Rodrigo Coelho
email: coelho@eit.uni-kl.de

Gerhard Fohler
email: fohler@eit.uni-kl.de

Jean-Luc Scharbarg
email: jean-luc.scharbarg@enseeiht.fr

Worst-case backlog for AFDX network with n-priorities

In most recent avionics systems, AFDX (Avionics Full Duplex Switched Ethernet) is the network used to replace the previously employed point-to-point networks. AFDX guarantees bandwidth reservations by means of virtual links which can be classified with two priority levels. AFDX compliant switches implement output buffers at each switch output port. The stored frames leave each output port according to a fixed priority FIFO policy. Overflow of these buffers must be avoided at all cost to prevent data loss. Although the AFDX standard determines the minimum buffer size dedicated to an output port, the actual length of each priority buffer, is a designer decision.

Previous works address the worst case backlog of ADFX buffers of one and two priorities. In this work we assume an extended AFDX network in which virtual links can be classified into n-priorities and present the problem statement to compute an upper bound on the worst case backlog faced by each buffer of each output port in each switch of the network.

I. INTRODUCTION

In most recent avionics systems, the switched ethernet network AFDX [START_REF]ARINC specification 664 P7-1. Aircraft Data Network Part-7 Avionics Full-Duplex Switched Ethernet Network[END_REF] (ARINC 664 Part-7), is chosen to substitute the point-to-point connections of previous avionics distributed systems. Currently, AFDX offers a network bandwidth of 100Mbps and allows for bandwidth isolation among all network traffic by employing the concept of virtual links (VL).

A VL defines a logical path from one source end-system (ES) to one or more destination ESs. The physical route for each VL is statically defined at design time and therefore the switches traversed by each VL are known before run-time. The predictable behavior of AFDX is further ensured by the parameters BAG and S max , respectively bandwidth allocation gap (minimum time interval between the transmission of two consecutive frames of a VL) and maximum frame size associated with a VL. AFDX further allows for the classification of VLs into two priority levels: high and low.

AFDX compliant switches perform store-and-forward. In order to cope with contention for the switches output ports, each output port offers one FIFO queue per priority level. Due to the fixed priority FIFO scheduling at the output ports, switches have to send all frames with high priority before the low priority ones. Considering the non-preemptive property of frame scheduling, a switch cannot abort the transmission of a lower priority frame in favor of a higher priority one.

The AFDX standard specifies the minimum number of frames that must be buffered on the switches output ports. However, the actual output port buffer size for each priority level is left as a design decision and is used in the configuration phase of the network (section 4.7.3.2 of [START_REF]ARINC specification 664 P7-1. Aircraft Data Network Part-7 Avionics Full-Duplex Switched Ethernet Network[END_REF]). Thus, in order to avoid buffer overflow at the output ports and consequently packet loss, the designer must compute an upper bound for the backlog of each priority buffer.

Previous works address the computation of upper bounds for the worst case backlog of ADFX buffers considering virtual links with one and two priority levels. In this work we consider an extended AFDX network where the number of priorities assigned to VLs is unlimited (n-priority levels instead of two) to present the problem statement and the challenges for the computation of an upper bound for the backlog of each priority buffer on each output port of each switch of the network. II. RELATED WORK [START_REF] Ridouard | Stochastic network calculus for buffer overflow evaluation in an avionics switched ethernet[END_REF] presents how to compute probabilistic bounds on buffer backlogs, based on stochastic network calculus (NC). [START_REF] Bauer | Worst-case backlog evaluation of avionics switched ethernet networks with the trajectory approach[END_REF] shows how NC leads to pessimistic results when compared to those achieved by trajectory approach (TA).

In contrast to NC, which considers each VL as a flow, TA analyzes the VL traffic at a finer granularity, accounting for the individual frames of the VLs. In [START_REF] Bauer | Applying and optimizing trajectory approach for performance evaluation of AFDX avionics network[END_REF] and [START_REF] Bauer | Applying trajectory approach with static priority queueing for improving the use of available AFDX resources[END_REF] the authors make use of TA to compute the e2e delay for FIFO output buffers with single priority and distinct static priority flows respectively. [START_REF] Bauer | Worst-case backlog evaluation of avionics switched ethernet networks with the trajectory approach[END_REF] computes the worst case backlog for FIFO output buffers with single priority flows. and extends the previously mentioned works presenting an analysis of the pessimism intrinsic to TA. Preliminary results for the buffer backlog of AFDX networks with two priority flows have been presented in [START_REF] Garikiparthi | Calculation of worst case backlog for afdx buffers with two priority levels using trajectory approach[END_REF]. We extend this analysis and present the analysis for the computation of an upper bound of the worst case buffer backlog assuming an AFDX network with n-priorities.

III. WORST CASE SCENARIO AND COMPUTATION OF BACKLOG UPPER BOUND

We compute an upper bound for the worst case backlog for any output port buffer in three steps: first, we identify all VLs competing for the output port, second we compute the number of frames of the competing VLs that impact the worst case backlog of the buffer under study, and third we determine the worst case arriving sequence for these frames and compute an upper bound for the worst case backlog.

In this paper we consider AFDX virtual links with npriorities. Consequently, we assume that n buffers, one buffer for each priority, exist on each switch output port (similarly to what [START_REF]ARINC specification 664 P7-1. Aircraft Data Network Part-7 Avionics Full-Duplex Switched Ethernet Network[END_REF] defines for two priorities).

Identifying the competing virtual links is straightforward: the routes used by the virtual links are computed off-line and do not change during run time. To compute the number of frames from the competing virtual links that impact the backlog encountered by each frame (f m ) with same priority as the buffer under study, we make use of the trajectory approach (TA) [START_REF] Bauer | Applying and optimizing trajectory approach for performance evaluation of AFDX avionics network[END_REF]. In principle, any other method that provides the number of competing frames can be used in our analysis.

Studying the worst case arriving sequence for the competing frames is the main contribution of this paper towards the worst case backlog computation. We first classify the competing virtual links into three sets: VL S , VL H and VL L , respectively with virtual links of same (sp), higher (hp) and lower (lp) priority than the buffer under study. Then we analyze the impact of each of these sets into each frame of VL S . We start our analysis presenting limit values for the worst case backlog upper bound. Further, we expand this analysis to compute a tighter upper bound.

A. Worst case backlog upper bound limits

If we create an imaginary scenario where the frames of the same priority set (VL S ) are the only frames on the network, an upper bound for the worst case buffer backlog of the buffer under analysis (Buffer S ) can be computed as presented in [START_REF] Bauer | Worst-case backlog evaluation of avionics switched ethernet networks with the trajectory approach[END_REF] and named here as bl onlyS max . Computing an upper bound for the worst case backlog considering all frames (of VL S , VL H and VL L ), can only lead a value larger than or equal to bl onlyS max . The backlog of Buffer S can never be larger than the sum of the sizes of all frames in VL S .

Thus, the computed upper bound for the worst case backlog for a buffer of a given priority is limited by:

bl onlyS max ≤ bl S max ≤ ∀f S k ∈VL S s(f S k ) (1)
where the function s(f ) represents the length of a frame f .

B. Worst case backlog scenario

Figure 1 presents the arrival of sp and hp frames and how they are scheduled at the output port according to the fixed priority FIFO policy. Figure 1 further shows the backlog of the buffer under analysis (Buffer S ). In this figure, frames arrive from four input links and compete for the same output port. Red frames have the same priority as the buffer under analysis (elements of VL S set) and blue frames are those with higher priority (elements of VL L set). The impact of lower priority frames will be considered later.

Next we present the meaning of the points in time and time lengths depicted in the figure: For the sake of simplicity and without loss generality, we assume that s(f ) units of time is the amount of time required to send a frame of length s(f ).

The worst case backlog faced by Buffer S occurs after all sp frames arrive and the access of sp frames to the output port suffers the largest delay. Therefore, we compute bl S max at θ sp and construct a scenario in which all frames from VL H and VL L delay the dispatch of sp frames the longest.

According to Figure 1, the computation of the worst case backlog for Buffer S can be presented as the sum of all frames in VL S minus the amount of data transmitted during the time interval ∆, i.e.:

bl S max = ∀f S k ∈VL S s(f S k )-| ∆ | (2)
An intuitive approach to construct a scenario that leads to the worst case backlog of Buffer S , based on previous analysis for single priority AFDX, is to assume that all hp frames arrive before the sp frames (see Figure 1). Further, frames arrive in decreasing order of size within the same set to avoid idle times at the output link [START_REF] Bauer | Applying trajectory approach with static priority queueing for improving the use of available AFDX resources[END_REF].

According to equation ( 2), in order to achieve the worst case backlog for Buffer S , the arrival sequence of the frames should be such that | ∆ | is minimum. Intuitively, in order to compute the shortest ∆ (which is equivalent to the longest β since θ sp is constant), we should compute the latest α (α max ) and assume that the output link will be busy with all hp frames until β, i.e.: However, neither the scenario presented in Figure1 nor the equation (3) leads to the largest β for every set of frames.

β = α max + ∀f H k ∈VL H s(f H k ) (3)
Figure 2 presents a scenario in which one frame of VL S is shorter than in Figure 1. In this case, the set VL H remains unchanged and so does the result of equation [START_REF] Bauer | Worst-case backlog evaluation of avionics switched ethernet networks with the trajectory approach[END_REF]. Nevertheless, in Figure 2, β is larger than 741, in fact β = 771 due to an idle time of 30 units of time at the output link.

Figure 3 presents a scenario with the same frames as in Figure 2, but a different arriving sequence for the second input link: one lp frame arrives before the sequence of sp frames. Again, the result of equation ( 3) is 741 but the actual β is equal to 812, even larger than the one of the scenario presented in Figure 2 because of the larger idle time at the output link.

