domain: math shs

text:
Luis Siero
email: lsiero@uabc.edu.mx

Avenilde Romo
email: aromov@ipn.mx

Arturo Abundez

Methodology for designing didactical activities as an engineering project of a tactile sensorial therapeutic ramp

Keywords: Mathematical models, linear algebra, engineering training, ATD. Thematic Working Group 06

The objective of this investigation is designing an activity research study in mathematical modeling for the training of future engineers in the subjects of linear algebra, and structural mechanics of composite materials, in an application of stress and strain calculation. Considering elements of the Anthropological Theory of Didactics (ATD) implementing a methodology within the mentioned theory. This allows the analysis of mathematical models in use, as basis for designing didactical activities, in order to create a link between these two subjects, showing the future engineers that mathematics can be used to solve problems in an extra-mathematical context.

Introduction

The objective of this work is to design modeling activities for a mathematical training of engineers. The design is based on mathematical modeling analysis on specialty training courses, focusing on laminated composite materials. This collaborative work involves aerospace, mechanical engineers and mathematicians that teach in the carriers of Aerospace engineering and mechanical engineering, having an opportunity to analyze a real context of modeling; namely Calculation of stress and strain of composite materials. This project was proposed on a structural mechanics of composite materials course; because the students wanted to know where they could use the mathematics they were learning. For this work we have considered elements of the Anthropological Theory of Didactics proposing a methodology associated to this theory that permits the analysis of mathematical models in use based on the design of didactic activities.

Elements of the Anthropological Theory of Didactic

The ATD is an epistemological model that allows the study of human activity in its institutional dimension. An institution is a stable social organization that defines the human activities generating resources that make them possible. These materials or intellectual resources, which are made available to the subjects, have been produced by communities along the confrontation of problematic situations with the objective of solve them in a regularly and effectively way [START_REF] Castela | Des mathématiques a l'automatique: étude des effets de transposition sur la transformée de Laplace dans la formation des ingénieurs[END_REF]. The classic praxeological model, proposed by [START_REF] Chevallard | La recherche en didactique et la formation des professeurs : Problématiques, concepts, problèmes[END_REF], recognizes the praxeology [𝑇, 𝜏, 𝜃, Θ] as a minimal unit of analysis of human activity. Its four components are: the task type (𝑇), the technique (𝜏), the technology (𝜃); and the theory (Θ). The 'task' refers to what is to be done; the 'technique' is how it is to be done; the 'technology' is a discourse that produces, justifies and explains the 'technique'; while the 'theory' produces, justifies and explains the 'technology'.

The training of engineers through institutions

The training of engineers can be seen through institutions, Romo (2009) distinguish three types: production of knowledge P(S), teaching of knowledge E(S) and use of knowledge or practices Ip. This distinction seeks to recognize the first vocation of the institutions and the production of knowledge correspond to scientific disciplines, such as mathematics or engineering sciences, are validated in these the existence of knowledge and the relations between them. The teaching of knowledge E(S) are responsible for displaying and disseminating praxeologies, meanwhile in the institutions of usage Ip, the praxeologies are used to solve problems of practice. This does not mean that within the institutions P(S) there is no teaching or usage praxeologies nor in teaching E(S) and practice Ip there is no production of knowledge. In this investigation linear algebra is considered an institution of teaching mathematics E(AL), and in teaching engineering, structural mechanics of composite materials E(MC) and connect them through a Study and Research Activity (SRA). The SRA constitute didactic devices for the construction of a praxeology through three didactic moments that are: first encounter with T, exploration of T and the emergence of the τ technique and construction of the technological-theoretical block, in this case coming from the structural mechanics of composite materials as outlined below In order to initiate the transition from the traditional paradigm in mathematics "visitinng works" (to the teaching of pre-existing mathematical objects) to questioning the world according to inquirybased mathematics education (IBE).

Didactical design methodology of an SRA

The methodology initially proposed in [START_REF] Macias | Uso de las nuevas tecnologías en la formación matemática de ingenieros[END_REF] allows the design of SRA that involves non mathematical elements. Its four stages are: 1) Election of an extra-mathematical context; 2) Praxeological analysis and identification of a mathematical model; 3) Analysis of the identified mathematical model and their relationship with E(M) and 4) design of the SRA.

Election of an extra-mathematical context

To choose an appropriate extra-mathematical context for the design of an SRA, the following elements were considered. Teachers. In any of your subjects taught in an engineering career you use mathematics? Have you used a mathematical model or adapted one in any of your courses? Which model and for what? 67 students from 3rd to the 7th semester were surveyed, finding that the most useful subjects are calculus and linear algebra. 54% recognizes the work with mathematical models. All teachers who teach subjects of MC, integral calculus CI and AL were surveyed, 92% of them recognize the use of matrices in their courses 2) Interviews with the coordinator of the career of mechanical engineering, who noted that the subject of MC required early in the course of a review of linear algebra.

3) Approach with MC teachers, who pointed out that in general, students do not recall procedures or operations previous courses, AL for example. 4) Joint work with engineers-researchers in the area of materials.

Praxealogical analysis and identification of a mathematical model

The praxeology that is identified and analyzed is the calculation of stress, strain and elastic modulus in laminated materials. The type of task is to calculate the stress or strain of a laminated material, the technique is associated with the use of the matrix S (strain) or matrix Q (stiffness), the technology is Hooke's law and the theory is the mechanics of materials. The analysis is based on a technical report of basic mechanics of laminated composite plates [START_REF] Nettles | Basic mechanics of laminated composite plates[END_REF], suggested by an engineerresearcher, who indicated it, as a heavily used reference material. In this section the theoretical technological-block is shown, which displays how the stiffness matrix associated with the technique of calculating the stress and strain is presented. This will provide the basis to show in the next section full praxeology from the analysis of an exercise presented in a class of structural mechanics of composite materials.

Technology: Generalized Hooe´s law for anisotropic materials 1 Nettles explains that the relationship between stress and strain is independent of the direction of the force, and is provided by the constant of elasticity (Young's modulus), this is for isotropic 2 materials. In nonisotropic materials it should use two elastic constants at least. The relationship stress / strain for isotropic materials appears as follows:

σ = Eε (1)
Where 𝜎: is the stress, E: Denotes the Youngs modulus and 𝜀: is the strain. For orthotropic materials 3 , the direction must be specified in the stress/strain relationship:

𝜎 1 = 𝐸 1 𝜀 1 ; 𝜎 2 = 𝐸 2 𝜀 2
(2) where 𝜎 1 : Denotes the stress in the longitudinal direction 𝐸 1 : Denotes the stiffness in the longitudinal direction (Young´s modulus) 𝜀 1 : Denotes the strain in the longitudinal direction 𝜎 2 : Denotes the stress in the transversal direction 1 Anisotropic materials: is the material that its mechanical properties differ according to the load direction 2 Isotropic materials: It is the material that has identical mechanical properties in all directions regardless of the direction of the load 3 Orthotropic materials: is the material in which mechanical properties are different in three perpendicular directions 𝐸 2 : Denotes the stiffness in the transversal direction (Young´s modulus) 𝜀 2 : Denotes the strain in the transversal direction 𝐸 1 = 𝐸 𝐿 Defines the stiffness in the longitudinal direction and 𝐸 2 = 𝐸 𝑇 is the stiffness in the transversal direction. This law produces different techniques, Nettles initiated by the special orthotropic plates and is why we analyze them below..

Stress and strain for special orthotropic plates

The author begins by explaining that on a plate, stress can be given in more than one direction. Immediately he defines Poisson's ratio as the strain perpendicular to a given loading direction, showing the relationship for different loads.

For loading along the fibers:

𝑃𝑜𝑖𝑠𝑠𝑜𝑛´𝑠 𝑟𝑎𝑡𝑖𝑜 = 𝜈 12 = 𝜀 𝑇 𝜀 𝐿 = 𝜀 2 𝜀 1 (3a)
For loading perpendicular to the fibers

𝑃𝑜𝑖𝑠𝑠𝑜𝑛´𝑠 𝑟𝑎𝑡𝑖𝑜 = 𝜈 21 = 𝜀 𝐿 𝜀 𝑇 = 𝜀 1 𝜀 2 (3b)
The strain is equal to the difference between the stretched component deformation due to an applied force and contraction of the Poisson´s effect due to other forces perpendicular to the applied force, thus:

𝜀 1 = 𝜎 1 𝐸 1 -𝜈 21 𝜀 2 𝑦 𝜀 2 = 𝜎 2 𝐸 2 -𝜈 12 𝜀 1 (4a)
aplicando la ecuación ( 2)

𝜀 1 = 𝜎 1 𝐸 1 -𝜈 21 𝜎 2 𝐸 2 𝑦 𝜀 2 = 𝜎 2 𝐸 2 -𝜈 12 𝜎 1 𝐸 1 (4b)
Subsequently, the author considers the presence of shear forces. The shear stress and shear strain are related by a constant called shear modulus, denoted by G.

