercivity was measured inside the MnFe2O4 core with lower values close to the interface region, while the inner core presents a substantial coercive field (0.54 T) and a very high remnant magnetization (90% of the magnetization at saturation).

Introduction

Ferrite nanoparticles with a spinel structure (γ-Fe2O3, Fe3O4, MnFe2O4, CoFe2O4) have become very popular for their original magnetic properties that are not shown by bulk materials. The reduced size combined to enhanced magnetic properties has led to numerous applications from biomedicine to high density storage devices, [1], [2], [3] although further developments and miniaturization of magnetic storage devices based on nanoferrites are now limited by the "superparamagnetic limit". [4] To overcome this, chemists have imagined new synthesis pathways to tune further the magnetic properties of nanospinels by playing on their structuration, leading to the emergence of core@shell nanostructures which rapidly focused interest. [5] Since they allow the possibility to combine a core and a shell from two different magnetic species (such as for example a ferro/ferrimagnetic phase with an antiferromagnetic one, [6], [7], [8] or a soft magnetic with a hard magnetic one [9], [10], [11] ), bimagnetic core@shell nanospinels offer a fertile ground to overtake the limits of conventional magnetization parameters (anisotropy constant K, blocking temperature TB, saturation magnetization Ms and coercive field Hc [12], [13], [14] ) or to observe strong exchange coupling between both magnetic phases, such as exchange bias [15], [16], [17] or exchange-spring [18], [19] coupling effects. In parallel, magnetic nanoparticles have found numerous applications when dispersed in liquids to form stable magnetic colloids. [20] Such systems are called ferrofluids when they are composed of ferro-or ferrimagnetic single domain nanoparticles, with a typical diameter in the 5 nm-30 nm range. [START_REF] Odenbach | Colloidal Magnetic Fluids: Basics, Development and Application of Ferrofluids[END_REF] In biomedicine, ferrofluids of core@shell nanoparticles have recently been considered for advanced applications, such as nanoheater-based therapies for cancer treatment (magnetic hyperthermia), which use radiofrequency fields. [START_REF] Hurley | [END_REF], [23], [24] The efficiency of the process relies on the magnetic anisotropy of the nanoparticle, which governs the high frequency losses, hence the idea to modulate magnetic anisotropy by designing novel core@shell architectures and inducing emergent magnetic properties.

In this work, we study the magnetic anisotropies in well-crystallized 7.0 nm core@shell nanoparticles, which combine a soft core of MnFe2O4 covered with a thin (0.5 nm) hard shell of CoFe2O4. Using an original experimental approach combining electron and x-ray techniques with different probing depths, we investigate separately the cationic distribution and magnetic anisotropies in the core and those in the shell, as well as their mutual influence.

Among the techniques dedicated to study magnetic nanoparticles, X-ray Absorption Spectroscopy (XAS) and X-ray Magnetic Circular Dichroism (XMCD) have now become standard techniques. [25], [26], [27], [28] XMCD is the difference in the absorption of circularly polarized X-rays for opposite sample magnetization direction. It can be measured in ferroand ferrimagnetic materials and provides information on the magnetic properties of the absorbing atom. [29], [30] XMCD measurements are mainly performed using soft X-ray photons (~600 and 780 eV at the Mn and Co L2,3 absorption edges, respectively) delivered by synchrotron radiation facilities. Thanks to element specificity, XMCD allows disentangling the magnetic signature of the core -by measuring the Mn absorption edge, from that of the shell -by measuring the Co absorption edge. The existence of a possible magnetic coupling between both components can be determined. [31], [32] Additionally, the cationic distribution of the different ions amongst crystallographic sites can be solved thanks to the sensitivity of the XAS and XMCD spectral shape to the local site symmetry of the absorbing atom and to its valence, which yields crucial information to rationalize the magnetic properties of these materials. However, for core@shell nanoparticles, where the magnetic structure of the outermost layers is usually very different from that of the inner core, a quantitative analysis based on XAS and XMCD is often limited by the short penetration depth of soft X-rays. Indeed, using the total electron yield traditional detection mode, the probing depth is rather small, with 60% of the signal coming from the top 2 nm. In order to circumvent this issue, hard X-rays with a much larger penetration depth can be used to probe the full nanoparticle depth. In addition to soft X-ray XAS and XMCD measurements, we have investigated MnFe2O4@CoFe2O4 particles using the RIXS-MCD technique, which consists in coupling hard X-ray MCD and Resonant Inelastic X-ray Scattering (RIXS) spectroscopy at the K-edge of 3d ions. [33], [34] RIXS-MCD was recently used to investigate the buried interface in bi-magnetic core@shell γ-Fe2O3@Mn3O4 nanoparticles. [35] Such a combined approach is, to our knowledge, the first investigation of its kind.

