Ameya Vaishnavi

Elie Murukutla

Benjamin Cattan

Remi Lecouteux

Olivier Ronfard

Palombi

Vaishnavi Ameya Murukutla

Elie Cattan

Benjamin Lecouteux

Remi Ronfard

Olivier Palombi
email: olivier.palombi@univ-grenoble-alpes.fr

Text-to-Movie Authoring of Anatomy Lessons

Keywords: Anatomy pedagogy, Domain Specific Languages, Finite state machines, Computer animation

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Introduction

Anatomy is one of the most essential yet challenging subjects in medical education. It is introduced very early in the curriculum, and the student must retain this knowledge throughout their practice. One of the significant challenges faced in anatomy pedagogy is presenting complex three-dimensional body parts in classes. Multimedia methods such as animated videos and 3D anatomical models are popular to visualise the human body. However, the issue of developing an easy-to-use software that enables teachers to create their own lessons without the need for design experts is ongoing. Anatomy professors create all the teaching materials used in lessons, either by themselves or by others following their close instructions. In France, until a few years ago, the teachers would draw the part they were teaching on the chalkboard in front of the class as they led the lesson. In this paper, we present a Text-to-movie authoring system for anatomy professors. Our goal is to have fast production, quick visualisation and easier editing giving teachers more freedom at different stages of content creation. We chose text as the method for input in our system as this fits seamlessly with the current way teachers create their lessons and it is also particularly favourable as the teachers did not have to learn a new and often intimidating user interface for design software, making our system intuitive and easy to learn. We present a domain-specific language called the Anatomy Storyboard Language (ASL) which forms the basis of input for our authoring system. The users have to write scripts in ASL in which they list all the anatomical parts that they want to show in the video and outline the directions for the camera movements and animation of the parts. Our system reads the script, and the resulting animation is played in a Unity player. Then we present the results of our evaluation of the software done with four anatomy professors. The teachers could make their own narrated, animated video lessons on the knee. The study showed that their limited experience in animation or video editing did not hold them back from creating an informative course.

State of the art

Anatomy learning has two parts: theory and practical. The lesson starts with classroom lectures with visual aids such as chalkboard drawings and slide presentations. These sessions are for the entire class, and then students are divided into smaller groups for practical dissection sessions on cadavers and prosections. The students have better access to the teachers to ask questions and are informally evaluated on their knowledge by oral questioning during the practical sessions. However due to increasing class sizes, reduced faculty and less time allotted for the anatomy curriculum it has become imperative that traditional methods of teaching be augmented with multimedia approaches. In our survey, presented further in the paper, teachers have stated that they are willing to invest their time and expertise to develop digital content by themselves.

Various multidisciplinary techniques have been introduced to make anatomical learning more engaging and effective [START_REF] Estai | Best teaching practices in anatomy education: A critical review[END_REF]. While there cannot be a complete digitalisation of anatomical lectures, it is beneficial to use multimedia tools as a part of the blended learning approach to better prepare the students for the hands-on practical sessions [START_REF] Majerník | Preparation of medical students for cadaveric anatomy using multimedia education tools[END_REF][START_REF] Stirling | An enriched multimedia ebook application to facilitate learning of anatomy[END_REF][START_REF] Khalil | Teaching of anatomical sciences: A blended learning approach[END_REF]. The use, for instance, of a virtual dissection table, with its straightforward handling, enables the instructor to create multimedia content that is easy to record. These live sessions are often highly valued by students [START_REF] Alasmari | Medical students' feedback of applying the virtual dissection table (anato-mage) in learning anatomy: A cross-sectional descriptive study[END_REF]. This approach is not only engaging for the students but also cost-effective [START_REF] Khalil | Teaching of anatomical sciences: A blended learning approach[END_REF][START_REF] Clifton | The importance of teaching clinical anatomy in surgical skills education: Spare the patient, use a sim![END_REF]. As anatomy involves the study of complex spatial structures, it can benefit from recent results in computer graphics and informatics [START_REF] Trelease | Anatomical informatics: Millennial perspectives on a newer frontier[END_REF]. The effectiveness of new methods has been confirmed in previous studies [START_REF] Hoyek | Effectiveness of three-dimensional digital animation in teaching human anatomy in an authentic classroom context[END_REF][START_REF] Pereira | Using videoclips to improve theoretical anatomy teaching[END_REF][START_REF] Vernon | The benefits of 3d modelling and animation in medical teaching[END_REF]. We can particularly mention immersive approaches based on 3D anatomy software like Z-anatomy software, which allow, among other things, the use of 2D sections with didactic value well known to anatomists. Student engagement and immersion can be further enhanced by using 3D glasses like Oculus, which can be simultaneously used by multiple users [START_REF] Uruthiralingam | Augmented and virtual reality in anatomical education -a systematic review[END_REF]. All of these tools can be combined with audio recording capabilities for course recording. However, the approaches can be more sophisticated. Pereira et al. use a mix of teacher-created recordings and previously available material, whereas Hoyek et al. use the extensive library of 3D animated videos created by Lyon 1 university in collaboration with a graphics team 1 . Both these studies highlight the need for a teacher authoring system for animated content creation. The videos made by the Anatomie 3D Lyon team have been integral in our process of designing and developing an authoring system. As these videos fit the standard practices of teaching anatomy in France we modelled our final output to be similar to them. Animated videos are the most commonly used teaching aids in anatomy [START_REF] Hulme | Videos in anatomy education: History, present usage and future prospects[END_REF]. Even beyond the course suggestions, students reported using video sharing sites such as YouTube to learn new concepts, clear doubts or better visualise anatomy [START_REF] Jaffar | Youtube: An emerging tool in anatomy education[END_REF].

Authoring high-quality 3D animation is a costly process requiring the skills of designers, animators and directors as seen in the workflow chart in Figure :1. Likewise, creating 3D animation for a new course in anatomy involves time, effort and money. Few authoring systems are easy enough for an anatomy teacher not trained in computer graphics to create 3D animation by herself. Recently, commercial game engines have started providing visual programming tools to facilitate the creation of 3D animation (Unreal Engine's blueprints2 , Unity's Cinemachine3 and Timeline Editor, Huttong Games' PlayMaker 4 and even spatial programming tools to facilitate the creation of mixed-reality content (Unity's EditorXR5 ). But they are better suited for expert game developers than medical anatomy teachers.

Clearly, there is a need for authoring tools providing better support for authors who are experts in their own field (human anatomy) and not in computer graphics and animation. In the field of web design, this problem has effectively been addressed by introducing a graphic style sheet, which clearly separates the roles of content experts and style experts. The corresponding notion of an animation stylesheet has been advocated by Ken Perlin [START_REF] Perlin | Toward interactive narrative[END_REF] but remains an open research question.

Text-to-movie (or text-to-scene) authoring is a general class of methods that have been proposed for automatically generating 3D graphics and animation from text written by a domain expert. An overview of these methods are given in [START_REF] Ronfard | Film Directing for Computer Games and Animation[END_REF]. Good results have been obtained in limited domains, such as generating 3D scenes from natural language accident reports [START_REF] Akerberg | Carsim: An automatic 3d text-to-scene conversion system applied to road accident reports[END_REF][START_REF] O'kane | Text-to-scene conversion for accident visualization[END_REF] or generating cartoon animation from scripted dialogue scenes [START_REF] Seversky | Real-time automatic 3d scene generation from natural language voice and text descriptions[END_REF]. Commercially available text-tomovie systems such as NawmalMAKE 6 and Plotagon Studio7 are specifically designed to generate dialogue scenes in selected cartoon styles. Xtranormal Technology Inc. first used the term text-to-movie to describe their authoring system that combined the previously used text-to-speech and text-to-scene concepts. Text-to speech enables text to be converted into speech signals that imitate human voice and intonation. This can be used to convert written dialogue in scripts into speech or as voiceovers in instructional videos. Text-to-scene enables the visualisation of natural language descriptions. This needs 3D models that are labelled and positioned in the 3D world depending on the text descriptions. A combination of these two technologies, along with the addition of animations and interactions of virtual actors with the 3D scene based on text descriptions, falls under the domain of text-to-movie authoring. In a system called State developed by Xtranormal, the input is "text" consisting of dialogues and a combination of markers that are similar to emojis. These markers direct the actor movement, facial expressions, posture, interactions and also the camera movements. In essence, the input method is a mix of natural language text and markers that directs both camera and actor actions. The output is synthetically generated and can be visualised immediately as the 3D scene is being built and directed (Figure :2). Generic text-to-movie authoring is also an active area of research. Ye and Baldwin described a system for automatically generating storyboards from natural language movie scripts [START_REF] Ye | Towards automatic animated storyboarding[END_REF]. More recently, Marti et al. described the CARDINAL system [START_REF] Marti | Cardinal: Computer assisted authoring of movie scripts[END_REF] which automatically generates 3D animation of movie scripts for pre-visualisation .

One key requirement for a text-to-movie application is the ability to associate graphic objects and animations with a large vocabulary of concepts expressed in natural language. In the field of medical anatomy teaching, My Corporis Fabrica (MyCF), provides an extensive ontology of human anatomy concepts linked to 3D models [START_REF] Palombi | My corporis fabrica: an ontology-based tool for reasoning and querying on complex anatomical models[END_REF], which have been used to demonstrate quick browsing and visualisation from high-level queries. In this work, we extend the idea by proposing a system that automatically translates anatomical queries into animated movies.

Another key requirement for a text-to-movie application is the ability to automatically place and control the virtual camera so that anatomical entities are shown appropriately and with the right timing. Previous work in intelligent cinematography and editing such as those done by Galvane et al. [START_REF] Galvane | Narrative-Driven Camera Control for Cinematic Replay of Computer Games[END_REF] and He et al. [START_REF] L.-W. He | The virtual cinematographer: A paradigm for automatic real-time camera control and directing[END_REF] must be carefully adapted for the special case of anatomy. In this work, we introduce a special purpose authoring language with a rich vocabulary of cinematographic terms relevant to medical anatomy as seen in the workflow chart in Figure:

Text-to-movie

Anatomy Storyboard Language

The input for our system is the text written in a formal language called the Anatomy Storyboard Language (ASL). It is a domain-specific language in which the video to be produced is written as a set of unique sentences. Each sentence describes all the visual elements, camera actions and animations seen from the start of the recording till the camera stops. It is an anatomical extension of the Prose Storyboard Language [START_REF] Ronfard | The prose storyboard language: A tool for annotating and directing movies[END_REF] that was designed for annotating and directing movies.As each sentence generates a complete shot, it must have all the information necessary to transition into the shot, build the composition, direct camera movements, record all the developments from the initial composition and finally describe the last composition before the camera stops. the detailed And/Or graph of the grammar for Anatomy storyboard language is presented in Figure : 4.

