Fanny Ducel
email: ducelfanny@gmail.com

Karën Fort
email: karen.fort@loria.fr

Gaël Lejeune
email: gael.lejeune@sorbonne-universite.fr

Yves Lepage
email: yves.lepage@waseda.jp

Do we Name the Languages we Study? The #BenderRule in LREC and ACL articles

Keywords: #BenderRule, language diversity, ethics

This article studies the application of the #BenderRule in Natural Language Processing (NLP) articles according to two dimensions. Firstly, in a contrastive manner, by considering two major international conferences, LREC and ACL, and secondly, in a diachronic manner, by inspecting nearly 14,000 articles over a period of time ranging from 2000 to 2020 for LREC and from 1979 to 2020 for ACL. For this purpose, we created a corpus from LREC and ACL articles from the above-mentioned periods, from which we manually annotated nearly 1,000. We then developed two classifiers to automatically annotate the rest of the corpus. We show that LREC articles tend to respect the #BenderRule (80 to 90% of them respect it), whereas only around half of ACL articles do. Interestingly, over the considered periods, the results appear to be stable for the two conferences, even though a rebound in ACL 2020 could be a sign of the influence of the blog post about the #BenderRule.

Introduction 1.Background

This article is positioned in the field of NLP4NLP, an acronym known from the series of articles by Mariani et al. (2019a) (part II in [START_REF] Mariani | The NLP4NLP Corpus (II): 50 Years of Research in Speech and Language Processing[END_REF]). NLP4NLP consists in applying Natural Language Processing (NLP) techniques and exploring the field of NLP itself, by mainly inspecting the content of published articles. The aims of NLP4NLP can be various. For example, exploring the progress in the field, e.g., identifying discoveries or hot topics [START_REF] Mariani | Rediscovering 25 years of discoveries in spoken language processing: a preliminary analysis of the ISCA archive[END_REF][START_REF] Mariani | Rediscovering 15 years of discoveries in language resources and evaluation: The LREC anthology analysis[END_REF][START_REF] Buitelaar | Hot topics and schisms in NLP: Community and trend analysis with saffron on ACL and LREC proceedings[END_REF], addressing ethical issues, e.g., reuse and plagiarism [START_REF] Mariani | A study of reuse and plagiarism in speech and natural language processing papers[END_REF] 1 , or even questioning the reliability of the methods used, e.g., in machine translation [START_REF] Marie | Scientific credibility of machine translation research: A meta-evaluation of 769 papers[END_REF] 2 . This article addresses a question with both ethical and scientific implications: we inspect the number and which languages are studied in two NLP conferences, LREC and ACL, that have already been contrasted in a previous work [START_REF] Buitelaar | Hot topics and schisms in NLP: Community and trend analysis with saffron on ACL and LREC proceedings[END_REF].

To that end, we firstly annotate a sample of articles by hand and then generalize our results to the entire collection of articles by using linguistic feature extraction and classification techniques from machine learning.

Topic Addressed in this Article

The problem of the number and which languages are studied in the field of NLP has been raised recently, although it had been identified earlier in the history of grammar or linguistics (see Section 1.3). As for NLP, even if the majority of research is still devoted to a limited number of languages, the situation seems to be evolving. This is at least what [START_REF] Joshi | The state and fate of linguistic diversity and inclusion in the NLP world[END_REF] suggest in their article on linguistic inclusion and diversity in the scientific community. Being concerned with language diversity implies that the languages studied matter and therefore should at the very least be stated. This might sound obvious, but, a decade ago, [START_REF] Bender | On achieving and evaluating language-independence in NLP[END_REF] noticed that many researchers simply do not cite the languages they work on. She consequently urged researchers to specify the name of the languages being studied, even (or especially) when it is English: "Do state the name of the language that is being studied, even if it's English. Acknowledging that we are working on a particular language foregrounds the possibility that the techniques may in fact be language specific. Conversely, neglecting to state that the particular data used were in, say, English, gives [a] false veneer of language-independence to the work." (Bender, 2011, p. 18) Eight years later, in a blog post [START_REF] Bender | The #BenderRule: On naming the languages we study and why it matters[END_REF], she reminded the community that "English is neither synonymous with nor representative of natural language". Specifying the languages studied thus became the #BenderRule, which can be summarized as follows:

"Always name the language(s) you're working on." [START_REF] Bender | The #BenderRule: On naming the languages we study and why it matters[END_REF] Indeed, the non application of the #BenderRule raises linguistic, sociological and ethical issues. Not naming the language studied (which is often the case for English), implicitly implies that the methods developed could work on any other language, as if English were a neutral and universal language. However, as argued by Bender herself, English is far from representing the diversity of languages.

Linguistic Issues

Examples of now well-identified biases towards something believed universal at some point in time can be given, from the history of grammar and linguistics. Let us take the example of word order. In the European history of grammar, the very influential Grammaire de Port-Royal [START_REF] Arnauld | Grammaire générale et raisonnée de Port-Royal. chez Prault fils l'aîné[END_REF] took the order of words in French as the "natural" one. It claimed its universality, with the consequence that the order in other languages, like German, had to be explained by casting it back to that of French. Fixity of word order is one of several traits listed by Bender to insist on the fact that English is not universal ("English has relatively fixed word order"). Earlier, in linguistics, it had already been pointed at as somewhat special. Specifically, Mel'čuk (1988, p. 4) warned that focusing on it might well lead to methodological issues: English is very exotic in that it uses constituency as its only expressive device in syntax, i.e., as the only device for encoding syntactic structure in actual sentences. [. . . ] Constituency is a MANIFESTATION of syntactic structure, not syntactic structure itself. But thinking or even simply working mostly with English lures the researcher into mistaking this idiosyncratic surface trait of a particular language [. . . ] for a universal mechanism of syntactic representation.

With the unfortunate habit of citing articles no older than a decade back, there exists a risk of forgetting about the (even relatively recent) history of sciences, and consequently a risk of repeating methodological mistakes from several decades or even centuries ago.

Sociological and Ethical Issues

In an article on meta-learning, [START_REF] Nooralahzadeh | Zero-shot cross-lingual transfer with meta learning[END_REF] take note of the fact that "although there is growing awareness in the field, as evidenced by the release of datasets such as XNLI (Conneau et al., 2018), most NLP research still only considers English [START_REF] Bender | The #BenderRule: On naming the languages we study and why it matters[END_REF]."

Reducing studies to only English might have sociological impacts and real consequences on the diversity of NLP works. This leads to a vicious circle: working only on English becomes more rewarding (i.e., chances to have an article accepted are higher) because "[w]ork [ing] on languages other than English is often considered 'language specific' and thus reviewed as less important than equivalent work on English." [START_REF] Bender | The #BenderRule: On naming the languages we study and why it matters[END_REF]. As stated by [START_REF] Hovy | The social impact of natural language processing[END_REF], "[p]otential consequences are exclusion or demographic misrepresentation. This in itself already represents an ethical problem for research purposes, threatening the universality and objectivity of scientific knowledge (Merton, 1973)." Along the same line, [START_REF] Bender | On the dangers of stochastic parrots: Can language models be too big?[END_REF] claim that "most language technology is built to serve the needs of those who already have the most privilege in society". Indeed, scientific research can result in the development of products, applications and software that will only be accessible to English (and, to a lesser extent, Chinese) speakers. Such concerns of unequal treatment of languages have been raised in various places. For instance, Wagner (2021) uses Bender's words in the introduction of her thesis on Icelandic to argue that NLP plays a role in the extinction or survival of languages.

Contributions of this Article

The extent to which the #BenderRule is applied or not is unclear and, to our knowledge, there is no published work that explicitly addresses this issue3 , apart from some studies focusing on linguistic diversity and representativeness, like [START_REF] Joshi | The state and fate of linguistic diversity and inclusion in the NLP world[END_REF] already mentioned above. This article is a first attempt at quantifying the application of the #BenderRule, i.e., to empirically examine whether and to what extent the #Bender-Rule is actually applied. To this end, we train two classifiers to automatically annotate research articles with whether they apply the #BenderRule, and which language(s) they study. The study reported in this article is conducted on two corpora in the field of NLP: articles from the successive occurrences of Language Resources and Evaluation Conference (LREC) and the Annual Meetings of the Association for Computational Linguistics (ACL). 4The contributions of this article are as follows:

• we built a corpus of nearly 14,000 articles from the LREC and ACL conferences;

• we manually annotated a sample of nearly 1,000 of these articles, i.e., we checked whether the #BenderRule is applied or not, and took record of the studied languages when possible;

• we trained classifiers on these manual annotations and applied them on the entirety of the articles in both conferences;

• we analyzed the results from a contrastive and diachronic perspective and extracted general trends.

We detail the methodology used to build the corpus and the classifiers in the following section (Section 2). We analyze the results obtained in Section 3. Finally, we discuss the limits of our work in Section 4.

Methodology

Corpus Building

We created a corpus of articles in English from LREC (from 2000 to 2020) and ACL (from 1979 to 2020).

To do so, we converted all the PDF articles from the LREC conferences (6,715 files) to text files using Dallas Card's scripts5 . We obtained the PDF files from the official website of the LREC conference 6 and from the ACL Anthology7 for the articles published after 2016. Some of the articles from ACL were already in text format. We obtained the files from 1979 to 2016 from the AAN Anthology Network Corpus [START_REF] Radev | The ACL anthology network corpus[END_REF] 8 . We performed a pdftotxt conversion on the ones from after 2016 using the scripts mentioned above. This ACL sub-corpus (text+pdftotxt) contains 7,262 articles. We then eliminated the articles which presented OCR issues and could not be used (46 LREC files and 24 ACL files). We detected these files thanks to a very simple string-matching test. The text files that do not include the token "the" can be considered unreadable due to OCR issues. Finally, we obtained a corpus of 13,931 articles, with two sub-corpora, one of 6,669 LREC articles and one of 7,262 ACL articles (see Table 1).

Manual Annotation

One of the authors of the present article manually annotated 550 randomly selected LREC articles (50 per edition of the conference, representing approximately 8% of the LREC sub-corpus) and 420 randomly selected ACL articles (10 per year, representing approximately 5% of the ACL sub-corpus), for a total of 970 articles. Each article was annotated with one of the four following categories, along with the languages studied:

• Applied: The #BenderRule is applicable and applied. The language studied is clearly mentioned somewhere in the article.

