Gea Z Hernández

Castro Zazil

Nicolas Kervyn

Rosa A Gómez Zúñiga

Sheila G Rojas Pérez

Developing a Blended-Method Dance Workshop for the Popularization of Science Through Choreography

Keywords: Dance, choreography, scientific communication, hybrid teaching, online teaching, art and science Danse, chorégraphie, communication scientifique, enseignement hybride, enseignement en ligne, art et science

This teaching practice report concerns a doctoral workshop developed by the authors in order to prepare Ph.D. students to participate in "Dance Your Ph.D." -an international contest of online videos, whereby doctoral students use dance to communicate their research. This workshop provides Ph.D. students with the theoretical and methodological basis, as well as choreographic tools, and the self-confidence necessary to take part in the contest. The first edition was organized fully online due to the COVID-19 lockdown. This initial constraint led to the development of a series of techniques that enabled holding a dance workshop remotely, using the Teams software. In this report, we describe how we adapted to organize the workshop online and how this led to pedagogical innovations that we continued to use in subsequent hybrid iterations of the workshop. Discussing the possibilities and challenges presented by our pedagogical approach, we position this text within related literature debates and identify directions for future research for both embodied and virtual pedagogies.

Introduction

"Dance Your Ph.D." is an international contest sponsored by the American Association for the Advancement of Science (AAAS) and Science Magazine. For this contest, Ph.D. students create online videos, using dance to communicate their research. The competition was created by John Bohannon in 2008 as a way of promoting more creative ways of communicating science through the performing arts [START_REF] Leavy | Method meets art: Arts-based research practice[END_REF][START_REF] Spry | Body, paper, stage: Writing and performing autoethnography[END_REF], specifically challenging scientists to pursue their research with/through movement and choreography [START_REF] Bohannon | Dance vs. powerpoint, a modest proposal[END_REF] and to communicate their findings through this embodied format once recorded in a video. Doing so, the contest offers space for rethinking research as an embodied practice, seeking to reconcile long-held binaries between mind and body in knowledge creation [START_REF] Mandalaki | Dancers as inter-corporeality: Breaking down the reluctant body[END_REF][START_REF] Mandalaki | It takes two to tango: Theorizing inter-corporeality through nakedness and eros in researching and writing organizations[END_REF].

To participate, Ph.D. students express their findings through solo or collective dance performances and share the YouTube or TikTok URL of their dance video (Science, 2023a). Entries are classified into one of four categories based on the scientific field of the thesis: Physics, Chemistry, Biology, and Social Sciences. The contest is organized in two phases. In the first phase, a panel of judges selected by AAAS choose a maximum of five dances in each field, based on both artistic and scientific merit. From this shortlist, a second panel of judges selects one winner in each of the four categories based on scientific merit, artistic merit, and creative combination of science and art [START_REF] Leavy | Method meets art: Arts-based research practice[END_REF]. The Category Winner receiving the highest total score awarded by the panel of judges wins the Grand Prize for that year's "Dance Your Ph.D." contest (Science, 2023b).

The contest itself is an interesting case study of how information technologies might be used in university teaching [START_REF] Li | Shifts in pedagogy and autonomy in virtual dance teaching and learning[END_REF]. Indeed, even though the contest was launched twelve years before the first COVID lockdown, from the start it was done remotely through dance videos posted online. This format makes the contest accessible to participants worldwide (assuming a stable internet connection, which we recognize is not the case for everyone), as it can be organized at a reasonable cost. It also allows for wide exposure of the winning dance videos, and those of the other participants, through social networks and online articles featuring the videos, enabling Ph.D. researchers to disseminate their research to different audiences.

Dance Your Ph.D. Doctoral Workshop

The doctoral workshop presented here was developed by the authors at UCLouvain (Belgium) in 2020, 2022 and 2024 with the objective of providing UCLouvain Ph.D. students with the theoretical and methodological basis, as well as choreographic tools, and the self-confidence necessary to take part in the "Dance your Ph.D." contest. The first edition of the workshop was organized completely online due to the COVID lockdown. Adapting to this initial constraint led to the development of a series of techniques that made it possible to hold a dance workshop remotely, in this case using Teams software. Some of these adaptations proved so successful that for the 2022 (and the planned 2024) version of the workshop we decided to use a hybrid format, structured around a mix of online and face-to-face sessions.

The first workshop was planned for four consecutive mornings in May 2020 and, for funding reasons, was restricted to humanities and social sciences faculty. Fifteen Ph.D. students from UCLouvain signed up. Each morning was supposed to start with one hour of theory in a classroom followed by two hours of workshop in a dance studio. In the first part of the week, the professors were going to give theoretical lectures presenting the contest, the objectives of the workshop and some of their own work combining art and science (Garoia et al., 2014, p. 14;Hernández Castro, n.d., 2022; Un desierto para la danza, 2015, p. 23, section I Love Buchomp). Through carefully crafted embodied exercises and discussions, the students would then move from oral to choreographic forms of sharing their research. For instance, students were asked to experiment with different qualities of movements (fast-slow, low-high), which gradually became more abstract (solid-liquid, active-passive) before finally trying to express totally abstract concepts with their movements (focus-distraction; democracy-totalitarianism). Towards the end of the week, the professors were going to adopt more of a coaching role, providing mentoring and feedback to the students as they worked on choreographing their individual dance videos. The goal of this workshop was not for the Ph.D. students to come out with a finished dance video. It rather sought to provide the students with useful tools while encouraging and supervising the first steps of their creative process. The aim was to nurture the participants' ability and confidence, enabling them to develop a "Dance your Ph.D." video in the following weeks and enter it in the contest the next January.

Although the 2020 lockdown (which started in mid-March in Belgium) began to ease at the end of April, it still prohibited us, the organizers, from holding an in-person dance workshop and made it impossible for Prof. R. Gomez Zuñiga to travel from Northern Mexico to Belgium. We postponed the workshop from May to late June, and then again to mid-December. By December 2020, Belgium was again under strict lockdown, so we decided to organize the workshop completely online. Due to the various changes and the shift to an online format, there were a few cancellations. In the end, we had six confirmed participants, all of them students from the UCLouvain faculties of history, humanities and business.

Online Dance Workshop

The program (see Table 1) was more-or-less the same as that for the workshop originally planned for May. However, having a fifth day allowed us to add the presentation of online choreography projects developed during the first lockdown (Gomez Zuñiga, 2021;[START_REF] Castro | VidéoconféDANSES. ChorégraKIDS[END_REF], and to make more time for individual coaching and feedback sessions with the participants. The remote dance workshop also offered space to explore the creative potential of pandemic-related restrictions that recent accounts in organization studies have discussed (e.g., [START_REF] Mandalaki | Writing memory work through artistic intersections[END_REF][START_REF] Pérezts | Three walls[END_REF]. Of course, the online format also had a number of disadvantages, such as the lack of physical and sensorial engagement between participants and the physical surroundings or informal encounters for finding common ground to build trust and mutual involvement in the creative process. However, there were also positive aspects which we sought to leverage. First, the online workshop proved to be in line with the objectives of the "Dance your Ph.D." contest, which from day one has taken place via dance videos rather than live performances. Navigating the pandemic restrictions forced us into a completely online format [START_REF] Parrish | Dancing the distance: iDance Arizona videoconferencing reaches rural communities[END_REF][START_REF] Plasson | La vidéo-danse sur Internet: l'exemple de Numeridanse[END_REF], although the initial project had been based on a more traditional approach (i.e., work involving physical interaction) with the unspoken assumption that the choreography would be prepared for the stage before being turned into a dance video. This challenged us creatively and had practical implications in terms of how we taught the online and subsequent hybrid workshops, as discussed next.

Two-Dimensional Dancing

Given the online format, we used language and techniques inspired by cinematography rather than stage performance. Specifically, instead of talking about stage left and stage right or upstage and downstage, as is common in physical stage performances, we talked about the bottom, top, right and left sides of the physical screen. On the first day, in addition to the exercises presenting the three dancing levels (low/floor, medium and high), we introduced and experimented with different shot sizes (extreme close-up, close-up, medium shot, long shot) and camera angles (eye level, low angle, high angle, overhead, ground level), as well as with the possibilities available with mobile cameras (e.g., point of view, sequence shot). Colleagues such as [START_REF] Li | Shifts in pedagogy and autonomy in virtual dance teaching and learning[END_REF] and [START_REF] Mcpherson | Making video dance: A step-by-step guide to creating dance for the screen[END_REF] have discussed the potential of digital means as innovative pedagogical practices for the creation of dance videos during the global pandemic. These studies specifically explain how recording embodied movement from different angles, in confined spaces, can be a form of dancing with the camera, and also point out some of the limitations of this approach [START_REF] Li | Shifts in pedagogy and autonomy in virtual dance teaching and learning[END_REF].

