Thanh Phuong Nguyen
email: nguyentp@loria.fr

Isabelle Debled-Rennesson

Curvature and torsion estimators for 3D curves ⋆

. These estimators can naturally work with disconnected curves.

Introduction

Geometric properties of curves are important characteristics to be exploited in geometric processing. They directly lead to applications in machine vision [START_REF] Poyato | A method for dominant points detection and matching 2d object identification[END_REF] and computer graphics [START_REF] Lewiner | Curvature and torsion estimators based on parametric curve fitting[END_REF]. In the planar case, many applications are based on the curvature property in domains such as curve approximation [START_REF] Salmon | A new method to detect arcs and segments from curvature profiles[END_REF], geometry compression [START_REF] Lewiner | Curvature and torsion estimators based on parametric curve fitting[END_REF], and particularly in corner detection after the pioneer paper of Attneave [START_REF] Attneave | Some informational aspects of visual perception[END_REF]. In 3D space, torsion and curvature are the most important properties that permit to describe how a spatial curve bends. Several methods have been proposed for torsion estimation. Mokhtarian [START_REF] Mokhtarian | A theory of multiscale, torsion-based shape representation for space curves[END_REF] used Gaussian smoothing to estimate it directly from torsion formula. Similarly, Kehtarnavaz et. al. [START_REF] Kehtarnavaz | A 3-d contour segmentation scheme based on curvature and torsion[END_REF] used B-spline smoothing technique; Lewiner et al. [START_REF] Lewiner | Curvature and torsion estimators based on parametric curve fitting[END_REF] utilized weighted least-squares fitting techniques. Raluben Medina et al. [START_REF] Medina | Curvature and torsion estimation for coronary-artery motion analysis[END_REF] proposed two methods to estimate torsion and curvature values at each point of the curve. The first one utilized Fourier transform, the second one is based on the least squares fitting. These methods are applied for description of arteries in medical imaging. We propose in this paper a novel method for the estimation of local geometric parameters of a spatial curve. It uses a geometric approach and relies on results of discrete geometry on decomposition of a curve into maximal blurred segments [START_REF] Feschet | Optimal time computation of the tangent of a discrete curve: Application to the curvature[END_REF][START_REF] Debled-Rennesson | Optimal blurred segments decomposition of noisy shapes in linear time[END_REF][START_REF] Nguyen | Curvature estimation in noisy curves[END_REF]. This paper presents an extension to 3D of these results. The 3D curvature estimator given in [START_REF] Coeurjolly | Estimation of curvature along curves with application to fibres in 3d images of paper[END_REF] is extended here with the notion of blurred segment and it permits to study curves possibly noisy or disconnected. We also propose a new approach to the discrete torsion estimation. We recall, in the Section II, 2D definitions and results [START_REF] Nguyen | Curvature estimation in noisy curves[END_REF] that we use. Section III presents how to extend these ideas into 3D space. Sections IV and V respectively propose a curvature and a torsion estimator. Last sections give experiments and conclusions.

Maximal 2D blurred segment of width ν

The notion of blurred segments relies on the arithmetical definition of discrete lines [START_REF] Reveillès | Géométrie discrète, calculs en nombre entiers et algorithmique[END_REF] where a line, with slope a b , is the set of integer points (x, y) verifying µ ≤ axby < µ + ω (a, b, µ and ω being integer and gcd(a, b) = 1). Such a line is denoted by D(a, b, µ, ω). The notion of 2D blurred segment extends the notion of segment of a discrete line and permits more flexibility in operations such as recognition, segmentation of discrete curves. Let us recall definitions [START_REF] Debled-Rennesson | Optimal blurred segments decomposition of noisy shapes in linear time[END_REF][START_REF] Nguyen | Curvature estimation in noisy curves[END_REF] that we use in this paper (see also the Fig. 1). Let C be a discrete curve and C i,j a sequence of points of C indexing from i to j. Suppose that the predicate "C i,j is a blurred segment of width ν" is denoted by BS(i, j, ν). Definition 2. C i,j is called a maximal blurred segment of width ν and noted M BS(i, j, ν) iff BS(i, j, ν), ¬BS(i, j + 1, ν) and ¬BS(i -1, j, ν) (see the In [START_REF] Nguyen | Curvature estimation in noisy curves[END_REF] an algorithm is proposed to decompose a planar curve into maximal blurred segments for a given width and the theorem below is proved. The algorithm relies on operations of insertion (or deletion) of a point to (or from) the convex hull of the current studied segment. We have proven that the decomposition of a planar curve with n points into maximal blurred segments of width ν can be done in time O(n log 2 n).

3D blurred segment of width ν

The notion of 3D discrete line (see the references [START_REF] Debled-Rennesson | Reconnaissance des droites et plans discrets[END_REF][START_REF] Coeurjolly | Segmentation and length estimation of 3d discrete curves[END_REF]) is defined as follows:

Definition 3. A 3D discrete line [43], denoted D 3D (a, b, c, µ, µ ′ , e, e ′
), with a main vector (a, b, c) such that (a, b, c) ∈ Z Z 3 , and a ≥ b ≥ c is defined as the set of points (x, y, z) from Z Z 3 verifying:

D µ ≤ cx -az < µ + e (1) µ ′ ≤ bx -ay < µ ′ + e ′ (2)
with µ, µ ′ , e, e ′ ∈ Z Z. e and e ′ are called arithmetical width of D.

According to the definition, it is obvious that a 3D discrete line is bijectively projected into two projection planes as two 2D arithmetical discrete lines. Thanks to that property, we naturally define the notion of 3D blurred segment by using the notion of 2D blurred segment and by considering the projections of the sequence of studied points in the coordinate planes (see the Fig. 2.a). Definition 4. Let Sf 3D be a sequence of points of Z Z3, Sf 3D is a 3D blurred segment of width ν with a main vector (a, b, c) such that (a, b, c) ∈ Z Z 3 , and a ≥ b ≥ c if it possesses a said optimal discrete line, named D 3D (a, b, c, µ, µ ′ , e, e ′ ), such that -D(a, b, µ ′ , e ′ ) is optimal for the sequence of projections of points of Sf 3D in the plane (O, x, y) and A linear algorithm of 3D blurred segment recognition may be deduced from that definition. Indeed, we only need to use an algorithm of 2D blurred segment recognition in each projection plane.

Maximal 3D blurred segment of width ν

In this section, we present an algorithm to obtain the sequence of 3D maximal blurred segments of width ν in time O(n log 2 n) for any noisy 3D discrete curve C. This sequence is noted

M BS ν (C) = {M BS i (B i , E i , ν)} i=0 to m-1 with B i (resp. E i ) the index
of the first (resp. last) point of the ith maximal blurred segment, M BS i , of C. This algorithm uses an algorithm to determine the 2D maximal blurred segment (see [START_REF] Nguyen | Curvature estimation in noisy curves[END_REF]) of the projections in the coordinate planes of the points of the studied curve. (*) To determine the optimal discrete line of the current 3D blurred segment S b , we consider the characteristic of the two 2D blurred segments obtained in the planes of projection and combine them to obtain the characteristics of the optimal 3D discrete line of S b . As the whole process is done in dimension 2, this algorithm has the same complexity as the one in dimension 2. So, we have the following result : The decomposition of a 3D curve into maximal blurred segments of width ν can be done in time O(nlog 2 n).

3D discrete curvature of width ν

In this section, we're interested in curvature estimation based on osculating circle. Deducing from [START_REF] Nguyen | Curvature estimation in noisy curves[END_REF], we present below the notion of discrete curvature of width ν at each point of a 3D curve (see the Fig. 2.b). This method extends the one proposed in [START_REF] Coeurjolly | Estimation of curvature along curves with application to fibres in 3d images of paper[END_REF] to the blurred segments and is adapted to noisy curves thanks to the width parameter. Let C be a 3D discrete curve, C k is a point of the curve. Let us consider the points C l and C r of C such that : l < k < r, BS(l, k, ν) and ¬BS(l -1, k, ν), BS(k, r, ν) and ¬BS(k, r + 1, ν). The points C l and C r for a given point C k of C are deduced from the sequence of maximal blurred segments of width ν of C. The estimation of the 3D curvature of width ν at the point C k is determined thanks to the radius of the circle passing through the points C l , C k and C r . To determine the radius R ν (C k ) of the circumcircle of the triangle [C l , C k , C r ], we use the formula given in [START_REF] Harris | Handbook of mathematics and computational science[END_REF] as follows:

Let

s 1 = || ---→ C k C r ||, s 2 = || ---→ C k C l || and s 3 = || ---→ C l C r ||, then R ν (C k ) = s 1 s 2 s 3 (s 1 + s 2 + s 3 )(s 1 -s 2 + s 3 )(s 1 + s 2 -s 3 )(s 2 + s 3 -s 1 )
Then, the curvature of width ν at the point

C k is C ν (C k ) = s Rν (C k ) with s = sign(det( ---→ C k C r , ---→ C k C l )) (it

indicates concavities and convexities of the curve).

Thanks to the sequence of maximal blurred segments of width ν (M BS ν ) of a 3D curve C, an algorithm for curvature estimation at each point of a 3D curve can be directly deduced from [START_REF] Nguyen | Curvature estimation in noisy curves[END_REF].

m = |M BSν |; E-1 = -1; Bm = n; for i = 0 to m -1 do for k = Ei-1 + 1 to Ei do L(k) = Bi; for k = Bi to Bi+1 -1 do R(k) = Ei; end for i = 0 ( * ) to n -1 ( * ) do Rν (Ci) = Radius of the circumcircle to [C L(i) , Ci, C R(i) ]; Cν(Ci) = s Rν (C i ) ; end end (*)
The bounds mentioned in the algorithm 2 are correct for a closed curve. In case of an open curve, the instruction becomes: for i = l to n -1 -l with l fixed to a constant value.

Discrete torsion of width ν

Preliminary

The 3D curvature is not sufficient to characterize the local property of a 3D curve. This parameter only measures how rapidly the direction of the curve changes. In case Fig. 2. From left to right: a. D3D(45, 27, 20, -45, -81, 90, 90) optimal discrete line of the grey points, b. The curvature at the red point is defined as the inverse of circumcircle (passing through both blue points and the red point) radius. of a planar curve, the osculating plane does not change. For 3D curves, torsion is a parameter that measures how rapidly the osculating plane changes. To clarify this notion, we recall below some definitions and results in differential geometry (see [START_REF] Oprea | Differential geometry and its applications[END_REF] for more details). Definition 5. Let r : I → R 3 be a regular unit speed curve parameterized by t.

i T (t) (resp. N (t)) a unit vector in direction r

′ (t) (resp. r ′′ (t)). So, N (t) is a normal vector to T (t). T (t) (resp. N (t)) is called the unit tangent vector (resp. normal vector) t. ii |T ′ (t)| = k(t) is called the curvature of r at t.
iii The plane determined by the unit tangent and normal vectors (T (t) and N (t)), is called the osculating plane at t. The unit vector B(t) = T (t) ∧ N (t) is normal to the osculating plane and is called the binormal vector at t. iv |B ′ (t)| = τ (t) is called the torsion of curve at t. Thanks to theorem 3, the torsion value at a point is 0 if the curvature value at this point is 0.

