Ḥassāniyya Arabic 1 . G E N E R A L I N F O R M A T I O N

Ḥassāniyya (or klām əl-Bīđ̣ ān 'language of the Whites') is the mother tongue of the Arabicspeaking population of the western Sahara, especially the Moors (Bīđ̣ ān) of Mauretania and the former Spanish Sahara (from the Sagya el-Ḥamra and the Rio del Oro). It is difficult to draw the precise geographical limits of this dialect, but its approximate borders are Goulimine in the north, Tindouf in the northeast, Tombouctou in the southeast and the Senegal River in the south. The percentage of Ḥassāniyya speakers is highest in the central regions. There are about 3 million speakers, around 2 million living in Mauretania, out of a total estimated population of 2,9 million. By the middle of the 20th century, most speakers still had a Bedouin lifestyle.

The origin of the Ḥassāniyya is linked (as indicated by its name) to the arrival of the Banū Ḥassān, a branch of the Maˁqil Arabs -who were themselves linked to the movements of territorial expansion of the Banū Hilāl and the Banū Sulaym. Ḥassāniyya is a Bedouin dialect that is part of the western dialect group (Maghrebi) and has developed in a Berberspeaking environment. Despite the influence of the substrate and because of its Bedouin nature, it has more in common with eastern Arabic dialects, most of which (though not all) are Bedouin, than with Maghrebi dialects like Moroccan and Algerian.

Ḥassāniyya is rarely used as a lingua franca, even if certain Black-African Mauretanians are more or less proficient in it. It is not used as a means of written communication; Literary Arabic (Classical or Standard) or a foreign language, in particular French, are used for writing. An important oral literature exists, most of it poetry, but the conditions that gave rise to this literature are rapidly changing (-> Mauretania).

Ḥassāniyya has been studied extensively, but no recent manuals exist, except in xeroxed form. A reference grammar was published by Cohen (1963). Although its subtitle is 'Dialect of the Gabla', it may be regarded as typical of the dialect variety that is spoken in Mauretania as well as in the former Spanish Sahara. An overview of the grammar is given in the introduction to Taine-Cheikh's Ḥassāniyya/French dictionary (1988:I-CIII). For grammatical details see Taine-Cheikh's articles, which appeared in particular in Materiaux Arabes et Sudarabiques.

Ḥassāniyya shows an exceptional unity, with a few exceptions. Genuine Ḥassāniyya speakers, as well as unilingual or, more often, bilingual people speaking a variety that differs from the standard dialect, can be found at the Moroccan, Malian, and probably Algerian borders. The dialects of these speakers are so different that they are virtually incomprehensible to the uninformed Ḥassāniyya speaker (Taine-Cheikh 1997; [START_REF] Heath | Jewish and Muslim dialects of Moroccan Arabic[END_REF]Heath , 2004)). Nowadays, this variation is largely individual, but formerly it was often linked to the history of certain tribal groups, such as the Tekna of Morocco or the Brābīš and the Kunta of Mali.

In the 1970s, under the influence of Arab nationalist movements and in the context of forced settlement, intellectuals and their students have developed a form of Arabic that was a mix of dialect and Modern Standard Arabic, used in political discussions [START_REF] Taine-Cheikh | L'arabe médian parlé par les arabophones de Mauritanie: Etude morphosyntaxique[END_REF]. This 'middle' Arabic has spread gradually, introducing many new words in the traditionally very rural dialectal lexicon.

. L I N G U I S T I C D E S C R I P T I O N

Ḥassāniyya is a relatively conservative language (at least when compared to other Maghrebi dialects) but it has also developed several innovations, in particular certain morphosyntactic patterns (Taine-Cheikh 1991). The Berber substrate does not seem to have had a large influence on the structure of Arabic. Its presence may be perceived in the lexicon, but since most loanwords receive a special treatment, their influence is usually limited. Where common forms exist between Ḥassāniyya and Mauretanian Berber (Zenaga), it is often hard to attribute the source to either language. In several cases, a parallel development may have taken place, facilitated by the remote genealogical relation between Arabic and Berber. This development may have been furthered by the progressive disappearance of Zenaga and a process of osmosis between Arabic-and Berber-speaking groups.

