Mariusz P Mitoraj

Rafal Kurczab

Marek Boczar

Ph.D Artur Michalak
email: michalak@chemia.uj.edu.pl

Rafal Kurczab

Rafał Kurczab

Keywords: oxalic acid dimer and trimer, resonance assisted hydrogen bonds, extended transition state, natural orbitals for chemical valence, ETS-NOCV

come

Theoretical description of hydrogen bonding in oxalic acid dimer and trimer based on the combined extended-transition-state energy decomposition analysis and natural orbitals for chemical valence (ETS-NOCV)

a relatively small interaction energy into four components, some of which are larger than the overall interaction energy by factors of 2 or 3. Such discrepancies in magnitudes cause me to question the meaningfulness of the procedure. Perhaps the authors could address this point. My recommendation is for the paper to be accepted after the authors address these comments and those specific ones below."

A detailed knowledge on the hydrogen bonding phenomenon can be useful not only in understanding the geometrical structures of bio-molecules but it can also provide useful information on the reactivity of a given hydrogen-bonded system. In addition, such a deep knowledge can provide useful information on the rational design of a new compounds exhibiting desired bond characteristics (Ref. [START_REF] Jeffrey | Hydrogen Bonding in Biological Structures[END_REF]2). Accordingly, many scientist have made a great effort over the years to determine a possible factors contributing to the total strength of hydrogen bonding (Ref. 19,20,21,24,56,57).

The main achievement of our work is to extract, the stabilizing (orbital interaction, Eorb, and electrostatic, Eelstat) terms and destabilizing (EPauli, Edist) factors contributing to the total strength (Eint) of hydrogen bonding in oxalic acid systems. In addition, stabilization originating from electronic factor (Eorb) is further decomposed into σ -and  -contributions based on ETS-NOCV scheme.

As it was pointed out by the Reviewer, Eelstat and Eorb are 2, up to 3 times larger in magnitude (in absolute values) than Eint (total bond energy). In our opinion, there is nothing unusual or wrong with the fact that the orbital interaction term or the electrostatic contribution is larger in magnitude than Eint. "Such discrepancies in magnitudes" are not unusual. It is common in quantum chemistry, that some of the chemically meaningful, relatively small values, are obtained by the addition/sustraction of other, relatively huge values. In particular in the case of ETS analysis, it is obviously common as well (eq. Ref. 35).

1) The number of acronyms introduced is unacceptable. There are 8, just on the first page, and some of them have as many as five letters.

We agree with the Reviewer that some of the acronyms were in fact not necessary.. Therefore, we have removed those that were not used furthere in the text.

2) On page 7, what is meant by a rho-* orbital?

We have clarified the meaning of orbital on page 8.

3) The term transition state has a well-established meaning in kinetics. To use it as the authors do here in another sense is misleading.

We agree that "Transition State term" is well established in kinetics. Therefore, we have added a clarifying statement on page 5 (Computational Methods) to avoid any confusion. In addition, the exact meaning of ETS abbreviation has been explained in a detailed way on Page 5.

"ETS (Extended Transition State)", as the name for the Ziegler-Rauk energy decomposition scheme was introduced in 1979th by Ziegler (Ref. 26,27). Afterwards, we have developed NOCV methodology independently, followed by ETS-NOCV approach. (Ref. [28][29][30][31][32][33][34][35][36]. As the ETS term has already been known and used in the literature for years, it is natural and appropriate to use of this original term.

4)

On page 2, the word "enable" should be "able". We have corrected the word.

Response to the comments of Reviewer 2 P. 7 Should σ(H??? HO) be σ(O??? HO)?

It should be O---HO interaction. We have corrected the error on page 7. P.14 An energy unit is missing in the table 1.

The units have been included in the caption of Table The errors in the References section have been corrected.

Introduction

Hydrogen bonding (HB) plays an important role in biochemistry [START_REF] Jeffrey | Hydrogen Bonding in Biological Structures[END_REF][2]. Accordingly, many attempts were made to describe HB-phenomenon in both experimental and theoretical laboratories [START_REF] Lippincott | [END_REF][4][5][6][7][8][9][10][11]. For the purpose of understanding the nature of hydrogen bonding, theoretical approaches appeared to be of significant importance [12][13][14][15][16][17][18][19]. Especially, Electrostatic-Covalent HB Model, derived by Gilli [20][21] from the systematic analysis of structural and spectroscopic data, allowed to make a comprehensive classification of H-bonds in a different chemical classes: e.g. charge assisted hydrogen bonds, low barrier hydrogen bonds, dihydrogen bonds, and resonance-assisted hydrogen bonds (RAHBs). It clearly shows that different factors can influence the nature of hydrogen connections.

For the purpose of HB description, a various energy decomposition analysis (EDA) were also used [22][23][24][25]45]. In such decomposition schemes, the total interaction energy between interacting sub-systems is divided up into a few chemically meaningful contributions (electrostatic, exchange, charge-transfer, relaxation, etc.). Despite of great usefulness of these methods, as they allow to extract stabilizing and destabilizing factors, none of them provides a direct, qualitative and quantitative, separated information about σ-and π-contributions. Such specific knowledge on the hydrogen bonding phenomenon can be useful in understanding the structures of bio-molecules and further in a description of reactivity.

We have recently developed the ETS-NOCV scheme by combining the extended transition state (ETS) [26][27] energy decomposition approach with the natural orbitals for chemical valence (NOCV) density analysis method [28][29][30][31][32][33][34]. It was shown that ETS-NOCV is able to extract and directly quantify the crucial components (σ,π,, etc.) that participate in various types of chemical bonds including predominantly donor-acceptor and covalent interactions [35][36]. Furthermore, we have shown that ETS-NOCV can also be useful in a description of weak bonds [35][36], including intra-and inter-molecular hydrogen bonding. For example, we have found, based on the analysis of intermolecular bonding in Adenine-Thymine base pair, that ETS-NOCV provides separated, qualitative and quantitative information not only about both σ(O---HN) and σ(N---HN) interactions, but also about the resonance assisted hydrogen bonding contribution () [35].

The main goal of this article is to extend our study on weak intermolecular interaction by characterization of hydrogen bonding in the dimeric and trimeric systems consisting of both σ(O---HO) and π-interactions between the monomers. We will consider in our analysis the oxalic acid dimers exhibiting two different conformations  and  (1 and 2), Fig. 1. In addition, we will investigate oxalic acid trimers, 3 and 4, based on two different bondpartitioning, as indicated in Fig. 1. The purpose of doing so is to obtain a compact description of each O---HO interaction in trimer of oxalic acid. To the best of our knowledge this is the first study reporting on the character of bonding in the oxalic acid systems based on charge and energy decomposition analysis. All structures discussed here were fully optimized after initial guesses taken from the Cambridge Structural Database [37][38]. For the sake of clarity of presentation we will use throughout our study the --dimer‖ and --dimer‖ terms to describe the oxalic acid dimers that exhibit two different conformations  and , respectively. <Figure1>

Computational details

All the calculations, within Density Functional Theory approach, were based on the Amsterdam Density Functional (ADF2009) program in which ETS-NOCV scheme was implemented [39][40][41][START_REF] Geurra | Methods and techniques in computational chemistry METACC-9[END_REF]. The Becke-Perdew exchange-correlation functional [START_REF] Becke | [END_REF][44] was applied (BP86). A standard triple-zeta STO basis containing two sets of polarization functions was adopted for all of the elements (TZ2P). Auxiliary s, p, d, f, and g STO functions, centered on all nuclei, were used to fit the electron density. It was proven previously for similar systems [35,45] that optimized structures and interaction energies based on BP86/TZ2P calculations are in line with the corresponding estimates obtained from the experiment and from high-level ab initio methods. The contours of deformation densities were plotted using ADF-GUI [46].

In our analysis each of the system is divided up into two individual fragments A and B, as shown schematically by a thick line in Fig. 1. Then we used the ETS-NOCV method to study the interaction between A and B, as they are brought together to form A•••B. Thus, our analysis is based on the bonding between the two close shell molecular fragments.

Computational methods

As we mentioned in the previous paragraph, our analysis is based on the ETS-NOCV approach which is a combination of the extended transition state (ETS) [26][27] method with the natural orbitals for chemical valence (NOCV) [28][29][30][31][32][33][34] scheme.

We shall start by presenting the basic concepts of the ETS method. In this scheme, the total bonding energy between the interacting fragments (E total ) is divided into four chemically meaningful components (Eq. 1):

E int = E dist + E elstat + E Pauli + E orb (1)
The first component, referred to as the distortion term E dist , represents the amount of energy required to promote the separated fragments from their equilibrium geometry to the structure they will take up in the combined molecule. The second term, E elstat , corresponds to the classical electrostatic interaction between the promoted fragments as they are brought to their positions in the final complex. The third term, E Pauli , accounts for the repulsive Pauli interaction between occupied orbitals on the two fragments in the combined complex. Finally, the last term, E orb , represents the stabilizing interactions between the occupied molecular orbitals on one fragment with the unoccupied molecular orbitals of the other fragment as well as the mixing of occupied and virtual orbitals within the same fragment (intra-fragment polarization) after the two fragments have been united. We can write the change in density that gives rise to E orb as

(1) (1) (1) P          (2)
where the sum is over all the occupied and virtual molecular orbitals on the two fragments. It now follows from the ETS scheme that the E orb term is given by [26][27] T orb

E P F        (3)
where TS F  is a Kohn-Sham Fock matrix element that is defined in terms of a (transition state) potential that is mid between that of the combined fragments and the final molecule. It should be emphasized that the meaning of "transition state" in the ETS approach is completely different from that commonly used in the context of the transition state theory.

Turning next to the NOCV approach, we note that historically the Natural Orbitals for Chemical Valence (NOCV) [28][29][30][31][32][33][34] have been derived from the Nalewajski-Mrozek valence theory.[47-53] However, from a mathematical point of view the NOCV's, i  , are simply defined as the eigenvectors, ,

M ii C      (1) (1)
that diagonalize the deformation density matrix  introduced in Eq. 2. Thus, ; =1,

i v i M     (5)
where M denotes the total number of molecular orbitals on the fragments and   is a column vector containing the coefficients that defines the NOCV i  of Eq. 4. It follows further [28- 34] that the deformation density  of Eq. 2 can be expressed in the NOCV representation as a sum of pairs of complementary eigenfunctions ( -k , k ) corresponding to the eigenvaluesv k and +v k with the same absolute value but opposite signs:

/2 /2 22 11 ( ) [ ( ) ( )] ( ) MM k k k k kk r v r r r              (6)
Expression ( 6) is the most important for the interpretation of NOCV, as it defines the chargeflow channels decomposing the overall deformation density. Therefore, in the present study we will not discuss the orbitals themselves, but only the respective deformation density contributions,  k . Examples of orbitals and their interpretation can be found elsewhere [28][29][30].