Discrete torsion

Discrete torsion was studied in [START_REF] Lewiner | Curvature and torsion estimators based on parametric curve fitting[END_REF][START_REF] Mokhtarian | A theory of multiscale, torsion-based shape representation for space curves[END_REF][START_REF] Medina | Curvature and torsion estimation for coronary-artery motion analysis[END_REF][START_REF] Kehtarnavaz | A 3-d contour segmentation scheme based on curvature and torsion[END_REF]. In this section, we propose a new geometric approach for the problem of torsion estimation that uses the definitions and results presented in the previous sections.

Definitions

Let ζ be a 3D discrete curve, C k is k th point of the curve. Let us consider the points C l and Cr of ζ such that : l < k < r, BS(l, k, ν)&¬BS(l -1, k, ν) and BS(k, r, ν)&¬BS(k, r + 1, ν). Let's recall that the curvature of width ν (see section The osculating plane (C l , C k , Cr) has two unit tangent vectors :

- → t1 = ---→ C l C k | ---→ C k Cr | and - → t2 = ---→ C k Cr | ---→ C k Cr |
. Therefore, we have the binormal vector at the k th point:

- . The traditional method for derivation estimation of discrete sequence is utilizing Gaussian kernel [START_REF] Worring | Digital curvature estimation[END_REF]. We propose below a geometric approach method to this problem. Let us consider the curve ζ1 = {P } n i=0 that is constructed by this rule:

→ b k = - → t1 ∧ - → t2
----→ PiPi+1 = -→ bi , i = 0, .., n -1 (see the Fig. 3).

- → b3 - → b1 P0 P1 - → b4 - → b5 - → b5 P3 P4 P5 - → b0 - → b0 - → b1 - → b2 - → b3 - → b4 P2 P6 - → b2
Fig. 3. The curve ζ1 is constructed from the sequence of binormal vector Proposition 1 The tangent vector at each point Pi of the curve ζ1 is -→ bi (i = 0, .., n -1).

Proof. In differential geometry, the tangent vector of a curve r(t) at the point Pt 0 = r(t0) is defined as: t(t0) = r ′ (t0) = lim h→0 r(t 0 +h)-r(t 0 ) h

= lim h→0

---→ Pt 0 P h . Therefore, in discrete space, the tangent vector at the point Pi = α(i) can be estimated as t(i) =

Definition 1 .

 1 Let S b be a sequence of digital points. A discrete line D(a, b, µ, ω) is said optimal for S b if each point of S b belongs to D and if its vertical width, ω-1 max(|a|,|b|) , is equal to the vertical width of the convex hull of S b (see the Fig. 1.a). S b is a blurred segment of width ν iff there exists an optimal discrete line D(a, b, µ, ω) of S b such that ω-1 max(|a|,|b|) ≤ ν.

Fig. 1 .

 1 b).

Fig. 1 .

 1 Fig. 1. From left to right: a. D(5, 8, -8, 11) (blue and grey points) is the optimal discrete line of the sequence of grey points, b. the set of black points is a maximal blurred segment (MBS) of width 2.

e ′ - 1

 1 max(|a|,|b|) ≤ ν, -D(a, c, µ, e) is optimal for the sequence of projections of points of Sf 3D in the plane (O, x, z) and e-1 max(|a|,|c|) ≤ ν.

Algorithm 1 :

 1 Algorithm for the segmentation of a curve C into maximal 3D blurred segments of width ν Data: C -discrete curve with n points, ν -width of the segmentation Result: M BSν -the sequence of maximal blurred segments of width ν of C begin k=0; S b = {C0}; M BSν = ∅; a = 0; b = 1; ω = b, µ = 0; while the widths of 2 blurred segments obtained by projecting the points of S b in the coordinate planes are ≤ ν do k++; S b = S b ∪ C k ; Determine D3D(a, b, c, µ, µ ′ , e, e ′ ), optimal discrete line of S b ; (*) end bSegment=0; eSegment=k-1 ; M BSν = M BSν ∪ M BS(bSegment, eSegment, ν); while k < n -1 do while the widths of 2 blurred segments obtained by projecting the points of S b in the coordinate planes are > ν do S b = S b \ C bSegment ; bSegment++ ; Determine D3D(a, b, c, µ, µ ′ , e, e ′ ), optimal discrete line of S b ; (*) end while the widths of 2 blurred segments obtained by projecting the points of S b in the coordinate planes are ≤ ν do k++ ; S b = S b ∪ C k ; Determine D3D(a, b, c, µ, µ ′ , e, e ′ ), optimal discrete line of S b ; (*) end eSegment=k-1; M BSν = M BSν ∪ M BS(bSegment, eSegment, ν); end end

Algorithm 2 :

 2 Width ν curvature estimation at each point of ζ Data: C 3D discrete curve of n points, ν width of the segmentation Result: {Cν (C k )} k=0..n-1 -Curvature of width ν at each point of C begin Build M BSν = {M BSi(Bi, Ei, ν)}i=0 to m-1 (See the Algo. 1 );

Theorem 1 i| 3 ii

 13 Let r : I → R 3 be a spatial curve parameterized by t. The curvature of r at t ∈ I: k(t) = |r ′ ∧r ′′ | |r ′ The torsion of r at t ∈ I: τ (t) = (r ′ ∧r ′′ ).r ′′′ |r ′ ∧r ′′ | 2

4 .Definition 6 .

 46 ) is estimated by circumcircle of triangle △C l C k Cr. If ---→ C k C l and ---→ C k Cr are colinear, the curvature value at C k is 0, therefore the torsion value at C k is 0. So, without loss of generality, we suppose that ---→ C l C k and ---→ C k Cr are not colinear. In addition, the plane defined by ---→ C l C k and ---→ C k Cr is noted (C l , C k , Cr), and we propose the definition below. The osculating plane of width ν at C k is the plane (C l , C k , Cr).

Definition 7 .

 7 = (bx, by, bz) So, we propose the following definition of discrete torsion of width ν. The discrete torsion of width ν at C k is the derivation of -→ b k Torsion estimator Our proposed method for torsion estimation is based on the definition 7. Let us remark that the set { -→ b k } n-1 k=0 can be constructed from the set of maximal blurred segments in O(n log 2 n) time. So, we can obtain torsion value by calculating the derivative at each position of { -→ b k } n-1 k=0

Proposition 2

 2 The torsion value at each point of curve ζ corresponds to curvature value of ζ1 curve.Proof. Thanks to definition 8, the discrete torsion at C k of ζ curve is the derivation of -→ b k . In addition, -→ b k is the tangent vector at the k th point of ζ1 curve. So,this value is also curvature value at the k th point of ζ1 curve.

Therefore, by using these two propositions, we can estimate torsion value at each point of ζ curve by determining curvature value at corresponding point of ζ1 curve. Our proposed method is presented in the algorithm 3, it uses the curvature estimator presented in section 3.3. (*) Same remark as for the Algorithm 2.

⋆ This work is supported by ANR in the framework of the GEODIB project, BLAN 06 -2 134999

We introduce some experiments of our method on some ideal 3D curves : helix, Viviani's, spheric, horopter and hyper helix curves. The tests are done after a process of discretisation of these 3D curves (see the Fig. 6). We have tested our methods on this computer configuration: CPU Pentium 4 with 3.2GHz of speed, 1Gb of memory RAM, linux kernel 2.6.22-14 operating system. We introduce three criteria for measuring error: mean relative error (meanRE), max relative error (maxRE) and quadratic relative error (QRE). Let's consider 2 sequences: {IRi} n i=1 (resp. {RRi} n i=1 ) the ideal result (resp. estimated result) at each position. So, we have:

Because the estimated result is not correct for the beginning and the end of the open curve (see the bounds mentioned in the algorithms 2 and 3), during the phase of error estimation we use the border parameter to eliminate this influence.

In most cases of the studied curves (see the Table 1 and Fig. 6), the mean relative errors do not overtake 0.15, and the quadratic relative errors do not overtake 0.015. If the ideal torsion of the input curve has a value which is close to 0 at some positions, the obtained result is not very good. Let's see the case of Viviani's curve in the Table 1. In this case, the maximal relative error is high (15.6036). In spite of that, the mean relative error is acceptable (0.628899). In particular cases, if most of input curves has a torsion value which is close to 0, the obtained result is the worst (see the Fig. 4). Let's consider the case of an hyper helix curve (see the Fig. 4). The problem is that the torsion approximation is not good at nearly-0 values. In spite of that, the approximation value is also close to 0 but relative rate between approximation value and ideal value is very high. In Fig. 5, we show the relation between approximation torsion and ideal torsion of an hyper helix curve from the index 15 and to the index 250. In this index interval, the ideal torsion is close to 0. So, the relative error between approximation torsion and ideal torsion is very high, in spite of that the approximation value does not overtake 0.006. So, we propose to consider only the points whose ideal torsion value is greater than the threshold. In the error calculus, n is replaced by number of points whose ideal torsion value is greater than a threshold. The table 2 shows the approximated error with a threshold equal to 0.0005.

Conclusions

We have presented in this paper 2 methods to estimate curvature and torsion of a 3D curve. These methods benefit from the amelioration of curvature estimator in planar case [START_REF] Nguyen | Curvature estimation in noisy curves[END_REF], so they're efficient. These estimators permit to discover local properties of spatial curve. We hope to identify and classify 3D objects by using these estimators. In the future, we will compare our methods with other existing methods [START_REF] Mokhtarian | A theory of multiscale, torsion-based shape representation for space curves[END_REF][START_REF] Coeurjolly | Estimation of curvature along curves with application to fibres in 3d images of paper[END_REF][START_REF] Poyato | A method for dominant points detection and matching 2d object identification[END_REF][START_REF] Lewiner | Curvature and torsion estimators based on parametric curve fitting[END_REF] for curvature and torsion estimation. Morover, we will work with real discrete data of biology or medical area. We intend to present these works in a journal version.

Curves No of points


Gaïa Lépine

François Mariotti

Jean-François Huneau

Nathalie Poupin

Marie Tremblay-Franco

Marie-Anne Verny

Nicolas Macian

Marion Courrent

Laetitia Guerin-Deremaux

Catherine Lefranc-Millot

Study Protocol: A 2-Month Cross-Over Controlled Feeding Trial Investigating the Effect of Animal and Plant Protein Intake on the Metabolome and Cardiometabolic Health

come

Context:

▪ A dietary shift from animal protein (AP) to plant protein (PP) advocated for sustainability and health reasons (lower cardiovascular and diabetes risk) ▪ Mechanistic bases underlying these associations still poorly characterized and involving numerous metabolic pathways.