- + t ṭ t y k q () continuant voiced - + v (ṿ) ḏ đ̣ z ẓ ž ġ ' continuant voiceless - + Ú s ṣ š x ḥ h nasal - + m ṃ n (ṇ ) n y lateral - + l ḷ vibrant - + r ṛ semivowel w y
/ġ/ is attested only among certain speakers (mainly in the west, southwest and northwest) when it is not geminated. /ġ/ and /q/ have merged, being realized as [q] by the other Ḥassāniyya speakers (centre, east and northeast Mauretania, Mali and Algeria) and, by all, in all case of gemination (/ġġ/ realised [qq] as in Zenaga).

The phonological system is rich. Ḥassāniyya tends to phonemicize the opposition emphatic/non-emphatic (especially for the vibrants, /ṛ/ being more frequent than /r/) and to preserve the pronunciation of some of the borrowed lexemes, hence the presence of phonemes from Classical Arabic (/ḍ/, /q/, or even //), Zenaga (/ẓ/, /d y /, /t y /, /n y /) and even from Black African languages (/q/ and the palatalized consonants).

Yet, certain phonemes remain marginal, especially in loanwords and the velarized ḅ, ṃ, ṇ , ṿ). It is often difficult to find minimal pairs, except for l ~ ḷ, g~ q and above all r~ṛ: dār/idīr 'to put' vs. dāṛ/idōṛ 'to want'; gās 'to go toward' vs. qās 'measure'; gām 'to get up' vs. qām 'to prepare the tea'; ḍall 'to err [in religion]' vs. đ̣ all 'to spend the day'; zanga 'to skirt around while going up' vs. ẓanga 'to make someone pay a tribute'; walla 'to come back' vs. waḷḷa 'or'; bəṛga 'shacks' vs. ḅəṛga 'bad quality tea'; tamātāya '(a) gum tree' vs. taṃātāya '(a) tomato'.

Even though the status of some phonemes is problematic (especially in the case of ṇ and ṿ), this does not challenge the existence of emphasis (for an opposite position see Zavadovskij 1981:26-27).

Historical remarks on the inventory

Ḥassāniyya is characterized by the realization of qāf as [g], the maintaining of the interdentals (/đ̣ / being the reflex of most words with ḍād in Classical Arabic) and the disappearance of the hamza (often compensated, at the end of the syllable, by lengthening of the preceding vowel).

2.1.1.3 Phonetic realization /j/ is realized as a palato-alveolar fricative [Z]. The labial spirant is realized preferentially as a voiced consonant [v], except in contact with a voiceless consonant or when it is geminate. This realization is particular to this dialect (with the exception of the Ḥassāniyya of the Mali).

Distribution

The emphasis of the vibrant varies sometimes according to the context. Emphasis may be lost or absent in the presence of /y/, /ī/ or even /ə/: 'ašṛa 'ten', ˁəšrīn 'twenty', ˁāšər 'to count by tens'. More generally, the assimilation of sonority and emphasis -in particular for /s/~/ṣ/, /z/~/ẓ/ -is frequent among the consonants, whether in contact or not: 'rīs 'bride groom', but 'ṛūṣ 'bride'. Extended to all the forms of the same root, this helps to explain certain differences with Classical Arabic (sometimes common to other dialects), e.g. t-f-l: dvəl 'to spit' (but in the east and in Mali: tfəl), q-t-l: ktəl 'to kill', ṣ-ġ-r: sġayyər 'small', ṣ--y: ẓwä 'to twitter'. There are also some conditioned alterations between sibilant and palatoalveolar fricatives belonging to the same root (assimilation with loss of the palato-alveolar fricative), thus n-s-ž: näzz 'to weave', ž-z-z: zäzz 'to shear'.