Results and Discussion

Structural properties of dried nanoparticles

Two types of nanoparticles were synthesized as ferrofluids in heptane (see Experimental Section): MnFe2O4 and MnFe2O4@CoFe2O4 core@shell (also labeled Mn@Co hereafter). The latter were synthesized using the seeded-growth approach, i.e., Mn@Co particles are built up from the pre-made nanoparticle core of MnFe2O4 sample on which a shell of CoFe2O4 is grown. Atomic Absorption Spectrometry measurements provide an estimate of the metallic molar concentration in each sample (Table 1), showing that the obtained stoichiometry is close to the expected one for a spinel, i.e. 1 divalent ion for 2 trivalent ions. The TEM micrographs illustrated in Figure 1 show that all particles are spherical. The respective particle size histograms are well-fitted using normal distributions, leading to a mean diameter of 6.0 nm (σ = 0.25) for MnFe2O4 and 7.0 nm (σ = 0.31) for MnFe2O4@CoFe2O4, respectively (Table 1). These values are consistent with the size of coherent domains obtained from the XRD patterns (Figure S1, Supporting Information), which also confirm the spinel structure of both samples. From the difference in particle size, the thickness of the outer cobalt ferrite layer in the MnFe2O4@CoFe2O4 particles can be estimated to be ~0.5 nm.

Figure 1. TEM images and distribution in particle size for the MnFe2O4 (top), and Mn@Co dried powder samples (bottom). Size distributions are determined using a Gaussian fit of histograms (red line).

High Resolution Scanning Transmission Electron Microscopy (STEM) and Electron Energy

Loss Spectroscopy (EELS) analysis were performed to assess the elemental distribution within a Mn@Co nanoparticle, using a 0.1 nm step size and a reduced electron probe with 0.2 nm width. Results are shown in Figure 2. The High Angle Annular Dark Field (HAADF) signal acquired in parallel to the spectral information shows that the particle has a slightly faceted morphology (Figure 2a). The spatial distribution of Mn, Fe and Co ions obtained from the STEM-EELS analysis is mapped in Figures 2b-2e. The core@shell structuration of the nanoparticle is clearly visible, with Co ions exclusively located in the shell region and Mn ions mostly confined in the core region. This confirms the existence of the (Co, Fe)-rich outer shell grown on the (Mn, Fe)-rich core, but the local presence of residual Mn in the outer shell cannot be totally excluded. Indeed, when comparing the EELS spectrum for two selected areas that correspond to the inner core and the outermost shell region (labeled (A) and (B), respectively, Figure 2f), the presence of a residual signal at the energy loss of the Mn L2,3 edges cannot be discarded for area B, but the signal of Co is clearly more intense. Intermixing between the MnFe2O4 core and the CoFe2O4 shell is therefore very likely but seems limited. The temperature dependence of magnetization was measured using a Vibrating Sample Magnetometer (VSM) magnetometer. The value of the superparamagnetic blocking temperature (TB) is estimated from the maximum of the Zero Field Cooled (ZFC) curve plotted in Figure 3 together with the Field Cooled (FC) temperature dependence of magnetization. [36], [37] The Mn@Co nanoparticles show a higher blocking temperature (TB = 150K) than the MnFe2O4 nanoparticles (TB = 20K). The blocking temperature measured for MnFe2O4 nanoparticles is typical for such particle size and composition. [38] Since TB = B k KV 25 (where K is the magnetic anisotropy, V the volume and kB the Boltzmann constant), the increase of TB for the Mn@Co nanoparticles with respect to the MnFe2O4 seeds may be attributed to the slight volume increase due to the growth of the shell and most likely to the increase in magnetic anisotropy due to the formation of the harder CoFe2O4 outer layer. In order to disentangle both effects, we have investigated the depth-dependence of magnetic anisotropies inside a nanoparticle. Table 1. Structural and magnetic properties of the MnFe2O4 and Mn@Co nanoparticles.

