Florent Poulhès

Dominique Mouysset

Gérard Gil
email: gerard.gil@univ-amu.fr

Michèle P Bertrand
email: michele.bertrand@univ-amu.fr

Stéphane Gastaldi
email: stephane.gastaldi@univ-amu.fr

Speeding-up Enzyme-Catalyzed Synthesis of Polyamides using ω-Amino-α-Alkoxy-Acetate as Monomer

Keywords: polyamides, enzymes, monomers

The design of ω-amino-α-alkoxy-acetate as monomer enablestodramaticallyspeeduptheformationofpolyamidescatalyzedbythesupportedlipase B ofCandida antarctica. Only 30 min areneededtoreach 93% conversionwhereas 240 h arenecessarytoobservethe same levelofconversionwith a monomer devoidedof an oxygenatom in position β relative totheelectrophiliccarbonylgroup. Yields, Mn, Mw, DPnand PDI reachverygoodvaluesfor an enzymaticsynthesisofpolyamides.

Introduction

For many years, users and developers from a broad range of industries particularly value the properties of polyamides. These include high mechanical and thermal stress resistance, excellent low-friction properties and abrasion resistance, good chemical stability, low stress crack resistance and good electrical insulating behavior [1]. These interesting properties result mainly from hydrogen bonds induced by the amide function. These multiple interactions among adjacent strands are energetically strong due to the high dipolar moment of the amide group. The symmetrical and regular character of the backbone of the polyamide and the strength of interchain interactions lead often to highly crystalline polymers. However, the low solubility associated to high melting and glass transition temperatures caused by theircrystallinityinherent to their backbone lead to difficulties in synthesis, characterization and processing of polyamides [2][3][4].

The bio-inspired synthesis of polymers is recognized as a powerful tool, mostly because enzymes react under mild conditions and provide sustainable chemical processes [5][6][7][START_REF] Cheng | Polymer biocatalysis and Biomaterials[END_REF][START_REF] Cheng | Polymer biocatalysis and Biomaterials[END_REF][START_REF] Baum | [END_REF].

Polyesters are readily available from bio-catalyzed lactones ring-opening, condensation of ωhydroxyesters, or copolymerization of diesters with diols [5][6][7][START_REF] Cheng | Polymer biocatalysis and Biomaterials[END_REF][START_REF] Cheng | Polymer biocatalysis and Biomaterials[END_REF][START_REF] Baum | [END_REF][11][12]. Comparatively, very few examples of enzymatic syntheses of polyamides have been reported, even though it is an attractive alternative to usual methodologies [13][14][15][16][17][18][19][20].

Regarding lipase-promoted lactam ring-opening strategy, only the unsubstituted β-lactam ring enables the formation of poly(β-alanine) [13]. Polycondensations leading to Nylon 6,12 and Nylon 6,6 have been successfully achieved by Gross [14]. The influence of the carbon chain-length on the polymerization degree was emphasized in this study. The faster propagation of Nylon 6,12 chains as compared to Nylon 6,6 ones is in fair agreement with the chemical properties of lipases designed by nature for the hydrolysis of long chain fatty esters.

Cheng and coworkers have also reported the lipase-catalyzed synthesis of polyamides, through an AA/BB strategy [15]. They coupled saturated diamines, heteroatom-containing diamines and even polyamines with various diesters according to an enzymatic process. This study clearly demonstrated that the use of various lipases as polymerization catalysts led to the obtention of polymers with molecular properties similar to those obtained by a chemical approach at 180°C, but with better polydispersity indexes. Moreover, the cationic resins obtained from these products showed physical properties close to those of polymers derived from chemical processes. Bio-catalyzed bulk polymerizations were expanded to the formation of polyamides and polyester/amides based on polydimethylsiloxane blocks [16][17], sometimes coupled with an aliphatic fluorinated backbone [18].

Our group has been involved during the last years in the improvement of the formation of amide bond catalyzed by lipases and proteases [21][22][23], with the goal to make it compatible with thiyl radical-promoted racemization of aliphatic amines in a dynamic kinetic resolution process (DKR) [24][25][26][27]. The obvious extension of this DKR was to synthesize optically active polyamides by coupling the free-radical racemization with the efficient enzymatic polymerization we have recently developed. This strategy was designed and successfully applied by Meijer and coworkers to the synthesis of chiral polyesters [28][29][30].

However, despite the efficiency of the enzymatic synthesis of polyamides, the reaction times needed to reach good monomer conversion levels were far too high compared to the time necessary to complete the radical-promoted racemization [19][20]. In this publication, we present an enzymatic polymerization involving the supported variant of the lipase B of Candida antarctica, Novozym 435 (N435), which leads to high degrees of conversion within only a few minutes of reaction. This improvement was made possible by the design and synthesis of "methoxyacetate-like" monomers.

Experimental

2.1.Materials

All chemicals were analytical grade and were used as received. Glyme was distilled before use from sodium/benzophenoneketyl. Candida antarctica Lipase B (CAL-B) immobilized on macroporous acrylic resin (Novozym 435 (N435)) was a gift from Novozymes (Denmark).

2.2.General Procedure

A/B type polymerization under reduced pressure

In a typical experiment, ester 1 (219 mg, 1 mmol) in Ph 2 O (1 g) was introduced in a 10 mL flask. After stirring for 15 min, dried Novozym 435 (100 mg) was added. The mixture was then heated at 80 °C under 10 mbars for 240 h. After completion, the mixture was cooled at 60 °C, 10 mL of chloroform were added. The hot solution was filtered to remove the enzyme, the latter was washed with hot chloroform and the solution was concentrated to give the product as a sticky oilwhich was then triturated in methanol and dried under vacuum at 40 °C for 48 h (155 mg). 1

AA/BB type copolymerization under reduced pressure

In a typical experiment, diester5 (58.5 mg, 0.25 mmol) and diamine4 (111 mg, 0.25 mmol) in Ph 2 O (340 mg) were added in a 10 mL flask. After stirring for 15 min, dried Novozym 435 (60 mg) was added. The mixture was then heated at 80 °C under 10 mbars for 240 h. After completion, the mixture was cooled at 60 °C, 10 mL of chloroform were added. The hot solution was filtered to remove the enzyme, the latter was washed with hot chloroform and the solution was concentrated up to 2 mL. The precipitate formed after addition of methanol was filtered, and then washed with the same solvent. The polymer was recovered after drying at 40 °C under vacuum for 48 h (130 mg). 1

Characterization methods

Gel Permeation Chromatography Analyses

The analyses and the monitoring of polymerization reactions were achieved by gel permeation chromatography (GPC) using Macherey-Nagel Nucleogel GPC 500, 100 and 50 (10 μm porosity; 300 x 7.7 mm) analytical columns fitted in series after a GPC (50 x 7. Weight-average molecular weight (Mw), (Mn) and polydispersity index (PDI= Mw/Mn) were calculated from the chromatograms.

Instrumentals

DSC analyses were effected with a Q200 DSC TA Instrument, the calorimeter under nitrogen (30 mL/min) being connected to a cryostat from the same manufacturer. DSC scans were run at temperatures ranging from -150 to 150 °C with a heating rate of 5 °C/min. A controlled cooling rate of -10 °C/min was applied between heating runs.

TGA analyses were recorded with TGA Q500 TA Instruments apparatus, under an air flux (60 mL/min). The measurements were carried out with a rate of 10 °C/min.

Results and discussion

The acceleration of lipase-catalyzed kinetic resolution using ethyl methoxyacetate as acyl donor has been known for several years [31]. It allows the completion of the enzymatic reaction with excellent yields in short reaction times, without loss of enantioselectivity. Park and coworkers have suggested [32] that the rate would be increased by the electronic effect of the α-oxygen atom enhanced by hydrogen bonding with the NH group in the transition state of the aminolysis. The introduction of such an oxygenated acylating moiety into monomers might lead to macromolecules with good average degrees of polymerization in reduced reaction times.

3.1.A/B Polymerization

To reach this goal, amino-ester 1 was synthesized. It presents an ethyl alkoxyacetate moiety suitable for the kinetic study of A/B polymerizations. The presence of the ethylene glycol moiety ensures the solubility of the resulting polymer in organic solvents.

Monomer 1 was obtained in eight steps from allylethoxyethanol (see supporting information).

A/B polymerizations were carried out at 80 °C [33],either in bulk or in solvents (Scheme 1).

All the reactions were run during 240 hours; this time was determinedas optimal to reach good DPn values in our previous work [19]. The main results are compiled in Table 1. The use of reduced pressure, either in a high-boiling solvent (entry c) or in bulk (entry d) confirmed the suitability of 1 for enzymatic reaction. In both cases, the isolated polymers presented very good average degrees of polymerization, higher than those already reported in diphenyl oxide for equivalent reaction times [19][20]. Reaction in bulk under 10 mbars led to a DPn higher than 18 units (entry d). This value, which ranges among the best DPn reported for related enzymatic synthesis of polyamides, has to be linked with the impressive yield in isolated polymer close to 90%. The 1.60 value of the polydispersity index is rather good for an enzymatic polymerization. All these results clearly underline the compatibility of 1 with the used enzymatic polymerization conditions. Futhermore, polymerization of 1 without catalyst failed to give a polymer with a DPn higher than 2.6 units (entry e).

3.2.Rate of reaction.

The polymerizations involving monomer 1 were monitored by 1 H NMR. The evolution of the conversion versus time was evaluated by the ratio of the integration of the signals of OCH 2 CO 2 Et terminations to that of the methylene protons in α -position relative to the amide nitrogen atom. This monitoring clearly highlighted the positive impact of the alkoxyacetate moiety in 1 on the rate of polymerization. The conversion of 1 reached 93% after only 30 min. The direct comparison with amino-esters 2 and 3 is even more representative of this dramatic effect, as conversion after 30 min reached only 36% for 2 and 31% for 3 that is three times less than for

1.

If the gap was reduced as the polymerization evolved, a major difference still remained.

Between 2 and 8 hours, the conversion of 1 varied from 95 to 98%, and remained near this level after 240 h. The rate of conversion of 2only reached 80% after 8 h, to gain 10% more at the end of the reaction. The evolution of the polymerization of 3 was slower, going from 66% conversion after 8 h to reach 93% at the end. The slight difference in reactivity between 2 and 3 could be correlated to the high methylene content of 2, depicted by Gross and coworkers as responsible of an important enhancement of the rate of polymerization in the case of oxygenated polyesters [35].

In light of these results, it became obvious that the use of 1 in an enzymatic polymerization procedure did not require long reaction times to be fully efficient and to lead to a wellfeatured polymer, in opposition to monomers 2 and 3.

This enhanced rate is not linked to the experimental conditions, as demonstrated in graph 2, which shows the evolution of the conversion of 1 during the first 8 h of the polymerization in bulk under atmospheric pressure, under reduced pressure, and in diphenyl oxide under reduced pressure.