𝜏 12 = 𝛾 12 𝐺 12 (5)
Where: 𝜏 12 : Shear stress, 𝛾 12 : Shear strain y 𝐺 12 : Shear modulus Equation ( 5) is similar to equation (1) it only considers shear stress and strain, where the indices 1-2 indicate shear in the 1-2 plane. The author mentions that a relationship exists between the Poisson constant and the Young's modulus in both directions, in the longitudinal direction and the transverse direction to the material, and then it holds that:

𝜈 21 𝐸 1 = 𝜈 12 𝐸 2 (6)
Equations ( 4b) and ( 5) can be written in their matrix form obtaining

where,

Here it is where we can see a relationship between the subjects of linear algebra and structural mechanics of composite materials as a matrix model for calculating stress or strain of laminated We can appreciate in equation 7 to 8, the calculation of the inverse matrix was made, to switch the stress matrix to the stiffness matrix, with basic operations taught in linear algebra. The author explains, broadly, the mathematical model based on the algebraic work, having left the reader the task of verifying the connections between the different equations, until reaching the mathematical model that relates the stress, strain and Young's modulus (which are the elastic properties of the material). All this is the technological component of the praxeology. To illustrate the types of tasks that can be solved and the associated techniques to the praxeology, we analyze below a classroom exercise from the subject of structural mechanics of composite materials.

Analysis of the identified mathematical model and its relationship with E(M)

In the class of structural mechanics of composite materials (MC), where we analyzed and identified the praxeology of the stress calculation of a laminate material, with a task type T, calculate the modulus of elasticity of a laminated material in a particular direction, with two tasks, t1 y t2: t1: Calculate the modulus of elasticity of a laminated material in the X direction. Determine the stiffness matrix of the lairs in 45°.

Using the expressions for the calculation of the stiffness matrix, for any fiber orientation it can be written by:

[𝑄] = [ 𝑄 ̅ 𝑥𝑥 𝑄 ̅ 𝑥𝑦 𝑄 ̅ 𝑥𝑆 𝑄 ̅ 𝑦𝑥 𝑄 ̅ 𝑦𝑦 𝑄 ̅ 𝑦𝑆 𝑄 ̅ 𝑆𝑥 𝑄 ̅ 𝑆𝑦 𝑄 ̅ 𝑆𝑆 ]
xy axes: global axes; 1,2 axes : material axes. Thus the stiffness matrix turns to be: For the -45° the stiffness matrix is [𝑄] = [ 17.03 11.43 -7.72 11.43 17.03 -7.72 -7.72 -7.72 -7.01 ] 𝐺𝑃𝑎

Stiffness matrix in the flat tension of the laminate material In the direction of the X axes the relation between the strain and stress is given by: 𝜀 𝑥 = 0.06362 (10 9 ) 𝜎 𝑥 , on the other hand the apparent elasticity modulus in the X direction is presented as 𝐸 𝑥 = 1.572 𝐺𝑃𝑎 Particularly in the Y axis direction the relationship between the stress and strain is: 𝜀 𝑦 = 0.08874 (10 9 ) 𝜎 𝑦 . Furthermore the apparent elasticity modulus in the X direction is given by 𝐸 𝑦 = 11.27 𝐺𝑃𝑎. The praxeological analysis of the technical report and the classroom exercise presented here very briefly, allowed to identify the matrix as a mathematical model for calculating strain of a laminate (isotropic, anisotropic and orthotropic) material. Also, the matrix operations and calculating the inverse of the matrix are used to determine the modulus of elasticity of a composite material, specifically laminated materials.

SRA design proposed for a tactile sensorial therapeutic ramp (RTEST)

For the SRA design, we considered the construction of a product that requested the use of a laminated material and calculating the stress and strain of the material to ensure the usefulness of the product. Therefore, it was intended that the use of the mathematical model appear the way it happens in engineering projects. Determining that for students in the early college years, more than a structure was necessary to think of a product consisting of plates, which also would be useful for the community. Thus, it came to the proposal of a tactile sensory therapeutic ramp (RTEST) with laminate material to help children from three to ten years to correct gait problems.

Moment of the first meeting with T. To design the ramp students must develop three basic tasks, t1) design the ramp, t2) choose the laminate and stress calculations, t3) determine the type of material with which the ramp will stimulate the sensory part. Moment of the exploration of T and technical emergency τ. Students should find the technique or techniques to solve t1, t2 y t3. For t1, they should investigate the types of ramps, analyze and choose one. For t2, they should investigate laminated materials, choose one and make the stress calculation (the technic, p. (3-5)). And for t3, investigate the materials that promote sensory stimulation and choose one, justifying the reasons for their choice.

Moment of construction of the technological-theoretical block. The third moment intersects with the second because here students must build the ramp, using drawings and stress calculations previously made, as well as the preparation of the composite material. In t2 students should know and apply Hooke's law, for the problem they are solving, they have to know how to build the stress matrix and determine the stiffness matrix. To perform these calculations students can use computer programs such as MathLab, Scilab and SolidWorks.

First implementation of the SRA

Presenting a first implementation of the evolution of the SRA of one team:

Conclusion

The SRA is proposed within the framework of the paradigm of questioning the world in the training of engineers. In this SRA unlike the commonly proposed projects of engineering the mathematical topics are highlighted. In addition the engineering topics are shown in a more important roll in the mathematics subjects. The SRA involved students and teachers from different specializations. The design of the material and the RTEST requires the calculation of stress -matrix model-, knowledge of materials and design. To do this the students must investigate and study elements from different disciplines as well as practical knowledge, students learn to do research, model, use available knowledge to create new, teamwork, communicate their ideas and justify the practice with theoretical elements of different levels. The analysis of the development of the SRA would allow us to understand the institutional necessary conditions for designing SRA in a more complex environment.

Figure 1 :

 1 Figure 1: The SRA as an element linking educatinal institutions

1)

  Generation of surveys aimed at teachers and students about mathematical needs of engineers in training: Students. Which of these subjects you thought more important, and why? Have you used or adapted a mathematical model? Which model and for what?

  where ℎ is the thickness of the material[𝐴] in the plane stress of the laminate is written as[𝐴 * The relation between the average stress and the strain of the material is given by: {𝜎} = [𝐴 * ] • {𝜀}; Calculating the strain state for the loading state.

  𝝉 𝟐 : Applying a tensile stress in the Y directionAnalogously the mean stress acting on the laminate material is

Figure 2 :

 2 Figure 2: Momentum schematics of the SRAThe implementation with students of the core curriculum of engineering, designers and aerospace engineering; Of three different semesters (2, 6 and 7), trying to mimic the form of work of the industry: as different specialties as well as novices and experts engineers. The SRA was proposed to each teacher of the course to see if the assignment was pertinent to their subject and if they could make it part of their class and grade it. The development of the SRA was parallel to the classes of the teachers that agreed to work on the project assigning a certain time in each course for doubts they might have. A report was requested for each of the three phases. Phase 1: Proposal of a design for a RTEST ramp (3 weeks), phase 2: Strain calculation of the laminate material (3 weeks) and phase 3: Elaborate and choose materials for the RTEST ramp: Laminate materials and for the tactile and

Proceedings of CERME10

This took place in the Research Center for Applied Science and Advance Technology


text:
Avenilde Romo
email: aromov@ipn.mx

Santiago Tolentino
email: santiago.tolentino1@outlook.com

Rebeca Romo-Vázquez
email: rebeca.romovazquez@gmail.com

Didactical activities to engineering training: Methodological proposal

Keywords: Beer industry, Pareto chart, training engineer, didactical activity

In this paper we present a methodology to design didactical activities for training engineers. One phase of this methodology is selecting an extra-mathematical context, allowing identification and analysis of mathematical models used by engineers. We selected an industry beer context and we identified the Pareto chart as a tool to solve different problems, for example faults in production lines. This work uses elements from Anthropological Theory of Didactics. We present a praxeological analysis as basis of didactical activities.

Introduction

Over the past 30 years, mathematical modelling and applications have been a subject of study and development in the field of Educational Mathematics, as in the ICMI 14 study, edited by Blum, Galbraith, Henn and Niss (2007). Mathematical modelling and applications are activities in which math is used to solve problems in various contexts (math, engineering, economy, medicine, etc.). They are based on developing competencies associated with the application of mathematics and constructing mathematical models in extra-mathematical contexts [START_REF] Niss | Introduction Part I[END_REF]. However, mathematical modelling and applications are not widely used in primary and secondary education, which give more importance to concepts and procedures. At the university level, modelling has emerged as a new educational paradigm [START_REF] Bissell | Telling tales: Models, stories and meanings[END_REF]. In the future, teaching professionals will be required not only to create but also adapt mathematical models to solve practical problems [START_REF] Bissell | Telling tales: Models, stories and meanings[END_REF] related to interpreting solutions proposed by other professionals and to employ technology to perform mathematical tasks. This demands teaching math in such a way that it includes the use, adaptation, handling, and interpretation of mathematical models in order to adequately deal with tasks in extra-mathematical contexts. One of perspectives that comes closest to this demand is known as the realistic perspective, which is described in the proceedings of the Topic Study Group 21 (TSG21) of the 11 th International Congress on Mathematical Education in 2008, as follows: "In this perspective, mathematical modelling is viewed as applied problem solving and a strong emphasis is put on the real life situation to be modelled and on the interdisciplinary approaches" (Blomhoj & Carreira, 2009). In this work, we consider the work of [START_REF] Kadijevich | Simple spreadsheet modelling by first-year business undergraduate students: Difficulties in the transition from real world problem statement to mathematical model[END_REF], which illustrates this perspective, is a very interesting approach to the mathematical modelling due to the type of problems chosen to address it and to the use of information technology that is proposed. However, we believe that is not clear, from a theoretical point of view: how do you choose the actual context? How is a modelling activity generated in the context chosen? How is analyzed their relevance to the classroom? In general, it seems that the question of choosing the extra-mathematical context in order to propose the modelling activity has been theoretically little worked, it appears as a little cloudy element, leaving the emphasis on the characteristics of didactic activities [START_REF] Galbraith | Framework for identifying student blockages during transitions in the modelling process[END_REF] and on the modelling cycles that allow to describe and analyse it [START_REF] Blum | Mathematical modelling: Can it be taught and learnt[END_REF]. In an effort to attend to this lack of theoretical framing of the choice of additional mathematical context for the design of didactic activities, we have proposed a methodology based on the extended praxeological model [START_REF] Castela | Des mathématiques a l'automatique: Étude des effets de transposition sur la transformée de Laplace dans la formation des ingénieurs[END_REF] for designing didactic activities based on modelling that incorporate elements of the use of models in real contexts.