We can conclude that, although the frames arrive in decreasing order of lengths per set (VL S , VL H , VL L ), they do not arrive in decreasing order of lengths if we consider all sets together. Therefore some idle time may be present at the output link and consequently equation (3) does not hold. We propose equation (4) to account for this idle time: According to equation (4), we must achieve two goals in order to compute β max :

β max = α max + ∀f H k ∈VL H s(f H k ) + idle max (4)
1) compute the maximum idle time at the output link due to the non-decreasing arrival order of frames idle max . 2) compute the latest point in time in which the set of hp frames start transmission (α max ) such that no sp frame is transmitted before α max . Additionally, we must take into account the impact of lower priority frames.

IV. CONCLUSION AND FUTURE WORK

In this paper we presented the problem statement and the challenges for the computation of an upper bound for the backlog of each priority buffer on each output port of each switch of an extended AFDX network with n-priority levels (instead of two).

The challenges to compute an upper bound for the worst case backlog of each buffer, within the limits presented in (1), is summarized as follows:

• prove that equation (4) holds for any sequence of arriving frames • compute α max • compute idle max • compute the impact of lp frames into the worst case backlog of Buffer S Obviously, the computation of an upper bound for the worst case backlog of output buffers for the current AFDX network (with two priorities) is a sub problem of the one presented in this paper and can, therefore, be achieved by assuming n = 2.

•

  α: start of transmission of hp frames • β: end of transmission of hp frames • θ sp : end of reception of sp frames • ∆: θ spβ

  671 and therefore β = 741.

Fig. 1 :Fig. 2 :

 12 Fig.1: Intuitive approach for the worst case scenario and computation of the worst case backlog for Buffer S .

Fig. 3 :

 3 Fig. 3: One sp frame arrives first on the second input link leading to larger idle time at the output link and thus larger values of β.


Christophe Tournassat

Sylvain

come

Please ensure that your abstract fits into one column on one page and complies with the Instructions to Authors available from the Abstract Submission web page.

Retention processes in clay-rocks

CHRISTOPHE TOURNASSAT 1 AND SYLVAIN GRANGEON 1 1 BRGM, 3 av. Claude Guillemin, Orléans, France. c.tournassat@brgm.fr; s.grangeon@brgm.fr Within the context of the clay barrier concept for underground nuclear waste storage, montmorillonite and bentonite have been widely used as reference materials for radionuclides (RN) retention studies. Associated modeling work aims at understanding and predicting the retention of RN in clay-rocks where clay minerals are assumed to be representative of the most reactive phases. This "bottom-up" approach relies on a good confidence in the mechanistic understanding of adsorption processes on clay mineral surfaces, and in the understanding of the reactivity of associated mineral (or organic) phases.

In the first part of the presentation, experimental and modeling works will be reviewed with a focus on divalent metals surface complexation on clay mineral surfaces, showing the limitations of current modeling approaches [1]. Missing information, which is necessary to model adsorption data in a mechanistic way, together with experimental features that cast doubt on the ability of surface complexation models to catch adequately the nature of divalent metal adsorption, will be highlighted.

In the second part of the presentation, "bottom-up" modeling approaches and related uncertainties will be detailed for the prediction of divalent metals retention in clayrocks. The example of Ni 2+ retention [2] will be used to show how the chemical reactivity of non-clay mineral phases can lower or enhance RN retention through time-dependant processes taking place in laboratory experiments as well as in storage conditions.


Isabelle Tellier

Zineb Makhlouf

Yoann Dupont

Sequential Patterns of POS Labels Help to Characterize Language Acquisition

Keywords: language acquisition, POS labeling, CRF, Sequential Data Mining, emerging patterns

In this paper, we try to characterize various steps of the syntax acquisition of their native language by children with emerging sequential patterns of Part Of Speech (POS) labels. To achieve this goal, we first build a set of corpora from the French part of the CHILDES database. Then, we study the linguistic utterances of the children of various ages with tools coming from Natural Language Processing (morpho-syntactic labels obtained by supervised machine learning) and sequential Data Mining (emerging patterns among the sequences of morpho-syntactc labels). This work thus illustrates the interest of combining both approaches. We show that the distinct ages can be characterized by variations of proportions of morpho-syntactic labels, which are also clearly visible inside the emerging patterns.

Introduction

The acquisition of their native language by children, especially how grammatical constructions are gradually mastered, is a process which largely remains mysterious. Some researches address this issue within a Natural Language Processing framework, for example by implementing programs trying to mimic the learning process [START_REF] Chater | Probabilistic models of language processing and acquisition[END_REF][START_REF] Alishahi | Computational modeling of human language acquisition (Synthesis lectures on human language technologies)[END_REF]. Our approach in this paper is different: we do not target to reproduce, but to mine children productions, from a morphosyntactic point of view. More precisely, we study the linguistic utterances of children of various ages, seen as sequences of part-of-speech (POS) labels, with sequential data mining tools.

Sequential data mining can be applied to any kind of data following an order relation. This relation is often related to time; for texts, it is only the linear order of words in sentences. Sequential data mining allows to extract sequential patterns, that is sequences or sub-sequences of itemsets that repeatedly occur in the data. This domain has given rise to many works [AS95,SA96,Zak01,NR07]. If the extracted sequences are contiguous portions of texts, patterns coincides with the older notion of repeated segments [START_REF] Salem | Segments répétés et analyse statistique des données textuelles[END_REF].

When data are composed of natural language texts, the itemsets are not necessarily reduced to words: lemmas and POS labels can also be taken into account. The use of sequential data mining technics in such a linguistic context has recently been tested for the extraction of Named Entities [START_REF] Nouvel | Fouille de règles d'annotation partielles pour la reconnaissance d'entités nommées[END_REF], the discovery of relations between entities in the biological field [START_REF] Charnois | Fouille de données séquentielles pour l'extraction d'information[END_REF][START_REF] Cellier | Sequential patterns to discover and characterise biological relations[END_REF][START_REF] Béchet | Discovering linguistic patterns using sequence mining[END_REF] or the study of stylistic differences between textual genres [START_REF] Quiniou | Fouille de données pour la stylistique : cas des motifs séquentiels émergents[END_REF]. As we look at the emergence of grammatical constructions in children, we are mainly interested here in patterns of morpho-syntactic labels. As a matter of fact, they are more general than words or lemmas and provide more abstract characterizations of a given age. We seek in particular to exhibit specific emerging patterns for different age groups.

The remaining of the article is as follows. First, we present the way our corpora of children's productions of different age groups have been collected. Then, we explain how we processed their morpho-syntactic analysis. Observing that usual POS taggers available for French made many mistakes on our data, we have built a new one, by training a machine learning device (a CRF model) on a reduced set of manually corrected data. We show that, despite this reduced set of manual corrections, the new tagger obtained behaves far better than the previous one on our data. Finally, the last part of the paper describes the technique used for the extraction of n-grams of morpho-syntactic labels of each specific age group and provides quantitative and qualitative analyses of the corresponding emerging patterns.

Corpora

The CHILDES Corpus

Several resources collecting children's productions exist online, as those available in the CNRTL 1 . But the best known and most widely used database is CHILDES 2 [Elm01], a multilingual corpus of transcriptions of recorded interactions between adults and children. In this article, we are only interested in the French part of these data. The recordings of a child cover several months or years, the age of the children may therefore vary from one record to another. Relying on the transcription manual 3 which explicits the meta-data associated with the corpus, we created six different sub-corpora corresponding to six age groups: from the "1-2 years" to the "6-7 years".

Pretreatments

In this corpus, children and parents communicate by speech turns. Each speech turn is transcribed and delimited by a period. In the following, we consider that each line corresponds to a "sentence". The transcriptions are annotated and are often followed by additional information in a (semi-)standard format allowing to describe elements of the situation (e.g. objects which are in the scene). We performed a preprocessing step to focus only on linguistic productions. We have removed all special characters related to standards of transcription, as well as all information of phonetic nature, which are not relevant for the analysis of syntactic constructions and prevent the use of a tagger. We have also eliminated from our data all adult utterances. The characteristics of each of our initial sub-corpora are presented in the table of Figure 1. There are differences between them: the corpus for the age of "6-7 years" is the smallest one. To balance the corpora of the different age groups, we have sampled them according to the number of words: this feature is more reliable than the number of sentences, because the length of the sentences is a key factor which significantly varies from one age to another (see the following).

To have comparable sub-corpora, the number of words is thus more reliable than the number of sentences.

Sampling

The smallest corpus in terms of words (the one of "6-7 years") is the reference sample for the other age groups. So, we chose to take 20,000 words per corpus, with a rate of 0.01% tolerance. To build our new corpora from the initial ones, we sampled sentences randomly until the sum of all words in all sentences reaches this size. After the sampling, we have six new corpora, whose properties are given in the table of Figure 2.

The corpora now have comparable size in terms of words. The number of sentences in each corpus have of course decreased, but we note that the average lengths of the sentences follow the same evolution than in the initial corpora. This is crucial because, as long as the children grow up, they tend to produce longer sentences. This is a well-known key feature of language acquisition [START_REF] Brown | A first language: the early stages[END_REF][START_REF] Miller | The relation between age and mean length of utterance in morphemes[END_REF]. To go further in our exploration, we will now label the productions of the children with morpho-syntactic labels.