Graph 2.Evolution of the conversion of amino-ester 1 during the first 8 h of polymerization under different conditions.

Reactions at 10 mbars, either in bulk or in Ph 2 O present similar profiles. The conversion of 1 reached a value close to 93% after only 30 min. The reaction in bulk under atmospheric pressure did not reach such a high level, but the observed conversion was 83% after 30 min, to end at 90% in 8 h. It seems so that the difference in term of conversion is only related to the benefit of working under 10 mbars pressure, and that the ensuing kinetic improvement could be observed whatever the experimental conditions.

3.3.AA/BB Polymerization

In parallel with this study, the possibility to transpose the rate enhancement observed in A/B reactions to a AA/BB sequence has been investigated using thediester5which presents two ethyl alkoxyacetate moieties. Diamine4 was selected as reactant. It presents an oxygenated moiety to insure the solubility of the resulting polymers, and a high methylene contents.Experimental conditions involving reduced pressure in presence of Ph 2 O were selectedto insure a good diffusion of the monomers all along the reaction and to prevent an inconvenient elevation of the viscosity [19]. The molecular properties of the products are depicted in Table 2. As in the previous examples, the use of reduced pressure dramatically increased the efficiency of the reaction [19].This time, a polymer with a DPn equal to 12.5, and a polydispersity index of 1.60 was isolated in very good yield. Despite the fact that these results did not reach the highest values of DPnand PDIthat had already been reported, the compatibility of diester5 with the enzymatic procedure makes no doubt. The polymer produced from 4 and 5 could be precipitated at the end of the reaction, due to the high hydrophobic character of diamine4.

The AA/BB reactions were also monitored. However, the NMR methodology could not be applied to the reactions of diamine4 with diesters5 and 6 due to the impossibility to determine the exact nature of the end-groups of the polymeric chains during the polymerization. The reaction was therefore followed by the injection in size-exclusion chromatography of small aliquots withdrawn from the flask to determine the average degree of polymerization of the sample. The plot of DPn versus time during the first hours of reaction is shown in Table 3. The major improvement observed in the case the A/B reaction was no more observed in the AA/BB experiments involving diamine4 and diester5as compared to 6. Indeed during the first hours, a slight difference could be detected as the copolymerization with 5 led to a polymer with a DPn equal to 2.8, whereas in the case of 6,the DPn was only 2.3. The small gap no longer remained after 8 h.The curves crossed again 12 hours later with the polymer issued from 5 showing a slightly better DPn value. The higherDPn observed in the case of diester6at the end of the polymerization could be attributed to the influence of the carbon chain-length on the polymerization degree, which is in agreement with Gross and coworkers [14].

It is difficult to rationalize this absence of acceleration. An inhibition of the "key and lock" interaction predicted by Park and co-workers, caused by the structure of one of the monomers might be suspected.

The polymer obtained from 4 and 5was purified by precipitation to give a solid residue,thelatterwas thermally characterized through an ATG analysis.

Fig2.Thermogravimetric degradation profile of polymer P(4+5).

The degradation of the polyetheramideP(4+5) took place in one step comprising between 290 and 510 °C (Fig 2). These data are in good agreement with already reported values [19][20] and confirmed the good thermal behavior of such a class of polymers, which can face those obtained from Nylons synthesized using more conventional pathways [36][START_REF] Preston | Encyclopedia of Polymer Science and Technology[END_REF].

The potential crystalline behavior of P(4+5) was then investigated by differential scanning calorimetry (Fig 3).

Fig3.DSC curves ofpolymersP(4+5)and P(4+6).

The presence of two melting temperatures (71.41 °C and 88.01 °C) is in agreement with a semi-crystalline character for P(4+5). This observation might also argue in favor of the coexistence of two types of supramolecular organizations, resulting from the presence of different intermolecular interactions between polymeric chains [START_REF] Porter | [END_REF]. Such behavior has been already observed in the case of P(4+6) [19], which DSC curves is also showed in the figure 3.

Conclusion

Inserting an ethyl alkoxyacetate moieties in the backbone of A/B type monomer dramatically accelerates the formation of polyamides catalyzed by the lipase B of Candida antarctica. The "activated" amino-ester monomer allows to reach in 30 min conversion rates three times higher than those observed with conventional monomers, whatever the experimental conditions. The efficient polymerization procedures give rise to soluble polymers whose molecular, thermic and crystalline properties reach the best standard reported in this field.With this improvement, enzymatic polymerization becomes kinetically compatible with free radical-mediated racemization. This is a step forward towards the development of an efficient ITC procedure for the synthesis of optically active polyamides starting from racemic monomer.

  7 mm) precolumn. The system was piloted by Azur (Jasco©) software, and equipped with a Waters 600 pump, a Varian oven (model 510) and a Waters differential diffractometer (model 410). Analyses used a 0.1 M solution of LiBr in N,N-dimethylacetamide (HPLC grade, degassed and filtered on a Millipore membrane before use) at 80 °C (0.6 mL/min rate of flow). Sample concentrations of 1-3 mg/mL and injection volumes of 200 µL were used. Toluene (500 μL/100 mL) was introduced in the mobile phase as elution marker. System calibration data and relative molar mass calculations were acquired and processed using PSS WinGPC (Polymer laboratories) software. Narrow distribution polyethylene glycol standards (Varian) with number-average molecular weight (Mn) values of 106, 194, 430, 615, 1010, 1970, 3390, 7980, 12140 and 21030 Da were used to establish the calibration curves.

  Scheme 1. A/B polymerization involving amino-ester.

1 Fig 1 .Graph 1 .

 111 Fig 1. Amino-esters structures.The polymerization in diphenyl oxide under reduced pressure was compared with the

Scheme 2 .

 2 Scheme 2.AA/BB polymerization involving diester5 and 6.

  CH 2 OCH 2 CH 2 OCH 2 ), 3.23 (m, CH 2 NHCO), 2.99 (m, CH 2 NH 2 ), 1.58 (bs, CH 2 CH 2 NHCO et OCH 2 CH 2 CH 2 NH), 1.21 (t, J=7.2, CH 3 CH 2 OCO, terminations). 13 C

	H NMR (300 MHz, D 2 O): 7.86 (bs, NH), 4.14 (q, J=7.2, CH 3 CH 2 O,
	terminations), 4.09 (s, OCH 2 COOEt, terminations), 3.92 (s, OCH 2 CONH), 3.59 (m,
	OCH 2 CH 2 O), 3.46 (m, NMR (75 MHz, D 2 O): 173.0 (CONH), 172.1 (COOEt), 72.4 (CH 2 ), 72.3 (CH 2 ), 72.2 (CH 2 ),
	71.7 (CH 2 ), 71.6 (CH 2 ), 71.5 (CH 2 ), 69.8 (CH 2 ), 62,3 (CH 2 ), 40.0 (CH 2 ), 28.4 (CH 2 ), 27.8
	(CH 2 ), 15.0 (CH 3 ).

  H NMR (400 MHz, CDCl 3 ): 6.70 (bs, NH), 4.16 (m, OCH 2 CH 3 , terminations), 4.13 (s, OCH 2 COOEt, terminations), 4.01 (s, OCH 2 CONH), 3.72 (s, OCH 2 CH 2 O), 3.67-3.56 (m, OCH 2 CH 2 OCH 2 CH 2 O), 3.44 (t, J=6.8, CH 2 OCH 2 CH 2 OCH 2 CH 2 OCH 2 ), 3.29 (dt, J=7.0, 6.8, CH 2 NHCO), 1.61-1.47 (m, CH 2 CH 2 O et CH 2 CH 2 NHCO), 1.36-1.26 (m, (CH 2 ) 7 ), 0.88 (t, J=6.8, CH 3 CH 2 O, terminations). 13 C NMR (100 MHz, CDCl 3 ): 170.4 (CONH), 71.6 (CH 2 ), 70.7 (CH 2 ), 70.6 (CH 2 ), 70.1 (CH 2 ), 39.0 (CH 2 ), 31.0 (CH 2 ), 29.7 (CH 2 ), 29.6 (CH 2 ), 29.5 (CH 2 ), 29.4 (CH 2 ), 26.9 (CH 2 ), 26.1 (CH 2 ).

Table 1

 1 Molecular properties of synthesized A/B-type polymers with monomer 1.

		Conditions	M n	b	M n c	M w	b	PDI b Yield	DPn b
	a	Bulk	2020	2530	3050	1.50	71%	11.7
	b	Glyme	1750	2320	3270	1.90	79%	10.1
	c	Ph 2 O, 10 mbars	2540	3250	4160	1.60	90%	14.7
	d	Bulk, 10 mbars	3160	3880	5140	1.60	89%	18.3
	e a	Bulk, 10 mbars	450	-	1260	2.80	-	2.6
	a blank experiment performed without catalyst. b determination by SEC. c determination by 1 H NMR.
	The SEC analyses clearly highlighted the fact that monomer 1 was a good candidate for
	enzymatic polymerization. The bulk reaction, at 80 °C under atmospheric pressure (entry a),
	enabled the isolation of a polymer showing good properties, with a DPn equal to 11.7. This
	good reactivity was retained in solvent, as illustrated by the experiment performed in glyme
	(entry b) (DPnclose to 10).

Table 2

 2 Molecular properties of synthesized AA/BB type polymers with monomers 4 and 5.

		Conditions	Yield	M n	M w	PDI	DPn
	a	Ph 2 O, 240 h, 10 mbars	89%	7350 11770	1,60	12.5
	b	Ph 2 O, 240 h, 10 mbars a	-	-	-	-	2.6

a Blank experiment in the absence of enzyme.

Table 3

 3 Evolution of the DPn of polymers resulting from the reaction between diamine4 and diester5 and 6 under reduced pressure in Ph 2 O.

	Monomers	DPn (1 h)	DPn (3 h)	DPn (8 h)	DPn (20h)	DPn a (24 0 h)
	4+5	2.8	3.1	3.3	6.3	12.5
	4+6	2.3	3.1	3.7	3.9	16.7

a Isolated product.

Acknowledgment

The authors thank Dr Tran Phan for DSC measurements and Dr Emilie Bloch for ATG measurements. A gift of Novozym 435 from Novozymes (Denmark) is gratefully acknowledged.