Elements of the Anthropological Theory of Didactic

Since the extended praxeological model [START_REF] Castela | Des mathématiques a l'automatique: Étude des effets de transposition sur la transformée de Laplace dans la formation des ingénieurs[END_REF] is based on the Anthropological Theory of Didactics (ATD), we present in this section some of its elements. The ATD is an epistemological model that allows the study of human activity in its institutional dimension. An institution is a stable social organization that defines the human activities generating resources that make them possible. These materials or intellectual resources, which are made available to the subjects, have been produced by communities along the confrontation of problematic situations with the objective of solve them in a regularly and effectively way [START_REF] Castela | Des mathématiques a l'automatique: Étude des effets de transposition sur la transformée de Laplace dans la formation des ingénieurs[END_REF]. The classic praxeological model, proposed by [START_REF] Chevallard | La recherche en didactique et la formation des professeurs: Problématiques, concepts, problèmes[END_REF], recognizes the praxeology [T, τ, θ, Θ] as a minimal unit of analysis of human activity. Its four components are: the task type (T), the technique (τ), the technology (θ); and the theory (Θ). The 'task' refers to what is to be done; the 'technique' is how it is to be done; the 'technology' is a discourse that produces, justifies and explains the 'technique'; while the 'theory' produces, justifies and explains the 'technology'. Mathematical praxeologies or mathematical organizations can be of different level and they serve to a hierarchy of levels of determination proposed by [START_REF] Chevallard | La recherche en didactique et la formation des professeurs: Problématiques, concepts, problèmes[END_REF]. Mathematical institution imposes a model of subjection to the mathematical praxologies: it rests on a structure that organizes praxeologies in different interlocked levels that are in increasing order of size as follows: specific, local, regional and global. The most basic level of a mathematical organization is the punctual [T, τ, θ, Θ] and it has only one technique for performing such tasks. The next level is the local, which groups all punctual mathematical organizations associated with the same θ technology. The regional level regroups all punctual organizations associated with same theory Θ, global or domain regroups certain regional mathematical organizations; discipline is the top level and combines all domains.

Codetermination of the mathematical and didactic [START_REF] Chevallard | La recherche en didactique et la formation des professeurs: Problématiques, concepts, problèmes[END_REF] develops the model presented below, in order to take into account the subjections that weigh on the didactic organization of the study of praxeologies. In this work, the author notes that didactic organizations cannot be developed if they are found far from higher levels, domain and discipline; reciprocally these levels cannot be imposed without considering the conditions of the educational institution. In that sense yields a co-determination of mathematical and didactic organizations.

[…] each level imposes, at a given moment during life of educational system a set of constraints and support points: ecology that results is determined both by what restrictions prohibit or drive, and the exploitation that actors make to the support points that different levels offers. (Chevallard, 2002, p.49) As you can see the fact that Chevallard be interested in teaching leads to extend the range of levels. He introduces three higher levels: society, school and pedagogy, noting that levels of domain and discipline are also subject to restrictions imposed by these three levels that complement the scale upward: Society → School → Pedagogy → Discipline → Domain →Sector → Local → Specific. This hierarchy makes us consider that the study of mathematical praxeology or modelling praxeology in an institution that is subject to various restrictions imposed by institutions of higher levels.

Moments of the study

In the frame of ATD, the study is seen as the construction or reconstruction of elements of a mathematical praxeology, in order to perform a troublesome task (a task type for which a mathematical praxeology does not exist or is not available). In order to finely describe this process of construction or reconstruction, the ATD proposes a model of study of a punctual mathematical praxeology. This model distinguishes six moments, which are also associated with groups of activities. A moment is a dimension of the activity, a phase in the process of study, which may appear several times but following an internal global dynamics. [START_REF] Chevallard | La recherche en didactique et la formation des professeurs: Problématiques, concepts, problèmes[END_REF] Group II (Synthesis) 4. Moment of institutionalization.

Group III (Exercises and problems) 5. Moment of work of the mathematical organization (specifically of the technique).

Group IV (Controls) 6. Moment of the evaluation. These moments are not detailed in this paper, but they are presented with the aim of showing how the ATD in the process of construction and reconstruction of a praxeology is conceived.

In particular, we are interested in considering group I and generating an SRA: a didactic device for students to construct, in this case, a modeling praxeology involving the Pareto Diagram. One of our questions is, what legitimizes the chosen mathematical modeling praxeology? Why is it important for students to build this praxeology? And in particular, future engineers. For us it is important that the mathematical modeling of the classroom is related to the mathematical modeling of its professional practice, which is seen as a relationship between institutions as shown below.

Training in mathematics and the professional world seen as institutions

In the framework of the ATD, analysis of mathematical activity is considered in its institutional dimension. Given that our proposal is to generate a methodology for designing activities based on mathematical modelling that links mathematical knowledge teach at the training institution and the one used in the professional field, we must identify institutions that are adequate to participate in this process, and their interrelations. [START_REF] Romo-Vázquez | Les mathématiques dans la formation d'ingénieurs[END_REF] argued that training engineers involves three types of institutions: Production (P), where praxeologies are produced, Teaching (E) responsible for transmitting the praxeologies. Use or users Ip, where the praxeologies are employed. By producing institutions we refer to mathematics P(M) seen as a discipline, together with its intermediate disciplines P(DI), which we will also call Engineering Sciences (e.g. signal processing, control theory, electrical circuits, etc.). Teaching institutions are represented by mathematics E(M), and the intermediate disciplines E(DI); while the practical institution is Ip. The latter is examined at two levels: the professional practice of engineers, and the devices that, upon approaching practice, are developed in schools; for example, a project to innovate a product or service. Also taken into account were three inter-institutional tours that can be followed by a mathematical praxeology by going from P(M) to Ip. These can be represented schematically as follows (Figure 2): The transpositive effects (changes that occur upon moving from one institution to another) can be so large that a mathematical praxeology in Ip may not be recognized as mathematical. Research by Hoyles, Noss and Pozzi (2000) shows that some professionals utilize techniques and strategies in their practice that are based on mathematical models, but that when automated are no longer recognized as such. This begs the question: what sorts of didactic activities can be generated so that the mathematical models used in Ip practice or E(DI) find a place in methods of teaching mathematics E(M) (see Figure 2)? To answer these questions we propose the following methodology.

Figure 2: News institutional relations between Ip-E(M) and E(DI)-E(M)

Methodology for designing didactic activities (SRA) based on mathematical modelling

This methodology for designing didactic activities based on mathematical modelling emerged from research by Macias (2012). Here, modelling activities are seen as praxeologies (mathematical and/or modelling) to be performed in E(M), but in relation to praxeologies of E(DI) and/or Ip. It consists of four phases: 1) Selecting an extra-mathematical context, 2) Praxeological analysis and identification of a mathematical model. 3) Analysis of the mathematical model identified and its relation to E(M) and 4) Design of the didactic activity (SRA) for E(M).

1) Selecting an extra-mathematical context. First, we must consider the educational level at which teaching will take place, then the contexts where the mathematical applications will be put to use. For example, if we consider teaching programs for engineers, the natural contexts of use are specialty training E(DI) and professional practice Ip. After that, one must identify some of the elements (resistance of materials, control theory, data structure, among others) that are of macro scale and may include various sub-institutions for the analysis of the modelling activity that occurs there. Selecting this context must be based on an approach to the institution or sub-institution through interviews with one or more of the subjects involved (e.g. professors, expert users, researchers), a review of relevant documentation (suggested by the aforementioned subjects, and/or one's own search), and visits aimed at identifying the type of mathematical and modelling activity that is used. Specifically, it is important to analyse whether the mathematical models identified as being in use correspond to those that are actually taught E(M); examples could include functions, vectors, matrixes, mathematical optimization, or differential equations, among others. In this way, one can determine whether the context chosen provides a suitable analytical basis for designing the didactic activity.

2) Praxeological analysis and identification of a mathematical model. In this phase mathematical modelling activity is analysed through praxeologies. Modelling activities in an extramathematical context may consist of mathematical praxeologies and/or mixed praxeologies.

3) Analysis of the mathematical model identified and its relation to E(M).

A mathematical model that is in use but that is also taught in E(M) is identified and then analysed through the functions of the technology practice; i.e., describe, validate, explain, facilitate, motivate and evaluate. Describing the model in use allows us to elucidate the reasons relative to context on the basis of which that particular model was chosen to resolve tasks in the extra-mathematical context. Identifying the elements that validate the use of the model, and under what conditions, makes it possible to understand what contextual elements must be considered in designing didactic activities. For example, many mathematical models are used in "ideal" conditions such that they make it possible to resolve certain tasks more easily, though the solutions obtained will later need to be adapted to reality. This adaptation is conducted on the basis of certain elements that validate it.