MnFe2O4

Mn@Co Particle mean diameter from TEM (nm) 6.0

Electronic structure and cationic repartition.

Two types of measurements were performed at low temperature using x-ray magnetic spectroscopies. In a first step, spectral signatures related to the structural, electronic and magnetic properties were recorded. In a second step, element-selective magnetization versus field curves were measured in order to access the values of coercive field and remnant magnetization. Both types of measurements were conducted with surface sensitive XAS/XMCD (soft x-rays) and then with bulk sensitive RIXS/RIXS-MCD, therefore providing information with different probing depths. RIXS is a photon-in, photon-out element selective X-ray spectroscopy that provides for the absorbing atom a unique bi-dimensional mapping of its spectral signature. [39] The latter is intimately related to its electronic structure, i.e., mainly its valence (number of 3d electrons) and site symmetry (tetrahedral or octahedral).

In Figure 4 are shown the RIXS and RIXS-MCD planes measured on the Mn@Co particles at low temperature (i.e., below the particle blocking temperature and the solvent freezing point) using an external magnetic field of 1.2T, at the Mn K-edge (panels (a) and (b)) and at the Co K-edge (panels (c) and (d)). This allows probing selectively the speciation of Mn ions (that are located in the MnFe2O4 core) and that of Co (that are expected in the CoFe2O4 shell).

At the Mn edge, the large intensity of the RIXS-MCD signal (~20% peak-to-peak) and its characteristic shape (one positive peak at lower incident energy and lower energy transfer, accompanied by one negative peak) reveal the dominant contribution from tetrahedral Mn 2+ ions, whose 2D spectral signature is also similar to that of tetrahedral Fe 3+ ions (which are isoelectronic to Mn 2+ ions) in Fe3O4 and -Fe2O3. [34] Using the same polarization and field conventions as in the latter experiments, the sign of the RIXS-MCD signal indicates that Mn 2+ ions have their magnetic moments opposite to the field direction, which further confirms their location in the tetrahedral sites of the MnFe2O4 direct spinel structure. At the Co edge, the small peak-to-peak intensity of the RIXS-MCD signal (~4%) and its shape, which is dominated by two negative features, indicate the presence of Co 2+ ions in the octahedral sites of the CoFe2O4 direct spinel structure. The soft XAS/XMCD measurements provide similar outcomes regarding the speciation of Mn and Co ions inside a Mn@Co particle, although the much smaller probing depth (2 nm vs a few µm) informs only on the upper part of the particle (i.e., the Co shell and the Mn ions at the core-shell interface): the XMCD spectra measured at the different metal L2,3 edges (Figure S2, Supporting Information) show that 100% of Co ions are divalent and in octahedral sites, [40] while Mn ions are found exclusively divalent and mainly in tetrahedral sites. [41] 2.3. On the intermixing and the magnetic coupling between the core and the shell.

The RIXS and RIXS-MCD signatures measured in the Mn@Co ferrofluid were compared with those measured in the parent MnFe2O4 particles at the Mn edge. In an attempt to compare the spectral signature measured at the Co edge with that of a known reference sample, 6 nm CoFe2O4 nanoparticles were prepared with a similar synthesis route and measured (See Experimental Section and Supplementary Information, S9, Supporting Information). The spectral signature of Mn ions in the Mn@Co particles, which arises from the MnFe2O4 core, is very similar to that of the Mn ions in the MnFe2O4 particles. Likewise, Co ions in the CoFe2O4 particles and those of the CoFe2O4 shell in the Mn@Co sample show MCD spectra that are identical within noise level (note that the signal from the 0.5 nm thin shell is very weak). This again confirms that in Mn@Co particles, Mn ions are mainly located in tetrahedral sites and Co ions in octahedral sites.

From the MCD spectra one can determine the magnetic coupling between the core and the shell. Since the magnetic moments of tetrahedral Mn 2+ ions are opposite to the external magnetic field direction and those of octahedral Co 2+ ions are along the field direction, we can conclude that the coupling between the core and the shell is ferromagnetic in nature. In other words, magnetic moments in the tetrahedral sites of the core and in those of the shell are all antiparallel to the external magnetic field, while those in the octahedral sites of the core and the shell are all parallel to the external field. (Note that due to experimental issues it was not possible to measure at the very same temperature for both edges).

