Pierre Jezequel

Guillaume Wille
email: g.wille@brgm.fr.

Claire Bény

Fabian Delorme

Véronique Jean-Prost

Roger Cottier

Jean Breton

Frédéric Duré

Jackie Despriée

Claire Beny

Veronique Jean-Prost

Frederic Dure

Jacky Despriee

Characterization and origin of black and red Magdalenian pigments from Grottes de la Garenne (Vallée moyenne de la Creuse-France): a mineralogical and geochemical approach of the study of prehistorical paintings

Keywords: parietal art, pigments, chemical analysis, mineralogy, origin

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Introduction

35 000 years ago, prehistorical men gave the first major art forms, stones or bones bearing simple incisions, followed by schematic geometrical and animals figures. Time after time, this prehistorical art is becoming increasingly sophisticated and precise to achieve the exceptional quality of wall paintings such as Lascaux cave paintings. This parietal art has been extensively studied by archaeologists, but inputs of sciences like chemistry, biology or geology can be fruitful for several reasons. Numerous studies have been conducted on paintings to understand the nature of the materials and for the understanding of the techniques used by prehistoric men. Thus different authors have used various analytical techniques to analyze prehistorical paintings such as optical microscopy, SEM-EDS [START_REF] Garate | Etude de la matiere colorante de la grotte d'Arenaza[END_REF][START_REF] Chalmin | Les matieres picturales de la grotte d'Ekain[END_REF]Chalmin et al., 2004a;[START_REF] Chalmin | Minerals discoveres in paleolithic black pigments by transmission electron microscopy and micro-X-ray absorption near-edge structure[END_REF][START_REF] Vignaud | Le groupe des "Bisons adosses" de Lascaux. Etude de la technique de l'artiste par analyse des pigments[END_REF], Gas-Chromatography coupled to Mass spectroscopy (GCMS) [START_REF] Pepe | Le liant des peintures paleolithiques ariegoises[END_REF], TEM-EDS (Chalmin et al., 2004a;[START_REF] Chalmin | Minerals discoveres in paleolithic black pigments by transmission electron microscopy and micro-X-ray absorption near-edge structure[END_REF], X-Ray diffraction (XRD) [START_REF] Chalmin | Minerals discoveres in paleolithic black pigments by transmission electron microscopy and micro-X-ray absorption near-edge structure[END_REF][START_REF] Vignaud | Le groupe des "Bisons adosses" de Lascaux. Etude de la technique de l'artiste par analyse des pigments[END_REF], IRTF and Raman spectroscopies [START_REF] Van Der Weerd | Identification of black pigments on prehistoric Southwest American potsherds by infrared and Raman microscopy[END_REF][START_REF] Edwards | Raman spectroscopic analysis of pigments and substrata in prehistoric rock art[END_REF], synchrotron radiation spectroscopies (XANES, EXAFS, XRF) [START_REF] Chalmin | Minerals discoveres in paleolithic black pigments by transmission electron microscopy and micro-X-ray absorption near-edge structure[END_REF][START_REF] Reiche | Synchrotron radiation and cultural heritage : combined XANES/XRF study at Mn-K-edge of blue, grey or black coloured palaeontological and archaeological bone material[END_REF][START_REF] Farges | Archeological applications of XAFS: prehistorical paintings and medieval glasses[END_REF]. Some authors have tried to determine how prehistorical men prepared and used these pigments, and they suggested for example that red pigments can sometimes be obtained by dehydrating goethite to transform it into hematite (Chalmin et al., 2004b;[START_REF] Pomies | XRD study of the goethite-hematite transformation: application to the identification of heated prehistoric pigments[END_REF][START_REF] Pomies | TEM observation of goethite dehydration: application to archaeological samples[END_REF]. In contrast, no thermal treatments have been detected on black pigments obtained from manganese-based minerals (Chalmin et al., 2004b).

The Grottes de la Garenne, located at Saint-Marcel, Indre -France (Figure 1) are a set of Magdalenian living sites [START_REF] Despriee | Bulletin de l'association pour la sauvegarde du site archeologique d'Argentomagnus et Amis du Musee -Le Coteau de la garenne[END_REF]. These caves take place in a karstic context, belonging to an entrocs-containing bajocian limestone level that overlooks the river Creuse valley. The caves are located on the southern shore of the river Creuse, in a hill called Coteau de la Garenne. Known prehistoric sites are located on the west part of the hill. The first parts of this site have been discovered in 1848, during the building of the Paris -Toulouse railway. Several caves have been opened since this date. This site is a set of 9 archaeological levels, opened in the middle of the hill, at an altitude of approximately 120 m. These levels seem to be a succession of spaces, living places and other places, successively occupied from the bottom to the top of the hill. The different levels have been dated ( 14 C dating) from 15290 BP to 11920 BP [START_REF] Despriee | Bulletin de l'association pour la sauvegarde du site archeologique d'Argentomagnus et Amis du Musee -Le Coteau de la garenne[END_REF]. Some of these places are sometimes well equipped (flooring, fireplaces...). An important set of objects has been discovered in the different levels, such as tools made with reindeer antlers, animals bones (reindeers, horses, wolves, rodents or fishes). The analysis of pollens and animals remains shows that this period corresponds to a cold steppes climate.

The walls of these caves are covered by a substantial set of paintings whose study may allow archaeologists to obtain information or confirm hypothesis about the techniques and tools used by men from the Magdalenian age [START_REF] Vignaud | Le groupe des "Bisons adosses" de Lascaux. Etude de la technique de l'artiste par analyse des pigments[END_REF]. The study of these paintings can also be used to link painting materials to local or external supply sources. Several objects that could have been used as "crayons" have been found on the floor during explorations of the caves. Moreover, the answers obtain from this study can help to understand the risk of degradation of prehistorical painted caves like Lascaux cave [START_REF] Allemand | Qui sauvera Lascaux[END_REF], Gargas cave [START_REF] Mangin | La conservation des grottes ornees : un probleme de stabilite d'un systeme naturel (l'exemple de la grotte prehistorique de Gargas, Pyrenees francaises)[END_REF] or Arcy-Sur-Cure cave [START_REF] Chalmin | Microanalysis and synthesis of calcite. Growth mechanisms on prehistoric paintings in the Large Cave, Arcy-Sur-Cure[END_REF]. A study of pigments samples taken on the prehistoric site of the Coteau de la Garenne at St Marcel allowed us to determine the nature of the main pigments listed during exploration of the Grotte Blanchard. The Grotte Blanchard has been discovered in 1956 by J. Allain during Grand Abri exploration. It opens in the South of the hill, a few tens of meters above the Creuse River. The cave is a small cavity of 4 x 3 m. Engravings can be noticed on a part of the north-east wall, on a surface of 3 x 1.5 m. It consists in several V-shaped or multiple lines incisions. The north-west wall is partially covered by red and black paintings, on a surface of 0.7 x 1 m, forming a set of geometrical shapes (rectangles, lines, U-shaped lines, spots) (Figure 2). This cave is protected from the outside, the only access being a metallic scale controlled by a concrete airlock. This cave looks fragile and unstable, with cracked walls. The railway line which passes a few meters down in the valley causes severe vibrations. This highly increases the risk of rocks falling and walls degradation.

Moreover, evidences suggest that this cave has been reworked by human activity throughout the time [START_REF] Despriee | Bulletin de l'association pour la sauvegarde du site archeologique d'Argentomagnus et Amis du Musee -Le Coteau de la garenne[END_REF]2001). The use of this set of elements from the Magdalenian period is therefore urgent to increase the knowledge of this prehistorical stage through archaeological studies and subsequent actions for heritage conservation.

A synthesis of the different works conducted by several archaeologists since the discovery of the cave has been done by a research group directed by J. Despriee (DRAC Centre) [START_REF] Despriee | Bulletin de l'association pour la sauvegarde du site archeologique d'Argentomagnus et Amis du Musee -Le Coteau de la garenne[END_REF]. This synthesis gives a large overview of the knowledge of the site during the Magdalenian period.

The aim of this study is to characterize the red and black pigments in order to determine if the crayons found on the floor of the cave are related to the wall paintings and also to determine the geological origin of the red and black pigments.

Materials and methods

Red and black pigments of 2-3 mm 3 volume have been collected by scraping (Figure 3) on the walls of the Grotte Blanchard using a scalpel. Wall fragments have been collected together with pigments then samples have been purified by manual sorting under binocular microscope in lab in order to remove carbonates and others particles from the wall of the cave. Samples which are supposed to have been used as crayons have been collected on the floor of the cave. More than 110 samples (paintings, crayons) have been studied [START_REF] Jezequel | Etude minéralogique de 116 echantillons en provenance du site prehistorique du coteau de "La Garenne[END_REF]. Samples have been ground to powder using a agate mortar.

Powder X-ray diffraction (XRD) patterns have been performed on a Siemens D 5000 diffractometer (Bragg-Brentano geometry, θ/θ goniometer) using Co-K  radiation (λ = 0.178897 nm) and operating at 40 kV and 30 mA at room temperature. The scans have been recorded from 4 to 84° (2) with a step of 0.02° and a counting time of 1s per step.

Diagrams have been indexed using DIFFRAC plus software and international database JCPDS. Samples have been deposited on a suitable support for XRD analysis. For samples in small quantity, powder has been dispersed in ethanol and then deposited on a glass slide.

Fourier-Transform Infrared spectroscopy spectra (FTIR) have been acquired using BRUKER EQUINOX IFS-55 equipped with a high energy source, a KBr separator and a DTGS detector. Spectral resolution is 4 cm -1 . Transmission spectra have been acquired on samples compacted in KBr: 0.5 -2 mg are mixed within 250 mg KBr and compacted during 2 min under a pressure of 10 tons. Spectra have been collected in the range of wavelength 4000 -350 cm -1 . A GOLDEN-GATE ATR (Attenuated Total Reflectance) accessory has been also used for spectra acquisitions. This kind of acquisition does not require specific sample preparation. Sample is plated against a diamond brazed with wolfram carbide through a sapphire anvil. The range 4000-550 cm -1 is explored. Scan number is 32 for FTIR transmission acquisition, and 20 scans for ATR acquisition.

Raman micro-spectroscopy analysis has been performed using a DILOR XY800 Raman spectrometer equipped with an OLYMPUS microscope. Excitation beam wavelength is λ = 514.5 nm and the laser output power vary from 10 to 50 mW. Samples have been observed and analyzed using the x100 and the x50 ULWD (Ultra Long Working Distance) objectives. Acquisition times vary from 60 to 300 seconds.

The Transmission Electron Microscopy (TEM) observations have been realized at 120

kV [Philips CM120 coupled to an EDAX EDS spectrometer (Electron Dispersive X-ray Spectroscopy)]. The TEM samples were prepared by dispersing the powdered samples in alcohol by ultrasonic treatment, dropping them onto a porous carbon film supported on a copper grid, and then dried in air.

ICP/MS (Induction Coupling Plasma / Mass Spectroscopy) quantification analyses have been conducted using a THERMO-OPTEK system equipped with a high sensitivity interface. The samples are dissolved in a high purity acid, then the solution is heated at 7000 °C in the core of the argon plasma and analyzed by mass spectroscopy.

