Benjamin Bois

Cornelis Van Leeuwen
email: k-van-leeuwen@enitab.fr

Philippe Pieri

Jean-Pierre Gaudillère

Etienne Saur

Daniel Joly

Lucien Wald

Didier Grimal

Jean-Pierre Gaudillere

Viticultural agroclimatic cartography and zoning at mesoscale level using terrain information, remotely sensed data and weather station measurements. Case study of Bordeaux winegrowing area

Keywords: Climate, Zoning, Bordeaux, GIS, Grapevine Climat, Zonage, Bordeaux, SIG, Vigne

published or not. The documents may come

Introduction

Amongst the environmental components of terroir, climate plays a major role. Rainfall and evapotranspiration are the main climate components of vineyard water balance, which has a large influence on grape ripening [START_REF] Matthews | Fruit ripening in Vitis vinifera L.: Responses to seasonal water deficits[END_REF][START_REF] Tregoat | Etude du régime hydrique et de la nutrition azotée de la vigne par des indicateurs physiologiques. Influence sur le comportement de la vigne et la maturation du raisin (Vitis vinifera L. cv. Merlot, 2000, Bordeaux)[END_REF]. Solar radiation drives evapotranspiration process [START_REF] Bois | Using remotely sensed solar radiation data for reference evapotranspiration estimation at a daily time step[END_REF]. It also affects anthocyanin, malate and sugar contents within the berry [START_REF] Kliewer | Influence of temperature, solar radiation and nitrogen on coloration and composition of Emperor grapes[END_REF]. Air temperature governs much of grapevine development and phenology as well as grape ripening kinetics [START_REF] Huglin | Nouveau mode d'évaluation des possibilités héliothermiques d'un milieu viticole[END_REF][START_REF] Lebon | Branch development controls leaf area dynamics in grapevine (Vitis vinifera) growing in drying soil[END_REF]. Hence, it also modifies berry composition, notably malate, sugar and anthocyanins content at ripeness [START_REF] Buttrose | Effect of the temperature on the composition of Cabernet-Sauvignon berries[END_REF]. Several studies have been performed in attempt to identify which variables amongst physical components of terroir (i.e. soil and climate) play a major role on grapevine development and berry ripening processes. Some of them focused mainly on soil characteristics [START_REF] Morlat | Characterization of viticultural terroirs using a simple field model based on soil depth -II. Validation of the grape yield and berry quality in the Anjou vineyard (France)[END_REF], landscape [START_REF] Vaudour | Caracterisation spatiale des terroirs et constitution des raisins en A.O.C. Cotes-du-Rhone meridionales (Bassin de Nyons-Valreas)[END_REF] or climate. Others study several factor of the terroir effect. Van [START_REF] Van Leeuwen | Influence of climate, soil, and cultivar on terroir[END_REF] show that climate has a greater impact on vine development and grape ripening than soil or cultivar. At mesoscale level, climate studies focus mostly on one climate component such as air temperature used for frost risk assessment [START_REF] Beltrando | Spatial interpolation-based mapping of the spring frost hazard in the Champagne vineyards[END_REF] or for ripening potential [START_REF] Bindi | Extension of crop model outputs over the land surface by the application of statistical and neural network techniques to topographical and satellite data[END_REF]Maselli, 2001, Jones et al., 2004), global radiation [START_REF] Failla | Spatial distribution of solar radiation and its effects on vine phenology and grape ripening in an alpine environment[END_REF] or wind circulation patterns [START_REF] Bonnardot | Diurnal wind, relative humidity and temperature variation in the Stellenbosch-Groot Drakenstein wine-growing area[END_REF]. Only few studies have attempted to account for several climate characteristics [START_REF] Pythoud | Etude des terroirs viticoles vaudois. 3. Modélisation des paramètres climatiques[END_REF][START_REF] Zufferey | Terroirs viticoles vaudois et alimentation hydrique de la vigne[END_REF]. Recent advances in computer technologies (such as Geographical Information Systems, GIS) and measurement devices (e.g. remote sensing and automatic weather stations) provide useful material for climatological studies. In this paper, several techniques based upon geostatistics, regression models, Digital Elevation Model (DEM), GIS and remotely sensed data are evaluated and applied to perform an agroclimatic zoning of Bordeaux winegrowing region (France).

Material and methods

The study area

The study area is the Bordeaux winegrowing region, located in the southwest of France (figure 1A). Bordeaux appellations of origin cover 2500 km 2 (figure 1B), with about half of the area actually planted to grapevines. The most common cultivars are Merlot, Cabernet-Sauvignon and Cabernet franc (red varieties) and Sauvignon Blanc and Semillon (white varieties). The climate is characterized by moderately warm and dry summers, and mild and wet autumn and winter seasons. The annual rainfall is 930 mm and the mean annual temperature is 13.8°C (1976-2005 normals).

Climate data

Weather station data A database was elaborated by gathering daily weather records of 91 stations (figure 1C). Automatic Cimel ® weather stations provided daily rainfall and minimum and maximum air temperature records. Rainfall measurements were completed by manual rain gauges data administrated by Météo-France. One automatic weather station, located in Villenave d'Ornon (44.79 N, 0.58 W), equipped with a class 2 pyranometer, a humidity sensor and a cup anemometer at 2 meters high, was used for solar radiation and reference evapotranspiration validation. Data was checked and 601 aberrant data series were eliminated. As weather station networks are currently developing, the number of recording sites increases during the study period: weather station number varied from 36 (temperature in 2001) to 85 (rainfall in 2005). For rainfall, interpolations were performed on the 1994-2005 period. For temperature, solar radiation and reference evapotranspiration, the 2001-2005 period was used.

Satellite sensed data

Solar radiation data was retrieved from the HelioClim-1 data base, available at www.soda-is.org. The HelioClim-1 data base has been obtained by the application of the Heliosat-2 method to Météosat satellite images [START_REF] Gschwind | Converting a successful research project into a sustainable service: The case of the SoDa Web service[END_REF]. The Heliosat-2 method is based on the construction of cloud and clearness indices [START_REF] Rigollier | The method Heliosat-2 for deriving shortwave solar radiation from satellite images[END_REF]. Daily solar irradiation data was downloaded at 418 points each sampled at an approximate distance of 20 kilometres inside and at neighbouring sites of Bordeaux winegrowing region.

Geographical information

In order to account for local topography effects on air temperature, geographical information was collected and computed using the GRASS open source Geographical Information System (GIS). Ten geographical co-variables were extracted from a 50 m resolution Digital Elevation Model (DEM) and from the CORINE Land Cover database: elevation, slope, aspect, terrain rugosity, distances to the Atlantic Ocean and to the Gironde estuary, urban proximity index, vegetation index, forest proximity and potential solar radiation. Slope and aspect were also used to account terrain effects on solar irradiation.

Spatialization methods

For each climate variable a different spatialization procedure was used (figure 2). Rainfall and temperature data were interpolated using weather station records. Rainfall data was interpolated using ordinary kriging. Co-variance models were fitted automatically using the Gstat package [START_REF] Pebesma | Multivariable geostatistics in S: the gstat package[END_REF] of R open source statistical software (R Development Core Team, 2007). Rainfall daily data was used to calculate rainfall amount cumulated during relevant periods (i.e. year, grapevine vegetative period and grape development and ripening, see next section 2.5 for details) prior to interpolation.Daily temperature was interpolated using a multiple regression process: each day, the covariables significantly correlated (at 1% of significance level) to temperature were selected and a linear multiple regression model was constructed using a forward stepwise regression algorithm. The model residuals, calculated for each station location, were then interpolated using ordinary kriging, and added to regression model prediction, in order to provide unbiased estimates at each weather station location. This procedure was used to spatialize daily miminum (T min ) and maximum (T max ) temperatures. The mean daily temperature was calculated as the average of minimum and maximum temperature. Solar radiation daily data collected from the HelioClim-1 database was interpolated at 1 km resolution using ordinary kriging. For each 1 km pixel, the r.sun geometrical model (Hofierka and Suri, 2002) was used to account for terrain effect on solar radiation interception: using elevation, slope and aspect derived from the DEM, 50 m resolution maps of solar radiation were produced.

Climatic and agroclimatic indices

Daily spatialized data was used to calculate, for each pixel, several indices in order to account for climate potential consequences on grapevine development and grape ripening. Average daily temperature was used to calculate heat summation (in growing degree.days, GDD) with a base temperature of 10°C, from January 1 st to September 30 th .It is hereafter referred as ST 10 . Reference evapotranspiration (ET 0 , given in mm) was calculated using daily average air temperature and daily solar radiation, using the Turc method [START_REF] Turc | Evaluation des besoins en eau d'irrigation, évapotranspiration potentielle[END_REF], re-adjusted to Bordeaux weather conditions:

( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + = 15 70 88 , 23 014 , 0 0 T T R ET s (1)
where R s is the daily solar radiation (MJ m -2 ), and T is the daily average temperature (°C). Rainfall height, solar radiation and evapotranspiration daily average were calculated on the following periods: the whole year, the "vegetative" period (from April 1 st to September 30 th ) and the "summer" period (from July 1 st to September 30 th , which approximately corresponds to the grape growing and ripening period).

Evaluation of spatialization procedures

Rainfall and air temperature daily interpolation errors were calculated at each weather station using a cross-validation process. Solar radiation spatialization procedure (using remotely sensed data and a terrain integration model) was assessed by comparing spatialized ("predicted") data at the 50 by 50 m pixel where the weather station of Villenave d'Ornon is located, to the weather station records (pyranometer solar radiation, the "observed" value). For reference evapotranspiration, estimated values at Villenave d'Ornon weather station were obtained by calculating ET 0 with predicted daily average temperature values (interpolated using cross validation process, i.e. without Villenave d'Ornon temperature records), and spatialized solar radiation. These estimates were compared to reference evapotranspiration computed with Penman-Monteith FAO-56 method [START_REF] Allen | Crop evapotranspiration: guidelines for computing crop water requirements[END_REF] using relative humidity, solar radiation, air temperature and 2 m wind speed records. Statistical indicators of spatialization error were calculated for each weather variable: the root mean squared error (hereafter referred as RMSE) and the mean error (bias).

Results and discussion

Climate variable spatialization accuracy

The statistics relative to spatialization of each climate variable is shown on table 1. Rainfall interpolation was slightly over-estimated (+2.6 mm). Interpolation errors were higher during the whole year (in absolute value) than for summer and vegetation periods. However, when considering the relative error (in percent), rainfall interpolation was more accurate for the whole year and during the vegetative period (9% and 11% of the mean value, respectively), than during summer (15% of the mean value). This could be the consequence of a higher frequency of spatially concentrated rain storms, which induce strong variations of rainfall amounts even over small distances [START_REF] Ciach | Analysis and modeling of spatial correlation structure in small-scale rainfall in Central Oklahoma[END_REF]. Temperature interpolation using environmental co-variables and residual kriging was more accurate for maximum temperature than for minimum temperature. The difficulties in interpolating minimum temperature are well known from climatologists, as this variable provides strong spatial variations, governed by local topography. Temperature error propagation within heat summation model provided an RMSE of 56.3 GDD, from April to September. Daily solar radiation and ET 0 errors are considerable (RMSE ranging from 16 to 21% of the mean value). The lowest relative RMSE were found during the July to September period. These errors reduce the relevance of spatial and temporal analyses of these variables, within Bordeaux winegrowing region. However, strong differences were noticed due to terrain local variations [START_REF] Bois | Cartographie agroclimatique à méso-échelle : méthodologie et application à la variabilité spatiale du climat en Gironde viticole. Conséquences pour le développement de la vigne et la maturation du raisin[END_REF], suggesting that solar radiation and reference evapotranspiration are mostly governed by slope and aspect in Bordeaux vineyards.

Rainfall (mm)

Bordeaux winegrowing area agroclimatic zoning

Due to the lack of accuracy concerning ET 0 and R S spatialization, it seemed reasonable to focus on the rainfall and temperature spatial and temporal variations. Thermal characteristics were investigated by means of ST 10 , during 5 years (2001)(2002)(2003)(2004)(2005). Maps were constructed at a 50 m resolution (i.e. pixels of 50 m by 50 m), in order to account for local temperature patterns due to local environmental variations. Maps of rainfall heights were constructed for each year of the 1994 to 2005 period (12 years), at a 250 m resolution.

