Amelia Carolina Sparavigna

Solar Alignments of the Planning of Angkor Wat Temple Complex

Keywords: Solar Orientation, Solstices, Azimuth and Zenith, Architectural Planning, Archaeoastronomy

This paper is discussing some solar alignments that can be observed in the planning of the Angkor Wat temple complex in Cambodia. This complex was originally constructed in the early 12th century as a Hindu temple for the Khmer Empire and gradually transforming into a Buddhist temple. To illustrate the solar alignments we use SunCalc.net software and Photographer's Ephemeris on Google Earth satellite images.

Angkor Wat is a large temple complex in Cambodia, originally constructed as a Hindu temple for the Khmer Empire, that became a Buddhist temple in the last part of 12th century. Khmer King Suryavarman II built the temple complex in the early 12th century [START_REF] Jarus | Angkor Wat: History of Ancient Temple, Live Science[END_REF][START_REF] Richter | Recycling Monuments: The Hinduism/Buddhism Switch at Angkor, CyArk[END_REF][START_REF] Higham | Early Mainland Southeast Asia[END_REF], dedicating it to Vishnu. Angkor Wat shows the two basic Khmer temple architectures, that is, the temple-mountain and the galleried temple. In fact, it is planned to represent the Mount Meru with its five peaks, but there are also three rectangular galleries, which are each raising above the other (see Figure 1 on the left) [START_REF] Higham | Early Mainland Southeast Asia[END_REF]. As remarked in [START_REF] Jarus | Angkor Wat: History of Ancient Temple, Live Science[END_REF], the building of Angkor Wat complex was an "enormous undertaking that involved quarrying, careful artistic work and lots of digging"; this undertaking required to solve several engineering problems for the stability of the structures. To support the temples, a tough material, the laterite, was used "which in turn was encased with softer sandstone that was used for carving the reliefs" [START_REF] Jarus | Angkor Wat: History of Ancient Temple, Live Science[END_REF]. Angkor Wat itself is surrounded by a moat that encompasses a perimeter of more than 3 miles; this moat was used for helping stabilize the temple's foundation, "preventing groundwater from rising too high or falling too low" [START_REF] Jarus | Angkor Wat: History of Ancient Temple, Live Science[END_REF] (Figure 1 on the right). On the right, the Angkor Wat surrounded by a moat used for helping stabilize the temple's foundation [START_REF] Jarus | Angkor Wat: History of Ancient Temple, Live Science[END_REF]. Courtesy: Google Earth.

From the Figure 1 on the right we see an evident alignment to cardinal E-W directions, coincident to equinoctial sunrise and sunset directions. However, as observed in [START_REF] Vv | [END_REF], unlike most Angkorian temples, Angkor Wat has its gate oriented to the west. In 1976, researchers of the Michigan University carried a comprehensive analysis of the role of astronomy and cosmology in the planning of the temple [START_REF] Stencel | Astronomy and cosmology at Angkor Wat[END_REF]. The Michigan researchers suggested in their paper that the architects of Angkor Wat had encoded calendrical, historical and cosmological themes into the architectural layout. As shown by [START_REF] Stencel | Astronomy and cosmology at Angkor Wat[END_REF], the researches demonstrated how Angkor Wat's architects had established solar alignments between the temple and a nearby mountaintop shrine that took place during the summer solstice (see Figure 2) [START_REF] Stencel | Astronomy and cosmology at Angkor Wat[END_REF]. In fact, twenty-two possible alignments had been identified and their relationship to bas relief and Hindu time cycles examined. The authors in [START_REF] Stencel | Astronomy and cosmology at Angkor Wat[END_REF] concluded that: 1. The rising sun appears aligned on equinox and solstice days with the western entrance of Angkor Wat; 2. The movements of the moon can be observed from a variety of positions within the temple, and lunar cycles may have been recorded in the three sets of libraries; 3. The bas reliefs of the third gallery can be understood in relation to the movements of the sun; 4. The measurements of the temple appear proportional to calendric and cosmological time cycles. Moreover, in [START_REF] Stencel | Astronomy and cosmology at Angkor Wat[END_REF], it was observed that, in the central tower, the topmost elevation has external axial dimensions of 189.00 Cambodian cubits east-west, and 176.37 Cambodian cubits north-south, with the sum of 365.37 [START_REF] Stencel | Astronomy and cosmology at Angkor Wat[END_REF][START_REF] Kak | The Solar Numbers in Angkor Wat[END_REF]. This last figure is "perhaps the most outstanding number (in the Angkow Wat complex), almost the exact length of the solar year" [START_REF] Stencel | Astronomy and cosmology at Angkor Wat[END_REF][START_REF] Kak | The Solar Numbers in Angkor Wat[END_REF]. However, as shown by Subhash Kak in his very interesting work [START_REF] Kak | The Solar Numbers in Angkor Wat[END_REF], also the other two figures are remarkable: they are evidence of Vedic roots for the division of the solar year in Angkor Wat into two unequal parts. Kak tells that the inequality of the two dimensions with figures corresponding to the two parts of the year, a fact that the Michigan researchers were not able to explain, was not a mere accident but based on the ancient Indian astronomical knowledge [START_REF] Kak | Indian Physics: Outline of Early History[END_REF].

Angkor Wat is also displaying a connection to the zenith passage of the sun, because the temple complex is located in the tropical zone [START_REF] Sparavigna | The Zenith Passage of the Sun and its Role in the Planning of Architectures[END_REF]. A very interesting paper is in fact discussing the importance of zenith passage at Angkor Wat [START_REF] Barnhart | The Importance of Zenith Passage at Angkor, Cambodia[END_REF]; the authors, Edwin Barnhart and Christopher Powell, University of Texas, Austin, in August of 2010 and 2011 investigated this astronomical event on the site discovering, besides specific alignments, that Angkor temples had vertical zenith sighting tubes to observed the zenith passage of the sun or of other bodies of the celestial vault. As we have observed for the planning of some architectures (Mughal gardens, ancient Chinese towns and Buddhist complexes [START_REF] Sparavigna | The Gardens of Taj Mahal and the Sun[END_REF][START_REF] Sparavigna | Solar Azimuths in the Planning of a Nur Jahan's Charbagh[END_REF][START_REF] Sparavigna | Observations on the Orientation of Some Mughal Gardens[END_REF][START_REF] Sparavigna | Sunrise and Sunset Azimuths in the Planning of Ancient Chinese Towns[END_REF][START_REF] Sparavigna | A Solar Orientation in the Town-Planning of Xanadu, Archaeogate[END_REF][START_REF] Sparavigna | The Solar Orientation of the Lion Rock Complex in Sri Lanka[END_REF][START_REF] Sparavigna | On the Alignment of Sanchi Monuments[END_REF]), the architects had created in the orientations and alignments of the layout of their works a reminder of the macrocosmic order, representing on a local symbolic horizon the passage of time and the changing of azimuth and altitude of the Sun at specific moments of the year. To investigate the passage of the sun, in the abovementioned cases and in some others too, we used Sollumis.com or SunCalc.net. The use of this last software (SunCalc) was first proposed in [START_REF] Pankovic | Was Lepenski Vir an Ancient Sun or Pleiades Observatory?[END_REF][START_REF] Pankovic | Giza Pyramids and Taurus constellation[END_REF].

Here we can repeat what we did on the Mughal gardens of Taj Mahal for Angkor Wat too, to find specific alignments. Let us start from conclusion (1) in [START_REF] Stencel | Astronomy and cosmology at Angkor Wat[END_REF], that is: The rising sun appears aligned on equinox and solstice days with the western entrance of Angkor Wat.

The alignment on equinox is evident from the Figure 1. The alignments of the gate to the towers at the corners of the complex on summer and winter solstices are shown in the Figures 3 and4, where we used SunCalc.net software. However, it is possible to observe also an alignment connected to the zenith passage of the sun. The alignment exists if we consider as locus of reference the central tower of the complex, the axis of which is in fact representing the axis Zenith-Nadir of the world. The alignment is given in the Figure 5. If we observe the Figure 6, which is obtained using the Photographer's Ephemeris, at the web page http://app.photoephemeris.com, we can see that it is also connected to corners in the pathway of the garden. As we have shown in [START_REF] Sparavigna | The Zenith Passage of the Sun and its Role in the Planning of Architectures[END_REF][START_REF] Sparavigna | The Solar Orientation of the Lion Rock Complex in Sri Lanka[END_REF][START_REF] Sparavigna | On the Alignment of Sanchi Monuments[END_REF], the zenith passage of the sun was quite important for people of Asia living in the tropical zone. We have discussed in [START_REF] Sparavigna | The Solar Orientation of the Lion Rock Complex in Sri Lanka[END_REF], that the

Figure 1 :

 1 Figure 1: On the left, aerial view of the central structure. Courtesy Shyam tnj, Wikipedia.On the right, the Angkor Wat surrounded by a moat used for helping stabilize the temple's foundation[START_REF] Jarus | Angkor Wat: History of Ancient Temple, Live Science[END_REF]. Courtesy: Google Earth.

Figure 2 :

 2 Figure 2: Alignment to sunrise on Summer Solstice of Angkor Wat and the Phnom Bok hill.Courtesy: SuncCalc.net and Google Earth. In SunCalc, a yellow straight line (here prolonged by the white line) gives the sunrise direction, a orange line the sunset. The curved yellow line is rendering the altitude of the sun.