Recognizing the explanations of use allows us to understand what each element of the model represents and to what degree the model used allows us to model the context (or part of it).

Analyzing the elements that facilitate the use of the model reveals the process of mathematical modelling, which entails not only assuring that the mathematical model chosen will make it possible to resolve a problem in an extra-mathematical context but also that the resolution reached will be the least complex one. Identifying precisely what it is that motivates the use of the chosen model is a medullar phase in designing didactic activities, but this analysis of use must be complemented by a didactic analysis of the model in the context in which it is taught.

4) Designing didactic activities (SRA).

Designing a didactic activity must be based on both the praxeological analysis of use (praxeologies present in an Iu) and the mathematical model identified; i.e., one must recognize the praxeologies of both use and teaching in order to perceive the relations between them. One must choose the types of tasks that, because they emerge from use, can be adapted for a type of school task; for example, studying the behavior of a continuous signal, determining the total cost of an inventory, or calculating the inverse of a mixed matrix, etc. These types of tasks require mathematical techniques that may be school-related (being part of curriculum) but are also used in the professional field, so mathematical and non-mathematical technologies will have to be built by the students (3 rd moment) in order to validate, explain and justify techniques that emerges when confronting the problematic tasks. Specifically, technologies of use (part of practical praxeologies) must be adapted in such way the students can build them in the first college courses. The objective of the didactic activities must be oriented towards the type of praxeologies that figure in the activity, whether this be constructing, mobilizing, or searching, for knowledge. It is intended that this activity may be an SRA which allows building a praxeology of modelling.

The context proposed: Pareto chart in beer industry

To make a praxeological analysis we are chosen a Beer industry that is constituted by ten production lines and produce about one million hectoliters of beer per month. Apart from domestic beer this industry produce lots of beer for export to countries in all continents. To meet domestic demand and shipments abroad, it requires each of the ten production lines, meet high levels of efficiency. However, in each production line problems requiring immediate attention in order to achieve the planned goals they are presented. Thus various problems for the maintenance of thousands of machines, components and parts that make up each of the production lines are also presented. And logistics to control the flow of materials required in the production and control of shipments. In [START_REF] Tolentino | Uso del principio de Pareto en la industria cervecera y su posible vinculación con la enseñanza de las matemáticas[END_REF], these questions are studied: What Mathematics used in this industrial environment? Are there some common mathematical tools to manage the wide range of problems in the brewing? Tolentino, was both a master's student of mathematics education program and he was working in the industry as an engineer, he found that a Pareto chart is used to solve problems in the industry. This chart is based on the principle that if 20% of the causes of problems is attended a solution of 80% is obtained in effect.

Praxeology faults problems

Type of task. Identify the most important causes of faults in production lines of an industry.Task. To solve faults (time) that arise in different production lines of beer production. Technique. Step 1. Data collection. It is recorded in a table stop time of production lines due to operational faults, faults in machinery or material defects. Step 2. Is ordered from highest to lowest the time column of this table, the percentages of stop time are obtained for each line, in relation to the accumulated from 157.62 hours. A column for cumulative percentage is added. Step 3. Two graphs are performed in Excel: in the bar graph the columns for each line indicate stop time. Graph of the Lorenz curve or cumulative percentage. On the right vertical axis measures the time from 0 to 41.02 h. For the Lorenz curve on the right vertical axis the percentage of 0 is measured at 100 percent (See figure 3). Step 4. Identify the diagram. Finally add a line from the 80 percent that intersects the graph of the cumulative percentage, and descends to the x axis, to separate lines that are to the left of this line it is the line 2, 8, 12, 6 and 10 (see figure 4). These are the lines that generate 80 percent of the total time of overall production stoppages during this time period. That is although the line 11 was stopped 11 hours during this week, is considered out of importance according to the analysis Pareto realized. In this case, the Pareto Diagram is used again to analyze lines 2, 8, 12 and 6, to make decisions about the elements that must be repaired urgently, in each of the lines. The Pareto Diagram is made by engineers and allows them to identify the main causes of the problem, however a deeper analysis of the industry is necessary to recognize how the practical and theoretical knowledge allows solving the identified causes. This analysis is repeated in the brewing industry several times to have elements to act and address problems optimally, using fewer resources and obtaining the greatest benefits.

Conclusion

We consider that this praxeological analysis (phase 2 of methodology), briefest presented, allow us to see the importance of Pareto chart on the beer industry. To design the SRA, it is necessary to analyze university courses likely to identify a local Praxeology the Pareto chart. However, analysis of the Pareto chart of the beer industry gives elements for SRA: from data of different problems ask propose a model that allows the company to identify the major causes of the problems. Considering the problem of faults, you can ask students a model to identify the line that causes the greatest wasting time or the lines that cause 50% of the strikes, then 75% and then 80%. The interesting thing about this proposal may lie not only that students can work with data from businesses, but they reach handle the Pareto Principle, the 80-20 ratio. The Pareto Diagram also involves mathematical elements that support them as the Lorenz Curve, which has hardly been mentioned here. In [START_REF] Tolentino | Uso del principio de Pareto en la industria cervecera y su posible vinculación con la enseñanza de las matemáticas[END_REF] the mathematical analysis of the origin and evolution of the Pareto Diagram is presented and this must also be considered in the design of the SRA, involving three institutions P(M), E(M) and Ip.

Figure 1 :

 1 Figure 1: Institutional tours of a mathematical praxeology to go from P(M) to Ip

Figure 3 :Figure 4 :

 34 Figure 3: Bars and Lorenz curve

Table 1 : Wasting time of production lines

 1

Proceedings of CERME10


Carine Bryselbout

Pascale Henner

Éric Lichtfouse
email: eric.lichtfouse@dijon.inra.fr

Fossil fuel biomarkers in plant waxes as pollution parameters

Keywords: plant wax n-alkane, petroleum, hopane, sterane, pollution

Hopane and sterane derivatives typical of highly mature sedimentary organic matter, e.g. petroleum, have been identified in several plant species growing near Nancy, France. Analyses of plant waxes by gas chromatography-mass spectrometry over a restricted mass interval (m/z 185-195) allows definition of pollution parameters based on the relative concentration of fossil hopanes versus modern plant n-alkanes. Indeed, such parameters are higher for Pinus nigra growing along a high traffic highway than for Pinus nigra growing in a less polluted suburb area. Molecular pollution parameters based on fossil molecules are thus promising tools for measuring the extent of fossil fuel input into plant and food.

INTRODUCTION

After deposition at the bottom of lakes and seas, organic debris is slowly buried over geological time. With increasing time of burial, biogenic molecules, e.g. bacterial hopanols, undergo chemical alteration under the effect of biodegradation, mineral catalysis, increasing pressure and temperature to yield derivatives, e.g. fossil hopanes [START_REF] Van Dorsselaer | Degraded and extended hopane derivatives (C 27 to C 35 ) as ubiquitous geochemical markers[END_REF][START_REF] Mackenzie | Molecular parameters of maturation in the Toarcian shales, Paris Basin, France. I. Changes in the configurations of acyclic isoprenoid alkanes, steranes and triterpanes[END_REF][START_REF] Mackenzie | Chemical fossils: the geological fate of steroids[END_REF][START_REF] Ourisson | Hopanoids. 1. Geohopanoids: the most abundant natural products on earth?[END_REF]. Then, these molecular fossils may reenter the biosphere by natural and anthropogenic processes such as sediment uplift and petroleum exploration. Thus, the occurrence of molecular fossils in soils, mud, recent sediments and airborne particulate matter can be used to assess an input of petroleum products [START_REF] Dastillung | Molecular test for oil pollution in surface sediments[END_REF][START_REF] Rohmer | Hopanoids from C 30 to C 35 in recent muds[END_REF][START_REF] Simoneit | Application of molecular marker analysis to reconcile sources of carbonaceous particulates in tropospheric aerosols[END_REF][START_REF] Hostettler | Use of geochemical biomarkers in bottom sediment to track oil from a spill, San Francisco Bay, California[END_REF][START_REF] Lichtfouse | Ancient polycyclic aromatic hydrocarbons in modern soils: 13 C, 14 C and biomarker evidence[END_REF]. Here we report the first identification of fossil fuel biomarkers in plants.

EXPERIMENTAL

Full experimental procedure and data will be reported elsewhere. Aerial plant parts (leaves, needles) were sampled on march 25, 1998 at two locations at Vandoeuvre-Lès-Nancy, a suburb of Nancy : 1) highly contaminated samples growing on soils at about 3 m from a high traffic highway, and 2) less contaminated samples growing on urban soils of the Brabois area near the Agronomy school. Plant species include Pinus nigra (Austrian pine), Taraxacum obovatum (dandelion), Viola silvestris (violet), Primula auricula (primrose) and Poa trivialis L. (meadow grass). Fossil fuel biomarkers were identified by comparison of their chromatographic and mass spectrometric behaviour with data from a petroleum fraction from the Paris basin and from a petroleum fraction kindly provided by Dr. P. Adam (Strasbourg University).