Much like in the aforementioned studies, in our case the camera was immediately included in the choreographic approach, allowing for embodied connections and involvement in the research process despite the lack of physical presence [START_REF] Mandalaki | Writing memory work through artistic intersections[END_REF]. In the following days, we built on this approach by encouraging students to experiment with additional possibilities of the online digital format, such as different lighting (e.g., natural vs. artificial, backlighting; See Figure 1) and camera filters. Previous dance videos in the contest have been recorded in lab or fieldwork contexts [START_REF] Bohannon | Dance vs. powerpoint, a modest proposal[END_REF]. In our case, given the lockdown measures, participants were limited to the confines of their private spaces. As such, each participant joined the workshop from home (rather than meeting in the classroom and dance studio we had booked in May), rearranging their personal space as needed to accommodate dancing, in accordance with the contest's instructions. On the second day, we presented an example and then participants experimented with the choreographic potential of various parts of their homes, figuring out how even confined and often small spaces (e.g., kitchen, bathroom, bedroom) might offer alternative movement possibilities not afforded in a dance studio or an office [START_REF] Li | Teaching introduction to dance studies online under COVID-19 restrictions[END_REF][START_REF] Spry | Body, paper, stage: Writing and performing autoethnography[END_REF]. We further experimented with how dancers can use props, including work-related objects, to enhance their choreography. Notwithstanding the limitations of a purely online format -such as the absence of touch, being aware of others' breathing or physical presence, or the atmosphere of the place (Kaasila-Pakanen et al., 2020), these exercises allowed us to turn some of the constraints of the lockdown into opportunities for creativity (Mandalaki et al., 2022;[START_REF] Pérezts | Three walls[END_REF].

Choreographing using Teams

The format of the workshop led us to explore how Microsoft Teams, the software used by UCLouvain, could be made into a pedagogical tool for a dance workshop [START_REF] Godoi | Les défis et les apprentissages des formateurs d'enseignants d'éducation physique pendant la pandémie de COVID-19 au Brésil[END_REF]. On the second day, we organized a "Choreographing using Teams" exercise, whereby each participant took a turn as choreographer for the eight other dancers (Ph.D. students and workshop professors).

Using chat or audio, the choreographer gave instructions to the dancers using the concepts learned during the first day (i.e., dancing levels, shot sizes, camera angles, screen borders). With these instructions to individual dancers and to the whole group, the choreographer created live simultaneous performances on the Teams mosaic display. Later in the week, during the individual choreography coaching, we started experimenting with the possibility of using a personalized background on Teams. This allowed the choreographer to assign the different dancers backgrounds that were meaningful to their process. The dancers then experienced how the background reacted when they started dancing.

Another difficulty that arose with the digital interface was that the Teams software is designed to detect talking heads: a head and shoulders without a lot of movement. When dealing with other shot sizes (e.g., extreme close-up or long shot) of a person in motion, the software often struggles to distinguish the person from the background, such that the dancer often blends in with the background image (see Figure 2). Although this feature can seem confusing or uncomfortable, it also creates novel and creative ways of interacting with the images used to communicate the researcher's process. In our case, this offered a multidimensional relationship to pictures and frames that differed substantially and creatively from the overused, monodirectional "person standing next to PowerPoint slides" presentation style of research outputs (Mandalaki et al., 2022). It also provided an almost "natural" example of an entanglement of dancing bodies and digital video technology, blending with its physical surroundings to co-create the final output in a posthumanistic way. Finally, the ease of recording Teams calls made it possible to record the Ph.D. students' dances and use them as inspirational material for the students' methodological research processes. At the end of the session, each participant was able to retrieve the video in order to use it as work-in-progress for review [START_REF] Li | Teaching introduction to dance studies online under COVID-19 restrictions[END_REF].

Blended Dance Workshop

While the innovations in question were implemented out of necessity in the context of a forced digital format which led us to sustainably reconfigure how we approached our pedagogical and teaching practices [START_REF] Kervyn | Transition numérique d'un cours d'introduction au marketing : conception d'un dispositif d'enseignement mixte adapté à la méthode des études de cas[END_REF], moving forward we recognized that they had significant potential. Conceptualizing this new way of teaching dance, the first author coined the term 'VideoConfeDance' [START_REF] Castro | VidéoconféDANSES. ChorégraKIDS[END_REF], which also inspired the next editions of the "Dance your Ph.D." workshop in UCLouvain. To keep the positive aspects of the online dance teaching and mitigate its limitations, in September 2022 the second "Dance your Ph.D. @ UCLouvain" workshop was organized using a blended/hybrid teaching method (See Table 2).

Figure 2 Dancing Experimentation Using Teams Background

As shown in Table 2, in 2022 we began with one morning session online followed by two inperson meetings. When physically present, in addition to choreographic and body exercises, we also left the dance studio to explore the choreographic potential of other places (hallways, classrooms, etc.). The fourth half-day was held online, taking advantage of the format to organize dance theories and exercises relevant to the online format. This involved presenting and experimenting with cinematographic language (framing, camera angles) and the possibilities of mobile cameras (selfie, point of view, sequence shot), interacting with the background image and using cinematographic language in choreography. Based on the students' feedback from the first workshop, we also took time to present various tools and examples of how ideas and models might be turned into choreographies. For instance, we discussed how to use storyboards as an intermediary step, and we introduced some online tools for creating storyboards (storyboardthat.com; readfold.com).

Discussion

Despite the numerous lockdown-related obstacles faced when organizing this workshop, we approached these obstacles as opportunities for creativity (Mandalaki et al., 2022;[START_REF] Pérezts | Three walls[END_REF].

Post-COVID, this has led us to keep some parts of the workshop online, leveraging the advantages of this format, while also seeking to navigate its limitations. The transition from a completely online workshop to a hybrid one, and the further adaptations we are planning to integrate for future workshops, were guided by the qualitative feedback we received from participants at the end of each workshop. Students' overall feedback was positive, with suggestions for desirable adaptations (UCLouvain, 2022). We next discuss how we believe this workshop supported Ph.D. students in their research processes and enhanced their self-confidence. What we learned adds to related literature discussions about the possibilities and limitations of innovative pedagogical tools during and after the pandemic [START_REF] Castro | VidéoconféDANSES. ChorégraKIDS[END_REF][START_REF] Li | Shifts in pedagogy and autonomy in virtual dance teaching and learning[END_REF][START_REF] Mcpherson | Making video dance: A step-by-step guide to creating dance for the screen[END_REF]. Garoia et al., 2014, p. 14;Hernández Castro (n.d.). (b) Hernández [START_REF] Castro | VidéoconféDANSES. ChorégraKIDS[END_REF].

Theoretical and Methodological basis

Ph.D. participants' feedback stated that the workshop gave them the space to take a step back and reflect on their own theoretical and methodological positions from new perspectives. Engaging with the performative aspects of body movement, dance and other forms of art in how we research, write and teach, embraces vulnerable moments of being in relation to others [START_REF] Spry | Body, paper, stage: Writing and performing autoethnography[END_REF], carrying the potential to meaningfully transform university and business school pedagogies [START_REF] Jääskeläinen | Research as reach-searching from the kinesphere[END_REF][START_REF] Jääskeläinen | Chapter 4. Embodied bordering: Crossing over, protecting, and neighboring[END_REF]Mandalaki et al., 2022). Embodied practice, even when facilitated by technology, makes the researcher more aware of the vulnerability of the body [START_REF] Mandalaki | Writing memory work through artistic intersections[END_REF]. In our case, experimenting with dance and artistic experimentation provided students with reflexive methodological tools for rethinking the conduct and writing of academic research as embodied, vulnerable practices with attention to the affective, relational body and the senses [START_REF] Jääskeläinen | Research as reach-searching from the kinesphere[END_REF][START_REF] Mandalaki | It takes two to tango: Theorizing inter-corporeality through nakedness and eros in researching and writing organizations[END_REF][START_REF] Pérezts | Three walls[END_REF][START_REF] Pérezts | Unlearning organized numbness through poetic synesthesia: A study in scarlet[END_REF][START_REF] Spry | Body, paper, stage: Writing and performing autoethnography[END_REF].