Figure 3 :

 3 Figure 3: Alignment to sunrise on Summer Solstice. Courtesy: SuncCalc.net and Google Earth.

Figure 4 :

 4 Figure 4: Alignment to sunrise on Winter Solstice. Courtesy: SuncCalc.net and Google Earth.

Figure 5 :

 5 Figure 5: Alignments on a day of zenith passage of the sun (25 April). Courtesy: SuncCalc.net and Google Earth. The azimuth is about 76.2 degrees.

Figure 6 :

 6 Figure 6: Alignments on a day of zenith passage of the sun (25 April). Courtesy: the Photographer's Ephemeris at http://app.photoephemeris.com and Google Earth. The azimuth is about 76.2 degrees.

archaeological complex of Sigiriya, the Lion Rock in Sri Lanka, has its axis oriented to the sunset of day of a zenith passage of the sun. Also the very important Buddhist religious center of Sanchi, has an alignment of stupas to the sunset direction of the day of the zenithal sun. In fact this alignment is to the sunset direction on the summer solstice [16,19], but because Sanchi latitude is close to the Tropic of Cancer, this is also the day of the zenith passage of the sun. Here we have seen that, also in the case of Angkor Wat, the zenith passage of the sun is marked by alignments and by the presence of specific structures for observations [9]. Let us note that, in the case of Angkor Wat, we observe alignments both for sunrise and sunset.


Zehra Gabillon
email: zehra.gabillon@upf.pf&rodica.ailincai@upf.pf

Rodica Ailincai

THE ROLE OF ARTEFACTS AND GESTURES IN CLIL LESSONS

This classroom-based study, which took place in a French Polynesian primary school context, attempts to highlight the role played by social artefacts on the quality of dialogic exchanges and mediation of learning in beginner level young foreign language learners in Content and Language Integrated Learning (CLIL) classes. The CLIL experience described in this study is based on a sociocultural framework and the principles of the Action-Oriented approach (AOA). The study employed classroom observations and video recordings to gather data, which were then analysed qualitatively using discourse analysis. The discourse was split into meaningful segments and studied by focusing on the role social artefacts and gestures played during dialogic exchanges. The analysed data indicated that socially-mediated activity designs which enable the use of artefacts and gestures facilitate the mediation of learning, extend dialogic exchanges and improve the communicative quality of classroom interactions.

Introduction

This study attempts to highlight how artefacts and gestures (hereafter A&G) could extend dialogic exchanges and improve the fluency and communicative quality of classroom interactions of beginner level young foreign language learners in Content and Language Integrated Learning (hereafter CLIL) classes. This classroom-based research study took place in Tahiti, French Polynesia. The participants of the study were primary school children from 9 to 10 years of age whose native language (L1) was French. French Polynesian primary schools implement the French National Curriculum with some adjustments to adapt for local needs and French is the medium of school instruction. In French Polynesian primary schools, English as a Foreign Language was first introduced in 2006 as a pilot project, and it was progressively extended to all French Polynesian elementary schools. The CLIL experience that we describe in this paper is one of a number of multilingual learning projects that are part of the French Polynesian primary school context.

Theoretical Stance and Literature Review

The CLIL experience described in this study is based on the principles of Action-Oriented Learning (AOL) and a sociocultural framework. Within this sociocultural perspective, activity theory [START_REF] Leontiev | The problem of activity in psychology[END_REF][START_REF] Leontiev | Activity, consciousness, and personality[END_REF], as an extension of Vygotsky's sociocultural theory [START_REF] Vygotsky | Mind and society: The development of higher mental processes[END_REF], provides a coherent framework for understanding the role A&G play in CLIL activities.

CLIL is a dual-focused educational approach that uses a foreign, regional or local language or another official state language to teach a school subject. The objectives of CLIL are to develop both language skills and build disciplinary content knowledge. The term CLIL was first introduced to educational literature with the bilingual/multilingual education movement prompted by the European Commission in the late '90s. The 1990s were a period when multilingualism and language education became a key issue in improving communication among European Union (EU) states (European Commission 1995[START_REF]Multilingualism: An asset for Europe and a shared commitment[END_REF], 2008;[START_REF] Network | Content and language integrated learning (CLIL) at school in Europe[END_REF].

Teaching school subject content through the medium of a foreign/second language is not a new approach. CLIL shares similarities with other bilingual education approaches such as content-based instruction (CBI) and immersion programs (French immersion) which are widely used in North American contexts. The success of integrating language and content teaching has been empirically supported by research [START_REF] Cummins | Bilingualism in education: Aspects of theory, research and practice[END_REF][START_REF] Stoller | Content-Based Instruction[END_REF]Stryker and Leaver, 1997). The reports on these positive results on bilingual education have affected CLIL practices positively. Thus, for the last decade, CLIL research and practices have been on the rise not only in Europe but also in other continents (e.g. [START_REF] Admiraal | Evaluation of bilingual secondary education in the Netherlands: Students' language proficiency in English[END_REF][START_REF] Dalton-Puffer | Content-and-language integrated learning: From practice to principles[END_REF][START_REF] Lasagabaster | Foreign language competence in content and language integrated learning[END_REF], Lasagabaster & De Zarobe 2010;[START_REF] Turner | CLIL in Australia: the importance of context[END_REF]. Although the CLIL approach is gaining popularity, it does not yet offer a comprehensive educational model based on a robust theoretical frame. Most of the CLIL practices are based on educational policies, pragmatic pedagogies, and curricular guidelines [START_REF] Dalton-Puffer | Content-and-language integrated learning: From practice to principles[END_REF].

CLIL also shares some of the principles of the Action-Based Learning (ABL) approach. ABL is an approach recommended by European Council publications, more specifically by the Common European Framework of Reference (CEFR) for Languages, which is a reference text for foreign language teaching in a European context (Council of Europe, 2001). This approach views language learning as a situated activity and places the learner at the centre of the learning process. This means that the learner is viewed as an active agent in a socioculturally mediated activity. The action-based approach is also related to other approaches, such as content-based and task-based instruction [START_REF] Adler | Improving the quality of accounting students' learning through actionoriented learning tasks[END_REF][START_REF] Lier | Action-based teaching, autonomy and identity[END_REF].

Since the 1990s there has been an increasing interest in applying sociocultural theories to L2 learning. Theorists who employ a sociocultural stance regard language learning principally in social terms [START_REF] Lantolf | Sociocultural theory and second language acquisition[END_REF][START_REF] Lantolf | Sociocultural theory and L2: State of the art[END_REF]. The key idea of sociocultural theory (SCT) is "…the centrality of language as a 'tool for thought' or means of mediation, in mental activity..." (Mitchell & Myles, 2004, p.194). From this stance, learning is viewed as a socially mediated activity that is contingent on face-to-face interaction and joint attention [START_REF] Mitchell | Second language learning theories[END_REF]. According to Vygotsky, knowledge is first constructed on social planes through collaborative interaction, and then it is appropriated on personal planes. [START_REF] Vygotsky | Mind and society: The development of higher mental processes[END_REF] proposed that engaging in full social interaction with more experienced others (such as parents, teachers, and peers) enables the child to construct knowledge that s/he is not capable of doing alone. To explain this phenomenon Vygotsky introduced the concept of the Zone of Proximal Development (ZPD). ZPD refers to the difference between what a learner is capable of doing without guidance and what s/he is capable of doing with guidance.

Within this sociocultural perspective, the process of supportive dialogue, which directs the others' (learners' or peers') attention to the key features of learning using successive steps, is known as scaffolding. Scaffolding in language learning helps learners to notice crucial language input. The term was first introduced to educational literature by Wood, Brunner and Ross (1976) and later developed by [START_REF] Bruner | The role of dialogue in language acquisition[END_REF] as an extension of Vygotsky's concepts of ZPD and social mediation.

Mitchel and Myles explained the ZPD as the domain of knowledge or skill in which the learner cannot function independently, and could only achieve the desired outcome if s/he receives scaffolded help. In the schematic representation of ZPD shown in Figure 1 we present a schematic representation of the notion of ZPD that we applied to a formal educational context. The amount of help each learner needs may vary depending on the domain. In some content or language areas, some learners may need more scaffolded help than others. That is to say, a learner may need little scaffolding in a foreign language class whereas the same learner may need a lot more scaffolding in a science class and it can be vice versa for another student [START_REF] Gabillon | CLIL: A Science lesson with breakthrough level young EFL learners[END_REF]. The difference between learners could be used in an efficient way by involving more competent learners in the scaffolding process. Classroom activities which enable collaborative exchange in classroom settings may allow learners to provide this help in a natural way. SCT views human made material and symbolic objects as artefacts.

According to [START_REF] Swain | Sociocultural theory in second language education: An introduction through narratives[END_REF] all artefacts, whether material (e.g. table, pencil, books) or symbolic (e.g. language, concepts, belief systems ) have the potential to become mediating means. Activity Theory (AT) [START_REF] Leontiev | The problem of activity in psychology[END_REF][START_REF] Leontiev | Activity, consciousness, and personality[END_REF][START_REF] Engeström | Learning by expanding an activity -theoretic approach to developmental research[END_REF], which is an extension of Vygotsky's sociocultural theory, provides a robust theoretical framework for researchers who seek to understand how artefacts mediate learning. [START_REF] Swain | Sociocultural theory in second language education: An introduction through narratives[END_REF] maintain that individuals interact with the symbolic and material world around them, and that all forms of mental activity are mediated by these artefacts.