POS labeling

Use of an existing tagger

As we want to characterize the acquisition of syntactic constructions, we need more information than simple transcriptions of words. Our experiments in this article rely on a morpho-syntactic tagging of children's productions: we must thus assign to each word in the sub-corpora a label corresponding to its grammatical category. Several tools are available to annotate plain text in French with "Part of Speech" (POS) labels, such as TreeTagger [START_REF] Schmid | Probabilistic part-of-speech tagging using decision trees[END_REF]. In our work, we have used SEM 4 [TDE + 12], which was obtained by training a linear CRF (Conditional Random Fields) model on the French Treebank [START_REF] Abeillé | Building a treebank for french[END_REF]. The set of labels adopted in SEM, similar to the one of [START_REF] Crabbé | Expériences d'analyse syntaxique statistique du français[END_REF], includes 30 different categories among which the main important ones for the following are: NC (for common nouns), V (for verbs), DET (for determiners), P (for prepositions), I (for interjections) and CLS (for subject clitic). SEM also integrates the external lexical resource Lefff [START_REF] Clément | Morphology based automatic acquisition of large-coverage lexica[END_REF] to help achieve a better labeling. SEM has been learned with labeled sentences extracted from the French newspaper "Le Monde". Our texts of children productions have very different properties, and we therefore expect many annotation errors. Indeed, the corpus CHILDES is composed of oral transcription, whose conventions differ from those of writing (especially concerning punctuations). Furthermore, children utterances are often far from standard French. It has already been observed that, even if SEM is supposed to reach 97% accuracy on texts similar to those on which it has been learned, it reaches 95.6% accuracy on more casual written texts from blogs, and only 81.6% on oral productions of adults.

To assess the quality of SEM on our data, we have randomly selected 200 sentences from each of our six corpora, tagged them with SEM and manually corrected the labeling errors, following the annotation conventions of the French Treebank. The accuracy of SEM on these samples (see table of Figure 4) ranges from 70% (2-3 years) to 87% (6-7 years). The detailed F-measures of the main categories for each age group can also be seen in the table of Figure 3: the label interjection (I), very rare in the French Treebank but very frequent in our corpora, are particularly not well recognized by SEM (the F-measures goes from 33.33 for the "1-2 years" age group to 0 for the the "6-7 years" one).

Learning a New tagger

As we want to perform statistical measures on the morpho-syntactic labels, labeling errors must be reduced as much as possible. In [START_REF] Tellier | Adapt a text-oriented chunker for oral data: How much manual effort is necessary[END_REF], it has been shown that to learn a good tagger by supervised machine learning, it is more efficient to have a small annotated corpus similar to the target data than to have a large too different training set. So, we decided to use the labelled sentences which have been manually corrected for the evaluation of SEM as training data to learn a new tagger adapted to our corpora.

For this, we have used the same tools as those used to learn SEM, that is CRFs (Conditional Random Fields), introduced by [LMP01] and implemented in the software Wapiti [START_REF] Lavergne | Practical very large scale CRFs[END_REF]. CRFs are graphical models that have proven their effectiveness in the field of automatic annotation by supervised machine learning [TTA09,TDE + 12]. They allow to assign the best sequence of annotations y to an observable sequence x. For us, the elements of x are words enriched with endogenous attributes (presence of caps, digits, etc.) or exogenous ones (e.g. associated properties in Lefff), while y is the corresponding sequence of morphosyntactic labels.

We trained our new tagger thanks to 200 * 6 = 1200 annotated and manually corrected sentences (which is a very small number to learn a POS tagger), and we tested it on 50 * 6 = 300 other independent sentences, equally sampled from the 6 distinct sub-corpora. The table of Figure 3 gives the F-measures of the main labels obtained by SEM and by the re-learned tagger for each age group, while the accuracy of both taggers are provided in the table of Figure 4. We observe that the relearning leads to a significant improvement of the accuracy of about 10% in average. SEM is better for only 4 cells out of 36 in the table of Figure 3, probably thanks to its better vocabulary exposure: the French Treebank on which SEM was learned was about ten times larger than our training corpus. The improvement brought by relearning is larger for oral-specific labels such as I. It is therefore very beneficial, despite a very small training corpus. This corpus SEM re-trained tagger 1-2 years 82% 85% 2-3 years 70% 80% 3-4 years 73% 88% 4-5 years 75% 90% 5-6 years 80% 92% 6-7 years 87% 90% average 77.83% 87.5% Fig. 4. Impact of the re-learning on the accuracy of the distinct age groups can be explained by the fact that the vocabulary used in our texts is relatively limited and redundant: few data are therefore sufficient to obtain a tagger which is effective on our corpus, even if it is not uniformly better than SEM on every label (it would obviously be much less effective on other types of data). In the following, we systematically use the new version of the tagger.

Analysis of POS labels

Figure 5 shows the distribution of the main morpho-syntactic categories in the different age groups. For example, we see that the curve of the label I (interjection) is decreasing (except for the 4-5 years age group): it seems that children use fewer and fewer interjections in their productions as long as grow up. In contrast, the label P (preposition) is strictly increasing, which is consistent with an acquisition of increasingly sophisticated syntactic constructions. Curves for the labels CLS (subject clitic) and V (verb) follow very similar variations, probably because they are often used together: they increase till the age of 4, then decrease from 4 to 6, and finally stabilize at the age of 6. Observing labels DET (determiner) and NC (common nouns), we notice that until the age of 4 years, NC is the most common label, but not yet being systematically associated with a DET. It is only at the age of 4 that both curves become parallel (most probably when most NC is preceded by a DET). We finally note that from the age of 5 years, the proportions of different labels stabilize. The residual errors of the tagger (there is more than 10% remaining labeling errors) lead us to be prudent with these observations. But it is clear that some of the phenomena observed here would not have been possible without re-learning: interjections, for example, were the words most poorly recognized by the original SEM, because they are very rare in newspaper articles. However, their production appears to be an important indicator of the child's age group. Example sentences like "ah maman" ("ah mom") or "heu voilà " ("uh there") were respectively labeled as "ADJ NC" and "ADV V" with the original SEM tagger. After the re-learning, the labels became "I NC " and "I V", which is at least more correct.

Although we can already draw some interesting conclusions from these curves, we cannot characterize the syntactic acquisition of children from single isolated

Sequential Patterns Extraction

General Definitions

Many studies have focused on the analysis of texts seen as sequential data. For example, the notion of repeated segment is used in textometrics [START_REF] Salem | Segments répétés et analyse statistique des données textuelles[END_REF] to characterize a contiguous sequence of items appearing several times in a text. Sequential data mining [START_REF] Agrawal | Mining sequential patterns[END_REF] generalizes such concept, with notions like sequential patterns of itemsets. In our case, itemsets can be composed of words and POS labels. A sequence of itemsets is an ordered list of itemsets. An order relation can be defined on such sequences: a sequence S 1 = I 1 , I 2 , ..., I n is included into a sequence S 2 = I 1 , I 2 , ..., I m , which is noted S 1 ⊆ S 2 , if there exist integers 1 ≤ j 1 ≤ j 2 ≤ ... ≤ j n ≤ m such that I 1 ⊆ I j1 , I 2 ⊆ I j2 , ..., I n ⊆ I jn (in the classical sense of itemset inclusion). The table of Figure 6 provides examples of sequences of itemsets found in our corpus labelled with the re-trained tagger.

The support of a sequence S, denoted sup(S), is equal to the number of sentences of the corpus containing S. For example, in the table of Figure 6, sup( (ADJ) (NC) ) = 2. The relative support of a sequence S is the proportion of sequences containing S in the base of initial sequences. It is worth 1 2 for the sequence in our example, because this sequence is present in 2 out of the 4 sequences of the database. Algorithms mining sequential patterns are based on a minimum threshold for extracting frequent patterns. A frequent pattern is thus a sequence for which the support is greater than or equal to this threshold. Other concepts are also useful to limit the number of extracted patterns.

Extraction of Sequential Patterns under constraints

In [START_REF] Yan | Mining closed sequential patterns in large databases[END_REF], was introduced the notion of closed patterns that allows to eliminate redundancies without loss of information. A frequent pattern S is closed, if there is no other frequent pattern S such S ⊆ S and sup(S) = sup(S ). In our example, if we fix minsup=2, the frequent pattern (DET) (NC) , extracted from Figure 6, is not closed because it is included in the pattern (le, DET) (NC) and they both have a support equals to 3. But the pattern (DET) (small, ADJ) (NC) is closed. A length constraint can also be used. It defines the minimum and maximum number of items contained in a pattern [START_REF] Béchet | Discovering linguistic patterns using sequence mining[END_REF].

Algorithm

There are several available tools for extracting sequential patterns such as GSP [START_REF] Srikant | Mining sequential patterns: Generalizations and performance improvements[END_REF] and SPADE [START_REF] Zaki | Spade: An efficient algorithm for mining frequent sequences[END_REF]. CloSpan [START_REF] Yan | Mining closed sequential patterns in large databases[END_REF] and BIDE [START_REF] Wang | Bide: Efficient mining of frequent closed sequences[END_REF] are able to extract frequent closed sequential patterns. SDMC5 , used here, is a tool based on the method proposed in [PHMA + 01]. It extracts several types of sequential patterns, where items can correspond to simple words, lemma and/or their morpho-syntactic category (the tagger is parameterized, which allowed us to use our tagger). In this work, we wanted to characterize grammatical constructions, and we thus focused only on sequences of POS labels. The algorithm of SDMC implements the pattern growth technic; it is briefly discussed in [START_REF] Béchet | Discovering linguistic patterns using sequence mining[END_REF]. It allows to extract sequential patterns under several constraints.

Emerging Patterns

[DL99] introduced the concept of emerging pattern. A frequent sequential pattern is called emerging if its relative support in a set of data set is significantly higher than in another set of data. Formally, a sequential pattern P of a set of data D 1 is emerging relatively to another set of data D 2 if GrowthRate(P ) ≥ ρ, with ρ > 1. The growth rate function is defined by:

∞ if support D2 (P ) = 0 supp D 1 (P ) supp D 2 (P ) otherwise
where supp D1 (P ) (respectively supp D2 (P )) is the relative support of the pattern P in D 1 (respectively D 2 ). Any pattern P whose support is zero in a set is neglected.

