P Bailly

F Delvare

M Biessy

D Picart

Dynamic tests on a cohesive and frictional material. Influence of high pressure and high strain rate on compaction and shear

Many of aggregate composites are made of small size particles mixed with a highly compliant rubbery polymer. When the loading leads to high pressures, the behaviour gets closer to this of cohesive, frictional, and compactable materials. When this kind of material is subjected to an impact, the loading domain is the case of high pressure (from 100 to 1000 MPa) and high strain rate (roughly 1000 s -1 ). An experimental study is led by using the technique of the SHPB. The specimens are small cylinders placed in a ring which creates a passive confinement by opposing to the radial displacements. The instrumentation of the ring allows to estimate the radial stress and the radial displacement. The main point is that it makes possible the follow-up of the spherical and deviatoric parts of the stress and strain tensors during the test, without restrictive hypothesis on the behaviour of the material. The results show a high sensitivity to the strain rate. The stresses evolutions show that the frictional phenomenon is sensitive even under high pressures.

INTRODUCTION

The aim of this study is to observe experimentally the behaviour of a material under high pressure and high strain rate. The material is an aggregate composite made of small size particle mixed with a highly compliant rubbery polymer [START_REF] Le | Modélisation et identification du comportement plastique visco-élastique endommageable d'un matériau agrégataire[END_REF]. The load domain corresponds to the one which is reached during shocks or during impacts of missiles (from 100 to 1000 MPa for the pressure and roughly 1000 s -1 for the strain rate). The material peculiarities determine the loading path which it will be desirable to follow. Confined compression tests are defined and are obtained using a Split Hopkinson Pressure Bar (SHPB) device. The confinement is made by metallic rings. The dimensions and the constitutive materials of the ring are defined to obtain the wished loading path.

THE EXPERIMENTAL DEVICE

The SHPB device, also called Kolsky's apparatus, is a commonly used experimental technique in the study of constitutive laws of materials at high strain rates. Dynamic compression testing under confinement pressure is not so frequent. For low confinement pressures, experimental data are obtained using a SHPB device combined with a pressure cell [START_REF] Christensen | Split Hopkinson bar test on rock under confining pressure[END_REF], [START_REF] Malvern | Dynamic testing of laterally confined concrete, Micromechanics of failure of quasi brittle materials[END_REF], [START_REF] Semblat | Dynamic response of sand using 3D Hopkinson bar[END_REF], [START_REF] Gary | Behaviour of a quasi-brittle material at high strain rate. Experiment and modelling[END_REF]. This technique does not allow high pressures. Following a few authors [START_REF] Masson | Dynamic compressive testing under confining pressure on a quasi-brittle material[END_REF][7], we propose a test where the specimen is confined by an instrumented metallic ring and axially loaded by a SHPB system (figure 1 a) specially designed for this purpose. This system is composed of long input and output bars with a short specimen placed between them. When the striker impacts the input bar at its free end a compressive longitudinal "incident" wave is created. Once this incident wave reaches the interface between the specimen and the bar, a reflected pulse appears in the input bar and a transmitted one in the output bar. Strain gages are glued on the input and output bars and allows to record these three basic waves. The bars, made of hard steel (1000 MPa), have a 20 mm diameter. The input bar is 3 m long, the output bar is 2 m long and the striker is 1.2 m long. The length of the striker and the strain gages positions allow test duration of 400 µs. Forces and particle velocities at both faces of the specimen can be deduced from incident, transmitted and reflected waves. A complete data processing of the waves is used, including wave dispersion correction and an assisted time shifting based on the elastic transient response of the specimen. This precise data processing is especially important for testing brittle materials that cannot support large strains without damage [START_REF] Zhao | On the use of SHPB technique to determine the dynamic behavior of the materials in the range of small strains[END_REF]. An additional transversal gage is glued on the metallic ring (figure 1b). This gage measures the radial expansion of the ring. From this measurement and with the assumptions of an elastoplastic behaviour of the metallic ring, the lateral pressure on the specimen can be estimated. In order to exploit the tests, we use the assumption of stresses homogeneity in the specimen. The homogeneity of the axial stress is checked since we have the same forces at the two ends of the sample. By opposition to simple compression tests on brittle materials, the inertia effects are removed by the confinement ring. In present tests, we have a strong confinement and a small radial expansion of the ring, in particular when the ring keeps an elastic behaviour. The stresses evolutions are increasing and monotone. It can then be assumed that localisation will not occur. Stress-strain relations and a loading path can then be derived.

THE LOADING PATHS

To precise the stress state, we shall use the pressure P, or hydrostatic stress, and Q, the octahedral shear stress [START_REF] Le | Modélisation et identification du comportement plastique visco-élastique endommageable d'un matériau agrégataire[END_REF]. The subscript 1 denotes the axial direction of the specimen and the subscript 2 denotes the transverse direction.

σ = trace 3 1 P ℑ - σ = P S S : S 3 1 Q = ( ) ( )        σ - σ = σ + σ = 2 1 2 1 3 2 Q 2 3 1 P (1)
Triaxial quasi-static tests on the material were worked out. They allow to determine a shear resistance criterion [START_REF] Christensen | Split Hopkinson bar test on rock under confining pressure[END_REF] and an 20 ° internal friction angle. The usual values of the internal friction angle for brittle materials is included between 30° and 40° . The lower value is probably due to the highly compliant rubbery polymer.

( )

MPa P 25 , 0 12 Q + = °= ϕ = 20 MPa 10 C (2)
This material has also a high Poisson ratio (about 0.42) and its Young modulus E is equal to 2 GPa. If we confine the cylindrical specimen with a metallic ring, the ratio between Young moduli is such that the radial displacement of the ring is negligible. The stress state is very close to the oedometric state (uniaxial strain). With the characteristic values of the material, the shear resistance criterion ( 3 ) can not be reached by the oedometric loading path.

      ν + ν - = 1 2 1 P 2 Q ( ) P 22 , 0 Q = (3)
When the ring behaviour remains elastic, the transversal stress grows faster than when the ring begins to plastify. When the ring is completely plastified, the transversal stress remains constant (this value is denoted by p c ) and the corresponding loading path is parallel to the simple compression one. On figure 2, the line OA corresponds to the beginning of the test when the ring remains elastic. The line AB corresponds to the end of the test, when the ring is totally plastified. The line OC is the loading path corresponding to a simple compression test. The beginning of our test is close to the simple compression (the dotted line shows the real loading path). In fact, there is always a very small space between the specimen and the ring. So, the beginning of the test is close to the simple compression (the dotted line shows the real loading path). In order to obtain several values of the confinement pressure, different rings are used. They are made of steel or of brass with different thicknesses (figure 2b).

RESULTS

In this section, two representative tests results are presented. The first test is realised with a ring made of steel. The thickness of the ring is 5 mm. The confinement pressure imposed at the end of the test is 620 MPa so that the ring is plastified. During this test, the strain rate reached the value of 1300 s -1 . The figure 3a presents the evolution of the axial stress and of the radial stress versus the axial strain. The figure 3b represents the state stress in the P-Q diagram. We can notice, at the beginning of the test, a disturbance due to a short phase of simple compression caused by the small gap between the specimen and the ring. During the elastic behaviour phase of the ring (when the axial strain is less than 0.15), the ratio between radial stress and the axial stress is equal to 2/3. This ratio depends on the specimen Poisson ratio which is equal to 0.42. The second test is realised with a ring made of brass. The thickness of the ring is 2 mm. The confinement pressure imposed at the end of test is 150 MPa and the ring is plastified. During this test, the strain rate reached the value of 1200 s -1 . The figure 4a presents the evolution of the axial stress and of the radial stress versus the axial strain. The figure 4b represents the state stress in the P-Q diagram. We can also notice, at the beginning of the test, a disturbance caused by the small gap between the specimen and the ring. The plastic behaviour phase of this ring is reached for a lower hydrostatic pressure and allows a greater plastic deformation of the specimen than the one obtained in the previous test.

0,0E+00 1,0E+08 2,0E+08 3,0E+08 4,0E+08 5,0E+08 6,
0E+08 0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40

Axial strain

Stress (Pa) Other tests were also worked out with the same device and with the same specimen. Only the thickness of the rings and the impact velocities were changed. These parameters have an influence on the strain rate and on the confinement pressure p c . It is well known that the strain rate has an influence on the octahedral shear stress Q. The parameters of the tests and the results are gathered in table 1. In figure 5, each test is also represented in the P-Q diagram by a little star. For the cases C and D, the line and the dotted line correspond to the quasi-static test

0,0E+00 5,0E+07 1,0E+08 1,5E+08 2,0E+08 0,0E+00 5,0E+07 1,0E+08 1,5E+08 2,0E+08 2,5E+08 3

CONCLUSION

Confined compression tests were worked out using a classical SHPB device. The passive confinement was realized using a ring made of an elasto-plastic material. The tested material is an aggregate composite made of small size particle mixed with a highly compliant rubbery polymer. The material presents an internal friction angle (roughly 20°) which decreases when the pressure becomes higher. This angle is small compared with that of concrete or stones (between 30° and 40 °). This technique allows us to reach a large loading domain which depends on the ring thickness and on its constitutive material (steel or brass). With a steel ring, it is possible to reach a high level of hydrostatic pressure, while with a brass ring, it is possible to reach a large plastic deformation of the specimen but a weaker hydrostatic pressure. One interest of this experimental device is to be able to separate, in the dynamic behaviour, the spherical part and the shear part of the stress tensor. We can notice that the material strength strongly depends on the strain rate. This sensitivity seems rather connected to the shear strain.

The shear stress and the shear strain have an influence which does not allow to consider a simple relation between the hydrostatic pressure and the volume variation in order to describe the spherical behaviour of this material.

Figure 1 .

 1 Figure 1.The SHPB test system (a), the specimen confinement device (b) and typical records of the input and output waves and of the extension strain of the metallic ring (c).

  ) " D é f o rm a t io n " (V )

Figure 2 .

 2 Figure 2. The loading paths (a). A ring made of steel and a ring made of brass (after the test) (b).

Figure 3 .

 3 Figure 3. Dynamic compressive test with a steel ring with 5 mm in thickness. Axial and radial stresses versus the axial strain (a) and stress state in the P-Q diagram (b).

Figure 4 .

 4 Figure 4. Dynamic compressive test with a ring made of brass with a 2 mm thickness. Axial and radial stresses versus the axial strain (a) and stress state in the P-Q diagram (b).