The Study Our study consists of several work packages. Prior to the present study, between 2012 and 2013we did preliminary research on CLIL. The aim of this experimental study was to investigate if CLIL was possible with breakthrough (beginner) level young learners through irregular 30-minute CLIL lessons. The results we obtained from this preliminary work package suggested that CLIL is possible with young beginner level foreign language learners but requires a rich extra-linguistic context and socially mediated activity designs (see [START_REF] Gabillon | CLIL: A Science lesson with breakthrough level young EFL learners[END_REF]. Following the results obtained from this preliminary work we designed a research activity to observe the role played by A&G in CLIL classes with beginner level learners, which we will describe in this paper.

The study described in this paper took place from 2012 to 2014 in two elementary state schools in Tahiti. The participants were 9-10 year-old elementary school children with breakthrough level English. The learners had a maximum of two years' English language learning experience and participated in science lab experiments that were carried out through the medium of English. A total of 30 children participated in our study. The study was carried out with the participation of two primary school teachers and two lecturers from the University of French Polynesia: an assistant professor who specialises in foreign language and second language acquisition and an associate professor in Education. For this study we observed three science lab lessons. These CLIL lessons used English as a medium for instruction. The lessons lasted between 25-30 minutes each and they were in the form of a group activity that involved 9 to 11 children in each group.

In our study we viewed classroom actions as social activity where learning is mediated by collaborative dialogue (see Figure 2).

Figure 2. The activity design used in our CLIL lessons

Thus, we conceived CLIL lessons that employed socially-mediated activity designs to facilitate collaborative dialogue and to enable learning through joint attention. The aim of the collaborative dialogue was to provide learners with mediation (teacher and peer) to support both language development and disciplinary content learning. The science experiment activities aimed at creating a naturalistic learning setting and had recourse to a variety of sensory input (e.g. seeing, touching, smelling etc.). The activities also emphasised providing learners with experiential learning and hands-on experience to help them make meaning from direct experience. The CLIL teachers used simplified language forms and vocabulary and they had recourse to A&G to scaffold learning. In our study we considered A&G as one variable. Most of the time A&G were used together and separate analysis seemed difficult to carry out.

As I mentioned earlier, our previous CLIL experiences indicated a mediating effect of A&G on learning. Thus in this study we decided to shift our focus of observation and analysis on the use of A&G. Our foremost aim was to observe the role played by A&G during dialogic exchanges. The aims of this study can be summarised as follows: a) How did A&G mediate learning? b) How did the use of A&G influence dialogic exchanges?

In CLIL classes, especially with young learners who have little foreign language experience, the teachers need to make linguistic modification to keep learners interested and to help them understand content related notions. The primary objective of a CLIL lesson is to carry out meaningful interaction through which learners can learn new notions. The focus in a CLIL lesson may alternate between content learning and target language use at different moments of the lesson. This constant shift on emphasis may affect the amount of attention the teacher pays to the linguistic adjustments. Using artefacts and gestures can provide the teacher with a tool to make up for the learners' insufficient language competence to scaffold learning new concepts.

Methodology

In our study, we used discourse analysis to examine the data obtained from the CLIL classes we observed. Today face-to-face classroom interaction is viewed as a pivotal element of knowledge construction and classroom dynamics. Discourse analysis has been exponentially applied to the analysis of classroom exchanges to further the understanding of student learning and the role of dialogic exchanges in classroom settings. The corpus for this study was collected from three identical 25-to-30-minute CLIL lessons from three different groups of learners. Video recordings were used to obtain data and lessons were recorded in their entirety. The fact that the group sizes were small (9 to 11 pupils in each), allowed the researchers to have an uninterrupted view of each learner and record not only the linguistic data but also the nonverbal elements, (e.g. use of A&G, and facial expressions) of the phenomena observed. The videotaped data were then transcribed and the transcribed data were analysed both qualitatively and quantitatively using descriptive statistics.

The analysis procedure consisted of a series of stages involving different qualitative data processing techniques. First, the transcribed data were examined and re-examined to look for patterns and links within and across utterances in order to understand how, how often and why A&G were used and the consequences they produced. The steps that we used during our data analysis procedure can be summarised as follows:

1. During the conceptualisation stage after careful assessment of the transcribed data, we observed some patterns and links between the use of A&G, knowledge-building and knowledge/information-sharing processes.

2. After the identification of persisting patterns, we coded the data into categories. In these categories, we looked at the occurrences of A&G, how they were used and the effects they produced.

3. Then we split the data into smaller, manageable meaningful segments in order to be able to explain the phenomenon in a more methodical way. We defined the boundaries of discourse segments through identification of an opening move, which marked the beginning of a topic or a new action, and through identification of a framing move, which indicated the end of an exchange.

4. We re-grouped and labelled the data segments utilising a conversation analysis model similar to the one offered by Kerbrat-Oreccioni (1998). Although the analysis model proposed by Kerbrat-Oreccioni views conversation analysis as probing only linguistic components, the analysis model that we employed also integrated extra linguistic elements such as A&G. After splitting and re-grouping the data, we labelled the data segments using the following categories:

Analysis and discussion

The analysis of the corpus we gathered indicated that A&G helped classroom exchanges to continue without a break and added a communicative quality to the dialogic exchanges. Almost all of the exchanges we analysed corresponded to the maxims proposed by [START_REF] Grice | Syntax and semantics[END_REF]. Due to space limitations, in this paper we can provide only a few of the analysed discourse segments, as examples.

Extract 1 illustrates how using A&G by the learners and the teacher) mediated learning, contributed to carrying out of tasks and extended the exchange.

Extract 1

Extended Exchange (EE) EE 1. T: Ok. Do the experiment again (passes the jar to P5). Take some soap. Put it in water.

- Note. T=Teacher, P5= Pupil 5, Ps=: Pupils, EE= Extended Exchange, A&G= artefacts and gestures.

In Extract 1 above we observe A&G mediated collaborative exchange through providing scaffolding during the moments of instruction giving (moves 1&3) and comprehension checks (move 5). This particular extract also illustrates how learners utilised A&G to provide scaffolding during their explanations to clarify and justify the meaning of their utterances (move 8). Explanations given by the learner in move 8 could probably not have been possible without the use of the artefacts available within the vicinity of the learner. All of the moves in this extract conform to the maxims proposed by [START_REF] Grice | Syntax and semantics[END_REF].

Extract 2 below has a clear pattern where a teacher is encouraging a shy learner to take part in a classroom activity. This short exchange is a good example to illustrate how a Truncated Exchange (TE) could be extended using A&G. In this particular example the learner was unable to respond to the teacher's question because of a language structure which the pupil apparently had difficulty understanding. The teacher re-launched the exchange by modifying her language and supporting the linguistic modification with the use of A&G (move 3). The videotaped data clearly illustrated that the teacher's instant recourse to the objects and gestures contributed to the learner's comprehension and the natural flow of the dialogic exchange. The time interval between the teacher and the learner's moves was natural and the learner's reaction was free from any frustration. Although the exchange was short and the linguistic content (lexical and grammatical) modest, the exchange was linguistically appropriate and corresponds to the maxims of a social interaction. This simple exchange illustrates how the robust pragmatic dimension of an exchange could make up for linguistic simplicity. The situation was appropriate for the use of short language forms and pragmatic strategies both functional and interactional. Next, Extract 3 exemplifies how the teacher used supportive dialogue to direct the learners' attention to the key concepts of learning using A&G in successive steps. In this particular exchange, the teacher used A&G to mediate student learning and to encourage learner participation while clarifying the 'solubility' concept. Although the exchange had non-verbal elements and one-word utterances, evaluation of the discourse using Grice's maxims illustrated that the exchange complies perfectly well with the cooperative principles of an exchange.

Results

The overall results obtained from this CLIL research data indicated that the activity design that we used contributed to fostering meaningful use of A&G, which in return: a) provided scaffolding for learning, b) extended dialogical exchanges, and c) contributed to the amelioration of communicative quality and the fluency of the dialogic classroom exchanges.

A post task discussion with the learners in their mother tongue plainly indicated that the learners understood the scientific concepts conveyed through collaborative exchanges, regardless of their breakthrough level English. The classroom tasks were executed almost without any need for recourse to the learners' mother tongue (four moves in single word translations) and this gap was filled with the extensive use of A&G. Our overall data analysis has demonstrated that 58% of the collaborative dialogue (478 moves) in three CLIL lessons observed was in the form of EE (see Figure 3). Whenever there was a communication break the exchange was re-launched through use of the objects within the vicinity and through the wide use of extra-linguistic elements such as gestures. The analysis of the data clearly indicated that without the use of A&G, the majority of the exchanges would have been truncated exchanges with constant communication breaks. Although the learners' target language level was very low, the activities were carried out via natural dialogic exchanges and new concepts were constructed by using successive scaffolding with an extensive support bestowed by A&G. In some cases, although the learners did not have the necessary language skills, they were able to respond to the demands of the exchange just giving a non-verbal response, such as demonstrating with the use of the artefacts (e.g. filling the jar when required following teachers' instructions and mixing ingredients), or just pointing, and nodding. They also answered the demands of the interactions by using A&G to support their language in situations where complex structures which the learner had not yet mastered were required (e.g. "bubbles here…" "…look not clear". The discourse analysis that we carried out indicated that in the majority of the cases, the learners had recourse to A&G to make their meanings clear.