Appendix. Supplementary material

Supplementary material associated with this article can be found, in the online version, atdoi:


Meiyi Wu

Karine Le Guen

Jean-Michel André

Vita Ilakovac

Ian Vickridge

Didier Schmaus

Emrick Briand

Sébastien Steydli

Catherine Burcklen

Françoise Bridou

Evgueni Meltchakov

Sébastien De Rossi

Franck Delmotte

Philippe Jonnard

Kossel interferences of proton-induced X-ray emission lines in periodic multilayers

Keywords: PIXE, periodic multilayer, Kossel line

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

Introduction

The interferences of characteristic x-ray emission produced within a crystal, by the crystal lattice itself was demonstrated experimentally by Walter Kossel in 1935 [START_REF] Kossel | Die Richtungsverteilung der in einem Kristall entstandenen charakteristischen Röntgenstrahlung[END_REF]. These interferences lead to the observation of what are now called Kossel lines, which are analogous to the Kikuchi lines observed in electron diffraction. Since the observation of a characteristic emission requires as a first step the ionization of a core level, different ionizing radiations can be used to observe Kossel lines [START_REF] Langer | Däbritz, 75 years of Kossel patterns -past and future[END_REF]:

 Electrons from an electron gun [START_REF] Lider | X-ray divergent-beam (Kossel) technique: A review[END_REF][START_REF] Jonnard | Modulation of x-ray line intensity emitted by a periodic structure under electron excitation[END_REF][START_REF] Jonnard | Soft-x-ray Kossel structures from W/C multilayers under various electron ionization conditions[END_REF][START_REF] André | Radiation emitted by an oscillating dipole embedded in a periodic stratified structure: A direct matrix analysis[END_REF] or a scanning electron microscope [START_REF] Langer | Der Einfluß von Kristallfehlern auf Kossel und Weitwinkel Interferenzen[END_REF];  X-ray photons from an x-ray tube [START_REF] Chauvineau | Analyse angulaire de la fluorescence du fer dans une multicouche périodique Fe/C[END_REF][START_REF] Chauvineau | Lignes de Kossel observées avec des multicouches périodiques Fe/C[END_REF][START_REF] Chuev | Direct observation of anomalous Kossel lines[END_REF] or synchrotron radiation [START_REF] Marchesini | Standing waves and Kossel line patterns in structure determination[END_REF][START_REF] Schetelich | Laue and Kossel diffraction on quasicrystals by means of synchrotron radiation[END_REF][START_REF] Jonnard | Spontaneous soft x-ray fluorescence from a superlattice under Kossel diffraction conditions[END_REF][START_REF] Tu | X-ray fluorescence induced by standing waves in the grazing-incidence and grazing-exit modes: study of the Mg-Co-Zr system[END_REF]; this case is analogous to the x-ray standing wave technique [START_REF] Tu | X-ray fluorescence induced by standing waves in the grazing-incidence and grazing-exit modes: study of the Mg-Co-Zr system[END_REF][START_REF] Gog | Kossel diffraction and X-ray standing waves: two birds of one feather[END_REF] used to study the interfaces of multilayers [START_REF] Bera | Ion irradiation induced impurity redistribution in Pt/C multilayers[END_REF] or x-ray waveguides [START_REF] Rajput | Swift heavy ion induced modification in Fe/Ti system studied using X-ray waveguide structure[END_REF] as well as superficial thin films [START_REF] Zargham | On revealing the vertical structure of nanoparticle films with elemental resolution: A total external reflection X-ray standing waves study[END_REF];  Rapid charged particles (proton or ion beam) from an accelerator [START_REF] Geist | The influence of the crystal lattice on the angular distribution of X-rays emitted from a GaP single crystal by fast proton bombardment[END_REF][START_REF] Roberto | Positive-ion-induced Kossel lines in copper[END_REF][START_REF] Geist | Kossel interferences of protoninduced Al Kα radiation[END_REF][START_REF] Geist | Investigation of lattice deformation in proton bombarded GaP and ZnSiP2 by means of the proton-induced kossel effect[END_REF][START_REF] Geist | Investigation of GaN heteroepitaxial layers by means of the kossel effect technique[END_REF][START_REF] Rickards | Crystallography using the diffraction of proton induced X-rays[END_REF][START_REF] Schetelich | Recording of Kossel patterns using monochromatic synchrotron radiation[END_REF].

The technique requires a periodic structure to diffract the emitted radiation, thus it has been applied to study crystals and interferential multilayers. However, to the best of our knowledge, Kossel lines have never been observed in multilayers upon particle excitation. It is the aim of this paper to show that it is possible to study multilayers having a nanoscale period upon proton irradiation through the observation of Kossel lines.

Experimental details

The multilayer used for this experiment was a Cr/B 4 C/Sc periodic multilayer, whose period was repeated 100 times. The samples were deposited by magnetron sputtering onto a silicon substrate. The thickness of the Cr, B 4 C and Sc layers are 0.60, 0.20 and 0.92 nm respectively, as deduced from x-ray reflectivity (XRR) measurements in the hard and soft xray ranges. Thus the period of the stack is 1.72 nm. Other reflectivity measurements also showed that Cr atoms are present inside the B 4 C layers [START_REF] Burcklen | Cr/B4C multilayer mirrors: Study of interfaces and X-ray reflectance[END_REF]. The thin B 4 C barrier layers were introduced to prevent the interdiffusion between Cr and Sc layers and can also improve the thermal stability of the stack [START_REF] Prasciolu | Thermal stability studies of short period Sc/Cr and Sc/B_4C/Cr multilayers[END_REF]. The multilayer was capped with a 2.5 nm-thick B 4 C layer in order to prevent it from oxidation. Working with this Cr/B 4 C/Sc multilayer enabled us to measure well-resolved Sc and Cr K lines and to detect them at reasonable grazing angles. Indeed, this kind of short-period multilayer is chosen for microscopy or spectroscopy applications in the water window range [START_REF] Schäfers | Cr/Sc multilayers for the soft-x-ray range[END_REF][START_REF] Le Guen | X-ray spectroscopic application of Cr∕Sc periodic multilayers[END_REF] and requires well defined layers. Protons of 2.0 MeV produced by the Van de Graaff accelerator of the SAFIR Platform of the Université Pierre et Marie Curie were used to excite the sample. With such energy, the protons ionise the Sc and Cr atoms in their K shell uniformly over the full multilayer thickness. The size of the beam on the sample was approximately 2 mm and the beam current maintained between 100 and 150 nA for the duration of the experiments.

Possible evolution of the multilayer composition was monitored simultaneously via the elastically backscattered protons detected in a passivated implanted planar silicon (PIPS) detector, placed at 165° with respect to the direction of the proton beam. We present in Figure 1 the spectrum so obtained for a total proton dose of 600 µC. Protons scattered from the Cr and Sc atoms appear in the peak near 1800 keV. The area of this peak was monitored, and did not change during the measurements, indicating no measurable loss of matter induced by the beam. As a further precaution, the proton beam was moved on the sample surface, from one pristine zone to another, so that the dose did not exceed 600 µC at a given location. The scattering cross sections on the light elements are highly non-Rutherford: in particular the 11 B(p,p) 11 B cross section is much greater than the Rutherford cross-section which explains why the 11 B peak is visible in spite of the very small quantity of boron, and the highly non-Rutherford 28 Si(p,p) 28 Si cross-section is manifested in the resonant structure obtained from the thick silicon substrate.

The setup for the Kossel experiment is shown in Figure 2. The angle between the incident proton beam and the x-ray detector (SDD, silicon drift detector) was fixed at 90°. In addition to the beryllium window protecting the detector from the atmosphere, a 60 µm thick Mylar film was placed in front of the detector to block the scattered protons. Overall transmission for the Sc K and Cr K lines is greater than 0.65 [START_REF]CXRO X-Ray Interactions With Matter[END_REF] and practically zero for the Sc and Cr L lines, and also for the B and C K lines.

We rotated the sample so that the angle  between the sample surface plane and the direction of the SDD, was varied around the Bragg angle, which is around 4° for the Cr K emission and 5° for the Sc K emission (see below). The distance between the sample and the detector was 115 mm and a slit of ≈0.2 mm width was placed in front of the collimator of the SDD, giving an angular resolution of about 0.1°.

Results and discussion

We show in Figure 4 the Kossel curves for the Sc and Cr K emissions toward the low angles and starting from zero. At each angle, the proton dose is 30 µC. The angular step is 0.2° except in the region of the strongly varying intensity where it is 0.05°. For both curves, the intensity increases first sharply and then smoothly. The intensity increase is due to the total internal reflection of the radiation emitted within the sample. The angular shift observed in Figure 4, between the onset of the Sc and Cr curves, about 0.15°, comes from the difference of the mean optical indices of the multilayer at the energies of Sc K and Cr K radiations. The sharp edge inflection point is the angle of the total internal reflection and is chosen to calibrate the angular scale. This last angle is calculated (with a typical uncertainty of ±0.01°), by optical simulation [START_REF]CXRO X-Ray Interactions With Matter[END_REF] of the reflectance at the two corresponding energies, as 0.56° for Sc and 0.42° for Cr. The uncertainty on the absolute value of the experimental angles is estimated to be 0.05° (the angular step around the inflexion point) while the relative uncertainty is lower than this value.

Conclusion

In conclusion, we have observed for the first time the Kossel interferences of protoninduced x-ray emission lines in a multilayer. This was achieved by using a Cr/B 4 C/Sc periodic stack and observing the Sc K and Cr K characteristic emissions. Owing to the low background radiation, the x-ray spectra lead directly to the Kossel curves. The agreement between the experiments and the simulation is excellent regarding the shape and position of the Kossel features, although it may be improved regarding the contrast of these features obtained in this pioneering experiment.

The counting statistics for a given incident proton fluence could be improved by changing the angle of incidence between the protons and the sample surface and thus increasing the apparent thickness of the sample seen by the proton beam, or by using a colour camera [START_REF] Ordavo | A new pnCCDbased color X-ray camera for fast spatial and energy-resolved measurements[END_REF] sensitive both to the location and energy of the detected x-rays. This second solution could provide 1 to 2 orders of magnitude greater x-ray yield per incident proton and could open the way for systematic use of proton-induced x-ray emission with Kossel interferences to probe the nanoscale layers and interfaces of periodic multilayers. Thus, it will become possible to obtain with proton-induced x-ray emission under Kossel interferences the same kind of results and informations as those obtained by the widespread x-ray standing wave technique [START_REF] Zegenhagen | The X-Ray Standing Wave Technique: Principles and Applications[END_REF].

Figure 1 :

 1 Figure 1: 2.0 MeV proton elastic backscattering spectrum obtained from the Cr/B 4 C/Sc multilayer. The surface energies of the various elements are indicated by the downward arrows.The scattering cross sections on the light elements are highly non-Rutherford: in particular the 11 B(p,p)11 B cross section is much greater than the Rutherford cross-section which explains why the 11 B peak is visible in spite of the very small quantity of boron, and the highly non-Rutherford 28 Si(p,p)28 Si cross-section is manifested in the resonant structure obtained from the thick silicon substrate.