In doing so, it allowed the bodies of the researcher and the researched to be reframed as sites of knowledge from where reflexive research with social and organizational significance can emerge [START_REF] Mandalaki | Invi(α)gorating reflexivity in research: (Un)learnings from α field[END_REF][START_REF] Mandalaki | It takes two to tango: Theorizing inter-corporeality through nakedness and eros in researching and writing organizations[END_REF].

Secondly, the first two workshops were restricted to Ph.D. students from the university's humanities and social sciences faculty, limiting the breadth of exchange across epistemic disciplines. The upcoming workshop in 2024 will be open to all of the university's faculties and academic levels (Ph.D. students, post-docs and professors). The outcome of the workshop will no longer be solely focused on the official "Dance Your Ph.D." contest but on different ways in which choreography might be used in/for scientific research, writing, communication, and the dissemination of academic work. Broadening the type of participants and outcomes will hopefully allow for a wider appeal as well as rich multi-disciplinary exchanges between diverse scientific domains, methodologies and epistemological approaches. Such reflection will be fuelled by examples of the authors' personal dance research engagements, as, for instance, when two authors of this paper (E.M. and M.P.) danced together and developed a video which served as complementary material to their related manuscript submission to an academic journal1 [START_REF] Mandalaki | It takes two to tango: Theorizing inter-corporeality through nakedness and eros in researching and writing organizations[END_REF]. Through such practices, we hope to enhance the impact of the workshop by targeting not only the "Dance your Ph.D." contest but also other types of scientific communication, research and pedagogical approaches using choreography and embodied and dance-based methods more broadly.

Choreographic Tools

In the tables above we listed the different choreographic tools, such as dancing levels and experimenting with tools, which we used and adapted to online teaching. Many of these tools were highlighted by the participants' feedback along with reflections on the physical modalities used and their interactions with the online format. Some of these adaptations dovetailed with those of Zihao [START_REF] Li | Teaching introduction to dance studies online under COVID-19 restrictions[END_REF]2023), who also had to move their dance teaching online during the pandemic. Specifically, after the first workshop, participants reported that they would have liked to receive more tools to help them turn their research and abstract ideas into choreography. For the second workshop, we therefore introduced storyboards and how to use them. We plan to further integrate students' feedback to provide the desired practical tools that allow students to choreograph their research through dance videos [START_REF] Mcpherson | Making video dance: A step-by-step guide to creating dance for the screen[END_REF].

Self-Confidence to take part in the Workshop and Contest

In their feedback (UCLouvain, 2022), Ph.D. participants described the caring and constructive mood they experienced during the workshop, allowing them to learn and feel psychologically safe enough to actively participate in the sessions, regardless of their level as either dancers or academics. For example, we used a series of body-consciousness and expressivity exercises that do not require any special training or physical abilities, instead of elaborate dance techniques.

1. This video did not end up as part of the academic publication in question, but allowed the authors to reflect differently on the creative, embodied potential of their dance research practices. These were performed in a climate of trust, constructing a safe space within which participants would feel comfortable to 'loosen up' their bodies and find ways of expressing their research. The feedback also identified some room for improvement. First of all, starting the workshop with an online session was seen as problematic as it made it harder for participants to fully commit themselves to the workshop from the start. Indeed, the topic and method of the workshop are unusual for this audience, especially in the online format. To leverage the benefits of a hybrid format, as mentioned above, in future we plan to start with two days of in-person sessions to give participants the opportunity to engage physically with the process and interact together before moving to online sessions. This might allow participants to get to know each other and start establishing some common ground for physical connection, trust in their abilities and in each other, as well as exchanges of support in relation to their varied research processes.

Finally, one of the most positive outcomes of our experience is that for the first time in the history of the "Dance your Ph.D." contest, three UCLouvain Ph.D. students entered dance videos in the Social Sciences category of the official contest [START_REF] Delanaye | Dance your Ph.D. 2020-2021: Roman and Byzantine weights from the Iberian peninsula[END_REF][START_REF] Solbreux | Education through social entrepreneurship: An integrated identity construction -Dance your Ph.D[END_REF][START_REF] Wu Mandal | Dance Your Ph.D. 2022: How do Chinese adolescents resist their parents when feeling over-controlled[END_REF], and more plan to do so in the future. This might serve as a demonstration of how participation in this workshop can enhance students' belief and confidence in their abilities to conduct and disseminate their academic research differently, creatively, and meaningfully.

Students feel supported and encouraged to engage with embodied learning, success and failure, which admittedly makes them aware of their embodied, physical, and intellectual potential, equipping them with the necessary qualities to engage confidently with their work.

Considering the above, the positive impact of this workshop gives us hope and motivates us to continue experimenting with these creative possibilities and integrating students' feedback to further develop the workshop internationally, across different audiences, in years to come.

Building on what we have learned, we encourage future researchers to further explore the possibilities and limitations of hybrid or fully virtual formats in choreographing academic research practices, not only among homogenous groups of students but also across geographies and institutions, in consideration of participants' differences and intersectional positionalities.

Figure 1

 1 Figure 1Online Backlight Dancing Exercise

Table 1

 1 Program for December 2020 Online Workshop

		Theory	8:30 -9:00 9:00 -9:30	Presentation of the "Dance your Ph.D." contest Presentation of DanScie project (a)
	Monday		10:00 -10:30	Warm-up
	14/12/20	Practice	10:30 -11:00	Dancing levels
			11:00 -11:30 11:30 -12:00	Shot sizes, camera angles & mobile cam
		Theory	8:30 -9:00 9:00 -9:30	Short presentation of each participant's Ph.D. project & first choreography ideas
	Tuesday		10:00 -10:30	Warm-up
	15/12/20	Practice	10:30 -11:00 11:00 -11:30	Choregraphing using Teams
			11:30 -12:00	Exploring spaces & dancing with props
		Theory	8:30 -9:00 9:00 -9:30	Presentation of two choreography projects based on ethnographic background research: I Love Buchomp (b) & Retorno (c)
	Wednesday 16/12/20 ONLINE	Practice	10:00 -10:30 10:30 -11:00	Warm-up Lighting: natural vs. artificial, filters
			11:00 -11:30	Individual choreographic research
			11:30 -12:00	& personal coaching
	Thursday	Theory	8:30 -9:00 9:00 -9:30	Presentation of two online choreography projects: VidéoConféDanse (d) & Diario de un cuerpo en quarantine (e)
	17/12/20	Practice	10:00 -10:30 10:30 -11:00	Warm-up Individual choreographic research & personal coaching
		Theory	8:30 -9:00 9:00 -9:30	Feedback on workshop experience Planning of next steps for the contest
	Friday 18/12/20		10:00 -10:30	Warm-up
	ONLINE	Practice	10:30 -11:00 11:00 -11:30
			11:30 -12:00

Presentation of individual choreographies & feedback

(a)

Garoia et al., 2014, p. 14; Hernández Castro (n.d.)

. (b) Un desierto para la danza, 2015, p. 23. (c) Hernández Castro (2022). (d) Hernández Castro (2021). (e) Lozano (2021).