Experiments

Parameters

The corpora used in our experiments are those described in section 2.3. We are interested here in sequences of itemsets restricted to POS labels without any gap (thus corresponding to n-grams, or repeated segments of labels), under some constraints (such as having a support strictly greater than a given threshold or pruning non-closed patterns), to limit their number. To set the lengths of sequences, we took account of the average size of sentences. So, we have decided to select patterns of length between 1 and 10. The minsup threshold is set to 2 and ρ = 1.001. To find the emerging patterns of a certain age group, we do as [START_REF] Quiniou | Fouille de données pour la stylistique : cas des motifs séquentiels émergents[END_REF] did for literary genres: each age group (D 1 ) is compared to the set of every other age groups (D 2 ).

Quantitative Results

Figure 7 shows the number of frequent and emerging patterns obtained under our constraints for each age group. For example, for the age of 4-5 years, there are 1933 frequent patterns but only 842 emerging ones (42.6%). A serious reduction has occurred, which will make the observation easier. The number of emerging patterns is relatively stable across ages from 3-4 years and is important in each age group. As these emerging patterns are defined relatively to every other age group, this suggests the existence for each age group of characteristic phases of grammatical acquisitions.

Figure 8 shows the average size of the frequent and emerging patterns for each age group. The curves are very similar, suggesting that emerging patterns have properties which are similar to frequent patterns. In both cases, the length is increasing and reaches its maximum at the age of 5-6 years old. This parameter seems very correlated to the one of sentence length (see Figure 1): not only utterances become longer as the children grow up, but also the grammatical patterns they instantiate.

Figures 9 and 10 show the distributions of the main morpho-syntactic labels in frequent and emerging patterns respectively for each age group. These results are consistent with those obtained on the entire corpus (cf. Figure 5). The proportion of interjections still regularly decreases, while the one of prepositions increases, which is consistent with syntactic constructions of increasing complexity. We also note that the CLS and V curves are parallel and that, before the age of 4 years, the NC label is very frequent without being associated with the label DET. These curves show that the proportions of labels in the frequent and emerging patterns of each age group are similar to those of the corpus. In this sense, these patterns seem to be representative of the different age groups.

Qualitative Results

The table of Figure 11 provides examples of emerging patterns of each age group, and some corresponding sentences. These examples show that a single pattern can correspond to various sentences, and that they have increasing complexity. We note that even before the age of 2, children can produce sentences with a CN preceded by a DET. We also note, for example, that the patterns "(DET) (NC)" and "(DET) (NC) (CLS) (V) (VINF)" respectively extracted of the age Fig. 9. Proportions of distinct labels in frequent patterns for each age group Fig. 10. Proportions of distinct labels in emerging patterns for each age group "1-2 years" and "4-5 years are included in "(P) (DET) (NC) "and" (DET) (NC) (CLS) (V) (VINF) (DET) (NC)" respectively, of the following age group. This is consistent with a gradual acquisition of complex syntactic constructions.

Conclusion

In this article, we have applied techniques from Natural Language Processing, machine learning and sequential Data Mining to study the evolution of children's utterances of different ages. The phase of morpho-syntactic labeling required the learning of a specific tagger, adapted to our data. It was a necessity, considering that current available taggers do not properly handle oral transcriptions, and even less those of children: interjections, for example, which are very specific of 1-2 years (P) (NC) -à maman ("to mom") -sac à dos ("backpack") (DET) (NC) -le ballon ("the ball") -des abeilles ("some bees") 2-3 years (P) (DET) (NC) -de la tarte ("some pie") -poissons dans l'eau ("fishes in the water") (ADVWH) (CLS) (V)

où il est ?

("where it is ?") -comment il marche ? ("how it works ?") 3-4 years (ADV) (CLS) (V)

non il est par terre (" no it is on the floor") -ici il pourra passer ("here it will be able to pass") 4-5 years (ADV) (CLS) (CLO) (V)

alors tu m'as vue ? ("so you saw me ?") -oui j'en fais souvent ("yes I do some often") (DET) (NC) (CLS) (V) (VINF) -les lapins ils vont rentrer ("the rabbits they will come in") -le chat il veut attraper l'oiseau ("the cat it wants to catch the bird") 5-6 years (DET) (NC) (CLS) (V) (VINF) -l'enfant il va chercher le chat (DET) (NC) ("the child he goes and fetch the cat") -le monsieur il va chercher les cerises ("the man he goes and catch the cherries") (CC) (DET) (NC) (CLS) (V) -la maman et le papa ils regardaient le garçon (DET) (NC) ("the mommy and the daddy they watched the boy") -et le chat il mange les cerises ("and the cat it eats the cherries") 6-7 years (P) (VINF) (DET) (NC) -les oiseaux les aident à ramasser les cerises ("the birds help them to pick up the cherries") -il y a un chat qui essaie de chasser des oiseaux ("there is a cat trying to catch birds") (DET) (NC) (PROPEL) (V) -il y a un chat qui suit la fille avec son panier (DET) (NC) (P) (DET) (NC) ("there is a cat which follows the girl with a basket") -et aussi un monsieur qui ramasse des cerises dans un arbre ("and a man picking up cherries in a tree") Fig. 11. Examples of emerging patterns in each age group oral productions, would have been poorly recognized without re-learning. This is crucial, as the curves of label proportions show that their frequency appears as an important way to characterize a child's age group.

Fig. 5 .

 5 Fig. 5. Proportions of each label for each age group

Fig. 6 .

 6 Fig. 6. Examples of sequences of itemsets (word, POS label)

Fig. 7 .

 7 Fig. 7. Number of frequent versus emerging patterns for each age group

Fig. 8 .

 8 Fig. 8. Average length of frequent versus emerging patterns for each age group

https://sdmc.greyc.fr, login and password to be asked

We currently restricted our research to n-grams of POS labels but further work could use richer itemsets of the type (word, lemma, POS tag). Our exploration seems to confirm that the extracted emerging patterns are representative of the age group in which they arise. The provided examples further confirm the intuition that (at least some of) the patterns of increasing age groups are included into each other, going in the direction of a grammatical sophistication.

As far as we know, these kinds of analyses had never been performed before. Of course, a detailed analysis of the patterns obtained remains to be done by specialists of language acquisition. They could for example allow to characterize typical evolutions of grammatical knowledge, or help to diagnose pathological evolution of a child's productions. We hope that they will provide valuable tools for the study of language acquisition

Aknowlegment

This work is supported by a public grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (reference: ANR-10-LABX-0083). The authors acknowledge Christophe Parisse, for his advice.


Marion Borderon

Camille Perchoux

Sébastien Oliveau

Frédéric Audard

Richard Lalou

Malaria risk in Dakar : socio-spatial inequalities and urban environment

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


L Miguel Rodrigues
email: rodrigues@math.univ-lyon1.fr

Kevin Zumbrun
email: kzumbrun@indiana.edu

PERIODIC-COEFFICIENT DAMPING ESTIMATES, AND STABILITY OF LARGE-AMPLITUDE ROLL WAVES IN INCLINED THIN FILM FLOW

A technical obstruction preventing the conclusion of nonlinear stability of large-Froude number roll waves of the St. Venant equations for inclined thin film flow is the "slope condition" of Johnson-Noble-Zumbrun, used to obtain pointwise symmetrizability of the linearized equations and thereby high-frequency resolvent bounds and a crucial H s nonlinear damping estimate. Numerically, this condition is seen to hold for Froude numbers 2 < F 3.5, but to fail for 3.5 F . As hydraulic engineering applications typically involve Froude number 3 F 5, this issue is indeed relevant to practical considerations. Here, we show that the pointwise slope condition can be replaced by an averaged version which holds always, thereby completing the nonlinear theory in the large-F case. The analysis has potentially larger interest as an extension to the periodic case of a type of weighted "Kawashima-type" damping estimate introduced in the asymptotically-constant coefficient case for the study of stability of large-amplitude viscous shock waves.

Introduction

The St. Venant equations of inclined thin film flow, in nondimensional Lagrangian form, are (1.1)

∂ t τ -∂ x u = 0, ∂ t u + ∂ x τ -2 2F 2 = 1 -τ u 2 + ν∂ x (τ -2 ∂ x u),
where τ = 1/h is the reciprocal of fluid height h, u is tangential fluid velocity averaged with respect to height, x is a Lagrangian marker, F is a Froude number given by the ratio between a chosen reference speed of the fluid and speed of gravity waves, and ν = R -1 e , with R e the Reynolds number of the fluid. The terms 1 and τ u 2 on the righthand side of the second equation model, respectively, gravitational force and turbulent friction along the bottom. Roughly speaking, F measures inclination, with F = 0 corresponding to horizontal and F → ∞ to vertical inclination of the plane.

An interesting and much-studied phenomenon in thin film flow is the appearance of roll-waves, or spatially periodic traveling-waves corresponding to solutions (1.2) (τ, u)(x, t) = (τ , ū)(x -ct)

of (1.1). These are well-known hydrodynamic instabilities, arising for (1.1) in the region F > 2 for which constant solutions, corresponding to parallel flow, are unstable, with applications to landslides, river and spillway flow, and topography of sand dunes and sea beds [START_REF] Balmforth | Dynamics of roll waves[END_REF].

Nonlinear stability of roll-waves themselves has been a long-standing open problem. However, this problem has recently been mostly solved in a series of works by the authors together with Barker, Johnson, and Noble; see [JZN11, BJRZ11, BJN + 10, JNRZ14, BJN + 15]. More precisely, it has been shown that, under a certain technical condition having to do with the slope of the travelingwave profile (τ , ū), spectral stability in the sense of Schneider [Sch98, Sch96, JZN11, JNRZ14], implies linear and nonlinear modulational stability with optimal rates of decay, and, moreover, asymptotic behavior is well-described by a system of second-order Whitham equations obtained by formal WKB expansion.