Conclusion and Future Research Directions

The results we obtained via this research study clearly indicated that the use of A&G can extend dialogic exchanges and improve the communicative quality of classroom interactions.

In this particular research study, we were limited to two elementary schools. In our future research projects, we intend to investigate this phenomenon on a broader scale through gathering larger CLIL data from different Polynesian elementary schools. Starting from January 2015 the CLIL project will become a constituent of a larger project, which will investigate multilingual practices within the French Polynesian primary school context. The project will involve the ensemble of French Polynesian primary schools in five archipelagos that span an area as large as Europe. This project is financed by the Ministere des Outre-Mers (MOM), which is a French administrative department responsible for coordinating government actions in the overseas territories. Our part in this research project consists in building a corpus from CLIL practices and regular English classrooms from different elementary school classes. This corpus will later be analysed and used to inform our future projects. We intend to employ the data obtained via this research project to carry out comprehensive analysis methods to better understand the foreign language teaching practices in the French Polynesian context. We also intend to share the classroom implications of these experiences during pre-service and in-service education programmes.

Figure 1 .

 1 Figure 1. A Schematic representation of the notion of Zone of Proximal Development (ZPD) in an educational context

a)

  Limited exchange (LE): This is a short exchange of a maximum of two moves that contains either two verbal interventions, or a verbal and a non-verbal intervention, which indicates that the message has been understood and the learner is responding to it. b) Truncated Exchange (TE): This type of exchange demonstrates that the learner has not understood the message and is unable to respond. A truncated exchange consists of an opening move and a failed move which results in closure of the exchange. c) Re-launched Exchange (RE): This sort of exchange describes an exchange type that attempts to restart a truncated exchange. d) Truncated-Re-launched-Failed Exchange (T-R-FE): Describes an exchange type which fails after an attempt to re-launch a truncated exchange. e) Extended Exchange (EE): An exchange type that contains more than two learner moves on the same topic which indicates that the interlocutors are able to communicate on the subject. f) Truncated-Re-launched-Extended Exchange (T-R-EE): describes an extended exchange type which is successfully re-launched after it is truncated. 5. Paul Grice's (1975) conversational maxims were also applied to evaluate conversational quality of the exchanges: a) The Maxim of Quantity: Giving only the necessary amount of information -not too much or too little. b) The Maxim of Quality: Only speaking the truth-not knowingly giving false information. c) The Maxim of Relation: Being relevant to the current topic of conversation. d) The Maxim of Manner: Avoiding ambiguity or obscurity in your speech.

  T: Ok! Shall we start with Ode? Which one did you do Ode? 2. P7: (No answer.) RE 3. T: Which one is yours? Which one did you do? Was it sand (shows the sand)? Was it sugar? (shows the sugar). -(A&G) scaffolding repair EE 4. P7: Rice 5. T: Rice, ok take it. Show it to your friends. 6. P7: (She takes the jar and shows it) -(A&G) non-verbal response Note. T=Teacher, P7=Pupil 7, TE=Truncated Exchange, RE=Re-Launched Exchange, EE= Extended Exchange, A&G= artefacts and gestures.

Figure 3 .

 3 Figure 3. Learners' use of A&G to scaffold dialogic exchanges CLIL classrooms

Figure 4 .

 4 Figure 4. Learners' use of A&G to mediate dialogic exchanges

  Figure5below provides an overall view of the teachers' use of A&G during which took place during the CLIL lessons we observed. The data indicated that in most of the cases A&G were used as a scaffolding tool to help the teacher elicit learner responses, to give instructions and to build new concepts.

Figure 5 .

 5 Figure 5. Teachers' use of A&G to mediate dialogic exchanges and learning in CLIL classrooms


Mohammed Benalili
email: mbenalili@mail.univ-tlemcen.dz

Ali Zouaoui
email: 2014zouaoui@gmail.com

SINGULAR ELLIPTIC EQUATION INVOLVING THE GJMS OPERATOR ON COMPACT RIEMANNIAN MANIFOLD

Keywords:

In this paper we consider a singular elliptic equation involving the GJMS (Graham-Jenne-Mason-Sparling) operator of order k on n-dimensional compact Riemannian manifold with 2k < n. Mutiplicity and nonexistence results are established.

Let (M, g) be an n-dimensional Riemannian manifold. The k-th GJMS operator (Graham-Jenne-Mason-Sparling, see ( [START_REF] Graham | Conformally invariant powers of the Laplacien. I.Existence[END_REF]) P g is a differential operator defined for any integer k if the dimension n is odd, and 2k ≤ n otherwise. In the following, we will consider the case 2k ≤ n. P g is of the form

P g = ∆ k + lot
where ∆ = -div g (∇) is the Laplacian-Beltrami operator and lot denotes the lower terms. One of the fundamental property of P g is its behavior with respect to conformal change of metrics: for ϕ ∈ C ∞ (M ) , ϕ > 0 and g = ϕ 4 n-2k g a conformal metric to g, ϕ n+2k n-2k P g u = P g (ϕu) . P g is self-adjoint with respect to the L 2 -scalar product. To P g is associated a conformal invariant scalar function denoted Q g and is called the Q-curvature. For k = 1, the GJMS operator is ( up to a constant ) the conformal Laplacian and the corresponding Q-curvature function is simply the scalar curvature. For k = 2, the GJMS operator is the Paneitz operator introduced in ( [START_REF] Paneitz | A quartic conformally differential operator for arbitrary pseu-Riemannian manifolds[END_REF]). For 2k < n, the Q-curvature is Q g = 2 n-2k P g [START_REF] Benalili | On the singular $Q$-curvature type equation[END_REF]. Many works was devoted the Q-curvature equation in the last two decades (see [START_REF] Benalili | On the singular $Q$-curvature type equation[END_REF], [START_REF] Djadli | Existence of conformal metrics with constant $Q$-curvature[END_REF], [START_REF] Hebey | Coercivity and Struwe's compactness for Paneitz type operators with constant coefficients[END_REF], [START_REF] Robert | Admissible Q-curvatures under isometries for the conformal GJMS operators[END_REF]). Many authors investigated the interactions of conformal methods with mathematical physic which led them to study the Einstein-scalar fields Lichnerowiz equations (see [START_REF] Hebey | A variational Analysis of Einstein-scalar Field Lichnerowicz Equations on compact Riemannian Manifolds[END_REF], [START_REF] Hebey | The Lichnerowicz equation in the closed case of the Einstein-Maxwell theory[END_REF], [START_REF] Ngô | Existence results for the Einstein-scalar field Lichnerowicz equation on compact Riemannian manifolds[END_REF], [START_REF] Premoselli | Effective multiplicity for the Einstein-scalar field Lichnerowicz equation[END_REF], [START_REF] Premoselli | The Einstein-scalar field equation constraint in the positive case[END_REF],). These methods have been extended to scalar fields Einstein-Licherowicz type equation involving the Paneitz operator, (see [START_REF] Maalaoui | On a fourth order Lichnerowicz type equation involving the Paneitz-Branson operator[END_REF]). In this work we analyze an Einstein-Lichnerowicz scalar field equation containing the k-th order GJMS operator on a Riemannian n-dimensional manifold with 2k < n. Our work is organized as follows: in a first step we show the existence of a solution to equation (1.1) obtained by means of the mountain-pass theorem. In the second step we prove, by the Ekeland lemma, the existence of a second solution to equation (1.1). In the last section we give a result of nonexistence of solution.

1. Existence of a first solution Let (M, g) be an n-dimensional compact Riemannian manifold and k ∈ N * , with 2k < n. Consider the following equation (1.1)

P g (u) = B (x) u 2 -1 + A(x) u 2 +1 + C(x) u p u > 0
where 2 = 2n n-2k , P g is the k-th GJMS operator and p > 1. In all the sequel of this paper we assume that the operator P g is coercive which allows us ( see Proposition 2, [START_REF] Djadli | Existence of conformal metrics with constant $Q$-curvature[END_REF]) to endow H 2 k (M ) with the following appropriated equivalent norm u = M u.P g (u)dv g . So we deduce from the coercivity of P g and the continuity of the inclusion

H 2 k (M ) ⊂ L 2 (M ), the existence of a constant S > 0 such that (1.2) u 2 ≤ S u
where 2 = 2n n-2k . We assume also that the GJMS operator has a positive Green function.

Proposition 1. Suppose that the metric g is Einstein with positive scalar curvature of dimension n > 2k, then the GJMS P g admits a Green positive function.