Figure 2 :

 2 Figure 2: Experimental setup for the Kossel experiment. The sample is rotated so that the detection angle  is around the first order Bragg angle calculated by considering the emission wavelength and multilayer period. The x-ray and proton detectors are fixed.

Figure 3 :

 3 Figure 3: X-ray emission spectrum of the Cr/B 4 C/Sc multilayer induced by 2 MeV protons.

Figure 4 :

 4 Figure 4: Wide scan of the Kossel curves for the Sc K (solid curve) and Cr K (dashed curve) emissions.

Figure 5 Figure 5 :

 55 Figure5shows the Kossel curves obtained in the range of the Kossel feature with a 0.05° angular step and using the angular calibration determined above. At each angular position, the total dose was 120 µC for the Sc K curve, and 150 µC for the Cr K curve. The error bars represent one standard error. Experimental curves are compared to simulations[START_REF] Chauvineau | Analyse angulaire de la fluorescence du fer dans une multicouche périodique Fe/C[END_REF][START_REF] Chauvineau | Lignes de Kossel observées avec des multicouches périodiques Fe/C[END_REF][START_REF] Chambet | Etude experimentale de la diffusion des rayons x rasants par les multicouches pour optiques X-UV[END_REF] that calculate the electric field generated within the periodic stack by the emitted radiation according to the reciprocity theorem. The parameters of the multilayer (thickness, roughness) introduced in the simulation are those deduced from XRR analyses.The simulations are convolved with an experimental function (a rectangle) having a 0.1° width representing the angular resolution of the experimental setup. Let us note that in the case of proton excitation, simulations are easier to perform than those in the cases of electron and x-ray excitations, where the depth-dependent ionization probability should be considered; electrons loose significant energy within the thickness of the stack while x-rays are either attenuated or even create standing waves within the periodic structure.

Figure 6 :

 6 Figure 6: Depth distribution of the electric field generated within the Cr/B 4 C/Sc multilayer by the emitted Sc K radiation. The calculation illustrated here is made for an angle -0.035° away from the Bragg angle. Only the capping layer and the first four periods of the multilayer are represented.

Acknowledgments:

The multilayer sample has been deposited as part of CEMOX (Centrale d'Elaboration et de Métrologie d'Optique X), a platform of LUMAT federation (CNRS FR2764). D. Vernhet and J.-J. Ganem from INSP are thanked respectively for providing us with the x-ray detector and helping during the experiment.


J-P Pernot

P Veron

Giannini F Falcidieno
email: falcidieno@ge.imati.cnr.it

AESTHETIC-ORIENTED CLASSIFICATION OF 2D FREE-FORM CURVES

Keywords: Aesthetic properties, 2D free-form curves, shape characteristics, shape classification, machine learning techniques (MLT)

Nowadays, it is commonly admitted that the aesthetic appearance of a product has an enhanced role in its commercial success. Therefore, understanding and manipulating the aesthetic properties of shapes in the early design phases has become a very important field of research. There exists an appropriate vocabulary for describing the aesthetic properties of 2D free-form curves that includes terms such as straightness, acceleration, convexity and tension, which are normally used by stylists when describing and modifying shapes. However, the relationships between this vocabulary and the geometric models are not well established. This work investigates the possibility of applying Machine Learning Techniques (MLT) to discover possible classification patterns of 2D free-form curves with respect to the so-called straightness of the curve. First, we verified that MLT can correctly (99, 78%) reapply the classification to new curves. In addition, we verified the abilities of the Attribute Selection methods to identify the most important attributes for the considered classification, among a larger set of attributes. As a result, it was possible to recognize as the most characterizing parameters the same curve attributes previously used to compute the measure of straightness (S). Moreover, Linear Regression (LR) was able to extract automatically an exact mathematical model, which can correlate the geometric quantities with the class of the curve, congruent to one we previously specified. This work indeed demonstrates that MLT are very suitable and can be efficiently used in this context. The work is a first step towards the characterization and classification of free form surfaces giving the general directions on how MLT can be exploited to characterize free-form surfaces with respects to the aesthetic properties.

INTRODUCTION

Nowadays, the point of view for analysing the shape of a new product has been moved from production feasibility toward users acceptance. Until recently, all products were designed taking into consideration production and cost constraints therefore designers could not pay enough attention to the shape they prefer, because they had to focus on the shape that can be reached with production equipment as well. It was a closed system (designers-manufacturers) where the customers were somehow excluded. Now, with the availability of new materials and development of production equipment, especially with the development of five-axis CNC machines, we do not pose anymore the question of which shape can be produced, but which shape do you want to produce. Thus, the designers have more freedom to design what they like. Additionally, in the current competitive market, being faster in reaching the customers expectations is becoming extremely important. Therefore, we can foresee a new product development process in which the customer is an important actor as the other experts in analysing and defining the product. Taking into consideration the fact that customers are more important than before, aesthetic aspects of the product are becoming significantly more important and they become key factors for its acceptance in the market.

Including the customers in the design process emphasizes the difference of the languages used in different activities of the design process [START_REF] Giannini | Aesthetic-driven tools for industrial design[END_REF]. Namely, Marketing Language is referred to as first language and Designer language as second language, and the aesthetic characteristics represent a bridge between both languages. The first one is adopted during the specification of the wished product and during the briefing activity. It represents an emotional and individual description of the aesthetic character. The second one represents its detailed specification during the creation and modification of the product model according to geometric properties and criteria. In order to identify the appropriate aesthetic characterization of shapes and to understand which feature properties stylists consider, the design activities carried out in different industrial field have been deeply analysed within the European Project FIORES II (Character Preservation and Modelling in Aesthetic and Engineering Design). One of the objectives of this project was to find mappings between verbal styling description and geometric parameters and characteristics. Such a mapping can allow the development of modelling tools closer to the stylists mentality. References [START_REF] Giannini | Aesthetic-driven tools for industrial design[END_REF] and [START_REF] Giannini | A survey of tools for understanding and exploiting the link between shape and emotion in product design[END_REF] present the identified terms and developed measurements for styling properties together with CAD tools based on them. Even though an appropriate aesthetic vocabulary is determined, there is still the need for a verification of the understanding of their meaning by the non-designers. Moreover, it might be more useful to allow a qualitative judgment (e.g. less, more, very, not or not at all) of these aesthetic properties than quantitative measures. In this work, the focus is on the straightness property of 2D curves. Using the results provided in [START_REF] Giannini | Semantic Evaluation and Deformation of Curves Based on Aesthetic Criteria[END_REF] and [START_REF] Giannini | A Survey to Evaluate how non Designers Perceive Aesthetic Properties of Styling Features[END_REF] we aim at analysing possible ways to automatically defining styling properties measures and qualitative categorization of curves according to such properties. Thus, this work introduces a new framework for classifying curves with respect to aesthetic property. In particular it has the following goals:

• Verifying the interrelation between the curve characteristic quantities and the aesthetic property • Exploring the capability for selecting the most meaningful geometric quantities for classifying curves with respect to the aesthetic properties. • Evaluating the ability of extracting a mathematical model (equation) which will correlate the straightness with the geometric quantities.

To do that, a set (ST) of curves has been generated by applying appropriate deformation vectors at two points of initial straight lines. Afterwards, for each curve from the set ST, the characteristic quantity values of all curves are computed.

Finally, having the curves quantities on one hand and the measure of straightness on the other hand, we have the suitable data for applying Machine Learning Techniques to discover some structural patterns and to select most meaningful curve quantities (figure 1).

Figure 1 Overall framework and data workflow

The following section details the formulation for the measure of the straightness of 2D free-form curves. Then, after a brief presentation of the Machine Learning Techniques in section 3, the section 4 describes how the experiments were conducted and what the results are. Finally, section 5 discusses our results with a conclusion and some perspective for future works.

STRAIGHTNESS OF 2D FREE-FORM CURVES

Before providing the definition of straightness, we note here the distinction between the terms aesthetic and styling properties. The aesthetic property refers primarily to the appearance of the overall shape of the product while styling property refers mostly to the shape of the geometric elements (curves and surfaces). However, styling properties seem to be the best way to build up the aesthetic character (appearance) of the product. In other words, with modifying the shape of its geometric elements, the related styling properties are changed so the overall appearance of the object is changed and it might lead to another aesthetic character.

Among the terms and measures for styling properties identified by the European Project FIORES II, in this paper we focus on the straightness property. While in engineering, a curve is either straight or not (a part some tolerance), for a stylist a curve can be more or less straight, depending on how visually it differs from the line segment which is somehow related to the dimension of the overall curvature radius in re-lation to the curve length. The bigger the radius is, the straighter the curve. Even curves having inflection points and consequently variable radius can appear somehow straight.

The FIORES II project proposed the following formulation for the straightness measure: Because of previously mentioned drawbacks, the work described in [START_REF] Giannini | A Survey to Evaluate how non Designers Perceive Aesthetic Properties of Styling Features[END_REF] proposes a revised measure for the straightness.

straightness(S) = 1 -d min /d max (1)
N S = non -straightness = C • A • L l 2 (2)
non-straightness (NS) ∈ [0, )

S = straightness = 1 1 + non -straightness (3) straightness (S) ∈ (0, 1]
Where the curve is given as a function of the parameter u:

P (u) = (x(u), y(u)), u ∈ [0, 1] (4)
C = 1 0 |k(u)|du (5)
is the integral of the absolute value of the curvature.

A = 1 0 y(u) • ẋ(u)du (6)
is the value of the area between the curve and the line that joins the two extremes of the curves.

L = 1 0 ẏ(u) 2 + ẋ(u) 2 du (7)
is the length of the curve

l = (x(1) -x(0)) 2 + (y(1) -y(0)) 2 (8)
is the length of the chord between the two end points of the curve.

This measure of straightness better represents the character of the curves than the initial one. The initial qualitative classification of the straightness property according to property range values has been further verified (see Table 1). From the work in [START_REF] Giannini | A Survey to Evaluate how non Designers Perceive Aesthetic Properties of Styling Features[END_REF], it has been concluded that there is a general convergence in the categorisation of curves according the very not, not, fairly and very judgments for the straightness property. Therefore, in this work the measure of straightness expressed with the equ. ( 3) and the classification given in the table 1 are used in the further analysis to create the model for straightness category prediction.

Table 1 Curve classification with respect to the straightness measure formulation in (eq. 3)

Straightness (S) Classification (0 0,7) very-not-straight(nS) [0,7 0,9) not-straight(ns) [0,9 0,98) fairly-straight(s) [0,98 0,999) very-straight(S) [0, 999 1] straight-line(SS)

USING MACHINE LEARNING TECHNIQUES

This section provides some basic notions related to Machine Learning Techniques, to their usage in applications, and to the environment adopted in the reported experiment.