As climate characteristics strongly vary from one year to another, the redundancy of spatial climate relative differences from were explored. The redundancy of spatial patterns observed for each year within Bordeaux winegrowing region was assessed by a classification method, which consists in the following steps: (a) for each year, the whole set of spatialized data (i.e. pixels of the map) was partitioned in 5 classes, using a regular interval clustering. A value from 1 to 5 was given to each pixel, according to its rank within the data set (rank 1 corresponds to the highest amount of rainfall or to the highest ST 10 value, and rank 5 corresponds to the lowest rainfall amount or to the lowest ST 10 value). An annual map of relative classification is thus obtained. (b) For each pixel, the mean rank is calculated on the whole period, as well as the standard deviation of the rank. (c) If the standard deviation is larger or equal to 1, which suggest a strong year-© ACW, Agroscope Changins-Wädenswil 2008 VII e Congrès International des terroirs viticoles / VII th International terroir Congress to-year variability, the pixel is classified as "variable". If the standard deviation is lower than 1, the pixel rank is obtained by rounding its average yearly rank. Six classes are thus obtained. Maps of ST 10 and rainfall amounts from April to September are shown on figure 3. For GDD, only a small area in north-eastern part of Bordeaux vineyards is classified as "variable". This indicates a high redundancy concerning temperature spatial patterns. Most of the Bordeaux winegrowing region is classified as "medium" (class 3, 64% of the total area). 21% of the total appellation area is classified as "early" (class 2). It corresponds to areas located on the hillsides of Dorgogne and Garonne valleys, and close to the Gironde estuary. 12% of the total area is classified as "late" (class 4). Most of this surface is located on the north-western part of the Bordeaux winegrowing region, close to the Atlantic Ocean. Small areas exhibit very low temperature sums. These vineyards are located in the northwest, and are classified as "very late" (class 5, 1% of the total area). Three zones benefits of high ST 10 values: a small part of Saint-Emilion vineyards, located in the eastern part of Bordeaux winegrowing region, the vineyards of the appellation of origin Pessac-Leognan, located within the suburbs of Bordeaux city, in the west, and some vineyards located near Sauternes, in the south. The difference in GDD between each class varies from a year to another, from 42 to 79 GDD. As the grape ripening kinetics strongly depends on GDD [START_REF] Huglin | Nouveau mode d'évaluation des possibilités héliothermiques d'un milieu viticole[END_REF], maturity differences from 6 to 11 days are expected between each class (considering that the mean daily temperature is about 17°C at the end of September). Thus, a maximum difference of 30 to 55 days in maturity date may occur in Bordeaux winegrowing region, between the coolest and the warmest areas. However, such differences may not be observed in reality, because several other variables govern grape ripening (cultivar, soil or agricultural practices).

Concerning April-to-September rainfall, two thirds of the Bordeaux winegrowing region is classified as "variable". The north-western part of the region (Médoc and most of the Haut-Médoc appellations) exhibits repetitive low rainfall height, whereas the west-middle part (mostly the Pessac-Léognan appellation) is redundantly wetter than the rest of the winegrowing region. However, these patterns were not observed during the summer period, during which water availability strongly determines grape development and ripening, especially for red grape cultivars: a zoning applied to the July to September rainfall indicates that spatial rainfall patterns are strongly modified from a year to another (i.e. the entire winegrowing region is classified as "variable").

Conclusion

In this paper, an approach to characterize climate variations at mesoscale level is proposed. Using remotely sensed data, weather stations records and geographical information, several methods of climate spatialization are assessed. Rainfall and temperature interpolations provide reasonable accuracy, whereas errors are large for solar radiation and reference evapotranspiration spatialization processes.

The Bordeaux winegrowing region exhibits a large diversity of type of wine. This is the result of spatial variations of cultural practices, cultivar, soil and climatic diversity. In this study it has been shown that rainfall, the only source of water for Bordeaux vineyards, is probably not a factor which can explain the redundant typical characteristics and differences that can be observed for a same type of product (i.e. red, white, dry or sweet wines). As a matter of fact, even if rainfall spatial distribution within the same year may induce strong spatial variations of grapevine water status, grape ripening, and thus grape quality, this particular pattern changes from a year to another. In contrast, temperature spatial variations are highly redundant, although the maximum differences observed within Bordeaux winegrowing region may be small or large, depending on the vintage. The zoning performed on the 2001-2005 period has shown differences of heat summation (from January to September) from 42 to 79 GDD between each of the 5 classes obtained. These differences might induce considerable lags of harvest date, for a given cultivar. Theses results provide useful information for cultivar adaptation to local climate condition. Using this zoning, it would certainly be of great interest to compare the behaviour of a given cultivar in two contrasted GDD zones as a tool for climate change anticipation.

Figure 1

 1 Figure 1 The study area and the weather station locations. A: Location of the Gironde province in France. B: Appellations of Controlled Origin (A.O.C.) of the Bordeaux winegrowing region. C: Weather stations. The symbols indicate which variable each station monitors (RR: rainfall ; T: air temperature ; R S : solar radiation ; ET0: reference evapotranspiration)

Figure 2

 2 Figure 2 The global spatialization procedure for climatic and agroclimatic variables, and zoning (Sat: Satellite data ; DEM : Digital Elevation Model ; RR: rainfall ; T min and T max : minimuim and maximum temperature ; R S : solar radiation ; ET 0 : reference evapotranspiration)

Figure 3

 3 Figure 3 Maps of thermal (GDD) and rainfall zoning. A: zoning based upon ST 10 , period 2001-2005. B: zoning based upon rainfall height from April to September, period 1994-2005

Table 1 Statistical indices of spatialization procedures. Veg. : vegetative period ; Sum.: summer period ; N : number of observations (number of situations by number of available stations). See text for further details.

 1

					Temperature		Daily solar
					(°C or GDD)		radiation (MJ m 2 )	Daily ET 0 (mm)
		Year Veg. Sum.	T min T max . ST 10	Year Veg. Sum.	Year Veg. Sum.
	Mean	807	363	178	8.7 18.9 1644	13.4 19.6 19.1	2.6	4.0	4.0
	Bias	2.6	0.6	0.5	0.1	0.0	4.4	-1.5 -2.0 -1.8	-0.1 -0.2	0.0
	RMSE	75.5	39.1	26.1	0.9	0.6 56.3	2.7	3.3	3.2	0.6	0.6	0.6
	(%)	9% 11% 15%	-	-	-	20% 17% 17%	21% 16% 16%
	N	691	691	691	89238 89032 226	1818 909 454	1818 909 454

© ACW, Agroscope Changins-Wädenswil 2008

Modelling & Software, 21(11): 1555-1561. © ACW, Agroscope Changins-Wädenswil 2008

Acknowledgements

The authors thank the Conseil Interprofessionnel du Vin de Bordeaux for their financial and technical supports.


Prof Jean-Marc Thiriet
email: jean-marc.thiriet@ujf-grenoble.fr

Dr Hamed Yahoui

Dr Hélène Frémont

International dimension to increase Lifelong Learning possibilities in Europe

Keywords: component; Lifelong Learning, recognition, mobility of students, harmonization, Recognition of Prior Learning (RPL), accreditation

Thanks to the European policy, mobility of students has been encouraged via ERASMUS, allowing universities to recognise, within their programmes, semesters or complete academic years spent abroad. Within the frame of global European projects, such as the ERASMUS Thematic Networks [www.elleiec.eu], Tuning approach was proposed as a way to facilitate both exchange of students during their studies and mobility of workers during their professional life. The philosophy behind that is not to focus too much about the actual courses followed by a student (which is extremely tricky when a student is sent abroad) but more on what he gets as a whole concerning knowledge, skills and competences, taking also account of soft or generic skills like internationalization, multiculturalism, teamgroup work, foreign language… Another interesting aspect is the use of tools like RPL (Recognition of Prior Learning) which is very useful, when we want to give a worker the possibility to pass a diploma, based on the partial or total recognition of his professional knowledge, skill and competences. The present paper develops the work achieved within the ELLEIEC project, relative to International Modules (IM) and International Curricula Networks (ICN), concepts proposed in the project and experimented practically, to facilitate the mobility of students and of citizens/workers.

INTRODUCTION

One of the aims of the ELLEIEC (Enhancing Lifelong Learning in Electrical and Information Engineering, http://greenelleiec.eu/) project (proposed by the EAEEIE, European Association for Education in Electrical and Information Engineering, www.eaeeie.org), Thematic Network launched in 2008 for a three and a half year-duration, was to propose a synthesis about how to ensure mobility of students during their studies and also of citizens (workers) during their professional lives, in a LLL (Life-Long Learning) approach.

Questionnaires have been designed and filled, all around Europe, in order to have an overview of the use of Lifelong Learning approaches (such as RPL for instance) in the various countries. The working group of the ELLEIEC project has also proposed "best practices" for the use of RPL approaches throughout Europe. Some benchmarks are proposed, in order to show various tools that can be used (ERASMUS, ERASMUS Mundus, Atlantis, Curriculum Development programmes, bilateral agreements) which are more or less "light", as far as the implementation is concerned, and may be used quite easily to implement dual degree or double diploma programmes.

The last aspect proposed in the project is the concept of ICN (International Curricula Network) which deals with group of experts, being able to work together on an international basis, and defining IM (International Modules) which are basically modules which are proposed in at least two accredited programmes, in two various countries. These International Modules can be provided for Lifelong Learning.

II. SOME CONSIDERATIONS ABOUT RPL, ACCREDITATION

AND LLL RPL (Recognition of Prior Learning) is an interesting approach for LLL since it allows to "open" the mobility of students and the mobility of citizens. This approach consists not only in measuring learning outcomes and knowledge, which is classically applied, but to measure for a "student" her/his actual knowledge, skill and competence (KSC). This approach was formalised in particular by the Tuning project [START_REF] Tuning | Tuning Educational Structure in Europe. Final Report Pilot Project -Tuning project[END_REF]. The interest of such an approach is manifold:

 For classical students leaving their home university (HU) to spend a semester in a partner university (PU), the KSC approach helps in building a specific programme for the student, using the available PU courses, and not too strictly follow the programme of the HU.  For classical students coming back from their mobility, it is easier to "recognise" from the HU the semester spent abroad, considering once again the whole KSC, preferably to specific exact comparable courses.  For every citizens, and in phase with the White book of the European Union on "mobility of citizens/workers" [START_REF]White book knowledge[END_REF], the KSC approach, used for instance in the European Qualification Framework ( [START_REF]Commission of the European Communities, The European Qualifications Framework (EQF)[END_REF], [START_REF] Hoffmann | A Sectoral Qualification Framework for Engineers and Computer Scientists[END_REF]) helps in considering the complete professional experience of citizens/workers at the time to consider either to actually deliver them a diploma or to give them the possibility to enter into a programme, with some partial recognition (using continuous education for instance).

Accreditation is an important aspect at this stage. To allow the mobility of students or citizens through an accredited programme, it means that the accreditation allows this:

 In some countries or types of institutions, the accreditation board requires that the programme is completely defined and fixed, for the complete duration of the accreditation: in this case it is difficult to send students abroad or accept a KSC recognition, because in this case, the programme does not fit exactly. This is a "strict" way to see accreditation, with no margin for mobility.  In some countries, after discussion with colleagues within European projects, it seems impossible to envisage a complete KSC (RPL) approach, because there are some risks that people "buy" their diploma. In this case, the accreditation process must guarantee the quality of the approach (for instance RPL).  In some countries or Higher Education Institutions, the accredited programmes are organised in a more "open" manner, by using "minor" (combination of a major field of study and a minor field of study, this "minor" can be obtained and validated through a mobility) or by using for instance "international minor term" (in this case, the university encourages explicitly students to spend some times abroad and their experience is validated and recognised). Some US universities are developing mobility through this approach and in France, the "Commission des Titres de l'Ingénieur" (CTI) encourages HEIs to have a high percentage of students going for one semester at least abroad during their studies.

Various mechanisms can be envisaged for Lifelong Learning, once we consider giving the possibility for citizens to learn new things and gain credits, to validate competences, all their life long. Three points should be taken into account:  The actual situation of the learner, a) normal registered student, b) student with a specific contract, c) worker from a company.  The way the courses are organised, a) classical class, b) evening and week-end courses, c) intensive courses, d) programmes specifically organised for people working in companies (suite of intensive weeks) e) synchronous e-programmes (programmes with tutor, with milestones), f) asynchronous e-programmes (with or without tutors), g) no organised courses (APEL).  The type of contract between the learner, the company (if any) and the Higher Education Institution.