In the combined ETS-NOCV scheme [35][36] the orbital interaction term (E orb ) is expressed in terms of NOCV's as

/2 ,, 1 [] M TS TS orb k k k k k k E v F F        (7)
where

TS k , k F   and TS k , k
F are diagonal Kohn-Sham matrix elements defined over NOCV's with respect to the transition state (TS). The advantage of the expression in Eq. 7 for E orb over that of Eq. 3 is that only a few complementary NOCV pairs normally contribute significantly to E orb . We see above (Eqs. 6, 7) that we for each complementary NOCV pair, representing one of the charge deformations  k , not only can visualize  k but also provide the energy

contributions orb k E 
to the bond energy from  k . [28][29][30][31][32][33][34] The total bonding enthalpies (D e = -E int ) reported here do not include zero point energy additions, finite temperature contributions or basis set superposition error corrections. Our interest here has been to study the electronic nature of hydrogen bond formation through trends in the (electronic) bond enthalpy as revealed by our recently proposed ETS-NOCV procedure [35][36].

Results and discussion

Let us start our discussion from a qualitative description of hydrogen bonding in -dimer of oxalic acid (2), which is more stable (by 9.9 kcal mol -1 ) compared to dimer exhibiting conformation (1), Table 1. It is clear from

<Figure 3>

The ETS results gathered in Table 1 lead to the conclusion that -conformation of oxalic acid dimer (1) is energetically less stable than the corresponding -structure (|E int | lower by 9.9 kcal mol -1 ), as a result of changes in all of the bonding contributions defined in Eq. 1. The main reason of a relatively small stabilization from E orb is that solely one O---HO bond is formed between the oxalic acid monomers. The qualitative picture of (O---HO) formation (see Fig. 3) is analogous to 2, with practically no participation (< 1 kcal mol -1 ) from the network. The lack of considerable RAHB contribution is related to the broken planarity in 1.

The lower destabilization from Pauli term, as well as lower stabilization from electrostatic contribution for 1 originate from the fact that the monomers are further apart (by 0.2 Å), compared to -dimer (2). We should remark at this point that the experimental lattice energy estimations for and -conformations of oxalic acid in the crystal show that these two forms are roughly similar in stability [54]. This is not consistent with our result. Apparently, the dimeric and trimeric structures are not appropriate as models of real, infinite, periodic crystals. However, it was not our intention. As it was stated in the Introduction section, the main goal of our investigation was to verify the applicability of our new ETS-NOCV scheme in a description of bonding in hydrogen bonded molecules that exhibit both (O---HO) and  (RAHB) interactions. It would be worthy to conduct in the future an similar energy decomposition analysis (ETS), as implemented in the BAND program [55], which is suitable for periodic calculations. Such an analysis could shed a light on the origin of stability for both oxalic acid conformations in the crystal [54].

It is obvious from three deformation density contributions and their corresponding energies, depicted in Fig. 4, that there is practically no difference in the picture of O-HO bonding, neither qualitative nor quantitative when comparing to dimer 2 (Fig. 2). Furthermore, the data collected in Table 1 lead to the conclusion that not only electronic factor E orb , but as well the remaining contributions, i.e. Pauli repulsion, electrostatic and distorsion terms, are of the same magnitude, as in the case of 2. Accordingly, the total interaction energy, E int , in trimer 3 is exactly the same as for dimer 2. The same qualitative picture of bonding formation in 2 and 3 lead to the conclusion that NOCV's exhibit a feature of localized orbitals, i.e. they reveal the same characteristics of a given bond when going from one molecule to another.

<Figure 4>

Up to this point we have analyzed the two O---HO interactions in trimer 3, i.e. the trimer was divided up into two subsystems (A and B), as indicated in Fig. 1. For the purpose of simultaneous analysis of all four O---HO interactions that are present in -oxalic acid trimer, we have considered another bonding situation 4. In such a bonding pattern we have used the middle -oxalic acid as one fragment and the remaining two monomers as the second fragment. In such a way the formation of all four O---HO bonds connected each of the monomer is described, Fig. 1.

It is clear from the deformation densities and the corresponding energies, presented in Fig. 5, that the total strength of four (O---HO) contributions is by -56.9 kcal mol -1 , whereas the resonance assisted hydrogen bonding component (RAHB) is by -6.5 kcal mol -1 . The remaining intra-fragment polarizations amounts to -8.2 kcal mol -1 . Further, an inclusion of other stabilizing (electrostatic) and destabilizing (Pauli repulsion and distorsion term) factors

gives raise to the total interaction energy by -31.6 kcal mol -1 , which is, in line with the intuition, nearly doubled in magnitude, compared to E int for system 3, Table 1. It confirms an equivalence of each O---HO interaction in trimer of oxalic acid.

<Figure5>

To the best of our knowledge there is no available data in the literature on the character of bonding in dimer or trimer of oxalic acid. However, a few theoretical studies were reported recently for familiar molecules, i.e. for pyrrole-2-carboxylic acid and formic acid dimers, containing as well the two equivalent O---HO interactions, supported further by RAHB contribution [56][57]. In these studies the electronic enthalpies (per one O---HO hydrogen bond) are -5.8 kcal mol -1 and -8.5 kcal mol -1 for formic and pyrrole-2-carboxylic dimers, respectively. We obtained the value of -7.9 kcal mol -1 for -dimer. reported for Adenine-Thymine and Guanine-Cytosine base pairs that the dominant contribution comes from the -component, whereas -contribution adds very little stabilization [45]. This is similar trend to our findings for oxalic acid molecules.

Concluding remarks

In the present study we have characterized hydrogen bonding in the dimer and the trimer of -

Figure captions

Fig. 2 1 .

 21 that three deformation density contributions, complementary NOCV's (Eq. 6), describe the formation of two equivalent O---HO interactions in 2. The dominant component σ(O---HO) originates from the donation of electron density from the lone pair of oxygen into the empty * HO   , characterizing anti-bonding orbital of H-O bond in the oxalic acid monomer. In addition, density is shifted from hydrogen of H-O group of one oxalic acid monomer to oxygen of the second fragment. The density deformation (    ) in the  -network of 2 gives rise to an orbital interaction stabilization of 28We note that the participation of * HO  in the hydrogen bonding leads to a stretch of the H-O bond by 0.06 Å, compared to non- bonded monomer. The second contribution to π-polarization, which apparently justifies assignment of oxalic acid to the group of resonance assisted hydrogen bonded (RAHB) systems. We observed that the density depletion in the  -network gives rise to density buildup in the  -frame work and visa versa. More specifically, we noted a depletion of electron density from the  -type orbital on the oxygen atom of the OH group. It leads to a contraction (by 0.05 Å) of the HO-C bond. At the same time charge is donated to the carbonyl oxygen by electron transfer to the * CO  orbital. The result is an elongation of the CO carbonyl bond by 0.02 Å. Finally, the last contribution, pol   , with the corresponding energy by -4.1 kcal mol -1 , shows the density changes within each oxalic acid monomer (so called intra-fragment polarizations). <Figure 2> The second stabilizing contribution in 2 comes from the electrostatic attraction (E elstat ) between the oxalic acid monomers. It amounts to -36.7 kcal mol -1 . Such an interaction is due to the stabilizing interaction between the HO    dipole of the HO bond in one monomer and the OC     dipole of the carbonyl group of the second oxalic acid fragment. Further stabilization comes from the fact that the electron density of one fragment only partially can shield the interaction of the nuclei on the same fragment from the electron density on the other fragment. The reduced shielding is due to inter-penetration of the two fragment densities. It should be noted that E orb and E elstat are of the same importance (in absolute value by ~ 36 kcal mol -1 ). In addition, it is worth emphasizing that σ(O---HO) component (78% of E orb ) is roughly eight times as important as -RAHB contribution (9.8% of E orb ). At the same time, the stabilizing effect (originating from E orb and E elstat ) is reduced by the repulsive interaction between the occupied carbonyl lone-pair orbital on one monomer and the occupied O-H bonding orbital on the other oxalic acid fragment, what is accordingly manifested by considerable Pauli repulsion contribution (E Pauli ), by 51.5 kcal mol -

  In addition, similar qualitative picture of bonding formation based on the total deformation density contours was found by Beck et al. and Góra et al. [56-57]. In the above studies, however, no information on the relative importance of and -contributions was provided. Recently, Guerra et al.

  and -oxalic acid, based on the recently proposed ETS-NOCV procedure. The main attention was paid to characterization of HB from both qualitative (by providing deformation density contributions,  k ) and a quantitative (by providing energetic estimations k E  for each  k ) points of view.We found based on the NOCV-deformation density contours that the formation of hydrogen bonding is associated with the electronic charge redistribution in both the σ-( σ ) and networks (  ). It was shown that -donation is realized by electron transfer from the lone pair of oxygen on one monomer into the empty * HO   orbital of the second oxalic acid fragment. In addition, a covalent contribution is observed by the density transfer from hydrogen of H-O group in one oxalic acid monomer to the oxygen atom of the second fragment. The resonance assisted component (  ), RAHB, is based on the transfer of electron density from the orbital localized on the oxygen of OH on one oxalic acid monomer to the oxygen atom of the other fragment. The σ(O---HO) component is roughly eight times as important as -contribution in -oxalic acid dimer (2) and trimers[START_REF] Lippincott | [END_REF]4). The electrostatic factor (E elstat ) is quantitatively as important as orbital interaction term (E orb ).Finally, comparing -dimer (2) with trimer of oxalic acid (3) we found practically no difference in the characteristics of each of the O-HO bonds, neither qualitative nor quantitative.

Fig. 1

 1 Fig. 1The hydrogen bonded systems studied in the present work[START_REF] Jeffrey | Hydrogen Bonding in Biological Structures[END_REF](2)[START_REF] Lippincott | [END_REF](4). A thick line represents the way of fragmentation used in a description of hydrogen bonding. In bonding analysis of 4 the middle monomer was treated as first fragment (A) and the terminal monomers as the second fragment (B)

Fig. 3 bonding between the monomers in 1 .

 31 Fig. 3The contours of deformation density contributions , RAHB    describing the hydrogen bonding between the monomers in 1. In addition the corresponding ETS-NOCV-based energies (in kcal mol -1 ) are shown. The numerically smallest contour values are 0.001a.u.

Fig. 4 hydrogen bonding in 3 .

 43 Fig. 4The contours of deformation density contributions ,, RAHB

Fig. 5 hydrogen bonding in 4 .

 54 Fig. 5The contours of deformation density contributions ,, RAHB

  Figure 1 Click here to download high resolution image

  .