• Deducible: The #BenderRule is applicable, not applied but deducible. That is, there is at least one monolingual resource that is mentioned, which is listed in the LREMap, thus helping the annotator in deducing the language(s) studied.9

• Non-deducible: the #BenderRule is applicable, not applied and not deducible, i.e., the language is not mentioned and cannot be deduced from a resource name.10

• Non-applicable: The #BenderRule is not applicable. For instance, many theoretical articles do not focus on any particular language and even if they may use English examples, the claims are usually of a more general nature.

We proceeded in the following way for the establishment of the four above categories. We first performed trial annotations, then identified the problems and consequently designed annotation guidelines, so as to ensure a consistent annotation. The three following rules were established:

• i) an article is annotated as "Applied" when it mentions the studied language, even if this mention appears only once at the end of the article,

• ii) the LREMap [START_REF] Calzolari | The LRE Map. Harmonising Community Descriptions of Resources[END_REF] 11 was selected as the reference for the "Deducible" class12 ,

• iii) we decided to exclude using the examples given in the article to deduce the language, as they can be in languages neither we nor a classifier can identify and because they are not reliable enough as they can be translations (usually into English) from the studied language.

The distribution of the different categories in the manual annotation for LREC and the ACL sub-corpora is illustrated in Figure 1. It shows that LREC articles apply the #BenderRule much more often than ACL articles: 90% of LREC articles in our sub-corpus apply the #BenderRule, against 55% in ACL. Said in another way, and following the definitions given above for the four categories, 45% of ACL articles in our manually annotated corpus do not cite any language. Now, considering that 13% in these 45% of articles have been categorized as "Non-applicable", there are 32% articles in which the #BenderRule has not been applied when it should have. This represents almost a third of the annotated ACL articles and cannot be considered negligible. This has to be compared with (100% -90%) -2% = 8% for LREC articles. To summarize this comparison, four times more articles do not apply the #BenderRule when they should in ACL than in LREC.

To check the quality and consistency of the annotations, the three other authors double-annotated a selection of the above articles, as shown in Table 2, so that approximately 7% of the annotations were duplicated.

The sampling was generally random, but we tried to balance the categories (the entire corpus itself is obviously not balanced) and we added some complicated cases on purpose. For LREC, three to four articles per conference were double-annotated and for ACL, one article per conference between 2000 and 2020, except for 2000, 2005, 2010, 2015 and 2020 for which two articles were in the selection. Moreover, one article every two conferences between 1980 and 1998 was double-annotated too. Duplicated annotations allowed the computation of the inter-annotator agreement. We computed the observed inter-annotator agreement between annotator 0 (A 0 , who manually annotated almost 1,000 articles) and each of the others. We obtain an agreement of 80% with annotator 1, 83% with annotator 2 and 63% with annotator 3. This last result can probably be explained by the fact that annotator 3 annotated less articles than the others, so the articles they got were especially complicated and the classification was questionable. If we merge the annotations from annotators 1, 2 and 3 and compare them to annotator 0, we obtain a Cohen Kappa [START_REF] Cohen | A coefficient of agreement for nominal scales[END_REF]) of 0.82, which can be considered good [START_REF] Artstein | Inter-coder agreement for computational linguistics[END_REF].

To extend the study and cover all available articles in both conferences, we developed two classifiers.

Classification using Pattern-Matching

A first, baseline classifier, Class PM (PM for patternmactching), relies on an approach based on stringmatching. We consider that an article applies the #BenderRule when at least one language name is found in the text.

LREC ACL Total

For that, we use a list of over 500 different ISO 639-2 language names in English 13 . For each individual article, we memorize the languages mentioned, along with their number of mentions. The languages are obtained by pattern-matching after standard preprocessing of the texts, i.e., removal of punctuation, lowercasing, and tokenization. If at least one of the tokens matches one of the language names in the list, the article is classified as applying the #BenderRule ("Applied").

The above method allows us to put aside the articles in which no language is cited. This might be because the authors did not apply the #BenderRule, or because the #BenderRule cannot be applied (theoretical or metalinguistic work). We consider an article as "Deducible" if it contains at least one name of a language resource listed in the LREMap. 14 The list of all the monolingual language resources present on the LREMap website was extracted by using Web scraping methods based on the Python library BeautifulSoup with various improvements 15 . Tokens in articles are matched against language resource names and paths. Articles citing at least one resource from the list are put in the "Deducible" category.

The remaining articles are classified as not applying the #BenderRule or the #BenderRule is not applicable (referred to as "N/A" in Table 3 and Figure 2) 16 . Note that the classifiers cannot differentiate between the articles based on linguistic data which do not cite the studied language (i.e. that should apply the #BenderRule but do not) from those that do not have to mention any language, for example because the topic is theoretical or meta-linguistic.

13 See: https://github.com/ISO639/2. 14 We added to the list the following language resources, which we consider popular enough (for English and French): "Le Monde, France info, France inter, NYT, PropBank, Washington Post, Wall Street Journal, Brown Corpus, Le Robert, Switchboard". 15 See T. Ujhelyi's tutorial: https://data36.com/ scrape-multiple-web-pages-beautifulsoup-tutorial/. 16 It should be noted that the number of classes in the automatic processing therefore becomes three, to be compared with four during manual annotation.

Classification using Machine Learning

The approach based on string-matching does not recognize the cases where a language is mentioned without being the one actually studied. For instance, authors tend to mention the languages they would like to study in future works in the conclusion, or the languages that have already been studied by colleagues, in the introduction or the state-of-the-art sections. Thus, we train a sentence classifier with the aim of correcting the above-mentioned issues. This sentence classifier uses supervised machine learning techniques. It is trained on 2,625 sentences extracted from the manually annotated ACL articles and which contain at least one language name. Those sentences coming from articles that were manually annotated as "Applied" were categorized as such, whereas the ones coming from articles that were considered as not applying the #BenderRule were annotated as not applied or not applicable, i.e., "N/A". We use CountVectorizer from scikit-klearn to vectorize the corpus, with default parameters, and LogisticRegression (with the saga solver) as the supervised classification algorithm as this proved to be the best combination. Using tf•idf weighting or working with n-grams longer than unigrams did not improve the results. Logistic Regression showed the best results among the classifiers that we tested (Decision Trees, Random Forests, Bayesian and SVM classifiers). The classifier for articles, Class ML (ML for machine learning), classifies an article as "Applied" if it contains at least one sentence classified as such by the sentence classifier. Typically, a sentence like "The morphological status of affixes in Chinese [. . . ]" [START_REF] Tseng | Computational modeling of affixoid behavior in Chinese morphology[END_REF] would be classified as "Applied", whereas "These annotations can demarcate components of signs, [. . . ] or a translation into another language like English." [START_REF] Bragg | Sign language recognition, generation, and translation: An interdisciplinary perspective[END_REF] would not be classified as "Applied".

Results

Performance of the Classifiers

We evaluate our classifiers by comparing their outputs with the manual annotation we performed. The results are presented in Table 3. Class ML performs better than Class PM on ACL articles in terms of precision and presents a better Spearman correlation with Annotator 0. However, their performance is comparable on LREC articles. Results are less accurate for the category "Deducible" for ACL articles. We could question our definition of "Deducible" and especially our choice to use the LREMap as the sole reference, as a number of corpora do not appear in it. Nonetheless, it is difficult to assert whether a resource can be considered reliable enough or not and the LREMap has the merit of being a well-established meta-reference. Furthermore, some articles are missing from the category "Deducible" because they remain falsely classi- fied as applying the #BenderRule ("Applied").

Results of the Experiments

Classification Results

Class ML classifies 40% of all the ACL articles in our corpus as not applying the #BenderRule, which means that they either do not mention any language, or mention some, which are not the ones studied. Similarly, among the 420 manually annotated articles, 45% do not apply the #BenderRule according to Class ML . We explain this high proportion by the longer period of time for ACL (1979( -2020( , 2000( -2020 for LREC) for LREC). Multilingual works were rarer in the past, it was maybe obvious to work on English at that time, and therefore authors did not specify the studied language. Articles from LREC seem to apply the #BenderRule more often. Class PM predicts that only 17% of articles do not apply it (14% according to Class ML ), as compared to 9% in manual annotation.

The difference between the two conferences can be explained by the fact that LREC, following the aims of the conference, presents more linguistic diversity. Languages that are usually less represented are (almost) always explicitly mentioned. We note that in the last category (N/A), there are more articles where the #Ben-derRule is not applicable than not applied. Therefore, even when there is no language mentioned, it is legitimate. Furthermore, we noticed during manual annotation that, in some LREC articles, not only the language was specified but also its particular variant.

Figure 2 shows the distributions of articles obtained by automatic classification using Class ML over the whole LREC and ACL sub-corpora. They reasonably reproduce the distributions obtained by manual annotation of a sample of articles in each sub-corpus (see Figure 1). For ACL, the manual annotation gives 55% Manual inspection reveals that some articles are mistakenly assigned to the class "Deducible", because some terms are erroneously considered resources names, some acronyms are polysemous, or because some authors cite the resources used in previous works.

As for LREC, automatic classification yields 94% of "#BenderRule Applied" (vs. 91%), 2% of "Deducible" (vs. 4%) and 4% for "N/A" (vs. 5%). Among the articles applying the #BenderRule, we notice that some only quote one language once. For ACL, it concerns 13% of the total of articles, against 4% of the manually annotated ones. LREC stands out with only 5% of articles containing a single mention of a language. As for the manually annotated articles, it concerns only 1% of them. We conclude that authors focus more attention on the languages they study in the LREC conference.

Diachronic Study

We show in Figure 3 the proportion of articles applying the #BenderRule per year, for each year of the studied period of time, in each conference.

There does not seem to be any chronological evolution in ACL, the proportion being almost always between 50% and 60%. Nonetheless, we observe an increase in 2020, after a decreasing trend in average from 1986 to 2019. One can ask whether the increase was caused by Bender's blog post in 2019.

For LREC, the proportion of articles applying the #BenderRule exhibits a slowly increasing trend over the years, especially between 2000 and 2010. However it is worth noticing that, the lowest number, which concerns 2000, is already above 80%, that is, greater than the maximum for ACL, reached in 1986 with 68%.

This confirms our hypothesis that, LREC being a conference focusing on language resources, the languages being studied are most of the time mentioned, even if it is English (Figure 5 supports this observation).