In turn, spectral stability has been characterized analytically in the weakly unstable limit F → 2 and numerically for intermediate to large F in terms of two simple power-law descriptions, in the small-and large-F regimes, respectively, of the band of periods X for which roll waves are spectrally stable, as functions of F and discharge rate q (an invariant of the flow describing the flux of fluid through a given reference point) [BJN + 15]. That is, apart from the technical slope condition, there is at this point a rather complete theory of spectral, linear, and nonlinear stability of roll wave solutions of the St. Venant equations. However, up to now it was not clear whether failure of the slope condition was a purely technical issue or might be an additional mechanism for instability.

Precisely, this slope condition reads, in the Lagrangian formulation (1.1)-(1.2), as

(1.3) 2ν ūx < F -2 ,
where ū is the velocity component of traveling wave (1.2). It is seen numerically to be satisfied for F 3.5, but to fail for F 3.5 [BJN + 15]. For comparison, hydraulic engineering applications typically involve Froude numbers 2.5 F 20 [START_REF] Mina | Approximate solution of gravity-affected flow from planar sluice gate at high froude number[END_REF][START_REF] Brock | Development of roll-wave trains in open channels[END_REF][START_REF] Brock | Periodic permanent roll waves[END_REF]; hence (1.3) is a real physical restriction. From the mathematical point of view, the distinction is between small-amplitude, slowly varying waves for which (1.3) is evidently satisfied and large-amplitude, rapidly-varying waves, such as appear in the spectrally stable regime for small and large F , respectively [BJN + 15].

The role of condition (1.3) in the stability analysis is to obtain pointwise symmetrizability of the linearized equations and thereby high-frequency resolvent bounds and a crucial nonlinear damping estimate used to control higher derivatives in a nonlinear iteration scheme. The purpose of the present brief note is to show, by a refined version of the energy estimates of [START_REF] Johnson | Nonlinear stability of viscous roll waves[END_REF][START_REF] Barker | Metastability of solitary roll wave solutions of the St. Venant equations with viscosity[END_REF], that the pointwise condition (1.3) can be replaced by an averaged version that is always satisfied, while still retaining the high-frequency resolvent and nonlinear damping estimates needed for the nonlinear analysis of [START_REF] Johnson | Nonlinear stability of viscous roll waves[END_REF][START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF], thus effectively completing the nonlinear stability theory.

The remainder of this paper is devoted to establishing the requisite weighted energy estimates, first, in Sections 2-3, in the simplest, linear time-evolution setting then, in Sections 4.1 and 4.2, respectively, in the closely related high-frequency resolvent and nonlinear time-evolution settings. The estimates so derived may be seen to be periodic-coefficient analogs of weighted "Kawashimatype" estimates derived in the asymptotically-constant coefficient case for the study of stability of large-amplitude viscous shock waves [Zum04, Zum07, GMWZ06], to our knowledge the first examples of such estimates specialized to the periodic setting. We discuss this connection in Sections 5 & 6. More, this seems to be the first instance of a proof of hypocoercive 1 decay where periodicity is used in a crucial way. We note, finally, the relation between these weights and the "gauge functions" used for similar purposes in short-time (i.e., well-posedness) dispersive theory [LP02, BGDD06, Mie15], a connection brought out further by our choice of notation in the proof. This indicates perhaps a potential for wider applications of these ideas in the study of periodic wave trains.

Preliminary observations

Making the change of variables x → x -ct to co-moving coordinates, we convert (1.1) to (2.1)

τ t -cτ x -u x = 0, u t -cu x + ((2F 2 ) -1 τ -2 ) x = 1 -τ u 2 + ν(τ -2 u x ) x ,
1 The reader interested in replacing Kawashima-type estimates in the more general context of hypocoercive decay estimates is referred to [Vil09, Remark 17] and references in [Rod13, Appendix A], especially [START_REF] Beauchard | Large time asymptotics for partially dissipative hyperbolic systems[END_REF].

and the traveling-wave solution to a stationary solution U (x, t) = (τ (x, t), u(x, t)) = (τ (x), ū(x)) convenient for stability analyis. We note for later that the traveling-wave ODE becomes

(2.2) -cτ x -ūx = 0, -cū x + ((2F 2 ) -1 τ -2 ) x = 1 -τ ū2 + ν(τ -2 ūx ) x ,
yielding the key fact that

(2.3) f (τ )ū x = cf (τ )τ x
is a perfect derivative for any function f (•), hence zero mean over one period. We note also as in [START_REF] Johnson | Nonlinear stability of viscous roll waves[END_REF] that c = 0, else u ≡ constant and the equation for τ reduces to first order, hence does not admit nontrivial periodic solutions. Linearizing about Ū = (τ , ū) gives the linearized equations

(2.4) τ t -cτ x -u x = 0, u t -cu x -(ατ ) x = ν(τ -2 u x ) x -ū2 τ -2ūτ u,
where

(2.5) α := τ -3 (F -2 + 2ν ūx ).
With this notation, the slope condition of [START_REF] Johnson | Nonlinear stability of viscous roll waves[END_REF] appears as τ 3 α > 0. We note that, by (2.3), the mean over one period of g(τ )α is positive for any positive g:

(2.6) g(τ )α = g(τ )τ -3 F -2 > 0.
That is, (any reasonable version of) the slope condition holds always in an averaged sense. 2 An approximate asymptotic diagonalization in the large spectrum regime-see [BJRZ11, BJN + 15], in particular [BJN + 15, Appendix A]-reveals that the sharp 3 relevant averaged conditions is

ατ 2 ν > 0 .
We shall show in the rest of the paper that this averaged condition is in fact sufficient for the nonlinear analysis of [START_REF] Johnson | Nonlinear stability of viscous roll waves[END_REF][START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF].

Linear damping estimate

Introduce now some 'gauge' functions φ 1 , φ 2 and φ 3 and define for U = (τ, u) the energy

(3.1) E(U ) := 1 2 φ 1 τ 2 x + 1 2 φ 2 τ 3 u 2 x + φ 3 τ u x .
A brief computation yields that solutions U of (2.4) satisfy

(3.2) d dt E(U (t)) = -c 2 (φ 1 ) x + αφ 3 τ 2 x -ν τ 2 φ 2 u 2 xx + φ 1 -αφ 2 + ν τ 2 φ 3 τ x u xx + O ( u H 2 + τ H 1 )( u H 1 + τ L 2 ) .
The original gaugeless strategy that works when α is positive may be achieved by choosing

φ 1 ≡ 1, φ 2 = φ 1 /α and 0 < φ 3 ≡ constant ≪ 1.
The possibility of choosing φ 2 = φ 1 /α while keeping both φ 1 and φ 2 positive is a direct manifestation of the fact that in this case the first-order part of system (2.4) is symmetrizable. For the general case, of interest here, we instead take

(3.3) c 2 (φ 1 ) x + ατ 2 ν - ατ 2 ν φ 1 = 0, φ 1 (0) > 0,
2 Here and elsewhere we use h to denote mean over one period of a function h. 3 In the sense that there exist curves of spectrum for the operator L-defined below-that are going to infinity and whose real parts converge to -ατ 2 /ν.

(3.4)

φ 1 -αφ 2 + ν τ 2 φ 3 = 0, 0 < φ 2 ≡ constant ≪ 1,
so that φ 3 is chosen to kill the indefinite cross-term and the fact that φ 1 is not constant and thus does not commute with the generator of system (2.4) is used to average and cancel the "bad" oscillating part of ατ 2 ν through the arising nontrivial commutator. With these choices, we obtain after another brief computation (3.5)

d dt E(U (t)) = - τ 2 α ν φ 1 -α 2 τ 2 ν φ 2 τ 2 x + ν τ 2 φ 2 u 2 xx + O u H 2 + τ H 1 )( u H 1 + τ L 2 ) ≤ -η 1 ( u xx 2 L 2 + τ x 2 L 2 ) + C 1 u H 2 + τ H 1 )( u H 1 + τ L 2 ) ,
for some positive η 1 and C 1 , whence, by interpolation inequality

u H 1 u 1/2 L 2 u 1/2 H 2 and the fact that E(U ) ∼ ( u x 2 L 2 + τ x 2 L 2 ) modulo τ 2 L 2 , (3.6) d dt E(U (t)) ≤ -ηE(U (t)) + C U (t) 2
L 2 , for some positive η and C, a standard linear damping estimate.

Note that, in the step

E(U ) ∼ ( u x 2 L 2 + τ x 2 L 2 ) modulo L 2 , we have used in a critical way that x ατ 2 ν - ατ 2 ν ,
hence φ 1 and 1/φ 1 , remains bounded, a consequence of periodicity plus zero mean.

To derive a first corollary from the key estimate (3.6), one may combine it with the standard L 2 bound d dt

1 2 τ 2 + 1 2 u 2 = - ν τ 2 u 2 x + O ( u H 1 + τ L 2 )( u L 2 + τ L 2
) to obtain the following lemma.

Lemma 3.1. There exist positive θ and C such that any U solving (2.4) satisfies for any t ≥ 0

U (t) H 1 ≤ C e -θ t U (0) H 1 + C t 0 e -θ (t-s) U (s) L 2 ds .

Applications

Lemma 3.1 by itself is not of much direct practical use. However, as we will now show, we can readily adapt its proof, and especially estimate (3.6), to obtain various useful forms of highfrequency damping estimates. The reader unfamiliar with these considerations may benefit from first having a look at [Rod13, Appendix A] for a terse introduction to this approach. Indeed what follows stems directly from the mere introduction in the classical strategy described there of gauges leading to (3.6). See also [START_REF] Zumbrun | Stability of large-amplitude shock waves of compressible Navier-Stokes equations[END_REF]Zum07] for related estimates in the shock wave case. 4.1. High-frequency resolvent bound. An important part of the proofs in [JZN11, JNRZ14] is dedicated to estimates of semigroups generated by linearization around a given wave, to be used in an integral formulation of the original nonlinear systems. These estimates are deduced from spectral considerations and the noncritical part of the linearized evolution is directly controlled by an abstract spectral gap argument that only requires uniform bounds on certain resolvents. Our claim is that a spectral version of Lemma 3.1 does provide these uniform bounds.