Cuticular waxes were extracted from ground dried samples (150 mg) by elution of CH 2 Cl 2 over a modified Pasteur pipette [START_REF] Henner | Pipette Pasteur extraction: a fast, convenient, exhaustive and environmentally friendly method for the extraction of solid samples[END_REF]. The extracts were fractionated into alkane and aromatic fractions by elution with n-pentane and n-pentane-CH 2 Cl 2 65/35 v/v on a modified Pasteur pipette loaded with silica gel. Alkane fractions were analysed with a Varian Star 3400 gas chromatograph coupled to a Saturn 2000 ion trap mass spectrometer. Conditions: helium pressure 11 psi; on column injector; 30 m x 0.25 mm i.d. fused silica gel column coated with 5%phenyl-, 95%methylpolysiloxane phase (0.25 µm thickness); oven temperature : 30°C isothermal 1 min, 30-150°C at 15.6°/min, 150-310°C at 3°/min then 310°C isothermal 30 min; EI 70 eV. For each alkane fraction, one injection was performed scanning in the m/z 187-195 range for hopane derivatives and n-alkanes, and another injection was performed scanning in the m/z 213-221 range for sterane derivatives and n-alkanes. Identification was conducted by co-elution of pure nalkanes and comparison of chromatographic characteristics with the petroleum sample and with the literature [START_REF] Hostettler | Use of geochemical biomarkers in bottom sediment to track oil from a spill, San Francisco Bay, California[END_REF][START_REF] Lichtfouse | A molecular and isotopic study of the organic matter from the Paris Basin, France[END_REF][START_REF] Peters | The Biomarker Guide[END_REF]. Maturity parameters based on relative concentration were calculated by area integration from the m/z 191 chromatogram.

RESULTS AND DISCUSSION

Fossil fuel biomarkers

Several plant species were sampled in both high-and medium-contaminated areas near Nancy, France. Cuticular waxes were extracted then fractionated into alkane and aromatic fractions. Since fossil fuels occur only at low concentrations relative to plant wax n-alkanes, gas chromatophic-mass spectrometric analysis of alkane fractions was optimised by scanning over a limited interval, e.g. m/z 187-195 (Fig. 1). Selective detection was achieved using reconstructed ion currents at 194 amu for n-alkanes and 191 amu for hopanes (Fig. 2), thus giving ion current responses in the same order of magnitude for both chemical classes. Indeed, the ion current at 194 allowed to detect selectively n-alkanes, though the molecular mechanism leading to this fragment is unknown. Other typical n-alkane fragments may be used such as m/z 183 and 197. On the other hand, the relative concentration of n-alkanes and hopanes can be measured using using m/z 191 because this ion current shows both classes of molecules. For Pinus nigra needles, the ion current at m/z 194 shows the typical fingerprint of plant waxes dominated by odd-numbered carbon n-alkanes [START_REF] Eglinton | Leaf epicuticular waxes[END_REF] (Fig. 1). Several hopane homologues ranging from C 27 to C 34 were identified using ion current at m/z 191. All hopanes have the 17α,21β configuration found only in mature fossil fuels such as petroleum and deep sedimentary rocks [START_REF] Mackenzie | Molecular parameters of maturation in the Toarcian shales, Paris Basin, France. I. Changes in the configurations of acyclic isoprenoid alkanes, steranes and triterpanes[END_REF][START_REF] Seifert | Steranes and terpanes in kerogen pyrolysis for correlation of oils and source rocks[END_REF][START_REF] Seifert | Applications of steranes, terpanes and monoaromatics to the maturation, migration and source of crude oils[END_REF]. Several molecular parameters based on hopane stereochemistry have been previously defined to measure the thermal maturity of sedimentary organic matter. For instance, the percentage ratios of Ts, 's' for 'stable', relative to Tm, 'm' for maturable, and the percentage ratios of 22S versus 22R αβ-homohopanes (e.g. C 31 ) increase with burial depth. For Pinus nigra, Ts/Tm and S/R percentages are 56% and 58% respectively. Similar values were found for all plant species both in the highway-contaminated and suburb areas. They are typical of mature fossil fuels such as petroleum [START_REF] Peters | The Biomarker Guide[END_REF]. Moreover, two other chemical classes of molecular fossils, C 27 -C 29 steranes and diasteranes bearing mature configurations (Fig. 3) were identified in all plant samples. These findings imply that ancient fuel products can be trapped on to the surface of plant waxes. Therefore, they could be used as molecular parameters of pollution as explained below.

Molecular pollution parameters

Observation of ion currents at m/z 191 for Pinus nigra shows that the relative concentration of hopanes versus n-alkanes is higher for the highway-contaminated plant than for the suburb plant. We therefore defined two pollution parameters (PP) to measure the extent of fossil fuel input (see Fig. 1): PP1 = 100.Ts/(Ts + 29) PP2 =100. 30αβ/(30αβ + 29) where Ts refers to fossil fuel 18α-trisnorneohopane, 29 to plant wax C 29 n-alkane, and 30αβ to fossil fuel C 30 17α,21β-hopane (Fig. 3). Pollution parameters were calculated by peak area integration at m/z 191. Values for PP1 and PP2 are significantly higher for the highway pine, 38% and 66% respectively, than for the suburb pine, 9% and 23% respectively. Such molecular parameters are thus promising, highly sensitive new tools which could be used to assess the impact of fossil fuel pollutants on the modern environment. Moreover, since a large part of food products are plant-derived and since food production is now a very complex process involving multiple pathways and transportation, then fossil fuel biomarkers may also help to establish relationships between food and a possible contaminated crop field. Further work is under way to apply molecular pollution parameters to various modern media.

Figure 1 .

 1 Figure 1. Plant wax n-alkanes and fossil fuel hopanes occurring in pine located near a highway. Gas chromatography-mass spectrometry : 187-195 amu scan range. Numbers refer to n-alkane carbons. Ts: 18α-22,29,30-trisnorneohopane; Tm: 17α-22,29,30-trisnorhopane; 29-34αβ : 17α,21β-hopanes. S and R refer to hopane configuration at position 22.

Figure 2 .

 2 Figure 2. Mass spectra of plant wax n-alkane and fossil hopane from the highway pine, recorded by gas chromatography-mass spectrometry over the 187-195 amu range.

Figure 3 .

 3 Figure 3. Molecular fossils identified in the highway pine waxes.


Xavier Margueron
email: xavier.margueron@leg.ensieg.inpg.fr

Jean-Pierre Keradec

Tuan Phung Anh

Olivier Chadebec

Gérard Meunier

Applying, Canceling and Identifying the Magnetic Field Received and Emitted by a 2D Plan System

Abstract-2D Multipolar expansion is extended to account for externally created field and numerous relations are derived. The most general field created outside or inside a cylinder can be created by a current density flowing axially on the surface of the cylinder. Such a surface current can also cancel the field generated either inside by external sources or outside by internal sources. Knowing 1 field component on 2 coaxial cylinders allows both internally and externally created fields to be identified.

I. INTRODUCTION 3D multipolar expansion is a technique widely used in

physics [START_REF] Lorrain | Electromagnetic fields and waves[END_REF]. Some years ago we adapted this technique to 2D problems [START_REF] Keradec | Multipolar Development of Vector Potential for Parallel Wires. Application of the study of Eddy Currents effects in Transformer Windings[END_REF] in order to study the field radiated by a unidirectional current density flowing through a bounded area.

In this paper, general solution of Laplace equation in cylindrical coordinates is reminded. Its emitted part, which coincides with multipolar expansion, is accompanied by a received field. As a first result, a field surrounding a 2D system can be depicted by two sets of coefficients: first one is related to the emitted field, second one to the received field. (2)

Expansion of received field is useful to study the effect of a field applied to a device. The simplest and often used representation for this field is a homogenous one. When a more refined representation is required [START_REF] Margueron | Minimization of Eddy Current Losses in a Straight Rectangular Cross Section Conductor[END_REF], following terms of the expansion show how to complete the description (Fig. 1). In figure 1, fields are all created by a cylindrical axial surface current. Each of them is related to a specific term of the Fourier expansion of the current density. One can see that, this way, a perfectly homogenous field can be generated inside a cylinder.

In final paper, it will be show that these successive fields are connected to 2D Mac Laurin expansion of vector potential.

In magnetostatic, one cylindrical axial current density suffices to create a given field either inside or outside the cylinder, while, in magnetodynamic, two such densities may be needed to gives two fields with a phase lag.

Because a cylindrical current source can generate a given field either inside or outside a cylinder, it can be used to cancel the field due to a given source. Field created by internal currents can be canceled outside the cylinder and reversely. III. FIELD IDENTIFICATION Expressions for induction B are derived from (1-2) and, of course, components of A and B are periodical with j. So, if we know one over a circle, we can develop it in Fourier series. It will be show that this gives a mean to identify coefficients of (1) which describe the field. When both emitted and received fields superpose [START_REF] Margueron | Minimization of Eddy Current Losses in a Straight Rectangular Cross Section Conductor[END_REF], component variation must be known on two coaxial circles to deduce all the coefficients of (1-2).

IV. POWER EXCHANGES Starting from (1-2), which describes the field surrounding a device, and using Poynting vector, complex power going through a cylinder is expressed in terms of coefficients of (1-2). This way, it is established that, if a straight conductive wire with an arbitrary cross section shape is submitted to a homogenous variable field, the power it dissipates is uniquely related to its dipolar momentum. It appears also that this is not true when applied field is not homogenous.

V. CONCLUSION This work has been done to study eddy currents effects in parallel wires, especially inside transformers. It gave us tools to generate arbitrary fields, to identify fields applied to wires in a complex environment, and several ways to properly define the dipolar momentum acquired by a wire (based on field expansion or on power exchange).

Fig. 1 .