3.2.3. Languages for which the #BenderRule is not Applied Figure 4 gives the number of articles per number of languages mentioned. It shows that a large number of ACL articles do not cite any language at all (0 languages): they match the "Deducible" and "N/A" (not applied or not applicable) categories. Among the articles which apply the #BenderRule, we notice that the languages are more cited when the study is not monolingual. Indeed, it is necessary to name and distinguish the different studied languages in order to compare them in a multilingual work. Figure 4 makes the proportion of cross-lingual works visible, since most multilingual studies apply the #BenderRule. For ACL, the system detects many more articles which do not apply the #BenderRule, the proportion of articles that do not cite any language is almost three times higher than the others. We can also see that monolingual studies are a little more represented than bilingual ones, and that trilingual and quadrilingual studies are even less present. Nonetheless, there are also articles that study up to 13 languages simultaneously. Figure 4 shows that in LREC, there are significantly less articles which do not cite any language than articles that cite any other number of languages. We even notice that there are more bilingual studies than monolingual ones. We can generally say that LREC articles are much more multilingual, with studies that can concern up to several dozens of languages at once. 17Figure 5 shows that the top 10 languages in both conferences are very similar, with only one language be- There is a long tail after 20 languages for both conferences, the number of articles is then almost always one article only. The maximal number of languages is 92 languages for ACL and 90 for LREC.

ing different (Korean is only present in ACL, while Farsi appears only in LREC). Furthermore, it should be noticed that these top languages are spoken in politically or economically important countries or have a relatively large number of speakers. As Bender already noticed, "many of the languages included are close relatives of each other" [START_REF] Bender | Linguistically naïve != language independent: Why NLP needs linguistic typology[END_REF]. Indeed, except for Chinese, Japanese and Korean, they all (including Farsi) belong to the Indo-European language family. Moreover, only three languages constitute more than 10% of the total of mentioned languages.

In the LREC corpus, German is the most cited language, while English constitutes less than 16% of the mentions. However, the majority class is in fact the one composed of the 239 other languages present in the corpus. This shows once again the large linguistic diversity offered by LREC.

As for ACL, we observe that Mandarin Chinese comes right after English, which reflects the fact that it is gradually gaining ground in the English-speaking and the scientific community in general. We anticipate that, in a few years, Mandarin Chinese may overtake English in the rankings. This does not mean that Mandarin Chinese will be more studied than English, but that it will be more mentioned. The results are of course biased because the non-application of the #BenderRule leads to some invisibility of the word "English" in articles.

If we extract the list of studied languages in the articles which do not apply the #BenderRule (corresponding to languages that are studied without being men- Language (ISO 639-1) Percentage of articles mentioning the language LREC ACL Figure 5: Percentage of articles mentioning a given language in each corpus, over all articles applying the #Ben-derRule (LREC in white, ACL in black). The languages are sorted by decreasing order of percentage in LREC (white bars). Only the 10 most mentioned languages in each corpus are shown (11 languages in total). tioned), we only get English. This confirms Bender's assertions that English remains considered as a "default language", "synonymous of natural language" and that there is, indeed, a real "habit of failing to name the language studied when it is English" [START_REF] Bender | The #BenderRule: On naming the languages we study and why it matters[END_REF].

Potential Limitations of the Study

Obviously, we could not read the entirety of all the articles that we manually annotated, therefore some annotations might be incorrect or questionable. However, the inter-annotator agreement results that we obtained are reassuring. Moreover, the list of languages that we used is nonexhaustive and we probably missed some. We identified such a case: [START_REF] Yu | If you Even don't have a bit of Bible: Learning delexicalized POS taggers[END_REF] worked on 107 languages but only list them with their ISO codes which we do not take into account and could not detect. Another issue that we did not take into account is the lack of precision concerning the language, even when mentioned. For example, when Chinese is mentioned, the authors should precise which Chinese (usually, as Bender puts it in her blog post, it is Mandarin Chinese), the same goes for almost all languages. More importantly, we had to exclude 70 articles due to OCR issues (unreadable files). As we did not manually correct the OCR results, there might be some remaining issues that we could not identify. Finally, we limited our study to articles up to 2020 (the last LREC). The #BenderRule was worded in 2019, that is to say, shortly before that. Therefore, there might be an influence of this proximity that is difficult to assess properly now. In order to evaluate the impact of the #BenderRule on the longer term, these experiments should be rerun a few years from now. However, our main goal was not to evaluate the impact of the blog post, but to assess the prevalence of the issue, hence, this does not influence our research as much.

Conclusion

We compared the application of the #BenderRule in two important conferences. We found that LREC articles mention much more the languages they study. We also found that LREC presents more linguistic diversity, more multilingual works, and that, in general, it puts more emphasis on the variety of languages than ACL does. We performed the same experiments using the same programs (with a few adaptations for French) on the TALN conference and obtained results that are more similar to the figures presented here for ACL. We diachronically examined the evolution of the application of the #BenderRule but did not observe any significant change over time. Finally, in both corpora, we observed that when the #BenderRule is not applied, the studied language is English, which confirms some of Bender's assertions. The code and resources used are freely available on GitHub18 for replication purposes. In order to assess the impact of Bender's blog post, the classifiers should be re-run in the coming years on the ACL (2021 and following) and LREC (2022 and following) proceedings. Another perspective is to replicate the experiment on the articles from other international and national NLP conferences. As mentioned above, we already started to work on the proceedings of the French national conference TALN and expect to publish the results rapidly.

Figure 1 :

 1 Figure 1: Results of the manual annotation of the articles of LREC and ACL, in percentage (LREC in white, ACL in black).

Figure 2 :

 2 Figure 2: Results, in percentage, of the automatic classification by Class ML of LREC and ACL articles into three categories: #BenderRule applied (Applied), language(s) studied deducible (Deducible), or none of the previous (N/A). LREC in white, ACL in black.

Figure 3 :Figure 4 :

 34 Figure 3: Number of articles applying the #BenderRule, in percentage in each edition of both conferences (LREC in white, ACL in black). LREC is held every two years and its proceedings are only available from 2000.

Table 1 :

 1 Composition of the corpus.

		LREC	ACL
	Text files	0 4,867
	Converted files	6,715 2,419
	Excluded files	46	24
	Total	6,669 7,262

Table 2 :

 2 Double-annotated articles per annotator and conference, and observed agreement with A 0 .

	Agreement
	with A 0

Table 3 :

 3 Performance of the classifiers and Spearman correlation with Annotator 0 on the two sub-corpora (N/A: not applied or not applicable), the best result for each class and each measure is in bold..

	of "Ap-

With the felicitous conclusion, at that time, that "plagiarism is very uncommon in our community."

2 With guidelines recommending the use of more than one evaluation metric, and the systematic computation of statistical significance, to cite only the first two recommendations.

For instance, the impressively extensive study of the LREC archive in[START_REF] Mariani | Rediscovering 15 years of discoveries in language resources and evaluation: The LREC anthology analysis[END_REF] mentions small communities working on specific languages, but does not provide any figure about the languages studied in LREC, nor does it give the names of these specific languages.

 4 The #BenderRule tells us that it does not go without saying that the language of the corpora is English.

www.github.com/dallascard/acl-papers

See: https://www.lrec-conf.org/.

See: https://aclanthology.org/.

See: https://aan.how/.

The use of a "deducible" category is disputable. We acknowledge that letting the language be inferred is different from mentioning it explicitly. Therefore, the #BenderRule can be considered unenforced in such cases, and the impact in terms of visibility of the studied language is not as important as when the #BenderRule is properly

enforced. 10 As counter-examples, it is obvious that the Wall Street Journal (WSJ) is for (

American) English. 11 See: http

://lremap.elra.info/. 12 We used this reference because the idea was to put ourselves in a reader's shoes who doesn't have access to databases such as ELRA's

During manual annotation we even came across an article dealing with 107 languages, listed by their ISO code, not their names, so that the classifiers could not identify them.

See: https://github.com/FannyDucel/ bender-rule-lrec-acl.


Olivier Bernard

Liu-Di Lu

Julien Salomon

Optimal periodic resource allocation in reactive dynamical systems: Application to microalgal production

Keywords: Resource Allocation, Nonlinear Problems, Periodic Control, Dynamical System, Microalgae production, Periodic System, Impulse Control, 1 Switched Systems, Permutation, Linear Approximation, Assignment Problem

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Considering a fixed amount of the resources and a set of activities, we look for a distribution strategy which optimizes a given objective function. This is the so-called resource allocation problem [START_REF] Ibaraki | Resource allocation problems[END_REF]. Due to its simple structure, this problem is encountered in a number of applications including load scheduling [START_REF] Zhang | A model predictive control approach to the periodic implementation of the solutions of the optimal dynamic resource allocation problem[END_REF], manufacturing [START_REF] Zhang | A new manufacturing resource allocation method for supply chain optimization using extended genetic algorithm[END_REF], portfolio selection [START_REF] Hennessy | The use of archimedean copulas to model portfolio allocations[END_REF] and computational biological problem [START_REF] Akhmetzhanov | Optimal life-history strategies in seasonal consumer-resource dynamics[END_REF]. Periodic versions have also been considered. The periodic scheduling problem was first addressed in [START_REF] Liu | Scheduling algorithms for multiprogramming in a hard-real-time environment[END_REF] the framework of operation research. Later on, the concept of proportionate fairness constraint has been introduced [START_REF] Baruah | Proportionate progress: A notion of fairness in resource allocation[END_REF] to design allocation algorithms which schedule the resources in proportion to task weight. Periodic resource allocation problems are also used in ecology, e.g., in [START_REF] Cominetti | Asymptotic convergence of optimal policies for resource management with application to harvesting of multiple species forest[END_REF][START_REF] Piazza | About optimal harvesting policies for a multiple species forest without discounting[END_REF] where the authors investigate long-term behaviour of harvesting policies for a forest composed of multiple species with different maturity ages. In such problems, the state can also be described in terms of dynamical systems. As an example, hospital resources (hospital beds) continuous allocation is studied in [START_REF] Abdelrazec | Modeling the spread and control of dengue with limited public health resources[END_REF] as a strategy to control the dengue fever, associated with a patient recovery rate. In the same way, a population of a single species with logistic growth in a patchy environment is considered in [START_REF] Nagahara | Maximizing the total population with logistic growth in a patchy environment[END_REF]. The problem here consists of the maximization of the total population by re-distributing the limited resources among the patches.