ASL is a context-free language with terminals such as anatomical entities and cinematographic terms, and non-terminals, such as initial compositions and subsequent developments. The terminals are either generic terms used for camera action or animation, or specific terms referring to the subjects described in the shot such as anatomical parts and regions. The nomenclature of these specific terminals is derived from My Corporis Fabrica (MyCF) [START_REF] Palombi | My Corporis Fabrica: a Unified Ontological, Geometrical and Mechanical View of Human Anatomy[END_REF][START_REF] Palombi | My corporis fabrica: an ontology-based tool for reasoning and querying on complex anatomical models[END_REF] , an extensive ontology that describes structural and functional relations of different parts of human body and their relations in 3D models. Compositions are descriptions of all the elements that are seen in a particular frame. They need to be comprehensive in detailing the angle, size, the plane of view, anatomical location or specification, profile and relative screen position of the subjects viewed. The subjects in our case are anatomical parts and regions in the complete 3D male Zygote model for the human body 8 . The Size term specifies the extent of the subject that is visible within the camera frame. Describing the size for anatomical subjects is more difficult as it has to be relevant for the complete composition. In Figure : 5(b) sizes are illustrated for the femur with the head of the femur as the default centre but in Figure : 8 it is written as a Close Up on Femur, Tibia and Patella thereby making Patella the centre of the composition. Screen describes the position of the subject in terms of screen coordinates. This is useful if there is more than one connected body part in the scene. For now we concentrate on teaching one anatomical region at a time and this region will automatically be centred on the screen.

The most important descriptive elements that are essential in building the composition are Plane, Anatomical specification, Profile. Plane refers to the hypothetical planes that divide the human body. In ASL they define the view in which we see the anatomical parts and direct camera position accordingly. The three principal planes are sagittal (divides the body into left and right halves), frontal (divides the body into anterior and posterior parts) and transverse(divides the body into upper and lower parts) as seen in Figure : 6. If a plane is not mentioned in a composition then the system will automatically assign a plane in the vertical axis (sagittal or frontal) based on the Profile information but if the desired composition is in horizontal axis (transverse) then it must be mentioned in composition.

It is also important to note, especially in transverse plane, whether the anatomical part is viewed from the proximal end (close to the centre of the body) or the distal (further away from the centre). These details are mentioned in Anatomical specification. Finally, Profile describes the orientation of the part

Animation

Sentences written in ASL are parsed via the Parsimonious9 parser in the python language. The parsed sentences are then translated to a hierarchical finite state machine (HFSM), with one state per composition or development. The HFSM is described in a XML format with specific tags to define each state of the machine. Finally, the HFSM is interpreted and executed at runtime to generate the desired animation. We now describe each of those steps separately.

Style sheets. Some of the descriptive terms of the ASL need to be converted to numerical values in HFSMs. For example, in all the lessons written in this paper, we specified that the ASL term "high angle" will be translated to a 45 degrees bird's eye view. This numerical value of 45 is defined in an animation style sheet [START_REF] Perlin | Toward interactive narrative[END_REF] along with other parameters such as the camera speed. The teachers can edit this stylesheet depending on their preferences, giving them more nuanced control over the video-making process. Some aspects are not included in the ASL grammar, such as time spent on a composition or default time taken for a camera movement. These are global values that change the total run-time of the video and can be edited directly in the animation style sheet.

My Corporis Fabrica. All the anatomical terms used in the ASL authoring system are derived from MyCF. It is an ontology connecting the details of anatomical parts with their functions and as MyCF is queryable, we can write queries to obtain the list of parts we need to include in our models.

The list of parts is added to the grammar under the Anatomical Parts nonterminal. They can be updated or changed according to the lesson. When transitioning to a new state of an HFSM, the application first analyses this list of components named after the MyCF ontology. The names of parts in MyCF need to match the terms of 3D objects in the Zygote model used in Unity player. If the names do not match, we use a dictionary to convert MyCF names to the 3D objects of the zygote model. This conversion is not trivial since the relation is not bijective. Some MyCF objects are divided into subparts represented as one unique entity in the 3D model or vice versa. For example, "head of right femur" exists in MyCF, but the femur is not subdivided in the zygote model, and only the whole bone can be displayed in this case and in the other case, "Right gastrocnemius" is listed as one part in MyCF and is divided into its component parts of "Right gastrocnemius medial head", "Right gastrocnemius lateral head" and "Right Achilles tendon" in Zygote model. In these cases, we need the dictionary to convert the terms used in ASL scripts to match the terms used to build the 3D model in Unity. The dictionary is also built by using the querying feature of MyCF. In our Gastrocnemius example, we query in MyCF to find all the parts that make up the muscle and then find their names in the Zygote model.

Unity player. Our collaborators in the Anatoscope startup developed an application using the Unity 3D game engine to generate the desired animation at runtime from the HFSM obtained from the ASL script. The application is thus an interpreter from a specific XML format to 3D videos of anatomy. The player first builds a 3D space with the zygote model at the center with appropriate lighting. Then it adds the camera to look at the anatomical parts that were mentioned in the script. These parts are listed under a specific anatomy tag in the XML file for each state of the HFSM. The placement of the camera is computed automatically using both internal scene data and data from the ASL script converted through the animation style sheet. We use the bounding box of the 3D models to encompass the whole composition when view angles are setup depending on the ASL information. As the shot develops and there is a camera movement, the application computes a path from its previous position to the next one. It adjusts its orientation according to new parameters. The time taken for this movement is set either in the style sheet if there is no narration or by the time set by the narration if there is. If an animation tag is present in a XML state, the 3D models get animated following a previously registered animation stored in the application, such as the flexion of the knee. This animation database has been manually built and can be expanded as per necessity using Anatoscope software.

Narration

We then extended the system to include narrated voiceovers for the video lessons. The teachers have a choice to add narrations to their lessons. Suppose they want to have a narrated lesson. In that case, they record it and pass that recording through an Automated Speech Recognition(ASR) software developed by our collaborators at Laboratoire d'Informatique de Grenoble. ASR systems convert audio files and convert them into a sequence of words without a prior transcript of the narration. It works perfectly for us as the teachers can record themselves directly without providing a script for that narration. The input for this system is a .wav audio file. The first step is to perform speech recognition to generate a transcription. Then we use the Kaldi system [START_REF] Povey | The kaldi speech recognition toolkit[END_REF] of forced alignment to align audio with the text. Kaldi is an open-source C++ toolkit for ASR and speech processing. Forced alignment is a process that takes the text from the transcript of audio and finds the time stamp of every word as they occur in the audio segment. At the end of this process, we have the narration transcript used to build the ASL script and time stamps for each spoken word used to retime the videos.

After we get the time for narrated segments and match them to their corresponding ASL counterparts for the video, divide the time between the time spent in that HFSM state and the time taken for the camera movement, if there is any. If there is no narration, the time spent in each state or the delay and the time for camera movement is set by the style sheet's values. It is the usual time taken for the state to progress in the absence of narration. If the time taken for narration is more than twice the sum of the delay and time for camera movements, we calculate the extra time taken by the narration. This extra time is divided and added to the delay and the time taken for camera movements accordingly. If the time taken for narration is more than twice the time taken for the usual state time, we add two-thirds of this extra time to the delay and the remaining one third to the camera movement. We add more to the delay as more narration signifies that the teacher wants to spend more time describing the anatomical parts seen in that state. We don't want to extend the camera movement time as that will interfere visually with the shot as that, unlike the delay, can be seen in the video. If the narration time is less than the usual shot time, the difference between these two values is divided and subtracted from the delay and camera times. If narration is associated with a lesson, but not narration is linked to a state, then that will be assigned the delay and camera time from the style sheet for videos with narration.

Experimental evaluation

After we completed the authoring system, we evaluated its ease of use with four anatomy professors from Laboratoire d'Anatomie Des Alpes Françaises (LADAF) surgical school. They were from varied surgical specialities such as orthopaedics and trauma surgery, pediatric and urologic surgery, cardiothoracic surgery and endocrinology and public health medicine. The evaluation was done over two sessions. In the first session we conducted a short interview of their current teaching practices and then introduced the Anatomy Storyboard Language. At the end of the session, they were given the outline for the second session and were asked to record four short audio narrations for lessons on the knee. The four lessons were on the articulations, ligaments, muscles and the movements of the knee joint. Teachers then had to record the audio on any of the devices they were comfortable with and send them to us before the second session. The audio files were then proceeded using Automatic Speech Recognition (ASR) alignment software to extract the time stamps for each spoken word and get the text transcript of the narration.

In the second session, the teachers built the videos based on their recorded narrations. They referred to the transcripts and created the scenes for their lessons one development at a time. Scripts are written in comma-separated values (CSV) files. The first column of the file has the narration and the second column has the ASL sentence fragment that describes the visual elements the teachers want to show for the given narration. Narration for each development or state was added from the transcript. In this way, our authoring system puts 'narration first' as the visual components are built off of the audio parts, and the time spent on the visual states is also based on the duration of the narration for each state. They first referred to and divided the transcript into parts, and for each part, they wrote the ASL fragments. They could visualise their videos and animations immediately and could edit them in real-time as they built them. They could also expand the list of anatomical parts in the grammar to include the parts they were interested in showing. The first session was an hour long and the second session lasted for two hours.

After the second session, we recorded the videos lessons and added the narrations and sent the completed animated, narrated anatomical lessons on the knee joint back to the professors. Then we organised a short, 15-minute video call to perform the NASA task load test and analysed the results.

Pre-test interviews.

The object of the pre-testing interviews was to understand the current system of teaching anatomy in LADAF and assess the teacher's familiarity with video or graphic authoring systems.

What is the typical structure of the anatomy classes?