Element specific magnetic properties of the shell and the core.

In Figure 6 are shown the magnetization vs magnetic field curves acquired at 26 K with a PPMS instrument for MnFe2O4 and Mn@Co particles. At this temperature, a very low coercive field of ~1mT is measured for MnFe2O4 nanoparticles while it is two orders of magnitude larger for Mn@Co nanoparticles (~0.5 T), which is in line with the soft nature of MnFe2O4. However, disentangling quantitatively the respective contributions of both magnetic components to the resultant average magnetization curve is tedious; hence our alternative approach to measure MCD detected magnetization curves that provide element selectivity.

Figure S10, Supporting Information

). This difference is due to the fact that the latter have a very different magnetic anisotropy from that of a CoFe2O4 shell with 0.5 nm thickness, hence the necessity to measure the specific magnetic properties of the shell in the Mn@Co particles rather than using those of reference samples.

The normalized remnant magnetization (remnant magnetization divided by magnetization at saturation) is only ~50%, which could be the result of strong magnetic spin canting for the Co ions in the Mn@Co nanoparticles. Indeed, the canting is expected to be rather large because of the high proportion of Co ions situated at the particle surface that have less magnetic neighboring atoms than inside the bulk. [42] In a magnetic spinel structure, the magnetic structure is mainly governed by the antiferromagnetic coupling between tetrahedral and octahedral sites. Nevertheless, in a series of spinels bearing Fe 2+ and Fe 3+ in octahedral sites, it was shown that the magnetic interaction between octahedral sites is antiferromagnetic between Fe 3+ and Fe 3+ (J=-1.4K) and between Fe 2+ and Fe 2+ (J=-3.3K) because of super exchange, but it is ferromagnetic between Fe 2+ and Fe 3+ due to double exchange (J=+1.6K). [43] Both super exchange interactions plus the double exchange yield an average antiferromagnetic coupling between octahedral sites. At the surface of the Mn@Co particles, if the dominant antiferromagnetic coupling between tetrahedral and octahedral sites is reduced due to the finite size of the particle, the average antiferromagnetic coupling between octahedral sites now competes efficiently with the antiferromagnetic coupling between octahedral and tetrahedral sites. This results in the breaking of the collinear spin configuration. [44], [45] Figure 7. Element selective magnetization curves measured in Mn@Co at low temperature by XMCD using soft X-ray MCD (green lines) and hard X-ray RIXS-MCD (red lines), at the Co edge (left panel) and at the Mn edge (right panel).

At T=26K, the coercivity measured with hard X-rays ( H c BS , where BS means Bulk Sensitive) at the Mn edge in the Mn@Co particles is 0.54 Tesla, while it is almost zero (0.013 Tesla) in the MnFe2O4 sample (Figure S11, Supporting Information). The normalized remnant magnetization is ~90 +/-5%, which is similar to the expected value of 83-87% for randomly spherical ferrite nanoparticles with cubic anisotropy. [46] Such a high value associated to the square (non-slanted) shape of the magnetization curve reveals the lack of magnetic spin canting for the Mn ions. This is consistent with most of the Mn ions being in the "bulk" of the particle rather than partially diluted in the shell. In addition, since Mn ions are on the Td sites, they tend to be much less sensitive to spin canting that is preferentially affecting the Oh sites. [45], [36] Nevertheless, the coercivity induced in the core by the shell (H c BS (Mn) = 0.54 Tesla at T= 26K) is much larger than the coercivity of the shell itself (H c BS (Co) = 0.18 Tesla at T = 39K, a value that is likely similar at T = 26K because of the low T-dependence of Hc measured at these temperatures by XMCD at the L2,3 edges, see Figures S12 andS13, Supporting Information). This seems at first surprising, because one would expect the coercive field in the shell to induce a coercivity of the same magnitude in the single domain core. Instead, when the magnetic field is set to the value of H c BS (Co), leading to zero magnetization on the shell, a significant remnant magnetization is still observed on the core. The difference between the coercivity measured in the shell and the one measured in the core is difficult to explain by the existence of several populations of magnetic Co ions. Indeed, attempts to reproduce the shape of the hysteresis loop measured at the Co edge in the Mn@Co sample using one population of Co ions that would follow the magnetization curve of the Mn ions, plus one population of paramagnetic Co ions (as given by a Brillouin curve for S=3/2), did not succeed. Instead, the difference between H c BS (Co) and H c BS (Mn) may arise from a gradient in coercivity inside the MnFe2O4 core. Indeed, using the surface sensitive XMCD probe, one finds that the coercivity ( H c SS , SS means Surface Sensitive) of the probed Mn ions Comparison with the coercive field measured for the entire particle at the Mn edge (H c BS (Mn) =0.54 Tesla) implies the existence of a strong gradient in coercivity inside the MnFe2O4 core: lower values of coercive field arise from the interface region, while higher values originate from the center of the core.