2.1.1.5 Sociolinguistic variables Some tendencies appear among the least educated groups, such as the marginalization of /ḍ/; in the southwest, the merger of /q/ and /ġ/ and, only in a limited region, the tendency to emphasize t in contact: [ṭṛa:b] 'earth'.

Vowels

Consonant clusters

The general rule for consonant clusters is to introduce epenthetic vowels after elision of short vowels in an open syllable, malḥafa > /malḥfa/ 'veil (of the women)' [realized [mael ə ḥfae]. In monosyllabic nouns, metathesis is regular, except in loans from Classical Arabic: [ṭfəl] 'boy ' (but [ṭəvl-u] 'his boy').

Stress

Stress is on the third mora from the end of the word, e.g., on the first syllable in mäktäb 'desk', on the second in mäktūb 'written'. It is strongly marked only on long vowels (accent of length rather than intensity). A lot of grammatical morphemes are clitics.

Morphology

Pronouns

The gender opposition is never marked in the 1st person. 3) fem.

Personal independent pronouns (Table

-ak -ək

-k -kum 3rd masc. -u (-ū-) -h -hum fem.
-hä (-hā-) 2.2.1.3 Demonstratives (Table 5) iii. Adverbs of time : đ̣ aṛk, đ̣ aṛkāti(yyä) 'now', l-yäwm 'today', ġdä 'tomorrow', əṣ-ṣəbḥ 'tomorrow morning', yāməs 'yesterday', əl-bārəḥ 'yesterday night', əl-läylä 'tonight'. iv. Adverbs of quantity: yāsər 'a lot', ḥattä 'very', šwäyy 'little/few'.

Particles

The l of the definite articles əl-assimilates to all 'sunletters' and to ž. There is no indefinite article and no particle of the genitive. The verbal negative form is mā in assertive sentences (without a second element), lā with the imperative. In a nominal sentence, the negative form is combined with the suffix pronouns (mā-ni, mā-n-ak, etc.). i.

Prepositions: və (vī-+ pronoun) 'in', 'lä (əˁlī-) 'on', bə (bī-) 'with', mən 'de', 'and 'by', šäwṛ 'toward', gäddām 'in front of', uṛa 'behind, after', sābəg 'before', taḥt 'under', väwg 'above'. ii.

Subordinating conjunctions: ˁan, änn 'that', mnäyn 'when', bī(h)əlli 'because', äyyāk, bbāš 'for', ilā, iḏa 'if' (condition), (yä)kān 'whether' (indirect interrogation). iii. Coordinating conjunctions : wə / u 'and', waḷḷa 'or', (ya)ġäyr/(ya)qäyr 'but'.

Nouns

The singular feminine form of nouns and adjectives ends in -a(t) with a few exceptions such as ˁanz 'goat', xādəm 'woman slave', ˁayn 'eye', dāṛ 'house', and xandūd 'good milker', ḥāməl 'pregnant' (but ḥāmlä '[who] wears').

Apart from the broken plurals there is an external plural: masculine -īn and feminine -āt; there are some -> pseudo-duals such as uḏnäyn 'ears, two ears'.

Numerous Berber loanwords have special affixes: prefixes in a(a)-/i(i)-for masculine, ta(a)-/ti(i)-for feminine nouns; suffixes -t for feminine singular and -ən for plural nouns.

The pattern C1aC2C2āC3 (nouns of habit, profession) is very frequent: käḏḏāb 'liar'.

Adjectives of color and defect: masc. sg. aCCaC, aḥmaṛ 'red', fem. sg. CaCCa, ḥamṛa, comm. pl. CəCC, ḥəmṛ.

aCCaC is also the pattern of the comparative form (invariable): akbaṛ 'taller', aḥmaṛ 'more red'.