	Please carefully check reference, such as Ref 17, should use journal
	abbreviation ? And Ref 25.

Acknowledgments

This work was supported by a research grant from the Polish Ministry of Science and Higher Education in Poland (N N204 227534).

Tables

The labeling corresponds to Fig. 1


Michel Bourdeau
email: bourdeau@ehess.fr

Comte's Lamarckian Heritage

Comte praised Lamarck highly as a biologist and attributed to him two main contributions: his seminal contribution to environmental theory and the way he conceived animal hierarchy. Given that the first point is now well documented, we are only dealing with the second one, which is less favorable to Comte, as biologists disagree today with Lamarck on this subject. We explain first how Comte uses biology in his sociology: the Lamarckian animal series is the first scientific example of progress and social progress is only an extension of it. We then present the Lamarck-Cuvier controversy about classification and show how Lamarck's insistence on the role of external circumstances was a way out of the difficulties he encountered in the construction of the animal series.

Classification, animal series, environment, biology and sociology.

L'humanité ne constituant, au fond, que le principal degré de l'animalité, les plus hautes notions de la sociologie, et même de la morale, trouvent nécessairement en biologie leur première ébauche, pour les esprits vraiment philosophiques qui savent les y saisir A. Comte

Introduction.

In the controversy concerning species fixity, Comte sided with Cuvier.

Nevertheless, he always considered Lamarck a better biologist than Cuvier, as shown by the fact that the former, not the latter, appears in the positivist calendar.

The reasons for such preference were twofold. On the one hand, Lamarck conceives organic hierarchy in a much more satisfactory way than Cuvier does. Indeed, for social static, i. e. sociology's section that deals with social order, classification is as important as it is for a biologist. On the other hand, Comte sees Lamarck's notion of « ambient environment » as a fundamental contribution to biology. However, his final judgment is balanced: Lamarck's ideas are good, but the way he explains them is unsatisfactory. The best example is to be found precisely in the controversy with Cuvier. As a matter of fact, Lamarck's error comes precisely from the fact that he gives external circumstances a role they do not have. He did not see that, besides the action of environment upon the organism, there is also an action of the organism upon its environment, -a fact that is crucial if we are to understand man's action upon nature. Nevertheless, in both cases, according to Comte, Lamarck made important contributions: he brought new ideas and opened new directions to be followed by biologists; this cannot be said about Cuvier.

In many respects, the second idea is the more interesting one, as it opens up an avenue to some of the most unexpected aspects of late positivism; namely, its concern with ecology, sustainable development and biotechnology. But the subject has already been dealt with [START_REF] Braunstein | Le concept de milieu, de Lamarck à Comte et aux positivistes[END_REF][START_REF] Braunstein | La philosophie de la médecine d'Auguste Comte[END_REF] and I do not want to revisit it.

For this reason, I shall focus instead on the alleged superiority of Lamarck concerning animal hierarchy. As we shall see, such superiority is far from evident, but the topic is very important for Comte as it has a bearing on the relationship of sociology with biology.

Regarding the Lamarck-Cuvier controversy, Comte unambiguously sided with the latter. While Lamarck's position allows for an endless variability of species, one of the tenets of positivism holds that, if laws do not exclude possible variation, there are always limits to the variation of any phenomenon. However, Comte's agreement with Cuvier on this point is only half of the matter. The very paragraph where this is stated ends with a sentence in which Comte acknowledges that "Lamarck showed by far the clearer and profounder conception of the organic hierarchy."1 From our point of view, however, Cuvier was right in this respect too. So, Comte's last assertion is also wrong, which means that to choose this as a subject is not the best way to give a positive idea of him. But there are two reasons for this choice: first, it deals with classification, and classification remains the main contribution of biology to the methodology of the sciences; second, it throws light on one of Comte's central ideas, the close relationship between biology and sociology, between the animal series and the social series.

Besides species fixity, there was indeed a second point of disagreement between Lamarck and Cuvier, relating to the existence or nonexistence of an "animal series." The biologist cannot be content with the establishment of the various species, kinds or classes of animals; he needs to go one step further and to coordinate the various classes he recognizes. Cuvier proposed to distinguish four branches (vertebrates, molluscs, articulated, zoophytes) and gave up on finding any continuity or hierarchy between them. Lamarck, on the contrary, was still, as it were, an adept supporter of the old idea of a scale of beings ("échelle des êtres", scala naturae). He thought it was possible to arrange all animal forms in a unique series, illustrating the progressive attainment of "Nature's Plan", i. e., according to various degrees of perfection.2 As to Comte, he took his view of Lamarck from his close friend, Ducrotay de Blainville, who was elected to the Chair of Cuvier in the Museum, but always showed a pronounced preference for Lamarck and for a unique, serial and ascending animal series. 3In what follows, I shall proceed in a regressive way, taking my point of departure in Comte and more specifically in what he says about the way the social series extends the animal one. The point belongs to a central theme in Comte's philosophy, namely, the close connection existing between sociology and biology. In order to understand this idea, we shall go back to biology itself and look at what Comte says about the animal series. The next step will then consist of going from Comte to Lamarck and to his controversy with Cuvier. In light of today's understanding, Lamarck, and consequently Comte, was wrong. But it would be unfair to cling to this conclusion. The animal series was not only a new version of the scala naturae, but it was also a way to introduce a genetic approach, which was totally missing in Cuvier. It is precisely in this context that the notion of the "influence of circumstances" is introduced, and we shall therefore end with some words about environment and transformism.

The two series.

Biology and Sociology.

From the sociological point of view we chose to start from, the main idea is the following: "This [i. e. the social] series is like the animal hierarchy, of which it is in fact a kind of special prolongation." 4 Its most explicit formulation is to be found in the first page of lesson fifty-one of the Cours de philosophie positive, entitled Social dynamics, or, theory of the natural progress of human society:

If we regard the course of human development from the highest scientific point of view, we shall perceive that it consists of educing, more and more, the characteristic faculties of humanity, in comparison with those of animality; and especially with those which Man has in common with the whole organic kingdom. It is in this philosophical sense that the most eminent civilization must be pronounced to be fully accordant with nature, since it is, in fact, only a more marked manifestation of the chief properties of our species; properties which, latent at first, can come into play only in that advanced state of social life for which they are exclusively destined. The whole system of biological philosophy indicates the natural progression. We have seen how, in the brute kingdom, the superiority of each race is determined by the degree of preponderance of the animal life over the organic. In like manner, we see that our social evolution is only the final term of a progression which has continued from the simplest vegetables and most insignificant animals, up through the higher reptiles, to birds and the mammifers, and still on to carnivorous animals and monkeys, the organic characteristics retiring, and the animal prevailing more and more, till the intellectual and moral tend towards the ascendancy which can never be fully obtained, even in the highest state of human perfection that we can conceive of. This comparative estimate affords us the scientific view of human progression, connected, as we see it is, with the whole course of animal advancement, of which it is itself the highest degree (Comte 1853, t. 2, p. 149).

The dependence of sociology on biology presents many aspects. Only one of them, namely progress, which is the subject matter of social dynamics, will retain our attention. The idea is so central to Comte's philosophy that it was already expressed in the Opuscule fondamental of 1822: "la civilisation est assujettie dans son développement progressif à une marche naturelle et irrévocable, dérivée des lois de l'organisation humaine."5 There is no mention of series, but the idea is the same. Not only is human nature the condition of social progress but there is no need to go to sociology to get the idea of progress, for it is already present in biology, and social progress is an extension of the progress we find in the succession of animal life.

Biology offers the scientific basis of the idea of progress.

From Sociology to Biology: the animal series, 1.

In order to understand social progress, we are thus driven back to biology. The main ideas were already present in the passage we just quoted. The first one, the very idea of series, may be seen as an outgrowth of the notion of the animal kingdom, or as Comte prefers to say, of animal life. Animals can not only be considered as forming one single class but they can also be arranged in one single series, which amounts to introducing a progression between the elements: they can be arranged in a pre-established order. In the second place, the progression is also an improvement (perfectionnement): the phenomenon being studied here, animality, becomes more and more complex, more and more perfect. That is the reason why biological evolution gives the first scientific examples of progress. This second component goes together with two other ideas: the idea of type and the idea of degree. The former is like limit in mathematical series: it denotes that element toward which the progression tends, without being always able to reach it. Moreover, the existence of a type (or type-species) plays a methodological role, because, as a rule, a phenomenon is best studied where it exists with the greatest intensity. In this case, the type is Man, humanity being the accomplishment of animality, as shown by the progresses of the nervous system. 6 On the other hand, once the type has been established, the animal series is also an animal hierarchy. There are lower and upper forms of animal life, according to the degree to which they exhibit the characteristic attribute of the species type. It is like a ladder with many rungs and, as with a ladder, we can go both ways, in ascending or descending order.

From Comte to Lamarck: the animal series, 2.

What Comte says is essentially Lamarckian, which comes as no surprise, if we consider that, on this point as on many others regarding biology, he follows Blainville, who sided with Lamarck against Cuvier (Blainville, 1845, t. 3: 371-411). The famous controversy between the two biologists was studied many years ago by Daudin. His book is still the best account available and, in what follows, I shall rely mainly upon volume two: Les classes zoologiques et l'idée de série animale, which is entirely dedicated to this quite intricate topic. What is at stake is clear: the principle of subordination of characters tells us how to form natural kinds, but the question remains: once you have set up the six main groups of animals, how do you coordinate them? Let us begin with Cuvier. Quite early on, he voiced his objection against the idea of a scale, or chain of beings (Daudin (1926-27, t. 2: 83-86). As an anatomist, he had noted that the degradation of organs could go in various directions, and not only one. Like his contemporaries, he had understood that zoologists could not be satisfied with merely establishing classes and he acknowledged they had also to be coordinated; however, he proposed to do it in an entirely new way that was incompatible with the idea of series. He claimed that the correct way to proceed was to apply to zoology the same principle of subordination of characters that Jussieu had applied in botany. Consequently, we have to change the hierarchical level of molluscs: they are not on the same level as a given class of vertebrates but of vertebrates in general. If it is so, we are compelled to give up the idea of a single series and recognize instead the existence of four independent divisions or branches, each one exhibiting an organization plan of its own. It may seem strange that, in our account, Lamarck comes after Cuvier, but it agrees with the fact that, in this controversy, he is essentially reacting against his younger colleague's new propositions. If the position Lamarck argues for is indeed an old one, he takes into account the new data his adversary had discovered in compared anatomy, and he brings new arguments in favor of the old ideas. The controversy is not about facts, but about the right way to interpret them. The animal series is conceived as the realization of Nature's plan. Lamarck takes the idea from Linnaeus, but he gives it new meaning, not only static but also dynamic. It is not the mere coordination of living beings; it is also a process of formation. Indeed, what he is after is not what Cuvier was seeking. He contests that the purpose of zoology could be reached by merely extending the principle of subordination of characters.