To be more specific let L denote the operator generating the linearized evolution around Ū , that is, such that system (2.4) reads U t -LU . The operator L is a differential operator with periodic coefficients but acting on functions defined on the full line. We do not apply directly spectral considerations to L but rather to its operator-valued Bloch symbols L ξ , associated with the Floquet-Bloch transform-see [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF][START_REF] Rodrigues | Asymptotic stability and modulation of periodic wavetrains, general theory & applications to thin film flows[END_REF] for instance. Explicitly, if Ξ denotes the fundamental period of Ū , for any Floquet exponent ξ in the Brillouin zone [-π/Ξ, π/Ξ), the operator L ξ acts on functions of period Ξ by L ξ := e -iξ• Le iξ• .

The operator ∂ x itself has Bloch symbols ∂ x + iξ. As a result, when dealing with L ξ , the

(equivalent) norm of interest on H s per (0, Ξ) is f H s ξ = s k=0 (∂ x + iξ) k f 2 L 2 (0,Ξ) 1/2
. Consider now the resolvent equation

(4.1) (λ -L ξ )U = F .
Letting •, • denote complex inner product, we find by computations essentially identical to those in Section 3, substituting λU for U t and ∂ x + iξ for ∂ x , that, defining φ j as in (3.3)-(3.4), and

E ξ (U ) := 1 2 φ 1 (∂ x + iξ)τ, (∂ x + iξ)τ + 1 2 φ 2 (∂ x + iξ)u, (∂ x + iξ)u + ℜ φ 3 τ, (∂ x + iξ)u , one derives (4.2) 2ℜ(λ) E ξ (U ) = ℜ(2λE ξ (U )) = ℜ( φ 1 (∂ x + iξ)u, (∂ x + iξ)(λu) + φ 2 (∂ x + iξ)τ, (∂ x + iξ)(λτ ) + φ 3 τ, (∂ x + iξ)(λu) + φ 3 (∂ x + iξ)u, λτ ) = -( c 2 (φ 1 ) x + αφ 3 )|(∂ x + iξ)τ | 2 -( ν τ 2 φ 2 )|(∂ x + iξ) 2 u| 2 + (φ 1 -αφ 2 + ν τ 2 φ 3 )ℜ (∂ x + iξ)τ (∂ x + iξ) 2 u + O u H 2 ξ + τ H 1 ξ )( u H 1 ξ + τ L 2 ) + O( U H 1 ξ F H 1 ξ ) ≤ -η E ξ (U ) + C( U 2 L 2 + F 2 H 1 ξ ).
for some positive η and C uniform with respect to λ and ξ.

Combining (4.2) with the easy estimate

U 2 L 2 = 1 |λ| | U, λU | ≤ C|λ| -1 ( U 2 H 1 ξ + F 2 L 2 ),
for some C, we obtain for ℜλ ≥ -η/2 and |λ| sufficiently large, the estimate

U 2 H 1 ξ ≤ C F 2 H 1 ξ
for some uniform C. Incidentally, since L ξ has compact resolvents hence discrete spectrum composed entirely of eigenvalues, the above estimate also implies that such λ do not belong to the spectrum of L ξ . More generally, by adapting the previous computations to higher-order estimates, along the lines of the method expounded in next subsection, one proves the following result required by the analysis of [START_REF] Johnson | Nonlinear stability of viscous roll waves[END_REF][START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF].

Proposition 4.1 (Resolvent bounds). For any positive integer s, there exist positive η, C and R such that if λ ∈ C is such that |λ| ≥ R and ℜ(λ) ≥ -η then, for any [-π/Ξ, π/Ξ), we have

λ / ∈ σ H s per (0,Ξ) (L ξ ) and (λ -L ξ ) -1 H s ξ →H s ξ ≤ C .
This offers a direct replacement for [JZN11, Appendix B] without assuming any condition on the background wave Ū .

Nonlinear damping estimate.

The other place where high-frequency estimates play a role in the arguments of [START_REF] Johnson | Nonlinear stability of viscous roll waves[END_REF][START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF] is in providing a nonlinear slaving bound that shows that high-regularity norms are controlled by low-regularity ones and enables us to close in regularity a nonlinear iteration. With the strategy implemented above we are also able to reproduce this bound without assuming the slope condition (1.3).

To be more specific let us first warn the reader that, because of the complex spatio-temporal dynamics that take place around periodic waves, the appropriate notion of stability is neither the standard one nor the simpler orbital stability but space-modulated stability, as recalled in the next subsection. For this reason, following [JZN11, JNRZ14], instead of directly estimating Ũ -Ū , where Ũ = (τ , ũ) is a solution of (2.1), we need to introduce (V, ψ) such that

(4.3) V (x, t) = Ũ (x -ψ(x, t), t) -Ū (x),
intending to prove that V and the derivatives of ψ remain small provided that they are sufficiently small initially. Mark that even if ψ is initially zero, as assumed in [START_REF] Johnson | Nonlinear stability of viscous roll waves[END_REF], one may not achieve the latter goal while imposing ψ ≡ 0. In other words a modulation in space, encoded by a spacetime dependent phase is in any case needed in the argument. See the detailed discussions in [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF][START_REF] Rodrigues | Asymptotic stability and modulation of periodic wavetrains, general theory & applications to thin film flows[END_REF]. Our new unknowns, which have to be determined together in a nonlinear way, are then V = (τ, u) and ψ, and a specific educated choice, that we shall not detail here, is then needed to obtain concrete equations for those. However, let us at least mention that in constructions of [START_REF] Johnson | Nonlinear stability of viscous roll waves[END_REF][START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF] the phase shift ψ is always slow so that only high regularity control on V remains to be proved. This is what we provide now.

To do so in a precise but concise way, we set

f (τ ) = (2F 2 ) -1 τ -2 , g(τ ) = ντ -2 and h(τ, u) = 1 -τ u 2 . Then Ũ in (4.3) solves (2.1) provided that V = (τ, u) and ψ satisfy (4.4) (1 -ψ x ) V t -L V = -ψ t ( Ū + V ) x + 0 -ψ x h( Ū + U ) + 0 -ψx 1-ψx g(τ + τ )(ū + u) x x + 0 (g(τ + τ ) -g(τ ))u x + (g(τ + τ ) -g(τ ) -g ′ (τ )τ )ū x x - 0 f (τ + τ ) -f (τ ) -f ′ (τ )τ x + 0 h( Ū + U ) -h( Ū ) -dh( Ū )(U )
.

Defining the modified energy

(4.5) E ψ (U ) := (1 -ψ x ) 1 2 φ 1 τ 2 x + 1 2 φ 2 τ 3 u 2 x + φ 3 τ u x ,
repeating the argument of Section 3, absorbing nonlinear terms into the linear ones and separating out ψ terms using Sobolev's embeddings in Gagliardo-Nirenberg form and Young's inequality, we obtain, in analogy to (3.6), that solutions to (4.4) satisfy the nonlinear estimate

d dt E ψ (V ) ≤ -η E ψ (V ) + C V 2 L 2 + (ψ t , ψ x ) 2 H 1 ,
for some positive C and η, provided that we know in advance some sufficiently small upper bound on (V, ψ t , ψ x ) H 1 and thus are allowed to use Lipschitz bounds for f , g and h and their derivatives on a fixed neighborhood of Ū . Differentiating the equations and performing the same estimate on ∂ k x V , with higher-order interpolation inequalities, we obtain likewise when k is a positive integer

(4.6) d dt E ψ (∂ k x V ) ≤ -η E ψ (∂ k x V ) + C V 2 L 2 + (ψ t , ψ x ) 2 H k ,
so long as V H 1 and (ψ x , ψ t ) H k remain sufficiently small. Applying Gronwall's inequality and recalling that

E ψ (∂ k x V ) ∼ ∂ k x V 2
L 2 modulo lower-order terms, with constants uniform with respect to ψ x satisfying constraints above, we obtain the following key estimate showing that higher Sobolev norms V H k are slaved to V L 2 and (ψ t , ψ x ) H k , the final nonlinear estimate needed for the analysis of [START_REF] Johnson | Nonlinear stability of viscous roll waves[END_REF][START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF]. This provides a result analogous to [JNRZ14, Proposition 2.5] and directly replacing [JZN11, Appendix A], without making any use of a pointwise symmetrization hence dropping the slope constraint (1.3). Proposition 4.2 (Nonlinear damping). For any positive integer s there exist positive constants θ, C and ε such that if V and ψ solve (4.4) on [0, T ] for some T > 0 and

sup t∈[0,T ] (V, ψ t , ψ x )(t) H s (R) ≤ ε then, for all 0 ≤ t ≤ T , (4.7) v(t) 2 H s (R) ≤ Ce -θt v(0) 2 H s (R) + C t 0 e -θ(t-s) v(s) 2 L 2 (R) + (ψ t , ψ x )(s) 2 H s (R) ds .
4.3. Asymptotic stability. As discussed with great detail in [JNRZ14, Appendix D], uniform resolvent bounds of Proposition 4.1 and nonlinear slaving estimates of Proposition 4.2 are the only structural conditions needed to apply almost word-by-word the arguments of [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF] to a periodic wave of a given 'parabolic' system. Our foregoing analysis shows that system (1.1) satisfies those around any given wave so that all conclusions of [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF] apply to any spectrally-stable periodic wave of (1.1). In particular, any spectrally-stable roll-wave is also nonlinearly-stable, provided that one uses definitions of stability adapted to periodic waves of parabolic systems, as we now briefly recall.