Results and discussions

Red pigments

Pigments from the Grottes de la Garenne -paintings and crayons

As noticed by several authors in numerous Magdalenian caves [START_REF] Chalmin | Les matieres picturales de la grotte d'Ekain[END_REF][START_REF] Pepe | Le liant des peintures paleolithiques ariegoises[END_REF][START_REF] Vignaud | Le groupe des "Bisons adosses" de Lascaux. Etude de la technique de l'artiste par analyse des pigments[END_REF][START_REF] Pomies | TEM observation of goethite dehydration: application to archaeological samples[END_REF], the main components identified by X-ray diffraction on red crayons are iron oxides (hematite Fe 2 O 3 ) and hydroxides, gehlenite (Ca 2 Al 2 SiO 7 ), quartz and microcline (KAlSi 3 O 8 ) (Figure 4).

Evidence of one or several amorphous phases is shown by XRD. The same composition has been noticed for the paintings pigments. Presence of carbonates in higher quantities in crayons than in paints can be highlighted by FTIR (bands at 1430, 877 and 713 cm -1 ).

It shows that several types of crayons are presents (Figure 5). Furthermore, the detritic origin of hematite is confirmed by its hydration and its association to a clay fraction (kaolinite, FTIR bands at 3696 and 3622 cm -1 ) and to gehlenite.

The presence of carbon matter in the paint is noticed by ATR-FTIR peaks at 2963, 2930, 2860 cm -1 in the wall paintings, but not in the crayons themselves (Figure 5). It could suggest that a mixture has been used rather than pure natural ore. Occurrence of a sub-layer of carbon matter before painting, or variations in the composition of the natural or manufactured crayons could be other hypotheses. Several authors have studied the techniques of the artists of the Magdalenian period and the use of organic matter and / or charcoal has already been reported (Chalmin et al., 2004a;Menu andWalter, 1996, Pepe et al., 1991).

The ICP/MS characterization of trace elements, especially Rare Earth Elements (REE), confirms that pigments from paints and from crayons have the same origin, their REE compositions being similar (Figure 6).

Study of the local sites with a compatible iron index

Iron-based minerals have been extensively studied in the area around Argenton/Creuse and reported in numerous publications [START_REF] Grossouvre | Etude sur les gisements de minerai de fer du centre de la France[END_REF]BRGM 1998;[START_REF] Laut | L'espace rural antique autour d'Argentomagus[END_REF][START_REF] Dieudonne-Glad | La metallurgie du fer chez les bituriges à l'epoque gallo-romaine[END_REF]. Numerous works have been conducted at the XIX th and XX th centuries for iron minerals extraction [START_REF] Grossouvre | Etude sur les gisements de minerai de fer du centre de la France[END_REF]BRGM, 1998). The study of Gallo-roman iron manufacturing and use around Argenton/Creuse shows that some iron ore (especially the Chaillac ore, known for a specific manganese and barium concentration) have been manufactured sometimes more than 30 km around there extraction site [START_REF] Laut | L'espace rural antique autour d'Argentomagus[END_REF][START_REF] Dieudonne-Glad | La metallurgie du fer chez les bituriges à l'epoque gallo-romaine[END_REF].

Possible extraction sites around La Garenne have been selected by crossing literature data [START_REF] Grossouvre | Etude sur les gisements de minerai de fer du centre de la France[END_REF]BRGM 1998;[START_REF] Laut | L'espace rural antique autour d'Argentomagus[END_REF][START_REF] Dieudonne-Glad | La metallurgie du fer chez les bituriges à l'epoque gallo-romaine[END_REF] and geological maps (Geological maps 1/50 000 n° 592 , 569, 570, 593 -BRGM). 23 sites have been selected by geological criteria and sampled for comparison with the red pigments from La Garenne. A schematic repartition of the sampling sites is shown on The study shows that the origin of the red pigments has to be found in ferruginous oolites emplaced in old alluvial deposits over the right bank of the river Creuse, in an area close to the cities of St Gaultier, Ciron, Scoury and Ruffec (Figure 7). Indeed, the efficiency of the hematite staining is much more important when associated with a clay fraction. A specific ICP/MS analysis of samples from a site upstream of the village of St Gaultier suggests that this site corresponds to the red pigments extraction site (Figure 8).

Black pigments

Pigments from the Grottes de la Garenne -paintings and crayons

Numerous studies on black pigments of Magdalenian paintings suggest that the most used black pigments are charcoal and manganese oxides [START_REF] Chalmin | Les matieres picturales de la grotte d'Ekain[END_REF]Chalmin et al., 2004a;[START_REF] Vignaud | Le groupe des "Bisons adosses" de Lascaux. Etude de la technique de l'artiste par analyse des pigments[END_REF].

XRD study of black pigments shows that the samples from the Grotte Blanchard are made of several manganese-containing phases such as cryptomelane (KMn 8 O 16 ) and pyrolusite (MnO 2 ), and a little part of quartz (Figure 9). Infra-Red analyses also shows the presence of clays, carbonates and carbonaceous matter.

In addition, the main band of cryptomelane shows frequency variations ( Figure 11) on the Raman spectra (639 to 643 cm -1 ). These variations are probably related to composition variations of this mineral. Indeed, barium substitution of potassium in some cryptomelane crystallites is noticed in EDS-TEM analysis (Figure 12). Chemical analysis results from ICP/MS for REE concentration are similar for pigments in the paintings and in some of the crayons and both present an anomaly for cerium (Figure 13). Another anomaly is also noticed for thallium (0.15 -0.20% Tl) both in the paint and in the crayons. TEM characterization of single grains allows demonstrating that Tl is systematically associated to cryptomelane (Figure 12). Finally, it is possible to conclude that some crayons found in the caves could have been used to produce the black-colored parietal paintings, regarding the different FTIR, Raman, EDS-TEM and ICP/MS results.

Study of local sites and sites from the French Massif Central with a compatible manganese index

In order to identify the extraction sites of the black pigments, a sampling has been performed after a pre-selection of the most compatible Mn indexes in an area of 30-40 km around the caves, and extended to the French Massif Central. This selection was based on chemical and geological data, but also on archeometry data regarding the Magdalenian age in this area.

The study of the samples by EDS-TEM has shown that Tl is associated to cryptomelane in the crayons and the paintings. Tl is not found as an element of other parts of the pigments matter. The association Mn -Tl is known since the 60's [START_REF] Vlasov | Geochemistry, mineralogy and generic types of occurrences of the rare earth elements[END_REF]. It is related to a substitution Mn -Tl in final hydrothermal phases. The presence of thallium in the pigments could indicate that the cryptomelane has been extracted from an ore coming from a late hydrothermal process. Moreover, this type of high oxidation degree -manganese ore is found in hydrothermal-type deposits, in hypogene veins, in vein fillings, or comes from the processing of the primary ore by supergene oxidation. It can be found as a residue in superficial ore. 16 sampling sites have been selected (Figure 14). Collected samples have been analyzed by EDS-SEM and ICP/MS and XRD. Local indexes of manganese reveal an association with iron and in a less extent barium. However, cave black pigments analyses do not reveal the presence of iron. This lack of iron, coupled to the presence of thallium, and the specificity of the REE spectra concentration profile (Figure 15, Table 1) indicate that probably no local ore can provide matter in good agreement with the pigments composition. Unlike for red pigments, it allows to conclude to an allochtonous origin: samples from the sampling sites have been analyzed and are not compatible with the black pigments. It is possible that the sampling area is too small and the hypothesis of a farthest supply site must be considered. Some mineralogical and geochemical investigations could be necessary towards the Atlantic (Poitou strait), the North (Morvan area) or even in a wider area (French Pyrenees, Spain ...).

The "non-local" origin of the black "manganese oxide-based" pigments has been reported by other authors in the pictorial materials of Magdalenian caves [START_REF] Chalmin | Les matieres picturales de la grotte d'Ekain[END_REF][START_REF] Chalmin | Minerals discoveres in paleolithic black pigments by transmission electron microscopy and micro-X-ray absorption near-edge structure[END_REF].

However, the analysis of some of the crayons collected in the cave, that have not been used for the wall paintings, is compatible with one of the sampled sites (La Redoutiere -Chaillac District, 30km from La Garenne). It shows that for these crayons manganese mineral could have been locally collected. New questions arise then: What was the use of these crayons as they were not used for the wall paintings, and why Magdalenian men did not use them to achieve the paintings of the Grotte Blanchard?

Conclusion

This study focused on the colors of the Grottes de la Garenne. It allowed us to determine the nature of the pigments and the variety of their origins.

Red pigments have been identified to be based on more or less hydrated hematite associated to a small part of clays (kaolinite), some different silicates (gehlenite, illitemicas...) and carbon matter. ICP/MS demonstrated that pigments from the paints and from the crayons have the same origin. It allowed showing that this pigment composition is compatible with local siderolithic facieses.

On the other hand, black pigments from paintings and some crayons are essentially made of manganese-containing phases, such as cryptomelane, pyrolusite, associated to a small part of quartz, phyllites, carbonates and carbon matter. The same origin of pigments from paints and some crayons has been established by ICP/MS analyses.

However, unlike the red pigments, their origin is not compatible with local facieses.

Indeed, an anomaly on cerium concentration, the presence of thallium, the trace or absence of iron are noticed in the pigments composition, while manganese is associated to iron and more or less barium in local indexes. Thallium-containing cryptomelane could indicate an ore coming from a late hydrothermal process, whose origin need to be determined. Extension of the sampling area to the all French Massif Central shows also no compatible facieses.

Besides, it could be useful to understand for which use was another type of black crayons whose origin has been locally discovered, and why these crayons have not been used for the paintings in the Grotte Blanchard. This case is not unique, we can cite works done in Lascaux, as an example (Chalmin et al, 2004a).

Finally, this works highlights the fact that a complete and precise geological approach is necessary for the identification of the nature and the supply sources of prehistorical paintings and pigments.

Figures

Figure 1 : Location of Saint-Marcel, France Figure 2 : The Grand Abri and the Grotte Blanchard [START_REF] Despriee | Bulletin de l'association pour la sauvegarde du site archeologique d'Argentomagnus et Amis du Musee -Le Coteau de la garenne[END_REF] Figure 3 : Red and black pigments on the wall of the Grotte Blanchard and sampling Figure 4 : XRD on a red crayon (* Hematite;◊ Quartz; □ Microcline; • Gehlenite) Figure 5 : FTIR spectra -red pigments from the paint and a crayon Figure 6 : rare earth elements concentration (ICP/MS) in red pigments from the paintings (full square) and the crayon (empty square) Figure 7 : map of the iron minerals sampling sites Figure 8 : rare earth elements concentration (ICP/MS) of the iron-compatible samples collected close to St Gaultier Figure 9 : XRD on black pigments Figure 10 : FTIR spectra of black pigments Figure 11 : Raman analysis of black pigments Figure 12 : EDS spectrum of a cryptomelane particle -Mn / Ba substitution Figure 13 : Rare earth elements concentration in black pigments from the paint (Full Square) and the crayon (Empty Square) Figure 14 : Map of the sampling sites around La Garenne for Mn ore identification Figure 15 Rare earth elements concentration in black pigments from the pigments (dotted line) and Mn ores (see Table 1 for description) Tables Table 1 Black Superficial manganese oxide deposits on pyroclastic flow, south of the Guery lake (Puy de Dôme) For samples ech2 to ech16, see Figure 14

Figure 7 .