Figure 5 .

 5 Figure 5. Results of several tests in the P-Q diagram.

Table 1 .

 1 Parameters of the tests

		Ring (thickness)	Strain rate (s -1 )	p c (MPa)	P (MPa)	Q (Mpa)
	A1	No ring	0	0	4	13
	A2		500		30	90
	B	Brass (1 mm)	1000	80	140	80
	C1	Brass (2 mm)	0	150	200	60
	C2		1100		270	110
	C3		1200		275	180
	C4		1300		270	140
	D1	Brass (3 mm)	1000	240	340	140
	D2		1400		360	160
	D3		1400		380	200
	E1	Steel (5 mm)	0	620	750	110
	E2		1300		750	190
	E3		1600		850	340


Vincent Levorato
email: vincent.levorato@univ-orleans.fr

MODELING GROUPS IN SOCIAL NETWORKS

Keywords: complex systems, modeling, simulation, social networks, group, pretopology

Studies in social network analysis consider the behavior of a group to be equivalent to the "sum" of individual behaviors. However, a group has to be considered as a complex system where emerging phenomena can appear. In this paper, a formalism is proposed to resolve this problematic by modeling groups in social networks using pretopology as a generalization of the graph theory. We also give an example of simulation using our formalism.

Introduction

Network modeling is used in several research domains (physics, biology, computer sciences, ...) and is mainly based on graph theory. Especially in social networks modeling, graphs are used to describe the links representing relationships or flows between entities. The studies consider in most cases individuals as single elements, a group being formed by several persons interacting with each other. Most work on modeling social networks consider a group as a combination of persons [START_REF] Everett | Extending centrality. In Models and methods in social network analysis[END_REF], not as a whole entity. As social networks are complex networks [START_REF] Newman | The Structure and Dynamics of Networks[END_REF], emergence of phenomena can occur, and the behavior of a group of persons can be different from the "sum" of each behavior of the persons composing the group. Thus, in our opinion, graph theory is inadequate to model all complex interactions occurring in a social network. As some group modeling approaches based on graphs have been proposed before (Everett andBorgatti, 1999, 2005), and more recently [START_REF] Saha | Group proximity measure for recommending groups in online social networks[END_REF], to our knowledge, there is no general framework for modeling a group of entities in a network. From this fact, we propose a mathematical framework for group modeling in social networks from a topological point of view. This paper is structured in two parts: first we introduce pretopology formalism, and we give some definitions and illustrations for group modeling in a social network. Then we show a simple simulation example of a model using our approach. The reader of this paper is supposed to be familiar with graph theory and some concepts from mathematical topology.

Modeling groups in social networks

Before giving theoretical definitions, we first illustrate the problem of a realistic group modeling by giving a simple example: we would like to represent interactions between four persons. A small social network with a simple symmetrical friendship relation shall help in illustrating our approach (Fig. 1).

John Tim

Ed Ben The operation we made in this small example is defining the neighborhood of an element, and more generally, of a set of elements. Thus, we need a theory able to precisely model the neighborhood of sets: here we thought of pretopology formalism.

Pretopology is a mathematical modeling tool for the concept of proximity in mathematical spaces [START_REF] Belmandt | Manuel de prétopologie et ses applications[END_REF]. It furthermore establishes powerful tools for structure analysis and automatic classification [START_REF] Le | Pretopology and a homogeneous method for data clustering[END_REF]. Pretopology theory generalizes topology theory and has weaker axioms than topology, allowing us to analyze transformation procedures step-by-step, and has interesting properties in discrete case. As pretopology formalism is based on set theory, a group of individuals is now considered as a set, allowing us to consider a group as a whole independent entity, which is the main problematic we want to answer.

Pretopological notions

The pretopology formalism comes from classical topology but has weaker axioms. The "closure" process is not necessary idempotent and only the extensivity axiom is required. [START_REF] Brissaud | Les espaces prétopologiques[END_REF] gave the first definition of a pretopological space. [START_REF] Fréchet | Espaces abstraits[END_REF] spaces, [START_REF] Kuratowski | Topologie[END_REF] closure axioms and [START_REF] Cech | Closed sets and closures in pretopology[END_REF] closure operator have been preponderant in definition of a pretopological space and of the pseudo-closure map (Def. 1 and Fig. 2). Some definitions of the formalism used in pretopology can be found in [START_REF] Belmandt | Basics of pretopology[END_REF].

Definition 1 (Pseudo-closure) Let consider a nonempty set E, and P(E) which designates all the subsets of E. A pseudo-closure map, denoted by a(.) : P(E) -→ P(E), is a map meeting the following conditions:

1. a(∅) = ∅ 2. ∀A ∈ P(E), A ⊆ a(A)
A pretopological space is a set E endowed with a pseudo-closure map.

Definition 2 (Pretopological space) A pretopological space is a couple (E, a) where E is a non-empty set and a is a pseudo-closure map a(.) : P(E) → P(E). Pseudo-closure operator is useful to model dynamic processes in the considered space (E, a). In social networks, we can say that elements belonging to a(A) are close to A, allowing us to determine direct neighbors. For each pseudo-closure, we absorb new elements (A ⊆ a 2 (A) ⊆ a 3 (A) ⊆ ... ⊆ a k (A)) which are more and more "distant". Hence, we are able to model complex dynamics like, for instance, information diffusion process. Next, we will briefly give a characterization summarizing the different pretopological spaces that can be encountered. Each type has interesting properties that brings its own corpus of theoretical results and can be used to model particular problems.

A a(A)

Definition 3 (Pretopological space types) A

pretopological space (E, a) can be of various types (considering, when needed, two subsets A and B of P(E)):

• a general pretopological space

• a V-type pretopological space if: A ⊆ B ⇒ a(A) ⊆ a(B) • a V D -type pretopological space if it is a V type and a(A ∪ B) = a(A) ∪ a(B) • a V S -type pretopological space if it is a V D type and a(A) = x∈A a({x})
• a topological space if it is a V D -type and it fulfills the following property :

a(A) = a(a(A)) (idempo- tence)
It can easily be proven that a pretopological V D -type structure is a V S one in the countable case. Habitually, the most interesting space types are the V-type and the general type, because of their weaker axioms. With these definitions, we are able to model emerging phenomena. We can model more general spaces, with more complex behaviors and more complex neighborhoods. To understand this approach, let us return to our network example with the four friends and define the problem by applying pretopology formalism. We build a pretopological space (E, a) with four elements in E and we build a(A) with A ∈ P(E) such as: a(A) = A ∪ people who are friends with each person of A.

With the same assumptions, we make the same operations as in the previous example:

• a({John}) = {John, T im, Ben, Ed} • a({Ed}) = {Ed, John, Ben}
• a({John, Ed}) = {John, Ed, Ben} (Fig. 3)

As we notice, a({John, Ed}) = a({John}) ∪ a({Ed}). The space is not a V D -type. We have also {John} ⊆ {John, Ed} but not a({John}) ⊆ a({John, Ed}). The space is not a V-type either. We observe a general pretopological space. Pretopology helps us to define neighborhoods which are not convenient to describe with other formalisms. In a network, diffusion process modeled by the pseudoclosure can stop its progression. Such a configuration is called a closure [START_REF] Cech | Closed sets and closures in pretopology[END_REF] (Fig. 4).

Definition 4 (Closed subset and Closure) Let (E, a) a pretopological space:

• a subset F of E such as a(F ) = F is called a closed subset of E for a(.). • an elementary closed subset, denoted as F x , is the closure of a one element set {x} of E.

Closure is very important because of the information it gives about the "influence" or "reachability" of a set, meaning, for example, that a set A can influence or reach elements into F (A), but not further.

Modeling group interactions

First, we have to define a social network with pretopology concepts. A social network is a social structure made of nodes (which are generally individuals or organizations) that are tied by one or more specific types of binary or valued relations [START_REF] Degenne | Between tool and theory, social networks. Social Networks of the Internet[END_REF].

In pretopology, we can generalize this definition by saying that a (social) network is a family of pretopologies on a given set E (Fig. 5). to another if they are friends, working together, and living close to each other (using a given distance threshold).

The following paragraphs give to the reader examples of how we can build pseudo-closures depending on the data or information we want to use for our modeling.

Example 1: Metric Space We represent the relations between people with an euclidean distance (the people are represented as nodes), considering for instance the geographical location of each person.

Let E be endowed with a metric defined by a distance d.

Let r be a positive real. For each element x of E, B(x, r) is a ball with the center x and a radius r defined by :

B(x, r) = {y ∈ E/d(x, y) ≤ r}
A pseudo-closure a(.) can be builded with B(x, r):

∀A ∈ P(E), a(A) = {x ∈ E/B(x, r) ∩ A = ∅}
The pseudo-closure a(A) is composed of all elements of A and all elements y / ∈ A such that y is within a distance of at most r from at least one element of A (Fig. 6).

A a(

A) = { x E / B(x,r) A } E a(A)
Figure 6: Pseudo-closure of A in a metric pretopological space.

Example 2: Binary Space In social networks, we are confronted with non-metric relations, representing qualitative information (friendship relation, etc.). In such a space, the elements of E are bound by n reflexive binary relations R i with i ∈ N * . R is not necessarily symmetric. We define ∀ i ∈ {1, ..., n}:

R i (x) = {y ∈ E/xR i y} R -1 i (x) = {y ∈ E/yR i x}
As the relation R is reflexive, x belongs to R(x) and to R -1 (x). We can construct two pseudo-closures:

∀A ∈ P(E), a(A) = {x ∈ E/∀i ∈ {1, ..., n}, Ri(x)∩A = ∅} ∀A ∈ P(E), a(A) = {x ∈ E/∀i ∈ {1, ..., n}, R -1 i (x)∩A = ∅}
These pseudo-closures are not equivalent when R is not symmetric. The pseudo-closure a(A) is composed of A (R is reflexive) and of all elements y which have R n (or R -1 n ) relations with, at least, one element of A. An illustration of this kind of space with two binary relations is illustrated in Fig. 7. In this example, an element Example 3: Valued space In order to model certain problems, binary relations are not sufficient. We therefore need to have a value (integer, real, function, ...) on the links. In this kind of space, the elements of E are bound by a valued relation. For instance, we can define an integer value v on relations as :

x of E belongs to pseudo-closure of A if x is in relation with an element of A according to relation R 1 and R 2 . A E a(A) = { x E / i {1,2} R i (x) A } R 2 R 1 R 2 R 1 R 2 R 1 a(A) R 2 R 1
• E × E → N • (x, y) → v(x, y)
We can build a pseudo-closure giving different results between singletons and sets (E -A and A c are equivalent notations corresponding to the complementary of A and s is an integer):

∀A ∈ P(E), a(A) = {y ∈ E -A / x∈A v(x, y) ≥ s}∪A
The pseudo-closure a(A) is composed of A and of all elements y where the sum of valued edges between some elements of A and y is greater than the threshold s. Figure 8 gives an illustration of this space with s = 4. This kind of modeling can be used in social networks where weighted relations are necessary, and illustrates the interest of the pretopology modeling. Indeed, this example shows that group behavior is different than the "sum" of individuals composing it (V-type). In Fig. 8, the bottom left person (red) is absorbed because he knows 2 persons (value = 1) a little, and better another one (value = 2), so he can be considered as a friend of the group A. If we take each individual of A saying this external individual, the bottom left person, is friend of A if the value of a link is superior to s = 4, he will not be taken in account.