The general anatomy curriculum in French medical education is divided over two years. The first-year courses have a strength of about 1800 students and are all held online. The first-year courses are similar for students from a wide range of medical and related fields of dentistry, physiotherapy, nursing and others. As the students are from diverse fields, the courses are pretty generalised. They also have in-person teaching sessions in the amphitheatre for smaller groups of about 100 students each. After the students finish the online courses, if they have any questions, they send them to the professors 3 to 4 days before the live teaching sessions. The professors incorporate these questions into their lessons and discuss problem areas in detail with the students. At the end of the first year, examinations enable the students to choose their field. Around 1/10 of the students from the first year pass the exams to join the medical track. These students then start their second year of anatomy courses. The second-year courses are more clinically oriented and focus on problem-based learning by introducing clinical case studies. 2. What are your prefered tools/ multimedia methods for teaching?

During the first-year courses, both online and in the amphitheatre sessions, the preferred teaching method is via PowerPoint slides. For the online classes, these slides have a voice-over from the professors. For the inperson sessions, they reuse the slides without the narration and can show either surgical data or animated videos and talk over them. 3. How were these videos/other media made?

The teachers created their slides based on the learning objectives they wanted them to achieve for that lesson.A learning objective is a clear and precise goal set before the lesson by the instructors or institutions for the students. At the end of a lesson, a student should successfully reach these objectives. All the teachers have experience in drawing their illustrations. They either draw on paper and scan them or on graphic tablets. In addition to their illustrations, they also use images from surgical procedures, dissections and 3D models. After they create the slides, they record their narrations using the sound stage in LADAF. These narrations are then processed and synchronised to the slides by an audio engineer. The use of videos and 3D models in teaching is based on personal preferences. As stated earlier, they are more commonly used during the amphitheatre sessions rather than the main online classes. One professor had also commissioned a series of videos for his lessons on the pericardium (the double-layered protective sac around the heart). The 3D model was built from computerised tomography (CT) scans of the heart, and then the labels and camera movements were added to this model. The work was done by an engineer under the direction of the professor and took six months to complete.

How familiar are you with the animation authoring pipeline?

All the professors that we interviewed stated that they did not have any experience with animation software or were familiar with creating an animation.

Have you used any authoring systems before?

None of the professors used an authoring system for creating animated videos before. They were interested in testing a system in which the input for the video was a text-based script. This method aligned with the pipeline they had already in place for creating their slides and narration.

After the interview, the teachers were introduced to ASL grammar using the And/Or tree. Then the professors were asked to record four short audio clips on four concepts of the knee joint, the articulations, ligaments, musculature and movements. They could record this when convenient, and on any system they liked. They were asked to send the audio clips to us before we conducted the second session. We analysed the audio files sent by the teachers using the ASR alignment software. The input for this software is the .wav file of the audio narration, and the output is a text file with the time stamp for each spoken word. We then create a text file of this audio transcription which serves as a guide and a script for the teachers to develop their ASL visual directions. The metrics for the lessons are presented in table 1. The storyboards for the lessons made by one of the professors on Articulation are provided in Figure : 10 and Performance assessment using the NASA Taskload test.

The mental workload for creating video lessons using our system was measured using the NASA Task Load Index or NASA-TLX. It is a commonly used multidimensional assessment tool for measuring subjective mental workload developed in the 1980s by Sandra Hart in the Human Performance Group at NASA's Ames Research centre [START_REF] Hart | Development of nasa-tlx (task load index): Results of empirical and theoretical research[END_REF]. It measures the mental workload of a task using six subscales or dimensions, mental demand, physical demand, temporal demand, performance, effort and frustration. The subscales are nearly independent of each other and are general. It is also important to note that all tasks are not influenced equally by six subscales. Some of the variables may be irrelevant in specific tasks. Still, this issue is addressed in the weighting procedure, where the user is presented with 15 pairwise comparisons of the six subscales. The user has to chose which of the pair of variables affected the task more. This way, the subscale that was not as important while performing the task will be assigned a lower weight, and their influence on the final score will reflect their contribution to the mental workload. We used software created by Dr Keith Vertanen to perform the test 10 .

Results.

We analysed the six subscales' raw scores from the rating stage and the final workload after the weighing process. We present the average for each subscale in Figure : 9. This is before the weights have been applied and represent the quantitative subjective rating for each subscale from 1 to 20 (raw scores are given in table 2). We find that our system scores well for physical demand, frustration and effort. It also scores favourably for performance. This means that the users found our authoring system to have minimal physical demand and that it did not have a high contribution to the overall mental workload. The physical components of our system were typing an ASL script and recording narrations. The teachers were very familiar with both of these actions. They compared this with other 3D software they were familiar with, in which they had to manipulate the 3D model using an onscreen navigation bar that required a lot of clicking. They found our system physically more intuitive. This was reflected in the low scores for frustration which meant the teachers were not discouraged or annoyed by the system. They did not have to learn any new actions, and for the effort they put in, they achieved the goal they were expecting. The average performance score was 8 out of 20.

The main contributors to the workload of the authoring task were the mental and temporal demands. The teachers had to learn the partonomy and taxonomy of a formalised cinematographic language, ASL. They are not used to thinking of anatomy in terms of the shot sizes or angles. The placement of a camera and its related views and movements were wholly new and took time to be internalised. Another factor was that this was also in English, which is not the native language for the professors. So, they had to think of the visual description of the video with two inbuilt levels of difficulty, a new cinematographic vocabulary and a nonnative language. They also specified that they do not have any prior experience in authoring to compare the mental demand of this system. The weighing system helps us in this regard as the teachers can choose which subscale contributed more to the workload, but the comparison is only within the subscales and not with another system. It would be ideal if the teachers used other approaches of creating videos to compare to ours. The temporal demand was high as a consequence of high mental demand. The task required new concepts that took a lot of trial and error to master. Again the teachers were comparing the time taken to create the videos using our system with the time taken to create slides. They found it challenging to unhook this comparison and to treat this task as a stand-alone entity. Despite these issues, the teachers were very enthusiastic about using the system in the future. They recognised the potential for creative freedom in designing their own digital content and were willing to invest the time and expertise to become more familiar with the authoring tool. They were very enthusiastic about creating their library of videos that can be edited and fine-tuned on a lesson to lesson basis. As the professors involved in our study were at different stages of their teaching careers ranging from veteran teachers with over twenty years of experience to relative newcomers, we had interesting and inclusive feedback. The teachers just starting their teaching tenure were significantly invested in learning and using our system and agreed to further studies to improve the software. They also decided to use the lessons made using our system in their actual classes to get feedback from the students. After the complete sessions of authoring and evaluating the new authoring systems, the teachers expressed interest in further testing the system to get used to this content creation method. The main areas where they faced issues were learning the cinematographic vernacular and estimating the speed of narration. They wanted to use the system more to get used to these new elements. Most of them do not use videos in their regular classes as they did not have the means to create the videos themselves. The teachers stated that they would like to include videos in their lessons if making them were easy. They were very enthusiastic about our authoring system and planned on using it further to create longer lessons.

Concerning the videos themselves, the teachers gave us a list of features they would like to incorporate in the future. These include controlling the opacity of specific parts to see through them, having animated labels of parts that moved as the camera moved and incorporating their current slides and figures into the video lessons. We are already working on these suggestions and chose not to include some of the features into the test version to concentrate more on the ASL authoring aspect. We will incorporate them in the later versions of our system. We plan to conduct the same experiment with the new features and the teachers with prior experience using our authoring system. We believe that will decrease the mental load for the task.

The authoring tool we propose is generic enough to be able to manipulate other 3D content libraries. The ASL language does not depend on the models used; it uses identifiers that are linked to anatomical entities. It is sufficient to provide a correspondence map between anatomical entities and the corresponding models. This flexibility also allows, for example, to associate a collection of reference anatomical cross-sections (2D) that could be called via ASL when needed.

Limitation of the project

Attempting to intervene in the production pipeline of animated videos for anatomy education is essential to facilitate the inclusion of 3D digital content in teachers' courses. Our solution introduces cinematic concepts that are unfamiliar to educators. To master the authoring tool, one must learn these concepts and become proficient in the ASL language. This can be a barrier to its use. This difficulty seems more easily overcome by younger anatomists who are more comfortable with computer languages and more inclined to seek assistance in designing their courses. This limitation is even more true as it appears much simpler to learn how to manipulate a virtual table or 3D anatomy software and improvise animations to illustrate discourse than to learn a formal language (ASL) that is constraining. However, the perspectives offered by ASL go far beyond the production of a single video. Beyond simplified content updates, transparent changes for authors of the source 3D library, the real added value lies in the concept of a formalised language. Indeed, ASL opens the doors wide to generative AI. The creation of a language model that includes ASL allows for the envisioning of high-level assistance in the use of 3D in anatomy courses. The power of generative AI in the service of anatomy education will inevitably require a formalisation of the visual dimension of training materials