What he purports to study is the relationships living beings have with one another; and such a study is quite different from the mere search for characters: it has to consider not only the internal organization, as required by Cuvier, but also the external conformation. Moreover, both are only means to the overall goal of this study: to bring together all animals into a systematic scheme, in such a way that each of them be, as it were, defined by its location in the scheme (Daudin 1926-27, t. 2: 90, 117-123). -However, the old ways of thinking have not disappeared. When Lamarck has to develop his conception, he often contents himself with translating the familiar static scheme into a dynamic language and we find him reverting to the ideas that were operative in the traditional classifications: continuity and gradual progress toward a type conceived as exhibiting animality in its most perfect form. 7However, Lamarck quickly saw that it was far from easy to put all the available facts into a single linear series. In order to solve the difficulty, he had to weaken his thesis and to introduce new considerations. Strictly speaking, only the bigger units of classification (groups or families) can be put into such a series. At the lower level, there is no such thing as a linear series; if we want to include species or even types, what we get looks rather like a ramified tree (une série rameuse). Such a position amounts to distinguishing between a progression we should be entitled to consider as regular and linear and the existence of anomalies which disturb regularity; and it is in order to account for those anomalies that Lamarck turns to the influence of external circumstances (Daudin, 1926-27, t. 2: 150-152).

The controversy between the two French biologists has long been settled.

Cuvier was right not only about species fixity but also in holding that, with the principle of subordination of characters, botanists had found the logical tool needed for the correct solution to the difficult problem of animal classification. Lamarck's error came from the confusion he made between two different questions: the proper demands of classification and the search for a genetic explanation of the various forms of animal life (Le Guyader, 2000: 346-53). As he sided with Lamarck regarding the animal series, Comte too was wrong and this shows one more instance of his lack of perspicacity in scientific matters, along with his condemnation of stellar astronomy or cellular theory. However, it would be unfair to quickly blame him. At the time, the situation was quite intricate: if the conceptual scheme of the linear series was the poorest one from a taxonomic point of view. It was also the most apt to suggest transformism, while a taxonomy based on subordination of characters, although a significant scientific progress seemed to support fixism. Comte was not the only philosopher of science to err. Under his influence, Mill, too, upheld a position in favor of animal series, and his arguments are not to be discarded without examination (Mill, 1843, Bk. III, ch. viii: Of classification by series).

Transformism and Environment Theory.

Until now, no reference has been made to transformism and to the hereditary transmission of acquired characters, which is a point at least as important as the perfectionnement non interrompu, une grande nappe incessante des êtres qui pourraient se former les uns à partir des autres » (Foucault, 1966, p. 288 ;cf. Foucault, 1970).

existence and the nature of the animal series. As the two points are interrelated and as the former allows a more positive look at Lamarck, as well as at Comte, some brief comments are in order.

As already mentioned, transformism was a way to meet the difficulties raised by the effective construction of the animal series. The variety of animal forms was not attributed solely to "Nature's plan," as in the first theory, but also to the joint action of two factors: Nature's plan, as before, which accounted for the approximate regularities in the progressive complexity of animal life, but also the influence of circumstances, which accounted for the various anomalies observed in the realization of this plan. This last factor is only "a fortuitous and consequently variable cause," which acts only to modify "the first and predominating cause,", namely, nature.8 With this new and fruitful insight, Lamarck may be seen as the forerunner of environment theory and what Comte says in this respect is a good example of the balanced way he appreciates Lamarck's place in history: he praises him for having been the first to stress the importance of environment in biology; but he blames him for not having given it a satisfactory formulation. On the one hand, Lamarck has clearly seen the fundamental dualism of the organism and its environment, which is constitutive of life. Long before Comte, he criticized Bichat's definition of life as the sum of the functions which oppose death. Such a conception assumes living beings to be, as it were, immortal: what lies outside the organism is supposed to be hostile and death is attributed to exogenous causes. Actually, death, natural death, is the result of endogenous processes. As to environment, without it, life would be unimaginable; far from destroying life, it is the main cause of its development.

Lamarck, however, went too far and greatly overestimated the action of circumstances. His mechanistic conception denies any spontaneous activity to the organism and gives nearly unlimited power to environment. A sound theory of environment must recognize that the action goes both ways: just as the environment modifies the organism, the organism modifies the environment, and this is already the case at the vegetal level. Between the organic and the inorganic world, there is a complex system of action and reaction, which aims toward an equilibrium. Industry, that is, man's action upon the external world, by which he substitutes a natural order by an artificial one more suitable to his needs, has to be understood in this framework, as an instance of a phenomenon which is co-extensive to life. In this case, as in many others, sociology depends on biology.

Conclusion.

This last aspect of Lamarck's contribution to biology is perhaps the most fruitful one but, as said in the introduction, it had already been studied and, in order to present the Lamarckian heritage in Comte, I chose instead to focus on the second point for which Comte credits Lamarck; namely, his understanding of the nature of the animal series. Such a topic was not the most appropriate to give a positive image of both authors, as it appears that Lamarck was mistaken twice: about species fixity of course but also about the principles of classification. This first conclusion is neither to Lamarck's nor Comte's advantage, but it would be unfair to stop there and two more points are worth considering. First of all, classification is a very difficult subject, which reappears periodically. If, on the one hand, Lamarck remains a man of the Eighteenth Century, sticking to ideas which had to be relinquished, on the other hand, the genetic, dynamic point of view for which he argued was far from sterile and one must remember that Cuvier, too, missed one dimension of the problem. Behind their controversy, there are two ways of thinking which are not limited to biology and are still alive today. Recently, as shown by cladistics, our taxonomic principles have been deeply modified and further revisions are not to be excluded. In the second place, by studying what Comte says about Lamarck, we get some idea of Lamarck's reception. It is sometimes said that his ideas were quickly forgotten and had to wait until Darwin's time to be rediscovered. Such an assessment does not seem quite accurate. Lamarck's influence on Comte is certain and quite extensive. If we consider the place of positivists among French biologists around 1850, we have some grounds to assume that, at that time, Lamarckism was still alive. A final word: Besides Darwinism, there is a social Darwinism. What I said shows, I hope, that the relationship between biology and sociology is as old as sociology itself, and, from the fact that Darwin is a better scientist that Lamarck, it does not follow that the sociology inspired by Darwin is more scientific than the sociology inspired by Lamarck.

Comte, 1853, t. I: 414; At the same time she translated the Course, H. Martineau "freely condensed" it, with Comte's authorization, who, later on, found this abbreviated version preferable to the original one. If it made the book more readable, it was at the cost of a loss of precision. For this reason, when necessary, I shall give the French text.

« Il est donc vrai de dire qu'il existe pour chaque règne des corps vivants une série unique et graduée dans la disposition des masses, conformément à la composition croissante de l'organisation, en s'élevant, dans le règne animal, des animalcules les plus simples aux animaux les plus parfaits » (Lamarck, as quoted inGuillo, 2003, p. 125).

Between the series as given by Blainville and the series as given by Lamarck, there are big differences, but the main ideas are the same. About Blainville's contribution to the subject, seeGuillo 2003 : 200-209.

Comte, A. (1853, t. 2: 356); translates: "la série sociale se présente rationnellement comme un prolongement spécial de la grande série animale"(Comte 1830, vol. 2 : 486).

"Civilization, in its progressive development, is subject to a natural and irrevocable march, derived from the laws of human organization"(Comte 1854, vol. 4, app. p. 89)

This explains the methodological revolution introduced by biology, which starts from man, as the best known of animals, and proceeds, not from simple to complex, as before, but the other way round, from complex to simple.

« Lamarck ne pensait la transformation des espèces qu'à partir de la continuité ontologique qui était celle de l'histoire naturelle des classiques. Il supposait une gradation progressive, un

Lamarck quoted in[START_REF] Daudin | Cuvier et Lamarck : les classes zoologiques et l'idée de série animale[END_REF], t.2: 224. This conception is the second of the three stages of Lamarck's conception of the role of circumstances, see


Romain Goffe
email: romain.goffe@greyc.ensicaen.fr

Luc Brun
email: luc.brun@greyc.ensicaen.fr

Guillaume Damiand
email: guillaume.damiand@liris.cnrs.fr

Tiled top-down pyramids and segmentation of large histological images

Keywords: Irregular pyramid, Topological model, Combinatorial map

Recent microscopic imaging systems such as whole slide scanners provide very large (up to 18GB) high resolution images. Such amounts of memory raise major issues that prevent usual image representation models from being used. Moreover, using such high resolution images, global image features, such as tissues, do not clearly appear at full resolution. Such images contain thus different hierarchical information at different resolutions. This paper presents the model of tiled top-down pyramids which provides a framework to handle such images. This model encodes a hierarchy of partitions of large images defined at different resolutions. We also propose a generic construction scheme of such pyramids whose validity is evaluated on an histological image application.

Introduction

The increasing amount of high resolution images raises new and major issues in the field of image analysis and processing. For instance, microscopic scanners have recently been improved to the point where whole slide imaging techniques may offer a ×40 000 magnification. However, the segmentation of such images requires to handle a hierarchy of large partitions defined on up to 18GB data volumes. A suitable model to encode segmentation of such images should be compatible with memory constraints induced by this large amount of data and should allow to design segmentation algorithms based on the same top-down analysis scheme than pathologists.

In this regard, we identified two key steps during pathologists' manual analysis of pyramids of histological images. First, an identification of histological components for each resolution is performed and regions of interest are determined according to topological or geometric features. Second, these regions of interest are used within a hierarchical scheme from the lowest to the highest resolution (top-down), each region being analyzed in the context defined by its ancestors.

As a consequence, a model with geometrical, topological and hierarchical features is required to provide a segmentation of large histological images. Usual non hierarchical models are devoted to few partitions' properties and do not provide a full encoding of geometrical and topological properties of a partition. For example, RAG data structures do not provide an efficient access to the geometry of regions' borders. This drawback is addressed by more sophisticated models such as topological maps [START_REF] Brun | Incremental modifications on segmented image defined by discrete maps[END_REF][START_REF] Damiand | Topological model for two-dimensional image representation: definition and optimal extraction algorithm[END_REF] which encode both geometric and topological properties of a partition.

Hierarchical models such as quadtrees or regular pyramids are commonly used for multi-resolution images representation and segmentation. Yet, both frameworks entail several drawbacks [START_REF] Bister | A critical view of pyramid segmentation algorithms[END_REF]. For example, they may not preserve adjacency of connected regions through different levels of a pyramid. Those drawbacks lead to the design of irregular pyramids [START_REF] Brun | Incremental modifications on segmented image defined by discrete maps[END_REF][START_REF] Damiand | Topological model for two-dimensional image representation: definition and optimal extraction algorithm[END_REF] in order to take advantage of the efficiency of graphs and topological maps for geometric and topological operations while keeping the advantages of their regular ancestors.