A E a(A) = { y E-A / v(x,y) 4 } A x A 2 1 2 1 1 5 3 4 a(A)

A Complex Group Interaction Model

We show how we can model social networks with complex interactions. The following example is taken from [START_REF] Sampson | A Novitiate in a Period of Change. An Experimental and Case Study of Social Relationships[END_REF]) study, a "toy" dataset which represents several relations between monks in a cloister, and has been often used in sociology studies. It concerns the social interactions which have been collected as numerous sociometric rankings. A lot of relations are coded, but we focus on two of them: esteem and disesteem. Relations are non-symmetric and weighted with three integer values, from 1 to 3 (3 indicates the highest or first choice and 1 the last choice). We builded our model on the following assumptions, that a person y is close to another person x if:

• there is an esteem relation from y to x according to a chosen threshold.

• there is no disesteem relation from x to y according to a chosen threshold.

This proximity definition makes sense if we consider group coalitions. We base our model on the following hypothesis: if a person asks other people "who wants to join my group ?", people who have the greatest esteem for him should join his group more than the others. But this person won't accept in his group people he doesn't esteem, even if these people have esteem for him. We set two valued non symmetric relations with E the set containing individuals :

• E × E → N • (x, y) → est(x, y) • (x, y) → disest(x, y)
with est the esteem relation and disest the disesteem relation.

We build our pseudo-closure as:

∀A ∈ P(E), a(A) = {y ∈ E -A / x∈A est(y, x) ≥ α ∧ x∈A disest(x, y) < β} ∪ A with α, β ∈ N.
With α = 3 and β = 1, we put a strong restriction on neighborhood: if we apply pseudo-closure on a set A, people would be in a(A) if they have esteem with one or several persons of A (according to α), and if no people of A have disesteem for them (see Fig. 9.). A typical question [START_REF] Everett | The centrality of groups and classes[END_REF] in social networks is: who is the person who has the best ability to gather a group around him ? Who is the most influent person ? In this model, finding the largest group following the rules of our neighborhood can be treated by building F e , the family of all elementary closed subsets of E (see Def. 4.). When closure is applied to each singleton of E, it reveals that VICTOR_8 is the person who can rally the biggest amount of people in the network (Fig. 10.). Results of our mathematic model has been simulated in JAVA using PretopoLib library which implements all pretopological concepts [START_REF] Levorato | Data structures and algorithms for pretopology: the java based software library pretopolib[END_REF].

Conclusion

In this work, examples of social networks models based on pretopology formalism have been presented. Building a pseudo-closure map depends on the network nature and problem representation, and can need several pseudo-closures to be combined to obtain specific results. Several previous works have investigated in modeling complex systems with pretopology and have shown some interesting results in: modeling the impact of geographic proximity on scientific collaborations [START_REF] Largeron | A pretopological approach for structural analysis[END_REF], structuring and clustering data [START_REF] Le | Pretopology and a homogeneous method for data clustering[END_REF], modeling pollution phenomena [START_REF] Benamor | A stochastic and pretopological modeling aerial pollution of an urban area[END_REF], or analyzing communities of the Web [START_REF] Levorato | Modeling the complex dynamics of distributed communities of the web with pretopology[END_REF]. We presented ideas and concepts which lead to the following contribution: we proposed a general mathematical framework in order to model groups in social networks. With definitions and examples, we have highlighted the interest to model complex interactions using the pretopology formalism, generalizing graph theory. Furthermore, the introduction of various pseudo-closures functions through examples should be quite helpful in understanding our approach, and we showed an example of how complex group interactions can be modeled. Future work can be axed on some interesting new methods using generalized measures to find groups which maximize a given centrality (degree, betweeness, closeness), to optimize the "efficiency" of a group, or to identify emergent groups in a network. Moreover, centralities measures could be generalized and based on the same mathematical framework.

Figure 1 :

 1 Figure 1: Example of a small social network with friendship relations.

Figure 2 :

 2 Figure 2: Pseudo-closure of A.

Figure 3 :

 3 Figure 3: Pseudo-closure of {John,Ed}.

Figure 4 :

 4 Figure 4: Iterated application of the pseudo-closure map leading to the closure.

Figure 5 :

 5 Figure 5: Social network defined with pretopology and composed of three different pseudo-closures.

Figure 7 :

 7 Figure 7: Pseudo-closure of A in a binary pretopological space.

Figure 8 :

 8 Figure 8: Pseudo-closure of A in a valued pretopological space.

Figure 9 :

 9 Figure 9: Pseudo-closure of A with α = 3, β = 1. Esteem relation is plain and disesteem relation is dashed.

Figure 10 :

 10 Figure 10: Simulation of our model in order to find most influent person according to our neighborhood definition.


Ridha Hambli

A multi-scale bone study to estimate the risk of fracture related to osteoporosis

Objective:

Osteoporosis is a disease marked by decreased density and quality of bone, leading to weakness of the skeleton and increased risk of fracture. Bone fractures caused by the osteoporosis become increasingly important goal for both clinicians and biomedical researchers who try to evaluate and to prevent the risk of fracture. In this study, a finite element model at the nanoscale level "mineralized collagen fibril" is proposed to investigate this problem.

Material and Methods:

The aim of this work is to investigate the effect of mineral density on fracture behaviour of mineralized collagen fibrils (Fig. 1).

Figure1. Mineralized collagen fibrils model

A two-dimensional finite element model of collagen fibril, which is a composite material containing an inorganic mineral hydroxyapatite reinforced by collagen microfibrils linked together by cross-links, with symmetric and periodic boundary conditions is considered in this study. To model the progressive development of damage of bone through the decrease in its elastic stiffness, a homogenized measure of damage is introduced in quasi-brittle damage law, which in the simplest case is represented by a scalar D, called the damage variable.

kl ijkl ij C D   ) 1 (   0 < D < 1 (1)
Where: the Cauchy stress components, kl  the linear strains and ijkl C the components of elasticity tensor.

Bone mineral density measurements are used to diagnose osteoporosis (a condition marked by decreased bone mass). In this work, the variable reflecting mineral density is given by the formula:

m i mp E ψ E  (2) m i I ψ    With 1 0   i  and 1 m   (3) ij 
Where m E Young's modulus of the pure mineral, mp E Young's modulus of the porous mineral and i  the mineral density of the whole structure.

Results:

The result of the proposed finite element simulations coupled to anisotropic damage behavior law is shown below.

Conclusion:

The proposed model coupling anisotropic behaviour with damage law allows for the simulation of the progressive fracture of the fibril in realistic way. Apparent local failure properties of bone can be predicted from nanoscale level. Such local properties can be assigned to whole bone in order to assess its risk of fracture.

Figure3.

  Figure3. Fracture stress of a collagen fibril as a function of fibril mineral density

Figure2.

  Figure2. Stress-strain curve with varying fibril mineral density

Figure4.

  Figure4. Fibril Young's modulus as a function of fibril mineral density

Abdelwahed BARKAOUI * , Awad BETTAMER * , Ridha HAMBLI *

	* PRISME Laboratory, EA4229, University of Orleans
	Polytech' Orléans, 8, Rue Léonard de Vinci 45072 Orléans, France

Acknowledgments:

This work has been supported by French National Research Agency (ANR) through TecSan program (Project MoDos, n°ANR-09-TECS-018).


Jean-François Coeurjolly
email: francois.coeurjolly@upmf-grenoble.fr

Hedi Kortas

Expectiles for subordinated Gaussian processes with applications

Keywords: expectiles, robustness, local shift sensitivity, subordinated Gaussian process, fractional Brownian motion

In this paper, we introduce a new class of estimators of the Hurst exponent of the fractional Brownian motion (fBm) process. These estimators are based on sample expectiles of discrete variations of a sample path of the fBm process. In order to derive the statistical properties of the proposed estimators, we establish asymptotic results for sample expectiles of subordinated stationary Gaussian processes with unit variance and correlation function satisfying ρ(i) ∼ κ|i| -α (κ ∈ R) with α > 0. Via a simulation study, we demonstrate the relevance of the expectile-based estimation method and show that the suggested estimators are more robust to data rounding than their sample quantile-based counterparts.

Introduction

In the statistic literature, there has been a tremendous interest in analysis, estimation and simulation issues pertaining to the fractional Brownian motion (fBm) [START_REF] Mandelbrot | Siam review[END_REF]. This is due to the fact that the fBm process offers an adequate modeling framework for nonstationary self-similar stochastic processes with stationary increments and can be used to model stochastic phenomena relating to various fields of research. A fractional Brownian motion (fBm), denoted {B H (t), t ∈ R} with Hurst exponent 0 < H < 1, is a zero-mean continuous-time Gaussian stochastic process whose correlation function satisfies E[B H (t)B H (s)] = σ 2 2 (|t| 2H + |s| 2H -|t -s| 2H ) for all pairs (t, s) ∈ R × R and σ 2 = E(B H (1) 2 ). The fBm is H-self-similar i.e., for all α > 0, B H (αt) d = α H B H (t), where d = means the equality of all its finite-dimensional probability distributions. The process corresponding to the first-order increments of the fBm is known as the fractional Gaussian noise (fGn) whose correlation function ρ H (i) is asymptotically of the order of |i| 2H-2 for large lag lengths i. In particular, for 1/2 < H < 1, the correlations are not summable, i.e. +∞ i=-∞ ρ H (i) = ∞. This property is referred to as long-range dependence or long-memory whereas the case 0 < H < 1/2 corresponds to short memory.