Conclusion

Emergent magnetic properties were measured in core@shell MnFe2O4@CoFe2O4 7.0 nm particles using a combination of element selective magnetic spectroscopies with different probing depths. A strong gradient in coercivity was measured inside the MnFe2O4 core with lower values close to the interface region (0.15 T at 26K), while the inner core presents a rather large coercive field (0.54 T at 26K). The coercivity of the inner core is larger than both the one of the thin 0.5 nm CoFe2O4 shell (~0.16 T) and the one of bare MnFe2O4 particles (0.01 T). In addition, our microscopic and spectroscopic findings show that only a limited interdiffusion is occurring, with no sign of cationic rearrangement that would explain such a spectacular change in anisotropy. This is clear evidence that the magnetic properties of the investigated core@shell particles arise from an emergent magnetic coupling between the core and the shell, which is due to the presence of the interface and is found to be ferromagnetic in nature. We expect that micromagnetic atomistic simulations could allow understanding the mechanism of the magnetic interaction between the core and the shell.

Experimental Section

Synthesis. Nanoparticles were prepared following the thermal decomposition process of metallic acetylacetonate precursors in the presence of oleic acid, 1,2-hexadecanediol, oleylamine and benzyl ether. Nanoparticles with controlled morphology and excellent crystallinity were obtained following the process of Sun et al. [47] The synthesis of MnFe2O4, CoFe2O4 and MnFe2O4@CoFe2O4 nanoparticles has been described elsewhere. [48] Transmission Electron Microscopy. TEM micrographs were obtained using a JEOL 100 CX2.

Size histograms were obtained from the analysis of the TEM micrographs over more than 10,000 nanoparticles. These histograms were fitted by a normal law with a least-square refinement that provides particles mean diameter d0 and their respective polydispersity σ.

X-Ray Diffraction. XRD θ/2θ patterns were recorded with a PANALYTICAL X'Pert Pro MPD diffractometer using Fe filtered the Co Kλ Å radiation from a mobile anode at 40 kV, 40 mA. The measurements were carried out in a range of 30-80° 2in steps of 0.02° and collection time of 7200s. Coherent domain sizes are calculated from the Scherrer equation. [49] Atomic Absorption Spectrometry. Measurements were performed using a Perkin Elmer Analyst 100 with an air-acetylene flame at a mean temperature of 2300°C. Experiments were repeated at least three times on each sample. Molar concentrations of cobalt, manganese and iron are determined after degradation of the nanoparticles in concentrated hydrochloric acid.

From the metallic concentrations species, the volume fraction of the nanoparticles was calculated and molar ratios XM are determined for each sample (Table 1):

XM = [M] / ([M] + [Fe]
) where [M] is [Mn] for MnFe2O4 and [Mn] + [Co] for Mn@Co.

EELS. High resolution Scanning Transmission Electron Microscopy and Electron Energy