The diminutive formation is very productive and very differentiated for nouns and adjectives: CCayC kläyb (< kälb 'dog'), CCayyəC ktäyyəb (< ktāb 'book'), CCayCəC ˁgäyrəb (<ˁagrəb 'scorpion'), CCayCīC bzäyzīl (< bäzzūl 'udder'), aCayCəC aḥaymər (< aḥmaṛ 'red'). The stem vowels are /a/ or /ə/; there are 3 subclasses, the most frequent being that of verbs with a harmony between the two vowels: type a: ktəb/yəktəb 'to write' (often root without back consonant); type b: vtaḥ/yavtaḥ 'to open'; type c (mixed): ṛgaṣ/yərgəṣ 'to dance'.

Numerals

The diminutive form aC1ayC2aC3/yaC1ayC2aC3: äkäytäb/yäkäytäb 'to write with a bad handwriting' is rare.

Pattern I : 4-radicals

This type is unique, with two stem vowels /a/: gaṛmaṣ/igaṛmaṣ 'to pinch'. Particular cases are: (R2=R4) baxbax 'to burst out laughing', (R3=R4) baḥšäš 'to strangle'.

Derived patterns (Table 6)

The derived forms are numerous and productive. In many cases, there is a correlation between active and reflexive (forms with/without t-or -t-) and a systematic relation between active and passive (forms with/without n-or u-). The internal passive has disappeared in Ḥassāniyya (as in most dialects), but a new system of formal oppositions has developed to denote the distinction between middle/reflexive/reflexive-passive, on the one hand, and a real passive form, on the other. Only form 'XI' (expressing a change of state) is isolated in this system.

The stem vowel of the derived forms is always that of the perfect of Form I. Form VIII (rare as middle, əštġal 'to work') is used as passive of Form I if the first radical is l, m, n, r, ṛ, w or an original hamza: (ə)rtdəm 'to be buried', (ə)ltˁan 'to be cursed'. Form VII is the regular passive of the theme I: (ə)nktəb 'to be written', ənvtaḥ 'to be opened'. Form II is very frequent as causative-factitive and iterative of Form I or denominative: gaṭṭaˁ 'to have it cut; to cut in small pieces', baṛṛag 'to make it shine'. Form V is frequent as middle or reflexive of Form II: tgaṭṭaˁ/yətgaṭṭaˁ 'to cut itself in small pieces'; u-II: ugaṭṭaˁ 'to be cut in small pieces; to be made to go across'.

Form III is quite frequent as extensive of Form I or causative-factitive of Form VI: vārəg 'to separate one from the other'. Form VI is frequent as reciprocal, middle or reflexive of From III: tvārəg 'to separate ourselves from one another'; u-III: uvārəg 'to be separated from one another'.

Form 'IV' has some causatives-factitives of form X): saˁṛab 'to arabize', saḥmaṛ 'to get it reddish'. Form X is quite frequent as reflexive, middle, and inchoative: staˁṛab 'to arabize itself', staḥmaṛ 'to become reddish'.

Form 'XI' is rare: gṣāṛ 'to become short'.

Examples of derived forms of quadriradical verbs are baˁṛaṣ 'to fluster', tbaˁṛaṣ 'to fluster oneself', ubaˁṛaṣ 'to be flustered'.

2.2.6.2 Inflection of aspects and moods 2.2.6.2.1 Perfect (Table 7) All derived verbs have a participial form in m-except VII and 'XI'. The participles form their plural form with the suffixes -īn and -āt. The suffix of the feminine (*-at) is in -a, except before a direct object clitic pronoun: hiyyä šārbä 'she drank', hiyyä šār ə bt-u 'she drank it'. Elatives (invariable) exist for all participles, with an identical form for the forms with/without t (Taine-Cheikh 1984:284-290): I (vāhəm) äfhäm mən 'understanding better than'; VIII (məštmaṛ) äšmaṛ mən 'demonstrating more courage than'; II/V (mˁalläm, mətˁalläm) äˁalläm mən 'teaching/studying better than'; III/VI (mvārəg, mətvārəg) ävārəg mən 'separating (themselves) from one another better than'; 'IV'/X (msaˁṛab, məstaˁṛab) äsaˁṛab mən 'arabizing (oneself) more than'; quadriradicals (I and with t-) (mbaˁṛaṣ, mətbaˁṛaṣ) abaˁṛaṣ mən 'flustering (oneself) more than'.