Proof. On n-dimensional Einstein manifold, the GJMS operator of order k (see [START_REF] Graham | Conformally invariant powers of the Laplacien. I.Existence[END_REF]) is given by

P g = k l=1 (∆ -c l Sc) where c l = (n+2l-2)(n-2l) 4n(n-1)
, Sc is the scalar curvature. If the scalar curvature is positive it is well known that the operator ∆ -c l Sc has a positive Green function. Denote by L l = ∆ -c l Sc, l = 1, ..., k; by definition of the Green function of L l we know that for all u ∈ C ∞ (M ),

(L l u) (x) = M G l+1 ((x, y) (L l+1 L l u) (y)dv g (y) . So u(x) = M G l (x, z) (L l u) (z) dv g (z) + 1 V ol(M ) M u(x)dv g (x) = M M G l (x, z)G l+1 ((z, y)dv g (z) (L l+1 L l u) (y)dv g (y)+ 1 V ol(M ) M u(x)dv g (x)
and letting

G l,l+1 (x, y) = G l * G l+1 (x, y) = M G l (x, z)G l+1 ((z, y)dv g (z).
And by induction, we get

u(x) = M G 1 * ... * G k (x, y)P g (y)dv g (y)
.

Thus P g admits a positive Green function.

In this section we establish the following theorem.

Theorem 1. Let (M, g) be a compact Riemannian manifold with dimension n > 2k and A > 0, B > 0, C > 0 are smooth functions on M . Suppose moreover that the operator P g is coercive and have a positive Green function.

If there exists a constant C (n, p, k) > 0 depending only on n, p , k such that

(1.3) ϕ 2 2 M A(x) ϕ 2 dv g ≤ C (n, p, k) Smax x∈M B(x) 2+2 2-2
and

(1.4) ϕ p-1 p -1 M C(x) ϕ p-1 dv g ≤ C (n, p, k) Smax x∈M B(x) p+1 2-2
for some smooth function ϕ > 0, then equation (1.1) admits a smooth solution.

To prove the existence of solutions to (1.1), we consider the following -approximating equations (1.5)

P g (u) = B (x) (u) 2 -1 + A (x) u (ε + u 2 ) 2 +1 + C(x)u (ε + u 2 ) p+1 2
where 2 = 2 2 , p > 1.Which gives us a sequence (u ε ) ε of solutions to (1.5). The solution of equation (1.1) is then obtained as the limiting of (u ε ) ε . Following Hebey-Pacard-Pollak ( [START_REF] Hebey | A variational Analysis of Einstein-scalar Field Lichnerowicz Equations on compact Riemannian Manifolds[END_REF]), to get rid of negative exponents, we consider the energy functional associated to (1.5) defined by, for any ε > 0

I ε (u) = I 1 (u) + I (2)
ε (u) where I (1) : H 2 k (M ) → R is given by

I (1) (u) = 1 2 M uP g (u)dv g - 1 2 M B (x) u + 2 dv g and I (2) ε : H 2 k (M ) -→ R is I (2) ε (u) = 1 2 M A (x) ε + (u + ) 2 2 dv g + 1 p -1 M C(x) ε + (u + ) 2 p-1 2 dv g .
It is easy to check the following inequality

(1.6) Φ ( u ) ≤ I (1) (u) ≤ Ψ ( u ) with Φ (t) = 1 2 t 2 -S max M B(x) t 2 2
and

Ψ (t) = 1 2 t 2 + S max M B(x) t 2 2 .
The Φ (t) is increasing on [0, t 0 ] and decreasing on ]t 0 , +∞[, where

(1.7)

t 0 = Smax x∈M B(x) 2k-n 4k and (1.8) Φ (t 0 ) = 1 2 - 1 2 Smax x∈M B(x) 2k-n 2k = k n t 0 . Lemma 1. Let θ > 0 such that a 2 2 2 < θ 2 < 1 2 (n -k) = a 2 2
where

a = 1 (2 (n -k)) 2 2
and put

t 1 = θt 0 .
Then we have the following inequality

(1.9) Ψ (t 1 ) ≤ θ 2 2 + 2 2 -2 Φ (t 0 ) < 1 2k Φ (t 0 ) .
Proof. In fact

Ψ (t 1 ) = 1 2 t 2 1 + Smax x∈M B(x) t 2 1 2 = θ 2 1 2 t 2 0 + Smax x∈M B(x)θ 2 -2 t 2 0 2 . Since (1.10) t 2 0 Smax x∈M B(x) = t 2 we get Ψ (t 1 ) = θ 2 1 2 t 2 0 + θ 2 -2 2 t 2 0 = θ 2 t 2 0 1 2 + θ 2 -2 2 ≤ θ 2 t 2 0 1 2 + n -2k 2n ≤ θ 2 t 2 0 2n -2k 2n
≤ (n -k) θ 2 t 2 0 n . And since 2 + 2 2 -2 = n -k k and Φ (t 0 ) = k t 2 0 n we infer that Ψ (t 1 ) ≤ θ 2 2 + 2 2 -2 Φ (t 0 ) < 1 2k Φ (t 0 ) .
Now we check the Mountain-Pass lemma conditions for the functional I ε .

Lemma 2. The functional I satisfies the following conditions i) There exit ρ > 0 and t 0 > 0 such that

I (u) ≥ ρ for every u ∈ H 2 k (M ) with u = t 0 ii)There exits is v ∈ H 2 k (M ) such that v = t 2 > t 0 and I (v) < 0.
Proof. Let ϕ ∈ C ∞ (M ), ϕ > 0 on M and without loss of generality we can assume ϕ = 1. Put

(1.11) C (n, p, k) = (2k -1) θ 2 +p 4n ≤ C 1 (n, k) = (2k -1) θ 2 4n .
The inequality (1.3) becomes

(1.12) 1 2 M A (x) (t 1 ϕ) 2 dv g ≤ 2k -1 4k Φ (t 0 ) .
Indeed, we have

(1.13) Φ(t o ) = k n t 2 o .
and by (1.11), we get

1 2 M A(x) (t 1 ϕ) 2 dv g ≤ C 1 (n, k) t 2 o θ 2 S. max M B (x) 2+2 2-2 = 2k -1 4k Φ(t o ).
Analogously and by putting

C 2 (n, p, k) = (2k -1) θ p-1 4n we obtain (1.14) 1 p -1 M C(x) (t 1 ϕ) p-1 dv g ≤ 2k -1 4k Φ(t o ).
Let us prove know (i ). By relations (1.6), (1.9), (1.12) and (1.14), we infer that

I (t 1 ϕ) ≤ Ψ ( t 1 ϕ )+ 1 2 M A (x) ε + (t 1 ϕ) 2 2 dv g + 1 p -1 M C(x) ε + (t 1 ϕ) 2 p-1 2 dv g (1.15) ≤ Ψ (t 1 )+ 1 2 M A (x) ε + (t 1 ϕ) 2 2 dv g + 1 p -1 M C(x) ε + (t 1 ϕ) 2 p-1 2 dv g ≤ Φ (t 0 ) .
Again from (1.6), we deduce that

I (t 0 ϕ) ≥ Φ (t 0 ) + 1 2 M A (x) ε + (t 0 ϕ) 2 2 dv g + 1 p -1 M C(x) ε + (t o ϕ) 2 p-1 2
dv g and since A and C are assumed with positive values, we obtain (1.16)

I (t 0 ϕ) ≥ Φ (t 0 ) .
Finally from (1.15) and (1.16), we get

I (t 1 ϕ) < Φ (t 0 ) ≤ I (t 0 ϕ) Now to have (i) just take u = t 0 ϕ and ρ = Φ (t 0 ) .
it remains to prove (ii); to do so we remark that lim t→+∞

I (tϕ) = lim t→+∞    1 2 tϕ 2 Pg - 1 2 M   B(x) (tϕ) 2 dv g - A(x) ε + (tϕ) 2 2    dv g    -lim t→+∞ 1 p -1 M C(x) ε + (tϕ) 2 p-1 2 dv g = lim t→+∞ t 2 1 2t 2 -2 - 1 2 M B (x) ϕ 2 dv (g) .
Since by assumption B > 0, we have

lim t→+∞ I (tϕ) = -∞
consequently there is t 2 such that t 2 > t 0 and I (t 2 ϕ) < 0 Hence, putting u 2 = t 2 ϕ, we get the assumption (ii) of Lemma 2.

Lemma 1 allows us to apply the Mountain-Pass Lemma to the functional

I . Let C = inf γ∈Γ max u∈γ I (u)
where Γ denotes the set of paths in H 2 k (M ) joining the functions

u 1 = t 1 ϕ and u 2 = t 2 ϕ. So C ε is a critical value of I ε and moreover C > Φ (t 0 )
and by putting γ (t) = tϕ, for t ∈ [t 1 , t 2 ], we see that C ε is uniformly when ε goes to 0, so we have

(1.17) 0 < Φ (t 0 ) < C ≤ C
for ε sufficiently small and C > 0 not depending on ε.