 1 Fig. 1. Magnetic field due to a cylindrical surface current: a) constant with j b) proportional to sin(j ) c) proportional to sin(2j ) II. ARBITRARY FIELD GENERATION AND CANCELATION


Philippe Jégou
email: philippe.jegou@lis-lab.fr

Hélène Kanso
email: hkanso@effatuniversity.edu.sa

Cyril Terrioux
email: cyril.terrioux@lis-lab.fr

On the Relevance of Optimal Tree Decompositions for Constraint Networks

Keywords: Constraint Networks, Graphical Models, Solving, Optimization, Tree-decomposition

For the study and the solving of NP-hard problems, the concept of tree decomposition is nowadays a major topic in Computer Science, in Artificial Intelligence and particularly in Constraint Programming. It appears as a promising field for the theoretical study of numerous graphical models like Bayesian Networks or (Weighted) Constraint Networks, since it can ensure, under some hypothesis, the existence of polynomial time algorithms. This concept is also used in a wide range of applications. Recently, a real improvement in the practical computation of optimal tree decompositions has been observed, allowing new promising applications of this concept in real applications.

In this paper, we first aim to analyze the real relevance of such optimal decompositions. We first set that a larger set of instances are now optimally decomposable in practice but using these algorithms on a practical level still constitutes a real difficulty. In a second time, we assess the impact of such optimal decompositions for solving these instances and note a discrepancy between the empirical results and what is expected from the complexity analysis. Finally, we discuss of the next investigations which are needed on this topic.

I. INTRODUCTION

Solving NP-hard problems is a central issue in Computer Science, in Artificial Intelligence and especially in Constraint Programming because the problems to solve are generally at least NP-complete, and frequently more difficult if we deal with problems of search, optimization, counting or enumeration. Among the proposed approaches to solve them, algorithmic techniques based on decomposition form sometimes an efficient approach especially when these problems are represented by graphs. Strategies of decompositions of problems are then based on the theoretical notion of structural decomposition of graphs. Used for a long time, at least since the 1960s [START_REF] Bertele | Nonserial Dynamic Programming[END_REF], these notions of decompositions were formalized mainly by the notion of tree decomposition introduced by Robertson and Seymour [START_REF] Robertson | Graph minors II: Algorithmic aspects of treewidth[END_REF] in the 1980s. Their work was not originally motivated by the solving of such problems, but it has yielded significant and numerous theoretical results. This kind of work has led to many theoretical results in various fields of Computer Science, including Artificial Intelligence.

In AI, these characterizations in terms of decomposition have in particular been studied at the level of Graphical Models. The notion of Graphical Model addresses a large collection of formalisms like Constraint Networks, Propositional Logic, Cost Function Networks, Bayesian Networks, or Markov Random Fields. In these frameworks, a joint function over a set of variables is described as a factorized combination of functions over few variables. Such formalisms have been intensively used in AI, but also in other domains in Computer Science. The name of graphical model [START_REF] Dechter | Reasoning with Probabilistic and Deterministic Graphical Models: Exact Algorithms[END_REF] is used to underline the fact that such a factorization defines a (hyper)graph whose vertices correspond to the variables and whose (hyper)edges include the variables of each factor. The expressiveness of these models makes them capable to represent a variety of real-world problems. In the case of finite domains, we can consider, for example, the existence of a variable assignment that satisfies the logical conjunction of a given set of Boolean functions (that is to say constraints) involving subsets of variables. This decision problem is known as the Constraint Satisfaction Problem (CSP [START_REF]Constraint processing[END_REF]) and is NPcomplete. It becomes NP-hard for the optimization of a sum and #P-complete for counting. Solving these problems thanks to graphical decompositions runs on two steps: the calculation of a decomposition of a graph and the solving that exploits this decomposition.

The complexity of such approaches is O(exp(w)) where w is the width of the considered decomposition. For the case of tree decompositions, this parameter is called treewidth. Unfortunately, the computation of an optimal tree decomposition is a NP-hard problem [START_REF] Arnborg | Complexity of finding embeddings in a k-tree[END_REF]. This negative result has also been observed on a practical level since, until very recently, the best algorithms could only process graphs of a few dozen of vertices at most. Moreover, the implementation of efficient solving methods based on decompositions is particularly difficult from a practical point of view. So, systems or solvers exploiting decompositions generally use heuristic approaches to calculate decompositions (e.g. MinFill [START_REF] Rose | A graph theoretic study of the numerical solution of sparse positive definite systems of linear equations[END_REF], [START_REF] Kjaerulff | Triangulation of Graphs -Algorithms Giving Small Total State Space[END_REF], MCS [START_REF] Tarjan | Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs[END_REF], or H-TD-WT [START_REF] Jégou | An Algorithmic Framework for Decomposing Constraint Networks[END_REF]). Of course, these heuristics do not provide any guarantee on the optimality of the width. Nonetheless, they make it possible, in some cases, to calculate the decomposition of real-world instances having a large number of vertices. Recently, new algorithms computing optimal decompositions have emerged, especially during the last PACE Challenge [START_REF] Dell | The PACE 2017 Parameterized Algorithms and Computational Experiments Challenge: The Second Iteration[END_REF]. They made it possible now to process larger graphs using reasonable runtime.

So, in this paper, we study the impact of these new algorithms by addressing Constraint Networks and its associated decision problems. The experiments have been realized on large classes of instances coming from XCSP3 database.

The first question we answer addresses the size of the CSP instances that can be processed by the exact decomposition algorithms. We show that instances of several hundred variables can now be processed in a reasonable amount of time. We then evaluate the difference between the treewidth and the widths obtained by the heuristic methods. This kind of results, to our knowledge, have never been established before. It makes it possible to better assess, on the one hand, the capabilities of new exact methods, but also the real quality of the heuristics, which until now remained a real unknown on more realistic networks. Unfortunately, even if, thanks to the improvement of their runtime, exact algorithms can be used for processing larger instances, we show that it still constitutes a real obstacle for their exploitation on an practical level and for their scalability.

The second question we answer addresses the estimation of the contribution of exact decompositions for solving CSP instances. We show that when optimal decompositions can be calculated, they are not necessarily the more relevant for the solving step. While it has already been observed that the value of w is not always the crucial parameter for the solving step of decomposition methods, such analyses have never been realized using optimal decompositions (unless perhaps on very small instances). Thus, such conclusions about the relative relevance of optimal decompositions could never been established previously since they were only assumptions but not observed facts.

On the basis of these observations, we then propose ways to study this problem which remains fundamental in Computer Science, and particularly for efficient systems in Constraint Programming.

This paper is organized as follows. In Section II, we give the definitions of tree decomposition and graphical models as well as the general principles to solve them. Section III presents the various approaches for calculating decompositions and the results on their evaluations while Section IV reports experimental results on the solving of graphical models expressed as CSP instances. The last section concludes this paper discussing the future investigations about such approaches.

II. TREE DECOMPOSITION FOR GRAPHICAL MODELS A. Tree Decomposition

While several different decompositions of (hyper)graphs have been proposed, tree decomposition is the one that has generated the most works and has been the most used in practice in Computer Science and more particularly in AI. Other decomposition approaches have been studied for Graphical Models (e.g. hypertree decomposition [START_REF] Gottlob | A Comparison of Structural CSP Decomposition Methods[END_REF]). However, if they extend the theoretical results, they do not seem to offer the same practical interest. So we focus our study on tree decomposition. A tree decomposition of a graph G = (V, E) is a pair (B, T ) with T = (I, F ) a tree and B = {B i : i ∈ I} a family of subsets of V , such that each subset (called bag or cluster) B i is a node of T and satisfies: (1) ∪ i∈I B i = V , (2) for each edge {x, y} ∈ E, there exists i ∈ I with {x, y} ⊆ B i , and (3) for all i, j, k ∈ I, if k is in a path from i to j in T , then B i ∩ B j ⊆ B k . The width w of a tree decomposition (B, T ) is equal to max i∈I |B i | -1. The treewidth w * of G is the minimal width over all the tree decompositions of G.

Note that tree decomposition can be defined in the case of hypergraphs, considering their primal graphs which are graphs defined with same vertices and such that two vertices are connected if they appear in the same hyperedge. So, tree decomposition can be considered to handle graphical models whose structure are hypergraphs.

B. Graphical Models

As indicated before, there are several types of graphical models. A general definition is given in [START_REF] Dechter | Reasoning with Probabilistic and Deterministic Graphical Models: Exact Algorithms[END_REF]:

Definition 1: A Graphical Model M is a tuple (X, D, F, ⊗) where (1) X = {X 1 , . . . X n } is a finite set of n variables, (2) D = {D 1 , . . . D n } is the set of their respective finite domains of values, (3) F = {f 1 , . . . f r }
is a set of positive real-valued discrete functions (called local functions), defined over scopes of variables S = {S 1 , . . . S r } where S i ⊆ X, and (4) ⊗ is a combination operator (e.g. ⊗ ∈ {Π, Σ, }). The graphical model represents a global function whose scope is X and which is the combination of all its functions ⊗ r i=1 f i . Based on this general definition, it is possible to define well known classical frameworks. The basic one is the Constraint Network model (also called CSP) which is a 4-tuple (X, D, C, ) where

D = {D 1 , . . . D n } is a set of discrete- valued domains while C = {C 1 , . . . C r } is a set of constraints. Each constraint C i is a pair (S i , R i )
where R i is a relation defined on the domains of the variables of its scope S i ⊆ X. Note that the Satisfiability Problem (SAT) in propositional logic can be viewed as a particular case of CSP. So, one can observe that relations can be given by tables (as originally for CSP) or functions (boolean functions expressed by clauses in SAT). Other classes of graphical models which can express combinatorial optimization problems are Cost Functions Networks such as ILP, WCSP, MAX-CSP (and then MAX-SAT). More sophisticated models as Bayesian Networks or Markov Random Fields are also graphical models.