Conclusion

We introduced and evaluated a new system for authoring anatomy video lessons with four professors of anatomy from the Grenoble Centre Hospitalier Universitaire (teaching hospital). The teacher had no prior experience with video creating tools. After a short introduction and hands-on tutorial session, the teacher made and manipulated the 3D scenes from the zygote model of the human body using our text-based input system. They learnt the Anatomy Storyboard Language, which combines cinematographic and anatomical terms that describes all the visual elements seen in the video. The teachers learnt to incorporate their audio narrations into the video lessons and found the authoring system intuitive and interesting. The learning curve is a bit steep as there were new terms and concepts from cinematography that they had to learn and use, but the overall mental workload for the complete authoring task was not high. This workload will only reduce with further use of the system as their familiarity with these new concepts improves. Based on this feedback and the enthusiasm shown by the teachers to learn and use our tool, we will continue to make improvements they suggested, such as including animated labelling and the ability to change the opacity of parts. We focus on the authoring tool as the current thesis aims to teach anatomy pedagogy. The logical next step is to use the lessons made using our system in the anatomy curriculum and get feedback from the students. These validation experiments are being planned for the future and in collaboration with the Anatosocpe start-up team. Given the positive feedback for the authoring tool, we are confident of improving its features and presenting a robust, easy-to-use system for anatomy teachers to use. = " d o l l y " ( " i n " / " o u t " ) ? Crane = " c r a n e " ( " up" / "down " ) ( " l e f t "/" r i g h t " ) ? Arc = " a r c " ( "up" / "down " ) ? ( " c l o c k w i s e " /" a n t i c l o c k w i s e " ) ? Zoom = "zoom" ( " i n " / " o u t " ) Speed = " s l o w " / " q u i c k " / ( " f o l l o w i n g " S u b j e c t ) T r a n s i t i o n = ( " c u t t o " / " d i s s o l v e t o " / " f a d e i n t o " ) C r o s s = " c r o s s e s " ( " o v e r " / " u n d e r " ) S u b j e c t ( S u b j e c t ) ? F l e x = " f l e x e s " ( " p a r t i a l l y "/" c o m p l e t e l y " ) R o t a t e = " r o t a t e s " ( " i n t e r n a l l y "/" e x t e r n a l l y " ) Angle = " low a n g l e " / " h i g h a n g l e " S i z e = "ECU" / "CU" / "MCU" / "MS" / "MLS" / "FS" / "LS" / "ELS" P r o f i l e = " v e n t r a l " / " d o r s a l " / " m e d i a l " / " l a t e r a l " / " v e n t r o m e d i a l " / " d o r s o m e d i a l " / " v e n t r o l a t e r a l " / " d o r s o l a t e r a l " A n a t S p e c s = ( " L e f t "/" R i g h t " ) ( " p r o x i m a l " /" d i s t a l " ) ? S c r e e n = " s c r e e n " ( " t o p " / " bottom " ) ? ( " l e f t " / " c e n t e r "/ " r i g h t " ) ? P l a n e = " f r o n t a l "/ " t r a n s v e r s e " / " s a g i t t a l " Time = ˜r " [ 0 -9 ] * " i ( " s e c o n d s " / " s " ) S t r i n g = ˜r " [ A-Z 0 -9] * " i A c t i o n = C r o s s / F l e x / R o t a t e E f f e c t = " a p p e a r s " / " d i s a p p e a r s " Event = S u b j e c t ( A c t i o n / E f f e c t ) A n a t o m i c a l P a r t = " P a t e l l a r l i g a m e n t " / "Femur" / " P a t e l l a " / " M e d i a l m e n i s c u s " / " F i b u l a " / " Hip bone " /" T i b i a " /" L a t e r a l m e n i s c u s " /" A r t i c u l a r c a p s u l e o f l e f t k n e e j o i n t " / " P o s t e r i o r c r u c i a t e l i g a m e n t " R e g i o n

= "Knee" s p a c e = ˜r "\ s " = ( s p a c e ) *

Figure 1 :

 1 Figure 1: Standard workflow for creation of narrated animation

Figure 2 :Figure 3 :

 23 Figure 2: Text-to-Movie workflow for creation of narrated animation -Xtranormal

  3.

Figure 4 :

 4 Figure 4: And/Or Graph representation of the Anatomy Storyboard Language grammar. ASL scenes are made of shots containing an initial composition and one or more optional developments.

Figure 5 :

 5 Figure 5: Angles and sizes in Anatomy Storyboard Language

Figure 6 :

 6 Figure 6: Anatomical planes

  (a) Specifications (b) Profiles in transverse plane

Figure 7 :

 7 Figure 7: ASL specifications and profiles.

  (a) FS Left Femur Tibia Patella ventral (b) CU Left Femur Tibia Patella ventral

Figure 8 :

 8 Figure 8: Sizes in HFSM

Figure 9 :

 9 Figure 9: Average score for each subscale before weighing

Figure 10 :

 10 Figure 10: Storyboard for lesson on articulation -Part 1

Figure 11 :

 11 Figure 11: Storyboard for lesson on articulation -Part 2

Figure 12 :

 12 Figure 12: ASL Grammar.

Table 1 :

 1 Metrics for the 4 lessons made by the 4 teachers given in minutes and seconds

	Lessons	Teacher 1 Teacher 2 Teacher 3 Teacher 4
	Articulation	1:42	1:05	0:33	1:22
	Ligaments	2:17	2:05	1:03	1:38
	Muscles	1:37	0:31	1:23	0:55
	Movements	2:00	0:32	1:09	0:17
	Total	7:36	4:13	4:08	4:12
	11.

Table 2 :

 2 Ratings for each subscales of NASA Taskload test before weighing

	Sub scale	Teacher 1 Teacher 2 Teacher 3 Teacher 4
	Mental Demand	14	10	16	13
	Physical Demand	2	1	2	2
	Temporal Demand	10	13	14	8
	Performance	6	5	10	11
	Effort	8	3	8	7
	Frustration	8	1	2	4

https://www.youtube.com/channel/UCHKlhyLLAxFA69nO0NK9wPA

https://docs.unrealengine.com/4.26/en-US/ProgrammingAndScripting/Blueprints/

https://unity.com/unity/features/editor/art-and-design/cinemachine

https://hutonggames.com/

https://github.com/Unity-Technologies/EditorXR

https://www.nawmal.com/

https://plotagon.com/

https://www.zygote.com

https://github.com/erikrose/parsimonious

https://www.keithv.com/software/nasatlx/nasatlx.html

Acknowledgement

This work was funded by the French National Research Agency through An@tomy2020 project (ANR-16-CE38-0011). A CC-BY public copyright license has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission, in accordance with the grant's open access conditions.


Frédérique Duyrat

Nowadays, the Internet, 3D scanning and interactive media have become useful tools for scientific research as well as for education. The website Orient Cunéiforme aims to use all these modern media to highlight important pieces of the Mesopotamian written heritage.

Orient Cunéiforme, http://archeologie.culture.fr/orient-cuneiforme/fr, is a section of the website Patrimoine du Proche Orient http://archeologie.culture.fr/ proche-orient/fr, led and funded by the Ministère de la Culture -Direction Générale des Patrimoines of France. The project of this website was

200

This portal aims to present selected pieces of the cuneiform heritage in an attractive way. The commentaries on the objects have to be comprehensible for the general public and at the same time reflect the latest advances in scientific research related to the topic. The website is now in French but translations in Eng-lish and Arabic will be provided. The first four objects selected and presented on the website are the following:

-The kudurru "caillou Michaux", discovered in 1784 by André Michaux. It is the first monument containing cuneiform writing to have been intro-duced in Europe and it is currently kept in the Cabinet des Médailles, Bibliothèque Nationale de France.

-The tablet of the 8 th campaign of the Assyrian king Sargon the second (AO 5372). This large clay tablet contains an exhaustive account of this Assyrian military campaign and is kept in the Louvre.

-The statuette of Pazuzu (MNB 467), also from the Louvre collection. This object was chosen because it represents a demon who has a certain notoriety in the current popular culture after the release of the horror movie "The Exorcist" in 1973 where he appeared as an evil character.

-The Esagil tablet, kept in the Louvre (AO 6555). This famous mathemat-ical text gives the ideal dimensions of the Etemenanki ziggurat, the symbol of the city of Babylon.

Through this paper, we hope to share our experience of the construction of the website Orient Cuneiform and to show the benefits of new technologies for the propagation of knowledge and scientific research.

Objectives and content of the website 1.The objectives

The website primarily intends to host inscribed objects of a high historical interest from Northern and Southern Iraq that are recognizable to the general public. The project has four main objectives. 1. First, to create a virtual museum, open to everybody, in order to make pieces of Mesopotamian history kept in French Museums accessible anywhere in the world, especially in the countries they come from. That is why special attention has been paid to the quality of the pictures of the objects: they are either 3D reconstructions or photographs with a very high resolution. 2. Second, Orient Cunéiforme, as a part of the website Patrimoine du Proche-Orient, supports its determination to show the outstanding value of Near-Eastern heritage in a time when war poses a continuous threat to it. It aims to show that the Near East stores parts of our shared past which the general public is rarely aware of. 3. In the continuity of the above-mentioned idea, this website will provide ma-terials for schoolteachers to teach Ancient Near Eastern History. The history of the ancient Near East is a part of the French school curriculum and is taught in 1st year of junior high school. 4. Finally, the website also intends to make advanced research available to and understandable for the general public.

The virtual objects

Our ambition is to get the users immediately immersed in the object and the cu-neiform writing, in order to arouse their curiosity at the first click. Therefore, the website's home page makes use of the interactive media and presents 3D scans or high definition pictures of the objects. Each media is interactive and signals points of interest on which the reader can click to have either the details of the iconogra-phy or the translation of the cuneiform text. The caillou Michaux has been scanned in 3D (fig. 1). The scanning was al-lowed and supported by the director of the Museum, Frédérique Duyrat, and was made by Yohann Janin (autoentrepreneur in virtual archaeology), with the pho-togrammetry technique. One of the difficulties was due to the Caillou itself. The uniformity and dark colour of the panels made it difficult to achieve the best possible results. During the image capture process, it was crucial to maintain good illumination of the object. The broken parts of the stele also posed difficulties, because their texture was different from the smooth parts of the stone. The final result is hosted on the site sketchfab and can also be seen on Orient Cunéiforme. For the three other objects, kept in the Louvre, high definition pictures are avail-able. As it is the case with the 3D scans, they are interactive. It is possible to zoom on the cuneiform writing and to click on the points of interest. They give access to the translation of the cuneiform text and quick explanations of the content, which might help to generate interest and comprehension among the general public.

The architecture of the website

After seeing the object, itself, the user is offered more detailed explanations. Each object is presented in five pages, according to the same model. The first page is the "cartel" and gives the technical data of the objects: museum number, place of conservation and links to the institutional websites of the museums, date and the place of origin of the object, dimensions, writing and language. Below these data, a short introduction to the object is provided, summing up its main characteristics and historical interest, in order to show at the outset, the reasons why this object is crucial for scientific research and our knowledge of Mesopotamian civilization. The second page tells the reader what the inscription on the object is about. The purpose is to go into more detail in the content of the text, and explain its nature. The last three pages are dedicated to the development of themes related both to the content of the tablet and to major aspects of Mesopotamian history and culture. For instance, the presentation of the tablet of the 8 th campaign of Sargon II highlights the originality and special interest of this tablet through three topics: the geography, the figure of Sargon II, king of Assyria (722-705), and the relation between Urartu and Assyria. Thus, the tablet of the 8 th campaign of Sargon II is an opportunity to present the Assyrian royalty, the tablet of Esagil -mathematics and religion, and Pazuzu -the beliefs of the Mesopotamian people and the practice of exorcism. The presentation of the Caillou Michaux explains the context of the discovery of the first large object, inscribed in cuneiform, to have arrived in Europe and shows its role in the very beginning of the decipherment of cuneiform writing. This site is intended for both scientists and readers unfamiliar with Assyriology. That is why the difficult terms or Akkadian words are linked to a definition in a glossary. Bibliography and credits are provided at the bottom of each page.