Consequently there exists a sequence (u m ) m of functions in

H 2 k (M ) such that (1.18) I (u m ) →

Proof. By (1.18) we get for any

ϕ ∈ H 2 k (M ) DI (u m ) ϕ = o (1) i.e. for any ϕ ∈ H 2 k (M ) one has (1.19) M ϕP g u m dv g = M B (x) u + m 2 -1 ϕdv g + M A (x) u + m ϕ ε + u + m 2 2 +1 dv g + M C(x)u + m ϕ ε + u + m 2 p 2 +1 dv g + o (1)
in particular, for ϕ = u m we have

M u m P g u m dv g - M B(x) u + m 2 dv g = M A(x) (u + m ) 2 ε + u + m 2 2 +1 dv g + M C(x) (u + m ) 2 ε + (u m ) 2 p 2 +1 dv g + o (1) . or - 1 2 M u m P g u m dv g + 1 2 M B(x) u + m 2 dv g (1.20) + 1 2 M A(x) (u + m ) 2 ε + u + m 2 2 +1 dv g + M C(x) (u + m ) 2 ε + u + m 2 p 2 +1 dv g = o (1) .
On the other hand it comes from (1.18

) that 1 2 M u m P g u m dv g - 1 2 M B(x) u + m 2 dv g (1.21) + 1 2 M A(x) (u + m ) 2 ε + u + m 2 2 +1 dv g + 1 p M M C(x) (u + m ) 2 ε + u + m 2 p 2 +1 dv g = C + o (1) .
So by adding (1.20) and (1.21) we get

(1.22) 2 n M B(x) u + m 2 dv g + 1 2 M A(x) (u + m ) 2 ε + u + m 2 2 +1 dv g + 1 2 M A(x) (u + m ) 2 ε + u + m 2 2 +1 dv g 1 2 M C(x) (u + m ) 2 ε + u + m 2 p 2 +1 dv g + 1 2 M C(x) (u + m ) 2 ε + u + m 2 p 2 +1 dv g = C + o (1) .
For sufficiently large m we deduce that

2 n M B(x) u + m 2 dv g ≤ 2C + o (1) or 1 2 M B(x) u + m 2 dv g dv (g) ≤ n 2 C + o (1)
and plugging this last inequality with in (1.21) we obtain

1 2 M u m P g u m dv (g) ≤ C + n 2 C + o (1) ≤ nC + n (n -4) 2n C + o (1) ≤ 2 (n -2) C + o (1) .
Hence for m large enough

M u m P g u m dv (g) ≤ 4nC + o (1) ≤ 4nC + 1 i.e. (1.23) u m 2 ≤ 4nC + 1
Thus we prove the sequence (u m ) m is bounded in H 2 k (M ) so we can extract a subsequence, still denoted (u m ) m which verifies:

1.

u m → u ε weakly in H 2 k (M ) . 2. u m → u ε strongly in L p (M ), ∀p < 2n n-2k 3. u m → u ε a.e. in M. 4. (u m ) 2 -1 → u 2 -1 ε weakly in L 2 2 -1 (M ). Furthermore, putting g (x) = 1
ε q , where ε > 0 and q > 0, we get by Lebesgue's dominated convergence theorem that ∀k ∈ N:

u + m 2 + ε -q < ε -q and ε -q ∈ L p (M ) ∀p ≥ 1 thus (u + m ) 2 + ε -q → u • 2 + ε -q
strongly in L p (M ) ∀p ≥ 1 and with (ii) we infer that

u + m (u + m) 2 +ε q → u + ε+(u + ε )
2 q strongly in L 2 (M ). So if we let m go to +∞ in (1.19) we obtain that u is a weak solution to the equation (1.24)

P g u ε = B (x) u + ε 2 -1 + A (x) u + ε ε + u + ε 2 2 +1 + C(x)u + ε ε + u + ε 2 p+1 2
where 2 b = 2 2 and p > 1. Our solution u is not identically zero; indeed by (1.21) we have

- 1 2 M B (x) u + m 2 dv (g) ≤ C ε + o (1) ≤ 2C ε + o (1) or - 2 n M B (x) u + m 2 dv (g) ≤ 4 n .2 C ε + o (1) .
In addition by (1.22), we get

1 2 M A (x) ε + (u • m ) 2 2 dv (g) ≤ C ε - 2 n M B (x) u + m 2 dv (g) ≤ C ε + 4 n 2 C ε + o (1) ≤ 2 -1 C ε + o (1) .
Letting k → +∞ and taking into account of (1.17) we infer that

(1.25) 1 2 M A (x) ε + u + ε 2 2 dv (g) ≤ 2 -1 C
where C is the upper bound of C ε . Now if for a sequence ε j → 0 j→+∞ ( with j > 0, ∀j ∈ N ); u ε j goes to 0, then it follows

(1.26) 1 2 (2 -1) ε 2 j M A (x) dv (g) ≤ C.
So if j → +∞, it leads to a contradiction since by assumption A > 0.

Finally, for ε sufficiently small, u ε is a solution not identically zero of the equation (1.5).

By writing the equation (1.24) in the form

∆ k g u + k-1 l=1 A l (x) ∇ l u, ∇ l u +     A 0 (x) - A (x) ε + u + 2 2 b +1 - C (x) ε + u + 2 p+1 2     u + ε = B u + ε 2 -1
where A l is a smooth tensor field of type T 0 2l and symmetric in the sense that A l (X, Y ) = A l (Y, X) for tensors X,Y of type T l 0 on M (see [START_REF] Robert | Admissible Q-curvatures under isometries for the conformal GJMS operators[END_REF]). Noticing that

A 0 (x) - A (x) ε + u + 2 2 b +1 - C (x) ε + u + 2 p+1 2 ∈ L ∞ (M )
we deduce by standard regularity theory that u ∈ C 2k (M ).

Now we are in position to prove Theorem 1.

Proof.

From what precedes u is a C 2k (M ) nontrivial solution to equation (1.5), moreover u is a weak limit of the sequence (u k ) k which allows us by the lower semicontinuity of the norm to write

u ≤ lim k→+∞ inf u k .
And by the inequalities (1.17), (1.23) we deduce that the sequence (u ε ) ε of the ε-approximating solutions is bounded in H 2 k (M ) for sufficiently small > 0 i.e.

(1.27)

u 2 ≤ 4nC + 1 thus we can extract a subsequence still labelled (u k ) k satisfying: i) u k -→ u weakly in H 2 k (M ) ii) u k -→ u strongly in L p (M ) for p < 2 iii) u k -→ u a.e. in M . vi) u 2 -1 k -→ u 2 -1 weakly in L 2 2 -1 .
Furthermore the sequence (u k ) k is bounded below: indeed as the functions u k are continuous, denote by x k their respective maximums on M and put x o = lim x k ( a subsequence of (x k ) k still labelled (x k ) k ). Since by assumption the operator P g admits a positive Green function, then we can write

u k (x k ) = M G (x k , y)     B (y) u + k (y) 2 -1 + A (y) u + k (y) ε + u + k (y) 2 2 +1 + C(y)u + k (y) ε + u + k (y) 2 p+1 2     dv g
and by the Fatou's lemma, we get

lim inf k u k (x k ) ≥ M lim inf k G (x k , y)     B (y) u + k (y) 2 -1 + A (y) u + k (y) ε + u + k (y) 2 2 +1 + C(y)u + k (y) ε + u + k (y) 2 p+1 2     dv g = M lim inf k G (x k , y)     B (y) u + (y) 2 -1 + A (y) u + (y) ε + (u + (y)) 2 2 +1 + C(y)u + (y) ε + (u + (y)) 2 p+1 2     dv g .
And since the functions A, B, C are positive, then lim inf k u k (x k ) = 0 implies that u + = 0. This contradicts relation (1.26). Thus, there exists δ > 0, such that u k ≥ δ. We can once again use Lebesgue's dominated convergence theorem to get 1

ε k + (u k ) 2 q → 1 (u) 2q strongly in L p (M ) , ∀p ≥ 1, ∀q ≥ 1. Since for k large enough u k > 0 there is ε > 0 such that 1 ε k + u 2 k q ≤ 1 εq with q > 0.
Thus by the Lebesgue's dominated convergence theorem, we infer that 1

ε k + (u k ) 2 q -→ 1 u 2q strongly in L p (M ) , ∀p ≥ 1, ∀q > 0.
Finally, with ii), it follows that

u k ε k + (u k ) 2 2 +1 -→ 1 u 2 +1 strongly in L 2 (M ) .
with u > 0. Letting ε k → 0 in (1.24) as k → +∞,we get that u is a weak solution of equation (1.1) and since the regularity of u is the one of the function γ(x) = |x| 4k n-2k and since u > 0, then u ∈ C ∞ (M ).

Existence of a second solution

According to the previous section our functional admits a local maximum C ε , this means the following inequalities

I ε (t 1 ϕ) < Φ(t 0 ) < I ε (t 0 ϕ) ≤ C ε
where t 0 , t 1 are real numbers satisfying 0 < t 1 < t 0 and ϕ ∈ C ∞ (M ) with ϕ > 0 and ϕ = 1.

On the other hand and as I ε (tϕ) tends to -∞ as t goes to +∞, there is t 2 >> t 0 such that I ε (t 2 ϕ) < 0. Now if we let t and ε tend both to 0 + , the functional la I ε tends to +∞. Indeed,

lim t→0 + lim ε→0 + I ε (t.ϕ) = lim t→0 + I (1) (t.ϕ) + I (2) 0 (t.ϕ) = lim t→0 + M (t.ϕ)P g (t.ϕ) - 2 2 B(x)(t.ϕ) 2 dv(g) + lim t→0 +   1 2 M A(x) (t.ϕ) 2 dv(g) + 1 p -1 M C(x) (t.ϕ) p-1 dv(g).   = +∞.
and it follows that for ε small enough, there is near 0 a real number 0 < t << t 1 such that I ε (t ϕ) > Φ(t 0 > I ε (t 1 ϕ). What let us see, from this effect, that our function has a local lower bound. We will give the necessary conditions for this lower bound to exist, then we show by Ekeland's lemma that this lower bound is reached.