AP-rich and PP-rich diets

▪ AP-rich diet → western omnivore diet (65% AP:35% PV) ▪ PP-rich diet → flexitarian diet including small amounts of meat and dairy (36% AP:64% PV) ▪ Both diets mixed all the major animal and plant sources, without any exclusion

Assessing metabolic flexibility following a high-fat meal

Postprandial follow-up after a high-fat challenge (identical in each crossover arm) allows to measure adaptation capacity under challenging postprandial conditions (high influx of saturated fatty acids and sugar) and is expected to reveal subtle differences induced by the dietary intervention which could not be visible at the fasted state, when tight homeostatic processes still prevail.

Challenge meal composition: 900kcal (80% lipid, 13% carbohydrate, 7% protein) Cream 30% fat (267g) + white sugar (20g) + whole milk protein isolate (11,5g)

Effects on cardiometabolic health

What is new about this study?

Overall health outcomes

Multi-omics phenotyping

The combination of omics approaches, providing a multilevel phenotyping, will give the possibility to characterize very subtle metabolic reorientations induced by dietary changes.

Study design: 2-month cross-over randomized trial

  Very little is known regarding the metabolomics signature associated to AP and PP intakes in humans (see poster P11-019-21) and up to day, no RCT has specifically investigated the effects of the AP→PP dietary shift using a combination of "omics" approaches. This study will bring new hypotheses regarding the metabolic pathways affected by AP and PP-rich diets and associated biomarkers and will characterize with high-end methods the effects on endothelial function. Study protocol: a 2-month cross-over controlled feeding trial investigating the effect of animal and plant protein intake on the metabolome and cardiometabolic health Gaïa Lépine 1,2 (presenting author, gaia.lepine@inrae.fr), François Mariotti 2 , Jean-François Huneau 2 , Nathalie Poupin 3 , Marie Tremblay-Franco 3,4 , Marie-Anne Verny 1 , Nicolas Macian 5 , Marion Courrent 5 , Laetitia Guerin-Deremaux 6 , Catherine Lefranc-Millot 7 , Didier Remond 1 , Hélène Fouillet 2 and Sergio Polakof 1 1: Université Clermont Auvergne, INRAE, UMR1019, Unité Nutrition Humaine, 63000, Clermont-Ferrand France; 2: Université Paris-Saclay, AgroParisTech, UMR PNCA, 75005, Paris, France; 3: Toxalim (Research Centre in Food Toxicology), Toulouse University, INRAE, ENVT, INP-Purpan, UPS, Toulouse, France; 4: Metatoul-AXIOM platform, MetaboHUB, Toxalim, INRAE, Toulouse, France; 5: University Hospital Clermont-Ferrand, Inserm CIC 1405, Clinical Investigation Center, 63000 Clermont-Ferrand, France; 6: Roquette Frères, rue de la haute loge, 62080, Lestrem, France; 7: Roquette Frères, 101 avenue de la République, La Madeleine, 59110, France

▪▪Figure 1 :

 1 Figure 1: Mean daily macronutrients contribution to total energy intake provided by ready-to-eat meals

Table 1 :

 1 Description of the sample already included

▪ Body composition measured by impedancemetry ▪ Glucose homeostasis, lipid and protein metabolism assessed by clinical biochemistry ▪ Inflammation status (cytokines)

Table 2 :

 2 Mean

dietary composition of ready-to-eat meals provided in AP and PP-rich diets


Keywords: Metabolomics, transcriptomics, fluxomics, animal protein, plant protein, cardiometabolic risk

est destinée au dépôt

Multi-omics phenotyping highlights organ-specific metabolic and inflammatory shifts associated with differential plant and animal protein intake in high fat fed rats

Gaïa Lépine, Hélène Fouillet, Marie Tremblay-Franco, Didier Remond, Nathalie Poupin, Veronique V. Mathe, Jérémie David, Dominique Hermier, Laetitia Guérin-Deremaux, Catherine Lefranc-Millot, et al.

To cite this version:

Gaïa

Introduction:

The dietary transition from less animal protein (AP) to more plant protein (PV) sources has been advocated mostly for sustainability reasons but has also been associated to beneficial health effects such as decreased cardiometabolic risk. AP and PP differential amino acid (AA) compositions could contribute to explain those health effects. With this study, we aimed at characterizing metabolic and inflammatory adaptations associated to AP and PP intake in the context of plethoric diets, which currently remains mostly unknown.

Methods: 44 Wistar male rats were fed with 4 iso-protein diets, where protein was provided by either a milk protein isolate (AP) or a blend of 50% pea protein isolate-50% wheat protein (PP), in a regular normal fat (NF) or a high-fat (HF) context. Plasma, liver, intestine, skeletal muscle and adipose tissue samples were collected after 4 months. Tissue targeted transcriptomics (TLDA) and fluxomics (D2O tracer) and plasma untargeted metabolomics (LC-MS) were used for a multi-omic phenotyping. Clinical biomarkers and natural 15 N enrichment (EA-IRMS) were also measured. Both multivariate (AMO-PLS) and univariate (ANOVA) statistical analyses were conducted.

Results: Distinct plasma metabolomics signatures were related to the energy level (HF vs NF), the protein type (AP vs PP) and their interactions, explaining respectively 12.7%, 11.5% and 4.7% of total variability. Natural 15 N enrichment analysis revealed a higher AA reshuffle associated to PP in muscle, intestine and liver, which was further confirmed by increased mRNA levels related to AA and protein metabolism (BCAA oxidation, AA transport, myogenesis and muscle atrophy) in liver and muscle. At the hepatic level, AP diets were associated to a higher fat content and increased lipogenesis, assessed by fluxomics and enzyme activities. mRNA results were more heterogenous, in particular in the adipose tissue, where a trend for increased lipogenesis and cholesterol synthesis-related targets was observed.

Regarding glucose homeostasis, the PP diets resulted in an improved whole-body insulin sensitivity (HOMA-IR), only in a HF context, and an up-regulation of insulin-related genes in the muscle at both energy levels (HF and NF). Several inflammation-related transcripts were also modified by the protein source, especially in the muscle and the liver, but the direction of the association remained unclear.

Discussion and conclusion: While the alterations observed at the adipose tissue level and some inflammatory markers (hepatic Tnfa) would deserve further attention, overall, our multi-omics approach highlighted several beneficial effects of PP intake: a reduced hepatic lipid accumulation through down-regulated lipogenesis, better whole-body insulin sensitivity associated with more

  Lépine, Hélène Fouillet, Marie Tremblay-Franco, Didier Remond, Nathalie Poupin, et al.. Multiomics phenotyping highlights organ-specific metabolic and inflammatory shifts associated with differential plant and animal protein intake in high fat fed rats. eNuGOweek 2021, Sep 2021, Online, France. hal-03338806

  Multi-omics phenotyping highlights organ-specific metabolic and inflammatory shifts associated with differential plant and animal protein intake in high fat fed rats Authors: Gaïa Lépine 1,2 , Hélène Fouillet 2 , Marie Tremblay-Franco 3,4 , Didier Rémond 1 , Nathalie Poupin 3 , Véronique Mathé 2 , Jérémie David 1 , Dominique Hermier 2 , Laetitia Guérin-Deremaux 5 , Catherine Lefranc-Millot 6 , Jean-François Huneau 2 , François Mariotti 2 , Sergio Polakof 1

	Affiliations :
	1 : Université Clermont Auvergne, INRAE, UMR 1019, Unité Nutrition Humaine, 63000, Clermont-Ferrand,
	France
	2 : Université Paris-Saclay, AgroParisTech, INRAE, UMR PNCA, 75005, Paris, France
	3 : Toxalim (Research Centre in Food Toxicology), Toulouse University, INRAE, ENVT, INP-Purpan, UPS,
	Toulouse, France
	4 : Metatoul-AXIOM platform, MetaboHUB, Toxalim, INRAE, Toulouse, France ;
	5 : Roquette Frères, Rue de la haute loge, 62080, Lestrem, France
	6 : Roquette Frères, 101 avenue de la République, 56564, La Madeleine, France

Title:


Diane Robert
email: diane-robert@live.fr

John Crisp

Protest movements against industry-related environmental burdens and territorial injustice in Gabès and Kerkennah (Tunisia)

Keywords: Tunisia, mobilisation, industrial pollution, environmental justice, community Résumé Tunisie, mobilisation, pollution industrielle, justice environnementale, communauté

The mobilisations about the environmental impacts of industries, which have increased in post-2011 Tunisia, are infused with conceptions of justice and injustice. They question the production of industrial space. This article offers a reflection in situ on the part played by community loyalties in mobilisation dynamics and the production of (in)justice. These loyalties can be levers for mobilisation. First, the community is the level at which industrial burden are felt; second, they form a mesh of social connections on which mobilisations are based; third, shared collective identities favour the cohesion of groups who implement collective action. However, they can also nurture fragmentation logics and be used by power devices to play off protesting groups against each other. Furthermore, the new arrangements that respond to mobilisations can in turn be disputed because of the new injustice they trigger. The production of fairer space actually depends on the ability for mobilisations to reshape power relations and on the ways state and industry powers respond to them.

Introduction

The protests that followed the revolutionary uprisings in Tunisia in 2010-2011 revealed problems that had remained unspoken under the authoritarian regime of Zine el-Abidine Ben Ali. The fact that the focal points of the protest had been regions within the country led to the "(re)discovery of the marginalisation and exclusion" of these regions (Hibou, 2015, p. 99), lifted the veil on a "social and spatial asymmetry" that seemed "constitutive of the state in Tunisia" and was "as much the outcome of a centuries-long process as of the protectorate and of capitalism" (Hibou, 2015, p. 148). However, one should be careful not to draw over-simplistic contrasts between the interior and coastal regions, since the latter are not homogeneous: sociospatial fractures also exist at other levels, for example between the big cities and their hinterlands, or between different districts of a single city [START_REF] Abdelkarim | La révolution tunisienne de janvier 2011 : une lecture par les déséquilibres du territoire[END_REF]. In addition, the national demands were accompanied by claims that were specific to the regions where they emerged. In certain areas, their particular focus was on environmental issues around access to resources and living conditions, once again raising the question of regional segregation.