Interactive media

On top of the written explanations, each page is illustrated by interactive media, through which the reader can explore some specific themes related to the object presented in more detail (fig. 3). These media take the form of pictures, chronologies and maps. For instance, the statuette of Pazuzu kept in the Louvre is presented alongside pictures of similar objects kept in the British Museum. These pictures illustrate the popularity of these apotropaic amulets of Pazuzu in Mesopotamia and the di-versity of the forms they took: a statuette, plates, beads, pins used to fasten a gar-ment. The photographs of these objects can be presented on the site thanks to the arrangements with the museums that preserve them. Each picture contains a description with its reference and characteristics, accompanied by a link to the web-site of the Museum where it is kept.

Interactive maps and chronologies are other types of media useful for the transmission of knowledge. We used the free software "StorymapJS" and "TimelineJS" to build them. For instance, we've built an interactive map illustrating the travels taken by the discoverer of the kudurru caillou Michaux (fig. 4). It gives all the stages of André Michaux's exceptional journey to the Orient between 1782 and 1785, and shows what an extraordinary adventure is was for that time. The data on his travels come from his diaries kept in the Library of Versailles, and they were compiled by Régis Pluchet, descendant of André Michaux. 3 André Michaux was a botanist working at the gardens of Versailles under the reign of Louis XVI. In 1782 he left France for the Levant, where his mission was to gather new plants for the gardens of Marie-Antoinette in the Trianon. He dis-covered a lot of new species during his long travels across Syria, and Iran, such as flowers Michanxia campanuloides and Dionysia Michauxii. On his way back, near Ctesiphon, he came across a kudurru, perfectly preserved, datable to Marduk-nādin-aXXē, (1099-1082), recording a gift of land from a father to his daughter on the occasion of her marriage. The object was officially acquired by the French government in 1800, after the Revolution, by Lucien Bonaparte, Na-poleon's brother, and minister of the interior at that time. The discovery of the Caillou Michaux is important for the deciphering of cunei-form. For this reason, a chronology of the stages of scientific advances in this domain is proposed in the website. It is accompanied by an interactive chronology of all the main kudurrus discovered in Mesopotamia, which places the Caillou Michaux in its historical context and provides comparison with other stelae of the same nature. Thus, 3D scanning, high definition pictures, interactive maps and chronologies allow the website Orient Cunéiforme to accomplish its purpose and make the cu-neiform heritage and its history available in an interactive way.

The benefits of new technologies for education and research

For Education

This website can supplement teaching Mesopotamian history in high school. The advent of agriculture and the first cities and the invention of writing are a part of the national curriculum in history and have to be taught to every 11-years-old child at junior high school in France. We have already tried out the presentation of the selected objects featured on the website in class. The website can be a vector to introduce cuneiform writing and mathematics to the pupils, as well as Mesopotamian religion (e.g. presenting the Pazuzu demon (fig. 5) and the Esagil tablet), geography, kingship and war (e.g. the 8th campaign of Sargon), and daily life (e.g. the caillou Michaux, telling about the wedding arrangement). Moreover, the translated passage of the Esagil tablet, where the measurements of the levels of the ziggurat are given, is an interesting means to teach current school children to decipher the writing of numbers in cuneiform. It can also allow them to practice mathematics by calculating the dimensions of the base and the height of the ziggurat, or convert units of measure. Recent interpretations of the Esagil tablet suggest that the nature of this text was to be a scribe's exercise. 4 This case study is therefore an ideal opportunity for the students of our time to take on the role of a Mesopotamian scribe learning maths. The aim of this website is to give tools and materials to the teachers willing to introduce Mesopotamian history to their pupils, with the role support of its content. Orient Cunéiforme and, more generally, the site it belongs to, Patrimoine du Proche Orient, can also be a good way of introducing the notion of heritage and issues related to its safeguarding to the general public. Indeed, one of the objectives of this portal is to show that an important part of the shared past of the humanity lies in the Near East. Writing, mathematics, measurement of time and myths such as the Tower of Babel are all examples of the legacy of the Mesopotamian civilization to the modern world, and all of them are developed on Orient Cunéiforme. The website Patrimoine du Proche Orient presents archaeological sites that are either currently under threat or have already suffered damage as a result of the Near East wars and conflicts of recent years. On the website, one can admire 3D reconstructions of the city of Palmyra or the palace of Khorsabad. In a similar way, Orient Cunéiforme aims to raise awareness of the pressing necessity to pro-tect the heritage of Mesopotamia and more specifically to show the crucial histor-ical importance of the written objects.

Mustapha Djabellaoui, Paris -Louise Quillien, Paris 2.2. For Research This website is not only designed for education but also for the general public. It aims to make aspects of the more recent scientific research accessible to every-body. For instance, the commentary on the Esagil tablet mentions the debates con-cerning the architectural concept and the measurements of the ziggurat. The segment on the 8 th campaign of Sargon II presents different itineraries proposed for the military campaign (Fig. 6) and offers a new solution (Fig. 7). The 3D scanning of the caillou Michaux proposes a new translation of the cuneiform text. New technologies can also raise new questions for the scholars. For example, when scanning the caillou Michaux, we discovered that no scholar agreed about its material: some proposed that it could be basalt, others diorite. Further scientific analysis is needed to determine its material. Another example is the presentation of the tablet of the 8 th campaign of Sargon II that initiated new case studies in geography, in particular, Urartu and Northern Assyria (Mesopotamia) geography. The website can also be useful for the researchers themselves, providing new highresolution pictures of outstanding cuneiform objects. In the near future, we are planning to offer filmed interviews with researchers on the website, thanks to the technical support of the Maison d'Archéologie et d'Ethnologie of Nanterre. The objective is to interview scholars and museum curators about the objects presented on the website, in order to introduce the scientific and conservation issues related to them to the public. With the help of these videos, we wish to go deeper into some of the topics discussed on the website, and also to make the general public aware of the methodologies of the scientific research and the most recent discoveries. Video is a medium made for the general public, students and scholars. Through interactive media, the web-site will contribute to making highlevel scientific research more accessible for the larger public.

Future developments 3.1. The Harradum project

Besides Orient Cunéiforme, we have started a new project that will have its place on the website Patrimoine du Proche-Orient: a 3D restitution of the city of Har-radum, using the technology of immersive virtual reality with 3D glasses (Oculus Rift and youtube 360). Harradum, modern Khirbet-ed-Diniye, is located in Western Iraq, near the Eu-phrates. It was excavated in 6 campaigns between 1981 and 1986, directed by Christine Kepinski, before the construction of a dam at Haditha. The site has pro-vided a rich archaeological material as well as cuneiform tablets, published by Francis Joannès. 5 Thanks to the 3D glasses technology, we would like to give the users a guided tour of the walled city of Harradum, letting them experience the day-to-day life of a small city in Mesopotamia. The level that will be reproduced is level 3B; it is dated to the old Babylonian period, at a time when the city was part of the Baby-lonian kingdom, after the conquest of Mari by Hammurabi. The virtual tour of the city will lead the users from the bank of the Euphrates to the main gate, then through the central street. They will enter some houses, in particular, that of the mayor and the temple dedicated to Adad. Thanks to this virtual reconstruction we want to show to the public how ar-chaeological artefacts and cuneiform tablets can together help to reconstruct the history of the city. It is also a tool for research. For the moment, only a prototype exists, made by Yohann Janin, showing the elevations of the city walls and buildings as well as its location near the river. The 3D reconstruction gives a distinct 208 impression of the narrowness of the streets. It may help to test different hypothe-ses about the height of the city walls and the access to these walls. The 3D reconstitution will be accompanied by a written commentary presenting the archaeo-logical site and its material and the campaigns of excavations. Some of the excavation archives, including photographs, have already been made available online.

Other objects and collaborations with museums

This website is indeed a platform for new developments. We would like to enrich this site with other objects kept in France or in other countries. We have already obtained the support of several museums and institutions, in France and abroad, in particular the British Museum, the Chicago Oriental Institute, the Staatsbiblio-thek zu Berlin. They all have given their permission to publish pictures of the objects they preserve on the website. The virtual museums can help to reinforce scientific collaboration between different countries, and in the future we hope to develop international collaborations. The 3D scanning of the objects held in France can be an interesting way to develop exchanges between the French and the Iraqi scientific communities. The convention of 3D scanning of the caillou Michaux stipulates that the files will be given to the Iraqi Museum of Baghdad. We hope that these exchanges will lead to scientific collaboration between two countries in the future.

Through this article we wish to share our experience of building a website dedi-cated to the written heritage of Mesopotamia. This site is intended for specialist and non-specialist public, and its ambition is to highlight remarkable objects of the Mesopotamian heritage, to show its historical interest, and to transmit the achievements of the research more widely in the society. 3D and high-resolution photography made it possible to obtain accurate reconstructions of the inscribed objects, which is difficult to achieve by conventional means (e.g. photographs). Interactive media like maps, chronologies, texts and pictures, videos, are interest-ing tools for the transmission of knowledge. New technologies can therefore be useful for education, research, and raising the public awareness of the Near Eastern heritage. Our wish now is to make this site expand over time and to develop new collaborations at the international level.

Fig. 1 :

 1 Fig. 1: Picture of the 3D scan of the Caillou Michaux.

Fig. 2 :

 2 Fig. 2: High definition picture of the Esagil tablet.

Fig 3 :

 3 Fig 3: Examples of media.

Fig. 4 :

 4 Fig. 4: The map of André Michaux's journey.

Fig. 5 :

 5 Fig. 5: Interactive picture of the statuette of Pazuzu.

Fig. 6 :

 6 Fig. 6: Chronology of the different hypothesis of the 8th campaign of Sargon II itinerary. Fig. 7: Proposition of an itinerary for the 8 th campaign of Sargon II.

[START_REF] Pluchet | L'extraordinaire voyage d'un botaniste en Perse[END_REF]

[START_REF] George | Babylonian Topographical Texts[END_REF] George 2005 George /2006, p. 75-95. , p. 75-95.