In general, resource allocation problems are related to the assignment of a resource to a sequence of two or more tasks. However, we focus in this paper on problems where N resources are assigned to N tasks. Additionally, we consider permanent regimes which are often relevant in the case of long term processes, as, e.g., crop harvesting, scheduling of appliances, etc. Moreover, here we also account for the dynamical evolution of the system between two re-allocations, further increasing the difficulty of the analysis. In this way, our work is related to the fields of switched systems [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF], impulse control [START_REF] Bensoussan | Contrôle impulsionnel et inéquations quasi variationnelles[END_REF][START_REF] Hespanha | Switching between stabilizing controllers[END_REF] and to periodic control [START_REF] Colonius | Optimal periodic control[END_REF]. These techniques are usually used to tackle stabilization issues. In this paper, we consider them in view of optimization issues.

In order to model the allocation process in the periodic system, we study the following allocation problem : Consider a system with N resources and N activities, each activity uses the allocated resource to evolve during a given time T > 0. At time T , an extra control is applied to re-allocate the resources according to a given permutation. It is proven that if the dynamics of the system is periodic, then it is one period corresponding to one allocation process whatever the order of the considered control strategy is. A nonlinear problem is then introduced in order to find the optimal control strategies. Since N ! permutations need to be tested in the general case, it can be numerically solved only for a limited number of N . To overcome this difficulty, we propose a second optimization problem -a typical assignment problem -associated with a suboptimal solution of the initial problem for which its optimal control can be determined explicitly. In addition, a sufficient condition is provided to characterize cases when the two problems have the same solution.

For the sake of concreteness, we illustrate our theory by an industrial biological application, namely the mixing of microalgae cells in a cultivation setup. This emerging application has a promising potential, ranging from food to renewable energy [START_REF] Wijffels | An outlook on microalgal biofuels[END_REF][START_REF] Hu | Microalgal triacylglycerols as feedstocks for biofuel production: perspectives and advances[END_REF][START_REF] Scott | Biodiesel from algae: challenges and prospects[END_REF] and is also involved in many high added value commercial applications such as pharmaceutical processes, cosmetics or pigments [START_REF] Eppink | From current algae products to future biorefinery practices: a review[END_REF][START_REF] Spolaore | Commercial applications of microalgae[END_REF]. Outdoor algal cultivation is mainly carried out in open raceway ponds exposed to solar radiation. This hydrodynamic system is set in motion by a paddle wheel which homogenizes the medium for ensuring an equidistribution of the nutrients and guarantees that each cell will have regularly access to the light [START_REF] Chiaramonti | Review of energy balance in raceway ponds for microalgae cultivation: Re-thinking a traditional system is possible[END_REF]. Microalgae then grow between two re-distributions depending on the light intensity received in their layer. Different strategies have been proposed to optimize the production of the biomass in this algal raceway system [START_REF] De La Hoz | Optimization of microalgal productivity using an adaptive, non-linear model based strategy[END_REF][START_REF] Cuaresma | Luminostat operation: A tool to maximize microalgae photosynthetic efficiency in photobioreactors during the daily light cycle?[END_REF][START_REF] Cornet | A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors[END_REF][START_REF] Cornet | Calculation of optimal design and ideal productivities of volumetrically lightened photobioreactors using the constructal approach[END_REF][START_REF] Amos | Principles for attaining maximal microalgal productivity in photobioreactors: an overview[END_REF][START_REF] Bernard | Controlling the bottom topography of a microalgal pond to optimize productivity[END_REF]. First studies about the mixing policy have shown that a well-chosen mixing strategy may improve the algal growth [START_REF] Bernard | Shape optimization of a microalgal raceway to enhance productivity[END_REF][START_REF] Bernard | Mixing strategies combined with shape design to enhance productivity of a raceway pond[END_REF]. These works focus on algal production in a non-flat raceway system and assume constant velocity of the fluid and periodicity of the photosynthetic activity. The influence of the mixing strategy on the algal productivity is investigated only numerically by identifying the paddle wheel as a mixing device and modeling it by permutation [START_REF] Bernard | Optimizing microalgal productivity in raceway ponds through a controlled mixing device[END_REF]. Finally an approximation of the functional to optimize which gives rise to an explicit solution, whereas the original problem can only be solved at high computational cost.

In the current study, we extend these preliminary works to a general class of resource allocation problems, identify the periodic solution of the underlying dynamical system as an asymptotic steady state and develop a complete theory of the proposed approximation. In particular, our analysis enables us to establish a criterion to compare the solutions of the original and approximate resource allocation problems. New numerical results complete this study.

The paper is organized as follows. We introduce our periodic resource allocation problem and the related dynamical system in Section 2. More precisely, the optimization problem together with a simplified version based on an approximate functional are introduced in Subsection 2.2. Some technical lemmas are given in Subsection 2.3 and a criterion to guarantee that the original problem and its approximation share the same solution is given in Subsection 2.4. Some implementation remarks conclude this section in Subsection 2.5. Section 3 is devoted to the application to algal production. We present the models associated with the biological and the mixing device in a raceway pond in Subsection 3.1. The considered parameters are given in Subsection 3.2. We illustrate the performance of our control strategies by numerical experiments in Subsection 3.3. Finally, we conclude with some perspectives of our work in Section 4.

Notation.

In what follows, N denotes the set of non-negative integers. The cardinal of a set E is denoted by #E. Given a matrix M , we denote by ker(M ) its kernel, by M the corresponding transposed matrix and by M i,j its coefficient (i, j). In the same way, W n denotes the n-th coefficient of a vector W , whereas W ∞ denotes its infinite norm, i.e. W ∞ := max n |W n |. The scalar product in R N , is denoted by •, • , and we denote by I N the identity matrix of size N .

The set of permutations of N elements, i.e, the set of bijections of {1, • • • , N }, is denoted by S N . The set of permutation matrices of size N × N is denoted by P N . Recall that a permutation matrix is a matrix which has exactly one entry equal to 1 in each row and each column with the other entries being zero. A permutation matrix P is associated to a permutation σ by the formula

P i,j = 1 if i = σ(j) and P i,j = 0 otherwise. As consequence, if W ∈ R N , (P W ) n = W σ -1 (n) for all n ∈ {1, • • • , N }.

Description of the control problem and optimization

Given a period T , an initial time T 0 and a sequence (T k ) k∈N , with T k := kT + T 0 , we consider the following resource allocation process: let (r n ) N n=1 ∈ R N representing a set of N resources which are assumed to be constant over each time interval [T k , T k+1 ) and renewed at each time T k . These resources can be allocated to N activities denoted by (x n ) N n=1 where x n = x n (t) consists of a real-valued function of time. Given a sequence of permutations (π k ) k∈N , with π k ∈ S N , suppose that on the time interval [T k , T k+1 ), the resource r π k (n) is assigned to the activity x n , the latter evolving according to a linear dynamics

ẋn (t) = -a(r π k (n) )x n (t) + b(r π k (n) ), (1)
where a : R → R + and b : R → R + are given.

In this paper, we focus on an allocation strategy of the form π k = σ k , where σ ∈ S N is fixed and σ k denotes the k-times repeated composition of σ with itself. Such an assumption expresses that the same allocation device is used at each period of time. In this setting, the resource assignment process is such that at the end of each time period [T k , T k+1 ), the resource allocated to the activity n is re-allocated to the activity σ(n), or equivalently, that at the end of each time period [T k , T k+1 ), the resource n is re-allocated to the activity σ -1 (n).

Because the resource (r n ) N n=1 are constant with respect to time, the solution of (1) can be computed explicitly. More precisely, denote by x(t) ∈ R N the time dependent vector whose components are given by x n (t), i.e., x(t) := (x 1 (t), • • • , x N (t)) . The process we consider reads

x(t) =∆(t)x(T k ) + τ (t), t ∈ [T k , T k+1 ) (2)
x(T k ) =P x(T - k ), (3)
where ∆(t) is a time dependent diagonal matrix with ∆ nn (t) := e -a(r σ k (n) )(t-T k ) , τ (t) is a time dependent vector with

τ n (t) := b(r σ k (n) ) a(r σ k (n) ) (1 -e -a(r σ k (n) )(t-T k ) ), (4)
and P ∈ P N the permutation matrix associated with σ. In this way, k ∈ N represents the number of re-assignments and T - k represents the moment just before re-assignment.

Remark 1. All the results presented in this paper also hold for non-constant but T -periodic resources (r n (t)) N n=1 ∈ R N . In the case of non-constant resources, the matrix ∆(t) and the vector b(t) cannot be expressed explicitly. Such a technical issue can easily be handled using numerical integration and have no consequences for the ideas involved in analysis developed in this work. Hence, we consider constant resources for the clarity of the presentation.

Periodic control regime assumption

Define D := ∆(T ) and v := τ (T ) and consider as a control the permutation matrix P ∈ P N involved in [START_REF] Amos | Principles for attaining maximal microalgal productivity in photobioreactors: an overview[END_REF]. According to (2-3), we have

x(T k+1 ) = P x(T - k+1 ) = P Dx(T k ) + P v. (5)
In the next sections of this paper, we focus on a T -periodic solution of (2-3). We will motivate this choice by two theorems. These require the following preliminary result. Lemma 1. Given k ∈ N and P ∈ P N , the matrix I N -(P D) k is invertible.

Proof. Assume I N -P D is not invertible, then there exists a non-null vector X ∈ ker(I N -P D), which means X = P DX. Let us denote d n = D nn , n = 1, . . . , N . Denoting by σ the permutation associated with P , we find that (DX) n = d n X n and X n = (P DX) n = d σ(n) X σ(n) . In the same way, we have

X n = (P D) k X n = d σ k (n) . . . d σ(n) X σ k (n) .
Denoting by K the order of σ, we have

X n = (P D) K X n = d σ K (n) . . . d σ(n) X σ K (n) = d σ K (n) . . . d σ(n) X n . Since, 0 < d n < 1 for n = 1, . . . , N , then 0 < d σ K (n) . . . d σ(n) < 1.
This implies that X n = 0, which contradicts our assumption. Therefore, I N -P D is invertible. That I N -(P D) k is invertible for all k > 0 can be proved in much the same way.

We can now state a convergence result about (x(T k )) k∈N .

Theorem 1. There exists a unique T -periodic solution x per (t) of (2-3), satisfying

x per (T k ) = (I N -P D) -1 P v.

Moreover, for any arbitrary initial condition x(T 0 ), we have lim k→+∞ x(T k ) =

x per (T k ).