C. Structural Methods for Graphical Models

Specific solving methods have been proposed to solve these problems but two general approaches can be considered. On the one hand, those which apply traditional search techniques based on backtracking or branch and bound. In the worst case, their time complexity is in O(exp(n)) while being generally linear in space. Nevertheless, such algorithms are sometimes very efficient in practice. On the other hand, we can find the methods that exploit the notion of decomposition of graphs and which are based on Dynamic Programming (DP). These methods make it possible to guarantee a time complexity in O(exp(w * )) but they need to handle large tables leading to a space complexity also in O(exp(w * )). Often this space complexity can be reduced to O(exp(sep)) where sep is the maximum size of intersections between two clusters (with sep ≤ w * ) [START_REF]Constraint processing[END_REF]. Nevertheless this worst case space complexity makes this type of approach unusable in practice, particularly when w * and sep have close values. As a consequence, DP is rarely used except in the case of very small treewidths. For example, recently, experiments show that a system combining DP on tree decompositions with ASP needs several hours for solving real-world instances with w * = 5 and n = 138 [START_REF] Abseher | Improving the Efficiency of Dynamic Programming on Tree Decompositions via Machine Learning[END_REF].

So, approaches making compromises between search and memorizing have been proposed. They make it possible to take advantage of the practical efficiency of search techniques, while guaranteeing a time-complexity in O(exp(w * )), without suffering from the problems of memory space inherent to DP. Indeed, they make it possible to memorize in a limited way, even if, in the worst case space complexity, remains O(exp(sep)), a bound which is rarely observed in practice. These approaches offer today the best compromise between DP and search. For instance, the algorithm BTD [START_REF] Jégou | Hybrid backtracking bounded by treedecomposition of constraint networks[END_REF], which performs a search guided by a tree decomposition, realizes such a trade-off in order to solve CSP instances. Later, a similar approach called AND/OR Search has been developed for graphical models in a series of papers which starts with [START_REF] Dechter | The Impact of AND/OR Search Spaces on Constraint Satisfaction and Counting[END_REF]. So, today, such algorithms seem to be a good compromise between search and DP to exploit structure in graphical models. Some of them are implemented in efficient real systems like ToulBar2 [START_REF] De Givry | Exploiting Tree Decomposition and Soft Local Consistency in Weighted CSP[END_REF], a state of the art tool to solve Cost Functions Networks [START_REF] Hurley | Multi-language evaluation of exact solvers in graphical model discrete optimization[END_REF].

While solving graphical models can take advantage from tree decomposition, up to now, only approximations of optimal decompositions have been considered in practice. The next section deals with the calculation of tree decompositions.

III. CALCULATION OF TREE DECOMPOSITIONS A. Algorithms for Decompositions

Because of the significance of the concept of tree decomposition, many studies have focused on their calculation. The proposed algorithms can be classified into two approaches: the exact algorithms that compute decompositions of optimal width (equal to the treewidth) and the heuristic methods, which do not offer a guarantee on optimality. The motivation of heuristic approaches is due to the fact that the optimal computation is a NP-hard problem [START_REF] Arnborg | Complexity of finding embeddings in a k-tree[END_REF]. Numerous approaches rely, for example, on metaheuristic techniques, but the approaches dedicated to handle graphical models often use triangulations (such as MinFill [START_REF] Rose | A graph theoretic study of the numerical solution of sparse positive definite systems of linear equations[END_REF], [START_REF] Kjaerulff | Triangulation of Graphs -Algorithms Giving Small Total State Space[END_REF] and MCS [START_REF] Tarjan | Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs[END_REF]) or traversals of graphs (like [START_REF] Jégou | An Algorithmic Framework for Decomposing Constraint Networks[END_REF], [START_REF] Koster | Frequency Assignment -Models and Algorithms[END_REF]). These heuristic approaches allow to process graphs of several thousand or even tens of thousands of vertices in reasonable time, but without guaranteeing the quality of the obtained decompositions in terms of deviation from the optimum w * .

In contrast, the exact methods have long been limited to only a few dozen vertices [START_REF] Shoikhet | A practical algorithm for finding optimal triangulation[END_REF], [START_REF] Gogate | A complete anytime algorithm for treewidth[END_REF] and thus proved unusable as soon as the instances of graphical models exceed a few hundred variables. A recently published survey on these exact algorithms confirmed these practical limitations [START_REF] Bodlaender | On exact algorithms for treewidth[END_REF]. However, these limitations are questioned by very recent works stimulated by the two first PACE challenges [START_REF] Dell | The PACE 2017 Parameterized Algorithms and Computational Experiments Challenge: The Second Iteration[END_REF] and which improve significantly these approaches. Indeed, between the two iterations of the PACE challenges (i.e. between 2016 and 2017), an improvement of two orders of magnitude for runtime has been observed. So much so that now, the best algorithms can process graphs of several hundreds of vertices in a few dozens of seconds.

These recent advances lead us to reconsider the solving of graphical models by trying to exploit these optimal decompositions. We must indeed now assess their scalability on real-world instances. This must be done according to two criteria: on the one hand, their ability to decompose real-world instances, and the runtime they need for that, and, on the other hand, the real impact of the use of optimal decompositions on the effective solving of these instances. This second point is also crucial because it has been observed experimentally that the quality of a decomposition method is not only related to the value of the width, but also to other structural criteria of the decomposition, as well as to the nature of the problem to solve [START_REF] Abseher | Improving the Efficiency of Dynamic Programming on Tree Decompositions via Machine Learning[END_REF]. Moreover, decomposition algorithms dedicated to the efficiency of solving graphical models have been recently introduced [START_REF] Jégou | An Algorithmic Framework for Decomposing Constraint Networks[END_REF], [START_REF] Jégou | Towards a Dynamic Decomposition of CSPs with Separators of Bounded Size[END_REF]. The first one called Connected, calculates decompositions guaranteeing that subgraphs induced by clusters are connected (it is called H 2 in [START_REF] Jégou | An Algorithmic Framework for Decomposing Constraint Networks[END_REF]). The second one we call Small-sep calculates decompositions minimizing simultaneously w and sep (it is called H 5 in [START_REF] Jégou | Towards a Dynamic Decomposition of CSPs with Separators of Bounded Size[END_REF]). So, we evaluate now the approach to decompose graphs considering also such algorithms.

B. Ability to Compute Decomposition and Quality

First we describe our experimental protocol. We exploit the three implementations of exact methods for computing optimal tree decompositions, proposed to the PACE 2017 challenge [START_REF] Dell | The PACE 2017 Parameterized Algorithms and Computational Experiments Challenge: The Second Iteration[END_REF]. We call them larisch, tamaki and bannach like in this challenge. Regarding the heuristic methods, we use MCS [START_REF] Tarjan | Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs[END_REF], MinFill [START_REF] Rose | A graph theoretic study of the numerical solution of sparse positive definite systems of linear equations[END_REF], [START_REF] Kjaerulff | Triangulation of Graphs -Algorithms Giving Small Total State Space[END_REF], Connected [START_REF] Jégou | An Algorithmic Framework for Decomposing Constraint Networks[END_REF] and Small-sep [START_REF] Jégou | Towards a Dynamic Decomposition of CSPs with Separators of Bounded Size[END_REF] that we have implemented.

For the benchmark, we select 7,597 CSP instances encoded in the XCSP3 format [START_REF] Boussemart | XCSP3: an integrated format for benchmarking combinatorial constrained problems[END_REF] as follow from the XCSP3 database 1 . First, as we aim to compare the decomposition methods, we limit our study to instances whose primal graph is not a complete graph. Then, the selection is refined according to solver restrictions. So, only instances with intention, extension, all-different, sum or element constraints are kept. The selected instances have between 6 and 28,161 variables (vertices) and between 5 and 285,685 constraints (edges or hyperedges) whose arity ranges between 2 and 931. It results that the primal graphs have between 5 and 12,547,224 edges.

The experiments are performed with Intel Xeon processors 2.4 GHz and we allocated a slot of 30 minutes for decomposing each instance as for the PACE 2017 challenge.

Figure 1 shows the cumulative number of decomposed instances for each decomposition method. Unlike PACE 2017, no exact method is able to decompose all the instances. Then tamaki, which finishes at the second place at PACE 2017, is clearly the best exact method with about 79% of the considered instances which are decomposed against 75% for bannach and larisch. Afterwards, the gap between exact methods and heuristic ones remains important. Indeed, heuristic methods are able to decompose significantly more instances since the percentage of decomposed instances ranges between 98% (MinFill) and 100% (Small-sep).

Regarding the runtime, we can note that all the methods are quite efficient. Indeed, all the methods (including exact ones) require less than 120 s to process most of the decomposed instances. Finally, the main drawback of exact methods is the scalability. If the exact methods treat 92% of the instances having at most 300 vertices, they only decompose 7% of the instances with more than 300 vertices.

We now consider the structural parameters of the produced tree decompositions. If, by construction, the width of a tree decomposition produced by Connected or Small-sep are often far than the treewidth, we can note that the obtained widths for MinFill or MCS are often close. For example, if we consider the 5,996 instances which are optimally decomposed by one of the exact methods, we observe that MinFill computes an optimal decomposition for 3,638 instances. For illustration, we provide in Figure 2 to be a good solution for computing a relevant approximation of the treewidth. Concerning the parameter sep of the largest intersection between two clusters, we notice that sep is equal to the width for many decompositions produced by exact methods (more than 60% of the instances), MinFill or MCS (about 30%). For Connected and Small-sep, this phenomenon may occur but is more seldom (e.g. 0.6% for Small-sep).