Kepinski-Lecomte 1992;[START_REF] Joannès | Haradum II. Les textes de la période paléo-babylonienne, Samsu-iluna -Ammi-saduqa, avec les contributions de Christine Ke-pinski[END_REF] Kepinski 2012.

initiated in March, 2015. It stems from the will to react to the ransacking of museums and archaeological sites in Iraq and Syria. Its objective is to disseminate knowledge on threatened and attacked sites in the Near East to the general public and scien-tists, in order to preserve the memory of this heritage. The website Patrimoine du Proche Orient is led by the Directeur général des patrimoines, edited by the Musée d'Archéologie Nationale -Domaine national de Saint-Germainen-Laye and managed by Thomas Sagory (Musée d'Archéologie nationale). 1 The section Orient Cunéiforme is specifically dedicated to the written heritage of Mesopotamia. It is the result of a collaboration between a scientific team, ArScAn (Archeology and Sciences of Antiquity) and two museums, the Louvre and the Cabinet des Médailles -Bibliothèque Nationale de France. 2 We were the creators of the content. 1 The collection "Grands sites archéologiques" is a publication of the Ministère de la Culture, Direction directeur général des patrimoines; publication manager: sousproject partners were: at the laboratory ArScAn (Archéologie The project had benefited greatly from the support of Bruno Favel,


Ana Djurdjevac
email: adjurdjevac@zedat.fu-berlin.de.

Armen Shirikyan
email: armen.shirikyan@cyu.fr.

Exponential stability of the flow for a generalised Burgers equation on a circle

Keywords: AMS subject classifications: 35B35, 35K58, 60G10, 93D23 Burgers equation, exponential stability, bounded trajectory

The paper deals with the problem of stability for the flow of the 1D Burgers equation on a circle. Using some ideas from the theory of positivity preserving semigroups, we establish the strong contraction in the L 1 norm. As a consequence, it is proved that the equation with a bounded external force possesses a unique bounded solution on R, which is exponentially stable in H 1 as t → +∞. In the case of a random external force, we show that the difference between two trajectories goes to zero with probability 1.

Introduction

Let us consider the generalised viscous Burgers equation

∂ t u -ν∂ 2 x u + ∂ x f (u) = h(t, x), x ∈ S.
(1.1)

Here S = R/Z is a circle of unit length, so that all the functions are assumed to be 1-periodic in x, ν > 0 is a fixed parameter, h is an external force, and f ∈ C2 (R) is a given function (called flux ) whose second derivative satisfies the inequality1 f ′′ (u) ≥ σ for all u ∈ R, (1.2)

where σ > 0 is a number. The function h is assumed to be locally bounded on R + × S, which ensures the well-posedness of the Cauchy problem for (1.1). Namely, for any initial condition u 0 in the Sobolev class H 1 (S), Eq. (1.1) has a unique solution u(t, x) in an appropriate function space such that u(0, x) = u 0 (x);

(1.3) see Section 2.1 for an exact statement. To simplify the presentation, we assume in this introduction that all the functions defined on S have zero mean value.

A simple consequence of the maximum principle is the non-expansion of the L 1 norm of the difference between two solutions of (1.1). Combining this with the regularising property of the resolving operator, it is easy to prove that the flow generated by (1.1), (1.3) is H 1 -stable in the following sense: if a sequence of initial conditions (u n 0 ) converges to u 0 in H 1 , then the corresponding solutions (u n ) and u are such that sup t≥0 %u n (t) -u(t)% H 1 → 0 as n → ∞.

The goal of this paper is to prove that the flow is in fact exponentially asymptotically stable as t → ∞. We shall show, in particular, that the two properties below are true (see Section 4 for more details):

Global attractor: If h is a bounded function of t ∈ R with range in H 1 , then there is a unique H 1 -bounded solution of (1.1) defined throughout the real line, and any other solution converges to it in H 1 exponentially fast as t → +∞.

Stochastic equation:

If h is a pathwise continuous H 2 -valued stochastic process (not necessarily bounded in time, but satisfying a mild growth condition), then the L 1 norm of the difference between two solutions of Eq. (1.1) converges to zero with probability 1 as t → +∞.

The proof of both results is based on a squeezing property of the flow established in Section 3. The latter uses an idea coming from the theory of positive operators and applied in [START_REF] Shirikyan | Global exponential stabilisation for the Burgers equation with localised control[END_REF] in the case of the Dirichlet boundary condition.

Let us mention that various results on the stability of the Burgers flow were obtained earlier in both deterministic and stochastic settings, for the periodic and Dirichlet boundary conditions, as well as for the whole space. One of the first mathematically rigorous results was obtained by Sinai [START_REF] Ya | Two results concerning asymptotic behavior of solutions of the Burgers equation with force[END_REF]. He studied the cases when h is a time periodic deterministic function or spatially smooth white noise and proved the almost sure convergence of trajectories with random initial data to a limiting periodic solution or a measure, both of them independent from the initial condition. Kifer [Kif97] obtained a similar result for the Burgers equation in R d under the hypothesis that both the solution and external force are potential fields. Hill and Süli [START_REF] Hill | Dynamics of a nonlinear convection-diffusion equation in multidimensional bounded domains[END_REF] studied viscous scalar conservation laws in a bounded domain of R d with a time-independent external force and proved the existence, uniqueness, and exponential stability of a stationary solution. Jauslin-Kreiss-Moser [JKM99] studied the Burgers equation on S with a time-periodic external force and proved the existence and asymptotic stability of a time-periodic solution. Boritchev [START_REF] Boritchev | Sharp estimates for turbulence in white-forced generalised Burgers equation[END_REF] investigated generalised Burgers equation on S and assuming that h is a spatially smooth white noise proved that there is a unique stationary measure and that any other solution converges to it with a rate independent of the viscosity ν > 0. Chung and Kwon [START_REF] Chung | Asymptotic behavior for the viscous Burgers equation with a stationary source[END_REF] considered the Burgers equation on R with a quadratic flux and a non-negative time independent external force h and proved the existence and uniqueness of a non-negative stationary solution and convergence to it with an algebraic rate. Kalita and Zgliczyński [START_REF] Kalita | On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions[END_REF] studied the same equation with periodic and Dirichlet boundary conditions and established the convergence to a limiting trajectory. They also proved the exponential rate of convergence in the case of the Dirichlet boundary condition. Bakhtin-Li [START_REF] Bakhtin | Thermodynamic limit for directed polymers and stationary solutions of the Burgers equation[END_REF] and Dunlap-Graham-Ryzhik [START_REF] Dunlap | Stationary solutions to the stochastic Burgers equation on the line[END_REF] constructed space-time stationary solutions for the stochastic Burgers equation on R and proved their stability as t → +∞. Finally, the recent article [START_REF] Djurdjevac | Synchronisation for scalar conservation laws via Dirichlet boundary[END_REF] proves exponential synchronisation of a scalar viscous conservation law in a bounded domain with the Dirichlet boundary condition, in both deterministic and stochastic settings.

The present article establishes the exponential rate of convergence in the periodic setting and simplifies some of the proofs obtained earlier. Let us also note that the results of this paper can be extended to viscous scalar conservation laws in higher dimension. However, since the corresponding proofs do not involve any new ideas and can be carried out using the methods of the articles2 [DS22, DR22], we do not discuss them in this paper.

The paper is organised as follows. Section 2 gathers various results on the Cauchy problem for (1.1) and on linear parabolic equations. In Section 3, we formulate and prove the main theorem of this paper on the strong L 1 -contraction of the flow of the Burgers equation. Finally, in Section 4, we present exact formulation of the two properties mentioned above and give their proofs.

Notation and conventions

We write S = R/Z and identify any function on S with a 1-periodic function on R. The usual Lebesgue and Sobolev spaces on a domain D are denoted by L p (D) and H s (D), and we write | • | p and % • % s for the corresponding norms and (•, •) for the L 2 inner product. In the case when D = S, we often drop S and write simply L p and H s . Given a function g ∈ L 1 (S), we set

〈g〉 = ! S g(x)dx.
For a closed (not necessarily bounded) interval J ⊂ R and a separable Banach space X, we denote by C b (J, X) the space of bounded continuous functions f : J → X endowed with the natural norm and by L 2 loc (J, X) the space of Borel measurable functions f : J → X such that, for any bounded interval I ⊂ J,

%f % 2 L 2 (I,X) := ! I %f (t)% 2 X dt < ∞. Let X (J) be the space of functions f ∈ L 2 loc (J, H 2 ) such that ∂ t f ∈ L 2 loc (J, L 2 ). In the case when J is bounded, X (J) is a Hilbert space with a norm % • % X defined by %f % 2 X = ! J " %f (t)% 2 2 + |∂ t f (t)| 2 2 # dt.
If J = J T := [0, T ], then we write X T instead of X (J T ), and if J = R + , then we write X . Finally, for a Banach space X, we denote B X (R) the closed ball in X of radius R centred at zero.

Preliminaries

Cauchy problem

Let us fix ν > 0 and T > 0, and consider Eq. (1.1) in the domain

Q T := J T × S.
The following result is well known and can be established by standard methods based on a priori estimates and a fixed point argument; see [START_REF] Lions | Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires[END_REF][START_REF] Evans | Partial Differential Equations[END_REF].

Theorem 2.1. Let f ∈ C 2 (R) be an arbitrary function, let u 0 ∈ H 1 (S), and let h ∈ L ∞ (Q T ). Then problem (1.1), (1.3) has a unique solution u ∈ X T , and there is a number

K T > 0 depending on ν, f , |u 0 | L ∞ , and |h| L ∞ (Q T ) such that 3 |u| L ∞ (Q T ) ≤ |u 0 | L ∞ (S) + T |h| L ∞ (Q T ) , (2.1) %u% X T ≤ K T " %u 0 % 1 + |h| L 2 (Q T ) # . (2.2)
Proof. For the reader's convenience, we sketch the derivation of the a priori bounds. To prove (2.1), we note that the function u(t, x) can be regarded as a solution of the linear equation

∂ t u -ν∂ 2 x u + b(t, x)∂ x u = h(t, x),
where b(t, x) = f ′ (u(t, x)). Applying the maximum principle to this equation (see Section 3.1 in [START_REF] Landis | Second Order Equations of Elliptic and Parabolic Type[END_REF]), we obtain (2.1).