G. Z. Hernández Castro et al. Dance Your Ph.D. in VideoConfeDance: Developing a Blended-Method Dance Workshop ... 2024 -Revue internationale des technologies en pédagogie universitaire, 21(1) ritpu.org 29

Table 2

 2 Program for September 2022 Blended Workshop

			8:30 -9:00	Introduction
			9:00 -9:30	Rules & examples from past competitions
	Friday 02/09/22	Online	10:00 -10:30 Warm-up 10:30 -11:00 Cinematographic language
			11:00 -11:30 Dancing levels: Low, medium, high
			11:30 -12:00 Mobile cam: selfie, point of view, sequence shot, …
		Classroom	8:30 -9:00 9:00 -9:30	Presentation of Ph.D. project & first choreography ideas
	Monday		10:00 -10:30 Warm-up
	05/09/22	Dance	10:30 -11:00 Group choreography: DanScie (a)
		studio	11:00 -11:30 11:30 -12:00	Improvisation exercises
		Classroom	8:30 -9:00 9:00 -9:30	Choreography theory: turning concepts into dance (storyboard, …)
	Tuesday		10:00 -10:30 Warm-up
	06/09/22	Dance	10:30 -11:00	Exploring places
		studio	11:00 -11:30	(classroom, halls, …)
			11:30 -12:00 Experimenting with props
			8:30 -9:00 9:00 -9:30	Online choreography: VideoConféDANSES (b)
	Wednesday 07/09/22	Online	10:00 -10:30 Warm-up 10:30 -11:00 Lighting: natural vs. artificial, filters, …
			11:00 -11:30 Choregraphing using Teams
			11:30 -12:00 Interacting with background images
		Classroom	8:30 -9:00 9:00 -9:30	Presentation of "Dance my Ph.D." storyboards
	Thursday 08/09/22	Dance studio	10:00 -10:30 Warm-up Individual choreographic research 10:30 -11:00 & personalized coaching
	Friday 09/09/22	Classroom Dance studio	8:30 -9:00 9:00 -9:30 10:00 -10:30 Warm-up Feedback on workshop experience Planning of next steps for the contest Presentation of individual choreographies 10:30 -11:00 & feedback
	(a)

  G. Z. HernándezCastro et al. Dance Your Ph.D. in VideoConfeDance: Developing a Blended-Method Dance Workshop ...

2024 -International Journal of Technologies in Higher Education, 21(1) ijthe.org 34

Acknowledgments

The corresponding author (N. K.) would like to thank Mrs. S. Baudine and the administrative team of the Louvain Research Institute of Management and Organizations for their support of this project and help in organizing the workshops.


Thanh Vuong Dang
email: dangthanh@math.univ-paris13.fr

On the boundary controlled wave equation

The HUM (Hilbert Uniqueness Method) is the method to find the boundary control satisfying the exact controllability of a partial differential equation such as heat equation, wave equation and Navier-Stoke equation. We study discretization of the one dimensional boundary controlled wave equation with finite difference method. Under appropriate assumption, we prove some theorems relating to the controllability of the discrete system as well as the observability inequality with some cure methods.

Introduction

The HUM (Hilbert Uniqueness Method) is firstly introduced by J.L.Lions [START_REF] Lions | Exact Controllability, Stabilization and Perturbations for distributed systems[END_REF] and explained rigorously in [START_REF] Lions | On the controllability of distributed systems[END_REF] and in his book [START_REF] Glowinski | Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach[END_REF] with R.Glowinski and J.He. R. Glowinski, C-H-Li and J.L.Lions [START_REF] Glowinski | A Numerical Approach to the Exact Boundary Controllability of Wave Equation (I) Dirichlet Controls: Description of the Numerical Methods[END_REF] gave theoretical and numerical implementation of this method for two dimensional wave equation. Enrique Zuazua [START_REF] Zuazua | Propagation, Observation, and Control of Wave Approximated by Finite Difference Methods[END_REF] make a summary of optimal control results with one dimension wave system. The research about optimal control of wave equation and other equation has became active with some remarkable papers of M.Asch and G.Lebeau [START_REF] Asch | Geometrical aspects of exact boundary controllability for the wave equation -a numerical study[END_REF] in 1998, J.A.Infante and E.zuazua [START_REF] Zuazua | Boundary observation for the spacediscretizations of 1-d wave equation[END_REF] in 1999, L.León and E.Zuazua [START_REF] Le Ón | Boundary controllability of the finite difference space semidiscretizations of the beam equation[END_REF] in 2002, M.Negreanu and E.Zuazua [START_REF] Negreanu | Convergence of a multigrid method for the controllability of a 1-d wave equation[END_REF] in 2004, P.Loreti and M.Mehrenberger [START_REF] Loreti | An Ingham type proof for a two-grid observability theorem[END_REF] in 2008 and S.Ervedoza and E.Zuazua [START_REF] Ervedoza | The Wave Equation: Control and Numerics[END_REF] in 2011. In this paper, we study the boundary controlled wave equation together with fully discrete scheme and finite difference method. We also prove some theorems related to this scheme compared to continuous controlled wave equation. Furthermore, we indicate the efficiency of some cure methods namely Tikhonov regularization theoretically and numerically.

Model description and HUM

Similarly in [START_REF] Zuazua | Propagation, Observation, and Control of Wave Approximated by Finite Difference Methods[END_REF], we consider the one dimensional controlled wave equation. For any (w 0 , w 1 ) ∈ L 2 (0, 1) × H -1 (0, 1), find control v(t) ∈ L 2 (0, T ) such that      w tt -w xx = 0, 0 < x < 1, 0 < t < T, w(0, t) = 0, w(1, t) = v(t), 0 < t < T, w(x, 0) = w 0 (x), w t (x, 0) = w 1 (x), 0 < x < 1, [START_REF] Brezis | Functional Analysis, Sobolev spaces and Partial Differential Equations[END_REF] satisfies null-controllability w(x, T ) = w t (x, T ) = 0. Wave equation with zero bounday condition.

     u tt -u xx = 0 0 < x < 1, 0 < t < T u(0, t) = u(1, t) = 0 0 < t < T u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x) 0 < x < 1, (2)
where (u 0 (x), u 1 (x)) ∈ H 1 0 (0, 1) × L 2 (0, 1). The total energy is defined as

E(t) = 1 2 1 0 |u x (x, t)| 2 + |u t (x, t)| 2 dx = E(0). (3)
The null-controllability of ( 1) is equivalent to the observability of (2).

Definition 2.1. System (2) is said to be observable if there exists a constant C(T ) such that

E(0) ≤ C(T ) T 0 |u x (1, t)| 2 dt. (4)
There is a fundamental result indicating that we have the observability as well as null-controllability when T ≥ 2.

Proposition 2.1 ([14]

). For an arbitrary T ≥ 2, system (2) is observable. Otherwise, when T < 2, we have

sup u E(0) T 0 |u x (1, t)| 2 dt = ∞, (5)
where sup takes over all solutions u of (2).

The Hilbert Uniqueness Method is decribed in detail in [START_REF] Glowinski | A Numerical Approach to the Exact Boundary Controllability of Wave Equation (I) Dirichlet Controls: Description of the Numerical Methods[END_REF].

3 Fully discrete scheme and some related results

Given two integers M and N , we define mesh point

{x i } N +1 i=0 and {t j } M +1 j=0 . Set ∆x = 1 N + 1 and ∆t = T M + 1
. We obtain the fully discrete system of ( 1) and (2) (change the notation of initial data (1) to e 0 (x), e 1 (x) and replace the initial data of (2) by zero final data).

           u k+1 j -2u k j + u k-1 j |∆t| 2 - u k j+1 -2u k j + u k j-1 |∆x| 2 = 0, k = 1, M , j = 1, N , u k 0 = 0, u k N +1 = 0, k = 0, M + 1, u 0 j = e 0 j , u 1 j -u 0 j ∆t = e 1 j , j = 0, N + 1, (6)
where (e 0 j , e 1 j ) ∈ H 1 0,∆x (0, 1) × L 2 ∆x (0, 1), and the backward equation

               w k+1 j -2w k j + w k-1 j |∆t| 2 - w k j+1 -2w k j + w k j-1 |∆x| 2 = 0, k = 1, M , j = 1, N , w k 0 = 0, w k N +1 = - u k N ∆x , k = 0, M + 1, w M +1 j = 0, w M +1 j -w M j ∆t = 0, j = 0, N + 1. (7)
The energy and norm of the control are defined as

E n ∆x,∆t = ∆x 2 N j=1 u n+1 j -u n j ∆t 2 + ∆x 2 N +1 j=1 u n+1 j -u n+1 j-1 |∆x| u n j -u n j-1 |∆x| , (8)
and

| |u N /∆x|| 2 l 2 (0,T ) = M +1 k=0 u k N ∆x 2 .|∆t|. (9)
3.1 Discrete nonuniform observability Proposition 3.1. The energy defined as

(8) is preserved in time, i.e E n+1 ∆x,∆t = E n ∆x,∆t ∀n ∈ N.
The energy in ( 8) is non-negative, which is crucial for proving nonuniform observability with fully discrete scheme.