Machine Learning

Machine Learning Techniques have been widely used to discover structural patterns for analysing problems in medical and biology application [START_REF] Lemm | Introduction to machine learning for brain imaging[END_REF][START_REF] Maddouri | A data mining approach based on machine learning techniques to classify biological sequences[END_REF][START_REF] Wang | Machine learning in radiology[END_REF][START_REF] Wang | Emotional state classification from EEG data using machine learning approach[END_REF][START_REF] Wasan | Application of statistics and machine learning for risk stratification of heritable cardiac arrhythmias[END_REF].

Next, wider application of the machine learning is in prediction of stock market [START_REF] Lee | Using support vector machine with a hybrid feature selection method to the stock trend prediction[END_REF][START_REF] Luo | Integration piecewise linear representation and weighted support vector machine for stock trading signal prediction[END_REF][START_REF] Ni | Stock trend prediction based on fractural feature selection and support vector machine[END_REF], where its application is used to discover which are the most relevant factors (features selection), that will influence the stock market. Furthermore, most common application of machine learning is in image processing [START_REF] Abreu | An Empirical Comparison of Individual Machine Learning Techniques in Signature and Fingerprint Classification[END_REF][START_REF] Conilione | Automatic localization and annotation of facial features using machine learning techniques[END_REF][START_REF] Lattner | A Combination of Machine Learning and Image Processing Technologies for the Classification of Image Regions[END_REF][START_REF] Motaal | Different Region Identification in Composite Strain-Encoded (C-SENC) Images Using Machine Learning Techniques[END_REF][START_REF] Negri | An innovative support vector machine based method for contextual image classification[END_REF][START_REF] Sajn | Image processing and machine learning for fully automated probabilistic evaluation of medical images[END_REF][START_REF] Schwenker | Pattern classification and clustering: A review of partially supervised learning approaches[END_REF], where machine learning is used to discover classification patterns for feature recognition and automatic annotation. Some similar usage is applied for 3D model classification and retrieval [START_REF] Desjardins | Learning User Preferences for Sets of Objects[END_REF][START_REF] Laga | 3D Shape Classification and Retrieval Using Heterogenous Features and Supervised Learning[END_REF][START_REF] Ip | Content-Based Classification of CAD Models with Supervised Learning[END_REF].

The later refers to discovering classification patterns of shape descriptors of already created 3D CAD models.

In design application, various works aim at understanding the association between the shape of a product and the evoked product image, i.e. the most appropriate description such as calm, feminine, aggressive, from customer interviews [START_REF] Lesot | Product shape and emotional design an application to perfume bottle[END_REF][START_REF] Lu | On Shape and the Computability of Emotions[END_REF][START_REF] Machwe | Integrating aesthetic criteria with evolutionary processes in complex, free-form design an initial investigation[END_REF][START_REF] Ren | Design Preference Elicitation, Identification and Estimation[END_REF]. These works are mainly devoted to provide tools able to foreseen the evoked product images for new products. Only very few works try to detect shape parameters directly usable for the shape manipulation in the early phases of design process [START_REF] Giannini | A survey of tools for understanding and exploiting the link between shape and emotion in product design[END_REF]. In FIORES-II, for instance, machine learning capabilities of CBR system have been applied to verify the emotional image of variants of product templates [START_REF] Giannini | Aesthetic-driven tools for industrial design[END_REF]. The the aim of this work is to investigate the possibility of applying machine learning in early phase of design process for mapping of aesthetic properties to geometric entity (2D free-form curves).

Machine learning is a core subarea of Artificial Intelligence (AI) that studies computer algorithms for learning rules from data to deal with some particular tasks.

The main domain where Artificial Intelligence (AI) has increasing application is in solving Data Mining (DM) problems. Different definitions of Data Mining exists, two of the most general state: Data Mining is defined as process of discovering patterns in data where the process must be automatic or semiautomatic and the patterns must be meaningful in that they lead to some advantages [START_REF] Witten | Data mining: Practical Machine Learning Tools and Techniques[END_REF]; Data Mining defines the automated extraction procedures of hidden predictive information from databases [START_REF] Negnevitsky | Artificial Intelligence: A Guide to Intelligent Systems[END_REF]. Overall, both interpretations designate that the capability of Data Mining is finding a way to mine (dig) in a huge set of data and getting some knowledge from it [START_REF] Shhab | A Study on Application of Machine Learning Techniques in Data Mining[END_REF]. In general, tasks that address Data Mining problems are divided into [START_REF] Witten | Data mining: Practical Machine Learning Tools and Techniques[END_REF]:

• A Predictive group that refers to Prediction, Classification, and Regression • A Descriptive group that refers to Association rules, Clustering, Summarization and Attribute Selection.

To get trustable rules form data it is important to have consistent data set on which to proceed in the learning process. Considering this, in our experiment we used set of curves ST consisting of 44.800 curves, which is an acceptable number to be candidate for a Data Mining problem solving.

In our work, the WEKA data mining workbench has been adopted. The WEKA (Waikato Environment for Knowledge Analysis University of Waikato, New Zealand) workbench is a collection of the most used machine learning algorithms and data pre-processing tools designed in a way that provides flexibility of their manipulation. Reference [START_REF] Witten | Data mining: Practical Machine Learning Tools and Techniques[END_REF] provides a comprehensive and detailed description of WEKA workbench.

In addition, WEKA performances and particular functions are described in various papers, such as [START_REF] Aher | Data Mining in Educational System using WEKA[END_REF][START_REF] Aksenova | Machine Learning with WEKA: WEKA Explorer Tutorial[END_REF][START_REF] Caruana | Greedy Attribute Selection[END_REF][START_REF] Garner | WEKA: The Waikato Environment for Knowledge Analysis[END_REF][START_REF] War | WEKA Machine learning Project Cow Culling[END_REF] illustrating its use in different applications.

Implementation of machine learning algorithms using WEKA workbench

There are three ways of using WEKA. The first one is to apply a learning method to a dataset and analyze its output in order to learn more about the data. The second way is to use a learned model in order to generate prediction of new instances and the third is to apply several different learners and compare their performance in order to choose the most suitable for prediction.

WEKA provides several different applications: Explorer, Experimenter, Knowledge Flow and Simple CLI. For our purposes, the most useful application is the Explorer that gives access, through menu selection, to quickly read a dataset from a ARFF file and build a decision tree from it or to any other algorithm, from all different Data Mining problems to explore.

The second WEKA application is Experimenter and it is designed to help to answer basic practical questions when applying classification or regression techniques -such as which methods and parameters values work best for the given problem. In addition, Experimenter provides a good environment to compare a variety of learning technics. The Knowledge Flow application allows the manipulation with boxes that represent learning algorithms and data sources around the screen in order to design the configuration wanted. In other words, it helps the design of the structure of the data flow with simple connection of box components that represent data source, appropriate tools, learning algorithms, evaluation methods and visualization modules.

In our case, the application Explorer is going to be used for reading and preparation of data, applying the learning algorithms and results visualization. The objective of this work is to create a classification model using Machine Learning algorithms and to investigate the possibility of using the Attribute Selection algorithm in order to select most meaningful geometric quantities of curve among of set of various basic quantities. The aim of the later objective is to see whether WEKA could be used for selection of most significant geometric quantities of surfaces in the view of defining styling properties and related measures also for surfaces and objects.

A more detailed analysis of previously mentioned objectives is presented in the next section.

Modeling of the straightness problem into WEKA

WEKAs native data file format is the ARFF (Attribute-Relation File Format). An ARFF file (fig. 4) has two distinct sections: the first section is the Header information which is followed by the Data information. The Header contains the name of the relation (@RE-LATION), a list of the associated attributes (@AT-TRIBUTE) with their types and if necessary comment lines starting with the character %. The Data section contains the data declaration (@DATA) line followed with the instance lines. Aiming at modeling the straightness, the measure expressed by the eq. 3 is adopted as a mathematical model of curve straightness and its parameters (C, A, L, l) represent the curves quantities. So as to create the model for straightness class prediction, as many as possible different curves have to be created and for each of them, the measure of straightness and curves quantities have to computed. Later, such computed values have to be stored in appropriate text data file (.arff) suitable for further use of WEKA. In order to get a suitable data an adequate program in Matlab has been implemented. It includes a function for the automatic generation of curves and a function that compute curves quantities and its associated curves class according with the eq. 3 and the table 1. At the end of this Matlab program, the ARFF file containing both Header and Data section of information is generated with the following Header section: % Header @RELATION straightness @ATTRIBUTE Area(A) REAL @ATTRIBUTE Curvature(C) REAL @ATTRIBUTE Curve length(L) REAL @ATTRIBUTE Base length(l) REAL @ATTRIBUTE Class {(SS), (S), (s), (ns), (nS)}

The Data section contains a line for each instance of curves with the computed values of the curves quantities and the corresponding class according to the table 1.

EXPERIMENTS

This section describes the experiments carried out in WEKA regarding the categorisation of curves according straightness property measures. First the method adopted for the data set creation is reported, and then the various experiments are reported together with a discussion on the obtained results.

Generation of an input set of curves (ST)

First, a set of significant number of instances of 2D curves is created. For this purpose, a deformation model (figure 4) has been defined and applied on an initial straight line to create many 2D free-form curves. The adopted deformation model uses three deformation modes ( 1 , 2 and 3 ) based on two displacement vectors (blue and yellow) applied on two control points of the parametric curve. 200 displacements are assigned to each vector direction. Figure 4 illustrates the different vector directions and the resulting types of shapes For example, in the 1 mode, in the first subset, we selected 11 different directions and then both vectors are equal and parallel with the 11 vector directions and 200 positions for each direction. In the second subset, only the vector modules are different and they are still parallel. The same principle is adopted for the 2 and 3 deformation modes with different number of vector directions. In these two cases we provided 6 different initial directions. To summarize, the deformation modes allow to create with 1 (11 directions * 200 positions * 2 cases) = 4400 curves, with 2 (6 directions * 200 positions * 2) = 2400 curves and with 3 4400 curves too. Collecting the curves obtained with the above described deformation modes gives us the set of 11200 curves applied on four different straight initial curve lengths (5, 8, 9 and 12 cm) to produce the global set ST of 44800 2D curves as illustrated in figure 5.

Classification using dimensional attributes

In terms of using machine learning algorithms, many different classification algorithms have been used but only the classification trees (particularly, tree J48) appeared to be most convenient in case of numerical attributes [START_REF] Shhab | A Study on Application of Machine Learning Techniques in Data Mining[END_REF]. The process of classification model creation consists of two steps (modeling and validation). In the first step, the classification algorithm is applied on the global input set ST of 2D curves, while in the second step, a subset S-ST (10-30%) of ST only is used to validate the model, giving the percentage of good classified instances (in this step the classification curves belonging to the set ST minus S-ST is considered as unknown). For the validation process, WEKA enables three different ways: cross-validation split test set and supplied test set, performing use of appropriate algorithms. Namely, cross-validation algorithm divides the set of instances in training set and testing set in a ratio that can be chosen by the user. For instance, if 10 folds cross-validation is chosen, it means that whole data set of instances is divided in 10 parts from which 9/10 are used as a training set (for creation of a model) and 1/10 is used as test set (for validation of the model) and this step is repeated 10 times. Unlike the cross-validation, split percentage algorithm splits the data set in training and test set by inserting the percentage (given by the user) of instances that will be contained in the test set and it is done only once. The third way is totally different: the user independently chooses the training and testing sets which are not required to be a part of a common data set.