To prove (2.2), we first take the L 2 inner product of (1.1) with 2u. Using the periodicity condition and the Cauchy-Schwarz inequality, we derive

∂ t |u| 2 2 + 2ν |∂ x u| 2 2 = 2(h, u) ≤ |h| 2 2 + |u| 2 2 .
(2.3) By the Gronwall inequality, it follows that

sup 0≤t≤T $ |u(t)| 2 2 + 2ν ! t 0 |∂ x u(s)| 2 2 ds % ≤ C(T ) " |u 0 | 2 2 + |h| 2 L 2 (Q T ) # ,
where C(T ) > 0 depends only on T . To obtain bounds on higher Sobolev norms, we take the L 2 inner product of (1.1) with -2∂ 2 x u. After some simple transformations, we obtain

∂ t |∂ x u| 2 2 + 2ν |∂ 2 x u| 2 2 ≤ 2|h| 2 |∂ x u| 2 + K 1 (T )|∂ x u| 2 |∂ 2 x u| 2 ;
here and henceforth we denote by K i (T ) some numbers that can be expressed in terms of the quantity |f ′ (u)| L ∞ (Q T ) (which is finite in view of (2.1)). Application of the Cauchy inequality results in

∂ t |∂ x u| 2 2 + ν |∂ 2 x u| 2 2 ≤ K 2 (T )|∂ x u| 2 2 + ν -1 |h| 2 2 .
Using again the Gronwall inequality, we obtain

sup 0≤t≤T $ |∂ x u(t)| 2 2 + ν ! t 0 |∂ 2 x u(s)| 2 2 ds % ≤ K 3 (T ) " %u 0 % 2 1 + |h| 2 L 2 (Q T ) # .
(2.4) Finally, resolving (1.1) with respect to ∂ t u and taking the L 2 norm over Q T , we derive

|∂ t u| 2 ≤ |h| 2 + K 4 (T ) |∂ x u| 2 + ν |∂ 2
x u| 2 . Using (2.4), we conclude that |∂ t u| 2 can also be estimated by the right-hand side of (2.2). This completes the proof of the theorem.

Dissipativity estimates

In this section, we assume that h : R + × S → R is a locally integrable function such that 〈h(t, •)〉 = 0 (2.5)

for almost every t ∈ R. Let us set Q = R + × S. Our first result concerns the dissipativity of the dynamics for (1.1).

Theorem 2.2. In addition to the above hypothesis, assume that h ∈ L ∞ (Q).

Then, for any c ∈ R, there exists a number C > 0 depending also on ν, f , and |h| L ∞ (Q) such that, for any initial condition u 0 ∈ H 1 (S) with 〈u 0 〉 = c, the solution u ∈ X of problem (1.1), (1.3) satisfies the following inequality for t ≥ C ln(|u 0 | 2 + 2):

%u(t)% 2 1 + ! t+1 t " %u(s)% 2 2 + |∂ s u(s)| 2 2 # ds ≤ C. (2.6)
Proof. Let us note that integrating (1.1) in t ∈ J T and x ∈ S, we obtain 〈u(T )〉 = 〈u 0 〉 = c for any T ≥ 0. Substituting u = c+v in (1.1), we obtain for v an equation of exactly the same form as for u, with the function f replaced by its translation. Therefore we can assume from the very beginning that 〈u 0 〉 = 0 and consider solutions with zero mean value. Moreover, since X 1 is continuously embedded into C(J 1 , H 1 ) (see Theorem 3.1 in [LM72, Chapter I]), it suffices to estimate the integral in the left-hand side of (2.6).

To this end, we note that the inner product in relation (2.3) does not exceed ν|∂ x u| 2 2 +C 1 ν -1 |h| 2 2 , where C i stand for some unessential numbers not depending on the solution. Combining this with (2.3), we derive

∂ t |u| 2 2 + ν |∂ x u| 2 2 ≤ C 1 ν -1 |h| 2 2 .
(2.7)

Recalling that u has zero mean value in x, applying the Poincaré inequality to |∂ x u| 2 2 , and using the Gronwall inequality, we derive

|u(t)| 2 2 ≤ e -γt |u 0 | 2 2 + C 2 sup s≥0 |h(s)| 2 2 .
Combining this with (2.7), we see that

! t+1 t |∂ x u(s)| 2 2 ds ≤ C 3 & e -γt |u 0 | 2 2 + sup s≥0 |h(s)| 2 2 ' .
In particular, we can find a number C 4 > 0 such that

! t+1 t |∂ x u(s)| 2 2 ds ≤ C 4 for t ≥ T := C 4 ln(|u 0 | 2 + 2).
Since u is a continuous function of time with range in H 1 , for any t ≥ T + 1 we can find t 0 ∈ [t -1, t] such that %u(t 0 )% 1 ≤ C 5 . Applying (2.2) to the Cauchy problem for (1.1) studied on the interval [t 0 , t 0 + 2], we see that the integral in (2.6) is bounded by an absolute constant C. This completes the proof of the theorem.

The next result is useful when the right-hand side of (1.1) is unbounded. It is related to the strong nonlinear dissipation of the Burgers equation with a strictly convex flux.

Theorem 2.3. In addition to the hypotheses of Theorem 2.2, let us assume that h ∈ C b (R + , H 2 ) and f satisfies (1.2). Then, for any c ∈ R, there is a number C > 0 depending also on ν, f , and |h| L ∞ (R+,H 2 ) such that, for any initial condition u 0 ∈ H 1 with 〈u 0 〉 = c, the solution u ∈ X of problem (1.1), (1.3) satisfies inequality (2.6) for t ≥ 1.

Proof. By the Kruzhkov maximum principle (see [START_REF] Kružkov | The Cauchy problem for certain classes of quasilinear parabolic equations[END_REF] or [Bor13, Section 4.1]), there is a number C 1 > 0 depending only on ν, f , and h such that, for any t 0 ≥ 1/2, we have

|u(t 0 )| ∞ ≤ C 1 .
In view of inequality (2.1) (which remains valid for L ∞ initial conditions), the norm of the solution in the space L ∞ ((t 0 , t 0 + 1) × S) is bounded by a constant depending only on h. Taking the L 2 inner product of (1.1) with the function -(t -t 0 )∂ 2 x u and carrying out some simple transformations, we easily obtain a universal bound on the H 1 norm of the solution at t = t 0 + 1/2, so that

sup t≥1 %u(t)% 1 ≤ C 2 ,
where C 2 > 0 depends on ν, f , and h. The proof can now be completed by exactly the same argument as for Theorem 2.2. Combining Theorems 2.2 and 2.3 with the regularising property of (1.1), we arrive at the following bound for higher derivatives of solutions. Its proof can be obtained by taking the L 2 inner product of (1.1) with the functions (t -t 0 )∂ 4

x u or ∂ 4

x u and carrying out some standard transformations. Corollary 2.4. Under the hypotheses of Theorem 2.2 (or those of Theorem 2.3), there is a number C > 0 not depending on the initial condition such that the solution u ∈ X of (1.1), (1.3) satisfies the inequality

%u(t)% 2 ≤ C, (2.8)
where t ≥ 0 varies in the same half-lines as in the above theorems. If, in addition, the initial condition u 0 belongs to H 2 , then (2.8) holds for all t ≥ 0 with some number C > 0 depending also on %u 0 % 2 .

Some properties of linear parabolic equations

A key point of our proof of the stability of the Burgers flow is a lower bound for positive solutions of parabolic equations. The corresponding result is expressed in terms of the Harnack inequality for the linear equation

∂ t w -ν∂ 2 x w + ∂ x " a(t, x)w # = 0, (t, x) ∈ Q T , (2.9)
where a : Q T → R is a given function. A proof of the following proposition can be found in [START_REF] Krylov | A property of the solutions of parabolic equations with measurable coefficients[END_REF] (see also Section IV.2 in [START_REF] Krylov | Nonlinear Elliptic and Parabolic Equations of the Second Order[END_REF]).

Proposition 2.5. Let ν, T , and ρ be positive numbers and let a ∈ L ∞ (Q T ) be a function such that

∂ x a ∈ L ∞ (Q T ) and |a| L ∞ (Q T ) + |∂ x a| L ∞ (Q T ) ≤ ρ.
(2.10)

Then, for any T ′ ∈ (0, T ), there is θ = θ(T ′ , T, ν, ρ) > 0 such that, for any non-negative solution w ∈ X T of (2.9), we have

θ max x∈S w(T ′ , x) ≤ min x∈S w(T, x).
(2.11)

We shall also need the property of non-expansion of the L 1 norm of solutions for (2.9). The latter follows immediately from the maximum principle by a duality argument; see Lemma 3.2.2 in [START_REF] Hörmander | Lectures on Nonlinear Hyperbolic Differential Equations[END_REF].

Proposition 2.6. Under the hypotheses of Proposition 2.5, for any solution w ∈ X T of (2.9), we have

|w(t)| 1 ≤ |w(s)| 1 for 0 ≤ s ≤ t ≤ T .
(2.12)

Main result

The main tool for proving the stability of the Burgers flow is the contraction of the L 1 norm of the difference between two solutions. The rate of contraction depends on the size of solutions, which in turn is determined by the initial conditions and the right-hand side. More precisely, we have the following result.