We have previously introduced the tiled maps model [START_REF] Goffe | Tiled top-down combinatorial pyramids for large images representation[END_REF], defined as a topological map decomposed into topological tiles to encode partitions of large images. In [START_REF] Goffe | A causal extraction scheme in topdown pyramids for large images segmentation[END_REF], we have proposed an efficient construction scheme of tiled top-down pyramids. Using such a construction scheme, each partition defined at a given resolution is initialised by the projection of the partition defined at the previous (lower) resolution and then further refined by split and merge operations. In this paper, we focus on three main points. We first provide an improved formalism to define our model of tiled top-down pyramid (Section 2). We propose in Section 3 a new projection step of each partition which takes into account the additional information provided by the current resolution. We finally show in Section 4 the efficiency of our model with the segmentation of an histological image.

Top-down framework

Topological maps

A topological map model [START_REF] Brun | Incremental modifications on segmented image defined by discrete maps[END_REF][START_REF] Damiand | Topological model for two-dimensional image representation: definition and optimal extraction algorithm[END_REF] encodes both topological and geometrical properties of a partition. It combines three distinct models: a 2-map that encodes topological relationships, a matrix of interpixel elements [START_REF] Kovalevsky | Finite topology as applied to image analysis[END_REF] that encodes the geometry of the partition and a tree of regions for inside relationships (Figure 1).

A combinatorial map is defined from a set of abstract basic elements called darts. Two operators denoted by β i , i ∈ {1, 2} that apply on darts allow to represent adjacency relationships between cells (Definition 1). Intuitively, a combinatorial map may be considered as a planar graph whose edges are decomposed into half edges called darts. Each dart is incident to a single vertex and belongs to a single edge and a single face. Sets of darts defining high level entities such as vertices, edges and faces are retrieved using cycles3 of β i permutations. In practice, the β 1 permutation connects a dart of a face to the next dart encountered when turning clockwise around it. The involution β 2 separates two adjacent faces and maps a dart of an edge to the only dart with an opposite orientation which belongs to the same edge. For instance, in Figure 1(a), β 1 (1) = 3 and β 2 (1) = 2. Vertices, edge and faces of a combinatorial map are respectively encoded by the cycles of the permutations β 1 • β 2 , β 2 and β 1 . All faces of a combinatorial map but one encode finite faces (faces corresponding to regions with a finite area). The face which does not satisfy this property encodes the background of the partition encoded by the combinatorial map and is called an infinite face. A combinatorial map is called minimal in number of cells if it does not contain any vertex with a degree lower or equal to 2 and removing any dart would change the topology. [START_REF] Kovalevsky | Finite topology as applied to image analysis[END_REF], the geometry of an an n×m image partition is encoded by an (n+1)×(m+1) array of marks. Each entry of this array encodes the existence of a given linel (or crack) element within the set of linels encoding image's boundaries and called bounding linels (Figure 1(b)). Note that both pointels incident to a same bounding linel are considered as bounding elements. We call embedding the association of bounding cells with a topological element (vertex, edge or face). In Figure 1(b), embedding of dart 1 is defined as the sequence of bounding elements (l 1 , p 1 , l 2 , p 2 , l 3 ). A region corresponds to the embedding of a face and is a set of adjacent pixels.

Definition 1 (Combinatorial map). In two dimensions, a combinatorial map

M is a triplet M = (D, β 1 , β 2 ) where: 1. D is a finite set of darts; 2. β 1 is a permutation 1 on D;

Matrix of interpixel elements Using interpixel elements

Tree of regions

The tree of regions describes inside/contains relationships: a region is the father of the regions it contains. In Figure 1 The root of the tree encodes the background of the image and is called the infinite region (denoted by r ∞ ).

(c), r 1 contains {r 2 , r 3 , r 4 }. (a) (b) (c)

Tiled topological maps

A tiled extension of the topological map model which decomposes a map into smaller elements called tiles has been proposed to overtake memory issues induced by the representation of large partitions [START_REF] Goffe | Tiled top-down combinatorial pyramids for large images representation[END_REF]. Indeed, tiles may be swapped on disk when they are not being processed. A topological tile is a geometric subdivision of a topological map whose interpixel matrix only encodes a subdivision of the image. Adjacency relationships between topological tiles are encoded by equivalent interpixel cells on tiles' shared borders: two pointels are equivalent if they have the same coordinates in the image referential and two linels are equivalent if their incident pointels are equivalent (Figure 2(b)). Darts' embeddings are encoded within the interpixel framework as sequences of pointels and linels. Consequently, two darts are said to have an equivalent embedding if their embeddings correspond to equivalent sequences of pointels and linels.

Let D t be the set of darts that belong to a tile t and D t,t ′ ∂ be the subset of D t composed of darts that are incident to the border of t and whose embedding has equivalent cells in t ′ . Connections between darts of two adjacent tiles t and t ′ are represented through a bijection ϕ t,t ′ :

D t,t ′ ∂ → D t ′ ,t ∂ that maps each dart d of D t,t ′ ∂ to a dart d ′ of D t ′ ,t
∂ such that d and d ′ have equivalent embeddings (Figure 2(c)). Note that such a bijection can always be obtained if we consider a decomposition of the tiles' borders into basic darts whose embedding is a single couple (pointel, linel).

Besides, the decomposition in tiles entails that some darts of a tile's border may encode the tile's decomposition of the image without encoding a real border between regions. In this case, the border is considered as a fictive border. Otherwise, the • δ 2 is an involution on D such as:

Q = 1 × 1/4 (b) Q = 2 × 2/4 (c) Q = 3 × 3/4
δ 2 (d) = ϕ t,t ′ (d) if d ∈ D t,t ∂ β 2 (d) otherwise
• δ 1 is a permutation on D such as:

δ 1 (d) = β 1 • (δ 2 • β 1 ) k (d) with k = min{p ∈ N | β 1 • (δ 2 • β 1 ) p (d) ∈ D }
Since δ 1 and δ 2 operators respectively define a permutation and an involution on D [START_REF] Goffe | A causal extraction scheme in topdown pyramids for large images segmentation[END_REF], the triplet M = (D, δ 1 , δ 2 ) is a combinatorial map.

Tiled top-down pyramids

Tiled maps allow to represent partitions from large images. However, such images usually contain different information at different resolutions hence requiring the design of a multi-resolution hierarchical data structure, such as the tiled top-down pyramidal model [START_REF] Goffe | A causal extraction scheme in topdown pyramids for large images segmentation[END_REF].

In this paper, we propose a novel approach for the definition of tiled top-down pyramids based on the combination of three hierarchical data structures: a pyramid of images encoding a multi-resolution image, a pyramid of tiles with a constant size and a pyramid of tiled maps.

A regular pyramid of images [START_REF] Bister | A critical view of pyramid segmentation algorithms[END_REF] is a sequence (I 0 , . . . , I n ) of images with exponentially increasing resolutions such that the reduction factor r encoding the ratio top-down pyramids and segmentation of large histological images between the size of two successive images remains constant along the pyramid. Each pixel in an image I k is related to a connected set of pixels in I k+1 called its reduction window and encoded as a M × N window. The value of a pixel in I k is deduced from the values of pixels within its reduction window using a reduction function. Different types of pyramids may be distinguished according to the ratio Q = M × N/r: holed pyramids (Q < 1), non overlapping pyramids without holes (Q = 1) and overlapping pyramids (Q > 1).

Such a pyramid may be implicitly associated to a stack of matrices of inter pixel elements, also defining a regular pyramid with a same reduction factor. The notion of reduction window on such a pyramid may be adapted as follows: the reduction window of a pointel p (or RW(p)) at level k (0 ≤ k < l) is an M ′ × N ′ window defined at level k + 1 and corresponding to the set of pointels encoded by p at level k. A hierarchical relationship is thus induced between pointels, each pointel being the father of the pointels within its reduction window. Different types of pyramids of pointels may be distinguished according to the ratio M ′ × N ′ /r (Fig. 3). In the following, use non overlapping pyramids without holes both for images and pointels pyramids.

Let us now consider a given tile's size and perform a tiling of each image of a non overlapping image pyramid without holes I = (I 0 , . . . , I n ). Each tile defined in an image I k is expanded into r × r tiles in I k+1 , where r denote the reduction factor of I. Hierarchical relationships established within an image pyramid induce thus hierarchical relationships between tiles decomposing each image of the pyramid. This last hierarchy is called a pyramid of tiles (Definition 3).

Definition 3 (Pyramid of tiles). Let I be an image. A pyramid of tiles associated

with I is a couple (I, T = (T 0 , . . . , T n )) where ∀k, 0 ≤ k < n:

• I = (I 0 , . . . , I n ) is a non-overlapping pyramid of images without holes associated with I;

• T k = {t(i, j, k)} (i,j)∈[0,l k /l [ ×[0,h k /h [ is a rectangular tiling of the l k × h k image I k by a set of l × h tiles t(i, j, k); • RW(t(i, j, k)) = {t(i ′ , j ′ , k + 1) | ⌊i ′ /r⌋ = i, ⌊j ′ /r⌋ =
j} is the reduction window of tile t(i, j, k), where ⌊.⌋ denotes the floor operator.

While both pyramids of images and tiles are regular hierarchical data structures, our aim is to define an irregular pyramid of tiled maps. A tiled top-down pyramid (Definition 4) is thus a stack of finer and finer partitions, defined at different resolutions and encoded by tiled topological maps. The hierarchy is represented by up/down relationships between darts and regions [START_REF] Goffe | Tiled top-down combinatorial pyramids for large images representation[END_REF]. Moreover, since each map G k+1 of the pyramid is finer than G k , a border defined at a given level of the pyramid cannot be removed at levels below. A tiled top-down pyramid is thus a causal structure [START_REF] Guigues | Scale-sets image analysis[END_REF]. Consequently, we propose a new definition for tiled top-down pyramids (Definition 4) where the tiled maps (levels) are based on the tiling of a given pyramid of tiles. • (I, T ) is a pyramid of tiles composed of n + 1 levels and associated with I;

• G k is a tiled map corresponding to the tiling of T k ;

• G k+1 is defined on (I k+1 , T k+1 ) and is deduced from G k by decomposition operations (e.g. splitting of faces of G k ) in order to be finer than G k .

The construction scheme of a tiled top-down pyramid may rely on the notion of focus of attention which allows to refine only regions of interest identified in the upper levels of the pyramid. The advantage of such a construction scheme is twofold since it may reduce memory usage and processing time while imitating experts' analysis scheme. Finally, note that within a given application field, only some specified levels of the pyramid may need to be explicitly encoded.