 7 Figure 7.

  pigments and Mn ores samples from Figure13-Description of samples from Figure14

Figure 1 :

 1 Figure 1 : Location of Saint-Marcel, France

Figure 3 :Figure 5 :

 35 Figure 3 : Red and black pigments on the wall of the Grotte Blanchard and sampling

FigureFigure 13 :Figure 14 :

 1314 Figure 11 : Raman analysis of black pigments

  Table 1 for description)

		10000,00
		1000,00
	Chondrites
	/
	Concentrations	100,00
		10,00
		1,00
		La	Ce	Pr	Nd	Sm	Eu	Gd	Tb	Dy	Ho	Er	Tm	Yb	Lu
								Eléments
			ech2	ech3		ech5	ech7	ech11		ech15	ech16		Grotte Blanchard ech2

Table 1

 1 Black pigments and Mn ores samples from Figure14-Description of samples from Figure 15 Lascaux and Objat (Dordogne), in altered schists characterized by Mn enrichment close to quartz lenses ech5 manganese oxide layer (centimeter sized), 2 km south from Brezon, direction of Pierrefort (Cantal), base of the volcanic flow ech7 iron, baryum and manganese veins, on the limit of an excavation, Lunel "la Fage" (Aveyron), vertical pipe and crosscut near Kaymar mines, South from La Fage ech11 Cournus vein : centimetric bed of psilomelane in quartz matrix, within metamorphic rocks (gneiss, schists), La Fouillade (Aveyron) ech15 Crusted psilomelane vein in fractures, Ste Colombe "La Vitarelle" Road D76 close to Bouxal (Lot) ech16

	Sample	Description
	Grotte	black pigments at the back of the cave, on a entrocs-containing
	Blanchard ech2	limestone wall
	ech2	manganese oxide, Chaillac "les Redoutieres" (Indre) -Collection
		(vein disappeared by mining) (Robert, 1972)
	ech3	manganese oxide, between

Acknowledgements

Pr Denis Vialou, Professor at the department of prehistory at the Museum National d'Histoire Naturelle (MNHM, Paris -France), is gratefully acknowledged for his help, especially for sampling the red and black pigments in the Grotte Blanchard.

This work has been funded by the DRAC Centre.


Miruna Stoicescu

Jean-Charles Fabre

Matthieu Roy

Towards a System Architecture for Resilient Computing

Keywords: reconfigurable, adaptive, pervasive, fault-tolerant

Nowadays, systems are not only becoming increasingly complex and heterogeneous but are also opened towards their environment by means of context-awareness. Furthermore, in the vast majority of cases, their specifications periodically evolve, leading to new versions. As resilient systems are expected to continuously provide trustworthy services, they must cope with changes coming from the environment or from the specifications and reconfigure in order to adapt to them. We propose a framework for designing and developing such systems.

I. PROBLEM STATEMENT

The evolution of systems is ineluctable and leads to the notion of resilient computing [START_REF] Laprie | From dependability to resilience[END_REF]. Among the evolution scenarios that can occur during the operational life of a fault-tolerant application, we focus on those that may have an impact on the fault tolerance mechanisms:

• one or several changes in the application assumptions which influence the choice of the fault tolerance mechanism, • changes in the requirements in terms of fault tolerance,

• changes in the system configuration (e.g. number of processors, memory resources, network bandwidth).

The essence of a resilient application lies in the fact that when faced with any of these changes or even with a combination of them, it must adapt itself while ensuring the observance of the safety criteria. * Corresponding author: miruna.stoicescu@laas.fr

We view the evolution of a system during its operational life as a trajectory in a space characterized by the parameters which can cause the aforementioned evolution. The three evolution scenarios can be aggregated into a frame of reference for this space, the three axes being: the application assumptions (e.g., whether it is deterministic or not), the fault tolerance requirements (e.g., the fault model) and the current system configuration (e.g., the resources it is using). We claim that a system's evolution can be represented in this vector space as shown in Figure 1. The combination of these three types of information indicates the most convenient fault tolerance mechanism to use, if any exists.

Given this frame of reference, we consider that each fault tolerance mechanism covers a certain region of space as shown in Figure 1 . Obviously, the union of all our fault tolerance mechanisms does not cover the entire space. A monitor keeps track of values in terms of the three axes and allows placing the state of the system for a given configuration in a certain region of this space. A change observed by the monitor triggers a transition towards a new region which might be covered by another fault tolerance mechanism or might be empty if there is currently no solution. In this case, we can still give some indications as to what must be adjusted in order to reach a region for which a fault tolerance mechanism exists.

Once our view of the problem stated, a certain number of issues arise, such as: how do we associate fault tolerance mechanisms with the combination of the three measures? how do we describe these fault tolerance mechanisms in a way which allows us to change them dynamically? how do we actually perform the transition from one mechanism to another? The following section presents our view on these matters.

II. OUR APPROACH

Separation of concerns is a generally accepted idea for introducing fault tolerance mechanisms in an application in a flexible way which allows subsequent modification and reuse. According to this principle, software architectures consist of two layers where the base provides the required functionalities and the top contains the fault tolerance mechanism(s). As we target the adaptation of fault tolerance mechanisms, we must manage the dynamics of the top layer, which can have two causes:

• The application layer remains unchanged but the fault tolerance mechanism must be modified either because the fault model changes or because an event in the environment, such as resource availability, makes it unsuitable. • Changes in the top layer are indirectly triggered by modifications in the application layer which make the fault tolerance mechanism unsuitable. In this case both layers execute a transition to a new state. In order to achieve the adaptation of the fault tolerance mechanisms in both scenarios, we build on the representation from the previous section and refine it in three steps.

A. Description of the frame of reference

The three parameters labeling our axes are actually multidimensional vectors but for the sake of visibility we have chosen an elementary representation.

• The fault tolerance requirements are part of the nonfunctional specifications of an application. Our main focus is on the fault model, i.e., the types of faults which must be tolerated by the application. We base our fault model classification on known types [START_REF] Avizienis | Basic concepts and taxonomy of dependable and secure computing[END_REF], e.g., physical faults, design faults. • The application assumptions regroup the characteristics of an application which have an impact on the choice of the fault tolerance mechanism but are not the same as the functional specifications of the application. These characteristics include: whether the application is stateless or stateful (i.e., if in case of failure we must rebuild its state for it to keep running), whether its state is accessible or not and whether the application is deterministic or not.

• The system configuration contains information such as the number of processors available, the available bandwidth, the memory resources.

B. Fault Tolerance Mechanisms

In order to place the mechanisms in the previously described frame of reference, we must first be able to classify them using the given criteria. In our approach, there are three steps in the selection process for finding the most adequate fault tolerance mechanism for an application: first, the fault model is identified, then the application assumptions and finally the current system configuration. The same process will be applied for classifying them.

C. System evolution

The state of a fault-tolerant application represents a point in the space given by our frame of reference. The application being fault-tolerant, this point must be in a region covered by a certain mechanism. A change (in terms of one of the three axes) is equivalent to a new state of the application, therefore a new point. As our purpose is to guarantee resilience when facing change, the application must always be "accompanied" by an adequate fault tolerance mechanism on its evolutionary trajectory. Therefore, we must either provide a fault tolerance mechanism encompassing a region which contains the new point or, should we fail to do so, "guide" the application towards a new region where a mechanism can be provided. We must design and build our fault tolerance mechanisms in view of such transitions and adaptations. The "distance" between two mechanisms as represented in our frame of reference should be equivalent to the difference between their implementations. If the new mechanism is close to the old one, we should be able to generate it through minor adjustments. The transition between distant mechanisms will most likely demand more complex modifications or even complete replacement.

The planned development process for achieving the smooth and safe transition between fault tolerance mechanisms consists of the following elements:

• a complete classification of the fault tolerance mechanisms we intend to use • a method for describing and storing configurations corresponding to the fault tolerance mechanisms • a set of tools allowing us to develop applications which are easily reconfigurable at runtime • one or several algorithms for performing the transition between mechanisms

III. CONCLUSION AND PERSPECTIVES

Our work lies at the intersection of three research domains, namely Fault Tolerant computing, Autonomic Computing [START_REF] Kephart | The vision of autonomic computing[END_REF] and Ubiquitous Systems. The fault tolerance mechanisms we are considering are well-established solutions for certain usecases/scenarios. This has led us to the concept of design patterns for fault tolerance, by drawing a parallel with the design patterns from software engineering. Each fault tolerance design pattern has an associated architecture. In order to operate a transition between fault tolerance mechanisms, we must be able to manipulate the configuration through its description file. A design principle that is commonly accepted in the area of reconfigurable systems is the use of componentbased technologies (CBSE) for developing the management framework. We will use component-based middleware as a basic layer for our adaptive architecture. Finally, a decision needs to be made concerning the granularity of the reconfiguration. We can imagine two approaches for replacing a fault tolerance mechanism with another one:

• an atomic approach, where the old mechanism is completely replaced by the new one • a differential approach, where we operate at a finer level by adding/removing the components corresponding to the difference between the two mechanisms The transitions between mechanisms will most likely be described through state-machines. The algorithms behind the transitions could lead to design patterns for system evolution.

Fig. 1 .

 1 Fig. 1. Context-aware fault tolerance computing

Supérieure d'Electronique, d'Electrotechnique, d'Informatique, de Télécommunications et d'Hydraulique de Toulouse, France in 2010 and the Certified Engineer degree from the Polytechnic University of Bucharest, Romania in 2009.


Jean B Lasserre

EXISTENCE OF GAUSSIAN CUBATURE FORMULAS

Keywords: 1991 Mathematics Subject Classification, 65D32 42C05 33C52, Key words and phrases, Orthogonal polynomials; Gaussian cubature formulas

We provide a necessary and sufficient condition for existence of Gaussian cubature formulas. It consists of checking whether some overdetermined linear system has a solution and so complements Mysovskikh's theorem which requires computing common zeros of orthonormal polynomials. Moreover, the size of the linear system shows that existence of a cubature formula imposes severe restrictions on the associated linear functional. For fixed precision (or degree), the larger the number of variables the worse it gets. And for fixed number of variables, the larger the precision the worse it gets. Finally, we also provide an interpretation of the necessary and sufficient condition in terms of existence of a polynomial with very specific properties.

Introduction

In addition of being of self-interest, Gaussian quadrature formulas for onedimensional integrals are particularly important because among all other quadrature formulas with same number of nodes, they have maximum precision. Indeed, when supported on p nodes, they are exact for all polynomials of degree at most 2p -1. Moreover, the nodes are exactly the zeros of the orthogonal polynomial of degree p and existence is guaranteed whenever the moment matrix of the associated linear functional is positive definite.