IV. EVALUATION FROM CSP SOLVING VIEWPOINT

A. Experimental Protocol

As indicated in the previous section, we used the same instances, that is to say, the benchmark 7,597 instances coming from XCSP3 database. Among these instances, some are realworld instances (e.g. RLFAP or Renault) or have a size similar to one of real-world instances.

We consider two solving algorithms, based on BTD, namely BTD-MAC+RST [START_REF] Jégou | Combining restarts, nogoods and bagconnected decompositions for solving CSPs[END_REF] and BTD-MAC+RST+Merge [START_REF] Jégou | Towards a Dynamic Decomposition of CSPs with Separators of Bounded Size[END_REF]. The latter is the solving algorithm exploiting tree decomposition which leads to solve the largest number of instances. However, to reach this goal, it may alter the considered decomposition by merging some clusters together. So, it may endanger the optimality of the initial decomposition (if so) during the solving. That is why we also assess the behavior of BTD-MAC+RST for which the decomposition remains unchanged during the search (except the root cluster which may be modified at each restart). Note that the setting of BTD (variable heuristic, root cluster heuristic, . . . ), except the tree decomposition method, is one described in [START_REF] Jégou | BTD and miniBTD[END_REF]. We exploit the implementation of BTD-MAC+RST+Merge and Small-sep [START_REF] Jégou | BTD and miniBTD[END_REF] available from the first XCSP3 Competition 2 . We add to it the ability to handle the optimal tree decompositions produced by larisch, tamaki and bannach methods.

As indicated before, the experiments are performed with Intel Xeon processors 2.4 GHz and for each pair decomposition/solving algorithm, we allocated a slot of 30 minutes for decomposing and solving each instance within the limit of 12 GB of memory. 2 See http://www.cril.univ-artois.fr/XCSP17 for more details.

B. Experimental Results

In this part, we assess the relevance of the computed decompositions with respect to the solving efficiency.

If our experiments are achieved with the two algorithms BTD-MAC+RST and BTD-MAC+RST+Merge, we only provide the results observed for BTD-MAC+RST by lack of place. On the one hand, the observed trends are the same for the two solving methods, except that, for each decomposition method, BTD-MAC+RST+Merge solves more instances than BTD-MAC+RST. On the other hand, BTD-MAC+RST has the advantage to preserve the considered decomposition and so its optimality (if so).

We first consider the cumulative number of instances which are solved by BTD-MAC+RST with each decomposition algorithm. As depicted in Figure 3, BTD-MAC+RST solves more instances when the decomposition is computed with a heuristic method than with an exact one. Such a result was foreseeable since heuristic methods are able to decompose more instances. For instance, BTD-MAC+RST with tamaki, which is again the best exact method, solves 5,221 instances among the 5,989 instances which are decomposed thanks to tamaki. In the same time, BTD-MAC+RST with a heuristic method solves between 960 and 1,018 additional instances (respectively for MCS and Small-sep), while the virtual best solver (VBS) treats successfully 1,159 additional instances. Note that BTD-MAC+RST with Small-sep and the VBS have clearly a close behavior.

When observing that BTD-MAC+RST with an exact de-composition method successes very often in solving the instance provided that the instance is decomposed, it seems to be natural to study the solving time (i.e. the runtime excluding the decomposition time). So Figure 4 compares the solving time of BTD-MAC+RST with tamaki against its solving time with Small-sep. Of course, by definition, such a figure excludes the instances which are not decomposed by tamaki. We remark that for 41% of the 5,873 considered instances, the solving with tamaki is as efficient as one with Small-sep, whilst, for 33% of the instances, BTD-MAC+RST with Small-sep performs better. Finally, the cumulative runtime for the instances solved by BTD-MAC+RST for both decomposition methods is 149,898 s for Small-sep against 172,571 s for tamaki. These results contradict the complexity analysis which claims that the smaller the width is, the more efficient the solving is. Indeed, solving a CSP instance thank to an optimal decomposition is often less efficient than exploiting a heuristic decomposition. This observation is even strengthened if we take into account both the decomposition step and the solving step. Indeed, in this case, exploiting a decomposition heuristic clearly appears to be the best solution. As shown in Figure 5, which compares the global runtime of BTD-MAC+RST with tamaki against its runtime with Small-sep, BTD-MAC+RST with Small-sep is clearly more efficient than with tamaki. As an illustration, 77% of the instances are solved more efficiently with Smallsep while BTD-MAC+RST with tamaki performs better only for 5% of the instances. Moreover, BTD-MAC+RST with tamaki requires 300,366 s to process all the instances solved for both decomposition methods, while it needs only 156,945 s with Small-sep. To conclude, note that the trends observed for WCSP or #CSP solving are similar (not reported here for lack of place), which allows to generalize our observations to a larger class of graphical models.

V. CONCLUSION AND DISCUSSION

Exploiting an optimal decomposition for solving graphical model instances was unthinkable two years ago. Thanks to the PACE challenge, it seems that it is now possible since, nowadays, algorithms like tamaki can deal with graphs of a few hundred vertices. E.g. tamaki has found optimal decompositions for 92% of the 6,409 graphs having at most 300 vertices within the 30 minutes timeout. We also observed that an approach like MinFill constitutes a very good heuristic in terms of approximation since it allows to obtain tree decompositions whose width is often close to the optimum as shown in Figure 2. But our experiments also showed that the runtime of the exact algorithms is a major drawback as soon as the graphs consist of several hundreds of vertices. In particular, it has been observed that only 7% of the graphs with more than 300 vertices can be optimally decomposed within 30 minutes. It ensues that, for the moment, exact methods cannot be exploited yet as a general tool for decomposing graphical models and so that the use of heuristic approaches remains necessary for the general case.

This observation is further strengthened when we consider the effective solving of graphical models (in the sense of the CSP in our experiments). Indeed, as soon as the instances of graphical models have a few hundred variables, the cumulated time for decomposing and solving is too important and constitutes a real obstacle to their use comparing to heuristic decompositions (as shown in Figure 3). Moreover, surprisingly, when optimal decompositions are computable by exact methods, they do not guarantee a more efficient solving. Indeed, the solving with such optimal decompositions is often outperformed by the solving with heuristic decomposition (see Figure 4). It had already been observed empirically that the width of a decomposition does not necessarily constitutes the crucial parameter for such solving methods [START_REF] Jégou | Combining restarts, nogoods and bagconnected decompositions for solving CSPs[END_REF]. However this could never be verified before on optimal decompositions. These results were obtained on the basis of a solving method (BTD) which exploits the tree decomposition and can be considered as an intermediate approach between search (backtracking or branch and bound) and the DP approach. This leads us to question the use of DP methods based on optimal decompositions. Indeed, very often, such solving approaches need memory space close to exp(w * ) because we observed that w * = sep for more than 60% of the instances of our benchmark. Moreover, [START_REF] Abseher | Improving the Efficiency of Dynamic Programming on Tree Decompositions via Machine Learning[END_REF] point out that the treatment of instances of small size (a real-world benchmark called "Vienna" with n = 138) with a limited treewidth (w * = 5 for these instances) can requires several hours of calculation to be solved when using a DP approach.

All these observations lead us to suggest different tracks for future investigations. Sure, the first one would be to improve the efficiency of exact methods, but it will most likely lead to decompositions for which sep will be very close to w * . Secondly, by observing that heuristics (like Small-sep) produce decompositions which are very relevant for the solving step while obtaining widths often far from w * , one can consider that such heuristics are a crucial object of study for the solving of real-world instances of graphical models. In addition, it seems relevant to study mixed approaches. Indeed, since from now on, it is possible to decompose optimally graphs a few hundred vertices, it could be useful to exploit the exact algorithms locally as routines in order to decompose the largest clusters so as to overcome the defects of the heuristics. This could lead to methods that are both time-efficient and more effective for solving.

Among the other tracks, one must mention an original work that uses machine learning techniques to identify the best decompositions to be used for the practical solving [START_REF] Abseher | Improving the Efficiency of Dynamic Programming on Tree Decompositions via Machine Learning[END_REF]. However, in conclusion, the authors highlight the current limitations of this kind of approach due to the preliminary training step necessary for its use.

Finally, we have now to better identify the properties that a decomposition must satisfy to ensure an efficient solving of a graphical model instance. Indeed, a heuristic like Smallsep appears to be efficient for both decomposing large graphs (all the considered instances have been processed), and, above all, solving associated instances. So, for future works, we have to highlight the parameters which allow us to produce the best decompositions in view of the solving. If some of them are probably related to the structure (like w and s, or local properties like for the heuristic Connected), taking into account the semantic of the instances now seems the most promising research track.

Fig. 1 .

 1 Fig. 1. The cumulative number of decomposed instances.

Fig. 2 .

 2 Fig. 2. The width obtained with MinFill vs the treewidth.

Fig. 3 .

 3 Fig. 3. The cumulative number of solved CSP instances.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Tamaki vs Small-sep w.r.t. the solving time (in s) for CSP instances.

ACKNOWLEDGMENT

This work has been funded by the french Agence Nationale de la Recherche, reference ANR-16-C40-0028.