Several methods aimed at estimating the Hurst characteristic exponent or long-memory exponent have been developed. Among these statistical methods figure the Fourier-based methods such as the Whittle maximum likelihood estimator (see e.g. [START_REF] Beran | Statistics for long memory processes[END_REF]; [START_REF] Robinson | Gaussian semiparametric estimation of long range dependence[END_REF]) or the spectral regression based estimator [START_REF] Beran | Statistics for long memory processes[END_REF]. The wavelet estimators have been also extensively investigated either with an ordinary least squares [START_REF] Flandrin | Wavelet analysis and synthesis of fractional brownian motion[END_REF], a weighted least squares (see e.g. [START_REF] Abry | Wavelets for the analysis, estimation, and synthesis of scaling data[END_REF][START_REF] Abry | Self-similarity and long-range dependence through the wavelet lens[END_REF]; [START_REF] Bardet | Wavelet estimator of long-range dependent processes[END_REF]; [START_REF] Soltani | Estimation of the self-similarity parameter using the wavelet transform[END_REF]) or a maximum likelihood (see e.g. [START_REF] Wornell | Estimation of fractal signals from noise measurements using wavelets[END_REF][START_REF] Percival | Wavelet Methods for Time Series Analysis[END_REF]) estimation schemes. [START_REF] Faÿ | Estimators of long-memory: Fourier versus wavelets[END_REF] present a deep analysis of Fourier and wavelet methods. Recently, the so-called discrete variations techniques (see e.g. [START_REF] Kent | Estimating the fractal dimension of a locally self-similar gaussian process using increments[END_REF]; [START_REF] Istas | Quadratic variations and estimation of the holder index of a gaussian process[END_REF]; [START_REF] Coeurjolly | Estimating the parameters of a fractional brownian motion by discrete variations of its sample paths[END_REF]) have been introduced. Within this class of estimators, [START_REF] Coeurjolly | Hurst exponent estimation of locally self-similar gaussian processes using sample quantiles[END_REF] proposed a new method based on sample quantiles to estimate the Hurst exponent in the more general setting of locally self-similar gaussian processes. This estimator has been proven robust when dealing with outliers [START_REF] Achard | Discrete variations of the fractional brownian motion in the presence of outliers and an additive noise[END_REF]. The latter, often encountered in real world applications, can induce a significant estimation bias. Actually, the advantage of quantiles is that they have a bounded gross-error-sensitivity [START_REF] Hampel | Robust Statistics: The Approach Based on Influence Functions[END_REF][START_REF] Huber | Robust statitics. Wiley series in probability and mathematical statistics[END_REF] allowing them to cope efficiently with the problem of outliers. Nevertheless, this is not the only problem faced when dealing with estimation issues. Indeed, data rounding is also a serious impediment.

Data rounding is common in finance [START_REF] Bijwaard | The effect of rounding on payment efficiency[END_REF][START_REF] Rosenbaum | Integrated volatility and round-off error[END_REF], economics [START_REF] Williams | The effects of rounding on the consumer price index[END_REF], computer science [START_REF] Matthieu | Semantics of roundoff error propagation in finite precision calculations[END_REF][START_REF] Bois | An algorithm for automatic round-off error analysis in discrete linear transforms[END_REF] and computational physics [START_REF] Vilmart | Reducing round-off errors in rigid body dynamics[END_REF] and can lead to several misinterpretations. Quantiles are unfortunately not robust against data rounding since their local shift sensitivity is unbounded, see [START_REF] Hampel | Robust Statistics: The Approach Based on Influence Functions[END_REF][START_REF] Huber | Robust statitics. Wiley series in probability and mathematical statistics[END_REF]. [START_REF] Newey | Asymmetric least squares estimation and testing[END_REF] have introduced the so-called expectile which, although similar to quantile, has a bounded local shift-sensitivity and thus can handle the rounding issue.

In this paper, we derive a Bahadur-type representation for sample expectiles of a subordinated Gaussian process with unit variance and correlation function with hyperbolic decay. This allows us to investigate the statistical properties of a new discrete variations estimator of the Hurst exponent of the fBm process. In constructing this estimator, we rely mainly on the scale and location equivariance properties of expectiles [START_REF] Newey | Asymmetric least squares estimation and testing[END_REF]. We will show via a simulation study the robustness of the proposed estimator against data rounding. The remainder of this paper is structured as follows: Section 2 deals with asymptotic properties of sample expectiles for a class of subordinated stationary Gaussian processes with unit variance and correlation function satisfying ρ(i) ∼ κ|i| -α (κ ∈ R) with α > 0. A short simulation study is conducted to corroborate our theoretical findings. In Section 3, we discuss a sample expectile-based estimator of the Hurst exponent and derive its statistical properties. We then perform a simulation study in order to confirm the effectiveness of the suggested estimation method.

2 Expectiles for subordinated Gaussian processes

A few notation

Given some random variable Z with mean µ, F Z is referred to the cumulative distribution function of Z and ξ Z (p) for p ∈ (0, 1) to its pth quantile. It is well-known that the pth quantile of a random variable Z can be obtained by minimizing asymmetrically the weighted mean absolute deviation

ξ Z (p) := argmin θ E |p -1 Z≤θ |.|Z -θ| .
In order to limit the local shift sensitivity of the pth quantile, [START_REF] Newey | Asymmetric least squares estimation and testing[END_REF] defined the notion of expectile denoted by E Z (p) for some p ∈ (0, 1). Rather than an absolute deviation (function), a quadratic loss function is considered:

E Z (p) := argmin θ E |p -1 Z≤θ |.(Z -θ) 2 .
(1)

We may note that the 50%-expectile if nothing else than the expectation of Z. [START_REF] Newey | Asymmetric least squares estimation and testing[END_REF] argued that providing E[Z] < +∞, then for every p ∈ (0, 1) the solution of ( 1) is unique on the set I FZ := {x ∈ R : F Z (x) ∈ (0, 1)}. The expectile can also be defined as the solution of

the equation E |p -1 Z≤θ |.(Z -θ) = 0.
A key property of the expectile is that it is scale and location equivariant [START_REF] Newey | Asymmetric least squares estimation and testing[END_REF]. The scale equivariance property means that for Y = aZ where a > 0, the pth expectile of Y satisfies:

E Y (p) = aE Z (p) (2)
The pth expectile is location equivariant in the sense that for Y = Z + b where b ∈ R, the pth expectile of Y is such that:

E Y (p) = E Z (p) + b (3)
Now, let Z = (Z 1 , . . . , Z n ) be a sample of identically distributed random variables with common distribution F Z , the sample expectile of order p is defined as:

E (p; Z) := argmin θ 1 n n i=1 |p -1 Zi≤θ | . (Z i -θ) 2 .

Main result

In order to derive asymptotic results for Hurst exponent estimates based on expectiles, we have to provide asymptotic results for sample expectiles of nonlinear functions of (centered) subordinated stationary Gaussian processes with variance 1 and with correlation function decreasing hyperbolically. This will be the setting of the rest of this section. Let {Y i } +∞ i=1 be such a Gaussian process with correlation function ρ(•) satisfying ρ(i) ∼ κ|i| -α for κ ∈ R and α > 0. Let Y = (Y 1 , . . . , Y n ) a sample of n observations and h(Y) = (h(Y 1 ), . . . , h(Y n )) its subordinated version for some measurable function h. We wish to provide asymptotic results for the sample pth expectile defined by

E (p; h(Y)) := argmin θ 1 n n i=1 p -1 h(Yi)≤θ . (h(Y i ) -θ) 2 . ( 4
)
Since the criterion is differentiable in θ, the sample pth expectile also satisfies the following

estimating equation ψ n E (p; h(Y)) ; h(Y) = 0 with ψ n (θ; h(Y)) := 1 n n i=1 p -1 h(Yi)≤θ . (h(Y i ) -θ) . (5)
In the following, we need the two following additional notation for Y ∼ N (0, 1)

ψ h(Y ) (θ; p) := E p -1 h(Y )≤θ .(h(Y ) -θ) ψ ′ h(Y ) (θ; p) := -E p -1 h(Y )≤θ = -p(1 -F h(Y ) (θ)) -(1 -p)F h(Y ) (θ),
the latter quantity corresponding to the derivative of ψ h(Y ) (•, p) if it is well-defined. Let us note that the pth expectile of h(Y ) satisfies ψ h(Y ) (E h(Y ) ; p) = 0. We now present the assumption on the function h considered in our asymptotic result.

[A(h,p)] h(•) is a measurable function such that Eh(Y ) 2 < +∞ and such that the function

ψ h(Y ) (•, p) is continuously differentiable in a neighborhood of E h(Y ) (p)
with negative derivative at this point.

Such an assumption is in particular satisfied under the following one:

[A ′ (h)] h(•) is a measurable function such that Eh(Y ) 2 < +∞, h is not "flat", i.e. for all θ ∈ R the set {y ∈ R : h(y) = θ} has null Lebesgue measure. Indeed, if h satisfies [A ′ (h)] then ψ(•, p) is differentiable in θ. And since, E h(Y ) (p) belongs to the set I h(Y ) = {x ∈ R : F h(Y ) (x) ∈ (0, 1)}, ψ ′ (E h(Y ) (p); p) is necessarily negative.
For the purpose of this paper, our main result will be applied with h

(•) = | • | β (with β > 0) or h(•) = log | • | which obviously satisfy [A ′ (h)].
The nature of the asymptotic result will depend on the correlation structure of the Gaussian process and on the Hermite rank, τ (p, θ) of the function

ψ(t; p, θ) := p -1 h(t)≤θ .(h(t) -θ) -ψ h(Y ) (θ; p).
We recall that the Hermite rank (see e.g. [START_REF] Taqqu | Law of the iterated logarithm for sums of non-linear functions of gaussian variables that exhibit a long range dependence[END_REF]) corresponds to the smallest integer such that the coefficient in the Hermite expansion of the considered function is not zero. For the sake of simplicity, assume that the Hermite rank of this function depends neither on θ nor p and denote it simply by τ . Again, this could be weakened since we believe that the next result could be proved with the following Hermit rank: inf θ∈V(E h(Y );p ) τ (p, θ). As an example, the Hermite

rank of ψ(•, p, θ) is 1 for h(•) = • and (p, θ) ∈ (0, 1) × R and 2 for h(•) = | • | β (β > 0) or log | • | for (p, θ) ∈ (0, 1) × R + \ {0}.
We now present our main result stating a Bahadur type representation for the sample pth expectile of a subordinated Gaussian process.