When its multi-dimensional analogue exists (then called Gaussian cubature formula), it shares those two important properties of support and precision. That is, the nodes of a cubature formula of degree (or precision) 2p -1 are the common zeros of all orthonormal polynomials of degree p. However, and in contrast to the one-dimensional case, its existence is not guaranteed even if the moment matrix of its associated linear functional is positive definite.

To the best of our knowledge, the only necessary and sufficient condition available is [START_REF] Dunkl | Orthogonal Polynomials of Several Variables[END_REF]Theorem 3.7.4] due to Mysovskikh, which states that an n-dimensional Gaussian cubature formula of degree 2m -1 exists if and only if the orhonormal polynomials of degree m have exactly s m-1 := n+m-1 n common zeros. So this condition requires computing those zeros, or, equivalently, all joint eigenvalues of n associated Jacobi matrices; see e.g. [START_REF] Dunkl | Orthogonal Polynomials of Several Variables[END_REF]Theorem 3.6.2]. Equivalently, this amounts to check whether some n multiplication matrices of size n+m-2 m-1 commute pairwise; see [START_REF] Dunkl | Orthogonal Polynomials of Several Variables[END_REF]Theorem 3.6.5].

On the other hand, in Cools et al. [START_REF] Cools | Cubature formulae and orthogonal polynomials[END_REF] one may find several lower bounds on the number of nodes required for a given specific precision; see e.g. [START_REF] Cools | Cubature formulae and orthogonal polynomials[END_REF]Theorem 9,10,11,12,13]. For more details on orthogonal polynomials and cubature formulas, the interested reader is referred to e.g. the excellent book of Dunkl and Xu [START_REF] Dunkl | Orthogonal Polynomials of Several Variables[END_REF] and the survey by Cools et al. [START_REF] Cools | Cubature formulae and orthogonal polynomials[END_REF].

Contribution. This paper is concerned with existence of Gaussian cubature formulas associated with a Borel measure µ on R n , with all moments finite. Our contribution is to provide a necessary and sufficient condition for existence, different in spirit from (and complementary to) Mysovskikh's theorem which requires solving polynomial equations, or, equivalently, computing all joint eigenvalues of n Jacobi matrices. Namely, we prove that a Gaussian cubature formula or precision 2m -1 exists if and only if a certain overdetermined linear system has a solution. The coefficient matrix of this linear system comes from expressing the product P α P β of any two orthonormal polynomials P α , P β of degree m, in the basis of orthonormal polynomials (P θ ), of degree at most 2m. In fact, in this expansion, only the constant coefficient and the coefficients of the degree-2m orthonormal polynomials matter. This linear system is always overdetermined and the larger n for fixed m (resp. the larger m for fixed n) the worse it gets. This shows that existence of a Gaussian cubature formula imposes severe restrictions on the associated linear functional.

Finally, we also provide an interpretation of the necessary and sufficient condition, namely that there exists a polynomial Q, which is a linear combination of orthonormal polynomials of degree 2m, and such that

P α (x)P β (x) dµ(x) = P α (x)P β (x) Q(x) dµ(x),
for all pairs (P α , P β ) of orthonormal polynomials of degree m..

Notation, definitions and preliminaries

Let x = (x 1 , . . . , x n ) and let R[x] be the ring of real polynomials in the variables x and R[x] d its subset of polynomials of degree at most d, which is a vector space of dimension s d := n+d d . The number of monomials of degree exactly d is denoted by r d := n+d-1 d . A polynomial f ∈ R[x] d can be identified with its vector of coefficients (f α ) =: f ∈ R s d , in the canonical basis (x α ), α ∈ N n . For any real symmetric matrix A, the notation A 0 (resp. A ≻ 0) stands for A is positive semidefinite (resp. positive definite).

Let us define the graded lexicographic order (Glex) α ≤ gl β on N n , which first creates a partial order by |α| ≤ |β|, and then refines this to a total order by breaking ties when |α| = |β| as would a dictionary with x 1 = a, x 2 = b, etc. For instance with n = 2, the ordering reads 1, x 1 , x 2 , x 2 1 , x 1 x 2 , x 2 2 , . . .. For every integer m ∈ N, let x m ∈ R[x] rm be the column vector of all monomials of degree m, with the < gl ordering. For instance, with n = 2 and k = 3, x 3 ∈ R[x] 4 and x 3 reads

x 3 = (x 3 1 , x 2 1 x 2 , x 1 x 2 2 , x 3
2 ) T . With any sequence y = (y α ), α ∈ N n , one may associate a linear function

L y : R[x] → R by (2.1) f (= α f α x α ) → L y (f ) = α f α y α , f ∈ R[x].
By a slight abuse of notation, and for any v ∈ R[x] p denote by L y (v) the vector (L y (v j )), j ≤ p. Similarly, for any matrix

V ∈ R[x] p×r denote by L y (V) the matrix (L y (V ij )), i ≤ p, j ≤ r. If µ is a finite and positive Borel measure with finite moments y = (y α ), α ∈ N n , then (2.2) L y (f ) = α∈N n f α y α = α∈N n f α x α dµ(x) = f dµ,
and µ is called a representing measure for y.

Moment matrix. The moment matrix M d (y) associated with a sequence y = (y α ), is the real symmetric matrix with rows and columns indexed by N n d , and whose entry (α,

β) is just L y (x α+β ) (= y α+β ), for every α, β ∈ N n d . Alternatively, M d (y) = L y ((1, x, . . . , x d ) T (1, x, . . . , x d )). Of course, if y has a representing mea- sure µ, then M d (y) 0 for all d ∈ N, because f , M d (y)f = L y (f 2 ) = f 2 dµ ≥ 0, ∀ f ∈ R s d .
However, the converse is not true in general, i.e., M d (y) 0 for all d, does not imply that y has representing measure.

2.1. Orthonormal polynomials. Most of the material of this section is taken from Helton et al. [START_REF] Helton | Measures with zeros in the inverse of their moment matrix[END_REF]. Given µ and y = (y α ), α ∈ N n 2d , as in (2.2), assume that M d (y) ≻ 0 for every d. Then one may define the scalar product

•, • y on R[x] d : f, g y := f , M d (y)g = f g dµ, ∀ f, g ∈ R[x] d .
With d ∈ N fixed, one may also associate a unique family of polynomials

(P α ) ⊂ R[x] d , α ∈ N n d
, orthonormal with respect to µ, as follows:

(2.3)

       P α ∈ lin.span {x β : β ≤ gl α} P α , P β y = δ α=β , α, β ∈ N n d P α , x β y = 0 if β < gl α P α , x α y > 0, α ∈ N n d ,
1 x 1 x 2 x 2 1 x 1 x 2       .
where the constant M is chosen so that P 2 (11) dµ = 1; eee e.g. [START_REF] Dunkl | Orthogonal Polynomials of Several Variables[END_REF][START_REF] Helton | Measures with zeros in the inverse of their moment matrix[END_REF].

Gaussian cubature formula

Given a finite Borel measure µ on K ⊂ R n with all moments finite, points x(i) ∈ R n , and weights γ i ∈ R, i = 1, . . . s, the linear functional

I : L 1 (µ) → R, (3.1) f → I(f ) := s i=1 γ i f (x(i)), f ∈ F ,
is called a cubature formula of degree (or precision) p, if I(f ) = K f dµ for all f ∈ R[x] p ; the points (x(i)) are called the nodes. In other words, the approximation of the integral f dµ by (3.1) is exact for all polynomials of degree at most p.

Cubature formulas are the multivariate analogues of quadrature formulas in the one dimensional case. By Tchakaloff's theorem, given µ on K with finite moments, and d ∈ N, there exists a measure ν finitely supported on at most r d points of K, and whose moments up to order at least d coincide with those of µ; see e.g. Putinar [START_REF] Putinar | A note on Tchakaloff 's theorem[END_REF]. And so there exists a cubature formula of degree d. In the one-dimensional case, the celebrated Gaussian quadrature formula based on d points (called nodes) of K has precision 2d -1 and positive weights. It exists as soon as the moment matrix of order d is positive definite, and all the nodes are zeros of the orthonormal polynomial of degree d. It is an important quadrature because among all quadrature formulas with same number of nodes, it is the one with maximum precision, When its multivariate analogue exists (then called a Gaussian cubature), the nodes are now the common zeros of all orthonormal polynomials of degree d, the precision is also 2d -1, and the weights are also positive. But its existence is not guaranteed even if all moment matrices are positive definite. For a detailed account on cubatures and orhogonal polynomials, the interested reader is referred to the very nice book of Dunkl and Xu [START_REF] Dunkl | Orthogonal Polynomials of Several Variables[END_REF], as well as the survey or Cools et al. [START_REF] Cools | Cubature formulae and orthogonal polynomials[END_REF].

3.1. Existence of Gaussian cubature. Mysovskikh's theorem [START_REF] Dunkl | Orthogonal Polynomials of Several Variables[END_REF]Theorem 3.7.4] states that a Gaussian cubature of degree m exists if and only if the orthonormal polynomials P m have exactly n+m-1 n common zeros. And so checking existence reduces to computing all joint eigenvalues of n so-called Jacobi matrices of size s m-1 described in e.g. [4, p. 114]. We here provide an alternative criterion for existence which reduces to check whether a certain overdetermined linear system has a solution.

Let µ be a finite Borel measure with support supp µ = K ⊂ R n , and with all moments y = (y α ), α ∈ N n , finite. We will adopt the same notation in [START_REF] Dunkl | Orthogonal Polynomials of Several Variables[END_REF]. Let (P α ), α ∈ N n , be a system of orthonormal polynomials with respect to µ, and with each m ∈ N, let P m = [P |α|=m ] ∈ R[x] rm be the (column) vector (P α (m 1 ) , . . . , P α (mr m ) ) T , where (α (mi) ) ⊂ N n m are all monomials of degree m, with α (m1) < gl α (m2) • • • < gl α (mr m ) . For instance with n = 2,

(3.3) A (0,m) + A (2m,m) u = 0, has a solution u ∈ R r2m .
Proof. Only if part. If there exists a Gaussian cubature (hence of degree 2m -1), there is a finite Borel measure ν supported on the s m-1 common zeros of P m ; see [START_REF] Dunkl | Orthogonal Polynomials of Several Variables[END_REF]Theorem 3.7.4]. Let z = (z α ), α ∈ N n 2m , be the moments up to order 2m, of the Borel measure ν. Since the cubature is exact for polynomials in R[x] 2m-1 ,

z α = L z (x α ) = L y (x α ) = y α , ∀α ∈ N n 2m-1 .
In addition, ν being supported on the common zeros of P m , we also have

L z (P 2 α ) = P α (x) 2 ν(dx) = 0, ∀α ∈ N n m with |α| = m,
that is, all diagonal elements of the positive semidefinite matrix L z (P m P T m ) vanish, which in turn implies L z (P m P T m ) = 0. Combining with (3.2) yields