Lemma 3.1 ([15]

). The energy [START_REF] Marthedal | Boundary Control of Linear Evolution PDEs -Continuous and Discrete[END_REF] satisfies inequality

E n ∆x,∆t ≥ ∆x 2 1 - ∆t ∆x 2 N j=1 u n+1 j -u n j ∆t 2 . ( 10
)
Lemma 3.2 ([5]). For any ∆x > 0 and any eigenvalue λ of discrete laplacian operator and eigenvector w, we derive the formula

(∆x) N j=0 w j+1 -w j ∆x 2 = 2 4 -λ(∆x) 2 w N ∆x 2 . ( 11
)
For continous case, we know that we have the observability when T ≥ 2 (propostion (1)). However, with fully discrete scheme, we obtain the nonuniform observability for all T .

Theorem 3.1. Given a number 0 < α < 1, under assumption ∆t = α∆x, we derive for any T > 0, when ∆x → 0

sup u E n ∆x,∆t | |u N /∆x|| 2 l 2 (0,T ) → ∞, ( 12
)
where u is solution of (6).

Proof. We shall construct a solution to the scheme (6) using the separation of variables. More specifically, we look for a solution u k j of the form u k j = a k w ∆x N,j where w ∆x N,j is defined in

u k+1 j -2u k j + u k-1 j (∆t) 2 + λ ∆x N u k j = 0. ( 13
)
Then, we see that a k satisfies the following recurrence equation

a k+1 + (∆t 2 λ ∆x N -2)a k + a k-1 = 0, (14)
whose associated characteristic equation is

y 2 + (∆t 2 λ ∆x N -2)y + 1 = 0. ( 15
)
We compute the delta for this quadratic equation

∆ = 4α 2 sin 2 N π∆x 2 -2 2 -4, = 4 2 α 2 sin 2 N π∆x 2 α 2 sin 2 N π∆x 2 -1
We can see that the ∆ < 0. Hence, we derive two roots as follow.

     y 1 = 2 -∆t 2 λ ∆x N + i √ -∆ 2 , y 2 = 2 -∆t 2 λ ∆x N -i √ -∆ 2 (16)
From that, we can obtain the formula a k as follow.

a k = e ( 2-∆t 2 λ ∆x N 2 )k∆t C 1 cos √ -∆ 2 k∆t + C 2 sin √ -∆ 2 k∆t , (17)
where C 1 and C 2 are real numbers (independence on k).

In this proof, we choose C 1 = 1 and C 2 = 0. We get

a k = e ( 2-∆t 2 λ ∆x N 2 )k∆t cos √ -∆ 2 k∆t . (18)
We consider the u k = a k w ∆x N . From this formula, we tend to compute the energy E n ∆x,∆t and | |u N /∆x||2 . From (10), we have

E n ∆x,∆t ≥ ∆x 2 1 - ∆t ∆x 2 N j=1 a 1 -a 0 ∆t 2 |w N,j | 2 (19)
and

| |u N (t)/∆x|| 2 = M +1 k=0 a k ∆x
Remind that, we have identity

- w ∆x N,j+1 -2w ∆x N,j + w ∆x N,j-1 (∆x) 2 = λ ∆x N w ∆x N,j , ∀j = 1, N. (21)
Multiplying both side by w ∆x N,j and summing from j = 1, N , we derive

- N j=1 w ∆x N,j+1 -w ∆x N,j (∆x) 2 w ∆x N,j + N j=1 w ∆x N,j -w ∆x N,j-1 (∆x) 2 w ∆x N,j = λ ∆x N N j=1 |w ∆x N,j | 2 . ( 22
)
Changing index of the left side, we indicate that

N j=0 w ∆x N,j+1 -w ∆x N,j ∆x 2 = λ ∆x N N j=1 |w ∆x N,j | 2 . ( 23
)
From the identity (24) and lemma 3.1, we derive

(∆x) N j=1 |w ∆x N,j | 2 = 2 (∆x) 2 λ ∆x N (4 -λ ∆x N (∆x) 2 ) |w ∆x N,N | 2 . ( 24
)
From ( 19) and (24), we have

E n ∆x,∆t ≥ 1 - ∆t ∆x 2 a 1 -a 0 ∆t 2 1 λ ∆x N (∆x) 2 (4 -λ ∆x N (∆x) 2 ) |w ∆x N,N | 2 . (25)
We imply

E n ∆x,∆t | |u N (t)/∆x|| 2 ≥ 1 - ∆t ∆x 2 a 1 -a 0 ∆t 2 1 |∆x| 2 λ ∆x N (4 -λ ∆x N (∆x) 2 )|∆t| M +1 k=0 a k ∆x 2 (26) or E n ∆x,∆t | |u N (t)/∆x|| 2 ≥ 1 α 2 1 -α 2 (a 1 -a 0 ) 2 1 |∆x| 2 λ ∆x N (4 -λ ∆x N (∆x) 2 )|∆t| M +1 k=0 (a k ) 2 (27)
From here, we need to indicate that lim

∆t→0 |∆t| M +1 k=0 (a k ) 2 < +∞. ( 28
)
Indeed, for any k ≥ 0 and k ≤ M + 1, we have

|a k | = e ( 2-∆t 2 λ ∆x N 2 )k∆t cos √ -∆ 2 k∆t ≤ e, (29)
which implies

|∆t| M +1 k=0 (a k ) 2 ≤ M + 2 M + 1 e. (30)
Hence, left side of (27) tends to +∞ when N tends to +∞ (|∆x| 2 λ ∆x N → 4 -).

Remark 3.1. We might not prove the nonuniform observability when ∆t = ∆x. In practice, when ∆t = ∆x, the algorithm is stable.

Property of operator Λ in discrete Hilbert Uniqueness Method

With fully discrete scheme, we consider the corresponding HUM. We have fully discrete operator Λ ∆t,∆x : H 1 0,∆x (0, 1) × L 2 ∆x (0, 1) → L 2 ∆x (0, 1) × H -1 ∆x (0, 1) as follow.

Λ ∆t,∆x (e 0 j , e 1 j ) =

w 1 j -w 0 j ∆t , -w 0 j ( 31
)
Lemma 3.3. Λ ∆t,∆x defined in (31) is positive semidefinite and symmetric.

Proof. For two data (e 0 j , e 1 j ) and (e 0 j , e 1 j ) corresponding to solution (u, w) and (u, w).

< Λ ∆t,∆x (e 0 , e 1 ), (e 0 , e 1 ) >=

N +1 j=0 w 1 j -w 0 j ∆t u 0 j -w 0 j u 1 j -u 0 j ∆t . ( 32
)
We introduce the following identity

w i+1 -w i ∆t u i+1 -w i+1 u i+1 -u i ∆t = w j+1 -w j ∆t u j -w j u j+1 -u j ∆t , + ∆t j k=i+1 u k+1 + u k-1 -2u k |∆t| 2 w k - w k+1 + w k-1 -2w k |∆t| 2 u k .
We have < Λ ∆t,∆x (e 0 , e 1 ), (e 0 , e 1 ) >=

N +1 j=0 w 1 j -w 0 j ∆t u 1 j -w 1 j u 1 j -u 0 j ∆t . ( 33
)
Applying the identity, we derive < Λ ∆t,∆x (e 0 , e 1 ), (e 0 , e 1 ) > = ∆t

N j=1 M k=1 u k+1 j + u k-1 j -2u k j |∆t| 2 w k j - w k+1 j + w k-1 j -2w k j |∆t| 2 u k j , = ∆t M k=1 M j=1 u k j+1 -u k j |∆x| 2 w k j - w k j+1 -w k j |∆x| 2 u k j - M -1 j=0 u k j+1 -u k j |∆x| 2 w k j+1 - w k j+1 -w k j |∆x| 2 u k j+1 , = ∆t M k=1 u k N +1 -u k N |∆x| 2 w k N - w k N +1 -w k N |∆x| 2 u k N - u k 1 -u k 0 |∆x| 2 w k 1 + w k 1 -w k 0 |∆x| 2 u k 1
Using the changing index technique, we obtain < Λ ∆t,∆x (e 0 , e 1 ), (e 0 , e 1 ) >= ∆t ∆x

M k=1 w k N +1 .w k N +1 (34)
From that, we imply the positive semidefinite < Λ ∆t,∆x (e 0 , e 1 ), (e 0 , e 1 ) >= ∆t ∆x

M k=1 (w k N +1 ) 2 . ( 35
)
Remark 3.2. With this property of discrete Λ ∆t,∆x , the conjugate gradient method might be applied to compute the discrete control (see [START_REF] Glowinski | A Numerical Approach to the Exact Boundary Controllability of Wave Equation (I) Dirichlet Controls: Description of the Numerical Methods[END_REF]).