Verbal nouns

With the exception of Form 'XI', verbal nouns usually exist for all verbs with a non-passive meaning. The forms vary for I (long or short vowel): vähm 'understanding', ḥsāb 'act of counting', ġrīg 'act of sinking', rsūl 'act of sending'. One form prevails for all other cases, often common to verbs with/without t: II/V təC1C2āC3, təbṛāg 'act of making something shine'; III/VI tC1āC2īC3, tvārīg 'mutual separation; act of separating from one another'; 'IV'/X staC1C2īC3, staˁrīb 'arabization, act of arabizing oneself'; quadriradicals (I and with t-) tC1aC2C3īC4, tbaḥšīš 'strangling, the fact of strangling (oneself)'.

The instance noun is generally in -a. It may be accompanied by a pattern change for I (räslä '(a) sending') and regularly so for II/V: təC1C2īC3a, təbṛīgä 'act of making something shine once'. The derived verbs are generally well attested. i. Geminated verbs: Derived forms include Forms III and VI: sātt/isātt 'to put per sixes'; tmāss mˁa 'to adjoin something'; X: with a joint form stäxaff 'not to take something seriously' or disjunctive stägläl 'regard as rare'. ii.

Weak verbs

I alif: In integrated borrowings from Classical Arabic, the first radical of this verbs is represented by ā : āḏən əl 'to authorize'; X: stāxaṛ 'to move back'. iii.

I w/y: Form VIII has a passive meaning: ūtzən yūtzən 'to be weighed'. iv.

II w/y: The alternation ā~ə attested in the perfect of Form I likewise occurs in the derived forms VII, VIII and IX; VIII: ḥtāl 'he was crafty', ḥtəlt 'I was crafty'; Form X: with ā, stävād 'to take advantage of'; with w or y, stälyän 'to become suppler'. v.

III w/y: The vowel of the 3rd person sg. is always /ā/ in the perfect. In the imperfect /ī/ is used in forms II, III, 'IV' and single quadriradicals verbs; /ā/ is used in forms V, VI, VII, VIII, X, quadriradicals verbs with t-and all the u-forms. vi.

Irregular verbs: There are two verbs with 'mixed' conjugation (perfect of II w, imperfect of I w) kāl yäwkäl 'to eat', participles wākəl and mäwkūl, and xāđ̣ /yäwxađ̣ 'to get out'. žä/iži 'to go', participle žāy.

Syntax

The syntax of the dialect shows only a few specificities when compared with the other Bedouin dialects of the Maghreb.

Noun phrase

The main characteristic of the noun phrase is the preservation of the synthetic construction.

Expression of definiteness and indefiniteness

The presence/absence of the definite article əl marks definiteness, except in the construct state and with certain masculine nouns borrowed from Berber (generally beginning with a(a)-or i(i)-): kälb '(a) dog' ~ əl-kälb 'the dog', ävūk '(a ~ the) veal'.

The demonstrative generally precedes the noun, which is always determined: ḏīk-əddāṛ 'this house'. Sometimes, it follows the noun, especially with proper nouns: ṭəvl aḥmäd ḏāk 'this son of Ahmad', 'the son of this Ahmad'. There is no indefinite article. The partitive is expressed with mən 'of': wāḥəd mən lə-ktūb 'one of the books', ktāb mən lə-ktūb 'any of the books'.