The occupation and picketing of landfill sites highlighted their terrible sanitary and environmental conditions, their impact on local neighbourhoods and the poor management of waste at institutional level [START_REF] Chiara | Local mobilisations and the formation of environmental networks in a democratizing Tunisia[END_REF]. The protests around water management structures in rural areas showed the inequalities in access to water [START_REF] Gana | Tunisie : les oubliés de la révolution. Protestations et conflits dans le monde agricole[END_REF]. Every year, there are multiple sit-ins and road blockades in protest against water supply cuts, publicising water management policies that negatively affect certain regions. In addition, public land has been occupied on grounds of historical legitimacy [START_REF] Gana | Mobilisations foncières en Tunisie : révélateur des paradoxes de l'après "révolution[END_REF], and there are demonstrations calling for the closure of industrial sites perceived as dangerous or for improvements to environmental practices.

These local protests tackle the environmental issue from perspectives that include social and territorial questions, demanding a change from the top-down approaches promoted by the state and certain NGOs. 1 The complaints about environmental hazards that affected their homes are combined with demands for respect and a condemnation of injustices experienced both collectively and geographically. 2 These protests relate to "conflicts of places" (Dechézelles and Olive, 2019) that bring into play and contribute to the redefinition of collective identities.

In this article, we look at two areas situated around the Gulf of Gabès, on the eastern part of the Tunisian coast, which were both arenas of protest against bad environmental practices by industrial complexes after 2011 (see figure 1). The first, Gabès, is the capital city of the governorate of the same name. A big port in southern Tunisia with a population of 150,000, it is host to a large industrial complex for the processing of phosphates that are extracted from the Gafsa mining basin and transported to Gabès by rail. The second, Kerkennah, is a group of islands off the city of Sfax and linked to it by a sea shuttle, with a permanent population of 16,000 people, which increases tenfold in summer. It has underground reserves of gas and oil that are exploited by two companies: TPS, which extracts oil through offshore platforms close to the north-eastern edge of the island group; and Perenco (which replaced Petrofac in 2018) which drills for gas on Chergui Island. The industrial activities were set up there at different times and are run by different combinations of public and private actors. In Gabès, there has been a chemicals industry since 1972. It is the product of a public policy of industrialisation fed by a modernising ideology and promoted as a pathway to decolonisation [START_REF] Pierre | Industrialisation, urbanisation et mutations de l'espace tunisien[END_REF], and is structured today around a state enterprise, the Tunisian Chemicals Group (GCT), which converts phosphates extracted from the mining basin into phosphoric acid and fertiliser. A few private companies have attached themselves to the complex. In Kerkennah, oil and gas drilling dates back to the 1990s. It is conducted by private multinational firms but the public company Tunisian Oil Activity Enterprise (ETAP) holds a half share in the operating permits, which provides a source of revenue for the government.

What these two areas have in common is the coexistence of industry with other activities, notably in the primary sector, as well as high levels of unemployment, particularly among young people. For a long time, the economy of Gabès was reliant on farming, fishing and its status as a hub. It has an unemployment rate of 25.4%, compared with 15.4% for the country as a whole. 3 In Kerkennah, most of the inhabitants live from fishing. Its population is ageing because a large percentage of the youngest people move to the mainland to study and find work.

Protests against the environmental damage caused by industrial activities in Gabès and Kerkennah existed under the authoritarian regime of Zine el-Abidine Ben Ali, although limited to certain sectors-fishing, farming-or to circumscribed areas. However, the fall of the regime in January 2011 opened up significant political opportunities for many protest groups, resulting in a broadening of the protest base, an intensification of collective action and the adoption of new modes of action.

In the Gabès region, occasional local protests about the pollution produced by the phosphate industry in the early months of 2011, combined with the emergence of citywide protests in subsequent years, launched a cycle of conflict. In Kerkennah, demands for job creation by unemployed youngsters and fishermen's protests against oil spills came together in sequences of activism of varying intensity interspersed with periods of calm.

These local actions raise questions about the distribution of pollution and profits from industrial activity, and drew on shared and territorialised representations of injustice and collective identities at multiple and overlapping scales. They are profoundly embedded in local contexts, founded on day-to-day practices. They rely on a variety of entangled organisational forms involving civil society and union structures, but also collectives rooted in community affiliations and acquaintanceship networks linked with day-to-day social relations. We therefore adopt "the hypothesis of a process of hybridisation between so-called 'community' ties and so-called 'citizenship' ties" within protest movements in the Mediterranean countries (Ben Néfissa, 2011, p. 12). 4 As used here, these notions do not refer to a moment of community that preceded a more evolved civil society or a moment of citizenship in the history of the organisations [START_REF] Jacquier | Qu'est-ce qu'une communauté ? En quoi cette notion peut-elle être utile aujourd'hui ?[END_REF]. They refer to a density of social ties at local level, and the attachment to place, which constitute both challenges and resources for protest movements. The definition of community we adopt here is Jean-François Médard's: "simultaneously a place, people living in that place, the interaction between those people, the feelings that arise from that interaction, the common life they share and the institutions that govern that life" (Médard, 1969, p. 18).

3. Data from the National Statistics Institute for the second quarter of 2018. 4. It should be noted that this hybridity is not new in Tunisia, where solidarity based on origin or religious belief played a role in the establishment of trade unionism [START_REF] Hamida | Syndicats, affinités et solidarités de type traditionnel dans la Tunisie coloniale (1924-1956)[END_REF].

By contrast with an approach that immediately postulates the "community logic against justice" (Lévy, Fauchille and Póvoas, 2018, p. 254), we are interested in how communities have taken up questions of justice in protesting against the environmental burdens they experience, and how their actions, as well as the reactions they generate, contribute to the reshaping of territorial disparities. The aim of this article is thus less to examine these issues in the light of theories that focus on the individual and lay claim to universality [START_REF] John | A Theory of Justice[END_REF], than to accept Iris Marion Young's invitation to explore justice and injustice in terms of particular social contexts and in relation to processes that generate inequalities [START_REF] Iris | Justice and the Politics of Difference[END_REF].

In the first part, we will see that the local level is the scale at which the burdens are perceived, and that the protests about them draw in large part on proximity networks. In the second section, we will examine how the denunciation of environmental damage is anchored in protest against regional segregation and is fed by narratives of dispossession and identity claims. Finally, we will look at how community membership is exploited in the responses to the protests, with the result that the movements tend to break down at microlocal level.

The article is based on field research conducted between 2017 and 2019 in Gabès and Kerkennah for a PhD thesis in geography. The data used in the article were collected through interviews (mainly with people involved in the protests, but also, to a lesser degree, officials in government agencies and businesses), through in situ observation of protests in Gabès, and from reading administrative reports and documents.

The role of proximity networks in activism against environmental hazards caused by industry in Gabès and Kerkennah

In both Gabès and Kerkennah, the very local nature of the environmental hazards caused by industrial activities explains why the protest movements against them are mainly-though not exclusively-driven by locally organised groups and proximity networks. Industrial pollution affects or even menaces localities, the material resources within them, the uses made of them, but also the symbolic and emotional attachments that link them to the people who use them. Exposure to environmental hazards is spatially differentiated: the different localities are not affected in the same way, which gives rise to local claims that are specific to different groups with roots in different areas.

The day-to-day practices and the relations, including hierarchical relations, between the members of the community constitute the bedrock of the organisation of collective action (which can also contribute to reshaping those practices and relationships). They give rise, for example, to a gendered division of activist roles: it is usually the men who organise protest actions; women are a minority (though present) at demonstrations, but their participation is encouraged on certain occasions. In addition, women have taken charge at sit-ins, or have taken initiatives that have prompted the rest of the community to join in.

In the Gabès region, multiple localised protests around the phosphate industry Industrial activity in Gabès generates different types of hazards that are perceived and opposed at the scale of the city, or even of different neighbourhoods or villages in and around the city. Gas emissions from the units have a substantial impact on localities situated around Gabès industrial zone, mainly Ghannouch, Bouchemma, Chenini and Chott Salem, with particularly high pollutant concentrations 5 that cause damage to the health of local people (cancers, respiratory disorders) and to crops. 6 They have triggered protest movements that arise in the different neighbourhoods but come together around common demands. The farmers in the oases went to court in the 1980s to obtain compensation from certain companies in the industrial zone, compensation that was distributed to holders of oasis plots in the surrounding localities (interviews with representatives of the firms Alkimia, ICF [October 2018] and officials of the Tunisian Chemicals Group [February 2019]). After 2011, reductions in polluting gas emissions were among the demands expressed in substantial demonstrations in the conurbation of Gabès, with participants numbering in the thousands on several occasions. 7 5. In particular fine particles (PM10), sulphur dioxide (SO2), hydrogen sulphide (H2S). 6. The damage to the oases of Gabès caused by the construction of the chemicals complex has been tackled in several doctoral theses (see in particular [START_REF] Maha | Les transformations socio-spatiales des oasis de Gabès (Tunisie) : déclin des activités agricoles, urbanisation informelle et dégradation de l'environnement à Zrig, des années 1970 à nos jours[END_REF][START_REF] Carpentier Irène | Les révolutions silencieuses des oasis du Sud tunisien : crise des modèles et réponses locales[END_REF][START_REF] Slaheddine | Mutations sociospatiales et modes de gouvernance de l'eau dans les oasis périurbaines du gouvernorat de Gabès (Sud-Est tunisien)[END_REF], as well as in the 2014 film Gabès labess by Habib Ayeb, which has been screened on many occasions in Tunisia. The latter is built around contrasting picture, taken up in subsequent publications [START_REF] Max | What lasted for 3000 years has been destroyed in 30": the struggle for food sovereignty in Tunisia[END_REF], between an industry-based model of development exemplified by the chemicals complex, and the traditional oasis model that it threatens. 7. The damage to health caused by gas emissions have been highlighted by the demonstrations held almost every year on the 5 June, the date set by the UN to celebrate World Environment Day. The Stop Pollution collective has played a leading role in this alongside organisations and groups based in the Jara and Chott Salem districts. The discharge of liquid and solid pollutants into the Gulf of Gabès more specifically effects the living environment of people living in coastal localities-Ghannouch and in particular Chott Salem (see figure 2)-which are adjacent to a mass outlet for the discharge of 12,000 tonnes a day of a gypsum sludge called phosphogypsum, a byproduct of the process of converting phosphate into phosphoric acid. This district has been heavily involved in protest dynamics in Gabès: several demonstrations have been held there to demand a stop to the discharge of phosphogypsum, in particular the Saker lemsab (Shut the Discharge) campaign in spring 2017, headed by a collection of local organisations 8 but structured around conversation in the cafes, a day-to-day meeting place for men. 9 These discharges into the sea also affect the revenues of the region's fisherperson who have taken action on 8. The fact that the organisers of the campaign, when interviewed, emphasised the "civic" nature of the organisations in the movement, the adoption of a charter governing the rights and duties of the members and their desire to "exercise participatory democracy" through civil society activity, seems to indicate a desire to legitimise their action by highlighting its civic character. 9. The core of the organisers of the Saker lemsab campaign consisted of men, mostly middle-aged, resident in Chott Salem and Jara (interview in July 2017, observation at the march on 30 June 2017). However, women's participation in this march was encouraged to emphasise its nonviolent "family" character.

several occasions to demand the end of waste discharges and compensation from the industrial firms. These fishermen join forces in the localities where they work. 10

Other protests arise in reaction to local incidents, which occur frequently when the GCT's production units restart. On 5 May 2017, following a gas leak in the industrial zone, children at a school in Bouchemma experienced significant breathing problems. The mothers then began blocking the road leading to Gabès as a protest against the health risks resulting from the proximity of the industrial zone, and a few days later initiated a general strike in Bouchemma alone. This climate of discontent led to the start of a sit-in, headed by young unemployed from the locality, in front of the site of the Nawara project (gas transfer) run by the firm OMV, to demand jobs or funding for entrepreneurial projects (interview with one of the initiators of the sit-in, April 2018). For their part, when farmers in the Bouchemma oasis observed "scorching" in their crops, they took action to obtain compensation.