A given periodic wave solution to (1.1) Ū , of period Ξ, is said to be diffusively spectrally stable provided that the generator L of the linearized evolution and its Bloch symbols L ξ , as defined in Subsection 4.1, satisfy

(D1) σ(L) ⊂ {λ | ℜλ < 0} ∪ {0}.
(D2) There exists θ > 0 such that for all ξ ∈ [-π/Ξ, π/Ξ) we have σ(L ξ ) ⊂ {λ | ℜλ ≤ -θ|ξ| 2 }. (D3) λ = 0 is an eigenvalue of L 0 with generalized eigenspace of dimension 2. (H) With respect to the Floquet exponent ξ, derivatives at 0 of the two spectral curves passing through zero are distinct. From the pioneering work [START_REF] Schneider | Nonlinear diffusive stability of spatially periodic solutions-abstract theorem and higher space dimensions[END_REF][START_REF] Schneider | Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation[END_REF] to the recent [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF], conditions (D1)-(D3) have slowly emerged as essentially sharp spectral stability conditions for periodic waves of dissipative systems. Some form of (H) is also needed but the present form could well be slightly relaxed in a near future, see precise discussion in [Rod13, Chapter 5]. All together, conditions (D1)-(D3) and (H) express that the spectrum of L is as noncritical and nondegenerate insofar as allowed by the presence around Ū of a two-dimensional family of periodic waves.

The spatial complexity of the periodic background Ū precludes any hope for a simple notion of nonlinear stability. Over the years there has arisen the concrete remedy implemented in (4.3), consisting in introducing a space-time dependent phase shift, though with various possible strategies in the prescription of separate-but coupled-equations for V and ψ. One obvious inspiration for introducing a phase in the nonlinear study comes from classical analysis of simpler, asymptoticallyconstant patterns such as fronts, kinks, solitary waves or shock waves, for which the relevant notion of stability-orbital stability-already requires the introduction of a time-dependent phase. As formalized in [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF] the corresponding notion of stability for periodic waves-space-modulated stability-is obtained by measuring proximity of a function u from a function v in a given functional space X with δ X (u, v) = inf

Ψ u • Ψ -v X + ∂ x (Ψ -Id) X .
and not with u -v X . At a given time this allows for a space-dependent phase synchronization provided that the synchronization differs from the identity by a sufficiently slow phase shift. The interested reader is again referred to [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF][START_REF] Rodrigues | Asymptotic stability and modulation of periodic wavetrains, general theory & applications to thin film flows[END_REF] for a detailed discussion of this concept. However, we stress again here that there is no hope for a better notion of stability unless the original system exhibits some nongeneric null conditions, denoted "phase uncoupling" in [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF]. With these definitions in hands, our analysis combined with the arguments of [JNRZ14] yield the following stability result.

Theorem 4.3 (Nonlinear stability). For any integer K, K ≥ 4, a diffusively spectrally stable periodic wave of (1.1) is nonlinearly asymptotically stable from L 1 (R) ∩ H k (R) to H K (R) in a space-modulated sense.

More explicitly, if Ū satisfies (D1)-( D3) and (H) then, for any K ≥ 4, there exist positive ε and C such that any Ũ0 such that δ L 1 ∩H K ( Ũ0 , Ū ) ≤ ε generates a global solution Ũ to (1.1) such that

∀ t ∈ R + , δ H K ( Ũ (•, t), Ū ) ≤ C δ L 1 ∩H K ( Ũ0 , Ū ) and δ H K ( Ũ (•, t), Ū ) t→∞ -→ 0 .
The actual proof provides a much more precise statement including, for instance, a bound of δ L p ( Ũ (•, t), Ū ) by C (1 + t) -1 2 (1-1/p) similar to those for L p -norms of a heat kernel or of self-similar solutions of viscous Burgers' equations. The reader is referred to [JNRZ14, Theorem 1.10] for such a precise statement. Once Theorem 4.3 is proved, the second part of the analysis [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF] may also be applied to (1.1). This yields a very precise description of the large-time asymptotic behavior in terms of a slow modulation in local parameters varying near constant parameters of the original wave, and obeying some averaged system of partial differential equations, as derived to various order of precision in [Whi74, Ser05, NR13, JNRZ14]. For the sake of conciseness, we do not state such a result here but rather refer the reader to [JNRZ14, Theorem 1.12] and accompanying discussions in [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF][START_REF] Rodrigues | Asymptotic stability and modulation of periodic wavetrains, general theory & applications to thin film flows[END_REF].

The shock wave case

As a sample of the potential wider use of the strategy expounded here, we next turn to the connection with viscous shock theory, showing that the same linear damping estimate (3.6) may be obtained by essentially the same argument in the asymptotically-constant, viscous shock wave case, thus recovering the bounds established in [Zum04, Zum07, GMWZ06] by related but slightly different weighted Kawashima-type energy estimates. 4 The equations of isentropic gas dynamics in Lagrangian coordinates, expressed in a comoving frame are (5.1)

∂ t τ -c∂ x τ -∂ x u = 0, ∂ t u -c∂ x u + ∂ x p(τ ) = ν∂ x (τ -1 ∂ x u),
where τ is specific volume, u is velocity, and p is pressure.

Traveling waves (τ, u)(x, t) = (τ , ū)(x) satisfy the profile ODE -c 2 τ -p(τ ) + q = cu ′ /τ, q = constant.

We note as in the periodic case that c = 0, else u, p(τ ) ≡ constant, yielding τ ≡ constant, a trivial solution. Assume that the shock is noncharacteristic, i.e., -p ′ (τ ± ) = c 2 , hence τ ± are nondegenerate equilibria and the shock profile decays exponentially to its endstates as x → ±∞.

The linearized equations are

(5.2) τ t -cτ x -u x = 0, u t -cu x -(ατ ) x = ν(τ -1 u x ) x ,
where α := p ′ (τ ) + ν ūx τ . Define I ατ 2 ν to be a smooth interpolant between ατ 2 ν | x=±∞ such that

(5.3) ατ 2 ν -I ατ 2 ν = O(e -θ|x| )
for some positive θ.

Taking as before E(U ) := 1 2 φ 1 τ 2 x + 1 2 φ 2 τ 3 u 2 x + φ 3 τ u x , we find again (5.4)

d dt E(U (t)) = -c 2 (φ 1 ) x + αφ 3 τ 2 x -ν τ 2 φ 2 u 2 xx + φ 1 -αφ 2 + ν τ 2 φ 3 τ x u xx + O ( u H 2 + τ H 1 )( u H 1 + τ L 2 ) .
Taking c 2 (φ 1 ) x + ατ 2 ν -I ατ 2 ν φ 1 = 0, φ 1 -αφ 2 + ν τ 2 φ 3 = 0, φ 1 (0) > 0, and 0 < φ 2 ≡ constant ≪ 1, we thus have (5.5)

d dt E(U (t)) = - I τ 2 α ν φ 1 -α 2 τ 2 ν φ 2 τ 2 x + ν τ 2 φ 2 u 2 xx + O (( u H 2 + τ H 1 )( u H 1 + τ L 2 )) ≤ -η ′ ( u xx 2 L 2 + |τ x 2 L 2 + C ′ ( u H 2 + τ H 1 )( u H 1 + τ L 2
), for some positive C ′ and η ′ , and thereby the same linear damping estimate as in the periodiccoefficient case:

(5.6)

d dt E(U (t)) ≤ -η E(U (t)) + C U (t) 2 L 2
for some positive η and C.

As in the periodic case, a crucial point is that x ατ 2 ν -I ατ 2 ν , hence φ 1 and 1/φ 1 , remains bounded, so that E(U ) ∼ ( u x 2 L 2 + τ x 2 L 2 ) modulo τ 2 L 2 , a property following in this case by exponential decay, (5.3).

The above may be recognized as exponentially weighted Kawashima-type estimates similar to those used in the study of viscous shock stability in [Zum04, Zum07, GMWZ06], reflecting the growing analogy between the periodic and asymptotically-constant cases. Actually, part of the recent activity of the authors, jointly with others, focused on dynamics around periodic waves, and culminating more or less in [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF], was motivated by the will to put its analysis on a par with classical ones on asymptotically-constant waves. With this respect the present contribution that provides analytical tools necessary to consider large-amplitude periodic waves of hyperbolicparabolic systems should be compared with [START_REF] Mascia | Stability of large-amplitude viscous shock profiles of hyperbolicparabolic systems[END_REF], where, motivated by some clever "transverse" energy estimates of Goodman [START_REF] Goodman | Remarks on the stability of viscous shock waves[END_REF] in the study of small-amplitude stability, the treatment of large-amplitude viscous shock waves was first carried out. In the reverse direction the Conjugation Lemma of [START_REF] Métivier | Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems[END_REF] on asymptotically constant-coefficient coordinate transformations from asymptotically constant-to constant-coefficient systems may be thought as analogous to the classical Floquet Lemma on periodic coordinate transformation of periodic-to constant-coefficient systems of equations.

Discussion

At the linear level, for a general second-order hyperbolic-parabolic principal part U t + AU x = (BU x ) x , ℜB ≥ 0, a Kawashima-type estimate is on an energy combining E(U ) := U x , A 0 U x +

The weights used in[START_REF] Zumbrun | Stability of large-amplitude shock waves of compressible Navier-Stokes equations[END_REF] Zum07,[START_REF] Olivier Guès | Navier-Stokes regularization of multidimensional Euler shocks[END_REF] are effectively φ1 = φ2 ≫ φ3, (φ1)x = -C ατ 2 ν -I ατ 2 ν φ1, C ≫ 1.