Observability inequality with Tikhonov regularization

There are some possibilities to eliminate the oscillations of discrete control. We mention here two methods which is discribed in [START_REF] Glowinski | A Numerical Approach to the Exact Boundary Controllability of Wave Equation (I) Dirichlet Controls: Description of the Numerical Methods[END_REF]. Theorem 3.2. For T > 4 + 2∆t, there exists a C(T, ∆t) > 0 such that

E 0 ∆x,∆t ≤ C(T, ∆t) (∆t) M k=1 u k N ∆x 2 +(∆x) 3 (∆t) M k=0 N j=0 u k+1 j+1 -u k j+1 -u k+1 j + u k j (∆x)(∆t) 2 + (∆x)(∆t) 3 4 M k=0 N j=0 u k+1 j+1 -u k+1 j -u k j+1 + u k j (∆x)(∆t) +(∆x)(∆t) 2 M k=0 N j=0 u k+1 j+1 -u k j+1 -u k+1 j + u k j (∆x)(∆t) 2   . (36)
Furthermore, lim ∆t→0 C(T, ∆t) = c(T ).

Proof. We have the following identity

|∆t| M k=1 N j=1 j u k+1 j -2u k j + u k-1 j |∆t| 2 u k j+1 -u k j-1 2 = |∆t| M k=1 N j=1 j u k j+1 -2u k j + u k j-1 |∆x| 2 u k j+1 -u k j-1 2 (37)
We have

M k=1 N j=1 j u k+1 j -2u k j + u k-1 j |∆t| 2 u k j+1 -u k j-1 2 = M k=1 N j=1 j u k+1 j -u k j |∆t| 2 - u k j -u k-1 j |∆t| 2 u k j+1 -u k j-1 2 , = N j=1 j M k=1 u k+1 j -u k j |∆t| 2 u k j+1 -u k j-1 2 - M -1 k=0 u k+1 j -u k j |∆t| 2 u k+1 j+1 -u k+1 j-1 2 , = N j=1 j M k=1 u k+1 j -u k j |∆t| 2 u k j+1 -u k j-1 2 - M -1 k=0 u k+1 j -u k j |∆t| 2 u k+1 j+1 -u k+1 j-1 2 , = N j=1 j M k=0 u k+1 j -u k j |∆t| 2 u k j+1 -u k j-1 2 - u k+1 j+1 -u k+1 j-1 2 + X ∆x , = M k=0 N j=1 u k+1 j -u k j |∆t| 2 u k+1 j+1 -u k j+1 2 + X ∆x ,
where

X ∆x = N j=1 j u M +1 j -u M j |∆t| u M +1 j+1 -u M +1 j-1 2|∆t| - u 1 j -u 0 j |∆t| u 0 j+1 -u 0 j-1 2|∆t| .
Moreover, we derive

M k=1 N j=1 j u k j+1 -2u k j + u k j-1 |∆x| 2 u k j+1 -u k j-1 2 = M k=1 N j=1 j (u k j+1 ) 2 -2u k j u k j+1 + 2u k j u k j-1 -(u k j-1 ) 2 |∆x| 2 , = M k=1 N j=1 j (u k j+1 ) -u k j ) 2 -(u k j -u k j-1 ) 2 2|∆x| 2 , = - 1 2 M k=1 N j=0 u k j+1 -u k j ∆x 2 + 1 2|∆x| M k=1 u k N ∆x 2 .
Hence, we derive the following identity

|∆x||∆t| M k=0 N j=1 u k+1 j -u k j |∆t| 2 u k+1 j+1 -u k j+1 2 + |∆x||∆t| 2 M k=1 N j=0 u k j+1 -u k j ∆x 2 + |∆x||∆t|X ∆x , = |∆t| 2 M k=1 u k N ∆x 2 .
We imply

T E 0 ∆x,∆t + |∆x||∆t| 2 M k=0 N j=0 (u k+1 j+1 -u k j+1 -u k+1 j + u k j ) u k+1 j -u k j |∆t| 2 - u k j+1 -u k j |∆x| 2 + |∆x||∆t|X ∆x = |∆t| 2 M k=1 u k N ∆x 2 + |∆x||∆t| 2 N j=0 u 0 j+1 -u 0 j ∆x 2 .
which is equivalent to

T E 0 ∆x,∆t + |∆x||∆t| 2 M k=0 N j=0 (u k+1 j+1 -u k j+1 -u k+1 j + u k j ) u k+1 j+1 -u k j+1 |∆t| 2 - u k j+1 -u k j |∆x| 2 + |∆x||∆t|X ∆x = |∆t| 2 M k=1 u k N ∆x 2 + |∆x||∆t| 2 N j=0 u 0 j+1 -u 0 j ∆x 2 + |∆x||∆t| 2 M k=0 N j=0 u k+1 j+1 -u k j+1 -u k+1 j + u k j |∆t| 2 .
Moreover,

|∆x||∆t| 2 M k=0 N j=0 (u k+1 j+1 -u k j+1 -u k+1 j + u k j ) u k+1 j+1 -u k j+1 |∆t| 2 - u k j+1 -u k j |∆x| 2 = |∆x||∆t| 2 M k=0 N j=0 u k+1 j+1 -u k j+1 |∆t| 2 - |∆x||∆t| 2 M k=0 N j=0 u k+1 j -u k j |∆t| u k+1 j+1 -u k j+1 |∆t| - |∆x||∆t| 2 M k=0 N j=0 u k+1 j+1 -u k+1 j |∆x| u k j+1 -u k j |∆x| + |∆x||∆t| 2 M k=0 N j=0 u k j+1 -u k j |∆x| 2
Applying Cauchy-Schwarz inequality, we have

u k+1 j+1 -u k+1 j |∆x| u k j+1 -u k j |∆x| + |∆x||∆t| 2 M k=0 N j=0 u k j+1 -u k j |∆x| 2 ≥ |∆x||∆t| 4 N j=0 u 0 j+1 -u 0 j ∆x 2 - |∆x||∆t| 4 N j=0 u M +1 j+1 -u M +1 j ∆x 2
Hence, we get the following inequality

T E 0 ∆x,∆t +|∆x||∆t|X ∆x - |∆x||∆t| 4 N j=0 u 0 j+1 -u 0 j ∆x 2 + |∆x||∆t| 4 N j=0 u M +1 j+1 -u M +1 j ∆x 2 ≤ |∆t| 2 M k=1 u k N ∆x 2 + (∆x) 3 (∆t) 2 M k=0 N j=0 u k+1 j+1 -u k j+1 -u k+1 j + u k j (∆x)(∆t) 2 (38)
We have

(∆x)(∆t)|X ∆x | ≤ 4E n ∆x,∆t + (∆x)(∆t) 2 2 M k=0 N j=0 u k+1 j+1 -u k j+1 -u k+1 j + u k j (∆x)(∆t) 2 (39)
Furthermore, (∆x)(∆t) 4

u 0 j+1 -u 0 j ∆x 2 = (∆x)(∆t) 3 4 u 0 j+1 -u 0 j -u 1 j+1 + u 1 j (∆x)(∆t) 2 + (∆x)(∆t) 2 (u 0 j+1 -u 0 j ) ∆x u 1 j+1 -u 1 j ∆x - u 1 j -u 1 j+1 ∆x 2 , ≤ (∆x)(∆t) 3 4 N j=0 u 0 j+1 -u 0 j -u 1 j+1 + u 1 j ∆x 2 + (∆t)E 0 ∆t,∆x . (40) Similarly, (∆x)(∆t) 4
N j=0 u M +1 j+1 -u M +1 j (∆x) 2 ≤ (∆x)(∆t) 3 4 N j=0 u M +1 j+1 -u M +1 j -u M j+1 + u M j (∆x)(∆t) 2 +(∆t)E M ∆t,∆x . ( 41
From ( 40) and (41),

- (∆x)(∆t) 4 N j=0 u 0 j+1 -u 0 j ∆x 2 + (∆x)(∆t) 4 N j=0 u M +1 j+1 -u M +1 j ∆x 2 ≤ M k=0 N j=0 (∆x)(∆t) 3 4 u k+1 j+1 -u k+1 j -u k j+1 + u k j ∆x 2 + 2(∆t)E 0 ∆t,∆x (42)
From ( 38), (39) and (42), we have

(T -4-2∆t)E 0 ∆x,∆t ≤ |∆t| 2 M k=1 u k N ∆x 2 + (∆x) 3 (∆t) 2 M k=0 N j=0 u k+1 j+1 -u k j+1 -u k+1 j + u k j (∆x)(∆t) 2 + (∆x)(∆t) 3 4 M k=0 N j=0 u k+1 j+1 -u k+1 j -u k j+1 + u k j (∆x)(∆t) 2 + (∆x)(∆t) 2 2 M k=0 N j=0 u k+1 j+1 -u k j+1 -u k+1 j + u k j (∆x)(∆t) 2 (43)
From (43), we obtain C(T, ∆t) = 1 2(T -4 -2∆t)

. When ∆t → 0, C(T, ∆t) → 1 2(T -4) .