0 = L z (P γ P β ) = A (0,2m) γβ + 2m-1 j=1 A (j,2m) γβ L z (P j ) =Ly(Pj )=0 +A (2m,2m) γβ L z (P 2m ) = A (0,2m) γβ + A (2m,2m) γβ L z (P 2m ) = A (0,2m) + A (2m,2m) L z (P 2m ) (γ, β) which is (3.3) with u = L z (P 2m ). If part.
Conversely, assume that (3.3) holds for some vector u ∈ R r2m . Consider the sequence z = (z α ), α ∈ N n 2m , with z α = y α for all α ∈ N n 2m-1 and L z (P α ) = u α for all α ∈ N n 2m , with |α| = 2m. (Equivalently, L z (P 2m ) = u.) And indeed such a sequence exists. It suffices to determine z α for all α ∈ N n 2m with |α| = 2m (equivalently, L z (x 2m )). Recall the notation x k = (x α ), α ∈ N n , with |α| = k, and recall that for each d ∈ N, (1, P 1 , . . . , P d ) is a basis of R[x] d , and so,

(3.4)       1 P 1 • • P 2m       =       1 0 • • 0 S 01 S 1 0 • 0 • • • • • • • • • • S 02m S 12m • S (2m-1)2m S 2m       =S       1 x 1 • • x 2m      
for some change of basis matrix S which is invertible and block-lower triangular. And so z solves

u = S 2m L z (x 2m ) + Θ L z (1, x 1 , • • • , x 2m-1 ) T =y2m-1 , where Θ ∈ R r2m×s(2m-1) is the matrix [S 02m | • • • |S (2m-1)2m ], and y 2m-1 = (y α ), |α| ≤ 2m -1. And so L z (x 2m ) = S -1 2m u -S -1 2m Θ y 2m-1 . Next, the moment matrix M m (z), which by definition reads           M m-1 (z) | L z (P T m ) L z (P 1 P T m ) • • L z (P m-1 P T m ) - - L z (P m ) L z (P m P 1 ) T • • • • • • L z (P m P T m-1 ) | L z (P m P T m )           , simplifies to (3.5) M m (z) =   I | 0 - - 0 | L z (P m P T m )   ,
because L z (P i P T j ) = L y (P i P T j ) = 0 for all i = j with i + j ≤ 2m -1. But then we also have L z (P m P T m ) = 0 because from (3.2),

L z (P γ P β ) = A (0,2m) γβ + 2m-1 j=1 A (j,2m) γβ L z (P j ) =Ly(Pj )=0 +A (2m,2m) γβ L z (P 2m ) =u = A (0,2m) + A (2m,2m) u (γ, β) = 0,
for all γ, β ∈ N n m with |γ| = |β| = m. And so M m (z) 0, and rank M m (z) = rank M m-1 (z). By the flat extension theorem of Curto and Fialkow, z is the moment sequence of a finite Borel measure ψ on R n , supported on rank M m (z) = s m-1 distinct points x(k), k = 1, . . . , s m-1 ; see e.g. Curto and Fialkow [START_REF] Curto | Flat extensions of positive moment matrices: recursively generated relations[END_REF][START_REF] Curto | The truncated K-moment problem[END_REF], Lasserre [START_REF] Lasserre | Moments, Positive Polynomials and Their Applications[END_REF]Theorem 3.7], or Laurent [START_REF] Laurent | Revisiting two theorems of Curto and Fialkow on moment matrices[END_REF]. That is, denoting by δ x the Dirac measure at x ∈ R n ,

ψ = sm-1 k=1 γ k δ x(k) ,
for some strictly positive weights γ k . From 0 = L z (P m P T m ) we obtain

0 = L z (P 2 α ) = R n P α (x) 2 dψ(x), ∀α ∈ N n m , with |α| = m,
which proves that each x(k) is a common zero of P m , k = 1, . . . , s m-1 . But by [START_REF] Dunkl | Orthogonal Polynomials of Several Variables[END_REF]Corollary 3.6.4], P m has at most s m-1 common zeros. Therefore, P m has exactly s m-1 common zeros and ψ is supported on all common zeros of P m . By [START_REF] Dunkl | Orthogonal Polynomials of Several Variables[END_REF]Theorem 3.7.4], µ admits a Gaussian cubature formula of degree 2m-1. Indeed, since z α = y α for all |α| ≤ 2m -1, then for every f

∈ R[x] 2m-1 , f (x) dµ(x) = L y (f ) = L z (f ) = f dψ = sm-1 k=1 γ k f (x(k)),
with γ k > 0 for every k.

By [1, Theorem 7], we also know that the x(k)'s belong to the interior of the convex hull conv K of K.

Observe that existence of a solution u ∈ R r2m for the linear system (3.3) imposes drastic conditions on the linear functional L y because (3.3) is always overdetermined. Moreover, for fixed m (resp. for fixed n) the larger n (resp. the larger m) the worse it gets. This is because t m (= r m (r m + 1)/2) increases faster than r 2m .

Interpretation. We next provide an explicit expression of the vector A (0,2m) as well as the matrix A (2m,2m) , which permits to obtain an interpretation of the condition (3.3).

For γ, β ∈ N m , applying the linear functional L y to both sides of (3.2) yields

(3.6) δ γ=β = L y (P γ P β ) = A (0,2m) γβ , ∀|γ| = |β| = m.
because L y (P k ) = 0 for all k. This provides an explicit expression of the vector A (0,2m) . Similarly, multiplying both sides of (3. LAAS-CNRS and Institute of Mathematics, University of Toulouse, LAAS, 7 avenue du Colonel Roche, 31077 Toulouse Cédex 4, France, Tel: +33561336415 E-mail address: lasserre@laas.fr

y

  where δ α=β denotes the usual Kronecker symbol. Existence and uniqueness of such a family is guaranteed by the Gram-Schmidt orhonormalization process following the 'Glex" order of monomials, and by positivity of the moment matrix M d (y). See e.g. [4, Theorem 3.1.11, p. 68]. Computation. Computing the family of orthonormal polynomials is relatively easy once the moment matrix M d (y) is available. Suppose that one wants to compute p σ for some σ ∈ N n d . Build up the submatrix M σ d (y) obtained from M d (y) by keeping only those columns indexed by β ≤ gl σ and with rows indexed by α < gl σ. Hence M σ d (y) has one row less than columns. Complete M σ d (y) with an additional last row described by M σ d (y)[σ, β] = x β , β ≤ gl σ. Then, up to a normalizing constant, P σ = det M σ d (y). For instance with n = 2, d = 2 and σ = (11), one has (2.4) x → P (11) (x) = M det 00 y 10 y 01 y 20 y 11 y 10 y 20 y 11 y 30 y 21 y 01 y 11 y 02 y 21 y 12 y 20 y 30 y 21 y 40 y 31

P 2 =∈ 2 . 3 . 1 .

 2231 (P 20 , P 11 , P 02 ) T . For any sequence z = (z α ) denote by M d (z) the moment matrix, with rows and columns index in N n d , and with entries M d (z)[α, β] = L z (P α P β ), α, β ∈ N n d , where L z : R[x] → R is the linear functional defined in (2.1). In other words, M(z) is the moment matrix expressed in the basis of orthonormal polynomials and can be deduced from M d (z) by a linear transformation. Notice that if z = y then M d (y) is the identity matrix I. Observe that for each γ, β ∈ N n m , with |γ| = |β| = m, R rj , j = 0, 1, . . . , 2m. Let t m := r m (r m + 1)/Theorem Let A (0,2m) be the vector (A (0,2m) γβ ) ∈ R tm , and let A (2m,2m) ∈ R tm×r2m be the matrix whose rows are the vectors A (2m,2m) γβ , defined in (3.2) for γ, β ∈ N n m with |γ| = |β| = m. There exists a Gaussian cubature formula of degree 2m -1 if and only if the linear system

(3. 8 )

 8 L y (P γ P β ) = L y (P γ P β Q), for all β, γ ∈ N n m with |γ|, |β| = m. Equivalently,(3.8) readsP γ (x)P β (x) dµ(x) = P γ (x)P β (x) Q(x) dµ(x),for all β, γ ∈ N n m with |γ|, |β| = m, or in matrix form:I = P m (x)P m (x) T Q(x) dµ(x),Proof. With u as in (3.3) and using (3.6)-(3.7), for all β, γ ∈ N n m with |γ|, |β| = m, one obtainsL y (P γ P β ) = A (0,2m) γβ = (A (2m,2m) u) γ,β = L y   P γ P β ( α∈N n 2m ,|α|=2m u α P α )   = L y P γ P β (u T P 2m ) , which is (3.8) with Q = u T P 2m . Conversely, if (3.8) holds with for some Q = u T P 2m then obviously u is a solution of (3.3).Final remark. One may obtain more information about the polynomial Q of Corollary 3.2. From the proof of Theorem 3.1, for any solution u ∈ R r2m of (3.3), one has u = L z (P 2m ) where z is the moment vector of the measure supported on the s m-1 nodes of the Gaussian cubature. That is,u = L z (P 2m ) = sm-1 k=1 γ k P 2m (x(k)),for some positive weights γ k . On the other hand, the polynomial Q of Corollary 3.2 is of the form u T P 2m where u solves (3.3). Therefore,x → Q(x) = u T P 2m (x) = sm-1 k=1 γ k P 2m (x(k)) T P 2m (x).Hence Q satisfies L y (P α Q) = u T L y (P α P 2m ) = 0, ∀|α| < 2m (in particular Qdµ = 0), andL y (P α Q) = u α = sm-1 k=1γ k P α (x(k)), ∀|α| = 2m.


Jamel Saadaoui
email: jamel.saadaoui@edu.univ-paris13.fr

Exchange Rate Dynamics and Fundamental Equilibrium Exchange Rates

Keywords: JEL Classification: F00, F31, F32, C23 Fundamental equilibrium exchange rates, Panel unit root tests, Global imbalances, Fully modified ordinary least square, Dynamic ordinary least square, Pooled Mean Group

The paper investigates if the most popular alternative to the purchasing parity power approach (PPP) to estimate equilibrium exchange rates, the fundamental equilibrium exchange rate (FEER) influences exchange rate dynamics in the long run. For a large panel of industrialized and emerging countries and on the period 1982-2007, we detect the presence of unit roots in the series of real effective exchange rates and in the series of FEERs. We find and estimate a cointegration relationship between real effective exchange rates and FEERs. The results show that the FEER has a positive and significant influence on exchange rate dynamics in the long run.

Introduction

The evolution of current account balances, which have considerably increased since the last fifteen years, has raised many questions. Among them, many observers (see e.g. [START_REF] Cline | Currency Wars?[END_REF] think that some countries (mainly South East Asian countries) have pursued aggressive exchange rate policy to maintain their currency at artificially low level in order to preserve their competitiveness on foreign markets. These politics of exchange rate undervaluation have allowed these countries to accumulate huge current account surplus. However as the misalignments (deviation from equilibrium exchange rate) offset each other at the world scale, if some countries are undervalued and run large current account surplus then some other countries are overvalued and run large current account deficits.