Construct state

Possession is expressed by the construct state: ktāb əṭ-ṭfəl 'the book of the boy', ktāb-u 'his book'. There is no genitive particle in Ḥassāniyya, except, infrequently, in Morocco: dyāl (Taine-cheikh 1999:98-99), ntāˁ (Heath 2002:7).

Numeral phrase

If the counted noun is indetermined, the numerals (from 3 upward) are always constructed as nouns in annexion: ˁašrīn žmäl 'twenty camels'. The form used is the one of the construct state: from 3 to 10 short and/or contracted (axməs äklāb 'five dogs''; with a -t suffix before some masculine forms with an original hamza, axməs-t äyyām 'five days'); from 11 to 19 long with -əṛ (axməsṭaˁšəṛ ktāb 'fifteen books').

If the counted noun is determined, the numeral is constructed as an adjective (invariable in gender from 2 upward): l-äklāb l-äṯnäyn 'the two dogs', ktūb aḥmäd əl-ˁašrīn 'the twenty books of Ahmed', žmāl-u l-miyyä 'his hundred camels'. The form used is the one of the absolute state: from 3 to 10 long in -a (l-äklāb l-xamsä 'the five dogs'); from 11 to 19 short without -əṛ (lə-ktūb l-axməsṭaˁš 'the fifteen books').

Adjectives phrases

The order is noun+adjective. Adjectives agree in gender and number with the noun they determine: ṛāžəl msäggäm '(a) fair man', ṛažžālä msäggmīn '(some) fair men', mṛa msäggmä '(a) fair woman', 'läy(y)āt msäggmāt '(some) fair women'. They are preceded by the definite article when the noun is determined, ən-nāgä l-bäyđ̣ a 'the white female camel', nyāg aḥmäd əl-bīđ̣ 'the white female camels of Ahmad', or when it is highly referential, maṛyäm əššäybāniyyä 'Maryem, the old woman'.

Elative constructions

Followed by mən (introducing the second term of the comparison), the elative expresses the comparative: äkbaṛ mən xū-h 'taller than his brother'. Followed by a determined noun (or pronoun), it expresses the relative superlative: äkbaṛ-hum 'the taller among them', äkbaṛ əṭṭavilāt 'the taller of the girls'. When it is definite, it expresses the absolute superlative: läkbaṛ 'the tallest', lə-mṛa l-äkbaṛ 'the tallest woman'.

Relative clauses

The relative pronoun does not appear with an undetermined antecedent: mšä mˁa ḥadd mā gaṭṭ šəfnā-h 'he is gone with someone we have never seen', but mšä mˁa əṛ-ṛāžəl lli vətnä šəfnä 'he is gone with the man we had already seen'. Note the absence of the referential pronoun in relative clauses with lli.

Verbal phrase

The direct object precedes the indirect object, which is introduced by əl: əˁṭa žmäl l-xāl-u 'he gave a camel to his maternal uncle', əˁṭā-h l-u 'he gave it to him'. However, the indirect object is expressed without əl when it is the only suffix: əˁṭā-h žmäl 'he gave him a camel'.

Verbal aspect: Time and tense

Innovations are very limited. There is no indicative prefix.

Future intent prefixes

The predicted future is expressed with the invariable particle lāhi (+ imperfect) which corresponds etymologically to the participle of lhä 'to keep oneself busy doing something': lāhi ngīs nwākšōṭ əš-šhaṛ əd-dāxəl (in-šā-aḷḷāh!) 'I'll go to Nouakchott next (God willing!)'. Combined with past modality (perfect of kān 'to be'), lāhi express the future in the past, kənt lāhi ngīs nwākšōṭ mnäyn ˁədt mäwžūˁ 'I was about to go to when I fall ill'. The imperfect of dāṛ (verb of desire) is also used, especially for the intended future: ndōṛ nəmši 'I am going (willing) to leave'.