A plan to store phosphogypsum in slag heaps has been under consideration since the late 1990s, under international pressure since the Gulf of Gabès is identified as a "pollution hotspot" by the UN Environment Programme. Funding was offered by the European Investment Bank in the form of a loan. It was still under the Ben Ali regime that a first site was earmarked near Oudhref (some 20 km from Gabès). However, immediately following the fall of the regime, the population of Oudhref several times manifested its strong opposition by means of general strikes and marches through the city. Here again, while these actions were initiated by local organisations (interview with the Mayor of Oudhref, a member of the Beit el kheir Association (["the house of charity"] and very strongly committed to opposing the phosphogypsum slag heap in the post-2011 period, April 2019), it was through the activation of family and neighbourhood networks that they were able to bring together a large proportion of the town's inhabitants.

On 5 December 2018, the government announced that it had chosen a site in the Menzel El Habib delegation belonging to the same Gabès governorate, which would now become home not only to the phosphogypsum slag heap, but also new Tunisian Chemicals Group production units intended to replace those in the Gabès industrial zone. The municipal council, civil society organisations and local union sections immediately manifested their rejection of the project but, according to the town's mayor (interview with the mayor of Menzel El Habib, elected in May 2018 partly because of her determination to oppose the planned new industrial zone in Menzel El Habib, April 2019), the population is divided: some hope to be able to sell their land 10. The fishermen of Ghannouch are organised in the Ghannouch's Fishing Group (GDP). Combined actions are conducted with the fishermen of the port of Gabès (interview with the Ghannouch GDP, March 2018).

or get jobs,11 whereas others are worried about the impact on the water table in a predominantly agricultural region. The municipal council and union and civil society organisations in the neighbouring delegation of El Hamma joined the opposition movement, once again organising gatherings, marches and a widely followed general strike on 10 December. 12The struggle against pollution from the phosphate industry in the Gabès region is not a unified movement. It is impelled by multiple movements reflecting specific demands linked with the localised nature of the environmental hazards and largely reliant on proximity networks and networks of day-to-day social relations.

In Kerkennah, interwoven conflicts around oil and gas industries In Kerkennah, the pollution caused by the oil and gas companies is less obvious and ubiquitous than in Gabès, and the target of protests against industrial hazards is mainly hydrocarbon leaks from the offshore platforms. Alongside overfishing and the use of techniques that damage the seafloor (trawling), these leaks are instrumental in threatening the revenues of the fisherpeople, as well as having a deleterious effect on tourism. In March 2016, oil deposits on the beach next to the Sidi Fredj tourist area triggered a protest movement by fishermen and residents of Mellita, Ouled Kacem and Ouled Yaneg (localities situated near the discharges, see figure 3), which blamed the oil company TPS. Small groups held gatherings in front of the firm's premises (interviews with several fishermen involved in protests against environmental damage caused by TPS, October 2018). Fishermen, supported by local organisations,13 tried to institute court proceedings, but eventually gave up because of the difficulty of paying the fees for legal and technical representation. This movement occurred at the same time as a sit-in in front of the Petrofac gas production site on the island of Chergui by unemployed graduates calling for the job allocation agreements made in the aftermath of the revolution to be honoured. 14 The two movements became interwoven as the conflict progressed. They shared the view that the oil and gas companies' contribution to the "development" of the island was inadequate, given the wealth they extracted and the damage they caused. They felt that the firms should give more to the Kerkennah area.

14. In April 2011, after demonstrations by unemployed people, an "environmental work programme" was set up: 248 beneficiaries (rising to 266 in 2012) were assigned to public structures in Kerkennah and paid by the oil company Petrofac. Petrofac stopped the payments in January 2015. The beneficiaries of the programme responded with an initial sit-in in front of the company's premises, which ended following a regularisation agreement. However, in December 2015 Petrofac again stopped the payments.

Taking action in a context background of regional segregation-the use of historical revision and collective identities

Threats to regional survival

The demands that have emerged from the protest movements against industrial hazards draw on the lexicon of survival and subsistence. For example, the rallying cry of the first march organised in Gabès, on 5 June 2012, was "Stop pollution - I want to live" (interview with a member of the Stop Pollution collective, October 2017), and the focus of the continuing Saker lemsab campaign in spring 2017, according to its spokesman, was to be able to "breathe pure air, to live" (interview jointly conducted with Irène Carpentier in April 2018, when he was a candidate in the Gabès municipal elections for the Attahadi [Challenge] group).

"Living" can be understood here in its literal sense: escaping from the cancers and other sometimes fatal conditions attributed to pollution. However, the word also covers a broader meaning, with material and symbolic dimensions: living a life of dignity. The rhetoric of dignity here is connected with the "semantic framing in terms of dignity" (Ayari, 2011) that underpinned the uprisings that led to the departure of President Zine el-Abidine Ben Ali on 14 January 2011.

However, the possibility of leading a life of dignity seemed conditional on obtaining a decent job, which was currently compromised by the lack of sources of employment in specialist industrial areas, 15 or on the possibility of continuing employment, whether in farming, tourism or fishing, sectors where development was compromised by industrial pollution. Phosphogypsum deposits on the seafloor of the Gulf of Gabès and the disappearance of the posidonia meadows have led to a deterioration in fishery resources (interview with a researcher at the National Institute of Marine Sciences and Technologies, March 2019) which has had a severe impact on the livings of the region's fishermen and forced them to take on debt (interview with the president of the Ghannouch Fishing Group, March 2018). Those fishing in Kerkennah also experience disruptions to the marine ecosystem and loss of earnings, 16 which they blame on oil leaks as well as the spread of trawl fishing.

15. This kind of specialist industry is not labour intensive. Since the start, few jobs have been created as a result of the investment in Gabès [START_REF] Adnane | L'industrialisation à Gabès et ses conséquences. Étude de géographie urbaine et économique[END_REF]. The International Monetary Fund (IMF) and the World Bank made restructuring a condition of the Structural Adjustment Plan in the 1980s, which led to job cuts. Nevertheless, there was an increase in workforce numbers after 2011 in response to the pressures exercised by the unemployed. 16. "The fishermen are at the end of their tether. In the 1990s, we used to catch 30 to 40 kg of fish without a net, now it is 2 kg" (interview with a fisherman from Ouled Yaneg who was very active in the movement against TPS, October 2018).

Industrial activities are therefore opposed for their impacts on the living environment, but also because they threaten the economic and social viability of the areas where they take place. "Oil will not keep the island alive," observes one hotelier in Kerkennah. 17 This observation has prompted some civil society and union activists to question the coherence of the development model on which industrial activity in their area is based.

Narratives of dispossession

Protest action over industrial hazards is often linked with narratives of mobilisation that draw upon regional history while making sharp distinctions between the time before and after the top-down imposition of industry. In the narrative, the latter plays the role of a disruptive event, one that shattered a doubtless idealised situation of initial equilibrium. 18 In Gabès, some 19 in hindsight reinterpret the decision to set up an industrial chemical hub in the town as a punishment inflicted on the region because of the purportedly large number of supporters there of Salah Ben Youssef, the nationalist leader who was a rival of the first president of independent Tunisia, Habib Bourguiba. And the story goes that the latter's home region, the Sahel, was for its part sheltered from pollution, becoming the site of less environmentally burdensome tourist infrastructures. 20 The submission of a "victim region" case on behalf of the region of Gabès to the Truth and Dignity Commission by civil society organisations, including the local section of the Tunisian Human Rights League, echoes this interpretation of the past by suggesting "intentional discrimination on the part of the state" (Gana, 2019, p. 127). This reading re-situates the antipollution movements within a decades-long history of resistance to domination by the central state. 21 "The Sahel or the North, they live off the wealth of the South, whether it be oil or mining, or even vegetables and 17. This hotel owner is facing a loss of business because of the tourist crisis (interview in October 2018). 18. In the interviews in Kerkennah, there are recurrent references to "tranquillity" and "simplicity" before the arrival of the oil and gas firms; in Gabès, several interviewees mention childhood memories of bathing in the sea or in the oasis rivers, activity that pollution or growing water shortages have now made impossible. 19. Others, from left-wing parties of opposition to the Ben Ali regime, are keen to underplay the region's opposition to Habib Bourguiba. 20. The fact that the phosphate processing hub was not built in the Sahel has not prevented substantial environmental degradation caused by urbanisation, artificialisation of the shoreline, and the inadequate treatment of discharged wastewater, notably from the textile industries. This is evidenced, for example, by the work of Racha Sallemi (2017), and by the reports of the Tunisian Forum for Economic and Social Rights. 21. It should be noted that this is a retrospective reinterpretation, which takes no account of the reactions at the time when the decision to create an industrial hub in Gabès was taken, but was forged gradually, when the role of the project as a driver of the regional economy was abandoned in favour of a shift to an export model, and when the environmental impacts became more and more severe.

fruit, it comes from the South," claimed the spokesperson of the Saker Lemsab campaign during the run-up to the municipal council elections.

Among the organisers of the movements in Kerkennah against TPS, the relations with central government tend to be presented as indicative of the marginalisation of the island group, reflected in the inadequacy of services and public investment, or else poor public oversight of illegal fishing practices (interview with association members, fishermen and inhabitants of Kerkennah involved in the protest movement against TPS, October 2018). At the same time, the state is described as "absent", content to maintain the continuity of oil and gas production, subject to foreign interests, incapable of expressing itself other than by means of repression, exemplified by the movements of 2016 or the legal proceedings against actions that took place in 2017 (interviews with several inhabitants of Mellita, fishermen from Ouled Yaneg and Remla, October 2018). This is a situation of dispossession that the actors of the movement against TPS condemn: "It's our land, our resources, and they give us nothing"; "They eat and they throw us the bones" (interview with a fisherman from Ouled Yaneg [Kerkennah] very active in the movements against TPS and Petrofac, when he described the environmental damage caused by the activities of the oil and gas companies, their wrongdoings and the failings of central government, October 2018). The distinction between "them" and "us" points to a contrast between those who benefit from the resources present in the area-the foreign firms and the state actors, presented as corrupt-and those, defined by their territorial and social belonging to Kerkennah, who enjoy none of the benefits that should come to them from those resources but suffer the negative impacts of extraction. Some claim to prize membership of their community over national attachment, or even-by way of provocation-hint at a desire for independence, 22 which suggests doubts about being full members of the national community due to experiences of rejection, the sense of not being recognised as fully Tunisian.