The paper investigates if the most popular alternative to the PPP approach to estimate equilibrium exchange rates, the FEER [START_REF] Williamson | The Exchange Rate System[END_REF][START_REF] Williamson | Estimating Equilibrium Exchange Rates[END_REF] influences exchange rate dynamics in the long run. If the FEER affects the long run exchange rate dynamics, it can be considered as the "true equilibrium exchange rate" and policy makers should pursue this rate to ensure global macroeconomic stability in a context of large imbalances.

As pointed out by [START_REF] Clark | Exchange Rates and Economics Fundamentals: A Methodological Comparison of BEERs and FEERs[END_REF], the notion of equilibrium in the FEER, which is a method of calculation of the equilibrium exchange rate consistent with external equilibrium (sustainable capital flows) and internal equilibrium (full utilization of productive potential), can be questioned if we consider that some variables which influence directly exchange rate behavior are often omitted in the calculations. [START_REF] Zhou | Fundamental Equilibrium Exchange Rates and Exchange Rate Dynamics[END_REF] tests if the FEER affects exchange dynamics thanks to unit root tests and bivariate cointegration tests. She found empirical evidence that real effective exchange rates (REERs) and FEERs are unit roots process however she does not found that REERs and FEERs are cointegrated and, then, she concluded that exchange rate dynamics is not affected by the FEER.

These results can be misleading because they concerned only two countries (Japan and Germany) and a relatively short span of time (1974)(1975)(1976)(1977)(1978)(1979)(1980)(1981)(1982)(1983)(1984)(1985)(1986)(1987)(1988) whereas the FEER approach is essentially a multilateral approach in which the global consistency, mentioned above, is crucial. Barisone et alii (2006) study this question by using data for the G7 countries on the period 1973-1997. They implemented unit root tests, panel unit root tests and panel cointegration tests [START_REF] Pedroni | Critical Values for Cointegration Tests in Heterogeneous Panels with Multiple Regressors[END_REF]. They found empirical evidence that REERs and FEERs are unit roots process and, contrary to [START_REF] Zhou | Fundamental Equilibrium Exchange Rates and Exchange Rate Dynamics[END_REF], they found that REERs and FEERs are cointegrated.

For a large panel of industrialized and emerging countries and on the period 1982-2007, we detect the presence of unit roots in the series of REERs and in the series of FEERs. We find and estimate a cointegration relationship between REERs and FEERs. The results show that the FEER has a positive and significant influence on exchange rate dynamics in the long run.

The paper is organized as follow. Section 2 presents panel unit root tests. Section 3 tests a long run relationship between REERs and FEERs by using Pedroni"s panel cointegration tests (1999). Section 4 estimates the long run relationship between REERs and FEERs by using the fully modified ordinary least square estimator (FMOLS) and the dynamic ordinary least square estimator (DOLS) introduced by [START_REF] Pedroni | Purchasing Power Parity Tests in Cointegrated Panels[END_REF]. We also use a Pooled Mean Group (PMG) estimator introduced by Pesaran et alii (1999). Section 5 concludes.

Panel unit root tests for REERs and FEERs

In this section, we implement a number of panel unit root tests for the series of REERs2 and for the series of FEERs of seventeen industrialized and emerging countries (the U.S., the U.K., the Euro area, Japan, Korea, China, Brazil, India, Mexico, Argentina, Chile, Colombia, Indonesia, Malaysia, Philippines, Thailand and Uruguay) and on the period 1982-2007.

The FEER is defined as the exchange rate prevailing when the economy simultaneously reaches the external equilibrium and the internal equilibrium for all the trading partners. This measure was derived from a standard world trade model in which all the variables are endogenous except the external equilibrium (sustainable current account determined by structural variables) and the internal equilibrium (full utilization of the productive potential). The external equilibrium is estimated with panel regression techniques. The internal equilibrium is reached when the output gap is closed3 . [START_REF] Zhou | Fundamental Equilibrium Exchange Rates and Exchange Rate Dynamics[END_REF], Barisone et alii, 2006).

Testing a long run relationship between REERs and FEERs

In order to test a long run relationship between REERs and FEERs, we can conduct either panel unit root tests on the difference between REERs and FEERs or panel cointegration tests [START_REF] Pedroni | Critical Values for Cointegration Tests in Heterogeneous Panels with Multiple Regressors[END_REF].

As we have found that the series of REERs and the series of FEERs are nonstationary then if we found that the difference between these two series is stationary, it would mean that these series are cointegrated with a (1; -1) coefficient. In this paper, we estimate the long run relationship between REERs and FEERs thanks to the FMOLS estimator and the DOLS estimator introduced by Pedroni ( 2001) so it is unnecessary to impose any value to the cointegration vector before the empirical estimation (Barisone et alii, 2006). As a robustness check, we use the Pooled Mean Group estimator PMG (Pesaran et alii, 1999). We test the following long run equation4 : Before to proceeding to the next step which will consist to estimate explicitly this long run relationship. We have to conduct panel cointegration test [START_REF] Pedroni | Critical Values for Cointegration Tests in Heterogeneous Panels with Multiple Regressors[END_REF] between the FEER and the REER since, as it is described in appendix B, the causal relationship is bi-directional i.e. the FEER causes the REER and the REER causes the FEER. We test the following long run equation:

, , , i t i i t i t feer reer       (2)
As we can see in table 3, the null hypothesis of no cointegration is strongly rejected in all of the tests. These results show that there is a long run relationship between FEERs and REERs.

Estimation of the long run relationship between REERs and FEERs

In this section, we implement the FMOLS estimator, the DOLS estimator and the PMG estimator to estimate the long run relationship between REERs and FEERs, as it is specified in equation ( 1) and equation ( 2). We use these econometric tools, introduced by Pedroni (2001) and Pesaran et alii (1999), in a context of nonstationarity of the series to avoid problems of fallacious regressions.

The results, in table 4 to 7, indicate that the FEER has a positive and significant influence on exchange rate dynamics in the long run6 . In order to check the sensitivity to the results at the type of countries included in the panel, we re-estimate the equation (1) and equation ( 2) for emerging countries only, the results are largely similar.

Conclusion

We provide empirical evidence that the most popular alternative to the purchasing parity power approach (PPP) to estimate equilibrium exchange rates, the fundamental equilibrium exchange rate (FEER) influences exchange rate dynamics in the long run. For a large panel of industrialized and emerging countries and on the period 1982-2007, we detect the presence of unit roots in the series of real effective exchange rates and in the series of FEERs. We find and estimate a cointegration relationship between real effective exchange rates and FEERs.

The results show that the FEER has a positive and significant influence on exchange rate dynamics in the long run. Therefore it can be considered as the "true equilibrium exchange rate" and policy makers should pursue this rate to ensure global macroeconomic stability in a context of large imbalances.

1 i i i     ;     1 1 2 1 i i i i       
Since the PMG estimator imposes long-run coefficients to be constant for all individuals, while it allows short run heterogeneity, the error correction model is written:

  , , 1
             (8)
We estimate two reciprocal equations for causality test with the PMG estimator (i.e. we test if the REER causes the FEER and if the FEER causes the REER). The decision rule is: when the error correction term is negative and significant then the null hypothesis of no causality is rejected.

Table B1: Panel Causality Tests

Error-corrections coefficient (ϕ) FEER→REER -0.24*** (-7.30) REER→FEER -0.38*** (-6.37)

Number of Observations 442

Notes: t-statistics in parentheses: *** significant at 1 percent. Source: authors" calculations. We can conclude that:

 The no causality hypothesis is strongly rejected in all cases  The error-correction coefficients (ϕ's) are negative and statistically significant, indicating the causal relationship is bi-directional.

 These relationships are robust to the types of countries.

Table 1 : Panel unit root tests

 1 we can see in table 1, we detect the presence of unit root in the series of REERs and in the series of FEERs. The series are nonstationary in level and stationary in first difference. The REERs and FEERs are nonstationary I(1) series. As a series is I(1) if it achieves stationarity after first differencing. This result is confirmed by other empirical studies

	Test:	LLC	Breit. F_ADF F_PP	LLC	Breit. F_ADF F_PP
	Difference:	No	No	No	No	Yes	Yes	Yes	Yes
	Exogenous variable:	Constant, Trend	Constant, Trend	Constant, Trend	Constant, Trend	None	None	None	None
	Null Hypothesis:	UR	UR	UR	UR	UR	UR	UR	UR
	Common UR: Yes	Yes	No	No	Yes	Yes	No	No
	reer i,t	0.5	2.7	27.3	27.9 -15.1*** -11.9*** 267.4*** 286.0***
	feer i,t	-1.1	1.4	38.4	62.2*** -17.4*** -13.7*** 312.8*** 363.9***

Notes: "UR" indicates the null hypothesis of the presence of unit root. The symbol *** indicates statistical stationarity at the 1 percent level. The table shows different panel unit root tests: Levin, Lin, and Chu (2002) (LLC);

[START_REF] Breitung | The local power of some unit root tests for panel data[END_REF]

;

[START_REF] Maddala | A comparative study of unit root tests with panel data and a simple new test[END_REF]

and

[START_REF] Choi | Unit root tests for panel data[END_REF]

Fischer-type panel unit root tests (F_ADF and F_PP). Source: authors" calculations.

As

  minuscule represents natural logarithms. As we can see in table 2, the null hypothesis of no cointegration is strongly rejected in all of the tests. These results show that there is a long run relationship between REERs and FEERs.

	, i t reer	i   	, i t feer µ i t , 	(1)
	where variables in

Table 2 : Panel cointegration tests 5

 2

	Pedroni residual cointegration test (1999)
	Null Hypothesis: No cointegration
	Included observations	442
	Cross-sections included	17
	Alternative hypothesis: common AR coefficients (within-dimension)
	Panel-v	2.51*
	Panel-rho	-3.01***
	Panel-PP	-3.65***
	Panel-ADF	-3.98***
	Alternative hypothesis: individual AR coefficients (between-dimension)
	Group rho-Statistic	-0.99
	Group PP-Statistic	-3.22***
	Group ADF-Statistic	-5.82***
	Notes: The symbols *, *** indicate statistical significance at the 10 percent and 1 percent levels, respectively. ***Reject null
	of nonstationarity even at the 1% level; *Reject null of nonstationarity at the 10% level.
	Source: authors" calculations.

Table 3 : Panel cointegration tests

 3

	Pedroni residual cointegration test (1999)
	Null Hypothesis: No cointegration
	Included observations	442
	Cross-sections included	17
	Alternative hypothesis: common AR coefficients (within-dimension)
	Panel-v	0.85
	Panel-rho	-2.50***
	Panel-PP	-3.38***
	Panel-ADF	-3.39***
	Alternative hypothesis: individual AR coefficients (between-dimension)
	Group rho-Statistic	-1.59*
	Group PP-Statistic	-4.06***
	Group ADF-Statistic	-5.42***

Notes: The symbols *, *** indicate statistical significance at the 10 percent and 1 percent levels, respectively. ***Reject null of nonstationarity even at the 1% level; *Reject null of nonstationarity at the 10% level. Source: authors" calculations.