Proof. The existence of a constant sequence (x per (T k )) k∈N satisfying (5) follows from Lemma 1, applied with k = 1. Solving (5) in this setting gives [START_REF] Bernard | Shape optimization of a microalgal raceway to enhance productivity[END_REF]. Let us then define the sequence (e k ) k∈N by e k := x(T k ) -(I N -P D) -1 P v. Since

e k+1 = (P D)e k , (7)
we find that

e k+1 ∞ = P De k ∞ = De k ∞ ≤ d max e k ∞ ,
where

d max := max n=1,...,N (d n ) < 1. The result follows.
This theorem shows that after a transient response, the system x(t) can be correctly approximated by x per (t). This steady state can be obtained in another way.

Theorem 2. We keep the notation of the previous lemma. Given k 0 > 0, assume that the state x is k 0 T -periodic in the sense that after k 0 times of re-assignment, the state of each activity returns to its initial state x n (T k0 ) = x n (T 0 ). Then x = x per .

Proof. Since x is assumed to be k 0 T -periodic, we have e 0 = e k0 = (P D) k0 e 0 . According to Lemma 1, I N -(P D) k0 is invertible, meaning that e 0 = 0. Combining this with (7), we get that e k = 0, for k ∈ N. The result follows.

A natural choice for k 0 would be the order K of the permutation associated with P , which is the smallest integer greater than one such that P K = I N . Indeed, in this case K is the minimal number of re-assignments required to recover the initial order of the components of x. The previous result shows that every KT -periodic evolution will actually be T -periodic. In the next section, we show that this property is decisive to formulate an optimization problem. In addition, the computations to solve the optimization problem will be reduced, since the CPU time required to assess the quality of a permutation will not depend on its order.

Objective function

We still consider an arbitrary control P ∈ P N and the vector of activities x(t) defined by [START_REF] Akhmetzhanov | Optimal life-history strategies in seasonal consumer-resource dynamics[END_REF][START_REF] Amos | Principles for attaining maximal microalgal productivity in photobioreactors: an overview[END_REF]. Assume that the mean benefit of the process on the time period [T k , T k+1 ), i.e., after k times of re-assignment, reads

f k := w, 1 T T k+1 T k x(t)dt , (8)
where w ∈ R N is a weighting vector expressing the relative importance of each activity.

Then the average benefit after K re-assignment operations is given by

J av := 1 K K-1 k=0 f k . ( 9
)
Such a formalization has been used by Cominetti et al. in the context of forest maintenance and exploitation [START_REF] Cominetti | Asymptotic convergence of optimal policies for resource management with application to harvesting of multiple species forest[END_REF]. In this work, an infinite sum is considered to study the total benefit of all the re-assignment operations. Replace now x(t) in the benefit [START_REF] Bernard | Mixing strategies combined with shape design to enhance productivity of a raceway pond[END_REF], by its expression (2-3). We get

f k = 1 T w, Dx(T k ) + ṽ = 1 T Dw, x(T k ) + w, ṽ ,
where Dnn =

T k+1 T k ∆ nn (t)dt and ṽn = T k+1 T k τ n (t)dt.
The only term which depends on the re-assignment process is x(T k ).

From now on, we focus of on the steady state introduced in Theorem 1, meaning that we assume that x = x per . Because of ( 6), one finds

Dw, x per (T k ) = Dw, (I N -P D) -1 P v ,
meaning that the benefit is the same for each re-assignment process. As a consequence, f k does not depend on k and that the average benefit J av (see ( 9)) satisfies J av (P ) = 1 T (J(P ) + w, ṽ ) ,

where

J(P ) := u, (I N -P D) -1 P v , (10)
with u = Dw. It follows that maximizing J av with respect to P is equivalent to maximizing J with respect to P . Since #P N = #S N = N !, an exhaustive test of all the possible controls is out of range for large value of N . Hence, the maximization of J cannot be tackled in realistic cases where a good numerical accuracy is required. To overcome this difficulty, we propose in this section an approximation of this problem whose optimum can be determined explicitly, with a negligible computational cost. For this purpose, we expand the functional (10) as follows u, (I N -P D)

-1 P v = +∞ l=0 u, (P D) l P v = u, P v + +∞ l=1 u, (P D) l P v ,
and consider as an approximation the first term of this series, namely

J approx (P ) := u, P v . ( 11
)
Without loss of generality (see Appendix B for the details), we assume that the entries of u are sorted in ascending order, meaning that u 1 ≤ . . . ≤ u N . Note that optimizing J approx amounts to solving an assignment problem [START_REF] Burkard | Assignment Problems[END_REF]. Indeed, we have for example

min P ∈P N J approx (P ) = min σ∈S N N n=1 u n v σ(n) .
The latter expression reads as an assignment problem associated with the cost matrix [11, p.5] [u i v j ] (i,j=1,...,N ) . To make our exposition self-contained, we give the solution of this problem in Section 2.4.

Remark 2. A fairly common approach to deal with permutation matrices in discrete or combinatorial optimization is to relax the problem by extending the optimization to the set of bistochastic matrices. As an example, this technique corresponds to the Kantorovitch relaxation considered in optimal transport [START_REF] Kantorovich | On the translocation of masses (in russian)[END_REF] (see also [START_REF] Bertsimas | Introduction to Linear Optimization[END_REF] for a more general presentation of the linear case, and [START_REF] Lieb | Variational principle for many-fermion systems[END_REF] for a similar strategy in the context of quantum chemistry). This approach allows the optimization to be performed by gradient-type methods. At the theoretical level, the goal is then to prove that the convergence takes place towards extremal points, i.e. permutation matrices. We have tested this approach on the nonlinear problem [START_REF] Bertsimas | Introduction to Linear Optimization[END_REF]. Our experiments indicate that the obtained limits are neither always permutation matrices nor optimal, which leads us to conjecture the existence of local (non-global) maxima for this extended form of J.

Some technical lemmas

Let us state some preliminary properties about the permutation set S N that we will use in the next section. Given k ∈ N, and two arbitrary permutations σ, σ ∈ S N , let us define

E k (σ, σ) := n = 1, . . . , N | σ k (n) = σ k (n) , G k (σ, σ) :={n = 1, . . . , N | ∀k ≤ k, σ k (n) = σ k (n)},
and m k := #E k (σ, σ). We have the following result.

Lemma 2. For k ∈ N, we have m k ≤ km 1 and #G k (σ, σ) ≥ max(N -km 1 , 0).
Proof. To shorten notation, we write in this proof

E k instead of E k (σ, σ), E k+1 instead of E k+1 (σ, σ), G k instead of G k (σ, σ), etc.
From the definition of E k , we have:

E k+1 = (({1, . . . , N } \ E 1 ) ∩ E k+1 ) ∪ (E 1 ∩ E k+1 ).
The first subset in the right-hand side satisfies

σ (({1, . . . , N } \ E 1 ) ∩ E k+1 ) = σ (({1, . . . , N } \ E 1 ) ∩ E k+1 ) ⊂ E k , so that # (({1, . . . , N } \ E 1 ) ∩ E k+1 ) ≤ #E k =: m k . On the other hand, (E 1 ∩ E k+1 ) ⊂ E 1 , hence #(E 1 ∩ E k+1 ) ≤ m 1 . As a consequence, m k+1 ≤ m k + m 1 . This implies m k ≤ km 1 .
As for G k , we have:

G k = (G k+1 ∩ G k ) ∪ (σ -k (E 1 ) ∩ G k ). ( 12
) Indeed, let n ∈ G k , i.e, σ k (n) = σ k (n). If σ k+1 (n) = σ k+1 (n), then n ∈ G k+1 . Otherwise, σ k+1 (n) = σ k+1 (n), meaning that σ k+1 (n) = σ(σ k (n)) which implies σ k (n) = σ k (n) ∈ E 1 , so that n ∈ σ -k (E 1 )
. This proves [START_REF] Chiaramonti | Review of energy balance in raceway ponds for microalgae cultivation: Re-thinking a traditional system is possible[END_REF], and we get as a by-product

(G k+1 ∩ G k ) ∩ (σ -k (E 1 ) ∩ G k ) = ∅. Moreover, since G k+1 ⊂ G k , we get G k+1 ∩ G k = G k+1 . It follows that #G k = #G k+1 + #{σ -k (E 1 ) ∩ G k }. Since #{σ -k (E 1 ) ∩ G k } ≤ #E 1 = m 1 , we obtain #G k+1 ≥ #G k -m 1 . The result follows.
In what follows, a transposition in S N between two elements i = j is denoted by (i j). By abuse of notation, (n n) denotes the identity for all n = 1, . . . , N . Given a permutation σ ∈ S N , we consider the sequence of permutations (σ n ) n=0,...,N defined by

σ 0 = σ σ n = (n σ n-1 (n)) • σ n-1 . ( 13
)
For all n ≤ N , it immediately follows from this definition that

σ n | {1,...,n} = Id| {1,...,n} and σ N -1 = σ N = Id,
where Id denote the identity permutation. Let us give two additional properties of this sequence.

Lemma 3. Let σ ∈ S N and (σ n ) n=1,...,N -1 defined by [START_REF] Colonius | Optimal periodic control[END_REF]. One has:

{i = 1, . . . , N | σ(i) = i} = {i = 1, . . . , N | ∀n = 1, . . . , N -1, σ n (i) = i} .
Proof. Given i with 1 ≤ i ≤ N , such that σ(i) = i, let us prove that σ n (i) = i by induction on n. Since σ 0 = σ, the result holds for n = 0. Suppose it holds at a rank n -1, meaning that σ n-1 (i) = i. By definition of (σ n ) n=1,...,N , one has:

σ n (i) = (n σ n-1 (n)) • σ n-1 (i) = (n σ n-1 (n))(i). If i = n, then (n σ n-1 (n))(i) = σ n-1 (n) = σ n-1 (i) = i. If i = σ n-1 (n), then i = σ n-1 (i) = σ n-1 (
n) and i = n, so that we conclude as in the previous case.

In the other cases, σ n (i) = σ n-1 (i) = i. The result follows.

Lemma 4. Let i, j ∈ {1, . . . , N }, with i < j. Let σ ∈ S N , with σ = (i j) • σ , where (i j) and σ ∈ S N have disjoint supports, i.e., σ (i) = i and σ (j) = j.

The sequence defined by (13) satisfies σ j = σ j-1 .

Proof. From ( 13), one has

σ j = (j σ j-1 (j)) • σ j-1 .
We need to prove that σ j-1 (j) = j. Since σ and (i j) are disjoint, then for n < i, σ n = (i j) • σ n , where σ n is defined by [START_REF] Colonius | Optimal periodic control[END_REF], with the initial term σ 0 = σ . In particular, σ n (i) = j for n < i.