Theorem 1 Let {Y i } +∞ i=1 a (centered) stationary Gaussian process with variance 1 and correlation function satisfying ρ(i) ∼ κ|i| -α (κ ∈ R), as |i| → +∞ with α > 0 and with a function h

satisfying [A(h,p)]. Let h(Y) = (h(Y 1 ), . . . , h(Y n )) a sample of n observations of the subordi- nated process, then, for all p ∈ (0, 1) E (p; h(Y)) -E h(Y ) (p) = - ψ n E h(Y ) (p); h(Y) ψ ′ E h(Y ) (p); p + o P (r n ), (6)
where the sequence r n = r n (α, τ ) is defined by

r n =          n -1/2 if ατ > 1 n -1/2 log(n) if ατ = 1 n -ατ /2 if ατ > 1.
Proof.

Let us simplify the notation for sake of conciseness: let

E = E(p; h(Y)), E = E h(Y ) (p), ψ n (E) = ψ n (E h(Y ) (p); h(Y)) and ψ ′ (E) = ψ ′ h(Y ) (E h(Y ) (p); p).
The first thing to note is that the sequence r n corresponds to the short-range or long-range dependence characteristic of the sequence ψ(h(Y 1 ); p, θ), . . . , ψ(h(Y n ); p, θ). More precisely r 2 n corresponds to the asymptotic behavior of Eψ n (E) 2 . Indeed, if (c j ) j≥0 denotes the sequence of the Hermite coefficients of the expansion of ψ(•; p, E) in Hermite polynomials (denoted by (H j (t)) j≥0 and normalized in such a way that E[H j (Y )H k (Y )] = j!δ jk ), we may have using standard developments on Hermite polynomials (see e.g. [START_REF] Taqqu | Law of the iterated logarithm for sums of non-linear functions of gaussian variables that exhibit a long range dependence[END_REF])

Eψ n (E) 2 = 1 n 2 n i,j=1 E ψ(Y i ; p, E) ψ(Y j ; p, E) = 1 n 2 n i,j=1 k1,k2≥0 c k1 c k2 k 1 !k 2 ! E [H k1 (Y i )H k2 (Y j )] = 1 n 2 n i,j=1 k≥τ c 2 k k! ρ(j -i) k = O 1 n |i|≤n |ρ(i)| τ =:ρn = O(r 2 n ). ( 7
)
Let us define

V n := r -1 n ( E -E) and W n (E) := -r -1 n ψ n (E)/ψ ′ (E).
We just have to prove that V n -W n (E) converges in probability to 0 as n → +∞. The proof is based on the application of Lemma 1 of [START_REF] Ghosh | A new proof of the bahadur representation of quantiles and an application[END_REF] which consists in satisfying the two following conditions:

(a) for all δ > 0, there exists

ε = ε(δ) such that P (|W n (E)| > ε) < δ.
(b) for all y ∈ R and for all ε > 0 1). (b) We consider only the first limit. The second one follows similar developments. We first state that the map ψ n (•) is decreasing. Indeed, let θ ≤ θ ′ and denote by Z i (θ) the variable

lim n→+∞ P (V n ≤ y, W n (E) ≥ y + ε) = lim n→+∞ P (V n ≥ y + ε, W n (E) ≤ y) = 0. (a) is in particular fulfilled if we prove that EW n (E) 2 = O(1) which follows from (7) since EW n (E) 2 = ψ ′ (E) -2 r -2 n Eψ n (E) 2 = r -2 n × O(ρ n ) = O(
|p -1 h(yi)≤θ) |.(h(Y i ) -θ). If h(Y i ) ≤ θ or h(Y i ) > θ, we obviously get Z i (θ) > Z i (θ ′ ) a.s. leading
to the decreasing of ψ n (•). And in the in between case,

Z i (θ) -Z i (θ ′ ) = p(θ ′ -θ) + θ ′ -h(Y i ) ≥ 0
which leads to the same conclusion. Let y ∈ R, then also using the fact that ψ n ( E) = ψ(E) = 0 and ψ ′ (E) < 0, we derive

{V n ≤ y} = { E ≤ y × r n + E} = {ψ n ( E) ≥ ψ n (y × r n + E)} = {ψ(y × r n + E) -ψ n (y × r n + E) ≥ ψ(y × r n + E) -ψ(E)} = {W n (y × r n + E) ≤ y n }, where y n = r -1 n ψ ′ (E) -1 (ψ(y × r n + E) -ψ(E)). Under the assumption [A(h,p)], y n → y as n → +∞. Now, let U n := ψ ′ (E) (W n (E) -W n (y × r n + E)), explicitly given by U n = 1 nr n n i=1 ψ(Y i ; p, E + y × r n ) -ψ(Y i ; p, E) .
Let c j,n the jth Hermite coefficient of the function r -1 n ψ(t; p, E + y × r n )ψ(t; p, E) , then under the assumption [A(h,p)] and from the dominated convergence theorem we can prove that

c j,n n→+∞ -→ y E |p -1 h(Y )≤E |H j (Y ) =: c j .
Therefore for n large enough,

E[U 2 n ] = 1 n 2 n i,j=1 k≥τ c 2 k,n k! ρ(j -i) k ≤ 2 n |i|≤n k≥τ c 2 k k! ρ(i) τ = O(ρ n ) = O(r 2 n )
which leads to the convergence of U n to 0 in probability. For all ε > 0, there exists n 0 (ε) such that for all n ≥ n 0 (ε), y n ≤ y + ε/2. Therefore for n ≥ n 0 (ε)

P (V n ≤ y , W n ≥ y + ε) = P (W n (y × r n + E) ≤ y n , W n ≥ y + ε) ≤ P (W n (y × r n + E) ≤ y + ε/2 , W n (E) ≥ y + ε) ≤ P (|W n (y × r n + E) -W n (E)| ≥ ε/2) n→+∞ → 0,
which ends the proof.

In the case of short-range dependence, i.e. ατ > 1 then, using the Bahadur type representation of expectiles, we derive immediately the following asymptotic normality for the sample expectile and some generalisations. This result is based on standard central limit theorem for means of subordinated Gaussian stationary processes [START_REF] Taqqu | Law of the iterated logarithm for sums of non-linear functions of gaussian variables that exhibit a long range dependence[END_REF][START_REF] Arcones | Limit theorems for nonlinear functionals of stationary gaussian field of vectors[END_REF].Therefore the proof is omitted.

Corollary 2

(i) Under the assumptions of Theorem 1 with p ∈ (0, 1) and ατ > 1, then as n → +∞

√ n E (p; h(Y)) -E h(Y ) (p) d -→ N (0, σ 2 (p)),
where (ii) Let {Y 1 i } +∞ i=1 and {Y 2 i } +∞ i=1 two (centered) stationary Gaussian processes with variances 1 and correlation functions (resp. cross-correlation functions) ρ 1 , ρ 2 (resp. ρ 12 , ρ 21 ) decreasing hyperbolically with exponents α 1 , α 2 (resp. α 12 , α 21 ). Let p ∈ (0, 1), h a function satisfying [A(h,p)] and let h(Y 1 ) and h(Y 2 ) be the samples of n observations of the two subordinated

σ 2 (p) = 1 ψ ′ E h(Y ) (p); p 2 i∈Z k≥τ c k (p) 2 k! ρ(i) k
samples. If min(α 1 , α 2 , α 12 , α 21 ) × τ > 1, then as n → +∞ √ n E p; h(Y 1 ) -E h(Y ) (p), E p; h(Y 2 ) -E h(Y ) (p) T d -→ N (0, Σ).
where Σ is the (2, 2) matrix with entries Σ ab for a, b = 1, 2 given by

Σ ab = 1 ψ ′ E h(Y ) (p); p 2 i∈Z k≥τ c k (p) 2 k! ρ ab (i) k . ( 8
)
As it was established for sample quantiles [START_REF] Coeurjolly | Hurst exponent estimation of locally self-similar gaussian processes using sample quantiles[END_REF], a non standard limit towards a

Rosenblatt process is expected in the other cases (ατ ≤ 1). This case is not considered here.

Simulations

To illustrate a part of the previous results, we propose a short simulation study in this section.

The latent stationary Gaussian process we consider here is the fractional Gaussian noise with variance 1, which is obtained by taking the discretized increments from a fractional Brownian motion. The correlation function of the fractional Gaussian noise with Hurst parameter (or self-similarity parameter) H ∈ (0, 1) satisfies the hyperbolic decreasing property required in Theorem 1 with α = 2 -2H. Discretized sample paths of fractional Brownian motion can be generated exactly using the embedding circulant matrix method popularized by [START_REF] Wood | Simulation of stationary gaussian processes in [0, 1] d[END_REF] (see also [START_REF] Coeurjolly | Simulation and identification of the fractional brownian motion: a bibliographical and comparative study[END_REF]) which is implemented in the R package dvfBm. of α (or H), p and for the three functions h considered. If we focus on h(•) = (•), we can also remark a higher variance of the sample estimates for α = 0.6 compared to α = 1.4. This is in agreement with the theory since for α = 0.6, ατ = 0.6 < 1 and the rate of convergence is lower than n -1/2 which means an increasing of the variance. For the two other functions considered, then ατ is always greater than 1 (it equals either 2.8 or 1.2 in our simulations) and we do not observe such an increasing of the variance.

To put emphasis on this last point, Figure 3 shows in log-scale the average (over the 9 order of expectiles considered in the simulation, i.e. p = 0.1, . . . , 0.9) of the empirical variances in terms of n for the three h functions and for the two values of α = 0.6 and α = 1.4. We clearly observe that as soon as ατ > 1, the slope of the curves is close to -1 which agrees with the result presented in Corollary 2 for example. When h(•) = 1 and α = 0.6, we observe that the slope is about -0.6 which seems to agree with the convergence in n -ατ which is expected from Theorem 1. 3 Estimation of the Hurst exponent using sample expectiles and discrete variations

Estimation method and asymptotic results

Let X = (X(i)) i=1,...,n be a discretized version of a fractional Brownian motion process and let a be a filter of length ℓ + 1 and of order ν ≥ 1 with real components i.e.: ℓ q=0 q j a q = 0, for j = 0, . . . , ν -1 and ℓ q=0 q ν a q = 0.