III. QUESTIONNAIRE ABOUT RPL

Two questionnaires have been elaborated, one for the "bachelor and master" (BM) levels and one for the Doctoral level (PhD) [START_REF] Bonnaud | PhD in Electrical and Information Engineering in Europe: towards a harmonization including LiveLong Learning-ITHET[END_REF]. We focus here on the questionnaire for BM which is composed of four sets of questions. The fourth set of questions is composed of the two following questions:

 Admission possibilities for students with foreign based APL/APEL?  Where compulsory internship is part of a programme, is it possible to get credits for previous practical work/experience?

The results of the questionnaires are available in the book published by the project [START_REF]Under the coordination of H. Yahoui -Lifelong learning in ELECTRICAL and INFORMATION ENGINEERING in Europe[END_REF].

IV. IMPLEMENTATION OF SOME BENCHMARKS

It was decided, at the origin of the project, to propose some methodologies or tools, making possible to enhance mobility and cross-recognition. Another idea is the fact that the ERASMUS exchange programme, very well known at the European level, is based on bilateral agreements, but we think we can work also with a multilateral approach. We have introduced [START_REF] Thiriet | Fooz -Toward an International Curricula Network for exchanges and LifeLong Learning -Electronics and Electrical Engineering[END_REF] in the ELLEIEC project approach the concept of International Curricula Network (ICN) (see Fig. 1) [START_REF] Thiriet | Fooz -Toward an International Curricula Network for exchanges and LifeLong Learning -Electronics and Electrical Engineering[END_REF] whose purpose is to define a "soft" or "light" dynamic frame for the setting of multilateral collaborations. The use of existing programmes of the European Union can be a very good support (we give as examples Curriculum Development Programmes, Intensive Programmes, ERASMUS, Atlantis…). Crossrecognition is a basic condition, which can under certain assumptions lead to a double-diploma. We can mention the fact that ICN 2 (Master in vision) is using ERASMUS-Mundus + ERASMUS exchanges and bilateral agreements [START_REF] Thiriet | Fooz -Toward an International Curricula Network for exchanges and LifeLong Learning -Electronics and Electrical Engineering[END_REF]. The following sub-sections will describe ICN 3 (an existing bachelor programme) and ICN 1 (a new master to be proposed).

A. An existing Bachelor in Wireless networks and security

The Professional Bachelor's Degree "Networks and Telecommunications" specialising in "wireless networks and security" started in Grenoble in September 2005 (Frenchspeaking programme). In the French system, this programme concerns the last year of a bachelor's degree level (3 years and 180 ECTS) which means the 5th and 6th semester of the bachelor's degree programme. In order to develop international exchanges, especially with European countries where French is not so widespread in the technological fields, we decided to set up an international group, reduced in numbers, 14 students, with the same modules taught in English [START_REF] Thiriet | Professional" Bachelor's Degree in Wireless Networks and Security[END_REF]. This programme began in September 2007, and concerns only, as mentioned before, the 5th and 6th semesters.

a. International Modules based on the cross-recognition thanks to mobility of students

The ICN which exists around this programme is composed of several universities among which two from the USA1. This first set of partners is concerned by student exchange, which means that students from these institutions attended the international programme in Grenoble. These exchanges were organised within the frame of ERASMUS exchanges for the European partners, within a specific bilateral agreement with CEFET-Minais Gerais, and within the Atlantis Programme (project DESIRE2: Dependable Systems International Research and Educational Experience, http://www.desire. agh.edu.pl) for the US partners. All the concerned students validated at least some of the courses passed in Grenoble for their home programme (so the concerned modules are International Modules) and most of them validated at the same time the French diploma, which gave them the opportunity to get a double-diploma.

Most students follow the complete year, some of them only one semester. It means that Grenoble's modules are recognised by approximately all partner universities (sometimes only parts of the Grenoble's modules are recognised by the partner university, depending also on the content of the programme of the partner university). In the case of one Finnish student, he came at Grenoble for the fifth semester only, and the sixth semester was composed of two courses: "Databases" and "Law of internet" followed by the training period. In order to be able to get the French diploma, we did a "reverse-ERASMUS", which means that we validated the Database module and the internship he passed in Finland. Concerning "Law for internet", we asked him to write a memoir. The US students came for one semester only, with a selection of "à la carte" courses, so they didn't get the French diploma. Another interesting aspect for the potential multilateral aspect of International modules is the use of staff-mobility which is actually used with 7 European partners 2 . These colleagues come to Grenoble to provide some courses to our international students. It means that the model we use in this case is: a student from a country A institution A (example: a student from Krakow) receives in Grenoble a course by a teacher coming from a country B institution B (example: a teacher from Vigo) which means that 3 countries/institutions are considered, the third being Grenoble (example: Krakow, Vigo and Grenoble). c. International Modules based on the cross-recognition thanks to common courses at the same place For the Academic projet of students, Grenoble have been since September 2010 part of an Intensive Programme called DOSSEE (Developing Open Source System Expertise in Europe) composed of 7 European partners 3 . The students (10 from each university for each of the 7 universities) had first to prepare the definition of a project, this was prepared at home universities from October to January and then, all these students spent altogether 2 weeks in March 2011 in Alcala (Spain) in order to develop their IT projects in an international and multicultural environment, because the teams in Alcala were composed of students coming from the various partner universities. Academic project is so evaluated in an international basis, with students coming from 7 institutions, by an international committee of teachers. This is considered as an international module (5 ECTS) part of the actual curriculum of the students in the 7 universities. This is actually a join module with the same evaluation done by the same set of international teachers for the students of the seven universities.

d. International Modules based on the cross-recognition thanks to common courses at one place, using e-distant possibilities

This has not been implemented yet exactly in this international programme (and by this ICN) but some developments are on the way, within the frame of other programmes such as the Curriculum Development Programme ESPANT (European Study Programme for Advanced Networking Technologies, http://www.advancednetworking.eu/) [START_REF] Rugelj | Competence based joint study program on advanced networking technologies for blended learning[END_REF] composed by 5 partners 4 . Courses have been developed, to be used on a distant 2 Universidade de Vigo (ES), Technical university of Kosice (SK), Helsinki Metropolia UAS (FI), Haute-Ecole de la Province de Liège (BE), Faculty of Law, Aristotle University (GR), Università degli Studi di Salerno (IT), Technical University of Brno (CZ) 3 Helsinki Metropolia University of Applied Sciences (FI), Universidad de Alcala (ES), Transport and Telecommunication Institute (Riga, LV), FH Joanneum (AT), Technical university of Kosice (SK), Danish Technical University, Copenhagen (DK). 4 Katholieke Hogeschool Sint-Lieven (BE), Universitat de Girona (ES), Instituto Superior de Engenharia -Instituto Politécnico do Porto (PT), Helsinki Metropolia University of basis. These courses can become potentially International Modules, when they will be actually used and recognised by at least two partner institutions, which is not the case yet.

B. A new master programme in Green Energy

In recent years, there has been an increased awareness in the European public and political arena of the need for reducing the dependence on fossil fuel by aiming towards a 100% green energy agenda by 2050. Indeed, the Directive on renewable energy sets ambitious targets for all Member States, such that the EU will reach a 20% share of energy from renewable sources by 2020 and a 10% share of renewable energy specifically in the transport sector. This agenda is not only governed by the fact that fossil energy is getting scared and by the volatile price of oil but also by the lifestyle of its population that are increasingly reliant on new technologies and comfort that are energy-intensive. It is therefore clear that in the years to come there will be an ever growing demand in electricity and an imperative need for saving and storing energy.

One of the benefits of moving towards a green energy future is the large number of jobs that are naturally created in the renewable energy and related sector. It is estimated that more than 2.7 million jobs will be created by 2020 if Europe adopts a full renewable energy agenda. HEI have to respond to this objective by proposing new diploma related to green jobs.

Currently, several ELLEIEC partners are responsible for undergraduate and postgraduate degrees in renewable energies and sustainable systems. They have decided to start the design of an ICN dedicated to sustainable electrical systems and renewable energy field. Involvement in the project will consist in implementing the result of the project in their own country applied to their corresponding degrees. They have a long experience and they are currently fully active in delivering these types of courses. Some of them will bring their own specificity due to their local environment, such as the Icelandic partner regarding geothermal energy.

V. CONCEPTS OF INTERNATIONAL MODULES (IM) AND INTERNATIONAL CURRICULA NETWORKS (ICN)

These experiments show some possibilities to implement multilateral partnership, using both combinations of bilateral actions (ERASMUS, bilateral agreements…) and existing multilateral programmes (IP, CD…). For the first ICN, the set of partners is 23, or 19 if we consider actual running International Modules. The modules are recognised or crossrecognised by at least 2 institutions, and so they can be displayed in the library of international modules provided for Lifelong Learning by the ELLEIEC project. We can mention the fact that since the benchmark programme proposed (Professional Bachelor in Wireless Networks and Telecommunications) is composed of International Modules and also because this is the last year of a programme leading to a diploma (3rd year of an 180 ECTS-bachelor programme), it Applied Sciences (FI), University of Ljubljana, Faculty of Education (SI). is possible to allow the students to get two diplomas, one from their home university and one from Grenoble.

For the second ICN the work is in the starting phase and will follow the objectives listed below:

 to define correctly the competences, skills and knowledge behind green jobs in the spirit of 'new skill for a new job',  to define a useful searching database of learning outcomes about renewable and efficient energy sector Taking advantage of the experience of the ICN in Wireless Network and Security, the new green consortium will apply for a part the methodology developed in the ILERT project [START_REF] Kornecki | ILERT, international learning environment for real-time softwareintensive control systems[END_REF] to design his own ICN.

VI. CONCLUSIONS

The discussions in the working group of ELLEIEC-partners and the analysis of the questionnaire have given us the picture of the present situation in Europe. There are different practices concerning recognition of diploma and recognition of obtained competencies in different countries and even in different universities. The work done by the working group of ELLEIEC-partners showed many differences in how APL and APEL are used in validation of the programme.

The implementation on some benchmarks, mainly a bachelor programme in Networking and Telecommunications and a master programme on Green Energy are proposed.

Figure 1 .

 1 Figure 1. Scheme presenting the concepts and International Modules and International Curricula Networks Within the ELLEIEC project, International modules (IM) have been defined as modules which are recognised and validated in at least two accredited programmes in two different countries. The following ICN have been defined in the project:  ICN.1 Master in Renewable and embedded energy,  ICN.2 Master in Vision,  ICN.3 Bachelor in Networking and Communication,  ICN.4 Bachelor in Electrical Engineering (Automatic control),  ICN.5 Master in Electronics and Telecom. ICN 2, 3, 4 and 5 are actually running existing bachelor (3 and 4) or master courses (2 and 5). ICN 1 is setting a new master programme.

  b. International Modules based on the cross-recognition thanks to mobility of teachers 1 Universidade de Vigo (ES), Technical university of Kosice (SK), AGH University of science and Technology (PL), Helsinki Metropolia UAS (FI), Universidad Politécnica de Valencia (ES), Haute-Ecole de la Province de Liège (BE), Silesian University of Technology (PL), Cracow University of Technology (PL), Technical University of Crete (GR), CEFET-Minais Gerais, Belo Horizonte (BR), Embry-Riddle Aeronautical University, Daytona Beach, Florida (USA), University of Central Florida (USA).

  The first set of questions discusses about credits:  Do you know what ECTS is? Are you using ECTS in your own university? Who is in charge to design the learning agreement related to ECTS?  Do you know what a Diploma Supplement (DS) is? Are you using the DS in your own university? Who is in charge to design the DS?  Do you know what ECVET is? Are you using ECVET in your own university? Who is in charge to record a prior learning with ECVET?  What kind of grant your university provide for an ongoing student: European one, National one, Regional one, self University funding?

 What kind of grant your university provide for an incoming student: European one, National one, Regional one, self University funding?

The second set of questions is relative to the national policy in LLL:

 Do you have any official governmental/ministerial policy on LLL?  In your country, does the LLL process include one of the following: Accreditation of Prior Learning (APL) or Accreditation of Prior Experiential Learning (APEL) A third set of questions concerns the programmes which are available through LLL  Types of programmes/courses for which officially recognized LLL credits are awarded in your country (recognition means "It gives credits (ECTS) to go to the next level or to obtain the final diploma"): Regular University Programs, Bachelor, Master, Single courses/modules, Vocational training.