Segmentation scheme

We propose a generic segmentation scheme for large multi-resolution images based on the tiled top-down pyramid framework. Our segmentation process relies on the definition of an Oracle that may combine several criteria of different natures to take advantage of geometrical, topological and hierarchical features. Since this segmentation step is part of the global construction scheme of the tiled top-down pyramid, it must neither impact the causality of the pyramid nor its refinement by focus of attention. The process assumes an initial partition G 0 which may be a single region or which may result from an extraction algorithm [START_REF] Brun | Incremental modifications on segmented image defined by discrete maps[END_REF][START_REF] Damiand | Topological model for two-dimensional image representation: definition and optimal extraction algorithm[END_REF] at resolution I 0 . In both cases, a partition at level k + 1 is deduced from level k by applying the following two steps procedure:

• a projection step of the regions' borders of G k onto G k+1 which preserves the topology of G k and expands the geometry of G k+1 borders according to the reduction factor. • an Oracle-based refinement step, restricted to regions whose father is a region of interest in the previous level (focus of attention).

Figure 5 illustrates the main steps of our projection procedure. The main issue is to project the former borders onto the current resolution while preserving the same topology. Intuitively, the projection of a border b = (p 1 , . . . , p m ) in G k is a border b ′ in G k+1 contained within the strip formed by the reduction windows of the pointels (p i ) i=1,m of b. This strip is a connected set since the pyramid of images contains no holes. Moreover, the pyramid is not overlapping and therefore, two projected borders cannot intersect, except on their extremities which correspond to the intersection of several borders. Let us consider a pointel p 1 in G k which corresponds to an intersection of borders and its projection P (p 1 ) in G k+1 . In order to keep the same topology in G k and G k+1 , we define the projection of each linel in G k incident to p 1 as an horizontal or vertical straight line of length r in G k+1 , incident to P (p 1 ), and with the same direction than the initial linel. These straight lines correspond to a standard linel's projection which guarantees that P (p 1 ) is the only intersection point of projected borders within p 1 's reduction window. This standard projection is also performed on p m and the connection between the two straight lines incident to P (p 1 ) and P (p 2 ) is achieved using a Dijkstra algorithm within the strip formed by the reduction windows of pointels (p i ) i=2,m-1 . Dijkstra algorithm is applied with an edge weight proportional to the inverse of the gradient computed in I k+1 , hence taking into account the additional information provided by I k+1 .

The refinement step relies on the definition of the Oracle proposed in Algorithm 1: according to a segmentation criterion (line 3), our scheme refines a region if its father in the previous level is a region of interest (line 2) while preserving the causality of the model (line 1).

Application for large histological images segmentation

We propose a practical use case4 of the top-down framework for the segmentation of large histological images. Our objective is to demonstrate the generic aspect of our model with the integration of an existing segmentation scheme. In this application, histological images are produced as a stack of three images with increasing resolutions. Their manual analysis by pathologists are performed using a focus of attention over regions of interest up to the full resolution image where cells' mitosis are enumerated. This estimation is then used for breast cancer diagnosis grading.

Our top-down segmentation algorithm uses the same scheme than [START_REF] Roullier | Mitosis extraction in breastcancer histopathological whole slide images[END_REF] with the additional use of a tiled top-down pyramid (Figure 6). Successive k-means based classifications [START_REF] Kanungo | An efficient k-means clustering algorithm: Analysis and implementation[END_REF] allow to label and refine regions of interest. At resolution 1, the background is separated from tissue. Then, a classification in two classes of tissue at resolution 2 allows to extract lesions and a last classification distinguishes the cancer in situ from stroma at resolution 3.

Table 1 presents experimental results for the segmentation of a representative histological image. Column 2, 3 and 4 respectively indicate the size of the images, the number of tiles and the number of regions of their associated partitions. In Column 5 and 6, we provide runtimes for both pyramid construction scheme (model) and our segmentation algorithm (segmentation). These results demonstrate that our model does not introduce an important additional cost while providing an efficient access to hierarchical, topological and geometrical features. Note that runtimes for

Conclusion

In this paper, we have proposed a new approach to the definition of tiled top-down pyramids which emphasizes the combination of regular and irregular hierarchical data structures. We have described a segmentation scheme with a new projection step which takes into account the additional information provided by each image of the pyramid. Finally, we have demonstrated the efficiency of our framework with an application on histological images.

This work opens interesting perspectives such as the definition of topological criteria that could be combined to the present segmentation process. Those criteria may enhance results by taking further advantage of our top-down framework. Finally, a medical evaluation should be performed to confirm the accuracy of the partitions provided by our method.

3. β 2

 2 is an involution 2 on D.

Figure 1 :

 1 Figure 1: Topological maps: a combination of three models for the representation of a partition. (a) Combinatorial map for adjacency relationships. Dotted darts belong to the infinite region. (b) Interpixel representation for geometrical borders: bounding pointels and linels are drawn as circles and bold segments. (c) Tree of regions for inside/contains relationships.

Figure 2 :

 2 Figure 2: Tiled maps: an extension to the topological map model designed for large partitions. (a) Original image decomposed into four tiles. (b) Interpixel representation: arrows map equivalent bounding cells. (c) Tiled map: each tile is encoded by a topological map. Darts are connected between tiles through δ 1 (arcs) and δ 2 (segments) operators.

Figure 3 :Definition 2 (

 32 Figure 3: Reduction windows of pointels within the interpixel representation of a pyramid of images. (a) Non-overlapping holed pyramid. (b) Non-overlapping pyramid without holes. (c) Overlapping pyramid.

Figure 4 :Definition 4 (

 44 Figure 4: Tiled top-down pyramids: a combination of three hierarchical models for the representation of large multi-resolution images. (a) Pyramid of images (r = 2). (b) Pyramid of tiles. (c) Pyramid of tiled maps. Dotted arrows illustrate up/down relationships between darts and regions. Darts that belong to r ∞ are not represented.

Algorithm 1 :

 1 Oracle Data: Two adjacent pixels p and p ′ of a tile t ∈ G k+1 . Result: true if p and p ′ belong to the same region. r ← region(up(p)); r ′ ← region(up(p ′ )); l ← incident linel to p and p ′ ; if l is a bounding linel then 1 return false; if up(r) is not a region of interest in G k then 2 return true; return segmentation_criterion(region(p), region(p ′ ));

Figure 5 :

 5 Figure 5: Projection of regions borders. (a) Strip composed by the reduction windows of the pointels that belong to the border of the previous level. (b) Restrictions on borders extremities. (c) Shortest path according to a given energy.

Figure 6 :

 6 Figure 6: Classification scheme of histological images. (a) Original image. (b) Resolution 1: distinction between tissue and background. (c) Resolution 2: distinction between lesions and tissue. (d) Resolution 3: distinction between cancer in situ and stroma.

Table 1 :

 1 Segmentation of an histological image within our top-down framework. are linear with the size of the image but slightly increase with the number of tiles due to disc access delays. Finally, tiled maps allow to preserve a constant memory usage around 100MB (Column 7) with a tile size of 625 × 625 pixels.

	level	size	number	number	runtime (s)		ram
		(pixels)	of tiles	of regions model segmentation (MB)
	1	2 500 × 2 500	16	57 435	9.8	35.2	104
	2	5 000 × 5 000	64	132 402	43.6	186.6	101
	3 10 000 × 10 000	256	225 672	267.8	645.0	106
	levels' extraction

A permutation is a one to one mapping from S onto S.

An involution f is a one to one mapping from S onto S such that f = f -1 .

A cycle of a permutation is defined as its restriction to the set of images of an element.

Our model is implemented in C++ and computations are carried out on an Intel E5300@2GHz with 2GB RAM.

Acknowledgements

This research is part of the FoGrImMi project, supported by the ANR foundation under grant ANR-06-MDCA-008-01/FOGRIMMI.


Luc Blanchet

Jerome Novak

Blanchet Luc

Grεco

Jérôme Novak

Testing MOND in the Solar System

1 The external field effect with MOND The Modified Newtonian Dynamics (MOND) has been proposed 1 as an alternative to the dark matter paradigm 2 . At the non-relativistic level, the best formulation of MOND is the modified Poisson equation 3 ,

∇ • µ g a 0 ∇U = -4πGρ , (1)
where ρ is the density of ordinary (baryonic) matter, U is the gravitational potential, g = ∇U is the gravitational field and g = |g| its ordinary Euclidean norm. The modification of the Poisson equation is encoded in the MOND function µ(y) of the single argument y ≡ g/a 0 , where a 0 = 1.2 × 10 -10 m/s 2 denotes the MOND constant acceleration scale. The MOND function interpolates between the MOND regime corresponding to weak gravitational fields y = g/a 0 ≪ 1, for which it behaves as µ(y) = y + o(y), and the Newtonian strong-field regime y ≫ 1, where µ reduces to 1 so that we recover the usual Newtonian gravity. An important consequence of the non-linearity of Eq. (1) in the MOND regime, is that the gravitational dynamics of a system is influenced (besides the well-known tidal force) by the external gravitational environment in which the system is embedded. This is known as the external field effect (EFE), which has non-trivial implications for non-isolated gravitating systems. The EFE was conjectured to explain the dynamics of open star clusters in our galaxy 1 , since they do not show evidence of dark matter despite the involved weak internal accelerations (i.e. below a 0 ). The EFE effect shows that the dynamics of these systems should actually be Newtonian as a result of their immersion in the gravitational field of the Milky Way. The EFE is a rigorous prediction of the equation (1), and is best exemplified by the asymptotic behaviour of the solution of (1) far from a localised matter distribution (say, the Solar System), in the presence of a constant external gravitational field g e (the field of the Milky Way). At large distances r = |x| → ∞ we have 3

U = g e • x + GM/µ e r 1 + λ e sin 2 θ + O 1 r 2 , ( 2
)
where M is the mass of the localised matter distribution, where θ is the polar angle from the direction of the external field g e , and where we denote µ e ≡ µ(y e ) and λ e ≡ y e µ ′ e /µ e , with y e = g e /a 0 and µ ′ e = dµ(y e )/dy e . In the presence of the external field, the MOND internal potential u ≡ Ug e • x shows a Newtonian-like fall-off ∼ r -1 at large distances but with an effective gravitational constant G/µ e . a However, contrary to the Newtonian case, it exhibits a non-spherical deformation along the direction of the external field. The fact that the external field g e does not disappear from the internal dynamics can be interpreted as a violation of the strong version of the equivalence principle.