April 9, 2015. Research of L.M.R. was partially supported by the ANR project BoND ANR-13-BS01-0009-01. Research of K.Z. was partially supported under NSF grant no. DMS-0300487.

U, KU x , with the lower-order U, A 0 U , where A 0 is symmetric positive definite and K is skew symmetric, chosen, where possible, so that (6.1) ℜ(A 0 B + KA) > 0.

When A and B are constant, as arising from linearization around a constant state, and the original nonlinear system admits a strictly convex entropy, (6.1) may be reduced to a simple-looking condition that is satisfied by most of systems of physical interest; see [START_REF] Kawashima | Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics[END_REF][START_REF] Liu | Large time behavior of solutions for general quasilinear hyperbolicparabolic systems of conservation laws[END_REF]. As a result, for small-amplitude waves, a suitable choice of K may typically be achieved globally with a constant K. However, for large-amplitude shocks, this can be done typically only near x → ±∞ where A is symmetrizable, and one needs to recover coercivity in the near field |x| ≤ C in a different way.

As exemplified here-for the first time in a periodic context, the key to the treatment of largeamplitude waves, is to choose the "symmetrizer" A 0 jointly with the "compensator" K so that one may use a clever choice for A 0 to relax constraints on K and vice versa. In the present case,

for either of the St. Venant or isentropic compressible Navier-Stokes equations. The issue in the latter case is that symmetrizability holds in general only in the limits x → ±∞, in the former that it holds only on average, but in any case not pointwise. However, we have seen that energy estimates can be recovered by modulating classical symmetrizers and compensators with appropriate asymptotically-constant, or periodic exponential weights. Mark that our analysis, while apparently quite robust, leaves widely open the question of determining, for general systems, what kind of notion of symmetrizability on average could lead to similar periodic-coefficient high-frequency damping.


Joël Querci
email: joel.querci@univ-amu.fr

Sébastien Oliveau
email: sebastien.oliveau@univ-amu.fr

Session: 07-02 -The growing number and size of cities: causes and consequences" "Growing Indian cities and towns throughout the 20 th century"

There were 35 million plus cities in India, in 2001, and 46 in 2011. Since a few decades, we are attending to a relative take-off of the Indian urbanization rate (16.5% in 1951 and 32.1% in 2011). Indeed, the Indian urban population, in 2011, is more numerous than the population of the USA in 2012. Within the story of the urban India, the historical urban framework had to face two disturbances. The arrival and the departure of the Europeans reorganized the urban framework. The entry of India in the globalization process made an evolution in the urban development logics and forced the country to open, partially, its economy. We observed, since the second half of the 20 th century, a quick growth of the number of cities and towns. Assess the causes and consequences of that growing number and size of cities appears essential for the understanding of the future Indian urban society. To address this issue, we will focus on the evolution of the number of towns before analyzing the evolution structure of the urban system. At last, we will consider the link between speed of demographic growth and systemic feature of the Indian urban system.

Throughout the 20 th century, the evolution of the number of towns:

In India, a town has a minimum population of 5,000, at least 75 per cent of the male main working population engaged in non-agricultural pursuits and a density of population of at least 400 persons per sq. km. (Census of India, 2011). In addition, census office distinguishes six classes of towns and cities: Class VI, less than 5000 inhabitants; Class V, between 5000 and 9999 inhabitants; Class IV : entre 10000 et 19999 inhabitants ; Class III, between 20000 and 49999 inhabitants ; Class II, between 50000 and 99999 inhabitants and Class I, 100000 inhabitants and above (Census of India, 2011). This typology allows us to assess the evolution of the number of cities and the evolution of the proportion for each class of towns and cities. The figure 1 presents you the evolution of the urban population for each class of cities (at left) and the evolution of the proportion of each of those classes (at right). 2

Considering the evolution of the Indian urban population since 1901, we can observe the population of the Class I concentrate the main part of the Indian urban population. Nonetheless, we have to highlight the loss of speed of that class since a couple of decades. This situation can be explained by the wish to develop in priority the small and medium cities since the 1970s. The expected effect is to develop those cities by filtering the migrants to the biggest cities. Moreover, the second central element is the quick growth of the urban population since the 1950s. Indeed, India accessed to its independence in 1947 and built up in federal republic in 1950. This national situation resulted in the internal migration of million peoples and the concentration of poor people in cities (Ramachandran R., 1989). Considering the evolution of the proportion of towns of each class since 1901, the outcomes led us to highlight a stronger growth for the middle cities (classes 2 and 3). This situation reflects the wish to mainly develop the middle cities since the 1970s. These analyses led us to consider that the evolution of the urban development logics transformed the structure of the Indian urban system.

The evolving structure of the Indian urban system:

By their politic of urban development, the British reorganized durably the former urban framework. Since their departure, the Indian urban system has modified its urban development logics. Figure 2 presents the Indian cities and towns in place in 1951 (at left) and in 2001 (at right). We represent the annual demographic growth rate (in percentage) for each year and for each city. The intersection of these two variables is crucial for the understanding of the evolution of the urban system. Indeed, this allows visualizing the major population centres by relating them to the annual demographic growth rate. This intersection gives us information about the speed of growth, in terms of urban population, for each city and for a period of ten years. This situation is completely reversed fifty years later. Indeed, in 2001, the situation led us to assume of a progressive resorption of the regional situations of urban primacy identified by the study of the urban hierarchy at the regional level. This situation means significant changes for the urban population: by developing the second and third orders cities, regional framework rebalanced its structure while at the national scale the situation seems to be balanced. Geographers identified, in 2007, the Indian urban system as an ancient system which undergone a disruption (Bretagnolle A., 2009). This change of urban system logics appears more precisely when we look at the Zipf's rank-size distribution of Indian towns and cities on the 20 th century (Figure 3). The evolution of the shape of the urban hierarchy between 1901 and 2001 allows us to assume the system switched from an urban system with a centralized territorial management and a planned economical system to an urban system with a regionalized territorial management and a liberal economical system (Moriconi-Ebrard F., 1993). This conclusion led us to assume the existence of a spatial resilience phenomenon.

Systemic feature and speed of growth:

Spatial resilience of the urban systems corresponds to the capacity capacity of a shocked urban system to recover its structural balance state. Due to the complexity of the urban system (which is a complex system) it is not possible to measure it yet. However, we can demonstrate the presence of such a phenomenon. Demographically, we can assume the existence of that resilience phenomenon analyzing the evolution of the annual demographic growth rate. In our case, this method allows us to prove that the Indian urban system tends to a relatively lapping along the century by the resorption of the regional urban primacy situations. The rest of the work will consist in the measure of that resilience phenomenon using exploratory statistics in order to create an index of spatial resilience.

References:

Figure 1 : 3

 13 Figure 1 : The evolution of each class of towns

Figure 2 :

 2 Figure 2: Indian cities and towns in 1951 and 2001.

  BRETAGNOLLE A., PUMAIN D., VACCHIANI-MARCUZZO C., (2009), LANE D., PUMAIN D., VAN DER LEEUW S., WEST G. (Ed.) "The organisation of urban systems", in Complexity perspective in innovation and social change, pp. 197-220. CENSUS OF INDIA, 2011, Provisional Population Totals Paper 2, Volume 2, online at the following URL: http://www.censusindia.gov.in MORICONI-EBRARD F. (1993), L'urbanisation du monde depuis 1950, Paris, Collection Villes, éd. Anthropos, Economica, 372p. RAMACHANDRAN R. (1989), Urbanization and urban systems in India, Oxford India Paperbacks, 364 p.

Figure 3 :

 3 Figure 3: Evolution of the Indian urban hierarchy across the 20th century.


Svetlana M Ilina

Nicole Baran

Danielle Slomberg

Nicole Sani-Kast

Jérôme Labille

Martin Scheringer

Patrick Ollivier

The role of pesticides in aggregation ofTiO 2 nanoparticles in aquatic environments

Keywords: TiO 2 nanoparticles, natural colloids, pesticides, anthropogenic organic molecules, aggregation

ABSTRACT

The fate and behavior of engineered nanoparticles (NPs) released in aquatic environments will be influenced by the water chemistry, as well as the pesticide load due to the potential for NP interaction with anthropogenic organic molecules (AOMs). As such, surface charge and aggregation of pure hydrophilic 100 % rutile and pure hydrophilic 100 % anatase titanium dioxide nanoparticles (TiO 2 NPs, 5-30 nm)were evaluated in a modeled water solutionin the presence of three common AOMs,glyphosate, aminomethylphosphonic acid (AMPA), and 2.4-D. Thesurfacecharge and size distribution were assessed over time as a function of various factors including surface chemistry of the NPs andAOMs, presence of mono-and bi-valent cations, pH, and ionic strength of the aqueous solution.Thepresence of AOMs(5µg/L) affectedTiO 2 NP(5 mg/L) homoaggregationin solutions (IS=10 -3 M -10 -2 M) with pH values below the NP point of zero charge (PZC) for the anatase NPs (pH=6.5)and with pH values above the NP PZC for the rutile NPs (pH=4.5). No changes in NP aggregationwere observed in very low(IS=10 -4 M) or very high(IS= 10 -1 M) ionic strength solutions. Passing through the PZCresulted inirreversibleaggregation of the NPs,even in the presence of AOMs.The presence of the pesticides also caused a significant modification of the NP surface charge (zeta potential) over a large range of salt concentrations (IS=10 -4 M -10 - 1 M). Compared to mono-valent cations, bi-valent cations (Ca 2+ ) favored NP aggregation and an increase in zeta potential. Finally, these results demonstrated that,among the studiedAOMs,glyphosate (with 4pKa-s from 0.8 to 11)affects NP aggregation/stabilization in a widerrange of physicochemical conditions.Overall, these results will aid in the evaluation of potential environmental risks posed by engineered NPs in the aquatic environments exposed to pesticide load.