In the case n = i, one has

σ i = (i σ i-1 (i)) • σ i-1 = (i j) • σ i-1 = (i j) • (i j) • σ i-1 = σ i-1 .
In particular, σ i (j) = j.

Finally, since σ i-1 (i) = i, we find that σ i = σ i-1 , and it follows by induction that for n > i, σ n = σ n , which means σ n (j) = j. In particular σ j-1 (j) = j. This concludes the proof.

The sequence (σ n ) n=0,...,N can be used to decompose J(I N ) -J(P ) for an arbitrary P ∈ P N , as stated in the next Lemma.

Lemma 5. Let σ ∈ S N and P ∈ P N the associated permutation matrix, we have:

u, (I N -P )v = N -1 n=1 (u n -u σ -1 n-1 (n) )(v n -v σn-1(n) ).
Proof. Given j ∈ {0, . . . , N }, define S j = N n=1 u n v σj (n) . Since σ j (n) and σ j-1 (n) might only differ for n = j and n = σ -1 j-1 (j), we have

S j -S j-1 = N n=j u n (v σj (n) -v σj-1(n) ) = u j (v σj (j) -v σj-1(j) ) + u σ -1 j-1 (j) (v σj (σ -1 j-1 (j)) -v σj-1(σ -1 j-1 (j)) ) = u j (v j -v σj-1(j) ) + u σ -1 j-1 (j) (v σj-1(j) -v j ) = (u j -u σ -1 j-1 (j) )(v j -v σj-1(j)
). The result then follows from u, (I N -P )v = S N -1 -S 0 .

Solutions of the optimization problems

The previous lemma enables us to solve the problems max P ∈P N J approx (P ) and min P ∈P N J approx (P ). Recall that the entries of u are sorted in ascending order. [START_REF] Abdelrazec | Modeling the spread and control of dengue with limited public health resources[END_REF] and P + , P -∈ P N , the corresponding permutation matrices. Then P + = argmax P ∈P N J approx (P ), P -= argmin P ∈P N J approx (P ).

Lemma 6. Let σ + , σ -∈ S N such that v σ+(1) ≤ v σ+(2) • • • ≤ v σ+(N ) and v σ-(N ) ≤ v σ-(N -1) ≤ • • • ≤ v σ-
Proof. Let P ∈ P N and σ ∈ S N the associated permutation, we have u, (P

+ -P )v = u, (I N -P P -1 + )w = N -1 n=1 (u n -u (σ n-1 ) -1 (n) )(w n -w σ n-1 (n) ), (14)
where w = (w n ) N n=1 := (v σ+(n) ) N n=1 and σ n is the sequence defined by ( 13) with σ := σ -1 + • σ the permutation associated with P P -1 + . Since (w n ) N n=1 by its definition is an increasing sequence, σ n-1 (n) ≥ n and (σ n-1 ) -1 (n) ≥ n, we find that u, (P + -P )v ≥ 0. The proof for the problem min P ∈P N u, P v is similar.

We immediately deduce from this lemma that once u and v are given, the matrix P + , P -of Lemma 6 can be determined explicitly. More precisely, P + is the matrix corresponding to the permutation which associates the largest coefficient of u with the largest coefficient of v, the second-largest coefficient with the second-largest, and so on. In the same way, P -is the matrix corresponding to the permutation which associates the largest coefficient of u with the smallest coefficient of v, the second-largest coefficient with the second-smallest, and so on.

Remark 3. The optimal matrices P + and P -are not unique as soon as either u or v contains at least two identical entries.

We focus now on the case where u as well as v have entries with a constant sign. Since the results in this section hold both for minimization and maximization problems, we can assume without loss of generality that u, v are both positive. Using the properties given in the previous section, we will show that in some cases, the problem max P ∈P N J(P ) (resp. min P ∈P N J(P )) and max P ∈P N J approx (P ) (resp. min P ∈P N J approx (P )) have the same solution.

We keep the notation of Lemma 6. Define for n = 1, . . . , N ,

p n := min i,j=1,...,N,i =n,j =n |(u n -u i )(v σ+(n) -v σ+(j) )|. (15)
Denote by i n and j n the solutions of the previous problem. Since u n , v σ+(n) are sorted in ascending order, we find immediately that if n = 1 (resp. N ), then i n = j n = 2 (resp. i n = j n = N -1). Otherwise, i n = n -1 or i n = n + 1, and the same result holds for j n . Sort ( p n ) N n=1 and denote by (p n ) N n=1 the resulting sequence, i.e., p 1 ≤, . . . , ≤ p N . Define then for m = 1, . . . , N

s m := m n=1 p n , (16)
and

F - m := min(m,N ) n=1 u n v σ-(N -m+n) , F + m := N n=max(1,N -m+1) u n v σ+(n) . (17)
From the definition of these sequences, we have F + m ≥ F - m . See Appendix C for the case where u or v negative. We are now in a position to give the main result of this section. Theorem 3. Assume that u and v have positive entries and define

φ(m 1 ) := 1 s m 1 2 +∞ l=1 d l max F + (l+1)m1 -d l min F - (l+1)m1 , (18)
where m 1 refers to the notation in Lemma 2, d max := max n=1,...,N (d n ) and d min := min n=1,...,N (d n ). Assume that:

max m1≥2 φ(m 1 ) ≤ 1. ( 19
)
Then the problem max P ∈P N u, (I N -P D) -1 P v (resp. min P ∈P N u, (I N -P D) -1 P v ) and the problem max P ∈P N u, P v (resp. min P ∈P N u, P v ) have the same solution.

Proof. We keep the notation in Section 2.3 and give the proof in the case of the maximization problem. The case of the minimization problem can be handled in the very same way. Let P ∈ P N and σ ∈ S N the associated permutation, we have u, (I N -P + D)

-1 P + v -u, (I N -P D) -1 P v (20)
From the definition E k (σ + , σ) and G k (σ + , σ), we have

E 1 (σ + , σ) G 1 (σ + , σ) = {1, . . . , N }. Let us denote by (w n ) N n=1 = (v σ+(n) ) N n=1
and by σ n the sequence defined by [START_REF] Colonius | Optimal periodic control[END_REF]

with σ 0 := σ -1 + •σ. From the definition of E 1 (σ + , σ) and G 1 (σ + , σ), we have σ(G 1 (σ + , σ)) = σ + (G 1 (σ + , σ)) and σ(E 1 (σ + , σ)) = σ + (E 1 (σ + , σ)), which implies σ 0 (E 1 (σ + , σ)) = E 1 (σ + , σ), and for any i ∈ G 1 (σ + , σ), σ 0 (i) = i.
Using these properties and ( 14), we have u, (P

+ -P )v = N -1 n=1 (u n -u (σ n-1 ) -1 (n) )(w n -w σ n-1 (n) ) = n∈E1(σ+,σ) (u n -u (σ n-1 ) -1 (n) )(w n -w σ n-1 (n) ) + n∈G1(σ+,σ) (u n -u (σ n-1 ) -1 (n) )(w n -w σ n-1 (n) ) = n∈E1(σ+,σ) (u n -u (σ n-1 ) -1 (n) )(w n -w σ n-1 (n) ). (22)
In the case where there exists a transposition

(i i ) with i < i in σ , Lemma 4 implies that u (σ i -1 ) -1 (i ) = u i and w σ i -1 (i ) = w i . The maximum number of transpositions in σ 0 is m1 2 if m 1 is even, m1-3 2
otherwise. Hence, the smallest number of non-zero terms present in the last sum of ( 22) is given by m

1 -m1 2 = m1 2 if m 1 is even, m1-1 2
otherwise. In other words, there exists at least m1 2 non-zero terms in the last sum of [START_REF] Han | A mechanistic model of algal photoinhibition induced by photodamage to photosystem-ii[END_REF], which implies u, (P

+ -P )v = n∈E1(σ+,σ) (u n -u (σ n-1 ) -1 (n) )(w n -w σ n-1 (n) ) ≥ s m 1 2 . (23)
For n ∈ {1, . . . , N } and l ∈ N * , let us denote by d σ,l,n :=

d σ l (n) d σ l-1 (n) • • • d σ(n) .
We have

+∞ l=1 d l max F + (l+1)m1 -d l min F - (l+1)m1 = l * l=1 d l max F + (l+1)m1 -d l min F - (l+1)m1 + +∞ l=l * +1 d l max F + (l+1)m1 -d l min F - (l+1)m1 = l * l=1 d l max F + (l+1)m1 -d l min F - (l+1)m1 + d l * +1 max 1 -d max F + N - d l * +1 min 1 -d min F - N .
It follows that the infinite sum involved in φ(m 1 ) actually reduces to a finite sum that can be computed numerically without any approximation. As for the evaluation of s m 1 2

, only N 2 terms need to be computed. Examples of behaviour of s m and F + m , F - m are presented in Figure 3, whereas examples of behaviour of the function [START_REF] De La Hoz | Optimization of microalgal productivity using an adaptive, non-linear model based strategy[END_REF] with respect to m 1 are shown in Figure 4.

Application to algal production

Algal raceway reactors are currently the most extended technology for microalgae growth, more than 90% of the worldwide microalgae production is performed by this technology. In this system, the algae are exposed to solar radiation and advected in a laminar regime. This regime holds as long as they remain far enough from the mixing device, that usually consists of a paddle wheel. Meanwhile, they evolve at a constant depth and one can consider that their vertical positions only change after passing through this device [START_REF] Demory | How do microalgae perceive light in a high-rate pond? towards more realistic lagrangian experiments[END_REF]. Two main phenomena have to be taken into account to study algal production. First, the photosynthetic activity of the algae close to surface may suffer from photoinhibition by which an excess of light decreases the speed of photosynthesis. Second, the algae at the bottom of the raceway may not receive any light since this quantity exponentially decreases with respect to depth. In this framework, it has been shown that a well-chosen mixing device can increase significantly the algal growth by balancing the access to the light resource [START_REF] Bernard | Optimizing microalgal productivity in raceway ponds through a controlled mixing device[END_REF][START_REF] Bernard | Mixing strategies combined with shape design to enhance productivity of a raceway pond[END_REF]. In this section, we analyse this result in the framework of the resource allocation process and apply the theory developed in the previous section to an algal production case. Finally, we provide some numerical results to evaluate the efficiency of the mixing strategies and their approximation.