ACKNOWLEDGMENT

The authors wish to thank the European Commission for the grant in the Life long Learning Programme, the Executive Agency for its help and all the partners of the project for their contribution. Project Number: ELLEIEC 142814-LLP-1-2008-FR-ERASMUS-ENW -Instrument type: ERASMUS NETWORK as well as the grants obtained by various consortia for: -ESPANT (European Study Programme for Advanced Networking Technologies), Curriculum Development Programme -« Intensive programme » DOSSEE, « Developing Open Source System Expertise in Europe ».

-ILERT Toward International Learning Environment for Real-Time Safety-Critical Control Systems (thanks also to the US Department of Education for this project).

-DESIRE2 Dependable Systems International Research and Educational Experience, http://www.desire.agh.edu.pl (thanks also to the US Department of Education for this project). Not all the persons are authors of this paper, but the authors wish to strongly thank all the partners of the above-mentioned projects, in particular colleagues from exchange programmes, partners from ELLEIEC involved in the task and the experiments, partners of ESPANT, DOSSEE, ILERT, DESIRE2 projects which are used as examples.


Pierre Martinot
email: pierremartinot@orange.fr

Emmanuel Pedon
email: emmanuel.pedon@univ-reims.fr

Alberto G Setti
email: alberto.setti@uninsubria.it

THE SHIFTED WAVE EQUATION ON DAMEK-RICCI SPACES AND ON HOMOGENEOUS TREES

Keywords: Mathematics Subject Classification. Primary 35L05, 43A85; Secondary 20F67, 22E30, 22E35, 33C80, 43A80, 58J45 Abel transform, Damek-Ricci space, homogeneous tree, Huygens' principle, hyperbolic space, wave equation, wave propagation

We solve explicitly the shifted wave equation

)u(x, t) on Damek-Ricci spaces, using Ásgeirsson's theorem and the inverse dual Abel transform. As an application, we investigate Huygens' principle. A similar analysis is carried out in the discrete setting of homogeneous trees.

Introduction

In the book [START_REF] Helgason | Groups and Geometric Analysis (Integral Geometry, Invariant Differential Operators, and Spherical Functions)[END_REF] Helgason uses Ásgeirsson's mean value theorem (see Theorem II.5.28) to solve the wave equation (1) ∂ 2 t u(x, t) = ∆ x u(x, t), u(x, 0) = f (x), ∂ t | t=0 u(x, t) = g(x), on Euclidean spaces R d (see Exercise II.F.1 and its solution pp. 574-575) and the shifted wave equation

(2) ∂ 2 t u(x, t) = ∆ x + [ d-1 2 ] 2 u(x, t), u(x, 0) = f (x), ∂ t | t=0 u(x, t) = g(x),
on real hyperbolic spaces H d (R) (see Exercise II.F.2 and its solution pp. 575-577). In this work we extend this approach both to Damek-Ricci spaces and to homogeneous trees. Along the way we clarify the role of the inverse dual Abel transform in solving the shifted wave equation.

Recall that Damek-Ricci spaces are Riemannian manifolds, which contain all hyperbolic spaces H d (R), H d (C), H d (H), H 2 (O) as a small subclass and share nevertheless several features with these spaces. Before [START_REF] Helgason | Groups and Geometric Analysis (Integral Geometry, Invariant Differential Operators, and Spherical Functions)[END_REF] the shifted wave equation ( 2) on H d (R) was solved explicitly in [24, Section 7]. Other hyperbolic spaces were dealt with in [START_REF] Bunke | The wave kernel for the Laplacian on the classical locally symmetric spaces of rank one (theta functions, trace formulas and the Selberg zeta function)[END_REF][START_REF] Intissar | Solution explicite de l'équation des ondes dans un espace symétrique de type non compact de rang 1[END_REF][START_REF] Intissar | Solution explicite de l'équation des ondes dans l'espace symétrique de type non compact de rang 1[END_REF] and Damek-Ricci spaces in [START_REF] Noguchi | The solution of the shifted wave equation on Damek-Ricci space[END_REF]. All these approaches are awkward in our opinion. On one hand, [START_REF] Lax | The asymptotic distribution of lattice points in euclidean and noneuclidean spaces[END_REF], [START_REF] Bunke | The wave kernel for the Laplacian on the classical locally symmetric spaces of rank one (theta functions, trace formulas and the Selberg zeta function)[END_REF] and [START_REF] Intissar | Solution explicite de l'équation des ondes dans un espace symétrique de type non compact de rang 1[END_REF][START_REF] Intissar | Solution explicite de l'équation des ondes dans l'espace symétrique de type non compact de rang 1[END_REF] rely on the method of descent i.e. on shift operators, which reduce the problem to checking formulae in low dimensions. Moreover [START_REF] Bunke | The wave kernel for the Laplacian on the classical locally symmetric spaces of rank one (theta functions, trace formulas and the Selberg zeta function)[END_REF] involves classical compact dual symmetric spaces and doesn't cover the exceptional case. On the other hand, [START_REF] Noguchi | The solution of the shifted wave equation on Damek-Ricci space[END_REF] involves complicated computations and follows two different methods : Helgason's approach for hyperbolic spaces and heat kernel expressions [START_REF] Anker | Spherical analysis on harmonic AN groups[END_REF] for general Damek-Ricci spaces. In comparison we believe that our presentation is simpler and more conceptual.

Several other works deal with the shifted wave equation (2) without using explicit solutions. Let us mention [START_REF] Branson | Huygens' principle in Riemannian symmetric spaces[END_REF] (see also [START_REF] Helgason | Geometric Analysis on Symmetric Spaces[END_REF]Section V.5]) for Huygens' principle and the energy equipartition on Riemannian symmetric spaces of the noncompact type. This work was extended to Damek-Ricci spaces in [START_REF] Astengo | Huygens' principle and a Paley-Wiener type theorem on Damek-Ricci spaces[END_REF], to Chébli-Trimèche hypergroups in [START_REF] Kamel | Huygens' principle and equipartition of energy for the modified wave equation associated to a generalized radial Laplacian[END_REF] and to the trigonometric Dunkl setting in [START_REF] Said | Huygens' principle for the wave equation associated with the trigonometric Dunkl-Cherednik operators[END_REF][START_REF] Ayadi | Equipartition of energy for the wave equation associated to the Dunkl-Cherednik Laplacian[END_REF]. The nonlinear shifted wave equation was studied in [START_REF] Tataru | Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation[END_REF][START_REF] Anker | The wave equation on hyperbolic spaces[END_REF][START_REF] Anker | The wave equation on Damek-Ricci spaces[END_REF], first on real hyperbolic spaces and next on Damek-Ricci spaces. These works involve sharp dispersive and Strichartz estimates for the linear equation. Related L p → L p estimates were obtained in [START_REF] Ionescu | Fourier integral operators on noncompact symmetric spaces of real rank one[END_REF] on hyperbolic spaces.

Our paper is organized as follows. In Section 2, we review Damek-Ricci spaces and spherical analysis thereon. We give in particular explicit expressions for the Abel transform, its dual and the inverse transforms. In Section 3 we extend Ásgeirsson's mean value theorem to Damek-Ricci spaces, apply it to solutions to the shifted wave equation and deduce explicit expressions, using the inverse dual Abel transform. As an application, we investigate Huygens' principle. Section 4 deals with the shifted wave equation on homogeneous trees, which are discrete analogs of hyperbolic spaces.

Most of this work was done several years ago. The results on Damek-Ricci spaces were cited in [START_REF] Rouvière | Espaces de Damek-Ricci (géométrie et analyse)[END_REF] and we take this opportunity to thank François Rouvière for mentioning them and for encouraging us to publish details. We are also grateful to Nalini Anantharaman for pointing out to us the connection between our discrete wave equation ( 16) on trees and recent works [START_REF] Brooks | Non-localization of eigenfunctions on large regular graphs[END_REF][START_REF] Brooks | Graph eigenfunctions and quantum unique ergodicity[END_REF] of Brooks and Lindenstrauss.

Spherical analysis on Damek-Ricci spaces

We shall be content with a brief review about Damek-Ricci spaces and we refer to the lecture notes [START_REF] Rouvière | Espaces de Damek-Ricci (géométrie et analyse)[END_REF] for more information.

Damek-Ricci spaces are solvable Lie groups S = N⋊A, which are extensions of Heisenberg type groups N by A ∼ = R and which are equipped with a left-invariant Riemannian structure. At the Lie algebra level,

s ≡ R m ⊕ z R k n ⊕ R a , with Lie bracket [(X, Y, z), (X ′ , Y ′ , z ′ )] = ( z 2 X ′ -z ′ 2 X, z Y ′ -z ′ Y + [X, X ′ ], 0) and inner product (X, Y, z), (X ′ , Y ′ , z ′ ) = X, X ′ R m + Y, Y ′ R k + z z ′ . At the Lie group level, S ≡ R m × Z R k N × R A , with multiplication (X, Y, z) • (X ′ , Y ′ , z ′ ) = (X + e z 2 X ′ , Y + e z Y ′ + 1 2 e z 2 [X, X ′ ], z + z ′
). So far N could be any simply connected nilpotent Lie group of step ≤ 2. Heisenberg type groups are characterized by conditions involving the Lie bracket and the inner product on n, that we shall not need explicitly. In particular Z is the center of N and m is even. One denotes by n = m + k +1 the (manifold) dimension of S and by

Q = m 2 + k the so-called homogeneous dimension of N.
Via the Iwasawa decomposition, all hyperbolic spaces

H d (R), H d (C), H d (H), H 2 (
O) can be realized as Damek-Ricci spaces, real hyperbolic spaces corresponding to the degenerate case where N is abelian. But most Damek-Ricci spaces are not symmetric, although harmonic, and thus provide numerous counterexamples to the Lichnerowicz conjecture [START_REF] Damek | A class of nonsymmetric harmonic Riemannian spaces[END_REF]. Despite the lack of symmetry, radial analysis on S is similar to the hyperbolic space case and fits into Jacobi function theory [START_REF] Koornwinder | Jacobi functions and analysis on noncompact semisimple Lie groups[END_REF].

In polar coordinates, the Riemannian volume on S may be written as δ(r)drdσ, where

δ(r) = const. 2 m+1 π n 2 Γ n 2 -1 sinh r 2 m (sinh r) k = 2 n π n 2 Γ n 2 -1 const. cosh r 2 k sinh r 2 n-1
is the common surface measure of all spheres of radius r in S and dσ denotes the normalized surface measure on the unit sphere in s. We shall not need the full expression of the Laplace-Beltrami operator ∆ on S but only its radial part

rad ∆ = ∂ ∂r 2 + n-1 2 coth r 2 + k 2 tanh r 2 δ ′ (r) δ(r) ∂ ∂r
on radial functions and its horocyclic part

(3) ∆f = ∂ ∂z 2 f -Q ∂ ∂z f
on N-invariant functions i.e. on functions f = f (X, Y, z) depending only on z. The Laplacian ∆ commutes both with left translations and with the averaging projector

f ♯ (r) = 1 δ(r) S(e,r) dx f (x) ,
hence with all spherical means

f ♯ x (r) = 1 δ(r) S(x,r)
dy f (y) .