Results:

When the modelling process is over, learning schemes are automatically generated by WEKA and depending on which validation method has been performed, we obtained a percentage of correctly classified instances varying from 98.63 % -cross validation -to 98.18 % -split test. Moreover, when analysing the incorrectly classified instances, we can observe that they are classified just in the neighboring classification class and the error of the classification is not more than 0.5 %. Having these results into account and the fact that the intervals of the classification class (table 1) are very strict from a numerical value point of view, a change of 0.5 % in the intervals will not make any visible changes in the appearance of curves which allows us to consider that this classification model is relevant.

Discussion: One important aspect is that the classification in the curve category should be somehow independent on the curve dimension, i.e. it should reflect only shape behaviour characteristics. Thus, to verify whether this model is independent form the initial line length and use it to predict the straightness class of any curve, an evaluation of the model has been performed using curves obtained from other initial lines having different lengths, i.e. 0.8, 80 and 240 cm. The results obtained from the evaluation, show that the model correctly classifies from 21.14 % to 28.42 % of all data instances so it demonstrates that this way of model creation does not guarantee to be size independent. Therefore, another approach for model creation has to be defined that will be more general and independent from initial curves or attributes. In the next section this new approach is presented together with the results obtained.

Classification using dimensionless attributes

As mentioned before a new approach has been explored in order to overcome the lack of low rate of correctly classified instances. After an analysis of the mathematical model of the straightness (eq. 3), it was concluded that whilst the final measure is independent of the curve size, each single parameter (attribute) in the equation is related to the curve dimension. Thus, this model gives good results in classification only for the range of curves used for its creation. We had there-fore to find a way to apply the mathematical model (eq.

3) and to get rid of such size dependency of the constituting parameters. To achieve it, all the parameters of the equation are divided by a dimensional parameter, which is constant for all the deformed curves obtained from a specific initial curve (see section 4.1). Knowing the fact that all curves are obtained by applying deformation model over a straight line, it means that the parameter (eq. l) remains the same for that set of curves.

In order to use the same equation (eq. 3) but not to have dimensional parameters in it, a transformation of the equation is obtained as following:

N S = C • A • L l 2 ( l l ) = (C •l)( A l 2 )( L l ) = Cr•Ar•Lr (9)
Using this transformed version of the equation (eq. 3) ensures the same values for the measure of straightness but using dimensionless quantities.

Results:

Based on this reformulation of the measure parameters, the new values have been computed for the curve data set ST and stored in an appropriate ARFF data file. Afterward, the same classification algorithm has been applied to all data file curve instances to create the classification model. In order to evaluate this model, the same set of curves obtained by deformation of initial lines (length of 0.8, 80 and 240 cm) has been used and the results show that this model can correctly classify 99.78 % of all curve instances.

Discussion: As before, the 0.22 % of instances, which are incorrectly classified, are considered belonging to the neighbor class. Considering that at the border of the class interval (the error is less than 0.5 %), and that the mathematical model (eq. 3) for class evaluation is very rigid, a change of 0.5 % in the intervals will not make any visible changes in the appearance of curves. Therefore, we can conclude that this classification model is reliable and provides good classification of curves.

Relevant attribute selection

Another objective of our experiments was to verify the capabilities of MLT in identifying the key elements for entity classification. Thus, we intend to use the attribute selection capability to solve the problem of characterizing free form surfaces with respect to their aesthetic properties. At present, there is no clear specification of styling and aesthetic properties and related measures for surfaces. Exploiting such selection capability, we want to find out which of the various computable surface characteristics are the most significant for the evaluation and modification of their aesthetic and styling properties.

In general, when there is a huge set of data characterised by many different attributes, whose importance in the further analysis or is not known, then Attribute Selection (AS) as a part of Data Mining methodology can be applied. AS allows the identification of which attributes can be omitted without affecting the results of further analysis.

To investigate the Attribute Selection capabilities of WEKA for shape classification, we used the measure of straightness of curves together with its parameters and additional other computable properties for curves.

The idea is to see if the Attribute Selection provided by WEKA will select the same parameters already used for computation of the straightness measure (S) among a larger list of parameters (attributes). In other words, if the Attribute Selection using WEKA proposes the same curve parameters used in the computation of (S), then we can consider this methodology (AS) as reliable and use it in further investigation of free form surfaces with respect to the aesthetic properties. For The parameters value of max 13 ks Local curvature maximum this purpose, using the same deformation model, applied onto an initial straight line, a set of 11200 curves (SET (I L8) , fig. 5) is created and for each curve, the 13 different curve parameters shown in table 2 are computed. The first three correspond to the relative (transformed) parameters (C r , A r , L r ) of the equation ( 9) used for the NS measure computation.

Con -corresponds to the measure of convexity which is related to the sign of the curvature along the curve, acceleration is a measure of acceleration that describes the rise of the curvature along the curve and acc is a measure just opposed to the acceleration measure.

WEKA provides two different methods for the AS regarding the attributes evaluation and their representation. The first method uses algorithms that provide independent evaluation of all attributes and then applies search algorithms to rank all attributes in a list. In this case, the InfoGainAttributeEval evaluation algorithm is used which calculates the mutual information (information entropy) of the attributes and classes; then such calculated values are ranked in decreasing order by the Ranker algorithm. The second method uses correlation based algorithm to evaluate a subset of attributes; then it applies appropriate search of algorithms to rank and propose the best subset of attributes. In this case, the CfsSubsetEval evaluation algorithm is used and then BestFirst search algorithm is applied to propose a subset of attributes which is highly correlated with the classes, but the attributes in the subset are more independent among themselves.

Results:

The following figures show the results of AS using WEKA in which all attributes associated with the values of the mutual information (Information Theory) are ranked (figure 6), and a subset of attributes is proposed (figure 7) as most significant with respects to the straightness. The results produced, shown in figure 7 confirm the assumptions previously made: the AS using WEKA has chosen the same transformed parameters that were used in the computation of the measure of straightness. Therefore the AS is very promising for

Linear Regression model

Creation of classification model using a decision tree based on the attribute values can be convenient for visualization of the results. Anyhow, it is appropriate only in the case the decision tree has a little number of branches and leaves, so that the class of the entity can be easily established by following the decision rules. The use of this classification method to classify the 2D free form curves results in generation of decision tree with more than 450 branches and leaves.Thus, we do not show it in this paper. Having this size of the decision tree makes very complex and unclear the determination the class of the curve. Furthermore, it is very difficult to find out the correlation of the shape quantity values with the class of the curves, i.e. how the change of one quantity will affect the class of the curve. In order to reach this correlation, an appropriate mathematical model has to be applied. In our case, i.e. classification of the 2D free form curves with respect to straightness, there is the mathematical model indicated by the equations (eq. 2) and (eq. 3). So, the aim of this part is not to define a mathematical model for straightness but to evaluate the ability of Machine Learning Techniques in WEKA to solve Data Mining problem using Regression. This will give us interesting results that could be then reapplied for surfaces.

After analyzing all available machine learning algorithms in WEKA, the linear regression algorithm has shown as the most suitable for handling only with numerical inputs (attributes) and outputs (classes). The idea of linear regression is to express the class as a linear combination of the attributes:

C l = w 0 +w 1 •A 1 +w 2 •A 2 +w 3 •A 3 +••••+w n •A n (10)
Where, C lnumeric value for the class, A 1 , A 2 , A 3 , ...., A nthe attribute values, w 0 , w 1 , w 2 , w 3 , ...., w nthe weights.

Regarding the equation for computing the straightness (non-straightness), we can see that the equation (eq. 2) for non-straightness is combination (not linear) of the attributes of curve and yells values form 0 to (infinity). So, if we want to transform this equation (eq. 2) into a linear combination (which is the way how the regression algorithm gives the results) of the attributes, each left and right side of the equation has to be used as an argument of a logarithmic function: [START_REF] Giannini | A survey of tools for understanding and exploiting the link between shape and emotion in product design[END_REF] this holds also for the transformed size independent version of the eq. 2 (eq. 9):

log(N S) = log(C)+log(A)+log(L)-2•log(l) (
log(N S) = log(C r ) + log(A r ) + log(L r ) (12)
The reason why a logarithmic function is used is the fact that if we want to compare the results of the regression model with the equation (eq. 2), they have to be in same form . This means, all values (of the class and the attributes) have to be transformed using logarithmic function and then to be recorded in an appropriate ARFF file so that will adjust the input data into a form adequate for the use of regression.

Results:

As it was mentioned before, the values of the curves quantities and the value of the non-straightness for each curve generated using the deformation model (figure 4), are transformed using logarithmic function.

After the ARFF file is created, WEKA workbench is ran to create the linear regression model. The obtained results are given in following figures 8 and 9:

Figure 8 linear regression model for the eq. ( 11)

Figure 9 linear regression model for the eq. ( 12)

Discussion:

To prove that using linear regression modeling, a reliable mathematical model can be extracted we have to obtain the same results as with the linearized functions [START_REF] Giannini | A survey of tools for understanding and exploiting the link between shape and emotion in product design[END_REF] and [START_REF] Giannini | A Survey to Evaluate how non Designers Perceive Aesthetic Properties of Styling Features[END_REF]. Thus, equation ( 11) is compared with result shown in figure 8 and equation ( 12) with figure 9. It is evident that regression model in figure 8 completely match with the eq. ( 11) because 2log(l) = 4.1589 ( where l = 8 cm is the length of the initial line used to generate the set of curves used). The same conclusion can be made if we compare the regression model in figure 9 with the eq. ( 12). In both cases, the correlation coefficients are 1 and relative absolute error is 0 %. Therefore, we can make a conclusion stating that Machine Learning Techniques using Linear Regression modeling can extract a reliable mathematical model (equation) that will show the correlation of the geometric quantities with the class of the curve with respect to an aesthetic property straightness. Such an approach can be then applied for surfaces.

CONCLUSION AND PERSPECTIVES

The goal of this work is an attempt to i) introduce Machine Learning Techniques (MLT) as a mean for discovering classification patterns with respects to the aesthetic properties of shapes on 2D free form curves; ii) use Data Mining (DM) methodology, to investigate which of the shape characteristics of a geometric entity (curve, surface) are the most significant with respects to a specific aesthetic property.