Biological dynamics and raceway mixing modeling

The growth of algae results from the activity of photosynthetic cells generated by solar radiation. This complex interaction is accurately described by the Han model [START_REF] Han | A mechanistic model of algal photoinhibition induced by photodamage to photosystem-ii[END_REF], which takes into account the above-mentioned phenomenon of photoinhibition. In this model, each light harvesting unit is assumed to have three different states: open and ready to harvest a photon (A), closed while processing the absorbed photon energy (B), or inhibited if several photons have been absorbed simultaneously leading to an excess of energy (C). Their dynamics is described by the following system

         Ȧ = -σ H IA + 1 τ H B, Ḃ = σ H IA - 1 τ H B + k r C -k d σ H IB, Ċ = -k r C + k d σ H IB.
Here A, B and C are the relative frequencies of the three possible states with A+B +C = 1, and I is a continuous time-varying signal representing the photon flux density. The coefficient σ H stands for the specific photon absorption, τ H is the turnover rate, k r represents the photosystem repair rate and k d is the damage rate.

Since the sum of these three states is equal to one, the latter system can be reduced to two equations by eliminating B as following:

Ȧ Ċ = • (M H A C + b H ), = 1 0 0 k d , M H = -(σ H I + 1 τ H ) -1 τ H -σ H I -( kr k d + σ H I) , b H = 1 τ H σ H I .
The dynamics of the open state A reaches its steady state following a process whose speed is much higher than the dynamics of the photoinhibition state C. This phenomenon is mainly due to the presence of the multiplicative parameter k d which is on the order of 10 -4 whereas the absolute value of the entries of M H and b H are on the order of 0.1 -6 (see Table 1, where an example of typical values for the Han parameters is given). We can then apply a slow-fast approximation using singular perturbation theory [START_REF] Lamare | Gradient-based optimization of a rotating algal biofilm process[END_REF]. This approach consists in replacing A by its pseudo steady state A steady := 1-C τ H σ H I+1 into the reduced evolution equation of C. The Han dynamics can then be reduced to a single evolution equation on C:

Ċ = -α(I)C + β(I), (25)
where

α(I) := k d τ H (σ H I) 2 τ H σ H I + 1 + k r , β(I) := k d τ H (σ H I) 2 τ H σ H I + 1 .
The specific growth rate is proportional to σ H IA, replacing A by its pseudo steady state A steady , the net specific growth rate is defined by

µ(C, I) := -γ(I)C + ζ(I), (26)
where

γ(I) := k H σ H I τ H σ H I + 1 , ζ(I) := k H σ H I τ H σ H I + 1 -R.
Here, R denotes the respiration rate and k H is a factor which relates the photosynthetic activity to the growth rate. We assume that the system is perfectly mixed so that the biomass concentration is homogeneous. Meanwhile, we also assume that the photosynthetic units grow slowly such that the variations of biomass concentration and background turbidity are negligible over one lap of the raceway. At this timescale, the turbidity and biomass concentration can be supposed to be constant. In this framework, the light intensity reads as a function of the depth z and can be modeled by the Beer-Lambert, i.e.,

I(z) = I s exp(εz), (27)
where I s is the light intensity at the free surface and ε is the light extinction coefficient. The average net specific growth rate over the domain is then defined by

μ := 1 T T 0 1 h 0 -h µ C(t, z), I(z) dzdt, (28)
where h is the depth of the raceway pond and T is the average duration of one lap of the raceway pond.

Let us now see how this model can be included in the framework of Section 2. In order to compute numerically [START_REF] Lamare | Gradient-based optimization of a rotating algal biofilm process[END_REF], we introduce a vertical discretization of the fluid, consisting of N layers uniformly distributed on a vertical grid. The depth of the layer n is given by

z n := - n -1 2 N h, n = 1, . . . , N. (29)
For a given initial photoinhibition state C n (0) associated with a photosystem located in layer n, let C n (t) be the solution of (25) at time t. In this semidiscrete setting, the average net specific growth rate in the raceway pond can be defined by

μN := 1 T T 0 1 N N n=1 µ(C n (t), I n )dt, (30)
where I n is the light intensity received in the layer n. The solution of ( 25) can be computed explicitly to get a formula that takes the form of (2). Denoting by C(t) the time dependent vector whose components are given by C n (t), it follows that μN satisfies

μN = 1 N T Γ, C(0) + 1, Z , (31)
where 1 is a vector of size N whose coefficients are equal to 1, and Γ, Z are two vectors such that

Γ n := γ(I n ) α(I n ) (e -α(In)T -1), Z n := γ(I n )β(I n ) α(I n ) 2 (1 -e -α(In)T ) - γ(I n )β(I n ) α(I n ) T + ζ(I n )T.
The detail of the computations giving rise to [START_REF] Liu | Scheduling algorithms for multiprogramming in a hard-real-time environment[END_REF] is presented in Appendix A.

Assume now that at each lap, the layers are permuted according to σ ∈ S N , meaning that the algae at the layer n 1 are supposed to be entirely transferred to the layer n 2 = σ(n 1 ) when passing through the mixing device, This mixing process is depicted schematically on an example in Figure 1.

T k T - k+1 T k+1 T - k+2

Layer four

Layer three

Layer two

Layer one The interest of such a device is to mix the algae to better balance their exposure to light and increase the production. In this way, though the light resource is considered to be constant, this mixing model and the constraints on the light level received at each layer make the process equivalent to a resource allocation problem.

σ z 0 -h z 1 = z σ(4) z 2 = z σ(1)
z 3 = z σ(2) z 4 = z σ(3)
At this step, we see that up to a change of notation, this model can be interpreted in terms of the resource allocation process described in Section 2. More precisely, the activity x n (t) corresponds to the photoinhibition state C n (t), the resource r n correspond to the light intensity I n and the functions a, b correspond to α, β.

Considering now the periodic regime, we get thanks to Theorem 1 and (6) that the steady state of this system satisfies C(0) = (I N -P D) -1 P V , where D and V are a diagonal matrix and a vector of size N × N and N respectively, whose components are given by

D nn := e -α(In)T , V n := β(I n ) α(I n ) (1 -e -α(In)T ). ( 32
)
Our goal is to find the control, i.e. a permutation σ, which maximizes the average growth rate μN . Since only C(0) in [START_REF] Liu | Scheduling algorithms for multiprogramming in a hard-real-time environment[END_REF] depends on the control P , we find that the objective function μN = Γ, (I N -P D) -1 P V has the form of J(P ) in [START_REF] Bertsimas | Introduction to Linear Optimization[END_REF] with u = Γ, v = V and D defined in [START_REF] Nagahara | Maximizing the total population with logistic growth in a patchy environment[END_REF].

Parameter settings

Consider a raceway whose water elevation h = 0.4 m, which corresponds to typical raceway pond setting. All the numerical parameters values considered in this section for Han's model are taken from [START_REF] Grenier | Worldwide perspectives of rotating algal biofilm up-scaling[END_REF] and recalled in Table 1. Recall that I s is the light intensity at the free surface. In order to fix the value of the light extinction coefficient ε in [START_REF] Kantorovich | On the translocation of masses (in russian)[END_REF], we assume that only a fraction q of I s reaches the bottom of the raceway pond, meaning that I b = qI s , where q ∈ [0, 1] and I b is the light intensity at the bottom. It follows that ε can be computed by ε = (1/h) ln(1/q).

In practice, this quantity can be implemented in the experiments by adapting the biomass harvesting frequency, or the dilution rate for continuous cultivation.

In what follows, the varying parameters are I s , the ratio q and T . We consider I s ∈ [0, 2500]µmol m -2 s -1 , q ∈ [0.1%, 10%] and T ∈ [1, 1000]s. The number of layers N remains small as we need to test numerically N ! permutation matrices for each triplet (I s , q, T ).

Numerical tests

As shown in [9, Section IV.B], Problem (10) admits non-trivial optimal permutation strategies which may significantly change according to the parameter settings. In this section, we study and compare the true and the approximated solutions as well as their efficiency with respect to the average net specific [START_REF] Lamare | Gradient-based optimization of a rotating algal biofilm process[END_REF]. We start by investigating some properties of the items defined in the previous sections. Recall that the two sequences u, v used in Section 2.2 correspond in our application to Γ, V respectively. We consider N = 20 layers and two parameters triplets, namely (I s , q, T ) = (2000, 5%, 1000) and (800,0.5%,1). Figure 2 shows the evolution of these two quantities as a function of I. Note that in both cases, V is positive with sorted entries, as it can be seen in ( 4). On the contrary, the discretized Γ is negative and not necessarily sorted. We refer to Appendix A for more details about V and Γ.

We then study the behaviour of the sequences F + m , F - m , s m and φ(m 1 ) defined in Section 2.4 for the same two parameters triplets. Note that since Γ is negative, F - m and F + m are defined as in Appendix C (and not as in ( 17)). We choose N = 7 and N = 20 to check the performance for two different discretisation numbers of layers. One can see in Figure 4 that the maximal value of φ(m 1 ) is always obtained for m 1 = 2, and that the maximal value φ(m 1 ) appears to be an increasing function of N . This makes the criterion given in Section 2.4 less efficient for many layers N . Further analysis is required to obtain a criterion that does not depend on N .

The next test is devoted to the convergence of the average growth rate μN with respect to the number of layers N . We keep the two triplets of parameters of the previous test. Due to the limit of the computer memory, the computation of μN (P max ) is tractable for small values of N , in our case lower than or equal to N = 11. Such an issue does not occur in the case of μN (P + ). Figure 5 presents the behaviour of μN . For the parameter triplet (2000, 5%,1000), the criterion is satisfied until N = 8 (green circle), which is confirmed in Figure 4 (Left) for N = 7 where the maximal value of φ(m 1 ) is already close to 1. Though the criterion is not satisfied for N > 8, we observe that P + = P max from N = 2 to N = 11. As for the triplet (800,0.5%,1), one can see that P + = P max until N = 3. Figure 6 shows the optimal control strategies for these two different parameter triplets in the case N = 11 and N = 100. It can be observed that for the parameter triplet (2000, 5%, 1000), the two controls P + , P max have the same form for N = 11 and N = 100 (Figure 6 Top). Hence, one can expect P max = P + for larger N which is the case until N = 11 (as shown in Figure 5 black square). However, this may not be the case for (800,0.5%,1) since P + , P max have already different forms for N = 11 (Figure 6 Bottom).