Thus (4) (∆f ) ♯ x = (rad ∆)f ♯ x .
Finally ∆ has a spectral gap. More precisely its L 2 -spectrum is equal to the half-line -∞, -Q 2 4 . Radial Fourier analysis on S may be summarized by the following commutative diagram in the Schwartz space setting [START_REF] Anker | Spherical analysis on harmonic AN groups[END_REF] :

S(R) even H ր ≈ ≈ տ F S(S) ♯ ≈ -→ A S(R) even
Here

Hf (λ) = S dx ϕ λ (x) f (x)
denotes the spherical Fourier transform on S,

Af (z) = e -Q 2 z R m dX R k dY f (X, Y, z)
the Abel transform,

Ff (λ) = R dz e iλz f (z)
the classical Fourier transform on R and S(S) ♯ the space of smooth radial functions

f (x) = f (|x|) on S such that sup r ≥0 (1+ r) M e Q 2 r ∂ ∂r N f (r) < +∞
for every M, N ∈ N. Recall that the Abel transform and its inverse can be expressed explicitly in terms of Weyl fractional transforms, which are defined by

W τ µ f (r) = 1 Γ(µ+M ) +∞ r d(cosh τ s) (cosh τ s -cosh τ r) µ+M -1 -d d(cosh τ s) M f (s)
for τ > 0 and for µ ∈ C, M ∈ N such that Re µ > -M. Specifically,

A = c 1 W 1/2 m/2 • W 1 k/2 and A -1 = 1 c 1 W 1 -k/2 • W 1/2
where c 1 = 2

3 m+k 2 π m+k 2 . More precisely, A -1 f (r) = 1 c 1 -d d(cosh r) k 2 -d d (cosh r 2 ) m 2 f (r)
if n is odd i.e. k is even, and

A -1 f (r) = 1 c 1 √ π +∞ r ds √ cosh s -cosh r -d ds -d d(cosh s) k-1 2 -d d(cosh s 2 ) m 2 f (s)
if n is even i.e. k is odd. Similarly, the dual Abel transform (5)

A * f (r) = f ♯ (r), where f (X, Y, z) = e Q 2 z f (z)
, and its inverse can be expressed explicitly in terms of Riemann-Liouville fractional transforms R τ µ , which are defined by

R τ µ f (r) = 1 Γ(µ+M ) r 0 d(cosh τ s) (cosh τ r -cosh τ s) µ+M -1 d d(cosh τ s) M f (s) for τ > 0 and for µ ∈ C, M ∈ N such that Re µ > -M. Theorem 2.1. The dual Abel transform (5) is a topological isomorphism between C ∞ (R) even and C ∞ (S) ♯ ≡ C ∞ (R) even . Explicitly, A * f (r) = c 2 2 sinh r 2 -m (sinh r) -(k-1) R 1 k/2 cosh • 2 -1 R 1/2 m/2 sinh • 2 -1 f (r) and (A * ) -1 f (r) = 1 c 2 d dr R 1/2 -m/2 • R 1 -k/2+1 sinh • 2 m (sinh •) k-1 f (r)
where

c 2 = 2 n-1 2 Γ( n 2 ) √ π = (n-1)! 2 n-1 2 Γ( n +1 2 )
. More precisely,

(A * ) -1 f (r) = 1 c 2 d dr d d(cosh r 2 ) m 2 d d(cosh r) k 2 -1 sinh r 2 m (sinh r) k-1 f (r)
if n is odd i.e. k is even, and

(A * ) -1 f (r) = 1 c 2 √ π d dr d d(cosh r 2 ) m 2 d d(cosh r) k-1 2 r 0 ds √ cosh r -cosh s sinh s 2 m (sinh s) k f (s) if n is even i.e. k is odd. Proof. Everything follows from the duality formulae R dr Af (r) g(r) = S dx f (x) A * g(x) , +∞ 0 d(cosh τ r) W τ µ f (r) g(r) = +∞ 0 d(cosh τ r) f (r) R τ µ g(r) ,
and from the properties of the Riemann-Liouville transforms, in particular

R τ 1/2 : r ℓ C ∞ (R) even ≈ -→ r ℓ+1 C ∞ (R) even
for every integer ℓ ≥ -1.

Remark 2.2. In the degenerate case m = 0, we recover the classical expressions for real hyperbolic spaces H n (R):

Af (r) = (2π) n-1 2 Γ( n-1 2 ) +∞ r d(cosh s) (cosh s -cosh r) (n-3)/2 f (s) , A * f (r) = c 3 (sinh r) -(n-2) r 0 ds (cosh r -cosh s) n-3 2 f (s) ,
where

c 3 = 2 n-1 2 Γ( n 2 ) √ π Γ( n-1 2 ) = (n-2)! 2 n-3 2 Γ( n-1 2 ) 2 , A -1 f (r) = (2π) -n-1 2 -d d(cosh r) n-1 2 f (r) , (A * ) -1 f (r) = 2 n-1 2 ( n-1 2 )! (n-1)! d dr d d(cosh r) n-3 2 (sinh r) n-2 f (r)
if n is odd and

A -1 f (r) = 1 2 n-1 2 π n 2 +∞ r ds √ cosh s -cosh r -d ds -d d(cosh s) n 2 -1 f (s) , (A * ) -1 f (r) = 1 2 n-1 2 ( n 2 -1)! d dr d d(cosh r) n 2 -1 r 0 ds √ cosh r -cosh s (sinh s) n-1 f (s)
if n is even.

dy ′ U(x ′ , y ′ ) = S(x,s) dx ′ S(y,r) dy ′ U(x ′ , y ′ )
for every x, y ∈ S and r, s > 0.

The proof is similar to the real hyperbolic space case [17, Section II.5.6] once one has introduced the double spherical means

U ♯,♯ x,y (r, s) = 1 δ(r) S(x,r) dx ′ 1 δ(s) S(y,s) dy ′ U(x ′ , y ′ )
and transformed (6) into

(rad ∆) r U ♯,♯ x,y (r, s) = (rad ∆) s U ♯,♯ x,y (r, s) .
Ásgeirsson's Theorem is the following limit case of Theorem 3.1, which is obtained by dividing (7) by δ(s) and by letting s → 0. Corollary 3.2. Under the same assumptions,

S(x,r) dx ′ U(x ′ , y) = S(y,r) dy ′ U(x, y ′ ) .
Given a solution u ∈ C ∞ (S ×R) to the shifted wave equation ( 8)

∂ 2 t u(x, t) = ∆ x + Q 2 4 u(x, t) on S with initial data u(x, 0) = f (x) and ∂ t | t=0 u(x, t) = 0, set (9) U(x, y) = e Q 2 t u(x, t
), where t is the z coordinate of y. Then ( 9) satisfies ( 6), according to [START_REF] Anker | The wave equation on Damek-Ricci spaces[END_REF]. By applying Corollary 3.2 to (9) with y = e and r = |t|, we deduce that the dual Abel transform of t → u(x, t), as defined in [START_REF] Ayadi | Equipartition of energy for the wave equation associated to the Dunkl-Cherednik Laplacian[END_REF], is equal to the spherical mean f ♯ x (|t|) of the initial datum f . Hence u(x, t) = (A * ) -1 f ♯ x (t). By integrating with respect to time, we obtain the solutions

u(x, t) = t 0 ds (A * ) -1 g ♯ x (s)
to [START_REF] Brooks | Non-localization of eigenfunctions on large regular graphs[END_REF] with initial data u(x, 0) = 0 and

∂ t | t=0 u(x, t) = g(x).
In conclusion, general solutions to the shifted wave equation ( 10)

∂ 2 t u(x, t) = ∆ x + Q 2 4 u(x, t) u(x, 0) = f (x), ∂ t | t=0 u(x, t) = g(x)
on S are given by

u(x, t) = (A * ) -1 f ♯ x (t) + t 0 ds (A * ) -1 g ♯ x (s).
By using Theorem 2.1, we deduce the following explicit expressions.

Theorem 3.3. (a) When n is odd, the solution to [START_REF] Bunke | The wave kernel for the Laplacian on the classical locally symmetric spaces of rank one (theta functions, trace formulas and the Selberg zeta function)[END_REF] is given by

u(x, t) = c 4 ∂ ∂t ∂ ∂ (cosh t 2 ) m 2 ∂ ∂ (cosh t) k 2 -1 1 sinh t S(x,|t|) dy f (y) + c 4 ∂ ∂ (cosh t 2 ) m 2 ∂ ∂ (cosh t) k 2 -1 1 sinh t S(x,|t|)
dy g(y) ,

with c 4 = 2 -3m+k 2 -1 π -n-1 2 . (b) When n is even, the solution to (10) is given by u(x, t) = c 5 ∂ ∂|t| ∂ ∂ (cosh t 2 ) m 2 ∂ ∂ (cosh t) k-1 2 B(x,|t|) dy f (y) √ cosh t -cosh d(y,x) + c 5 sign(t) ∂ ∂ (cosh t 2 ) m 2 ∂ ∂ (cosh t) k-1 2 B(x,|t|) dy g(y) √ cosh t -cosh d(y,x)
,

with c 5 = 2 -3m+k 2 -1 π -n 2 .
Remark 3.4. These formulae extend to the degenerate case m = 0, which corresponds to real hyperbolic spaces H n (R) : (a) n odd :

u(t, x) = c 6 ∂ ∂t ∂ ∂ (cosh t) n-3 2
1 sinh t S(x,|t|) dy f (y) + c 6 ∂ ∂ (cosh t) n-3 2
1 sinh t S(x,|t|)
dy g(y) ,

with c 6 = 2 -n+1 2 π -n-1 2 . (b) n even : u(t, x) = c 7 ∂ ∂|t| ∂ ∂ (cosh t) n 2 -1 B(x,|t|) dy f (y) √ cosh t -cosh d(y,x) + c 7 sign(t) ∂ ∂ (cosh t) n 2 -1 B(x,|t|) dy g(y) √ cosh t -cosh d(y,x) , with c 7 = 2 -n+1 2 π -n 2 .
As an application, let us investigate the propagation of solutions u to the shifted wave equation [START_REF] Bunke | The wave kernel for the Laplacian on the classical locally symmetric spaces of rank one (theta functions, trace formulas and the Selberg zeta function)[END_REF] with initial data f, g supported in a ball B(x 0 , R). The following two statements are immediate consequences of Theorem 3.3. Firstly, waves propagate at unit speed. Corollary 3.5. Under the above assumptions,

supp u ⊂ {(x, t) ∈ S | d(x, x 0 ) ≤ |t|+ R } .
Secondly, Huygens' principle holds in odd dimension, as in the Euclidean setting. This phenomenon was already observed in [START_REF] Noguchi | The solution of the shifted wave equation on Damek-Ricci space[END_REF].

Corollary 3.6. Assume that n is odd. Then, under the above assumptions,

supp u ⊂ {(x, t) ∈ S | |t|-R ≤ d(x, x 0 ) ≤ |t|+ R } .
In even dimension, u(x, t) may not vanish when d(x, x 0 ) < |t|-R, but it tends asymptotically to 0. This phenomenon was observed in several settings, for instance on Euclidean spaces in [START_REF] Strichartz | Asymptotic behavior of waves[END_REF], on Riemannian symmetric spaces of the noncompact type [START_REF] Branson | Huygens' principle in Riemannian symmetric spaces[END_REF], on Damek-Ricci spaces [START_REF] Astengo | Huygens' principle and a Paley-Wiener type theorem on Damek-Ricci spaces[END_REF], for Chébli-Trimèche hypergroups [START_REF] Kamel | Huygens' principle and equipartition of energy for the modified wave equation associated to a generalized radial Laplacian[END_REF], ... Our next result differs from [START_REF] Branson | Huygens' principle in Riemannian symmetric spaces[END_REF][START_REF] Kamel | Huygens' principle and equipartition of energy for the modified wave equation associated to a generalized radial Laplacian[END_REF][START_REF] Astengo | Huygens' principle and a Paley-Wiener type theorem on Damek-Ricci spaces[END_REF] in two ways. On one hand, we use explicit expressions instead of the Fourier transform. On the other hand, we aim at energy estimates as in [START_REF] Strichartz | Asymptotic behavior of waves[END_REF], which are arguably more appropriate than pointwise estimates. Recall indeed that the total energy ( 11)

E(t) = K(t) + P(t)
is time independent, where

K(t) = 1 2 S dx |∂ t u(x, t)| 2
is the kinetic energy and

P(t) = 1 2 S dx -∆ x -Q 2 4 u(x, t) u(x, t) = 1 2 S dx |∇ x u(x, t)| 2 -Q 2 4 |u(x, t)| 2
the potential energy. By the way, let us mention that the equipartition of ( 11) into kinetic and potential energies was investigated in [START_REF] Branson | Huygens' principle in Riemannian symmetric spaces[END_REF] and in the subsequent works [START_REF] Kamel | Huygens' principle and equipartition of energy for the modified wave equation associated to a generalized radial Laplacian[END_REF][START_REF] Ayadi | Equipartition of energy for the wave equation associated to the Dunkl-Cherednik Laplacian[END_REF][START_REF] Astengo | Huygens' principle and a Paley-Wiener type theorem on Damek-Ricci spaces[END_REF] (see also [18, Section V.5.5] and the references cited therein).

Lemma 3.7. Let u be a solution to [START_REF] Bunke | The wave kernel for the Laplacian on the classical locally symmetric spaces of rank one (theta functions, trace formulas and the Selberg zeta function)[END_REF] with smooth initial data f, g supported in a ball B(x 0 , R). Then

u(x, t), ∂ t u(x, t), ∇ x u(x, t) are O e -(Q/2)|t|
for every x ∈ S and t ∈ R such that d(x, x 0 ) ≤ |t|-R-1.