With bigrid algorithm, we predict that the observability inequality satisfies when T > 2 √ 2 (see [START_REF] Loreti | An Ingham type proof for a two-grid observability theorem[END_REF]).

Conjecture 1. For N is an odd number and T > 2 √ 2, there exists a constant C(T ) such that

E n ∆x,∆t ≤ C(T )| |u N /∆x|| 2 l 2 (0,T ) . (44)
for all solution u obtained by bigrid algorithm with system (6) and (7).

Numerical experiment

We consider two numerical examples. Numerical experiment 1: We take y 0 is a piecewise constant function and y 1 is a zero function. With the mesh size in space N = 41, 81, 101, 121, we have We observe that when N is increased, there are more and more oscillations with discrete control and it seems that they will blow up.

With Tikhonov regularization, the discrete control convergences to the exact control. From this example, we predict that with continous data (u 0 (x), u 1 (x)) or (y 0 (x), y 1 (x)), there are not many oscillations and they do not blow up.

Conclusion

We use the finite difference scheme to discretize wave equation and compute the discrete control. Theoretically, we prove that the discrete observability inequality does not hold for any value T > 0, which is similar to [START_REF] Zuazua | Propagation, Observation, and Control of Wave Approximated by Finite Difference Methods[END_REF]. Furthermore, we also have the symmetry and positive defined properties of discrete operator Λ ∆x,∆t which is claimed by R.Glowinski, C-H.Li and J.L.Lions in [START_REF] Glowinski | A Numerical Approach to the Exact Boundary Controllability of Wave Equation (I) Dirichlet Controls: Description of the Numerical Methods[END_REF]. With conjugate gradient method, we compute the discrete control with blowup oscillations when we have nonsmooth data. This phenomenon is not occured with smooth data. Furthermore, we implement the Tikhonov regularization to eliminate the oscillations when T = 3. However, theoretical results show that this cure method works efficiently when T > 4.

Figure 1 :Figure 3 :

 13 Figure 1: N = 41 Figure 2: N = 81

Figure 5 :Figure 7 : N = 101 Figure 8 : N = 121 Remark 4 . 2 .Numerical experiment 2 :

 571018121422 Figure 5: N = 41 Figure 6: N = 81

Figure 9

 9 Figure 9: N = 41

.|∆t|.|w ∆x N,N | 2 . (20)


N Montès

Vilao B ; Badri

T ; & Gauquelin

Bertaudière-Montès

DIVERSITY OF JUNIPERAS THTIRIFERA L. TREE SHAPES: ORIGINS AND CONSEQUENCES ON BTOMASS ALLOCATTONS

Keywords: Juniperus thurifera, Architecture des arbres, Climat montagnard, Allocation de biomasse

In any points of its distribution area, Junipen' a result ôf interactions between endogenous and exo( identified 4 simple morphological types on the basis of c number of stems (ranging lrom 1 to 12). We discriminate (MS) in the case of (i) overall crowned trees with the c icrown length ratio < 1) with a visible trunk (PC)' Tre (crown repartition along the trunk x number oJ trunk; e' àtemmed trees and overallcrowned trees while single-s The determinism of the multi-stemmed trait is probably Mutti-stemmed tnr"âiri.rgri itràn singte-stemmed tiees with a higher biomass allocation in woody parh (trunks ànd branches) ano a nignàrïood prodrËtivity. The overall crownedlhape determinism cnuld be related to the low iàmpe,,atures at nign àlftude, trees' shape béing close to those of xerophytic shrubs and allocating more energy t0 the photosYnthetic organs. Keywordsi Juniperué thurifera,Tree architecture, Mountain climate, Biomass allocation

DIVERSTDAD MORFOLÔGrcA DE LAS SABINAS Y ENEBRALES: ORIGEN Y CONSECUENCIAS EN LAS ASIGNACIONES DE BIOMASA

Resumen En el coniunto de su ârea de repartici6n, la I resultado de la interacciôn de los factores end6genos 1 ' localizan con mayor lrecuencia en fuertes pendientes' t ra artitud estâ en retacùn frozrotemente ai debiritamiento,* â.0'#lr1: ij:l'.:ïJl$iii:i:ii,fri.iff:

;ién de las formas de la copa alsol puede estar ligada a las i ârboles adoptan ei porte caracterfstico de los xerétitos otosintéticos. nérbotes

INTRODUCTION

Thuriferous juniper is only found in isolated parts of the 'Westem Mediterranean Basin : France (Alps, Pyrenees and Corsican highlands), Spain, Algeria and Morocco [START_REF] Barbero | Les apports de la phytoécologie dans I'interprétation des changements et perturbations induits par I'Homme sur les écosystèmes forestiers méditerran éen[END_REF][START_REF] Blanco Castro | Sabinares y enebrales: Los bosques ibericos, una interpretacion geobotanica[END_REF]Ch:.rro and Blanc, 1997).In Morocco, these semi arid mountain stands, where thuriferous juniper trees grow in low-density open woodland, are seriously endangered as a result of the intensive wood removal and livestock activity in these densely populated areas (Gauquelin et a1.,1999).

On almost all of its natural range, Juniperus thurifera L. is characteizedby a large architectural diversity, as a result of an interaction betrnreen endogenous and exogenous factors. In France, about 8 architectural types were described (Polidori, 1986;[START_REF] Lathuillière | Le Genéwier thurifère (Juniperus thurifera L.). Monographie. Etude de la Thuriferaie de St-Crépin. Le Genéwier thurifère dans le Sud-Est de la France[END_REF], most of them in Alps related to the diversity in ecological parameters (climate, slope, altitude, exposure, soil, inter-specific competition). In Spain, some additional morphological types can be found in response to silvicultural and pastoral activities [START_REF] Blanco Castro | Sabinares y enebrales: Los bosques ibericos, una interpretacion geobotanica[END_REF]. [n Morocco, we also found particular shapes due to strong human impact (wood removals and overgrazing) and severe climatic constraints. We will focused in this paper on Moroccan stands, considering only few morphological types that could be objectively discriminated. Other shapes will only be mentioned.

MATERIALS AND METHODS

Thuriferous juniper is a dioecious bush or tree, with scaly leaves (we will employ the word "foliage" for these chlorophyllous organs) and black-bluish cones when mature. In Morocco, Juniper stands are found at the tree line mainly between elevations of 2,000 and 3,000m [START_REF] Gauquelin | DynaSnique de la végétation et des fonnations superficielles dans les montagnes du bassiô.occidental de la Méditerranée:Etude des formations à genévrier thurifère et à xérophyqgs épineuses en coussinet des Atlas marocains[END_REF]. Isolated specimens may be found above 3,000m (the lowestlevel stands are at 1,700m in the western High Atlas).

The site we hâve studied is the Juniperus thurifera woodland of Azzadenvalley, in the High Central Atlas, south of Marrakesh (Morocco). tn the upper level of this valleyjthis juniper (which can grow to 18m with a diameter of 5m) is the only arborescent woody species adapted to the severe high mountain Mediteranean climate, and represents the sole wood resource of the local tserber populations . The tree grows in low-density woodlands (about 78 trees.ha-' (Montès,,I999) with fairty old trees (probably more than several hundred years old), which are,.often scarrecl, and with a steppic shrù strata (Bupletmtrn spinosumL., Cytisus purgans L. Benth. ssp. balansae (Boiss.) M.,Alyssum spinosumL.).