Abnormal influence of the Galaxy in the Solar System

In two recent papers [START_REF] Milgrom | Mond effects in the inner solar system[END_REF][START_REF] Blanchet | External field effect of modified newtonian dynamics in the solar system[END_REF] it was shown that the imprint of the external galactic field g e on the Solar System (due to a violation of the strong equivalence principle) shows up not only asymptotically, but also in the inner regions of the system, where it may have implications for the motion of planets. This is somewhat unexpected because gravity is strong there (we have g ≫ a 0 ) and the dynamics should be Newtonian. However, because of the properties of the equation ( 1), the solution will be given by some non-local Poisson integral, and the dynamics in the strong-field region will be affected by the anomalous behaviour in the asymptotic weak-field region.

We assume that the external Galactic field g e is constant over the entire Solar System. b The motion of planets of the Solar System relatively to the Sun obeys the internal gravitational potential u defined by

u = U -g e • x , (3)
which is such that lim r→∞ u = 0. Contrary to what happens in the Newtonian case, the external field g e does not disappear from the gravitational field equation (1) and we want to investigate numerically its effect. The anomaly detected by a Newtonian physicist is the difference of internal potentials,

δu = u -u N , (4)
where u N denotes the ordinary Newtonian potential generated by the same ordinary matter distribution ρ, and thus solution of the Poisson equation ∆u N = -4πGρ with the boundary condition that lim r→∞ u N = 0. We neglect here the change in the matter distribution ρ when considering MOND theory instead of Newton's law. This is in general a good approximation because the gravitational field giving the hydrostatic equilibrium (and thus ρ) is strong and MOND effects are very small. Hence u N is given by the standard Poisson integral.

A short calculation shows that the anomaly obeys the Poisson equation ∆δu = -4πGρ pdm , where ρ pdm is the density of "phantom dark matter" defined by

ρ pdm = 1 4πG ∇ • (χ∇U ) , (5)
a Recall that in the absence of the external field the MOND potential behaves like U ∼ -√ GM a0 ln r, showing that there is no escape velocity from an isolated system 4 . However since no object is truly isolated the asymptotic behaviour of the potential is always given by (2), in the approximation where the external field is constant.

b For the Milky Way field at the level of the Sun we have ge ≃ 1.9 × 10 -10 m/s 2 which happens to be slightly above the MOND scale, i.e. η ≡ ge/a0 ≃ 1.6.

where we denote χ ≡ µ -1. The phantom dark matter represents the mass density that a Newtonian physicist would attribute to dark matter. In the model [START_REF] Blanchet | Model of dark matter and dark energy based on gravitational polarization[END_REF][START_REF] Blanchet | Dipolar dark matter and dark energy[END_REF] the phantom dark matter is interpreted as the density of polarisation of some dipolar dark matter medium and the coefficient χ represents the "gravitational susceptibility" of this dark matter medium.

The Poisson equation ∆δu = -4πGρ pdm is to be solved with the boundary condition that lim r→∞ δu = 0; hence the solution is given by the Poisson integral

δu(x, t) = G d 3 x ′ |x -x ′ | ρ pdm (x ′ , t) . (6)
We emphasise that, contrary to the Newtonian (linear) case, the knowledge of the matter density distribution does not allow to obtain an analytic solution for the potential, and the solution has to be investigated numerically. We can check that the phantom dark matter behaves like r -3 when r → ∞, so the integral ( 6) is perfectly convergent.

In the inner part of the Solar System the gravitational field is strong (g ≫ a 0 ) thus µ tends to one there, and χ tends to zero. Here we adopt the extreme case where χ is exactly zero in a neighbourhood of the origin, say for r ε, so that there is no phantom dark matter for r ε; for the full numerical integration later we shall still make this assumption by posing χ = 0 inside the Sun (in particular we shall always neglect the small MOND effect at the centre of the Sun where gravity is vanishingly small). If ρ pdm = 0 when r ε we can directly obtain the multipolar expansion of the anomalous term ( 6) about the origin by Taylor expanding the integrand when r = |x| → 0. In this way we obtain

c δu = +∞ l=0 (-) l l! x L Q L , (7)
where the multipole moments near the origin are given by

Q L = G r>ε d 3 x ρ pdm ∂ L 1 r . ( 8
)
Because the integration in ( 8) is limited to the domain r > ε and ∂ L (1/r) is symmetric-trace-free (STF) there [indeed ∆(1/r) = 0], we deduce that the multipole moments Q L themselves are STF. This can also be immediately inferred from the fact that ∆δu = 0 when r ε, hence the multipole expansion (7) must be a homogeneous solution of the Laplace equation which is regular at the origin, and is therefore necessarily made solely of STF tensors of type xL . Hence we can replace x L in (7) by its STF projection xL . It is now clear from the non-local integral in (8) that the MONDian gravitational field (for r r 0 ) can influence the near-zone expansion of the field when r → 0. An alternative expression of the multipole moments can also be proved, either directly or by explicit transformation of the integral (8). We have

Q L = -u N (0) δ l,0 + (-) l ( ∂L u)(0) , (9)
where the Newtonian potential u N and the STF derivatives of the internal potential u are to be evaluated at the centre 0 of the Sun. The multipole expansion (7) will be valid whenever r is much less than the MOND transition distance for the Solar System, defined by r 0 = GM/a 0 with M the mass of the Sun and a 0 the c Our notation is as follows:

L = i1 • • • i l denotes a multi-index composed of l multipolar spatial indices i1, • • • , i l (ranging from 1 to 3); ∂L = ∂i 1 • • • ∂i l is the product of l partial derivatives ∂i ≡ ∂/∂x i ; x L = x i 1 • • • x i l is the product of l spatial positions x i ; similarly n L = n i 1 • • • n i l = x L /
r l is the product of l unit vectors n i = x i /r; the symmetric-trace-free (STF) projection is indicated with a hat, for instance xL ≡ STF[x L ], and similarly for nL and ∂L. In the case of summed-up (dummy) multi-indices L, we do not write the l summations from 1 to 3 over their indices. MOND acceleration scale. This radius corresponds to the transition region where the Newtonian acceleration becomes of the order of the MOND acceleration a 0 and therefore, MOND effects become dominant. We have r 0 ≃ 7100 AU so the results ( 7)-( 9) hold in a large volume around the Sun including all the planets (recall that Neptune's orbit is at 30 AU).

3 Results for the induced quadrupole moment in the Solar System So far we have elucidated the structure of the multipole expansion of the anomaly δu near the origin. Next we resort to a numerical integration of the non-linear MOND equation ( 1) in order to obtain quantitative values for the multipole moments. d The Sun being assumed to be spherically symmetric, since all the multipole moments are induced by the presence of the external field g e in the preferred direction e, the situation is axisymmetric and all the moments Q L will have their axis pointing in that direction e. Thus we can define some multipole coefficients Q l by posing Q L = Q l êL , where êL denotes the STF part of the product of l unit vectors e L = e i 1 • • • e i l . The multipole expansion (7) reads then as

δu(r, θ) = +∞ l=0 (-) l (2l -1)!! r l Q l (r) P l (cos θ) , (10)
where P l (z) is the usual Legendre polynomial and θ is the angle away from the Galactic direction e. Although from the previous considerations the multipole coefficients Q l should be approximately constant within the MOND transition radius r 0 , here we compute them directly from the numerical solution of (1) and shall obtain their dependence on r. With our definition the quadrupolar piece in the internal field is given by

δu 2 = 1 2 r 2 Q 2 (r) cos 2 θ - 1 3 . ( 11
)
The radial dependence of the anomaly ( 11) is ∝ r 2 and can thus be separated from a quadrupolar deformation due to the Sun's oblateness which decreases like ∝ r -3 . As a first result, we show in Fig. 1 the profile of the quadrupole induced by the MOND theory through the function Q 2 (r) defined in Eq. (11). We find that this quadrupole is decreasing from Table 1: Numerical values of the quadrupole Q2 together with the associated dimensionless quantity q2 defined by Eq. (12). All values are given near the Sun. We use different choices of the function µ(y) defined in Eqs. (13).

MOND function

µ 1 (y) µ 2 (y) µ 20 (y) µ exp (y) µ TeVeS (y) Q 2 [s -2 ]
3.8 × 10 -26 2.2 × 10 -26 2.1 × 10 -27 3.0 × 10 -26 4.1 × 10 -26 q 2 0.33 0.19 1.8 × 10 -2 0.26 0.36 the Sun's neighbourhood to zero, on a typical scale of 10000 astronomical units (AU). However, we check numerically that Q 2 (r) is almost constant in a large sphere surrounding the Solar system, as it has a relative variation lower than 10 -4 within 30 AU (see the zoomed region in Fig. 1). We shall therefore refer to the quadrupole as a simple number, noted Q 2 (0) or simply Q 2 , when evaluating its influence on the orbits of Solar-system planets.

On dimensional analysis we expect that the quadrupole coefficient Q 2 should scale with the MOND acceleration a 0 like

Q 2 = a 0 r 0 q 2 (η) , (12)
where r 0 = GM/a 0 is the MOND transition radius and where the dimensionless coefficient q 2 depends on the ratio η = g e /a 0 between the external field and a 0 , and on the choice of the interpolating function µ. Our numerical results for the quadrupole are given in Table 1, for different coupling functions µ(y). e Here we consider various cases widely used in the literature:

µ n (y) = y n √ 1 + y n , (13a)
µ exp (y) = 1 -e -y , (13b)
µ TeVeS (y) = √ 1 + 4y -1 √ 1 + 4y + 1 . (13c)
The function µ 1 has been shown to yield good fits of galactic rotation curves 9 ; However because of its slow transition to the Newtonian regime it is a priori incompatible with Solar System observations. The function µ 2 is generally called the "standard" choice and was used in fits 10 . We include also the function µ exp having an exponentially fast transition to the Newtonian regime. The fourth choice µ TeVeS is motivated by the TeVeS theory 11 . One should note that none of these functions derives from a fundamental physical principle. We have used several functions of type µ n , as defined in Eq. (13a). One can notice that the value of Q 2 decreases with n, that is with a faster transition from the weak-field regime where µ(y) ∼ y, to the strong field regime where µ(y) ∼ 1. We have been unable to determine numerically a possible limit for Q 2 as n goes to infinity.

Effect on the dynamics of the Solar System planets

We investigate the consequence for the dynamics of inner planets of the Solar System of the presence of an abnormal quadrupole moment Q 2 oriented toward the direction e of the galactic centre. Recall that the domain of validity of this anomaly is expected to enclose all the inner Solar System (for distances r r 0 ≈ 7100 AU), with the quadrupole coefficient being constant up to say 50 AU (see Fig. 1). As we have seen, the anomaly induces a perturbation on the Newtonian gravitational potential, namely u = u N + δu, where u N = GM/r and the perturbation function R ≡ δu is given for the quadrupole moment by Eq. (11).