Proof. Assume t > 0 and consider the second part

(12) v(x, t) = ∂ ∂ (cosh t 2 ) m 2 ∂ ∂ (cosh t) k-1 2 B(x,t) dy g(y) √ cosh t -cosh d(y,x)
of the solution u(x, t) in Theorem 3.3.b. The case t < 0 and the first part are handled similarly. As B(x 0 , R) ⊂ B(x, t), we have

B(x,t) dy g(y) √ cosh t -cosh d(y,x) = B(x 0 ,R) dy g(y) √ cosh t -cosh d(y,x)
and thus it remains to apply the differential operator

D t = ∂ ∂ (cosh t 2 ) m 2 ∂ ∂ (cosh t) k-1 2
to cosh t -cosh d(y, x)

-1 2 . Firstly ∂ ∂ (cosh t) k-1 2 cosh t -cosh d(y, x) -1 2 = const. cosh t -cosh d(y, x) -k 2 and secondly ∂ ∂ (cosh t 2 ) m 2 cosh t -cosh d(y, x) -k 2 = 0≤j ≤ m 4 a j cosh t 2 m 2 -2j cosh t -cosh d(y, x) -m+k 2 +j ,
for some constants a j . As cosh t -cosh d(y, x) = 2 sinh t+d(y,x)

2 sinh t-d(y,x) 2 ≍ e t ,
we conclude that D t cosh t-cosh d(y, x)

-1
2 and hence v(x, t) are O e -Q 2 t . The derivatives ∂ t v(x, t) and ∇ x v(x, t) are estimated similarly. As far as ∇ x v(x, t) is concerned, we use in addition that sinh d(y, x) = O(e t ) and |∇ x d(y, x)| ≤ 1. This concludes the proof of Lemma 3.7.

Theorem 3.8. Let u be a solution to [START_REF] Bunke | The wave kernel for the Laplacian on the classical locally symmetric spaces of rank one (theta functions, trace formulas and the Selberg zeta function)[END_REF] are O e -Q R(t) and hence tend to 0 as t → ±∞.

The shifted wave equation on homogeneous trees

This section is devoted to a discrete setting, which is similar to the continuous setting considered so far. A homogeneous tree T = T q of degree q +1 > 2 is a connected graph with no loops and with the same number q +1 of edges at each vertex. We shall be content with a brief review and we refer to the expository paper [START_REF] Cowling | An overview of harmonic analysis on the group of isometries of a homogeneous tree[END_REF] for more information (see also the monographs [START_REF] Figà-Talamanca | Harmonic Analysis on Free Groups[END_REF][START_REF] Figà-Talamanca | Harmonic Analysis and Representation Theory for Groups Acting on Homogeneous Trees[END_REF]).

For the counting measure, the volume of any sphere S(x, n) in T is given by

δ(n) = 1 if n = 0, (q +1) q n-1 if n ∈ N * .
Once we have chosen an origin 0 ∈ T and a geodesic ω : Z → T through 0, let us denote by |x| ∈ N the distance of a vertex x ∈ T to the origin and by h(x) ∈ Z its horocyclic height (see Figure 1).

The combinatorial Laplacian is defined on Z by

L Z f (n) = f (n) -f (n+1) + f (n-1) 2 , 0 0 1 2 -1 h ω Figure 1. Upper half-space picture of T 3
and similarly on T by ( 13)

L T f (x) = f (x) -1 q +1 y∈S(x,1) f (y). The L 2 -spectrum of L T is equal to the interval [1-γ, 1+ γ ],
where

γ = 2 q 1/2 + q -1/2 ∈ (0, 1). We have (14) L T f (n) = f (0) -f (1) if n = 0 f (n) -1 q +1 f (n -1) -q q +1 f (n +1) if n ∈ N * on radial functions and L T f (h) = f (h) -q q +1 f (h -1) -1 q +1 f (h +1) = γ q h 2 L Z h q -h 2 f (h) + (1-γ) f (h) (15)
on horocyclic functions.

Again, radial Fourier analysis on T may be summarized by the following commutative diagram

C ∞ (R/τ Z) even H ր ≈ ≈ տ F S(T) ♯ ≈ -→ A S(Z) even Here Hf (λ) = x∈T ϕ λ (x) f (x) ∀ λ ∈ R
denotes the spherical Fourier transform on T,

Af (h) = q h 2 x∈T h(x)=h f (|x|) ∀ h ∈ Z
the Abel transform and

Ff (λ) = h∈Z q iλh f (h) ∀ λ ∈ R
a variant of the classical Fourier transform on Z. Moreover τ = 2π log q , S(Z) even denotes the space of even functions on Z such that

sup n∈N * n k |f (n)| < +∞ ∀ k ∈ N,
and S(T) ♯ the space of radial functions on T such that

sup n∈N * n k q n 2 |f (n)| < +∞ ∀ k ∈ N.
Consider finally the dual Abel transform

A * f (n) = 1 δ(n) x∈T |x|=n q h(x) 2 f h(x) ∀ n ∈ N.
The following expressions are obtained by elementary computations.

Af (h) = q |h| 2 f (|h|) + q -1 q +∞ k=1 q |h| 2 +k f (|h|+ 2k) = +∞ k=0 q |h| 2 +k f (|h|+ 2k) -f (|h|+ 2k + 2) ∀ h ∈ Z
and the dual Abel transform by

A * f (n) = 2 q q +1 q -|n| 2 f (±n) + q -1 q +1 q -|n| 2 -|n| < k < |n| k has same parity as n f (±k) if n ∈ Z * , resp. A * f (0) = f (0). (

b) The inverse Abel transform is given by

A -1 f (n) = +∞ k=0 q -n 2 -k f (n + 2k) -f (n + 2k + 2) = q -n 2 f (n) -(q -1) +∞ k=1 q -n 2 -k f (n + 2k) ∀ n ∈ N
and the inverse dual Abel transform by

(A * ) -1 f (h) = 1 2 q h 2 f (h) + 1 2 q -h 2 f (1) + 1 2 h-1 2 k=1 q h 2 -2k+1 f (h -2k + 2) -f (h -2k) = q 1/2 + q -1/2 2 q h-1 2 f (h) -q -q -1 2 q -h 2 0< k odd < h q k f (k) if h ∈ N is odd, respectively (A * ) -1 f (0) = f (0) and (A * ) -1 f (h) = 1 2 q h 2 f (h) + 1 2 q -h 2 f (0) + 1 2 h 2 k=1 q h 2 -2k+1 f (h -2k + 2) -f (h -2k) = q 1/2 + q -1/2 2 q h-1 2 f (h) -q 1/2 -q -1/2 2 q -h-1 2 f (0) -q -q -1 2 q -h 2 0< k even < h q k f (k) if h ∈ N * is even.
We are interested in the following shifted wave equation on T :

(16) γ L Z n u(x, n) = L T x -1+ γ u(x, n), u(x, 0) = f (x), {u(x, 1) -u(x, -1)}/2 = g(x).
As was pointed out to us by Nalini Anantharaman, this equation occurs in the recent works [START_REF] Brooks | Non-localization of eigenfunctions on large regular graphs[END_REF][START_REF] Brooks | Graph eigenfunctions and quantum unique ergodicity[END_REF]. The unshifted wave equation with discrete time was studied in [START_REF] Cohen | Explicit solutions for the wave equation on homogeneous trees[END_REF] and the shifted wave equation with continuous time in [START_REF] Medolla | The wave equation on homogeneous trees[END_REF].

We will solve ( 16) by applying the following discrete version of Ásgeirsson's mean value theorem and by using the explicit expression of the inverse dual Abel transform. Theorem 4.2. Let U be a function on T such that

(17) L T x U(x, y) = L T y U(x, y) ∀ x, y ∈ T. Then x ′ ∈S(x,m) y ′ ∈S(y,n) U(x ′ , y ′ ) = x ′ ∈S(x,n) y ′ ∈S(y,m) U(x ′ , y ′ )
for every x, y ∈ T and m, n ∈ N. In particular [START_REF] Helgason | Geometric Analysis on Symmetric Spaces[END_REF] x ′ ∈S(x,n)

U(x ′ , y) = y ′ ∈S(y,n) U(x, y ′ ) .
In order to prove Theorem 4.2, we need the following discrete analog of (4).

Lemma 4.3. Consider the spherical means

f ♯ x (n) = 1 δ(n) y∈S(x,n) f (y) ∀ x ∈ T, ∀ n ∈ N. Then (L T f ) ♯ x (n) = (rad L) n f ♯ x (n), where rad L denotes the radial part (14) of L T .
Proof of Lemma 4.3. We have

(L T f ) ♯ x (n) = f (x) -f ♯ x (1) if n = 0, f ♯ x (n) -1 q +1 f ♯ x (n -1) -q q +1 f ♯ x (n +1) if n ∈ N * .
Proof of Theorem 4.2. Fix x, y ∈ T and consider the double spherical means

U ♯,♯ x,y (m, n) = 1 δ(m) x ′ ∈S(x,m) 1 δ(n) y ′ ∈S(y,n) U(x ′ , y ′ ) ,
that we shall denote by V (m, n) for simplicity. According to Lemma 4.3, our assumption (17) may be rewritten as

(19) (rad L) m V (m, n) = (rad L) n V (m, n) .
Let us prove the symmetry 20) is trivial if ℓ = 0 and (20) with ℓ = 1 is equivalent to [START_REF] Intissar | Solution explicite de l'équation des ondes dans un espace symétrique de type non compact de rang 1[END_REF] with m = n = 0. Assume next that ℓ ≥ 1 and that (20) holds for m + n ≤ ℓ. On one hand, let m > n > 0 with m + n = ℓ +1 and let 1 ≤ k ≤ m -n. We deduce from [START_REF] Intissar | Solution explicite de l'équation des ondes dans un espace symétrique de type non compact de rang 1[END_REF] at the point (m -k, n + k -1) that ( 21)

(20) V (m, n) = V (n, m) ∀ m, n ∈ N by induction on ℓ = m + n. First of all, (
V (m -k +1, n + k -1) -V (m -k, n + k) = = q { V (m -k, n + k -2) -V (m -k -1, n + k -1)} .
By adding up ( 21) over k, we obtain

(22) V (m, n) -V (n, m) = q { V (m -1, n -1) -V (n -1, m -1)} ,
which vanishes by induction. On the other hand, we deduce from [START_REF] Intissar | Solution explicite de l'équation des ondes dans un espace symétrique de type non compact de rang 1[END_REF] at the points (ℓ, 0) and (0, ℓ) that V (ℓ +1, 0) = (q +1) V (ℓ, 1) -q V (ℓ, 0), V (0, ℓ +1) = (q +1) V (1, ℓ) -q V (0, ℓ).

Hence V (ℓ +1, 0) = V (0, ℓ +1) by using [START_REF] Koornwinder | Jacobi functions and analysis on noncompact semisimple Lie groups[END_REF] and by induction. This concludes the proof of Theorem 4.2.

Let us now solve explicitly the shifted wave equation ( 16) on T as we did in Section 3 for the shifted wave equation [START_REF] Bunke | The wave kernel for the Laplacian on the classical locally symmetric spaces of rank one (theta functions, trace formulas and the Selberg zeta function)[END_REF] on Damek-Ricci spaces. Consider first a solution u to [START_REF] Figà-Talamanca | Harmonic Analysis on Free Groups[END_REF] with initial data u(x, 0) = f (x) and {u(x, 1)-u(x, -1)}/2 = 0. On one hand, as (x, n) → u(x, -n) satisfies the same Cauchy problem, we have u(x, -n) = u(x, n) by uniqueness. On the other hand, according to [START_REF] Figà-Talamanca | Harmonic Analysis and Representation Theory for Groups Acting on Homogeneous Trees[END_REF], the function U(x, y) = q h(y) 2 u(x, h(y)) ∀ x, y ∈ T satisfies [START_REF] Helgason | Groups and Geometric Analysis (Integral Geometry, Invariant Differential Operators, and Spherical Functions)[END_REF]. Thus, by applying [START_REF] Helgason | Geometric Analysis on Symmetric Spaces[END_REF] to U with y = 0, we deduce that the dual Abel transform of n → u(x, n) is equal to the spherical mean f ♯ x (n) of the initial datum f . Hence

u(x, n) = (A * ) -1 f ♯ x (n) ∀ x ∈ T, ∀ n ∈ N.
Consider next a solution u to [START_REF] Figà-Talamanca | Harmonic Analysis on Free Groups[END_REF] with initial data u(x, 0) = 0 and {u(x, 1)-u(x, -1)}/2 = g(x). Then u(x, n) is an odd function of n and v(x, n) = u(x,n +1) -u(x,n -1) 2 is a solution to [START_REF] Figà-Talamanca | Harmonic Analysis on Free Groups[END_REF] with initial data v(x, 0) = g(x) and {v(x, 1) -v(x, -1)}/2 = 0. Hence

u(x, n) =    2 0<k odd <n v(x, k) if n ∈ N * is even, g(x) + 2 0<k even <n v(x, k) if n ∈ N * is odd, with v(x, n) = (A * ) -1 g ♯ x (n)
. By using Lemma 4.1.b, we deduce the following explicit expressions.