Use of active participle

In its predicative use, the active participle has the meaning of a concomitant action. Depending on the verb, it expresses a concomitant action in the unaccomplished [= progressive present or past] (huwwä ṭāləˁ əl-kədyä 'he is going up the mountain', hiyyä kānət ṭālˁa əl-kədyä 'she was going up the mountain') or in the accomplished [= resultative perfect] (huwwä mətġaddi 'he had lunch' [=he is not hungry anymore]). The participle is sometimes used for an imminent action, as if the agent was already engaged in the action: ānä māši (ṣ-ṣəbḥ) 'I am going (tomorrow)'.

Negation

The usual negative form is mā, but lā is used in several cases: for prohibitions, with the imperfect (lā təbki! 'don't cry!'); after the coordinator wə/u 'and' (mā žä u lā ktəb 'he didn't come and he didn't write'; for a negative wish, with the perfect, in some expressions (lā qərzu nyāg-hum! 'I wish the milk of your camels doesn't dry up!'); sometimes, as an 'expletive' negation after xāf 'to fear'.

The negation is not discontinuous but the affix pronoun appears regularly in the absence of a conjugated verbal form (non-verbal predicate or presence of lāhi): mān-ak gaṛṛāy 'you are not a teacher', mā-ni xāyəv 'I am not afraid', (aḥmäd) mā-hu lāhi yəbki '(Ahmad) he won't cry'.

Word order SVO, VSO

In the absence of thematization, the order of the verbal sentence is VSO. This is the only possible order if the subject is totally undetermined (Taine-Cheikh 1998). The interrogatives mən 'who?' and əš 'what?' come at the beginning of the sentence.

Existential sentences

The dialect has several 'pseudo-verbs' consisting of a preposition and an affix pronoun, in particular ˁand-(possession, hence 'to have'), l-(attribution and alienable possession), mˁa-'with ', vī-(location). The order is VO (SVO with a thematized 'subject' noun) and the negative form is mā: (ˁayšä) mā ˁand-hä vađ̣ đ̣ a '(Aïsha) she doesn't have money'. The present participle of xləg 'to exist' expresses existence: (mā-zāl) xāləg mbūṛu 'there is (still) bread'.

. L E X I C O N

The Ḥassāniyya lexicon is rich and well-structured around a relatively limited number of high frequency schemes (cf. Taine-Cheikh, Dictionnaire Ḥassāniyya-Français, 1988-...). It seems to have always had a tendency to enrich itself, either through borrowings (from Berber, closely related African languages, Literary Arabic, French, etc.) or through internal development. However the majority of the lexicon, at least 80% of the lexical items and maybe 90% of the roots (if one only takes into account those who correspond to families of names), is still of an Arabic origin. A quite important part of this vocabulary is more or less characteristic of the Maghrebi dialects, especially of the Bedouin type, e.g. ˁatrûs 'goat', bġa 'to desire', gḏəv 'to vomit' or yāməs 'yesterday'.

The Arabic core is very stable and can be found, for its essential parts, across the whole Ḥassāniyya-speaking area. Borrowings, calques, neologisms, semantic shifts and other innovations are often less stable and more localized (for the eastern region, cf. the lexicons of [START_REF] Pierret | Etude du dialecte maure des régions sahariennes et sahéliennes de l'Afrique Occidentale française[END_REF]Heath 2004). The history of those forms, hard to reconstruct, is often more autochthonous than it seems. The Ḥassāniyya dialect has certainly borrowed more to the Berber (especially the Zenaga) than to any other non-Arabic language, but many creations seem to have appeared, if not first in Ḥassāniyya, at least simultaneously in Ḥassāniyya and in Zenaga. Indeed, they are often absent from the Berber and Arabic lexicons.

  B I B L I O G R A P H I CA L R E F E R E N C E SCohen, David. 1963. Le dialecte arabe ḥassānīya de Mauritanie. Paris: Klincksieck. ---. 1970. "Le système des voyelles brèves dans les dialectes maghrébins". Etudes de linguistique sémitique et arabe, 172-178. The Hague and Paris: Mouton.