Loss Spectroscopy (EELS) analysis were performed using a Cs aberration-corrected STEM, the NION UltraSTEM200 operated at 100 kV and coupled with a high-sensitivity EEL spectrometer. The transmitted electron beam at each probed position has been analyzed with the spectrometer, to obtain a spectrum-image. The total dataset is formed by 109x96 pixels (spectra) with a beam spot size of 0.2 nm, and the acquisition time of each spectrum is 10 ms. Element-specific magnetization curves were measured using the EMPHU-65 undulator which allows the fast switching (1 Hz) of the X-ray helicity. [50] RIXS and RIXS-MCD spectroscopies. Experiments were carried out at ID26 beamline of the European Synchrotron Radiation Facility (Grenoble, France). Measurements were performed at the Co and Mn K-edges at low temperature on the frozen phase of ferrofluids using a dedicated liquid cell. The uncertainty on temperature is estimated to be +/-5K. The incident energy was selected using a pair of Si(311) crystals. The intensity of Mn and Co K emission lines (of the inelastically scattered beam) was analyzed using a set of four spherically bent Ge(111) and Si (531) crystals, respectively, arranged with an Avalanche Photo Diode in the Rowland Geometry with a scattering angle of 90°. The overall resolution was measured at 0.7 eV and 0.8 eV for Mn and Co respectively. For both Co and Mn edges, 1s2p RIXS planes were recorded as a set of Constant Energy Transfer scans over the energy of the Kα1 line and of the K pre-edge. Additionally, absorption spectra were measured using High Energy Resolution Fluorescence Detection (HERFD) by setting the emission energy to the maximum of the Kα line (6.9313 keV for Co, 5.8998 keV for Mn), namely HERFD-XAS. HERFD-MCD and RIXS-MCD experiments were carried out with the same setup as for HERFD-XAS and RIXS measurements, the differences being that (i) the incident beam is circularly polarized (instead of linearly polarized in the case of HERFD/RIXS), (ii) samples are kept in magnetic saturation using an electromagnet allowing to reach a magnetic field of 1.5 T, for which a detailed calibration curve was measured. The circular polarization was obtained using a 500 µm thick diamond (111) quarter wave plate set downstream the Si(311) monochromator, with a circular polarization degree estimated to be 75 %. RIXS-MCD planes were recorded by reversing the photon helicity at each incidence energy. RIXS-MCD and HERFD-MCD spectra were recorded in the region of the Co and Mn K pre-edge, after freezing the samples with no external magnetic field Spectra were normalized such that the maximum of the polarization-averaged absorption spectrum is equal to 1 in the pre-edge region. Since (i) selfabsorption corrections were found to be weak in the pre-edge range, (ii) all samples were measured in the same conditions and (iii) all MCD spectra are normalized to the pre-edge maximum, we assume that self-absorption effects do not impact our analysis.

Figure 2 .

 2 Figure 2. STEM-EELS analysis on an individual core@shell nanoparticle: (a) HAADF image; (b) superposition of the elemental maps of Mn and Co; (c) elemental map of Mn ; (d) elemental map of Fe; (e) elemental map of Co; (f) EELS spectra corresponding of selected areas in the core (spectrum A) and in the outer shell regions (spectrum B).

Figure 3 .

 3 Figure 3. Field Cooled (with a probing field of 70 Oe) and Zero Field Cooled curves measured on MnFe2O4 (blue squares) and Mn@Co nanoparticles (red squares). 2.2. Electronic structure and cationic repartition.

Figure 4 .

 4 Figure 4. Low temperature RIXS and RIXS-MCD planes measured on Mn@Co particles at the Mn edge (panels (a) and (b)) and at the Co edge (panels (c) and (d)). For both edges only the part of the planes corresponding to the K1 emission region is shown.

Figures

  The comparison of spectral signatures can be made on the 2D planes (Figure4, FiguresS6 and S7, Supporting Information) which contain the full information, or more conveniently, on 1D spectra extracted from the 2D planes, either as Constant Emission Energy (CEE) scans (a diagonal cut in the plane, Figure5) or as Constant Incident Energy (CIE) scans (a vertical cut in the plane, Figures S8 and

Figure 5 .

 5 Figure 5. Low Temperature absorption and MCD spectra measured at Constant Emission Energy at the Co edge (panels (a) and (b), T=39K, CEE =6.9313 keV) and at the Mn edge (panels (c) and (d), T=26K, CEE=5.8998 keV). (Note that due to experimental issues it was not possible to measure at the very same temperature for both edges).

Figure 6 .

 6 Figure 6. Magnetization versus field measurements with PPMS at T=26K for MnFe2O4 and Mn@Co particles.

(

  Figure S13, Supporting Information, which is also close to H c BS (Co)=0.18 Tesla at T=39K).