Theorem 4.4. The solution to [START_REF] Figà-Talamanca | Harmonic Analysis on Free Groups[END_REF] is given by

u(x, n) = 1 2 q -|n| 2 d(y,x)=|n| f (y) -q -1 2 q -|n| 2 d(y,x)< |n| |n|-d(y,x) even f (y) + sign(n) q -|n|-1 2 d(y,x)< |n| |n|-d(y,x) odd g(y) ∀ x ∈ T, ∀ n ∈ Z * ,
In other words,

(23) u(x, n) = C n M |n| -M |n|-2 2 f (x) + S n sign(n) M |n|-1 g(x) ,
where

(24) M n f (x) = q -n 2 d(y,x)≤ n n-d(y,x) even f (y) if n ≥ 0 and M -1 = 0.
Remark 4.5. Notice that the radial convolution operators C n and S n above correspond, via the Fourier transform, to the multipliers cos q nλ and sinq nλ sinq λ , where cos q λ = q i λ + q -i λ 2 and sin q λ = q i λ -q -i λ 2 i

.

As we did in Section 3, let us next deduce propagation properties of solutions u to the shifted wave equation [START_REF] Figà-Talamanca | Harmonic Analysis on Free Groups[END_REF] with initial data f, g supported in a ball B(x 0 , N). Corollary 4.6. Under the above assumptions,

(a) u(x, n) = O q -|n| 2 ∀ x ∈ T, ∀ n ∈ Z, (b) supp u ⊂ {(x, n) ∈ T×Z | d(x, x 0 ) ≤ |n|+ N } .
Obviously Huygens' principle doesn't hold for [START_REF] Figà-Talamanca | Harmonic Analysis on Free Groups[END_REF], strictly speaking. Let us show that it holds asymptotically, as for even dimensional Damek-Ricci spaces. For this purpose, define as follows the kinetic energy

K(n) = 1 2 x∈T u(x,n+1) -u(x,n-1) 2
2
and the potential energy ( 25)

P(n) = 1 4 q x,y∈T d(x,y)=2 u(x,n) -u(y,n) 2 2 -(q -1) 2 8 q x∈T |u(x, n)| 2 = q +1 8 x∈T L x -γ u(x, n) u(x, n)
for solutions u to [START_REF] Figà-Talamanca | Harmonic Analysis on Free Groups[END_REF]. Here

Lf (x) = f (x) -1 q (q +1) y∈S(x,2) f (y)
is the 2-step Laplacian on T and γ = (q -1) 2 q (q +1) ∈ (0, 1).

Lemma 4.7. (a) The L 2 -spectrum of L is equal to the interval γ, q +1 q . Thus the potential energy [START_REF] Noguchi | The solution of the shifted wave equation on Damek-Ricci space[END_REF]

is nonnegative. (b) The total energy E(n) = K(n) + P(n) is independent of n ∈ Z.
Proof. (a) follows for instance from the relation

L = q +1 q L T (2 -L T ) and from the fact that the L 2 -spectrum of L T is equal to the interval [1-γ, 1+ γ ]. (b) Notice that the shifted wave equation γ L Z n u(x, n) = L T x -1+ γ u(x, n) amounts to u(x, n +1) + u(x, n -1) = 1 √ q y∈S(x,1) u(y, n) . As x∈T y,z∈S(x,1) u(y, n) u(z, n) = (q +1) x∈T |u(x, n)| 2 + y,z∈T d(y,z)=2 u(y, n) u(z, n) ,
we have on one hand ( 26)

K(n) = q +1 8 q x∈T |u(x, n)| 2 + 1 2 x∈T |u(x, n ±1)| 2 + 1 8 q x,y∈T d(x,y)=2 u(x, n) u(y, n) -1 2 √ q x,y∈T d(x,y)=1 Re u(x, n) u(y, n ±1) .
On the other hand, ( 27)

P(n) = 3q -1 8 q x∈T |u(x, n)| 2 -1 8 q x,y∈T d(x,y)=2 u(x, n) u(y, n) .
By adding up ( 26) and ( 27), we obtain

E(n) = 1 2 x∈T |u(x, n)| 2 + 1 2 x∈T |u(x, n ±1)| 2 -1 2 √ q x,y∈T d(x,y)=1 Re u(x, n) u(y, n ±1)
and we deduce from this expression that

E(n) = E(n ±1).
This concludes the proof of Lemma 4.7.

Remark 4.8. Alternatively, Lemma 4.7.b can be proved by expressing the energies K(n), P(n), E(n) in terms of the initial data f, g and by using spectral calculus. Specifically,

K(n) = 1 8 x∈T (C n+1 -C n-1 ) 2 f (x) f (x) + 1 8 x∈T (S n+1 -S n-1 ) 2 g(x) g(x) + 1 4 Re x∈T (C n+1 -C n-1 )(S n+1 -S n-1 )f (x) g(x)
and

P(n) = 1 4 x∈T (1-C 2 ) C 2 n f (x) f (x) + 1 4 x∈T
(1-C 2 ) S 2 n g(x) g(x)

+ 1 2 Re x∈T (1-C 2 ) C n S n f (x) g(x) .
Here we have used the fact that

q +1 8 ( L -γ ) = 1 8 (3 -M 2 ) = 1 4 (1 -C 2 )
. Hence By considering the corresponding multipliers, we obtain

E(n) = x∈T U + n f (x) f (x) + x∈T V + n g(x) g(x) + 2 Re
U + n = 1 4
(1-C 2 ) , V + n = 1 2 , W + n = 0 , and we conclude that

E(n) = 1 4 x∈T (1-C 2 )f (x) f (x) + 1 2 x∈T |g(x)| 2 = E(0) .
Let us turn to the asymptotic equipartition of the total energy E = E(n).

Theorem 4.9. Let u be a solution to [START_REF] Figà-Talamanca | Harmonic Analysis on Free Groups[END_REF] with finitely supported initial data f and g. Then the kinetic energy K(n) and the potential energy P(n) tend both to E/2 as n → ±∞.

Proof. Let us show that the difference K(n) -P(n) tends to 0. By resuming the computations in Remark 4.8, we obtain

K(n) -P(n) = x∈T U - n f (x) f (x) + x∈T V - n g(x) g(x) + 2 Re x∈T W - n f (x) g(x) , with U - n = 1 8 (C n+1 -C n-1 ) 2 -1 4 (1-C 2 ) C 2 n = -1 4 (1-C 2 ) C 2n , V - n = 1 8 (S n+1 -S n-1 ) 2 -1 4 (1-C 2 ) S 2 n = 1 2 C 2n , W - n = 1 8 (C n+1 -C n-1 ) (S n+1 -S n-1 ) -1 4 (1-C 2 ) C n S n = -1 4 (1-C 2 ) S 2n . As C 2n f ℓ ∞ ≤ q -1 2 q -|n| f ℓ 1 and (1-C 2 )f ℓ 1 ≤ q -q -1 2 + 2 f ℓ 1 ,
the expression

S 2n f (x) (1-C 2 )f (x)
are handled in the same way. This concludes the proof of Theorem 4.9.

Let us conclude with the asymptotic Huygens principle. The proof is similar to the proof of Theorem 3.8.

3 .

 3 Ásgeirsson's mean value theorem and the shifted wave equation on Damek-Ricci spaces Theorem 3.1. Assume that U ∈ C ∞ (S ×S) satisfies (6) ∆ x U(x, y) = ∆ y U(x, y) . Then (7) S(x,r) dx ′ S(y,s)

Lemma 4 . 1 .

 41 (a) The Abel transform is given by

  n+1 -C n-1 ) 2 + 1 4 (1-C 2 ) C 2 n , V + n = 1 8 (S n+1 -S n-1 ) 2 + 1 4 (1-C 2 ) S 2 n , W + n = 1 8 (C n+1 -C n-1 )(S n+1 -S n-1 ) + 1 4 (1-C 2 ) C n S n .

C

  2n f (x) (1-C 2 )f (x) tends to 0. The expressions x∈T V - n g(x) g(x) = 1 2 x∈T C 2n g(x) g(x)andx∈T W - n f (x) g(x) = -1 4 x∈T

Theorem 4 . 10 .

 410 Let u be a solution to[START_REF] Figà-Talamanca | Harmonic Analysis on Free Groups[END_REF] with finitely supported initial data and let (N n ) n∈Z be a sequence of positive integers such thatN n → +∞ N n = o(|n|)as n → ±∞.Then the expressionsx∈T |x|<|n|-Nn | u(x, n)| 2 , x,y∈T |x|,|y|<|n|-Nn d(x,y)=2 | u(x, n)-u(y, n)| 2 , x∈T |x|<|n|-Nn | u(x, n +1)-u(x, n -1)| 2tend to 0 as n → ±∞. In other words, the energy of u concentrates asymptotically inside the spherical shell{ x ∈ T | |n|-N n ≤ |x| ≤ |n|+ N n } .

  with initial data f, g ∈ C ∞

			c (S) and let R = R(t)
	be a positive function such that
		R(t) → +∞ R(t) = o(|t|)	as t → ±∞.
	Then
	d(x,e)<|t|-R(t)	dx |u(x, t)| 2 + |∇ x u(x, t)| 2 + |∂ t u(x, t)| 2
	tend to 0 as t → ±∞. In other words, the energy of u concentrates asymptotically inside the spherical shell

{ x ∈ S | |t|-R(t) ≤ d(x, e) ≤ |t|+R(t)} .

Proof of Theorem 3.8. By combining Lemma 3.7 with the volume estimate vol B e, |t|-R(t) ≍ e Q{|t|-R(t)} as t → ±∞, we deduce that the three integrals

d(x,e)<|t|-R(t) dx |u(x, t)| 2 , d(x,e)<|t|-R(t) dx |∇ x u(x, t)| 2 , d(x,e)<|t|-R(t) dx |∂ t u(x, t)| 2

and by a French-Italian research project (PHC Galilée 25970QB VAMP 2011-2012).


Paul Byrne
email: paul.byrne@univ-rennes1.fr

Redouane Ghoubali

Jacques Miriel

DEVELOPMENT OF A SCROLL COMPRESSOR MODEL FOR PROPANE

Hydrocarbons are today considered as promising alternative substances because of their low environmental impact and their easy implementation in heat pumps. The heat pumping technology is similar to that of HFCs. However, some precautions have to be taken to thwart their flammability. European regulations impose stringent certifications (ATEX, Atmosphères Explosives) on components and an installation outside the buildings for devices having the refrigerant charge of small heat pumps for space heating.

In the frame of a research project on heat pumps for simultaneous heating and cooling (HPS), an R407C prototype of air-source HPS was built and tested. A second prototype is today being designed for propane. A simple and thermodynamically realistic model was developed using EES software. It is an adaptation of an R407C model validated with experimental data. The development of this scroll compressor model working with R407C and its adaptation to thermodynamic properties of propane is presented in this article. A comparison is finally carried out between R407C and R290 scroll compressor models.

INTRODUCTION

The study presented in this article is part of a project on heat pumps for simultaneous heating and cooling (HPS) [START_REF] Byrne | Design and simulation of a heat pump for simultaneous heating and cooling using HFC or CO 2 as a working fluid[END_REF]. A first prototype providing 15 kW heating capacity was built and tested (Byrne et al. 2011a). It can produce hot and cold water in two different water tanks at the same time (simultaneous mode), only hot water (heating mode) or only cold water (cooling mode). The main characteristic of the HPS is to operate using an alternation of modes. During a heating mode in winter, the HPS produces heat and also stores some energy recovered by subcooling of the refrigerant in the cold water tank (not used for cooling during winter). The water tank temperature then rises and the simultaneous mode is launched. During the operation under the simultaneous mode, the heat pump performance is increased because of a higher evaporating temperature using a temperature-controlled water source than using ambient air. Moreover, during that mode, the air evaporator is unused and free for defrosting. A previous study has demonstrated the mechanism of a two-phase thermosiphon defrosting technique (Byrne et al. 2011b). The thermosiphon is created between the water evaporator, being the evaporator of the thermosiphon, and the air evaporator, being the condenser of the thermosiphon. This under-development technique carries out defrosting without stopping the heat production.