To verify that MLT could be suitable and useful for shape classification, we have analyzed its behavior in the case of the straightness of 2D curves. We based our work of the mathematical formula for the straightness measure defined in our previous work. We verified that MLT can correctly reapply the classification to new curves. In addition, we verified the abilities of the Attribute Selection methods to identify the most important attributes for the considered classification: among a larger set of attributes. As a result, it was possible to recognize as the most characterizing parameters the same curve attributes previously used to compute the measure of straightness (S). Moreover, Linear Regression (LR) was able to extract automatically an exact mathematical model, which can correlate the geometric quantities with the class of the curve, congruent to one we previously specified.

Finally, this work indeed demonstrates that MLT is very suitable and can be used efficiently in this kind of mechanical engineering and aesthetic applications. This offers good perspectives for solving the same problem on free form surface.

At present, there is no classification of surfaces with appropriate aesthetic properties. This requires at first the identification of the most meaningful free form sur-face characteristics (parameters) and reciprocal relations with respects to the aesthetic properties and then their classification patterns to be discovered.

Therefore, this work is considered to be a first step towards the characterization and classification of free form surfaces with respect to their aesthetic properties.

Figure 2

 2 Figure 2 Straightness of a curve

Figure 3

 3 Figure 3 Structure of an ARFF input file for WEKA

Figure 4 Figure 5

 45 Figure 4 The deformation model used to generate a set containing a variety of 2D curves

Figure 6

 6 Figure 6 Ranking of the attributes Discussion:The results produced, shown in figure7confirm the assumptions previously made: the AS using WEKA has chosen the same transformed parameters that were used in the computation of the measure of straightness. Therefore the AS is very promising for

Figure 7

 7 Figure 7 Selection a subset of attributes

Table 2

 2 List of curve quantities used for AS

	No. Parameters	Description
	1	Lr	Relative curve length
	2	Ar	Relative area
	3	Cr	Curvature
	4	centrox	x -coordinate of barycenter
	5	centroy	y -coordinate of barycenter
	6	Ix	Moment of inertia over x-axis
	7	Iy	Moment of inertia over y-axis
	8	Con	The measure of Convexity
	9	acceleration	The measure of acceleration
	10	acc	Measure of non-acceleration
	11	j	Number of local maximums
	12	s

ACKNOWLEDGMENT

This work has been partially supported by the VISION-AIR project funded by the European Commission under grant agreement 262044.


Simon Carladous
email: simon.carladous@irstea.fr

Jean-Marc Tacnet
email: jean-marc.tacnet@irstea.fr

Jean Dezert
email: jean.dezert@onera.fr

Guillaume Dupouy
email: guillaume.dupouy@irstea.fr

Mireille Batton-Hubert

A New ER-MCDA Mapping for Decision-Making Based on Imperfect Information

Keywords: fuzzy sets, possibility theory, belief functions, mapping, ER-MCDA

come

Introduction

Evidential Reasoning for Multi-Criteria Decision-Analysis (ER-MCDA) is a multicriteria decision method which is able to take into account both imperfect evaluation of quantitative and qualitative criteria and multiple more or less reliable sources [START_REF] Tacnet | AHP and uncertainty theories for decision making using the ER-MCDA methodology[END_REF]. The principle is first to represent imperfect evaluation of each criterion through a possibility distribution. As it uses fusion, those evaluations must be transformed into a common Frame of Discernment (FoD). Therefore, a mass of belief is assigned to consonant intervals with a confidence level which correspond to the possibility distribution. The mapping process establishes the link between the basic belief assignment BBA m(•) on quantitative intervals with a BBA in the common FoD. It can be interpreted as a function from the possibility distribution to a BBA on the common FoD. The initial principle of mapping is based on a geometrical projection. It induces several issues such as its theoretical justification but also its limited ability to provide positive masses of belief on singletons only. Our approach aims at improving this mapping process.

We consider an application case where ER-MCDA is used to assess the stability (high, medium, low) of check dams against torrential risk in mountains.

This evaluation is based on several criteria, such as the scouring rate (expressed in percentage) which is a damageable loss of foundation support5 [START_REF] Carladous | Belief function theory based decision support methods: Application to torrent protection work effectiveness and reliability assessment[END_REF]. In this application context, the Fig. 1 shows how an expert often provides an imperfect evaluation of check dam stability according to its foundation's scouring rate. Indeed, a precise and direct measure would be somewhere too dangerous. In this paper, our scope is not to develop a theoretical proof of the mapping process but to highlight some theoretical issues to propose another method. In section 2, we briefly recall some basics of fuzzy sets, possibility and belief function theories. We introduce the classical transformation and we propose a new method in section 3. In section 4, we compare the two methods on a sample decision context of our application case.

Fuzzy sets, possibility and belief function theories

In the fuzzy set theory [START_REF] Zadeh | Fuzzy sets[END_REF], U is the universe of discourse of individual elements u. µ A is the membership function which associates each u ∈ U to the class (fuzzy set) A with the grade of membership µ A (u) ∈ [0, 1]. A fuzzy set A is normal when there is an element u ∈ U such as µ A (u) = 1. We use trapezoidal functions defined by the quadruplet {a, b, c, d} (Eq. (1), Fig. 2) given their simplicity to approximate fuzzy intervals [START_REF] Dubois | Fuzzy sets, probability and measurement[END_REF]. Intervals [a, d] and [b, c] are respectively the fuzzy set's support (supp A ) and its core (A c ) [START_REF] Dubois | Fuzzy sets, probability and measurement[END_REF]. We denote Ā the complement of A for u ∈ U , defined by the membership function µ Ā of the Eq. ( 2) and shown in the Fig. 2. Given µ B = µ A which represents another fuzzy set B for u ∈ U , the membership function µ A∪B of the Eq. ( 3) represents the union of A and B while µ A∩B of the Eq. ( 4) represents their intersection [START_REF] Zadeh | Fuzzy sets[END_REF] (see Fig. 2). In the possibility theory framework [START_REF] Zadeh | Fuzzy sets as a basis for a theory of possibility[END_REF], F is the fuzzy set of possible values of u ∈ U . The possibility distribution π is given by µ F (u) π(u) ∈ [0, 1] also defined by a quadruplet {a, b, c, d} [START_REF] Dubois | Properties of measures of information in evidence and possibility theories[END_REF]. Given X a subset of U and X its complement, the possibility measure is Π(X) sup u∈X π(u) [START_REF] Zadeh | Fuzzy sets as a basis for a theory of possibility[END_REF] and the necessity is [START_REF] Dubois | Possibility Theory -An approach to Competurized Processing of Uncertainty[END_REF] N (X)

µA(u)          0 if u / ∈ supp A u-a b-a if u ∈ [a, b] 1 if u ∈ A c u-d c-d if u ∈ [c, d] (1)
µ Ā(u) 1 -µA(u), u ∈ U (2) µA∪B(u) max u∈U (µA(u), µB(u)) (3) µA∩B(u) min u∈U (µA(u), µB(u)) (4)
1 -Π( X), ∀X, X ⊆ U as shown in the Fig. 3. Considering the fuzzy set µ A and a possibility distribution µ F , the Eq. ( 5) gives the possibility measure of A [START_REF] Zadeh | Fuzzy sets as a basis for a theory of possibility[END_REF].

Considering F as a nested family of α-cuts

{F α |α ∈ (0, 1]}, one has F α {u|µ F (u) α} [5]. Viewing F α as a uniformly distributed random set from α ∈ (0, 1] to F α ⊆ U , one has µ F (u) Π(A) sup u∈U µA∩F (u) (5) π(u) u∈Fα l m l (6)
Fig. 3. Possibility distribution, possibility and necessity measures, α-cuts.

In the belief function theory, a basic belief assignment (BBA), or mass of belief m(•), represents the imperfect knowledge of a body of evidence (or source) on a given FoD denoted as Θ = {θ 1 , θ 2 , . . . , θ q }. In the classical Shafer's model [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF], all elements θ k , k = 1, . . . , n are assumed exhaustive and mutually exclusive. The powerset 2 Θ is the set of all subsets of Θ, empty set ∅ included. For X ∈ 2 Θ , we denote X ⊆ Θ its complement. For each source, the mass function m(•) : 2 Θ → [0, 1] must satisfy m(∅) = 0, and X⊆Θ m(X) = 1, ∀X = ∅ ∈ 2 Θ . Considering the universe of discourse U as the FoD and assuming a normal fuzzy set [START_REF] Baudrit | Postprocessing the hybrid method for addressing uncertainty in risk assessments[END_REF]. Considering that the universe of discourse U is the FoD Θ, the plausibility measure Pl (X) is a possibility measure Π(X),

F , m(•) is extracted considering F α l ∈ 2 U , l = 1, . . . , l max . Given m(•) on 2 Θ , the belief of the hypothesis Y ∈ 2 Θ is defined by Bel (Y ) X⊆Y |X∈2 Θ m(X). Its plausibility is defined by Pl (Y ) X∩Y =∅|X∈2 Θ m(X)
∀X ⊆ Θ = U [7].
Shafer's assumption of exhaustivity of the FoD means that it is considered as a "closed-world" (c.w.). In some practical problems, this assumption is too strict and it is more convenient to consider the original FoD as an "open-world" (o.w.). Dealing with it can be done in two manners as shown in the Fig. 4.

1. In Smets' Transferable Belief Model (TBM) [START_REF] Smets | The transferable belief model[END_REF], Θ o.w. {θ 1 , . . . , θ q } and ∅ = Θo.w. . One has X∈2 Θ m(X) = 1, and one allows m(∅) 0. 2. In Yager's approach [START_REF] Yager | Hedging in the Combination of Evidence[END_REF], the open-world is closed by an hedge element θ c , so that Θ c.w. Θ o.w. ∪ {θ c }. Setting m c.w. (θ c ) = 0 and m c.w. (∅) = 0, one has Bel c.w. (A) = 0 and m c.w. (A) = 0 for each subset A ⊆ Θ o.w. . For each X ⊆ Θ c.w. , one computes Bel c.w. (X) and m c.w. (X) in the Eqs. ( 7) and [START_REF] Dubois | Possibility Theory -An approach to Competurized Processing of Uncertainty[END_REF].

Bel c.w. (X) 0 if X = A ⊆ Θ o.w. ⊂ Θ c.w. 1 -Pl c.w. ( X) if X = A ∪ θ c ⊆ Θ c.w. ( 7
) m c.w. (X)      0 if X = A ⊆ Θ o.w. ⊂ Θ c.w. 0 if X = θ c ⊂ Θ c.w. Y ⊆A (-1) |A-Y | Bel c.w. (Y ∪ θ c ) if X = A ∪ θ c ⊆ Θ c.w. (8)
Fig. 4. TBM and hedging models under an "open-world" with exclusivity, for q = 3.