We will need the following version of the Ekeland's lemma Lemma 3. Let V be a Banach space, J be a C 1 lower bounded function on a closed subset F of V and c = inf

F J. Let u ε ∈ F such that c ≤ J(u ε ) ≤ c+ε. Then there is u ε ∈ F such that    c ≤ J(u ε ) ≤ c + ε u ε -u ε V ≤ 2 √ ε ∀u ∈ F , u = u ε , J(u) -J(u ε ) + √ ε u -u ε V > 0.
If moreover, u ε is in the interior of F , then

DJ(u ε ) V ≤ √ ε.
We can consider the sequence ( u ε ) ε in the interior of F . Indeed if u ε is on the border of F then by the continuity of J there is u ε belonging to interior of

F such that |J(u ε ) -J(u ε )| < ε. Which gives, for ε sufficiently, c -ε < J(u ε ) < c + 2ε and J(u) -J(u ε ) + √ ε u -u ε V = J(u) -J(u ε ) + J(u ε ) -J(u ε ) + √ ε u -u ε + u ε -u ε V ≥ J(u) -J(u ε ) -ε + √ ε u -u ε V - √ ε u ε -u ε V > J(u) -J(u ε ) + √ ε u -u ε V -2ε > 0.
So we can speak about the differential DJ(u ε ). Before stating the main result of this section we will establish some preliminary lemmas.

Lemma 4. Let θ > 0 such that (2.1) a 2 2 2 < θ 2 < a 2 2
where a = 1

(2 (n -k))

2 and put

t 3 = a 2 1 2 t 0
then we have the following inequality

(2.2) Φ(t 3 ) > a 2 Φ(t 0 ).
Proof. Since (t 1 being defined as in Lemma 1),

(2.3)

t 3 = a 2 1 2 t 0 < θt 0 = t 1 . and (2.4) a 2 2 2 > a 2 .
by (1.7), we get

Φ(t 3 ) = 1 2 t 2 3 -S. max M B(x) t 2 3 2 = 1 2 a 2 1 2 .t 0 2 - 1 2 t 2 0 a 2 = 2n k 1 2 a 2 2 2 - a 2 n -2k 2n
k 2n t 2 0 .
Knowing by (1.13) that

k 2n t 2 0 = 1 2 Φ(t 0 )
we deduce

Φ(t 3 ) = n k a 2 2 2 -n a 2k + a 1 2 Φ(t 0 ) = n k a 2 2 2 - a 2 + a 1 2 Φ(t 0 ) > a 2 Φ(t 0 ).
Where we used the inequality (2.4) in the last line.

Lemma 5. Given a Riemannian compact manifold (M, g ) of dimension n > 2k, k ∈ N and 3 < p < 2 + 1. So there is a constant λ * > 0 such that: ∀ε ∈ ]0, λ [ the following inequalities take place

(2.5) M A(x) (ε + (t 3 ϕ) 2 ) 2 2 dv g ≥ 2 a .   M A(x)dv g   2 1 t 0 a 1 2 and (2.6) M C(x) (ε + (t 3 ϕ) 2 ) p-1 2 dv g ≥ 2 a p-1 2   M C(x)dv g   2 1 t 0 a 2 p-1
where a 1 , a 2 are positive constants and t 3 , a are chosen as in Lemma 4.

Proof. Let ϕ ∈ C ∞ (M ), ϕ > 0 in M with ϕ = 1. Put (2.7) β 1 = a 2V (M ) 2 2 t 0 a 1 √ 2 2 and
(2.8)

β 2 = 1 V (M ) 2 p-1 a 2 2 2 t 0 a 2 √ 2 2
where V (M ) denotes the volume of M . Let

λ = min (β 1 , β 2 ) .
By Hölder's inequality, we get

  M A(x)dv g   2 =   M A(x) [ε + (t 3 ϕ) 2 ] 2 4 ε + (t 3 ϕ) 2 2 4 dv g   2 ≤   M A(x) (ε + (t 3 ϕ) 2 ) 2 2 dv g   .   M ε + (t 3 ϕ) 2 2 2 dv g   (2.9) ≤ I ε + (t 3 ϕ) 2 2 2 2 2
where

I = M A(x) (ε + (t 3 ϕ) 2 ) 2 2 dv g .
Independently, the Minkowski's inequality can be written

ε + (t 3 ϕ) 2 2 2 ≤ ε 2 2 + t 2 3 ϕ 2 2 2 consequently (2.10) ε + (t 3 ϕ) 2 2 2 2 2 ≤ ε 2 2 + t 2 3 ϕ 2 2 2 2 2 . Notice that ε 2 2 = ε. [V g (M )] 2 2
and ϕ 2

2 2 = ϕ 2
By continuity of inclusion H 2 k (M ) ⊂ L 2 (M ), we deduce the existence of a positive constant S 1 such that

ϕ 2 ≤ S 1 ϕ H 2 k
and as the operator P g is coercive, the norms . H 2 k and . are equivalent (this results from Proposition2, reference [START_REF] Robert | Admissible Q-curvatures under isometries for the conformal GJMS operators[END_REF]) and the exists an another constant C > 0 such that

ϕ H 2 k ≤ C ϕ Pg ∀ϕ ∈ H 2 k (M ) and therefore (2.10) becomes ε + (t 3 ϕ) 2 2 2 2 2 ≤ ε (V (M )) 2 2 + t 2 3 (S 1 C) 2 2 2
where we have used the fact that ϕ Pg = 1. Taking account of

t 3 = a 2 1 2 t 0 and letting a 1 = √ 2S 1 C, (2.9) becomes 
 M A(x)dv g   2 ≤ I ε [V g (M )] 2 2 + a ε 2 2 2 t 2 0 a 1 √ 2 2 2 2
.

And since 0 < ε < λ ≤ β 1 , we get

  M A(x)dv g   2 ≤ I 2 a ε 2 2 2 t 0 α 1 √ 2 2 2 2
.

Finally we deduce

I ≥ 2 a ε   M A(x)dv g   2 1 t 0 .α 1 2 .
Let us now prove the second inequality; again with the Hölder's inequality, we have

  M C(x)dv g   2 =   M C(x) [ε + (t 3 ϕ) 2 ] p-1 4 . ε + (t 3 ϕ) 2 p-1 4 dv g   2 (2.11) ≤   M C(x) (ε + (t 3 ϕ) 2 ) p-1 2 dv g   .   M ε + (t 3 ϕ) 2 p-1 2 dv g   ≤ J. ε + (t 3 ϕ) 2 p-1 2 p-1 2
(2.12)

where

J = M C(x) (ε + (t 3 ϕ) 2 ) p-1 2 dv g .
Again with Minkowski's inequality, we have

ε + (t 3 ϕ) 2 p-1 2 ≤ ε p-1 2 + t 2 3 ϕ 2 p-1 2
(2.13)

≤ ε. [V g (M )] 2 p-1 + t 2 3 ϕ 2 p-1 (2.14)
and since the embedding

H 2 k (M ) ⊂ L p-1 (M ) is continuous, there is a posi- tive constant S 2 such that ϕ p-1 ≤ S 2 ϕ H 2 k . The norms . H 2 k , . being equivalent, there is a positive constant C 1 such that ϕ H 2 k ≤ C 1 ϕ , ∀ϕ ∈ H 2 k (M ). consequently (2.11) becomes   M C(x)dv g   2 ≤ J εV g (M ) 2 p-1 + t 2 3 S 2 2 C 2 p-1 2 .
Replacing t 3 by his expression and posing

a 2 = √ 2S 2 C 1 it comes that   M C(x)dv g   2 ≤ εJ.V (M ) 2 p-1 + a 2 2 2 t 0 a 2 √ 2 2 now since 0 < ε < λ where λ ≤ β 2 , then it follows that   M C(x)dv g   2 ≤ J 2 a 2 2 2 t 0 a 2 √ 2 2 p-1 2
i.e.:

M C(x) (ε + (t 3 ϕ) 2 ) p-1 2 dv g ≥   M C(x)dv g   2 2 a p-1 2 1 t 0 a 2 p-1
. Now we are able to prove the existence of a second solution to equation (1.1) Theorem 3. Let (M, g) be a compact Riemannian manifold of dimension n > 2k, (k ∈ N ). Suppose the operator P g is coercive, has a Green positive function and there is a constant C (n, p, k) > 0 which depends only on n, p , k such that:

ϕ 2 2 M A(x) ϕ 2 dv g ≤ C (n, p, k) Smax x∈M B(x) 2+2 2-2 and ϕ p-1 p -1 M C(x) ϕ p-1 dv g ≤ C (n, p, k) Smax x∈M B(x) p+1 2-2
for some smooth function ϕ > 0 on M . If moreover for every ε ∈ ]0, λ [ , where λ is a positive constant, the following two conditions occur

(2.15) 2 a   M A(x)dv g   2 1 t 0 a 1 2 > 2 k t 2 0 4n (2 -a)
and

(2.16) 2 a p-1 2   M C(x)dv g   2 1 t 0 a 2 p-1 > (p -1)k t 2 0 4n (2 -a)
where a 1 , a 2 are positive constants, 2 = 2n n-2k , 3 < p < 2 + 1. Then the equation (1.1) admits a second smooth solution.