Define also X a to be the series obtained by filtering X with a, then:

X a (i) = ℓ q=0 a q X (i -q) , for i ≥ ℓ + 1
and Xa as the normalized vector of X a , i.e.:

Xa = X a E((X a (1)) 2 ) 1/2 .
It should be noticed here that the filtering operation allows to decorrelate the increments of the discretized version of the fractional Brownian motion process. Indeed, it may be proved (see e.g. -2ν as |i| → +∞. Consider the sequence (a m ) m≥1 defined by:

Coeurjolly (2001)) that: ρ a H (i) ∼ k H |i| 2H
a m i =    a j if i = jm 0 otherwise for i = 0, . . . , mℓ ,
which is the filter a dilated m times. It has been shown in [START_REF] Coeurjolly | Estimating the parameters of a fractional brownian motion by discrete variations of its sample paths[END_REF][START_REF] Coeurjolly | Hurst exponent estimation of locally self-similar gaussian processes using sample quantiles[END_REF] that:

Xa m = X a m σ m where σ 2 m = m 2H σ 2 κ a H and κ a H = -1 2 ℓ q,q ′ =0 a q a q ′ |q -q ′ | 2H .
The following proposition allows us to construct an ordinary least squares (OLS) estimator of the Hurst exponent H of a fBm process based on sample expectiles.

Proposition 3 Let E p; h(X a m ) and E p; h( Xa m ) be the pth order sample expectiles for the filtered series h(X a m ) and h( Xa m ) respectively. Here two positive functions h(•) are considered, namely:

h(•) = | • | β for β > 0 and h(•) = log | • |. We have: E p; |X a m | β = σ β m E p; | Xa m | β ) (9)
and

E p; log |X a m | = 1 2 log(σ 2 m ) + E p; log | Xa m | . ( 10
)
Proof.

We have:

E p; |X a m | β = argmin θ 1 n -mℓ n-1 i=mℓ |p -1 {|X a m (i)| β ≤θ} |.(|X a m (i)| β -θ) 2 = argmin θ 1 n -ml n-1 i=ml |p -1 {| Xa m (i)| β ≤ θ σ β m } |.(| Xa m (i)| β - θ σ β m ) 2 . Setting θ ′ = θ σ β m
, the proof of the first relation ( 9) follows easily. Using the same methodology, we can demonstrate the result given by equation ( 10).

Remark 1 It should be stressed here that the scaling relationship relating the theoretical pth expectiles for the series h(X a m ) and h( Xa m ) can be obtained directly using the scale equivariance Now applying the logarithmic transformation to both sides of ( 9), we get:

log E p; |X a m | β = βH log(m) + log σ β (κ a H ) β/2 E |Y | β (p) + log   E p; | Xa m | β E |Y | β (p)   . ( 11
)
On the other hand, (10) can be reformulated in the following way:

E p; log |X a m | = H log(m) + 1 2 log(σ 2 κ a H ) + E log |Y | (p) + E p; log(| Xa m |) -E log |Y | (p)) . (12)
It is noteworthy here that we expect that log

E p; | Xa m | β /E |Y | β (p) and E p; log(| Xa m | β ) - E log |Y | (p)
to converge towards 0 as n → ∞. Hence, based on equations ( 11) and ( 12), we opt for an OLS regression scheme. This allows to derive the two following estimators of the hurst index defined by:

H β = A T β||A|| 2 log E p; |X a m | β m=1,...,M , (13)
and

H log = A T ||A|| 2 E p; log |X a m | m=1,...,M , ( 14
)
where A is the vector of length M with components A m = log m-1 M M m=1 log(m), m = 1, . . . , M for some M ≥ 2 whereas ||z|| for some vector z of length d designates the norm defined by

d i=1 z 2 i 1/2
. Notice here that H β and H log do not depend on σ 2 . We would like to put the stress on the fact that ( 13) and ( 14) are really similar to the ones developed in [START_REF] Coeurjolly | Estimating the parameters of a fractional brownian motion by discrete variations of its sample paths[END_REF][START_REF] Coeurjolly | Hurst exponent estimation of locally self-similar gaussian processes using sample quantiles[END_REF]. Indeed, the standard procedure developed in [START_REF] Coeurjolly | Estimating the parameters of a fractional brownian motion by discrete variations of its sample paths[END_REF] simply consists in replacing the sample expectile by the sample variance (this method will be denoted by ST in Section 3.2). To deal with outliers, the procedure developed in [START_REF] Coeurjolly | Hurst exponent estimation of locally self-similar gaussian processes using sample quantiles[END_REF] consists in replacing the sample expectile by either the sample median of (X a m ) 2 or the trimmed-means of (X a m ) 2 . These two last methods are denoted by MED and TM in Section 3.2. Now, we state the asymptotic results for these new estimates based on expectiles.

Proposition 4 Let a a filter with order ν ≥ 2, p ∈ (0, 1), β > 0 then as n → +∞, H β and H log converge in probability to H. Moreover, the following convergences in distribution hold

√ n H β -H d -→ N (0, σ 2 β ) and √ n H log -H d -→ N (0, σ 2 log ),
where

σ 2 β = 1 E |Y | β (p) 2 × A T Σ β A β 2 A 4 and σ 2 log = A T Σ log A A 4
and where the (M, M ) matrices Σ β and Σ log are defined by (8).

Proof. We only provide a sketch of the proof. We claim that once Theorem 1 and Corollary 2 are established, the obtention of convergences stated in Proposition 4 are semi-routine. First of all, let us notice that

H β -H = A T β A 2   log   E p; | X a m | β E |Y | β     m=1,...,M (15)
and

H log -H = A T β A 2 E p; log | X a m | -E log |Y | (p) m=1,...,M . (16)
Since the functions | • | β and log | • | have Hermite rank 2 and since the correlation function of the stationary sequence X a m decreases hyperbolically with an exponent α = 2ν -2H then for any m ∈ {1, . . . , M }, Theorem 1 holds with r n = n -1/2 (since ατ > 1 for all H ∈ (0, 1)). This ensures the convergence in probability of the new estimates.

The cross-correlation between X a m 1 and X a m 2 is defined by

ρ a m 1 ,a m 2 H (j) = π a m 1 ,a m 2 H (j) π a m 1 ,a m 1 H (0) 1/2 π a m 2 ,a m 2 H (0) 1/2 with π a m 1 ,a m 2 H (j) = ℓ q,r=0
a q a r |m 1 qm 2 r + j| 2H .

Lemma 1 in [START_REF] Coeurjolly | Hurst exponent estimation of locally self-similar gaussian processes using sample quantiles[END_REF] states that for all m 1 , m 2 the correlation function ρ a m 1 ,a m 2 H is also decreasing hyperbolically with an exponent α = 2ν -2H, then Corollary 2 may be applied to prove that

E p; | X a m | β -E |Y | β m=1,...,M d -→ N (0, Σ β ) (17) and E p; log | X a m | -E log |Y | m=1,...,M d -→ N (0, Σ log ), (18)
where according to (8), the (M, M ) matrices Σ β and Σ log are respectively defined by 17) and ( 18) combined with ( 15) and ( 16) and the use of the deltamethod (for the convergence of H β ) end the proof.

Σ β m1m2 = 1 ψ ′ E |Y | β (p); p 2 i∈Z k≥2 c β k (p) 2 k! ρ a m 1 ,a m 2 H (i) k (19) Σ log m1m2 = 1 ψ ′ E log |Y | (p); p 2 i∈Z k≥2 c log k (p) 2 k! ρ a m 1 ,a m 2 H (i) k , (20)

A short simulation study

In this section, we investigate the interest of the new estimators based on expectiles. We consider three different models in our simulations. (c) rounded fBm: we assume the data are given by the integer part of a discretized sample path of an original fBm.

To fix ideas, Figure 4 provides some examples of discretized sample paths of standard and contaminated fBm. The simulation results are presented in Tables 1 and2. For these simulations, as suggested in [START_REF] Coeurjolly | Estimating the parameters of a fractional brownian motion by discrete variations of its sample paths[END_REF], we chose the filter a = d4 corresponding to the wavelet Daubechies filter with order 4 (see [START_REF] Daubechies | Ten lectures on wavelets[END_REF]) and the maximum number of dilated filters M = 5. Also, in other simulations not presented here, we have observed that the estimates H β perform better than H log and, among all possible choices of β, the value β = 2 seems to be a good compromise. Therefore, we present only the result for this latter estimator, that is H β with β = 2.

In a first step, we had observed a quite large sensitivity to the value of the probability p defining the expectile. In order to have an efficient data-driven procedure, we propose to choose the probability parameter p via a Monte-Carlo approach as follows:

1. Estimate the parameters H and σ 2 using the standard method (the estimation of σ 2 is not described here but it may be found for example in [START_REF] Coeurjolly | Estimating the parameters of a fractional brownian motion by discrete variations of its sample paths[END_REF]). Denote these estimates H 0 and σ 2 0 .

2. Generate B = 100 contaminated fBm with Hurst parameter H 0 and scaling coefficient σ 2 0 , define a grid of probabilities (p 1 , . . . , p P ). For each new replication, we estimate H 0 with expectiles for all the p i . The optimal p, denoted in the tables by p opt , is then defined as the one achieving the smallest mean squared error (based on the B = 100 replications).

The procedure based on expectiles, denoted E(p) in the results, is compared to the standard method (ST) and to methods which efficiently deal with outliers, that is methods MED and TM (the last one is calculated by discarding 5% of the lowest and the highest values of (X a m ) 2 at each scale m).

The standard fBm model is used as a control to show that all methods perform well. As seen in the first two columns of Tables 1 and2, this is indeed true. All the methods seem to be asymptotically unbiased and have a variance converging to zero. We can also remark that in this situation whatever the value of H, estimates based on expectiles exhibit a performance which is very close to the one of the standard method (wich can also be viewed as the method based on expectile with p = 0.5). Several types of expectiles are investigated. When the discretized sample path of the fBm is contaminated by outliers, we recover the results already shown in [START_REF] Coeurjolly | Hurst exponent estimation of locally self-similar gaussian processes using sample quantiles[END_REF], [START_REF] Achard | Discrete variations of the fractional brownian motion in the presence of outliers and an additive noise[END_REF] or [START_REF] Kouamo | Central limit theorem for the robust log-regression wavelet estimation of the memory parameter in the gaussian semi-parametric context[END_REF]: methods based on medians or trimmed-means are very efficient which is in agreement with the fact that quantiles have a finite gross error sensitivity. The inefficiency of expectiles for such a contamination is also coherent since expectiles have infinite gross error sensitivity. Finally, the interest of the expectile-based method can be seen with the rounded fBm corresponding to the last two columns of each table.