"If there is an equilibrium between the regions, if everyone has the same opportunities, if everyone has a decent social situation, if there is a little bit of wealth in each governorate, we can have the same Tunisia. We can have that united Tunisia," argues a young activist from Gabès, a former member of a left-wing party converted to the cause of pollution.

The complaint about regional segregation that underpins the movements focuses not only on better spatial distribution of wealth and environmental burdens: it 22. "For me, Kerkennah comes before Tunis" (interview with a fisherman from near Remla, October 2018). also reveals symbolic and moral dimensions, which are revealed through stigmatisation and have real-world implications.

In response to stigmatisation, collective identity is a driver of mobilisation

The stereotypes about the South are a result of a social construction of the otherness of Tunisia, which was the basis of Bourguiba's political project, but has earlier roots. In order to achieve full independence, the nation had to move away from a certain state of "backwardness" (Bras, 2004, p. 296) and embrace modernity. However, this dual representation of two moments in the process of modernisation-the past to be left behind and the future to which to aspire-was territorially embodied [START_REF] Jean-Philippe | L'autre Tunisie de Bourguiba. Les ombres du Sud[END_REF]. The South belonged to the "other Tunisia" (Bras, 2004, p. 309)-backward, tribal, conservative. And Bourguiba would bring it into the modernisation process through aspirational policies of development. It might be said that the opposition between North and South has been transmuted into a division between East and West [START_REF] Amor | La fracture territoriale. Dimension spatiale de la Révolution tunisienne[END_REF], which contrasts the coast or the "useful Tunisia" with the "Tunisia of the interior". However, stereotypes about the South remain strong.

During the series of protests in 2016, derogatory images associated with the inhabitants of Kerkennah were also revived to explain the population's rebelliousness by its intrinsic characteristics and to justify the crushing of the movements, which was the state's preferred method of conflict management. These stereotypes emerged in the interviews conducted during the field survey: Kerkennians are described as "pigheaded", "narrow-minded" or "slow-witted" by members of the state agency, who were finding it hard to gain popular assent for their projects in Kerkennah, or even by prominent citizens from the area itself (interviews with officials of the Coastline Protection and Development Agency and with the Mayor of Kerkennah, April 2019).

In response to these stereotypes, the protesters express a certain pride in their identity as a source of mobilisation. They counter the insults by presenting themselves, for example, as "revolutionaries", as "warriors", as "resistance fighters", drawing on collective memory to reclaim collective value. The desire to oppose the stigma takes on a particular dimension in the movements against environmental hazards. Siad Darwish has written of a "moral geography of waste" in Tunisia, whereby places are labelled as dirty or clean: the presence of waste sullies places and the people who live in them, outlining sociospatial differences with strong moral foundations (Darwish, 2018, p. 66). The refusal to have a landfill site nearby arises both from its physical impact on its surroundings and from the moral taint that it might leave on the neighbourhood and its inhabitants. During the general strike in El Hamma in response to the announcement of the decision to locate the chemical processing plants and the slag heap for storing the phosphogypsum waste in Menzel El Habib, the protest slogans focused on the second of these entities. "No to phosphogypsum!", "Menzel El Habib is not a garbage pile!" proclaimed the banners. The march signalled a rejection of the territory being identified with a garbage tip by accepting waste refused by the district of Oudhref a few years earlier, rather than the health and environmental risks associated with the waste and the pressure on water resources. In addition, the speeches to the crowd emphasised tribal identity, referring to the courage of the Beni Zid as an emblem, a "resource that can be mobilised to assert group identity" (Camau, 2018, p. 214).

Fluctuation between demands for an end to the hazards and for compensation Two types of claims coexist within the movements against industrial hazards, with strong local roots: first, demands for an end to the environmental and health hazards caused by the industrial activities, or more rarely an end to the activities themselves; and second, demands for compensation-job creation, financial or material compensation, etc. In the first case, what is sought is the cessation of harm; in the second, the harm is accepted and the objective is to get something in return.

Civil society organisations and activist groups, which focus the area's necessary transition to other economic activities, tend to disregard the self-interested aspect of approaches in which the aim is to obtain some crumbs from the cake. However, the coexistence of these two types of demand within the same activist groups is understandable if one considers the material conditions of survival, the economic practices and hopes for justice of local actors, from the perspective of moral economybased approaches [START_REF] Edward | The Moral Economy of the English Crowd in the Eighteenth Century[END_REF][START_REF] James | The moral economy of the peasant: Rebellion and subsistence in Southeast Asia[END_REF][START_REF] Didier | Les économies morales revisitées[END_REF][START_REF] Siméant | Économie morale et protestation -détours africains[END_REF]Allal, Catusse and Emperador Badimon, 2018). The communities of Gabès and Kerkennah face industrial hazards, high levels of unemployment and falling revenues from farming and fishing, problems akin to "subsistence crises" (Scott, 1976, p. 17). In response, they claim moral rights, starting with the right to survival. Moreover, they are entangled in dependency relations with the industries responsible for the hazards because of the scarcity of alternative local sources of jobs and services, with the capacity to bring development funds and to distribute individual compensation to set against the pollution. The movements internalise the promises of a "state paternalism" (Camau, 2018, p. 228-230) on which the expectations of the actors depend and which forms the basis of their claims. They often arise when agreements are broken, when accidents threaten the fragile balance of subsistence or when the distribution of compensation seems unfair with respect to a moral order. For example, in Bouchemma in May 2017, the gas leak from the Tunisian Chemicals Group may be seen as the final straw that prompted the inhabitants take to the streets and call for compensation for the burnt harvests, for better hospitals and jobs for unemployed youngsters, including from other companies in the industrial zone, in the name of a shared conception of law and justice. One of the initiators of the sit-in at OMV explains it as follows: "People catch a big dose of pollution, but they are ground down by unemployment, by poverty. There needs to be a social equilibrium that gives us the right to live decently. You can live in polluted conditions, but only if you earn a decent wage, can enjoy a good weekend from time to time. People are not against the gas coming to Bouchemma, but they want the companies to accept their social responsibility." This interview (April 2018) extract shows the role of the moral register in the protest movement, but it also suggests that this movement may be instigated less to restore an old order than to grasp an opportunity and in the wake of initiatives taken by people who see it as an "art of the possible" (Fioroni, 2018, p. 162).

The exacerbation of conflicts at micro-local level

Community belonging and territorial networks of conflict While collective identities, because of their unifying power, provide foundations for movements of protest against environmental hazards (and are also, in certain respects, produced by them), they can also contribute to the breakdown of protest dynamics, triggering or reactivating tensions and conflicts of multiple kinds that (re)produce instances of otherness at a microlocal level. One sees the emergence of what could be called "territorial networks of conflict" (Beuret and Cadoret, 2014, p. 223): chronic conflicts and tensions that affect and are reactivated by these movements.

In Gabès, the tensions between shifting groups originating, on the one hand, from the districts of Menzel and Chenini and on the other from Jara and Chott Salem, revived old rivalries linked with conflicts around the water supplied to the oases but also with the French colonial conquest [START_REF] Abdelmajid | La résistance de Gabès à l'occupation française en 1881[END_REF], which Menzel-Chenini reportedly resisted heroically whereas Jara-Chott Salem purportedly made peace with the colonists. However, the tensions between these two groups are also fed by different political roots and sociological foundations. 23 These struggles of influence have been exacerbated by the incorporation of the local actors into national and transnational networks and initiatives intended to generate media coverage of the problem of pollution and apply pressure for action, but also to improve their position in local power relations.

At the height of the social conflict in Kerkennah in 2016, the people holding the sit-in at the Petrofac gas production site on Chergui Island demanding that it meet its previous undertakings on job creation, moved to Mellita, counting on the "protection" of the inhabitants of Mellita and their readiness to face down the police (interviews with one of the organisers of the sit-in for jobs and with the inhabitants of Mellita, October 2018). Following discussions with the protesters in Mellita, and in particular with a group involved in the movement against TPS, they agreed to include demands for contributions to development in addition on top of the call for regularisation of the situation on jobs for the unemployed. However, while the agreement signed in September 2016 between the protesters' representatives, the national authorities and the oil companies implemented job regularisation for the unemployed, not much was achieved regarding the "development" outcomes. This explains a certain bitterness in Mellita towards the group of unemployed graduates, which has reactivated resentment towards Remla, where many government departments are located (interviews with inhabitants of Mellita involved in the movement against the oil and gas companies, October 2018). Conversely, the inhabitants of Mellita are blamed for illegal trawler fishing and recurrent blockades that prevent access to the Sidi Youssef ferry. "They use us as hostages" (interview with an official in a civil society organisation in Kraten, March 2018).

Response to the protest movements: avoidance, displacement, "buying" of social peace These rivalries may be stirred up by the crisis management measures put in place by the authorities. Such measures range from short-term emergency adjustments that then become permanent-environmental development firms, increased compensation within a Corporate Social Responsibility (CSR) framework, diversions to the routes used by hydrocarbon trucks, etc.-to substantial projects that are never implemented (relocation of industrial units).

In the Gabès region, it is the dismantling of GCT's units in Gabès and their transfer to Menzel El Habib that the authorities present as the main response to discontent linked with environmental pollution. The project, which has met with opposition from the inhabitants of El Hamma and Menzel El Habib (although others, hopeful of new jobs and land purchases in this neglected rural area, have welcomed the project) has fallen behind. The government had undertaken to end the discharge of phosphogypsum in Chott Salem on 30 June 2017. In response to the demands of the inhabitants of Chott Salem, who want to return to decent living conditions and blame the authorities for not keeping their promises, the latter attribute the responsibility to the protesters in El Hamma and Menzel El Habib (and previously in Oudhref), thereby heightening tensions between the communities of Chott Salem and of El Hamma and Menzel El Habib (interviews with members of a civil society organisation in Chott Salem, March 2018).

In Kerkennah, since 2016, the trucks carrying Petrofac's condensate (an inflammable gas) have been diverted from their original route. In the past, they used the island's main road to reach the port of Sidi Youssef, passing through the village of Mellita. Since the conflict, they have been loaded onto boats that carry them to Sfax, where the landing stage for the tourist area of Sidi Fredj has been adapted for this purpose, to the great annoyance of the hoteliers (interviews with hotel managers in Sidi Fredj, October, April and May 2018). In this way, the company is able to reduce the risk of blockades mounted by the protest movements.