Theorem 3.1. Let f ∈ C 2 (R) be an arbitrary function. Then, for any c ∈ R and any positive numbers ν, R, and T , there is q ∈ (0, 1) such that the following property holds: for any external force h ∈ L ∞ (Q T ), with an L ∞ norm not exceeding R and mean value satisfying (2.5) for a. e. t ∈ J T , and any initial conditions u 0 , v 0 ∈ B H 1 (R) with 〈u 0 〉 = 〈v 0 〉 = c, the corresponding solutions satisfy the inequality

|u(T ) -v(T )| 1 ≤ q |u 0 -v 0 | 1 . (3.1)
Proof. Let us write w for the difference u -v and note that it satisfies Eq. (2.9), in which

a(t, x) = ! 1 0 f ′ (v(t, x) + τ w(t, x)) dτ.
Moreover, we have 〈w(t)〉 = 0 for any t ∈ J T , and in view of inequality (2.8) applied to u and v, there is ρ > 0 depending only on ν, f , R, and T such that (2.10) holds. In particular, the Harnack inequality (2.11) is valid for any non-negative solution of (2.9). Let us set w 0 = u 0 -v 0 and write w + 0 and w - 0 for the positive and negative parts of w 0 ; that is, w ± 0 = max{±w 0 , 0}. We denote by w ± the solution of (2.9) that is equal to w ± 0 at time t = 0, so that w(t) = w + (t) -w -(t) for any t ∈ J T and w ± are non-negative functions. If w + (T /2, x) ≤ 1 4 |w 0 | 1 for any x ∈ S, then

|w + (T /2)| 1 ≤ 1 4 |w 0 | 1 .
Since the mean value of w(t) is zero, the L 1 norms of w + (t) and w -(t) are the same, so that |w -(T /2)| 1 ≤ 1 4 |w 0 | 1 . Combining this with (2.12), we derive

|w(T )| 1 ≤ |w(T /2)| 1 ≤ |w + (T /2)| 1 + |w -(T /2)| 1 ≤ 1 2 |w 0 | 1 .
Exactly the same argument applies when w -(T /2, x) ≤ 1 4 |w 0 | 1 for any x ∈ S. We now assume that

max x∈S w ± (T /2, x) ≥ 1 4 |w 0 | 1 .
In view of the Harnack inequality (2.11), there is a number θ > 0 such that

min x∈S w ± (T, x) ≥ θ max x∈S w ± (T /2, x) ≥ θ 4 |w 0 | 1 .
Combining this with (2.12), we derive

|w(T, x)| 1 = ! S ( ( w + (T, x) -w -(T, x) ( ( dx = ! S ( ( (w
+ (T, x) -θ 4 |w 0 | 1 ) -(w -(T, x) -θ 4 |w 0 | 1 ) ( ( dx ≤ ! S " w + (T, x) -θ 4 |w 0 | 1 # dx + ! S " w -(T, x) -θ 4 |w 0 | 1 # dx = |w + (T )| 1 + |w -(T )| 1 -θ 2 |w 0 | 1 ≤ |w + 0 | 1 + |w - 0 | 1 -θ 2 |w 0 | 1 = " 1 -θ 2 # |w 0 | 1 .
We thus obtain the required inequality (3.1) with q = max{ 1 2 , 1 -θ 2 }.

Applications

Uniqueness and stability of a bounded trajectory

Let us consider Eq. (1.1), in which f ∈ C 2 (R) is an arbitrary function, and h belongs to the space C b (R, H 1 ) and satisfies (2.5) for all t ∈ R. The following result describes the large-time behaviour of solutions for (1.1).

Theorem 4.1. Under the above hypotheses, for any c ∈ R, there is a unique solution v(t, x) of (1.1) in the space

X (R) ∩ C b (R, H 1 ) such that 〈v(t, •)〉 = c for all t ∈ R. (4.1)
Moreover, there exists γ > 0 such that, for any R > 0, a sufficiently large C R > 0, and any initial condition u 0 ∈ B H 1 (R) with 〈u 0 〉 = c, the corresponding solution u(t, x) satisfies the inequality

%u(t) -v(t)% 1 ≤ C R e -γt |u 0 -v(0)| 2/5 1 , t ≥ 1. (4.2) Proof.
Step 1: Exponential stability. We first prove that, for any R > 0 and any initial conditions u 01 , u 02 ∈ B H 1 (R) with 〈u 01 〉 = 〈u 02 〉, the corresponding solutions satisfy the inequality

%u 1 (t) -u 2 (t)% 1 ≤ C R e -γt |u 01 -u 02 | 2/5 1 , t ≥ 1, (4.3)
where γ > 0 does not depend on the initial conditions. To this end, we first consider the case when the solutions satisfy inequality (2.8) for all t ≥ 0. By a well-known interpolation inequality (see Section 15.1 in [START_REF] Besov | Integral Representations of Functions and Imbedding Theorems[END_REF]), for any u ∈ H 2 we have %u% 1 ≤ C 1 %u%

3/5 2 |u| 2/5
1 . Since the H 2 norm of the difference u = u 1 -u 2 is bounded, inequality (4.3) will be established for t ≥ 0 once we prove the exponential decay of the L 1 norm |u(t)| 1 .

To see this, let us note that the difference u(t, x) satisfies the linear equation (2.9). In view of the non-expansion of the L 1 norm (see Proposition 2.6), it suffices to prove that |u(k)| 1 ≤ q k |u(0)| 1 for any integer k ≥ 0, where q ∈ (0, 1) does not depend on k. The latter estimate is an immediate consequence of Theorem 3.1.

To prove (4.3) for arbitrary initial conditions in B H 1 (R), note that the solutions u i satisfy (2.8) for t ≥ 1 with some constant C = C(R). Using the interpolation inequality and the non-expansion of the L 1 norm, we see that

%u(t)% 1 ≤ C 2 (R)|u(0)| 2/5 1
for t ≥ 1. On the other hand, the solutions u i ∈ X satisfy (2.8) for t ≥ t 0 = C 3 ln(R + 2), so that

%u(t)% 1 ≤ C 4 e -γ(t-t0) |u(t 0 )| 2/5 1 ≤ C 5 e -γt |u(t 0 )| 2/5 1
for t ≥ t 0 . Combining this with the above estimate for %u(t)% 1 valid for t ≥ 1, we arrive at (4.3).

Step 2: Bounded solution. We now fix c ∈ R and construct an H 1 -bounded solution v ∈ X (R) of (1.1) defined throughout the real line such that (4.1) holds. Once it is done, the stability inequality (4.2) will follow from (4.3).

The construction of v is based on a standard argument using solutions issued from initial times going to -∞. Namely, let us denote by u n the solution of (1.1) satisfying the initial condition u n (-n) = c. Then, for any m < n and T ≥ 1-m, inequality (4.3) applied on the half-line [-m, +∞) implies that

sup |t|≤T %u m (t) -u n (t)% 1 ≤ C 6 e -γ(T +m) .
Letting m → +∞, we see that (u n ) converges in H 1 to a limit v ∈ C b (R, H 1 ), uniformly on any bounded interval. Furthermore, the sequence (u n ) is bounded in X ([-T, T ]) for any T > 0, and the weak compactness of a unit ball in a Hilbert space implies that v ∈ X (R). It is straightforward to see that v is a solution of (1.1) and that (4.1) holds. This completes the proof of the theorem.

Equation with random external force

We now consider the case when the external force h is a stochastic process. Namely, we assume that almost every trajectory of h is a continuous function of time with range in H 2 such that relation (2.5) holds for t ≥ 0, and the random variable (%u(t)% 2 + %v(t)% 2 ) ≤ C.

K := lim sup T →∞ 1 T ! T 0 max t≤s≤t+1 %h(s)% 2 dt (4.
Using now Theorem 3.1, we can construct a random variable q ∈ (0, 1) such that |u(t k + 2) -v(t k + 2)| 1 ≤ q |u(t k + 1) -v(t k + 1)| 1 for any k ≥ 1. Combining this with (2.12), we conclude that, with probability 1,

|u(t k + 2) -v(t k + 2)| 1 ≤ q k |u 0 -v 0 | 1 for all k ≥ 1.
Using again (2.12), we see that (4.6) holds almost surely.

Remark 4.4. Theorem 4.2 does not specify the rate of convergence in (4.6), and it is unlikely that one can say something more about (4.6) without any additional information. On the other hand, if h is a mixing stationary process such that (4.5) holds, then the random variable K in (4.4) is almost surely equal to M (by the Birkhoff theorem). In this case, any information about the rate of mixing will provide some quantitative estimates for the random times (t k ) used in the above proof, and this allows to describe the rate of convergence in (4.6).

  The research of the first author has been partially supported by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114 Scaling Cascades in Complex Systems, Project Number 235221301, Project C10 -Numerical analysis for nonlinear SPDE models of particle systems. The research of the second author has been supported by the CY Initiative through the grant Investissements d'Avenir ANR-16-IDEX-0008 and by the Ministry of Science and Higher Education of the Russian Federation (megagrant agreement No. 075-15-2022-1115).

  Let f ∈ C 2 (R) be an arbitrary function satisfying (1.2), let the above hypotheses be fulfilled for h, and let u 0 , v 0 ∈ H 1 be arbitrary initial conditions such that 〈u 0 〉 = 〈v 0 〉. Then, with probability 1, the corresponding solutions belong to X and are such that However, combining dissipativity of the dynamics with the Markov property, it is not difficult to construct an increasing sequence of stopping times (t k ) such that the difference t k -t k-1 has a finite exponential moment, and the solutions are bounded inH 2 on [t k , t k + 1].Repeating then the argument used in the proof of Theorem 4.2, one can establish the exponential convergence in (4.6) and the uniqueness of a stationary measure. Since the approach described above is well known and is carried out in detail in the paper [DR22, Section 3], we skip the corresponding argument.Proof of Theorem 4.2. Let us fix two initial conditions u 0 , v 0 ∈ B H 1 (R) with the same mean value and denote by u and v the corresponding solutions. In view of Theorem 2.1, they are well defined and belong to X with probability 1. It follows from (4.4) that there is an increasing random sequence (t k ) going to +∞ with probability 1 such that supt k ≤t≤t k +2 %h(t)% 2 ≤ 2K + 1.Applying Corollary 2.4 to the solutions u and v on the intervals [t k , t k + 2], we find an almost surely finite random constant C > 0 such that supt k +1≤t≤t k +2

			4)
	is almost surely finite. For instance, in view of the Birkhoff theorem, condi-
	tion (4.4) is certainly satisfied if h is a stationary H 2 -valued stochastic process
	with almost surely continuous trajectories such that
	M := E max 0≤t≤1	%h(t)% 2 < ∞.	(4.5)
	Theorem 4.2.

|u(t) -v(t)| 1 → 0 as t → ∞.

(4.6)

Remark 4.3. Theorem 4.2 does not apply to the case when the external force in (1.1) has the form h + η, where h = h(x) is a deterministic function, and η = η(t, x) is a spatially regular white noise.

Inequality (1.2) on the flux is not needed if the external force is a bounded function of time. In particular, Theorems 3.1 and 4.1 are valid for any C

-smooth flux f .

The preprint[START_REF] Djurdjevac | Stabilisation of a viscous conservation law by a one-dimensional external force[END_REF] was withdrawn from publication, since the main result obtained there is essentially covered by the papers[START_REF] Hill | Dynamics of a nonlinear convection-diffusion equation in multidimensional bounded domains[END_REF][START_REF] Djurdjevac | Synchronisation for scalar conservation laws via Dirichlet boundary[END_REF].

Note that we did not impose any condition on the mean value of h with respect to x, so that the norm of solutions may indeed grow with T .

Acknowledgments