Hydrocarbons are positioned as natural alternative refrigerants to halogenated substances. Their environmental impact is very low in terms of global warming. For instance, the GWP of propane (R290) over a 100-year horizon is equal to 3. Performance of hydrocarbons is comparable to that of the fluorinated fluids in the same range of pressures. They are already widely used in household applications like domestic refrigerators. These features imply that hydrocarbons are also envisaged as drop-in substitutes to HFCs in heat pumps using the same technology. A nuance can be formulated concerning precautions to avoid contact between the refrigerant and components likely to generate sparks such as rotating pieces of the compressor or solenoid valves. However, scroll compressors are usually of a higher power range than the compressors for domestic refrigerators. Nevertheless, a study presented by [START_REF] Adamson | Application of hydrocarbon refrigerants with scroll compressors in low temperature cascade systems[END_REF] proves that ATEX-certified scroll compressors for hazardous areas exist. The ATEX certification is necessary to use R290 as a refrigerant.

A second HPS prototype, simplified and industry-oriented, is currently under design. The chosen refrigerant is R290. A simulation study has to be executed to evaluate the operating parameters of the HPS using this new fluid. The complete model describing the components in detail was presented in a paper presented at the 2008 Gustav Lorentzen Conference [START_REF] Byrne | Design and simulation of a heat pump for simultaneous heating and cooling using HFC or CO 2 as a working fluid[END_REF]. A special focus is made here on the compressor model that has been improved since that previous publication in the sense that it is now thermodynamically realistic.

Several studies were already carried out on more or less detailed compressor modelling. Chen et al. (2002a and2002b) worked on a mathematical modelling of scroll compressors. A geometry study was conducted for the suction, compression and discharge chambers of the compressor. It combines the models of refrigerant flows in the suction and discharge processes, radial and flank leakage and heat transfers between the refrigerant and the scroll wraps. [START_REF] Blunier | A new analytical and dynamical model of a scroll compressor with experimental validation[END_REF] developed a similar geometry description of the scroll compressor model. [START_REF] Tseng | Family design of scroll compressors with optimization[END_REF] and [START_REF] Sun | Simulation research on scroll refrigeration compressor with external cooling[END_REF] also propose a design optimization of the scroll compressor using mathematical models. The objective of this paper is to present the development of a simplified compressor model for future performance evaluation of a heat pump for simultaneous heating and cooling working with R290. It is an adaptation of the numerical model developed by Winandy et al. (2002a) using special assumptions, some of which are inspired by [START_REF] Duprez | Modelling of reciprocating and scroll compressors[END_REF]2010). The model was first constructed for a scroll R407C compressor and validated using experimental results and then adapted to refrigerant R290. A comparison is finally carried out to evaluate the mass flow rate and input power discrepancies and discharge temperature differences between R407C and R290 models.

MODEL DEVELOPMENT

This section describes the modelling procedure we used to develop our scroll compressor model and the validation steps by comparison with experimental data obtained on an Emerson Copeland ZB38KCE scroll compressor. It is based on the model of Winandy et al. (2002a) in which the refrigerant state is decomposed. The simulation software is EES [START_REF] Klein | EES -Engineering Equation Solver[END_REF]. The model calculates the refrigerant mass flow rate, the input power and the discharge temperature using the suction temperature and high and low pressures as input variables and internal heat exchange coefficients as parameters.

General equations

Transformations applied to the refrigerant through the compressor can be defined following figure 1. Three steps are involved: suction heating-up (from su to su1), polytropic compression (from su1 to ex1) and discharge cooling down (from ex1 to ex). W loss corresponds to electromechanical losses and W in to the internal compression power delivered to the refrigerant. The refrigerant mass flow rate is given by equation 1 where V s in m 3 /s is the constant swept volume provided by the manufacturer and v su1 is the specific volume at the end of the suction heating-up process.

1 su s v V m = & (1)
Figure 1. Refrigerant transformations through the compressor model

The fictitious wall is in steady state at a constant temperature, like in the publication of [START_REF] Duprez | Modelling of reciprocating and scroll compressors[END_REF], at a mean value between temperatures at points ex and su1 following equation 2.

2 1 su ex w T T T + = (2)
Heat transfers with the fictitious wall are supposed to be balanced and equation 3 can be stated.

0 = - - + amb su ex loss Q Q Q W & & & & (3)
Heat exchange at suction heating-up can be modelled using the DTLM method (Equation 4)

            ⋅ = 1 1 ln su w su w su su su su -T T -T T -T T UA Q & (4)
This thermal capacity is also defined by equation 5.

(

)

su su su h h m Q - ⋅ = 1 & & (5)
The same set of equations is used for discharge cooling down (Equations 6 and 7).

            - - ⋅ = w ex w ex ex ex ex ex T T T T -T T UA Q 1 1 ln & (6) ( ) ex ex ex h h m Q - ⋅ = 1 & & (7)
Heat transfer towards the environment can be calculated using equation 8, where T amb is assumed to be constant and equal to 20 °C. ( )

amb w amb amb -T T UA Q ⋅ = & (8)
During the internal compression, the refrigerant state evolution follows equation 9 using a constant polytropic exponent n.

          -       ⋅ ⋅ ⋅ - = - 1 1 1 1 n n su in LP HP v m LP n n W & & (9)
Finally, the compressor shaft power is given by equation 10.

loss in W W W & & & + = (10)
The other hypotheses made in this model are the following.

• Pressure drops are neglected inside the compressor crankcase.

• Superheat is constant and equal to 5 K.

• Heat transfer coefficients are constant.

• Polytropic exponent is constant and equal to 1.4.

Determination of parameters

In a first simulation, UA su , UA amb and UA ex are unknown. These coefficients are assumed to be constant parameters of the model. Their determination for R407C is carried out using values of the mass flow rate, the shaft power and the discharge temperature measured on our R407C prototype for nominal operating conditions. Table 1 shows the nominal values of main variables and parameters for an evaporating temperature of 0 °C and a condensing temperature of 50 °C at dew points.

Model validation

The model validation is carried out using experimental results obtained on the R407C HPS prototype. The ranges of evaporating and condensing temperatures are respectively -7 °C to +10 °C and 40 °C to 55 °C. They correspond to the temperatures used in the applications of the HPS, which are domestic hot water preheating and space heating and cooling in oceanic climate regions. The following figures show satisfactory accordance between experimental and modelled values.

Figure 2 shows a correct agreement between experimental and modelled mass flow rates. The maximum discrepancy is less than 9.2%.

Figure 3 shows the comparison between experimental and modelled shaft power at the compressor. Lower values of input power are underestimated by the model down to -14.7%. This discrepancy can come from an underestimation of the constant part of electromechanical losses described by Winandy et al. (2002) for the lower values of compression ratio. The maximal discrepancy is 6.2%. In addition, since scroll compressors have fixed built-in volume, the trend shown in figure 3 can be explained by the fact that the isentropic efficiency of the compressor effectively changes in the range of the considered pressure ratios (between 2.25 and 5.52). However we considered a fixed polytropic exponent to calculate the compression work. In a future version, correcting this assumption will participate to the improvement of the model.

Figure 4 shows a correct agreement between experimental and modelled discharge temperature. Temperature differences are included in an interval from -5.2 K to +4.2 K. The discharge temperature will

MODEL ADAPTATION FOR R290

The new model corresponds to the same compressor adapted to thermodynamic properties of propane. In this section, the model adaptation is presented and a comparison between R407C and R290 models is led.

Assumptions for model adaptation to R290

The adaptation of the model consists in recalculating the heat transfer coefficients at suction UA su and discharge UA ex for nominal conditions of evaporation at 0 °C and condensation of 50 °C. The U value is assumed to be calculated using the Nusselt number following equation 11. Nusselt number Nu and conductivity λ depend on the fluid. Nusselt number is given by Dittus-Boelter correlation (equation 12), where parameter m is equal to 0.4 for the suction heating process and 0.3 for the discharge cooling process.

Reynolds number Re depends on the viscosity ν of the fluid (equation 13). In equations 11 to 13, characteristic length L, velocity V (linked to swept volume V s ) and hydraulic diameter D are geometric parameters of the scroll compressor and are the same for R290 as for R407C. The adaptation to propane properties results in equation 14 for UA su and UA ex values. UA amb is independent on the refrigerant and has the same value as for R407C. A second adaptation is carried out on the polytropic exponent n R290 used in equation 9. It is recalculated depending on the ratio of isentropic exponents γ R290 and γ R407C (equation 15).

L U λ ⋅ = Nu (11)
⋅ ⋅ = (12) ν D V ⋅ = Re (13)         ⋅         ⋅         ⋅ = C R R m C R R R C R C R R UA UA
        ⋅ = C R R C R R n n 407 290 407 290 γ γ (15)
The results of the adaptation to propane are reported in table 2.

Comparison of R407C and R290 models

A comparison of mass flow rates, input powers and discharge temperatures is finally carried out between R407C and R290 in the same range of evaporating temperatures (-7 °C to +10 °C) and condensing temperatures (40 °C to 55 °C).

Figure 5 shows R290 mass flow rate compared to the one of R407C using the same compressor. The ratio between R407C and R290 values is nearly constant. The R290 mass flow rate is 45.4% lower than the R407C one. This discrepancy acts in an unfavourable manner for R290 scroll compressor because the heating or cooling capacities are proportional to the mass flow rate. Nevertheless the ratio of latent heat recoverable by condensation at 50 °C of R407C over the one of R290 is in the same proportion than the inverse ratio of mass flow rates. So the heating capacity recoverable by implementing R290 instead of R407C should be somewhat equivalent.

Figure 6 presents the comparison between input powers. The average decrease is 12.4%. This discrepancy acts in a favourable manner concerning the heat pump COP.

Figure 7 shows the difference between discharge temperatures using R407C and R290 models. The difference varies from 6.4 K to 12.0 K. It is a good result in terms of longevity of the compressor and prevention of oil deterioration. Besides, the desuperheating process will be decreased compared to an operation with the same discharge temperatures as the ones of R407C. However, the specific heat of R290 is twice the one of R407C in vapour phase at these pressures. So the heating capacity recoverable during the desuperheating process using R290 will be equivalent or even higher than that of R407C. Q thermal capacity (W) Re Reynolds number (-)

T temperature (K) U heat transfer coefficient (W⋅m -2 ⋅K -1 )

V velocity (m⋅s -1 ) V s swept volume (m 3 ⋅s -1 ) W mechanical work (W)

  Gustav Lorentzen Conference on Natural Refrigerants, Delft, The Netherlands, 2012 4

  Gustav Lorentzen Conference on Natural Refrigerants, Delft, The Netherlands, 2012 5 be used in the global heat pump model as an input variable for the condenser model. Therefore achieving satisfactory accuracy on this output variable is also very important.

Figure 2 .

 2 Figure 2. Comparison between experimental and modelled mass flow rate

Figure 4 .

 4 Figure 4. Comparison between experimental and modelled discharge temperature

Figure 5 .

 5 Figure 5. Comparison between R407C and R290 modelled mass flow rate

Table 1 .

 1 Main variables and parameters at nominal conditions for R407C model

	T ev	T cd	T w	T su	T su1	T ex1	T ex	m &	W &	UA su	UA amb	UA ex
	(°C)	(°C)	(°C)	(°C)	(°C)	(°C)	(°C)	(kg/s)	(W)	(W/K)	(W/K)	(W/K)
	0	50	46.06	5	12.57	83.29	79.56 0.07217 4204	12.38	26.73	8.094

Table 2 .

 2 Values of thermodynamic properties at nominal conditions for R407C and R290 models

	Refrigerant	Pr su (-)	λ λ λ λ su (W.m -1 .K -1 )	ν ν ν ν su (m 2 .s -1 )	Pr ex (-)	λ λ λ λ ex (W.m -1 .K -1 )	ν ν ν ν ex (m 2 .s -1 )	γ γ γ γ (-)	n (-)
	R407C	0.7763	0.01366	7.263•10 -7 0.8832	0.01844	1.955•10 -7	1.187	1.4
	R290	0.8240	0.01818	9.006•10 -7 1.029	0.02377	2.705•10 -7	1.195	1.409

th IIR Gustav Lorentzen Conference on Natural Refrigerants, Delft, The Netherlands, 2012