Transforming possibility distributions to BBAs on Θ

The FoD Θ = {θ 1 , . . . , θ q } of decision gathers q qualitative labels. For each criterion, scoring results from an evaluation scale which is the specific universe of discourse U . To take into account its imprecise and uncertain evaluation, a possibility distribution π : U → [0, 1] is given by the expert according to Eq.( 6).

To combine all the possibility distributions provided by several sources for several criteria, each one must be represented by a BBA m Θ (•) established on 2 Θ for the common FoD Θ as shown in the Fig. 5. A mapping model [START_REF] Tacnet | AHP and uncertainty theories for decision making using the ER-MCDA methodology[END_REF] is therefore based on q membership functions µ θ k : U → [0, 1] for k = 1, . . . , q according to Eq.( 1), corresponding to each θ k of the FoD. The construction of the q fuzzy sets respects the condition q k=1 µ θ k 1 for mutual exclusivity: if an element u ∈ U totally belongs to a label θ k (µ θ k (u ) = 1, k ∈ [1, q]), it cannot partially belongs to another label of Θ (µ θm (u ) = 0, m = k ∈ [1, q]). By construction, there is at least u ∈ U for which µ Θ= {θ k |k∈[1,q]} (u ) 1: it corresponds to an "open-world" assumption.

Classical transformation

Given the possibility distribution π and the q functions µ θ k for k = 1, . . . , q, the classical mapping [START_REF] Tacnet | AHP and uncertainty theories for decision making using the ER-MCDA methodology[END_REF] consists of the three following steps (shown in Fig. 6):

1. Given l max α-cuts with F α l ∈ 2 U , l = 1, . . . , l max , m U (•) is extracted for the FoD U from π as shown in Fig. 3. Each F α l is a focal element and its corresponding mass is denoted m U (F α l ). The BBA m U (•) thus represents the imperfect evaluation of a quantitative or qualitative evaluation of u ∈ U .

Each focal element is an interval which represents evaluation imprecision. Each mass of an interval takes into account the confidence level or uncertainty. By definition of m U (•), one has

lmax l=1 m U (F α l ) = 1. 2. For each F α l = u - α l ; u + α l , l = 1, . . . , l max , the area A k l under each curve µ θ k is given by A k l u + α l u - α l µ θ k (u)du. One has A l q k=1 A k l and computes the mass m Θ l (θ k ) A k l
A l for each F α l and each θ k . Thus, by definition of A l , one has

q k=1 m Θ l (θ k ) = 1. 3. One then builds the Bayesian 6 BBA m Θ (•) on 2 Θ for the FoD Θ = {θ 1 , θ 2 , . . . , θ q } with m Θ (θ k ) lmax l=1 m U (F α l ) × m Θ l (θ k ), k = 1, . . . , q.
This equation can be justified as follows.

From the two previous points, one has This principle of simple geometric transformation is however not fully satisfying since it only assigns BBAs to singletons and arbitrarily shares a BBA on several elements of the FoD regardless of partial or total ignorance. Moreover, from a practical point of view, this classical method needs an arbitrary setting of the value of l max but also of all the values of α l . That is why, it was first proposed that experts give their imprecise evaluation through intervals with a confidence level. It corresponds to a possibility distribution after α-cut, each α l being the confidence level given by the expert [START_REF] Tacnet | AHP and uncertainty theories for decision making using the ER-MCDA methodology[END_REF]. Nevertheless, in practice, we think that it is easier to give only the quadruplet {a, b, c, d} of Fig. 3 than several intervals with a confidence level.

q k=1 m Θ l (θ k ) × lmax l=1 m U (F α l ) = 1. Thus q k=1 lmax l=1 m Θ l (θ k ) × m U (F α l ) = 1. By definition, q k=1 m Θ (θ k ) = 1. As a consequence, m Θ (θ k ) lmax l=1 m U (F α l ) × m Θ l (θ k ), k = 1, . . . , q.

Application to the scouring rate of a check dam

In practice, the check dam stability is assessed through qualitative labels of the FoD Θ = {θ 1 = High, θ 2 = Medium, θ 3 = Low} [START_REF] Carladous | Approche quantiative pour élaborer une échelle d'évaluation floue d'un indicateur de stabilité structurale. Application l'affouillement d'ouvrages de stabilisation torrentielle[END_REF]. The lower is the scouring rate, the higher is check dam structural stability. The FoD cardinality |Θ| is here q = 3. For each check dam, the choice of the stability label notably depends on the scouring rate u of its foundation, with u a continuous value in U = [0%, 100%]. µ θ k is the membership function linking the scouring rate u ∈ U to each stability label θ k . Each µ θ k of the mapping model is represented in the Table 1 through a quadruplet {a, b, c, d} according to the Eq. ( 1). The practical way of defining such a mapping model has been developed in [START_REF] Carladous | Approche quantiative pour élaborer une échelle d'évaluation floue d'un indicateur de stabilité structurale. Application l'affouillement d'ouvrages de stabilisation torrentielle[END_REF]. It is based on a civil engineering analysis of the check dams stability given several hypotheses of scouring rate. They respect the exclusivity and "open-world" conditions given in the third paragraph of the section 3.

In practice, experts imperfectly measure scouring rate as shown in Fig. 1. Therefore, we compare evaluations through possibility distributions provided by six different experts. Each evaluation is represented by a quadruplet {a, b, c, d} in the Table 1, from a very imprecise evaluation (expert 1) to very precise ones (experts 5 and 6). The evaluation illustrated in Fig. 1 is given by expert 3. To apply the classical transformation on the Table 1, we assume l max = 4 with α-cuts α 1 = 1, α 2 = 0.7, α 3 = 0.4, and α 4 = 0.1 to extract the BBAs m U (•) [START_REF] Baudrit | Postprocessing the hybrid method for addressing uncertainty in risk assessments[END_REF]. One gets the final Bayesian BBAs m Θ (•) in columns "1" in the Table 3.

After steps 1 and 2 using Table 1 values, one gets the values of Pl c.w. (A ∪ θ c ), ∀A ∈ 2 Θ o.w. -{∅}, with Θ o.w. = {θ 1 , θ 2 , θ 3 }. Using them in the step 4, one gets the belief functions Bel c.w. (•) in the Table 2 and the BBAs m c.w. (•) in columns "2" in the Table 3. The latter gathers only focal elements with a positive mass.

Fig. 1 .

 1 Fig. 1. A real example of imperfect evaluation of check dam stability according to its foundation's scouring rate.

Fig. 2 .

 2 Fig. 2. Given U , trapezoidal fuzzy sets A, Ā, A ∪ B and A ∩ B.

Fig. 5 .

 5 Fig. 5. The general principle and objective of the mapping process.

Fig. 6 .

 6 Fig. 6. Steps of the ER-MCDA's classical mapping process: α-cuts choice, BBA m U (•) calculation, and projection on the mapping model to get a final BBA m Θ (•).

  ,c,d 20,30,80,90 30,40,60,80 40,50,50,60 30,40,40,50 40,40,40,40 45,45,45,45

Table 2 .

 2 Intermediary Bel c.w. (A ∪ θ c ) for the new mapping.

Table 1 .

 1 3 fuzzy sets µ θ k (u) and expert possibility distributions π(u), ∀u ∈ U .

Scouring is a process due to which the particles of the soil or rock under the check dam's foundation gets eroded and removed over a certain depth called scour depth and over the foundation area called scouring rate. Scouring often occurs in torrent because of the velocity and energy of the flowing in steep slopes.

0 µ Fα (u)dα. Considering a finite nested family of subsets {F α1 = u - α1 ; u + α1 ⊆ . . . ⊆ F α l = u - α lmax ; u + α lmax }, the set {α 1 > . . . > α l > . . . > α lmax } and m l α l -α l+1 ∈ [0,[START_REF] Tacnet | AHP and uncertainty theories for decision making using the ER-MCDA methodology[END_REF] with α l+1 = 0 (Fig.3), one has the Eq. (6).

The focal elements of a Bayesian BBA are only singletons of 2 Θ .

Acknowledgments. The authors thank the support of both French Ministry for Agriculture, Forest (MAAF) and Environment (MEEM).

New transformation

Our new method is based on four steps to get a BBA m Θ (•) on all focal elements X ∈ 2 Θ and not only on singletons of Θ = {θ 1 , . . . , θ k , . . . , θ q }. Given the construction of the q fuzzy sets, it assumes mutual exclusivity without exhaustivity. Thereafter we detail the method using Yager's model with hedging even if a similar approach has been also studied using the TBM. One thus considers Θ o.w. = {θ 1 , . . . , θ k , . . . , θ q } and Θ c.w. = Θ o.w. ∪ {θ c }. Different steps are:

1. Construction of 2q -1 functions µ c.w.

A∪θ c for all A = ∅ ⊆ Θ o.w. (see Fig. 7): given q functions µ o.w. θ k : U → [0, 1] for k = 1, . . . , q, the Eq. (3) gives the 2 q -1 functions µ o.w.

A

: U → [0, 1]. One uses the Eq. ( 9) to close the world and get the 2 q -1 functions µ c.w.

Step 1 of the new mapping for q = 3.

2. The Eq. ( 2) gives the 2 q -1 functions µ c.w. 3. The Eq. ( 7) first provides Bel c.w. (•). The Eq. ( 8) then gives m c.w. (•) on 2 Θ c.w. . 3 show that the new method affects a positive mass to total (experts 1 and 2) or partial ignorance with m c.w. (θ 1 ∪ θ 2 ∪ θ c ) > 0 (experts 3 and 4) according to the imprecision degree of initial evaluations. The less precise is the initial evaluation of the scouring rate, the less informative is the mapped BBA. It improves the classical transformation and it propagates the imperfection of the initial information on Θ c.w. . 0.24 0 0.15 0 0.05 0 0.5 0 0.5 0 0.25 0 θ2 0.44 0 0.75 0 0.95 0 0.5 0 0.5 0 0.75 0 θ3 0.32 0 0.10 0

Results given in the Table

Conclusion

This paper proposes a new mapping process in the ER-MCDA methodology. It allows to get a belief mass on non singletons elements of the powerset of the FoD. Unlike the initial approach [START_REF] Tacnet | AHP and uncertainty theories for decision making using the ER-MCDA methodology[END_REF], the possibility distribution is represented by its support and core and setting of values of l max and α l are not needed for an α-cut approach. It is based on the conjunction between the possibility distribution and fuzzy sets but also takes into account the relationship between the possibility and the plausibility measures. To relax the hypothesis of exhaustivity of the classical Shafer's model, we have chosen Yager's hedging model instead of Smets' TBM model to prevent from introducing an ambiguity in the interpretation of the empty set. According to application cases, the hypothesis of exclusivity should be relaxed. It will be studied in forthcoming publications.