These relocations, adaptations and displacements have redrawn the distribution map of environmental hazards: facilities are transferred to areas where the power balance is considered (wrongly or rightly) more favourable to the continuation of industrial activities without hindrance from roadblocks. The result is that the people living in these areas complain about new injustices between localities at regional level.

The other instrument employed by the industrial firms (with state support) to reduce conflicts around demands by community movements is to provide compensation in kind 24 or in cash, 25 subsidies for local organisations and festivals, or compensation in the form of jobs. 26 The way these are distributed among the different localities is often a source of suspicion over the fairness of the process, and maintains or reactivates rivalries, intensifying the fragmentation of the movements across the Gabès region or the Kerkennah island group. The dispersal of energy in conflicts between localities, together with a waning in the protests against the industries, has prompted a degree of discouragement and loss of motivation in the Gabès and Kerkennah movements (interviews with participants in the movement against TPS, October 2018; interviews with Stop Pollution activists in Gabès, May 2019).

24. GCT funded equipment for the fishermen in the Ghannouch Group (interview, March 2018). 25. Such as those distributed to the farmers on the oases adjacent to the Gabès industrial zone. 26. In Gabès, for example, the Société d'environnement, de plantation et de jardinage (SEPJ-Environment, Planting and Gardening Company), funded by GCT, employs some 2,600 people, most of whom do not work (interview with the CEO of SEPJ, May 2018).

Conclusion

This article has analysed the protest movements against environmental hazards caused by industrial activities in Gabès and Kerkennah as initiatives undertaken by the communities affected by these hazards, which they perceive as injustices. The regions studied are situated on the coast and not in the more marginalised or exploited areas in the country's interior. Nonetheless, they reveal expectations of justice relating in particular to the living environment, to resources and more generally to survival or a decent life for the regions and their inhabitants.

Community memberships play a role here on many levels: on the one hand, industrial hazards are experienced at the level of the community; on the other hand, communities form a network of social ties on which the protest movements are based; finally, collective identities linked with the locality foster cohesion in the groups that undertake collective actions.

However, while community affiliations can be breeding grounds for protest and for the organisation of collective movements against injustices, they can also feed into processes of fragmentation and be used by the power structures to turn communities anchored in different areas against each other. The production of (in)justices does not seem to depend so much on whether or not the protest movements are communitarian in nature as in the power relations that they are able to establish and on the ways in which the state and industrial forces respond to them.

Note

This article received support from the European Research Council (ERC) as part of the European Horizon 2020 Tarica programme (No. 695674).

To quote this article

Robert Diane, « Contestations croisées des nuisances environnementales des industries et des injustices territoriales à Gabès et Kerkennah (Tunisie) » ["Protest movements against industry-related environmental burdens and territorial injustice in Gabès and Kerkennah (Tunisia)"], Justice spatiale | Spatial Justice, n o 16, 2021 (http://www.jssj.org/article/contestations-croisees-des-nuisances-environnementalesdes-industries-et-des-injustices-territoriales-a-gabes-et-kerkennah-tunisie/).

Figure 1 :

 1 Figure 1: Map of research sites (illustration by author).

Figure 2 :

 2 Figure 2: The city of Gabès (illustration by the author).

Figure 3 :

 3 Figure 3: Map of Kerkennah (illustration by the author).

  23. For example, with former activists in the Progressive Democratic Party, a pre-2011 social-democrat opposition party, within the Stop Pollution collective, well embedded in Menzel and Chenini, and a greater presence of activists from the Islamist party Ennahdha in the movements in Chott Salem and Jara.

In particular for graduates, of whom 40.2% are unemployed in Menzel El Habib, as compared with 34.9% in the governorate as a whole (2014 census data).

Young activists at the General Union of Tunisian Students (UGET) ran a social media campaign for this event, in concert with the local section of the General Tunisian Labour Union (UGTT) (interviews and observations in December 2018).

Such as the Ouled Yaneg Development Association.


Jacob Szeftel

Nicolas Sandeau

Michel Abou Ghantous

Muhammad El-Saba

Towards room-temperature superconductivity

Keywords: numbers: 74.25.Bt, 74.25.Jb, 74.62.Bf

By taking advantage of a stability criterion established recently, the critical temperature Tc is reckoned with help of the microscopic parameters, characterising the normal and superconducting electrons, namely the independent-electron band structure and a repulsive two-electron force. The emphasis is laid on the sharp Tc dependence upon electron concentration and inter-electron coupling, which might offer a practical route toward higher Tc values and help to understand why high-Tc compounds exhibit such remarkable properties.

The BCS theory [START_REF] Schrieffer | Theory of Superconductivity[END_REF], despite its impressive success, does not enable one to predict [START_REF] Matthias | [END_REF] superconductivity occurring in any metallic compound. Such a drawback ensues from an attractive interaction, assumed to couple electrons together, which is not only at loggerheads with the sign of the Coulomb repulsion but in addition leads to questionable conclusions to be discussed below. Therefore this work is intended at investigating the T c dependence upon the parameters, characterising the motion of electrons correlated together through a repulsive force, within the framework of a two-fluid picture [3] to be recalled below.

The conduction electrons comprise bound and independent electrons, in respective temperature dependent concentration c s (T ), c n (T ), such that c 0 = c s (T ) + c n (T ) with c 0 being the total concentration of conduction electrons. They are organized, respectively, as a many bound electron [4] (MBE) state, characterised by its chemical potential µ(c s ), and a Fermi gas [START_REF] Ashcroft | Solid State Physics[END_REF] of Fermi energy E F (T, c n ). The Helmholz free energy of independent electrons per unit volume F n and E F on the one hand, and the eigenenergy per unit volume E s (c s ) of bound electrons and µ on the other hand, are related [START_REF] Ashcroft | Solid State Physics[END_REF][START_REF] Landau | Statistical Physics[END_REF], respectively, by E F = ∂Fn ∂cn and µ = ∂Es ∂cs . Then a stable equilibrium is conditioned [START_REF] Szeftel | [END_REF] by Gibbs and Duhem's law

E F (T, c n (T )) = µ(c s (T )) , (1)
which expresses [START_REF] Landau | Statistical Physics[END_REF] that the total free energy F n + E s is minimum provided ∂EF ∂cn + ∂µ ∂cs > 0. Noteworthy is that ∂µ ∂cs < 0 has been shown to be a prerequisite for persistent currents [START_REF] Szeftel | [END_REF], thermal equilibrium [4], the Josephson effect [8] and a stable [3] superconducting phase. Likewise, Eq.( 1) reads [4,[START_REF] Szeftel | [END_REF] for

T = T c E F (T c , c 0 ) = µ(c s = 0) = ε b /2 , ( 2
)
with ε b being the energy of a bound electron pair [4]. Note that Eqs.(1,2) are consistent with the superconducting transition being of second order [START_REF] Landau | Statistical Physics[END_REF], whereas it has been shown [4] to be of first order at

T < T c (⇒ E F (T, c 0 - c s ) = µ(c s ))
, if the sample is flown through by a finite current.

The binding energy [4] of the superconducting state E b (T < T c ) has been worked out as

E b (T ) = Tc T (C s (u) -C n (u)) du ,
with C s (T ), C n (T ) being the electronic specific heat of a superconductor, flown through by a vanishing current [4] and that of a degenerate Fermi gas [START_REF] Ashcroft | Solid State Physics[END_REF]. A stable phase

(⇒ E b > 0) requires C s (T c ) > C n (T c
), which can be secured [3] only by fulfilling the following condition

∂E F ∂c n (T c , c 0 ) = - ∂µ ∂c s (0), ρ ′ (E F (T c , c 0 )) > 0 , (3)
with ρ (ǫ) , ǫ being the independent electron density of states and one-electron energy, respectively, and ρ ′ = dρ dǫ . Since the remaining analysis relies heavily on Eqs. [START_REF] Matthias | [END_REF]3), explicit expressions are needed for E F (T c , c 0 ), ∂EF ∂cn (T c , c 0 ), ε b , ∂µ ∂cs (0). Because the independent electrons make up a degenerate Fermi gas (⇒ T << E F /k B with k B being Boltzmann's constant), applying the Sommerfeld expansion [START_REF] Ashcroft | Solid State Physics[END_REF] up to T 2 yields

E F (T c , c 0 ) = E F (0, c 0 ) -ρ ′ ρ (πkB Tc) 2 6 ∂EF ∂cn (T c , c 0 ) = ρ + ρ ′′ (πkB Tc) 2 6 -1 , ( 4
)
with ρ = ρ(E F (0, c 0 )), ρ ′ = dρ dEF (E F (0, c 0 )), ρ ′′ = d 2 ρ dE 2 F (E F (0, c 0 )).
As for ε b , ∂µ ∂cs (0), a truncated Hubbard Hamiltonian H K , introduced previously [9][START_REF] Szeftel | Electron Correlations and Material Properties 2[END_REF][START_REF] Szeftel | [END_REF], will be used. The main features of the calculation [4] are summarised below for self-containedness.

The independent electron motion is described by the Hamiltonian H d

H d = k,σ ǫ(k)c + k,σ c k,σ .
ǫ(k), k are the one-electron energy (ǫ(k) = ǫ(-k)) and a vector of the Brillouin zone, respectively, σ = ± is the electron spin and the sum over k is to be carried out over the whole Brillouin zone. Then c + k,σ , c k,σ are creation and annihilation operators on the Bloch state |k, σ

|k, σ = c + k,σ |0 , |0 = c k,σ |k, σ ,
with |0 being the no electron state. The Hamiltonian H K reads then

H K = H d + U N k,k ′ c + k,+ c + K-k,-c K-k ′ ,-c k ′ ,+ ,
with N >> 1, U > 0 being the number of atomic sites, making up the three-dimensional crystal, and the Hubbard constant, respectively. Note that the Hamiltonian used by Cooper [12] is identical to H K=0 , but with U < 0. H K sustains [4] a single bound pair eigenstate, the energy ε b (K) of which is obtained by solving

1 U = 1 N k 1 ε b (K) -ε(K, k) = tK -tK ρ K (ε) ε b (K) -ε dε. ( 5
)
±t K are the upper and lower bounds of the two-electron band, i.e. the maximum and minimum of

ε(K, k) = ǫ(k)+ǫ(K -k) over k, whereas ρ K (ε) is the corresponding two-electron density of states, taken equal to ρ K (ε) = 2 πt K 1 - ε t K 2 .
The dispersion curves ε b (K) are plotted in Fig. 1. Though Eq.( 5) is identical to the equation yielding the Cooper pair energy [12], their respective properties are quite different :