Table 4 : Long run relationship between REERs and FEERs

 4

		Long Run Coefficient (β)	T-Statistic
	FMOLS 1	0.66***	24.06
	DOLS 2	0.65***	29.37
	PMG 3	0.68***	11.05
	Hausman test	0.15	0.70
	Cross-section included	17
	Number of Observations	442

Notes: (1) FMOLS is the Fully Modified OLS estimation; (2) DOLS is the Dynamic OLS estimation; (3) PMG is the Mean Group estimation. The symbol *** indicates statistical significance at the 1 percent level. The null hypothesis in the Hausman test is homogeneity of the long run coefficient in the PMG estimation. Source: authors" calculations

Table 5 : Long run relationship between REERs and FEERs (emerging countries only)

 5

		Long Run Coefficient (β)	T-Statistic
	FMOLS 1	0.68***	22.28
	DOLS 2	0.66***	26.72
	PMG 3	0.66***	10.04
	Hausman test	1.73	0.18
	Cross-section included	13
	Number of Observations	338

Notes: (1) FMOLS is the Fully Modified OLS estimation; (2) DOLS is the Dynamic OLS estimation; (3) PMG is the Pooled Mean Group estimation. The symbol *** indicates statistical significance at the 1 percent level. The null hypothesis in the Hausman test is homogeneity of the long run coefficient in the PMG estimation. Source: authors" calculations

Table 6 : Long run relationship between FEERs and REERs

 6 FMOLS is the Fully Modified OLS estimation; (2) DOLS is the Dynamic OLS estimation; (3) PMG is the Pooled Mean Group estimation. The symbol *** indicates statistical significance at the 1 percent level. The null hypothesis in the Hausman test is homogeneity of the long run coefficient in the PMG estimation.

		Long Run Coefficient (θ)	T-Statistic
	FMOLS 1	0.62***	23.26
	DOLS 2	0.62***	27.64
	PMG 3	0.65***	15.36
	Hausman test	1.43	0.23
	Cross-section included	17
	Number of Observations	442
	Notes: (1)

Source: authors" calculations

Table 7 : Long run relationship between FEERs and REERs (emerging countries only)

 7 FMOLS is the Fully Modified OLS estimation; (2) DOLS is the Dynamic OLS estimation; (3) PMG is the Pooled Mean Group estimation. The symbol *** indicates statistical significance at the 1 percent level. The null hypothesis in the Hausman test is homogeneity of the long run coefficient in the PMG estimation.

		Long Run Coefficient (θ)	T-Statistic
	FMOLS 1	0.66***	22.09
	DOLS 2	0.66***	25.69
	PMG 3	0.71***	16.34
	Hausman test	0.85	0.35
	Cross-section included	13
	Number of Observations	338
	Notes: (1)

Source: authors" calculations

Table B2 : Panel Causality Tests (emerging countries only)

 B2

		Error-corrections coefficient (ϕ)
	FEER→REER	-0.27*** (-7.74)
	REER→FEER	-0.42*** (-6.05)
	Number of Observations	338
	Notes: t-statistics in parentheses: *** significant at 1 percent.
	Source: authors" calculations.

Source: Bank for International Settlements for the real effective exchange rate basis 100 in 2000 (annual average of monthly data).

See Jeong et al. (2010),[START_REF] Aflouk | Exchange Rate Misalignments and World Imbalances: A FEER Approach for Emerging Countries[END_REF]. The methodology used is a synthesis of previous works on the FEER[START_REF] Borowski | The Exchange Rate Macroeconomic Balance Approach: New Methodology and Results for the Euro, the Dollar, the Yen and the Pound Sterling[END_REF],[START_REF] Jeong | Exchange Rate Regimes and Equilibrium Exchange Rates in East Asia[END_REF]) and of the Symmetric Matrix Inversion Method (SMIM) recently proposed by[START_REF] Cline | Estimating Consistent Fundamental Equilibrium Exchange Rates[END_REF].

We do not include temporal trend as we want to test if REERs and FEERs are cointegrated without any divergence(Barisone et alii, 2006). We use the natural logarithms of these variables in order to obtain directly the elasticities.

In appendix A, we provide panel unit root tests[START_REF] Pesaran | A simple unit root test in the presence of cross section dependence[END_REF] and panel cointegration tests[START_REF] Westerlund | Testing for error correction in panel data[END_REF] which allows for cross section dependencies (i.e. existence of common shocks).

The FMOLS, DOLS and PMG estimations give similar results with a long run coefficient close to 0.6 for equation (1) and (2).

Autoregressive Distributed Lags

Appendix A: Panel unit root tests and panel cointegration tests

We use the CADF test introduced by [START_REF] Pesaran | A simple unit root test in the presence of cross section dependence[END_REF] to test the unit root properties of the variables in presence of cross section dependencies. By subtracting cross section averages of lagged levels in addition to the standard ADF equation, this test is robust to cross section dependencies. The series are nonstationary I(1) series. To test cointegration, the panel and group mean statistic suggested by [START_REF] Westerlund | Testing for error correction in panel data[END_REF] are applied. The existence of a negative and significant error correction term is taken as proof for cointegration. In case of cross section dependencies between members of the panel, critical values need to be obtained through bootstrapping. Results indicate that these variables are cointegrated.

The error correction equation yield:

 


Di Lonardo

A Predoi-Cross

A Ibrahim

M Herman

L Fusina

G Nivellini

G Di Lonardo

Prof Di Gianfranco

Lonardo

The high resolution infrared spectrum of the ν ν ν ν 4 band of 1,1,1-trifluoroethane

Keywords: CH3CF3, 1, 1, 1-trifluoroethane, high resolution, infrared spectrum, ν4 fundamental

The ν 4 band of 1,1,1-trifluoroethane (CH 3 CF 3 ) has been studied using synchrotron radiation at the far-infrared beamline, Canadian Light Source. The spectra were recorded at a resolution of 0.00096 cm -1 in the spectral range 800 to 860 cm -1 using a Bruker IFS125 FT spectrometer. Altogether 2785 lines were assigned to q P K (J) and q R K (J) transitions with J" up to 71 and K up to 45. The q Q K (J) sub-branches are too congested and overlapped to other hot-band Q branches to be analyzed. A Watson-type Hamiltonian containing the usual terms up to the 6 th power in the angular momentum operators was used in the analysis. The spectroscopic parameters obtained from the fit reproduce 2636 transitions with a standard deviation of about 0.00016 cm -1 .

The 1,1,1-trifluoro -ethane, CH 3 CF 3 , has been the subject of several spectroscopic investigations [1][2][3][4][5][6][7]. It attracted the interest of spectroscopists because of the presence of torsional splitting in the ground and excited vibrational states. Besides, the low energy torsional motion affects the whole infrared spectrum allowing the observation of many hot bands involving vibrationally excited torsional states. This molecule (HFC-143a) is also interesting from the environmental point of view since it could replace HCFC-22 (CHClF 2 ), eventually in mixture with other less flammable HFC [1]. The microwave and millimeterwave rotational spectra in the ground and excited torsional states up to v 6 = 3 have been reported in the literature [2][3][4]. The high resolution infrared spectra of several fundamental and combination bands have been recorded and analyzed. In 1993 Fraser et al. [5] studied the molecular-beam spectrum of the Fermi triad 2ν 6 +ν 11 , ν 5 +ν 12 and ν 10 located near 970 cm -1 .

The ν 2 and ν 8 fundamentals and the combination bands ν 4 + ν 5 and ν 6 + ν 9 were analyzed by Duxbury et al. [6]. Later, Wang et al. [4] studied the high resolution infrared spectrum of the first overtone of the ν 6 at 423 cm -1 together with the millimeter-wave rotational spectra in the four lowest torsional states. Wang et al. [7] also recorded and analyzed the ν 5 band together with the (ν 5 + ν 6 ) -ν 6 hot band. In this paper we report the analysis of the high resolution infrared spectrum of the ν 4 fundamental band, namely the C-C stretching, centred at 831 cm - 1 , recorded using synchrotron radiation. This band was already reported by Nivellini et al. [8] at a resolution of 0.004 cm -1 , but no K structure was observed (see figure 7 in Ref. 8). It was obvious that a detailed information could have been obtained only using a high resolution spectrometer equipped with a very brilliant source. These experimental conditions can only be fulfilled by a synchrotron radiation facility equipped with a very high resolution infrared spectrometer.

Experimental

For this study we have recorded four absorption spectra at an unapodized resolution of 0.00096 cm -1 in the 510 -1000 cm -1 spectral range using the Bruker IFS 125 Fourier transform spectrometer located at the far-infrared beamline, Canadian Lightsource. A summary of the experimental conditions for each spectrum is given in Table 1. All the 1,1,1trifluoroethane, CH 3 CF 3 , spectra were recorded using a multipass coolable absorption cell with a 2 m base length. Figure 1 shows an overview of the vibrational band of interest. The cell was equipped with wedged KBr windows. The radiation generated by either the synchrotron source (SR) or an infrared globar (GB) was detected using a Ge:Cu detector fitted with a cold filter. The diameter of the iris was 1.15 mm. A 5 kHz electronic filter was used in the experiments. The CH 3 CF 3 sample of 98% purity was supplied by the Cambridge Isotope Laboratories and was used without further purification. Besides the CH 3 CF 3 lines, a few lines belonging to residual water vapour present in the evacuated spectrometer were identified in the spectra. The gas sample temperature and pressure were monitored continuously during the data recording. A T-type thermocouple mounted inside the cell was used to monitor the gas sample temperatures. The gas pressures ranged between 85 to 301 mTorr and were measured using a 0-1 Torr capacitance gauge (Baratron). The calibration of the wavenumber scale of each spectrum was performed relative to well determined positions of the ν2 band carbon dioxide lines [9]. These lines appeared in the spectra due to residual amounts of carbon dioxide in the evacuated spectrometer tank. The gain in S/N by adopting a SR instead of a GB source can be demonstrated by comparing spectra 3 and 4 (see Table 1).