Proof. The proof will be done in three steps 1 th -step. The functional I ε has a local lower bound. This consists to find a strictly positive real number λ such ∀ε ∈ ]0, λ [ one has the following inequality

I ε (t 3 ϕ) > Φ(t 0 ) ∀ϕ ∈ C ∞ (M ), ϕ = 1, with t 3 < t 1 .
Indeed, according to Lemma (4) inequality (2.2), one has

I ε (t 3 ϕ) = I (1) (t 3 ϕ) + I (2) (t 3 ϕ) (2.17) > a 2 Φ(t 0 ) + 1 2 M A (x) (ε + (t 3 ϕ) 2 ) 2 b dv g + 1 p -1 M C (x) (ε + (t 3 ϕ) 2 ) p-1 dv g .
and as by assumption

2 a .   M A(x)dv g   2 1 t 0 a 1 2 > 2 k t 2 0 4n (2 -a) and λ = min (β 1 , β 2 )
where β 1 , β 2 are given by (2.7) and (2.8) and knowing that

Φ(t 0 ) = k n t 2 0 it follows by Lemma5 that, ∀ε ∈ ]0, λ [ (2.18) 1 2 M A(x) (ε + (t 3 ϕ) 2 ) 2 2 dv g > 1 2 - a 4 Φ(t 0 ).
Similarly, one has

2 a p-1 2   M C(x)dv g   2 1 t 0 .a 2 p-1 > (p -1)k t 2 0 4n (2 -a)
which implies with Lemma (5

) that ∀ε ∈ ]0, λ [ (2.19) 1 (p -1) M C(x) (ε + (t 3 ϕ) 2 ) p-1 2 dv g > 1 2 - a 4 .Φ(t 0 ).
Finally, by combination of (2.17), (2.18) and (2.19) we get

I ε (t 3 ϕ) > a 2 Φ(t 0 ) + 2 1 2 - a 4 Φ(t 0 ) (2.20) > Φ(t 0 ).
Hence our result.

2 th -step. The infimum of the functional I ε is reached. Denote by B(0, t 1 ) = u ∈ H 2 k (M ) : u ≤ t 1 the closed ball centred at the origin 0 of radius t 1 in H 2 k (M ). In this section we will show that c ε = inf B(0,t 1 ) I ε ( c ε < Φ(t 0 ) ) is reached. By Ekeland's Lemma, there exists a sequence (u m ) m∈N in B(0, t 1 ) such that I ε (u) → c ε = Inf B(0,t 1 ) I ε and DI ε (u m ) → 0 strongly in the dual space of H 2 k (M ). That is to say (u m ) is a Palais-Smale sequence, so by the same arguments as in Theorem2 and Theorem1, we get that equation (1.1) has a smooth solution v. Since the ε-approximating solutions are obtained as weak limit of sequences of functions from B(0, t 1 ), it follows by the weak lower semi-continuity of the norm that these ε-approximating solutions are in B(0, t 1 ). As in turn v is obtained as a limit of a sequence of ε-approximating solutions, v ∈ B(0, t 1 ).

3 th -step. The two solutions are distinct. Indeed, we have on the one other hand the energy of the solution v given by c = lim ε→0 + c ε ≤ Φ(t 0 ) and secondly the energy of the solution u obtained in the first section is C = lim ε→0 + C ε where C = inf Thus the Lebesgue's dominated theorem, we get

C = lim ε→0 + C ε ≥ Φ (t 0 ) + 1 2 M A (x) (t 0 ϕ) 2 dv g + 1 p -1 M C(x) (t 0 ϕ) p-1 dv g > Φ (t 0 ) .
So the solutions u and v are of different energies and therefore are distinct.

Nonexistence of solution

In this section we will be placed in a closed ball B(0, R) of H 2 k (M ) centered at the origin 0 and of radius R > 0, we prove that under some condition (inequality 3.1) that the equation (1.1) admits no solution.

m→+∞C

  and DI (u k ) → m→+∞ 0 By Lemma 2 the sequence (u m ) m∈N of H 2 k (M ) is a Palais-Smale sequence (P-S) for the functional I . Theorem 2. The Palais-Smale sequence (u m ) m∈N is bounded in H 2 k (M ) and converges weakly to nontrivial smooth solution u ε of equation (1.5).

I

  (u) and Γ denotes the set of paths joining the functions u 1 = t 1 ϕ and u 2 = t 2 ϕ. Now since any path joining u 1 and u 2 intersects the sphere centred at the origin and of radius t 0 in H 2 k (M ) thenC = inf γ∈Γ max u∈γ I (u) ≥ (I ε (t 0 ϕ)) = I (1) (t 0 ϕ) + I (2) ε (t 0 ϕ) ≥ Φ (t 0 ) + I (2) ε (t 0 ϕ)

Theorem 4 .M 1 p -1 2 2 2 +p- 1 .

 4121 Given (M, g) a compact Riemannian manifold of dimension n > 2k, (k ∈ N ) and A, B, C are positive smooth functions on M and 2 < p < 2 + 1. Assume that Bdv g > (SR) 2 where S, R are positive constants and C(n, p, k) = 2 + p -1 p -Then the equation (1.1) has no smooth positive solution u with energy u H 2 k (M ) ≤ R. Proof. Suppose that there exists a smooth positive solution u ∈ H 2 k (M ) such that u H 2 k (M ) ≤ R. By multiplying both sides of equation (1.1) by u end integrating over M , we getM uP g (u)dv g = M B (x) u 2 + A (x) u 2 + C (x) u p-1 dv g .And since u pg = M uP g (u)dv g is a norm equivalent to u H 2 k (M ) , there exists a constant S > 0 such that u ≤ S u H 2 k (M ) .


D Hauser

L Delaye

T Grelier

C Tourain

C Tison

P Castillan

Wave spectrum retrieval from SWIM data: speckle spectrum estimation Session: "Wave retrieval and applications"

Introduction

The Chinese and French Space Agencies propose to jointly carry out an innovative mission, CFOSAT (China France Oceanography Satellite project) devoted to the monitoring of the ocean surface and its related science and applications. CFOSAT will embark both a wind and a wave scatterometers, enabling a simultaneous measure of the wind and the wave vectors with a global coverage for the first time. The launch is planned for mid-2018. The satellite embarks two payloads; both are Ku-band radar scanning around the vertical axis:

-the wave scatterometer SWIM, a rotating 6-beams radar at small incidence (0 to 10°) [1, 2], -the wind scatterometer SCAT, a fan-beam radar at larger incidence angles (30° to 50°) [3].

This paper focuses on the processing of the SWIM data for the retrieval of the 2D wave spectrum and, especially the estimation of the speckle spectrum. The SWIM data are of three kinds [4]: the normalized radar cross-section profiles from 0° to 11°, the SWH and wind speed from the nadir beam and the wave spectrum from the 6°, 8° and 10°. The processing steps are following:

Overview of the SWIM data processing

-inversion of the measured power to normalized radar cross section (inversion of the radar equation) and geolocalization of each radar gate, -estimation of averaged normalized radar cross section per step of 0.5° in incidence and 15° in azimuth, -retracking of the nadir waveform to estimate the SWH and the wind speed, -estimation of the modulation spectrum from the  0 fluctuations and correction of the spectrum from speckle and impulse response, -computation of the wave spectrum from modulation spectrum through the computation of the modulation transfer function. There are two major issues: the estimation of the speckle spectrum and the estimation of the modulation transfer function. The paper addresses the issue of the speckle spectrum.

Figure 1 .

 1 Figure 1. Overview of the SWIM L1 and L2 products and the associated processing.

L2•

  Computation of wave spectra: -Estimation of the Modulation Transfer Function -Grouping of 1D wave spectra into 2D wave spectra per box of 70x90 km² • Partitioning and geophysical parameters: -Automatic partitioning through watersheding to recognize the different sea statesL2• Averaging of σ 0 per cells of 0

Figure 2 .

 2 Figure 2. Illustrations of the SWIM products for wave spectrum computation.

The speckle spectrum depends on the real number of independent pulses [5]. The dependency of the spectrum to the sea state conditions is still under investigation. Three main methods are used:

-analytical methods, -estimation from the noise floor, -use of cross-spectrum computation. The three methods will be detailed and discussed.

Results on simulated data and airborne data

The three methods have been tested on simulated data and on real airborne data. The results will be shown and discussed.

[1] Enjolras V., L. Rey, T. Amiot, C. Tison, P. Castillan, SWIM, a state of the art multi-incidence beams Ku-band waves scatterometer to go beyond current radar systems, in IGARSS'09, July 2009 [2] Hauser, D., Soussi, E., Thouvenot, E., and Rey, L.,"SWIMSAT: a real aperture radar to measure directional spectra of ocean waves from space main characteristics and performance simulation". Jour. Atmos. Oceanic Tech., 18, 2001 [3] Zhu J., X. Dong, W. Lin, X. Xu, Calibration and estimation of attitude errors for a rotating fan-beam scatterometer using calibration ground stations, IEEE JSTARS, PP( 9