  Bulk magnetic measurements. They were performed using a Vibrating Sample Magnetomer Quantum Design PPMS. Blocking Temperatures were estimated from the Zero-Field Cooled and Field Cooled (ZFC/FC) temperature dependence of magnetization measured under a 70 Oe field on the particles dispersed in paraffin. Paraffin is used here to avoid the critical fusion point of heptane. Magnetization curve vs. Field measurements were performed at 26K on the ferrofluids of the MnFe2O4, CoFe2O4 and MnFe2O4@CoFe2O4 particles.Soft XAS and XMCD measurements. XAS and XMCD signals were recorded at the Mn and Co L2,3 edges on the DEIMOS beamline at the French synchrotron, SOLEIL. The resolving power E / E is better than 5000. Spectra were measured in Total Electron Yield at 50K and in High Vacuum conditions (10 -8 mbar) on a dropcast of Mn@Co particles. XMCD signals were recorded by flipping both the circular polarization (either left or right helicity) and the external magnetic field (either +1.5 Tesla or -1.5 Tesla). Circularly polarized X-rays are provided by an Apple-II HU52 helical undulator for both XAS and XMCD measurements.

Figure S6 .

 S6 Figure S6. RIXS (panel (a)) and RIXS-MCD planes (panel (b)) measured at T=26K on the frozen phase of MnFe2O4 ferrofluid reference.

Figure S7 .

 S7 Figure S7. RIXS (panel (a)) and RIXS-MCD planes (panel (b)) measured at T=39K on the frozen phase of CoFe2O4 ferrofluid reference.

Figure S8 .

 S8 Figure S8. Low Temperature XES (panel (a)) and XES-MCD spectra (panel (b)) measured at Constant Incident Energy on the frozen phases of CoFe2O4 ferrofluid and Mn@Co ferrofluid.

Figure S9 .

 S9 Figure S9. Low Temperature XES (panel (a)) and XES-MCD spectra (panel (b)) measured at Constant Incident Energy on the frozen phase of MnFe2O4 ferrofluid and Mn@Co ferrofluid.

Figure S10 .

 S10 Figure S10.Element selective magnetization curves measured at 39K by RIXS-MCD using bulk sensitive hard X-rays at the Co edge for the 6 nm CoFe2O4 reference sample (green line) and the Mn@Co nanoparticles (red line).

Figure S11 .

 S11 Figure S11.Element selective magnetization curves measured at 26K by RIXS-MCD using bulk sensitive hard X-rays at the Mn edge for the MnFe2O4 reference sample (blue line) and the Mn@Co nanoparticles (red line).

Figure S12 .

 S12 Figure S12. Element selective magnetization curves measured by XMCD using surface sensitive soft X-rays at the Mn edge at 26 K (blue line) and at 39 K (orange line).

Figure S13 .

 S13 Figure S13. Element selective magnetization curves measured by XMCD using soft X-rays at the Co edge at 26 K (blue line) and at 39 K (orange line).

Acknowledgements

The authors thank Hugo Vitoux from the Sample Environment Support Service of the ESRF for his technical support and Jacques Borrel from the mechanical engineering pool of ESRF for his support in the assembly of the experimental setup. The beamlines ID24/BM23 of ESRF are also acknowledged for borrowing their electromagnet. The authors are very grateful to Dr. Edwige Otero from SOLEIL who has implemented on the DEIMOS beamline the 2 Tesla electromagnet (MK2T) end station used for the soft XMCD measurements. Benoit Baptiste is thanked for fruitful discussions on XRD. This work was supported by French state funds management by the ANR within the Investissements d'Avenir programm under reference ANR-11-IDEX-004-02, and more specifically within the framework of the Cluster of Excellence MATISSE. M.S. acknowledges support from the National Science Centre of Poland (2014/14/E/ST3/00026). A.J. and N.D. initiated and conceived the project. A.J. supervised the project. MV.G. and S.N. implemented the synthetic and analytical experiments. V.G. performed the TEM imaging. N.D performed the powder X-ray diffraction. N.D., V.G. and V.D. performed the magnetomety measurement and N.D. analyzed the magnetometry measurements. A.J., M.S., M.R., Ph.S., F.C., N.B., and N.D. performed the RIXS-MCD measurements. M.S. and A.J. analyzed the RIXS-MCD measurements, M.S. implemented the code for the RIXS-MCD map analysis. N.D, F.C, M.A.A and Ph. O. performed and analyzed the soft-XAS and soft-XMCD measurements. D. T. and A. G. performed the EELS analysis. A.J. and N.D. wrote the manuscripts with the help of all authors. All authors have given approval to the final version of the manuscript.

Supporting Information

XRD patterns, soft XAS/XMCD data, RIXS/RIXS-MCD data of reference samples, magnetization curves measured by VSM and soft XMCD.

Supporting Information