• the data in Fig. 1 have been calculated with U > 0, rather than U < 0 favoured by Cooper [12] and BCS [START_REF] Schrieffer | Theory of Superconductivity[END_REF], because, due to the inequality[4] U ∂µ ∂cs < 0, choosing U < 0 entails ∂µ ∂cs > 0, which has been shown not to be consistent with persistent currents [START_REF] Szeftel | [END_REF], thermal equilibrium [4], the Josephson effect [8] and occurence [3] of superconductivity. As a further consequence of U > 0, ε b (K) shows up in the upper gap of the two-electron band structure (⇒ ε b (K) > t K ) rather than in the lower gap (⇒ ε b (K = 0) < -t K ) in case of the Cooper pair [12]. Nevertheless the bound pair is thermodynamically stable, because every one-electron state of energy < E F (T c , c 0 ), is actually occupied, so that, due to Pauli's principle, a bound electron pair of energy ε b (K) = 2E F (T c , c 0 ), according to Eq.( 2), cannot decay into two one-electron states

ǫ(k) < E F , ǫ(K -k) < E F ; • a remarkable feature in Fig.1 is that ε b (K) → t K for U → t K /2
, so that there is no bound pair for U < t K /2 (accordingly, the dashed curve is no longer defined in Fig. 1 for Ka π < . 13), in marked contrast with the opposite conclusion drawn by Cooper [12], that there is a Cooper pair, even for U → 0. This discrepancy results from the threedimensional Van Hove singularities, showing up at both two-electron band edges ρ K (ε → ±t K ) ∝ t K -|ε|, unlike the two-electron density of states, used by Cooper [12] which is constant and thence displays no such singularity. Likewise the width of Cooper's two-electron band is equal to a Debye phonon energy 2t K=0 = ω D ≈ 30meV << E F ≈ 3eV . Hence the resulting small concentration of superconducting electrons, cs(T =0) c0 ≈ ωD EF ≈ .01, entails that London's length should be at least 10 times larger than observed values [13][14][15][16];

• at last Cooper's assumption U < 0 implies ε b /2 = E F (T c
), which is typical of a first order transition but runs afoul at all measurements, proving conversely the superconducting transition to be of second order (⇒ ε b /2 = E F (T c ) in accordance with Eq.( 2)).

The bound pair of energy ε b (K) turns, at finite concentration c s , into a MBE state, characterised by µ(c s ). Its properties have been calculated thanks to a variational procedure [4], displaying several merits with respect to that used by BCS [START_REF] Schrieffer | Theory of Superconductivity[END_REF] :

• it shows that µ(0) = ε b /2;
• the energy of the MBE state has been shown to be exact for |U | → ∞;

• an analytical expression has been worked out for ∂µ ∂cs (K, c s = 0) as :

∂µ ∂c s (K, c s = 0) = - tK -tK ρK (ε) (ε b (K)-ε) 3 dε 2 tK -tK ρK (ε) (ε b (K)-ε) 2 dε 2 . ( 6
)
The T c dependence on c 0 will be discussed by assigning to ρ(ǫ) the expression, valid for free electrons

ρ(ǫ) = η √ ǫ -ǫ b ⇒ c 0 = 2 3 η (E F (0, c 0 ) -ǫ b ) 3 2 , ( 7
) with η = √ 2m 3 2 V π 2 3
, whereas ǫ b , m, V = 17Å 3 stand for the bottom of the conduction band, electron mass and volume of the unit-cell, respectively. With help of Eq.(4), Eqs.(2,3) can be recast into a system of two equations

E F (0, c 0 ) -ρ ′ ρ (πkB Tc) 2 6 -ε b (K) 2 = 0 ρ + ρ ′′ (πkB Tc) 2 6 -1 + ∂µ ∂cs (K, c s = 0) = 0 , ( 8
)
to be solved for the two unknowns c 0 (T c ), t K (T c ) with T c being dealt with as a disposable parameter. To that end, starting values are assigned to U, t K , which gives access to ε b (K), ∂µ ∂cs (K, c s = 0)) and thence to E F (0, c 0 ) , ǫ b and finally to c 0 , owing to Eqs. [START_REF] Matthias | [END_REF]3,[START_REF] Szeftel | [END_REF]. Those values of c 0 , t K are then fed into Eqs. (8) to launch a Newton procedure, yielding the solutions c 0 (T c ), t K (T c ). The results are presented in table I. Since we intend to apply this analysis to high-T c compounds [17], we have focused upon low concentrations c 0 < 0.2, which entails, in view of Eqs. (4,[START_REF] Szeftel | [END_REF], that ∂µ ∂cs takes a high value. This requires in turn ε b (K) → t K (see Eq.( 6)) and thence [4]

U → tK 2 , in agreement with tK U ≈ 2 in table I.
A remarkable property of the data in table I is that c 0 , t K are barely sensitive to large variations of T c , i.e. |δc 0 | < 10 -3 , |δt K | < 10 -5 for δT c ≈ 400K. This can be understood as follows : taking advantage of Eqs. [START_REF] Matthias | [END_REF]4,[START_REF] Szeftel | [END_REF] results into

2E F (0, c 0 ) ε b (K) -1 = π 2 12 k B T c ∆(T c ) 2 ,
which, due to dtK dTc ≈ 0, ∆(T c ) ≈ 1eV, T c = 400K, yields indeed δc 0 = c 0 (400K) -c 0 (1K) ≈ 10 -3 , in agreement with the data in table I. Such a result is significant in two respects, regarding high-T c compounds, for which c 0 can be varied over a wide range :

• because of dc0 dTc ≈ 0, the one-electron band structure can be regarded safely as c 0 independent, which enhances the usefulness of the above analysis;

• the large doping rate up to ≈ 0.2 is likely to give rise to local fluctuations of c 0 , which, in view of the utmost sensitivity of T c with respect to c 0 , will result

U = 2.2
into a heterogeneous sample, consisting in domains, displaying T c varying from 0 up to a few hundreds of K. Thus the observed T c turns out to be the upper bound of a broad distribution of T c values, associated with superconducting regions, the set of which makes up a percolation path throughout the sample. However, if the daunting challenge of making samples, wherein local c 0 fluctuations would be kept well below 10 -4 , could be overcome, this might pave the way to superconductivity at room temperature.

The T c dependence upon U will be analysed with

ρ(ǫ) = 4 πt 1 -1 - ǫ t 2 ,
where 2t stands for the one-electron bandwidth. Our purpose is to determine the unknowns t K (E F , T c ), U (E F , T c ) with E F = E F (T = 0, c 0 ) and c 0 = EF 0 ρ(ǫ)dǫ. To that end, Eq.(3) will first be solved for t K by replacing ∂EF ∂cn (T c , c 0 ), ∂µ ∂cs (0) by their expressions given by Eqs. (4,[START_REF] Landau | Statistical Physics[END_REF], while taking advantage of Eq.( 2). Then the obtained t K value is fed into Eq.( 5) to determine U . The results are presented in Fig. 2.

It can be noticed that there is no solution for c 0 > .75, because ∂EF ∂cn (T c , c 0 ) ≈ 1 ρ (E F (0, c 0 )) and ∂µ ∂cs (0) > U 2 decrease and increase, respectively, with increasing c 0 , so that Eq.( 3) can no longer be fulfilled eventually. But the most significant feature is that δU is almost insensitive to large T c variation, except for E F → 0, i.e. for E F close to the Van Hove singularity, located at the bottom of the band, which has two consequences :

• c 0 cannot be varied in most superconducting materials, apart from high-T c compounds, so that U is unlikely to be equal to U (c 0 ), indicated in Fig. 2.

FIG. 2. Plots of EF (Tc, c0), tK (Tc, c0), U (Tc, c0) calculated for Tc = 1K and t = 3eV ; the unit for c0 is the number of conduction electrons per atomic site; δf with f = EF , tK , U is de-

fined as δf = 1 -f (300K,c 0 ) f (1K,c 0 )
; the scale is linear for EF , tK , U but logarithmic for δEF , δtK, δU .

Conversely, since high-T c compounds allow for wide c 0 variation, c 0 can be tuned so that U = U (c 0 );

• the only possibility for a non high-T c material to turn superconducting is then offered at the bottom of the band, because δU becomes large due to

ρ ′ ρ (E F → 0) ∝ 1 EF in Eq.( 4
). Such a conclusion, that superconductivity was likely to occur in the vicinity of a Van Hove singularity in low-T c materials, had already been drawn [4] independently, based on magnetostriction data.

It will be shown now that ρ(ǫ), ρ K (ε) cannot stem from the same one-electron band. The proof is by contradiction. As a matter of fact ρ(ǫ) should read in that case

ρ(ǫ) = 4 πt 1 - ǫ t 2 .
Hence U > 0 entails, in view of Fig. 1 and Eq.( 2), that there is ε b 2 = E F > 0, which implies ρ ′ (E F ) < 0 in contradiction with Eq.( 3). Accordingly, since the two different one-electron bands, defining respectively ρ(ǫ), ρ K (ε), display a sizeable overlap, they should in addition belong to different symmetry classes of the crystal point group, so that superconductivity cannot be observed if there are only s-like electrons at E F or if the point group reduces to identity. Noteworthy is that those conclusions had already been drawn empirically [START_REF] Matthias | [END_REF].

The critical temperature T c has been calculated for conduction electrons, coupled via a repulsive force, within a model based on conditions, expressed in Eqs. [START_REF] Matthias | [END_REF]3). Superconductivity occurring in conventional materials has been shown to require E F (T c ) being located near a Van Hove singularity, whereas a practical route towards still higher T c values has been delineated in high-T c compounds, provided the upper bound of local c 0 fluctuations can be kept very low. The thermodynamical criterions in Eqs. [START_REF] Matthias | [END_REF]3) unveil the close interplay between independent and bound electrons in giving rise to superconductivity. At last, it should be noted that Eqs.(2,3) could be applied as well to any second order transition, involving only conduction electrons, such as ferromagnetism or antiferromagnetism.

FIG. 1 .

 1 FIG.1. Dispersion curves of tK as a dashed-dotted line and of ε b (K) as solid, dashed and dotted lines, associated with various U values, respectively; those data have been obtained with tK = t cos (Ka/2), where t, a are the one-electron bandwidth and the lattice parameter, respectively.

TABLE I .

 I Solutions c0(Tc), tK (Tc), ∆(Tc) (∆(Tc) = EF (0, c0(Tc)) -ǫ b ) of Eqs.(8); tK , ∆, U are expressed in eV , whereas the unit for c0 is the number of conduction electrons per atomic site.

	Tc(K)	c0	tK	∆
	1	0.10215	6	1.1976	U = 3.39
	400 0.10225 5.9999 1.1984
	Tc(K)	c0	tK	∆
	1	0.14897	2	1.5402	U = 1.04
	400 0.14906 1.9999 1.5407
	Tc(K)	c0	tK	∆
	1	0.19158	4	1.8214
	400 0.19167 3.9999 1.8219