We apply the standard linear perturbation equations of celestial mechanics [START_REF] Brouwer | Methods of celestial mechanics[END_REF] . The unperturbed Keplerian orbit of a planet around the Sun is described by six orbital elements. For these we adopt the semi-major axis a, the eccentricity e, the inclination I of the orbital plane, the mean anomaly ℓ defined by ℓ = n(t -T ) where n = 2π/P (n is the mean motion, P is the orbital period and T is the instant of passage at the perihelion), the argument of the perihelion ω (or angular distance from ascending node to perihelion), and the longitude of the ascending node Ω. We also use the longitude of the perihelion defined by ω = ω + Ω.

The perturbation function R = δu 2 is a function of the orbital elements of the unperturbed Keplerian ellipse, say {c A } = {a, e, I, ℓ, ω, Ω}. The perturbation equations are generated by the partial derivatives of the perturbation function with respect to the orbital elements, namely ∂R/∂c A . We express the planet's absolute coordinates (x, y, z) (in some absolute Galilean frame) in terms of the orbital elements {a, e, I, ℓ, ω, Ω} by performing as usual three successive frame rotations with angles Ω, I and ω, to arrive at the frame (u, v, w) associated with the motion, where (u, v) is in the orbital plane, with u in the direction of the perihelion and v oriented in the sense of motion at perihelion. The unperturbed coordinates of the planet in this frame are

u = a (cos U -e) , ( 14a
) v = a 1 -e 2 sin U , (14b)
w = 0 , ( 14c
)
where U denotes the eccentric anomaly, related to ℓ by the Kepler equation ℓ = U -e sin U . The perturbation equations provide the variations of the orbital elements dc A /dt as linear combinations of the partial derivatives ∂R/∂c B of the perturbation function. We are interested only in secular effects, so we average in time the perturbation equations over one orbital period P . Denoting the time average by brackets, and transforming it to an average over the eccentric anomaly U , we have

dc A dt = 1 P P 0 dt dc A dt = 1 2π 2π 0 dU (1 -e cos U ) dc A dt . (15)
In the following, to simplify the presentation, we shall choose the x-direction of the absolute Galilean frame to be the direction of the galactic centre e = g e /g e . That is, we assume that the origin of the longitude of the ascending node Ω lies in the direction of the galactic centre. Furthermore, in order to make some estimate of the magnitude of the quadrupole effect, let us approximate the direction of the galactic centre (which is only 5.5 degrees off the plane of the ecliptic) as being located in the plane of the orbit; consequently we choose I = 0. In this case ω = ω + Ω is the relevant angle for the argument of the perihelion. We then find the following non-zero evolution equations:

de dt = 5Q 2 e √ 1 -e 2 4n sin(2ω) , (16a)
dℓ dt = n - Q 2 12n 7 + 3e 2 + 15(1 + e 2 ) cos(2ω) , (16b)
dω dt = Q 2 √ 1 -e 2 4n 1 + 5 cos(2ω) . (16c)
We recall that ω is the azimuthal angle between the direction of the perihelion and that of the galactic centre (approximated to lie in the orbital plane). Of particular interest is the secular precession of the perihelion dω/dt due to the quadrupole effect henceforth denoted by [START_REF] Fienga | Planetary and lunar ephemerides, inpop10a[END_REF] 0.4 ± 0.6 0.2 ± 1.5 -0.2 ± 0.9 -0.04 ± 0.15 -41 ± 42 0.15 ± 0.65 --

∆ 2 = Q 2 √ 1 -e 2 4n 1 + 5 cos(2ω) . (17)
The precession is non-spherical, in the sense that it depends on the orientation of the orbit relative to the galactic centre through its dependence upon the perihelion's longitude ω. The effect scales with the inverse of the orbital frequency n = 2π/P and therefore becomes more important for outer planets like Saturn than for inner planets like Mercury. This is in agreement with the fact that the quadrupole effect we are considering increases with the distance to the Sun (but of course will fall down when r becomes appreciably comparable to r 0 , see Fig. 1).

Our numerical values for the quadrupole anomalous precession ∆ 2 are reported in Table 2. As we see the quadrupolar precession ∆ 2 is in the range of the milli-arc-second per century which is not negligible. In particular it becomes interestingly large for the outer gaseous planets of the Solar System, essentially Saturn, Uranus and Neptune. The dependence on the choice of the MOND function µ is noticeable only for functions µ n (y) defined by (13a) with large values of n, where the effect decreases by a factor ∼ 10 between n = 2 and n = 20.

We then compare in Table 2 our results to the best published postfit residuals for any possible supplementary precession of planetary orbits (after the relativistic precession has been duly taken into account), which have been obtained from global fits of the Solar System dynamics [START_REF] Pitjeva | [END_REF][START_REF] Fienga | Gravity tests with inpop planetary ephemerides[END_REF][START_REF] Fienga | Planetary and lunar ephemerides, inpop10a[END_REF] . In particular the postfit residuals obtained by the INPOP planetary ephemerides [START_REF] Fienga | Gravity tests with inpop planetary ephemerides[END_REF][START_REF] Fienga | Planetary and lunar ephemerides, inpop10a[END_REF] use information from the combination of very accurate tracking data of spacecrafts orbiting different planets. We find that the values for ∆ 2 are smaller or much smaller than the published residuals except for the planets Mars and Saturn. Very interestingly, our values are smaller or grossly within the range of the postfit residuals for these planets. In the case of Saturn notably, the constraints seem already to exclude most of our obtained values for ∆ 2 , except for MOND functions of the type µ n and given by (13a) with rather large values of n.

However let us note that the INPOP ephemerides are used to detect the presence of an eventual abnormal precession, not to adjust precisely the value of that precession [START_REF] Fienga | Gravity tests with inpop planetary ephemerides[END_REF][START_REF] Fienga | Planetary and lunar ephemerides, inpop10a[END_REF] . On the other hand the postfit residuals are obtained by adding by hands an excess of precession for the planets and looking for the tolerance of the data on this excess [START_REF] Fienga | Gravity tests with inpop planetary ephemerides[END_REF][START_REF] Fienga | Planetary and lunar ephemerides, inpop10a[END_REF] . But in order to really test the anomalous quadrupolar precession rate ∆ 2 , one should consistently work in a MOND picture, i.e. consider also the other effects predicted by this theory, like the precession of the nodes, the variation of the eccentricity and the inclination, and so on -see Eqs. (16). Then one should perform a global fit of all these effects to the data; it is likely that in this way the quantitative conclusions would be different.

Finally let us cautiously remark that MOND and more sophisticated theories such as TeVeS 11 , which are intended to describe the weak field regime of gravity (below a 0 ), may not be extrapolated without modification to the strong field of the Solar System. For instance it has been argued 9 that a MOND interpolating function µ which performs well at fitting the rotation curves of galaxies is given by µ 1 defined by (13a). However this function has a rather slow transition to the Newtonian regime, given by µ 1 ∼ 1 -y -1 when y = g/a 0 → ∞, which is already excluded by Solar System observations. Indeed such slow fall-off -y -1 predicts a constant supplementary acceleration directed toward the Sun δg N = a 0 (i.e. a "Pioneer" effect), which is ruled out because not seen from the motion of planets. Thus it could be that the transition between MOND and the Newtonian regime is more complicated than what is modelled by Eq. ( 1). This is also true for the dipolar dark matter model [START_REF] Blanchet | Model of dark matter and dark energy based on gravitational polarization[END_REF][START_REF] Blanchet | Dipolar dark matter and dark energy[END_REF] which may only give an effective description valid in the weak field limit and cannot be extrapolated as it stands to the Solar System. While looking at MOND-like effects in the Solar System we should keep the previous proviso in mind. The potential conflict we find here with the Solar System dynamics (notably with the constraints on the orbital precession of Saturn [START_REF] Fienga | Gravity tests with inpop planetary ephemerides[END_REF][START_REF] Fienga | Planetary and lunar ephemerides, inpop10a[END_REF] ) may not necessarily invalidate those theories if they are not "fundamental" theories but rather "phenomenological" models only pertinent in a certain regime.

In any case, further studies are to be done if one wants to obtain more stringent conclusions about constraints imposed by Solar-system observations onto MOND-like theories. More precise observations could give valuable informations about an eventual EFE due to the MOND theory and restrict the number of possible MOND functions that are compatible with the observations. More generally the influence of the Galactic field on the Solar-system dynamics through a possible violation of the strong version of the equivalence principle (of which the EFE is a by-product in the case of MOND) is worth to be investigated.

Figure 1 :

 1 Figure 1: Left panel: profile of Q2(r) in the Solar System, for a standard choice of function µ1(y) [see Eq. (13a)], a0 = 1.2 × 10 -10 m.s -2 and ge = 1.9 × 10 -10 m.s -2 . The MOND transition radius is shown by a dash-dotted line at r0 ≃ 7100 AU. Right panel: zoom of the central region (r 50 AU), where the quadrupole is almost constant.

Table 2 :

 2 Results for the precession rates of planets ∆2 due to the quadrupole coefficient Q2. We use the values for Q2 for various MOND functions as computed in Table1. Published postfit residuals of orbital precession (after taking into account the relativistic precession). All results are given in milli-arc-seconds per century. ± 2 • 10 4 0 ± 2 • 10 4 Fienga et al.

			Quadrupolar precession rate ∆ 2 in mas/cy
	MOND function Mercury	Venus	Earth	Mars	Jupiter	Saturn	Uranus	Neptune
	µ 1 (y) µ 2 (y) µ 20 (y) µ exp (y) µ TeVeS (y)	0.04 0.02 2 × 10 -3 0.03 0.05	0.02 0.01 10 -3 0.02 0.02 Postfit residuals for ∆ = dω/dt in mas/cy 0.16 -0.16 -1.12 0.09 -0.09 -0.65 9 × 10 -3 -9 × 10 -3 -0.06 0.13 -0.13 -0.88 0.17 -0.17 -1.21	5.39 3.12 0.3 4.25 5.81		-10.14 -5.87 -0.56 -8.01 -10.94	7.93 4.59 0.44 6.26 8.56
	Origin	Mercury	Venus	Earth	Mars	Jupiter	Saturn		Uranus	Neptune
	Pitjeva 13 Fienga et al. 14 -10 ± 30 -3.6 ± 5 -0.4 ± 0.5 -0.2 ± 0.4 -4 ± 6 0 ± 0.016	0.1 ± 0.5 0 ± 0.2	-142 ± 156 -10 ± 8 0 -6 ± 2	-	-

d Our numerical scheme is based on the very efficient integrator of elliptic equations lorene, available from the website http://www.lorene.obspm.fr.

e Note that the quadrupole coefficient Q2 is found to be always positive which corresponds to a prolate elongation along the quadrupolar axis.