In this study, 102 trees rilere sampled in a vaiety of topographic conditions. The slopes ranged from 10 to 37 degrees, and the elevation ranged from l,9l5m to 2,350m.

The dendrometric parameters measured were height of the trees, basal circumference, crown projection area, crown length, number of living stems and number of cut stems.

The estimation of tree biomass was obtained in a previous study (Montès et al., 2002). The estimation of biomass of junipers is based on a non destructive method (Montès et a1.,2000b;Montès et a1.,2000a) very briefly described here: it consists of photographic sampling of trees to reconstruct the different component volumes of the trees (trunks/branches, branchlets (>20mm), and leaves). In view of the axial asymmetry of these junipers, two pictures of each tree were taken with orthogonal views. Then biomass is obtained (after the estimation of the bulk density of each component) by the formula: B : V x d (B: Biomass; V: Volume estimated for each component; d: Estimated density for each component). Similarly, the biomass of cones was measured on female trees.

On the basis of crown length ratio (ratio of crown length to tree height) and number of stems (ranging from 1 to l2). we identified four simple morphological types (Figure 1):

-Single-stemmed trees (SS): this morphological type correspond to trees with only one trunk and possible cut trunk. -Multi-stemmed trees (MS): trees with two or more trunks. Independently of the number 6f ffunks, we also discriminated:

-Trees with a crown length ratio: I,that mean overall crowned trees (OC).

-Trees with a crown length ratio < l, that mean the trunk(s) is(are) visible(s) (partially crowned trees, PC).

One way analysis of variance (ANOVA) were used to compare tree shapes according to the studied factors (descriptive factors, explicative factors and biomass allocations). Significant level was considered to be 95 oÂ. The software Statview@ (version 4.55, Abacus concepts, Inc., Copyright 1992-1996) was used.

RESULTS

Descriptive factors:

Even if the size was not used to discriminate overall crowned trees from partially crowned trees, the results show that partially crowned trees are significantly taller and have a higher basal circumference (Table 1) than overall crowned trees. Despite a low mean height (OC : 3.6 m and PC :4.3 m), trunk of trees is very large (diameter higher than 0,8 m). The differences between single-stemmed trees and multi-stemmed trees relate not only on the trunk, but also on trees' crown. For all measured variables (crown projection area, crown length, height and basal circumference), overall crowned and multi-stemmed trees (OC-MS) present values significantly higher than single-stemmed trees (OC-SS).

Explicative factors:

Overall crowned trees are located at significantly higher altitudes than partially crowned trees ( Table 2).The numbergf overall crowned trees increases along an altitudinal gradient (altitudinal sections of 125 m\whereas partially crowned trees are completely missing at 2,500 m (Figure . 2). The effect*of altitude is also significant on the distribution of multistemmed trees of which the proportion increases with the altitudinal gacfièn1. The distribution of single-stemmed trees and multi-stemmed trees along the altitudinal gradient is however less regular than it is for overall crowned trees and partially crowned trees (Figuer. 3), but differences between exheme altitudes are very marked. There is a similar proportion of single-stemmed and of multi-stenrrned trees at the lower altitude whereas there are only multi-stemmed trees at the higher altitude. Single-stemmed trees are also located on slopes significantly more pronounced than multi-stemmed trees. This is particularly tnre for the overall crowned trees. The percentage of cut stems by individual is significantly higher for partially crowned trees than for overall crowned trees. On the other haqd, no significant difference between single-stemmed trees and multi-stemmed trees could be highlighted.

Biomass allocation:

Table 3 reveals that biomass of multi-stemrned trees are higher than biomass of single stemmed-trees. However this difference is only significant in the case of the overall crowned trees. No significant difference in the weight of the individuals was evidenced between partially crowned trees and overall crowned trees. For the same cover, partially crowned trees represent, in a very significant way, more biomass than overall crowned trees (respectively 28.1and 18.4 kg.m-t). The observation is similar for multi-stemmed trees with an overall crown.

For all morphological types, the relative importance of the biomass of the various compartments is the same one: Trunk-Branches >> Leaves > Twigs. However, the strategies of biomass allocations between the various compartments of the piant differ according to the number of trunks and the importance of the crown.

The multi-stemmed and overall crowned trees (OC-MS) and partially crowned trees allocate significantly more energy than single-stemmed trees and overall crowned trees to the trunk-branches comparfment than to the branches and photosynthetic compartments (Table 3). In the multi-stemmed overall crowned trees (OC-MS), the preferentiâl way of biomass allocation in direction of the ligneous comparfment induces a higher productivity. No significant difference between morphological types was evidenceà conceming the energy invested in the reproduction (females' cones). Within partially crowned tr"es, whatever the parameter considered (descriptive factor, explicative factor or biomass allocation), there is a very strong homogeneity in the results (except for basal circumference) between single stemmed trees and multi-stemmed trees.

DISCUSSION.C ONCLUSION

The dominance of overall crowned trees at higher altitudes participates to an homogenization of architectural diversity. Trees and xerophytic shrubs present a dense canopy isolating the stem from very low temperatures in winter (Badri ei a1.,1994). The cambium is thus protected from frost damages.

At high altitude, these overall crowned trees are often multi-stemmed. The presence of multi-stemmed trees seems to be linked to the loss of apical dominance due to frost events on apical buds (Bertaudière et al.,200I). However, if climatic events could be responsible from the grcwth of secondary shoots developing from adventive buds [START_REF] Emberger | Aperçu général sur la végétation du Maroc. Commentaire de la carte phytogéographique du Maroc au 1/500 000ème[END_REF], the presence of multi-stemmed junipers càuld also be related to anthropogenic (browsing and cutting) factors.

Even if no significant dif[erences were observed between multi and single stemmed trees concerring the number of dead stems (as a measure of cutting impact), the grazing impact was not quantified but may be an additive factor especially on iaplings.

The presence of multi"stemmed junipers (trees but also saplings È.r ihutr 3 years old) in well preserved areas (French Pyreneàs ""4 alpg w-her.e trumalaisturUances (grazrïrg a1d, cutting) are limited, show never'theless that genetic factors could be involved in multiCaulis structure determinism.

The higher percentage of cut stems observed on partially crowned trees reflect a human impact that occured on old trees. Conversely, on overall crowned trees, stems could have been cut on saplings and are not visible today.

The higher total biomass of multi-stemmed trees seems to be related to their architecture: they could be compared to a sum of single-stemmed trees with their own stem and crown. Indeed, both crown projection area and basal circumference of these trees are higher than of single-stemmed. Even if competition for water and nutrient supply between stems of the same tree can occur, these competition phenomenon's appear nevertheless not to be disadvantageous for the radial growth of the trees with a low number of trunks. As shown by Bertaudlère et al. (2001), the comparison of the radial growth rate between multiand single-stemmed trees reveals no important differences. Thereby, wood productivity is significantly higher for overall crowned multi-stemmed trees (OC-MS). T-he multicaulis strucfure of Juniperus thurifera could be considered as an adaptation to severe environment (hard topographical, edaphic and climatic conditions and overgrazing).

The higher biomass allocation to woody compartments (trunk and branches) of partially crowned trees and overall crowned multi-stemmed trees (OC-MS) is expressed by a higher basal circumference. At high altitude, the biomass allocation strategy favours foliage particularly in the case of single-stemmed trees. In spite of the factors affecting architecture of the trees, none of them has affected the biomass allocation strategy conceming reproductive organs.

Figuer. 3 :

 3 Figuer. 3: Distribution of morphological types with altitude

Table 1 :

 1 Montès, N., Bertaudière-Montès, V., Badri, 'W., Zaoui, E.H., Gauquelin, T., 2002. Biomass and nutrient content of a semi-arid mountain ecosystem: the Juniperus thuriftra L. woodland of AzzadenValley (Morocco). For Ecol Manag 166,35-43. Polidori, J.L., 1986. Guide des conifères du Mercantour. Centre Régional de Documentation pédagogique de Nice Descriptive factors (bold means sigrrificant)

	TABLES A}[D FIGT]RES
	Padially crowned	Ovqall crowaed
	tÎee	ttr¤s

Table 3 :

 3 Biomass allocations (bold rireans signifrcant)