In fact, an increase of at least a factor of four in the S/N can be observed in the spectrum recorded with the SR source as shown in Figure 2. The CH 3 CF 3 molecule is a prolate symmetric top close to the spherical limit, γ = (A-B)/ ½(A+B) = 0.06. The v 4 = 1 state (C-C stretching) is of A 1 symmetry. As a consequence, the ν 4 band appears as a well behaved parallel band of medium-weak intensity, with a strong Q branch centred at about 830 cm -1 and a series of q P K (J) branches in the lower wavenumber side, and the corresponding q R K (J) branches in the higher wavenumber side, as can be seen in Figure 1. The P and R branches appear as a comb structure with a separation between successive J of about 0.345 cm -1 (2B) and could be analyzed up to J" = 69 in the R side and up to J" = 71 in the P side. Because of the small size (0.0005726 cm -1 ) and of the positive sign of the difference [(A'-A")-(B'-B")] the K structure of all the R and P branches appears very congested and blue degraded. In fact, up to q P K (12) on the low energy side, and up to q R K (13) on the high energy one, the K structure cannot be resolved at all. Even at high J values, K = 12 is usually the lowest K resolved transition, and below K = 9 the branch generally appears as a strong unresolved band feature. For the same reason the central Q branch appears completely unresolved and overlapped with other Q branches belonging to many hot bands appearing in the spectrum. The only characteristic feature in this branch is the very sharp edge on the high wavenumber side at 831.177 cm -1 which denotes the band origin. Starting from this point, by measuring the average spacing between the q R branches, a tentative assignment of the J numbering of the q R K (J) branches was attempted. Because of the presence of two groups of three identical fermions in the molecule an intensity alternation can be seen in the K structure of branches. The transitions with K = 3n with n = 0, 1, 2,….have twice the intensity of those with K ≠ 3n owing to the statistical weight of the corresponding K levels. Figure 3 illustrates the q R K (46) in the region 846.70-846.83 cm -1 where the intensity alternation is evident. Since K cannot be larger than J and because of the blue degradation of the branches, the line at higher wavenumber in a q R branch is the one with J" = K", which is a strong line when J" = 3n. In this way we were able to confirm the J numbering of the q R branch and assign the K structure. The J numbering was performed analogously in the q P K (J)

region, but with less confidence for the first few J values because of the strong overlap with the q Q branches. In order to assist the assignments, the method of the ground state combination differences (GSCD) was adopted. Satellites branches accompany each P, Q and R branch of the ν 4 fundamental band. They arise from hot bands involving the first few vibrational states ν 6 , ν 12 and 2ν 6 with origin at 223, 366 and 423.15 cm -1 , respectively [4,8].

The large majority of the assignments were made in spectrum 4, (see Table 1) with a few exceptions for high J high K values where the spectrum 2 was used.

Spectroscopic parameters for the ground state were already reported by Wang et al. [4,7] in studies of the microwave and millimeter-wave rotational spectrum in the ground and in excited torsional states up to v 6 = 4 adopting an effective torsional-rotational Hamiltonian.

No torsional splitting was ever observed in the ν 4 band and a torsional splitting of the order of 0.01 MHz was reported in the distortion moment-allowed Q branch transitions in the pure rotational spectrum [3]. The spectroscopic parameters for the ground state could thus be determined, by neglecting the splitting, from the fit of all the data available in the literature.

The usual Hamiltonian for a prolate symmetric top with terms up to the 6 th power in J, K was adopted. In addition, the ∆k = ±3 interaction terms leading to the K=3 split transitions observed in the forbidden rotational Q branches [3] and, in a few cases, in the millimeterwave spectrum at very high values of J [2] was taken into account.

The diagonal matrix elements for a prolate rotor, including terms up to the sixth power in the angular momentum, are given by

E(J,k)/hc = B [J (J + 1 ) -k 2 ] + Ak 2 -D J [J ( J + 1 )] 2 -D JK [J ( J + 1 )]k 2 -D K k 4 + H J [J ( J + 1 )] 3 + H JK [J ( J + 1 )] 2 k 2 + H KJ [J ( J + 1 )]k 4 + H K k 6 , (1)
<J, k  H/hc J, k ± 3> = { [ε + ε J J(J+1)](2k ± 3) + ε K [k 3 + (k ± 3) 3 ] }F ±3 (J, k), (2)
where

F ±3 (J, k) = {[J(J + 1)] -(k(k ± 1)] [J(J + 1) -(k ± 1) (k ± 2)][J(J + 1) -(k ± 2) (k ± 3)]}½
and the signed quantum number k = ± K.

The data in Table 2 of Ref. [2] were treated according to the suggestions of Ozier [3].

It was also decided not to consider the GSCD obtained from the data of the ν 4 band presently analyzed, since their accuracy is of the order of 5 MHz while that of microwave and millimetre-wave measurements is 0.2 MHz, in the worst case. The spectroscopic parameters for the vibrational ground state obtained from the fit are reported in column 2 of Table 2.

They are in good agreement with those obtained taking into account the torsional splitting [7].

After completing the analysis of the q P K (J) branches, all the ν 4 transitions were fitted using for the v 4 = 1 state the same Hamiltonian adopted for the ground state. Columns 3 and 4

in Table 2 list the spectroscopic parameters obtained from transition wavenumbers whose corresponding observed -calculated values do not exceed a chosen limit set to 0.0002 ( fit I ) and 0.0004 cm -1 ( fit II ), respectively. In the former case 694 out of 2785 fitted transitions (about 25%) exceeded the rejection limit and the obtained standard deviation of the fit was 0.00010 cm -1 . When a tolerance of 0.0004 cm -1 was adopted only 150 data (about 5%) were discarded but the standard deviation raised to 0.00016 cm -1 . It should be pointed out that the discarded transitions do not follow any specific trend but are scattered over all the assigned J and K values. Generally, in a spectrum taken at a resolution of 0.001 cm -1 with a S/N ≥ 50 one would expect, with a well behaved Hamiltonian, to fit the large majority of the transitions ( ≥ 90% ) with a limit for rejection of 0.0001 cm -1 , obtaining a standard deviation of the fit of the order of 0.00006-0.00009 cm -1 (i.e.2-3 MHz ). If this is not the case, as in the present analysis, it means that either in the Hamiltonian some important interaction terms are not considered or in the spectrum most of the transitions are overlapped, or both. The refinement of higher order coefficients ( L K and P K ) which could effectively account for the effect of perturbations, due to anharmonic or Coriolis interactions, did not improve significantly the quality of the fit. In fact, by adopting a rejection limit of 0.0002 cm -1 , the number of fitted transitions increased from 2091 to 2256 and the standard deviation of the fit remained almost unchanged. Moreover, the additional 165 transitions which were retained in the fit are randomly distributed over all J and K quantum numbers. This observation suggests that the high density of the lines, which leads to a large number of overlapping features in the spectra, is responsible of the quality of the fit. Although perturbations cannot be excluded a priori, they should be extremely small. All the diagonal parameters for the v 4 = 1 state have been determined, with the exception of H JK which has been set equal to the corresponding value of the ground state. Also the ∆k = ±3 interaction constants, ε and ε J , have been fixed to the corresponding ground state values, since no K = 3 transition has been identified in the spectrum due to the congestion of the low K lines.

The spectroscopic parameters in columns 3 and 4 of Table 2 are very similar, most of their values being within 3σ or less. Moreover, their values are close to the corresponding ones in the ground state, further supporting the hypothesis that the v 4 = 1 state is practically unperturbed. All line wavenumbers from the ro-vibrational analyses can be made available from the Bologna author. characterized through the observation and the analysis of the ν 4 -GS band at 831 cm -1 . An improved set of ground state spectroscopic parameters, however neglecting torsional splitting, was obtained from the analysis of all the data available in the literature. Two sets of spectroscopic parameters for the excited state have been evaluated. They differ for the chosen rejection limit (0.0002 or 0.0004 cm -1 ), for the number of data excluded in the final fit (694 or 150), leading to a standard deviation of the fit of 0.00010 and 0.00016 cm -1 , respectively. Table 1. Summary of experimental conditions of CH 3 CF 3 spectra.

Conclusions

Table 2. Spectroscopic parameters ( in cm -1 ) for the ground state and the v 4 = 1 state of CH 3 CF 3 .

Figure captions

Figure 1.

The infrared spectrum of CH 3 CF 3 in the region between 800 and 860 cm -1 showing the ν 4

fundamental band centered at 831 cm -1 . The J numbering of the q P K (J) and q R K (J) branches is indicated. Experimental conditions, see Table 1, spectrum 4.

Figure 2 .

A portion of the CF 3 CH 3 ν 4 band recorded at the far-infrared beamline using the (A) globar source and (B) the synchrotron. See Table 1, spectra 3 and 4, for more detailed experimental conditions. showing the K structure of the q R K (46) branch. Experimental conditions, see 1.42 x 10 -7 cm -1 1.56 x 10 -4 cm -1 1.00 x 10 -4 cm -1 a) Tolerance for rejection = 0.0004 cm -1 . b) Tolerance for rejection = 0.0002 cm -1 . c) Fixed to the ground state value.

3 .

 3 Description of the spectrum and analysis

  state (C-C stretching) of CH 3 CF 3 has been spectroscopically

AcnowledgmentsL.

  Fusina, G.Nivellini and G. Di Lonardo acknowledge the Università di Bologna and the Ministero della Ricerca e dell'Università ( PRIN " Trasferimento di energia e di carica: dalle collisioni ai processi dissipativi') for financial support. A. Predoi-Cross acknowledges the support she received from the National Sciences and Engineering Research Council of Canada. In Brussels this work was supported by the Fonds National de la Recherche Scientifique (F.R.S.-FNRS, contracts FRFC and IISN) and the « Action de Recherches Concertées de la Communauté française de Belgique ».

Figure 3 .

 3 Figure 3. Section of the infrared spectrum of CH 3 CF 3 in the region between 846.70 and 846.83 cm -1

Table 1

 1

	,

Page 12 of 18 URL: http://mc.manuscriptcentral.com/tandf/tmph

Table 1 .

 1 Summary of experimental conditions of CH 3 CF 3 spectra.

					Molecular Physics
	Source	Pressure (mTorr)	Temperature (K)	Pathlength (m)	Scanning speed (KHz)	Resolution (cm -1 )	Optical Filter (cm -1 )	Detector Filter (cm -1 )	Detector Gain	No. Scans
	1 SR a	85	298.25	24.15	40	0.00096 510-1000	490-1190	64	104
	2 SR a	200	298.25	72.15	60	0.00096 510-1000	490-1190	64	142
	3 GB b	200	298.45	24.15	40	0.00096	none	490-1190	64	350
	4 SR a	301	216.75	24.15	40	0.00096 300-1250	300-1200	16	374
	a synchrotron source.

b globar source. Page 14 of 18 URL: http://mc.manuscriptcentral.com/tandf/tmph

Table 2 .

 2 Spectroscopic parameters (in cm -1 ) for the ground state and the v 4 = 1 state of CH 3 CF 3 .

		ground state	υ 4 = 1 fit I a	υ 4 = 1 fit II b
	ν 0	_	831.1769618 (191)	831.1769049 (147)
	A	0.18342224765 (679)	0.1831688643 (668)	0.1831692008 (482)
	B	0.172957554120 (378)	0.1726383363 (289)	0.1726383572 (217)
	D J x 10 9	42.184614 (487)	42.2685 (130)	42.27015 (962)
	D JK x 10 9	66.78631 (169)	66.8795 (194)	66.9199 (138)
	D K x 10 9	-65.783 (114)	-65.6094 (688)	-65.3255 (485)
	H J x 10 15	18.336 (682)	17.38 (162)	17.32 (119)
	H JK x 10 15	-0.866 (282)	-0.866 c	-0.866 c
	H KJ x 10 12	0.148304 (820)	0.1350 (115)	0.15254 (810)
	H K x 10 12	2.480 (615)	2.3751 (234)	2.4458 (165)
	ε x 10 9	8.32004 (442)	8.32004 c	8.32004 c
	ε J x 10 15	-14.782 (868)	-14.782 c	-14.782 c
	total number of data	478	2785	2785
	number of data retained in the fit	447	2635	2091
	weighted standard
	deviation