In the following tests, we focus only on two special cases: large lap duration time (T = 1000 s) and small lap duration time (T = 1 s). In practice, the former corresponds to typical time required to complete one lap in a raceway pond system, whereas the latter rather corresponds to photobioreactor [START_REF] Lamare | Gradient-based optimization of a rotating algal biofilm process[END_REF]. In the small lap duration time case, we observe the so-called flashing effect. This phenomenon corresponds to the fact that the growth rate is an increasing function of the light exposition frequency. It can be observed in Figure 7, where μN (P max ) decreases with respect to T for all considered light intensities. This phenomenon has already been reported in literature, see, e.g. [START_REF] Lamare | Gradient-based optimization of a rotating algal biofilm process[END_REF].

The next test is dedicated to the efficiency of the criterion [START_REF] Demory | How do microalgae perceive light in a high-rate pond? towards more realistic lagrangian experiments[END_REF]. More precisely, we evaluate the function μN defined by [START_REF] Lieb | Variational principle for many-fermion systems[END_REF] for the optimal control P max which solves Problem [START_REF] Bertsimas | Introduction to Linear Optimization[END_REF] and for the control P + which solves the approximated Problem [START_REF] Burkard | Assignment Problems[END_REF]. We consider two different discretisation values N = 5 and N = 9. Figure 8 shows the results for T = 1 s and T = 1000 s. We see that for large values of T , the optimum approximation almost always coincides to the true optimum. Nevertheless, we observe that the criterion (19) becomes less efficient for larger N . Note that the case corresponding to I s = 0 µmol m -2 s -1 is particular since no light is available in the system, implying that Γ, V equal to zero. In this case the value of the objective function do not depend on the control P . Hence μN (P max ) = μN (P + ) when I s = 0 µmol m -2 s -1 . the approximation becomes less efficient in the case of short lap duration time. This can be observed in Figure 9 (Left) and in Figure 10 (Left). However, the maximal values of r 1 , r 2 are still preserved by their approximations r1 , r2 .

Conclusion

We have studied a periodic resource allocation problem combined with a dynamical system. The periodicity of the problem enables us to reduce the computation to one assignment process. A significant computational effort is still required when dealing with larger number of N . We overcome this difficulty by defining a second optimization problem which has an explicit solution that coincide with the true solution when a given criterion is satisfied.

This developed theory is then applied to a microalgal production system with a mixing device. Non-trivial optimal mixing strategies can be obtained and the proposed second optimization problem provides a reliable approximation for large time duration T . Besides, our experimental results show the significance of the choice of the mixing strategy: the relative ratio between the best and the worst case reaches 30% in some cases. We also observe a flashing effect meaning that better results are obtained when T goes to zero.

Further works will be devoted to the improvement of the function φ used in Theorem 3 in order to improve our approach for large number of N . An approximation problem for small lap duration T can also be considered with an appropriate criterion to evaluate this approximation.

A Explicit Computations

In this appendix, we provide the computational details to solve [START_REF] Hu | Microalgal triacylglycerols as feedstocks for biofuel production: perspectives and advances[END_REF] and [START_REF] Lieb | Variational principle for many-fermion systems[END_REF] for an arbitrary number n ∈ {1, . . . , N }. Given two points t 1 , t 2 ∈ [0, T ]. Since I n is constant, Equation ( 1) can be integrated and becomes C n (t 2 ) = e α(In)(t1-t2) C n (t 1 ) + β(I n ) α(I n ) (1 -e α(In)(t1-t2) ).

The time integral in [START_REF] Lieb | Variational principle for many-fermion systems[END_REF] can be computed by Remark that Γ and V always have the opposite sign. Note also that I → β(I) α(I)

is increasing on [0, +∞), which is not the case for I → γ(I) α(I) . It follows that V increases on R + and Γ is not monotonic on R + (see Figure 2).

B Optimization problem with arbitrary vectors

Let ũ, v ∈ R N two arbitrary vectors. Let Q ∈ P N such that u := Qũ has entries sorted in ascending order. Since Q is a permutation matrix, we have Q T = Q -1 . For any P ∈ P N , let us denote by P := Q -1 P Q, we have P ∈ P N a permutation matrix. Let us denote by ṽ := Q -1 v and by D = Q -1 DQ. Note that D is still

  P + D) l P + -(P D) l P v = u, (P + -P )v + +∞ l=1 u, (P + D) l P + -(P D) l P v .

Figure 1 :

 1 Figure 1: Schematic representation of the mixing process over two laps. Here, the vertical discretization number N = 4 and the mixing device corresponds to the cyclic permutation σ = (1 2 3 4).

Figure 2 :

 2 Figure 2: Γ and V with respect to the light intensity I (Blue curve). Discretisation points (Red point) chosen for (I s , q, T ) = (2000, 5%, 1000) (Left) and (800,0.5%,1) (Right).

Figure 3 :

 3 Figure 3: Example of sequences F + m , F - m (Top) and s m (Bottom) with respect to m for the two parameters triplets. Left: N = 7. Right: N = 20.

Figure 4 :

 4 Figure 4: Example of behaviour of φ(m 1 ) with respect to m 1 for two parameters triplets and two different N . Left: N = 7. Right: N = 20.

Figure 5 :

 5 Figure5: Average growth rate μN obtained with P max and P + as a function of N for the two parameters triplets. The green circles mark the case when the criterion is satisfied. The black squares mark the case when P max = P + is observed.

Figure 6 :

 6 Figure 6: Pattern of the optimal matrix P max for Problem (10) and N = 11 (Left) and P + for Problem (11) and N = 100 (Right) for the two parameters triplets. The blue points represent non-zero entries, i.e., entries equal to 1.

Figure 7 :

 7 Figure 7: Average specific growth rate in the case q = 0.1% and N = 7 for four different light intensities I s .

Figure 8 :

 8 Figure 8: Average net specific growth rate μN for T = 1 s (Top) and for T = 1000 s (Bottom). Left: N = 5. Right: N = 9. The red surface is obtained with the control P max and the blue surface is obtained with the control P + . The purple stars represent the cases where P max = P + or, in case of multiple solution, μN (P max ) = μN (P + ). The green circle represent the cases where the criterion (19) is satisfied.

Figure 10 :

 10 Figure 10: Two ratios (36)-(37) for T = 1 s (Left) and for T = 1000 s (Right). In each figure, the red surface represents r1 , the blue surface represents r2 .

0 C 0 e

 00 n (t), I n )dt = T 0 -γ(I n )C n (t) + ζ(I n )dt = -γ(I n ) T 0 C n (t)dt + ζ(I n )T.Replacing x n by C n , t 2 by t and t 1 by 0 in[START_REF] Zhang | A new manufacturing resource allocation method for supply chain optimization using extended genetic algorithm[END_REF] and integrating t from 0 to T givesT n (t)dt = T -α(In)t C n (0) + β(I n ) α(I n ) (1 -e -α(In)t ) dt = C n (0) α(I n ) (1 -e -α(In)T ) + β(I n ) α(I n ) T -β(I n ) α 2 (I n ) (1 -e -α(In)T ).Using notations given in Section 3.1, we haveΓ = γ(I) α(I) (e -α(I)T -1), V = β(I) α(I) (1 -e -α(I)T ).From the definition of α(I), β(I), γ(I), we findβ(I) α(I) = β(I) β(I) + k r = k d τ (σ H I) 2 k d τ (σ H I) 2 + k r τ σ H I + k r , γ(I) α(I) = k H σ H I k d τ (σ H I) 2 + k r τ σ H I + k r .

Table 1 :

 1 Parameter values for Han Model

	k r	6.8 10 -3	s -1
	k d	2.99 10 -4	-
	τ	0.25	s
	σ H	0.047	m 2 µmol -1
	k H	8.7 10 -6	-
	R	1.389 10 -7	s -1

Considering now the second term of the right-hand side of (21), we get < u, (P D) l P v >=

Using this notation and Lemma 2, we find

This result combined with [START_REF] Demory | How do microalgae perceive light in a high-rate pond? towards more realistic lagrangian experiments[END_REF], gives

≤s m 1 2

.

Considering now [START_REF] Hennessy | The use of archimedean copulas to model portfolio allocations[END_REF], we obtain

It follows that the first term of ( 21) dominates the second one. As a consequence, the former has the same sign as [START_REF] Eppink | From current algae products to future biorefinery practices: a review[END_REF]. The result follows.

Implementation remarks

In this section, we give details on the practical computation of the infinite sum in [START_REF] De La Hoz | Optimization of microalgal productivity using an adaptive, non-linear model based strategy[END_REF]. Given m 1 ∈ {2, . . . , N }, define by l * such that

We finally evaluate the efficiency of various mixing strategies. Define

where P min ∈ P N is the matrix that minimizes J, (see ( 10)), i.e., that corresponds to the worse strategy. We consider N = 9 layers. Figure 9 presents the results for T = 1 s and T = 1000 s. Better performance is in most cases obtained for a small lap duration T = 1 s. In this way, we observe that the relative improvement between the best and the no mixing strategy may reach 15%, whereas the relative improvement between the worst and the best strategy may reach 30%. In both two cases, a better improvement can be obtained with high values of I s and low values of q.

To compare the efficiency of the approximation P + with respect the true optimal mixing strategy P max , we define two extra ratios:

r2 := μN (P + ) -μN (P min ) μN (P min ) .

Figure 10 presents the results for T = 1 s and T = 1000 s. As already mentioned, for a large lap duration time, the optimization problem [START_REF] Burkard | Assignment Problems[END_REF] provides a good approximation. This can be observed with the blue and red surface in Figure 9 (Right) and in Figure 10 (Right), both surfaces have the same behaviours. As expected, a diagonal matrix with a different order of the diagonal coefficients. Using this notation, we find for the objective function [START_REF] Bertsimas | Introduction to Linear Optimization[END_REF] satisfies

For the objective function [START_REF] Burkard | Assignment Problems[END_REF], we get

Therefore, these problems can still be treated similarly in the general case.

C Remark on F + m , F - m Let u, v ∈ R N such that the entries of u are sorted in ascending order. One should be careful when defining the two sequences F + m and F - m in Section 2.4, since the sign of u and v plays an important role in the definition of these two sequences. For instance, assume that u is now negative and v is positive. Let ũ := -u, since u is assumed to be sorted in ascending order, ũ is positive and sorted in descending order. Using the definition in [START_REF] Cuaresma | Luminostat operation: A tool to maximize microalgae photosynthetic efficiency in photobioreactors during the daily light cycle?[END_REF], one has The case where u is positive and v is negative, or both u, v are negative can be treated similarly.