In this situation, expectiles are shown to be more efficient in terms of bias and its variance seems to be not too much affected by this type of strong contamination. We also put the stress on the interest and efficiency to choose the p value based on a Monte-Carlo approach. Hurst parameter H = 0.8 and sample size n = 500, 5000 are given between brackets. Methods based on expectiles, quantiles and trimmed-means as well as the standard method are considered.

The filter a correspond to the Daubechies wavelet filter with order 4 (two zero moments) and we set M 1 = 1, M 2 = 5. According to a sample size and a model, the method achieving the lowest mean squared error is printed in bold.

  and where c k (p) is the kth Hermite coefficient of the expansion of the function ψ(h(•); E h(Y ) (p); p) in Hermite polynomials.

Figures 1

 1 Figures 1 and 2 illustrate the convergence of the sample expectiles. Three h functions are considered: h(•) = (•), (•) 2 and log | • |. The related Hermite rank of the function ψ is respectively 1,2 and 2 for these three h functions. The sample size of the simulation is fixed to n = 500. We can claim the convergence of the sample expectile E(p; h(Y )) towards E h(Y ) (p) for all the values

Figure 1 :

 1 Figure 1: Boxplots of sample expectiles for expectiles of order p = 0.1, . . . , 0.9 based on m = 500 replications of fractional Gaussian noise with length n = 500 and with Hurst parmeter H = 0.3 (left, α = 1.4) and H = 0.7 (right, α = 0.6). The h functions considered here is the identity function (with Hermite rank 1). The curves correspond to the theoretical expectile functions for Y ∼ N (0, 1).

Figure 2 :

 2 Figure 2: Boxplots of sample expectiles for expectiles of order p = 0.1, . . . , 0.9 based on m = 500 replications of fractional Gaussian noise with length n = 500 and with Hurst parmeter H = 0.3 (left, α = 1.4) and H = 0.7 (right, α = 0.6). Two h functions with Hermite rank 2 have been considered here: h(•) = (•) 2 (top) and h(•) = log | • | (bottom). The curves correspond to the theoretical expectile functions for Y 2 (middle) and log |Y | (bottom) where Y ∼ N (0, 1).

Figure 3 :

 3 Figure 3: Means of empirical variances of sample expectiles in terms of n in log-scale based on m = 500 replications of fractional Gaussian noise with parameters H = 0.3 (left, α = 1.4) and H = 0.7 (right, α = 0.6). More precisely, we consider the vector of probability (0.1, . . . , 0.9) for the orders of the expectiles and we compute σ 2 n = 1/9 × 9 i=1 σ 2 i,n where σ 2 i,n is the empirical variance for the expectile with order i/10 for the sample size n. Three choices of h functions have been considered: h(•) = (•), (•) 2 and log | • |.

  property (2) for h(•) = | • | β and the location equivariance property (3) for h(•) = log | • |.

  where (c β k ) k≥2 and (c log k ) k≥2 are respectively the Hermite coefficients of the functions | • | β and log | • |. The convergences (

  (a) standard fBm: non-contaminated fractional Brownian motion.(b) fBm with additive outliers: we contaminate 5% of the observations of the increments of the fractional Brownian motion with an independent Gaussian noise such that the SNR of the considered components equals -20Db.

Figure 4 :

 4 Figure 4: Examples of discretized sample path of standard fBm (top), fBm with additive outliers (middle) and rounded fBm (bottom) for n = 500 and with Hurst parameters H = 0.2 (left) and H = 0.8 (right).

Table 1 :

 1 Empirical means and standard deviations of H estimates based on m = 500 replications of non-contaminated and contaminated fractional Brownian motions with scale parameter σ = .5, Hurst parameter H = 0.2 and sample size n = 500, 5000 are given between brackets. Methods based on expectiles, quantiles and trimmed-means as well as the standard method are considered.The filter a correspond to the Daubechies wavelet filter with order 4 (two zero moments) and we set M 1 = 1, M 2 = 5. According to a sample size and a model, the method achieving the lowest mean squared error is printed in bold.

		Standard fBm	fBm with additive outliers	Rounded fBm
		n = 500	n = 5000	n = 500	n = 5000	n = 500	n = 5000
	E(p = 0.2)	0.198 (0.033)	0.200 (0.011)	0.280 (0.062)	0.298 (0.024)	0.334 (0.036)	0.337 (0.011)
	E(p = 0.4)	0.198 (0.032)	0.200 (0.010)	0.288 (0.068)	0.309 (0.026)	0.298 (0.034)	0.300 (0.011)
	E(p = 0.6)	0.199 (0.032)	0.200 (0.010)	0.298 (0.076)	0.323 (0.029)	0.284 (0.035)	0.287 (0.011)
	E(p = 0.8)	0.199 (0.033)	0.200 (0.010)	0.311 (0.086)	0.349 (0.034)	0.275 (0.037)	0.277 (0.011)
	E(p = p opt )	0.199 (0.035)	0.200 (0.011)	0.314 (0.085)	0.368 (0.033) 0.249 (0.040) 0.240 (0.012)
	MED	0.197 (0.048)	0.200 (0.016)	0.227 (0.050)	0.227 (0.016)	0.451 (0.158)	0.361 (0.119)
	TM	0.206 (0.034)	0.201 (0.011) 0.222 (0.052) 0.225 (0.016)	0.294 (0.038)	0.289 (0.012)
	ST 0.199 (0.032) 0.200 (0.010)	0.293 (0.072)	0.315 (0.027)	0.290 (0.034)	0.292 (0.011)

Table 2 :

 2 Empirical means and standard deviations of H estimates based on m = 500 replications of non-contaminated and contaminated fractional Brownian motions with scale parameter σ = .5,


Aymen Laadhari

Pierre Saramito

Chaouqi

Chaouqi Misbah

Vesicle tumbling inhibited by inertia

Keywords: numbers: 87, 16, D-83, 50, Ha 87, 17, Jj 83, 80, Lz 87, 19, rh

published or not. The documents may come

I. INTRODUCTION

The study of models of red blood cells (RBCs), such as capsules and vesicles, has known recently an upsurge of interest in various communities, such as physics, applied mathematics, mechanical engineering, and so on. A systematic experimental analysis 1 of RBCs under shear flow has revealed that RBCs can either undergo a tanktreading (TT) or tumbling (TB) motion. In the TT regime the main axis makes a given angle (less than π/4) with respect to the flow direction whereas the RBCs membrane undergoes a tank-treading motion. Upon increasing the viscosity contrast λ (ratio of the internal over the external fluid viscosities) the RBC exhibits TB (or flipping) motion. There are two classes of model systems that are used to mimic RBCs: (i) capsules and (ii) vesicles. Capsules are made of an extensible polymer membrane which is endowed with an in-plane shear elasticity (mimicking the elasticity of the cytoskeleton of the RBCs) [START_REF] Barthès-Biesel | Capsule motion in flow: deformation and membrane buckling[END_REF] . Vesicles are, like RBCs, made of an inextensible phospholipid bilayer membrane which is purely fluid (thus devoid of shear elasticity) [START_REF] Vlahovska | Vesicles and red blood cells in flow: from individual dynamics to rheology[END_REF] . The inextensible character has proven to confer to vesicles rich dynamics [START_REF] Vlahovska | Vesicles and red blood cells in flow: from individual dynamics to rheology[END_REF] , since inextensibility, triggers, even to leading order, high order nonlinearities.

An early model to understand the TT-TB transition of RBCs has been presented by Keller and Skalak (KS) [START_REF] Keller | Motion of a tank-treading ellipsoidal particle in a shear flow[END_REF] . They adopted a vesicle-like model (fluid inextensible membrane), and imposed a fixed shape (only orientation in the flow is permitted). The shape of the vesicle is assumed to be ellipsoidal for which a solution for the Stokes flow was available. KS reported that the transition depends both on the viscosity contrast λ and on the reduced volume v (the actual volume over the volume of a sphere having the same area), which is given in 2D (the situation in which we are interested here) by v = (A/π)/(P/2π) 2 , where A is the area occupied by the internal liquid and P is the vesicle perimeter.

The analytical as well as numerical calculations [START_REF] Keller | Motion of a tank-treading ellipsoidal particle in a shear flow[END_REF][START_REF] Rioual | Analytical analysis of a vesicle tumbling under a shear flow[END_REF][START_REF] Beaucourt | Steady to unsteady dynamics of a vesicle in a flow[END_REF] for the TT-TB transition have been restricted to the Stokes limit: the inertia was neglected. Available experimental data on vesicles [START_REF] Kantsler | Transition to tumbling and two regimes of tumbling motion of a vesicle in shear flow[END_REF][START_REF] Mader | Dynamics of viscous vesicles in shear flow[END_REF] correspond to the very small Reynolds numbers limit. However, in the blood circulatory sys-tem, especially in the arterioles, the Reynolds number evaluated at the scale of the RBC may be of order unity. For example, in the arterioles 9 the mean velocities is of about U ∼ 20cm/s. Defining the Reynolds number as Re = ρ o RU/η o where R is the RBC radius (the radius of a sphere having the same area as the RBC; it is equal to about 3 µm for human RBCs), η o the plasma viscosity and ρ o its density, leading to Re ∼ 1. From this rough estimate (see later for more precise estimate) it follows that the inertial effect is of the same order as the viscous one.

Our main objective in this study is to analyze the dynamics of vesicles under a shear flow by taking into account the inertial effects. Our study reveals that a Reynolds number of order unity is capable of inhibiting TB in favor of TT motion. We analyze the phase diagram in the relevant parameter space, and discuss some implications together with the experimental feasibility to test this prediction. We consider a 2D vesicle under a linear shear flow. The legitimacy of a 2D geometry is supported by the fact that the 2D dynamics obtained in numerical simulations [START_REF] Beaucourt | Steady to unsteady dynamics of a vesicle in a flow[END_REF] are in good agreement with results obtained in 3D by KS. Hereafter r will denote a two dimensional position vector having the Cartesian components x, along the plates, and y, in the perpendicular direction (see Fig. 1). The vesicle is immersed in a fluid occupying a square domain of lateral length 2L. Let R be the radius of a circle having the same perimeter as the vesicle. The aspect ratio is given by R/L.

The velocity and pressure fields obey in the two fluid domains (inside and outside the vesicle)

ρ ∂u ∂t + u.∇u -div (2η D(u)) + ∇p = 0 div u = 0
where D(u) = (∇u + ∇u T )/2 is the deformation rate tensor, with η = η i is the internal viscosity and η = η o is the external one. At the vesicle membrane the hydrodynamic stress is balanced by the the membrane force

-κ ∂ 2 H ∂s 2 + H 3 2 n + Hζn - ∂ζ ∂s t + [2ηD(u)-pI] n = 0
where κ is the membrane bending rigidity, H is the curvature, n and t are the normal and tangential unit vectors, and ζ is a Lagrange multiplier enforcing locally constant arclength. It is fixed by requiring the surface divergence of the velocity field to vanish div s u = 0

The two plates at y = ±L move horizontally with opposite constant velocity V (shear flow). At y = ±L, we impose no-slip condition u = (±V, 0), while on the lateral size x = ±L we impose stress-free surface (see Fig. 1). The Navier-Stokes equations are nonlinear, so that the quite precise method based on boundary integral formulation [START_REF] Beaucourt | Steady to unsteady dynamics of a vesicle in a flow[END_REF][START_REF] Veerapaneni | Analytical and numerical solutions for shapes of quiescent twodimensional vesicles[END_REF] can not be used. We resort here to the level set approach. Very briefly, in this method the membrane location is defined by a level set function φ(r, t) depending on the 2D vector r = (x, y) and time. The membrane position is taken to be (this is an implicit representation of the membrane) φ = 0, and the level set function obeys a transport equation

∂φ ∂t + u.∇φ = 0
Because φ is defined everywhere in the entire domain, the fluid/membrane interaction is solved in a fully Eulerian scheme. The normal and the tangent vectors, as well as the curvature are defined in terms of φ: n = ∇φ/|∇φ|, H = div n, so that the full membrane force can be defined in the entire domain. The Lagrange multiplier is also defined in the entire domain. Note that while the membrane force is defined in the whole domain, its action is localized to the membrane region only.

The above set of equations has been reformulated in terms of a variational representation (the so-called weak formulation) and has been implemented in a finite element scheme. We use an improvement of the classical level set method that takes care of solving exactly the volume and area constraints. Details of the numerical study and benchmark tests are published elsewhere [START_REF] Laadhari | Computing the dynamics of biomembranes by combining conservative level set and adaptive finite element methods[END_REF] , while we focus here on the main physical results only.

We adimensionalize the equations by choosing R as a length scale, U = V R/L as a velocity scale, T = R/U as a unit of time and η o U/R as a unit of pressure. This leads us to three dimensionless physical parameters

Re = ρ o V R 2 η o , Ca = η o R 3 γ κ , λ = η i η o
This set has to be supplemented with two geometrical parameters, namely the vesicle confinement Cn = R/L and the reduced area v.

Here we shall take the same densities inside and outside the vesicle, and we shall keep the confinement to a given value (typically 0.4). Moreover, in 2D, the TT-TB transition is quasi-insensitive 6 to Ca. We set Ca = 100, and have checked that higher or smaller values do not affect the results. We are thus left with three free parameters v, λ and Re. In the exploration of the inertial effect, we shall prescribe a given value for v and vary only λ and Re. However, in order to compare our results obtained in the quasi-Stokes regime to existing numerical results in the Stokes regime we have also examined several values of Cn and v, as described below. In order to test and validate our analysis, we compare our results with available numerical data and set Re = 10 -2 . In our computations, a small confinement Cn = 0.25 has been chosen, so that the influence of boundaries be weak enough (see also [START_REF] Laadhari | Computing the dynamics of biomembranes by combining conservative level set and adaptive finite element methods[END_REF] ). We have varied λ, v and have determined the transition line separating the regime of tank-treading (TT) from that of tumbling (TB). The results are presented in Fig. 2 showing a good agreement with the phase field method (see Fig. 5 in Ref. [START_REF] Beaucourt | Steady to unsteady dynamics of a vesicle in a flow[END_REF] ). Remark that the 2D KS theory 4 , based on some simplification hypothesis, is able to capture the main behavior of the phase diagram. Next we examine the role of the Reynolds number on the two dynamical regime TT and TB. A first noticeable effect is that upon increasing Re in the TB regime the period of oscillation increases significantly (Fig. 3.a and 4) until it diverges (Fig. 3.b) for a critical value of Re. The interesting fact is that this behavior occurs for quite moderate values of Re. The divergence of the period means that the TB is suppressed in favor of a TT regime. Another impact of the inertial effect is that the vesicle adopts in the TT regime a terminal angle that significantly depends on Re; the terminal angle can have values which may be twice as large as compared to those obtained in the Stokes regime (see Fig. 5). Furthermore, the effect of inertia causes a stronger deformation of the vesicle, as shown on Fig. 1.

III. RESULTS AND DISCUSSION

Our current intuitive understanding is as follows. In the absence of inertia TB occurs [START_REF] Rioual | Analytical analysis of a vesicle tumbling under a shear flow[END_REF] when the viscosity contrast reaches a critical value such that the torque due to the applied shear flow can not anymore be efficiently converted into the membrane tank-treading torque because the internal fluid is so viscous that it precludes tanktreading. Therefore the vesicle behaves as almost quasirigid, and TB takes place. In other words, in the TB regime, the injected power due to shear is predominantly transferred to dissipation of the surrounding fluid. TB in the presence of inertia must, besides dissipation in the surrounding fluid, be accompanied with kinetic energy transfer to the surrounding fluid, a cost that increases with Reynolds number so that TB becomes unfavorable.

We have performed a systematic analysis (at fixed reduced area v = 0.82) on the occurrence of TT and TB as a function of the viscosity contrast λ and Reynolds number Re. The results are reported on Fig. 6. In the absence of inertia (Re = 0) the TT-TB bifurcation occurs at about λ = 6. As Re is increased TB is delayed. For example, at Re ≃ 5 the critical λ has approximately doubled (it is about 12). It is interesting to note that the separation line in Fig. 6 continues to increase, but still for Re ∼ 16, TT continues to prevail even though the viscosity contrast is quite large (λ = 80). At such a large contrast one might be tempted to expect the vesicle to behave like a rigid particle (meaning that TB prevails). However, inertia is still capable of enforcing membrane tank-treading, enabling the vesicle to enjoy its fluidity, making it quite distinct from a rigid particle. Some concluding remarks might be useful. In human arterioles 9 the wall shear rate is of about 8000 s -1 . Using the Reynolds number definition adopted here,

Re = ρ o V R 2 η o , with R ≃ 3 µm, and η o /ρ o ≃ 10 -2 cm 2 /s
(plasma kinematic viscosity), one obtains Re ∼ 0.1. When geometry suddenly changes (bifurcations, etc.) velocity gradients are enhanced resulting in much higher transient inertial effects. Furthermore, it is believed that the viscosity contrast of RBCs in vivo 4 is close to (in the Stokes limit) the critical value for the TT-TB transition. This idea would mean that inhibition of TB should be favored by moderate inertial effects (Re ∼ 1). Why should it be beneficial for RBCs to perform TT rather than TB at all? Our speculation is as follows. Under an alteration of the flow due, for example, to a tissue injury, or caused by abnormal widening of blood vessels (the so-called sites of aneurysms), RBCs tend to scrape along the blood vessel lining. In the TT regime cells experience a lift force of hydrodynamical (viscous) nature [START_REF] Mader | Dynamics of viscous vesicles in shear flow[END_REF][START_REF] Seifert | Hydrodynamic lift on bound vesicles[END_REF][START_REF] Cantat | Lift force and dynamical unbinding of adhering vesicles under shear flow[END_REF] . This force is essential to keep cells away from potential undesirable adhesion. However, RBCs which undergo TB experience practically no lift force (due to the quasi-up-down symmetry over a period of TB) [START_REF] Danker | Vesicles in Poiseuille flow[END_REF] . Transient suppression of TB (at special sites undergoing sudden changes of flow condition, where inertial effect are transiently amplified) would enable RBCs to be efficiently pushed away from blood vessel walls. A systematic study dealing with 3D simulations including shear elasticity (to mimic the RBC cytoskeleton) is an essential step in order to be more quantitative.

Vesicles would be a more adequate system in order to test the present finding, but also to check the above speculations. Vesicles may be produced with large enough size, for example, with radius in the range 20 -40 µm. Imposing wall shear rates of about 8000 s -1 one finds Re ≃ 3 -15, where we expect suppression of TB to be quite favorable. In a microfluidic channel of diameter d ∼ 100 µm, with maximum velocity V max of the imposed Poiseuille flow, the wall shear rate is equal to 8V max /d. Imposing a maximum velocity of order 10 cm/s (for which high speed camera are well adapted to capture vesicle dynamics with a good precision), one easily obtains shear rates of the order of 10 4 s -1 .

C.M. and A.L. Acknowledge financial support from CNES and ANR (MOSICOB project).

Figure 1 .

 1 Figure 1. (color on-line) The set-up and notations. Actually the vesicle shapes, in the TB regime, shown here are obtained by numerical simulations. Left: a vesicle with a weak inertia (Re = 0.01). Right: a vesicle with inertia (Re = 10) showing ample vesicle deformation as compared to that in the Stokes limit. The finite element network and the flow lines are shown.

Figure 2 .

 2 Figure 2. Phase diagram and comparison with the phase field method 6 (circles and crosses) where ε denotes the interface width.

2 Figure 3 .

 23 Figure 3. The behavior of the angle of the main axis of the vesicle as a function of time for different Reynolds numbers, showing that inertia slows down the motion, until suppressing TB (v = 0.82, λ = 20).

Figure 4 .Figure 5 .

 45 Figure 4. The behavior of the TB period T vs Re for various viscosity ratio λ, showing divergence of the period at a critical Reynolds number.

Figure 6 .

 6 Figure 6. The phase diagram of the TT and TB motion as a function of the viscosity contrast λ and the Reynolds numbers (v = 0.82, Ca = 100, Cn = 0.5).