Table 2

 2 The merger of /i/ and /u/ in closed syllables is characteristic of the nomadic dialects(Cohen 1970). The short phonemes /a/ and /ə/ are realized variably, according to context. In a neutral context, /a/ undergoes 'imāla and is realized more centralized (transcribed ä). Long vowels have variable length: long under the accent, average apart from the accent, and short in final position. Final long vowels are lengthened again before a suffix: šāvu 'they have seen', šāvū-h 'they have seen him/her/it'. When /ā/ is realized as a short vowel, it undergoes 'imāla: žä 'he came', žā-h 'he came to him'.2.1.3 DiphthongsThe four former diphthongs are preserved: /ay/, /aw/, /iy/, and /uw/. However, the realization of /ay/ and /aw/ sometimes tends towards [e:] and [o:].2.1.4 SyllableBecause of the general preference for closed syllables, short vowels in open syllables are rare, apart from loanwords and in final position. They are found, however, in several initial

	. Vowels
	long	short
		opened syllable	closed syllable
	/ī/	/i/	/ə/
	/ū/	/u/
	/ā/	/a/	/a/

syllables in which the short syllable represents a first radical w/y (uvä 'he is over') or plays an important morphological role (aḥmaṛ 'red', ikättäb 'he makes [them] write', udägdäg 'to be broken').

The most frequent syllabic type is CVC and CVV, but syllables with double coda CVCC or double-onset CCVC are frequent. Closed syllables with long vowels (CVVC) are attested, especially in the participles, kātbīn 'writing [pl.]'. Several open syllables have a secondary origin (3rd radical w/y): žäru < žərw 'young dog' (fem. žərwä).

Table 3 .

 3 Personal pronouns Possessive/object suffixes (Table4) The clitic pronoun of the 1st person singular has two different forms, after a verb (šāv-ni 'he saw me') or after a preposition (vī-yä 'in me') or noun (ktāb-i 'my book'). In certain special contexts (e.g., after mā-'not', mən-'who?'), a short form of the independent pronoun is used for the 3rd person singular: masc. -hu, fem. -hi. The possessive pronouns consist of a base that varies according to gender in the singular (masc. līl-fem. līlt-; pl. lwāyl-) and of an affix pronoun: līl-i 'mine'.

		singular	plural
	3rd masc.	huwwä, hūwä	hūmä
	fem.	hiyyä, hīyä	hūmāti
	2nd masc.	(ə)ntä	(ə)ntūmä
	fem.	(ə)nti, (ə)ntiyyä	(ə)ntūmāti
	1st	ānä	(ə)ḥnä, nəḥnä

Table 5

 5

		. Demonstratives
			masc. sg.	fem. sg.	pl.
	proximity or neutral use 'this'	ḏä	ḏi	ḏu
	proximity (hā-) 'this one, this'	hāḏä	hāḏi	hāḏu
	distance (-k) 'that one, that'	ḏāk	ḏīk	ḏūk
	2.2.1.4 Presentatives
					one?'
	(äyy-kum 'which one of you?').
	2.2.2 Adverbs
	i.	Interrogative adverbs: mnäyn et wäyn 'where?', äyntä 'when?', kämm 'how much?', əˁl-
		āš 'why?', škīv 'how?'.
	ii.

The presentatives consist of an independent personal pronoun (in the sg,, short or long form), preceded by a demonstrative or a particle with a verbal origin: ḏähu(wwä) 'here', ḏākhu(wwä) 'there', (a)ṛahu(wwä) 'there he is': ḏīkhi(yyä) maṛyäm 'there is Mariem'.

2.2.1.5 Relative pronoun

The relative pronoun is invariable in gender and in number: lli (sometimes əl) 'who, what'.

2.2.1.6 Interrogative pronouns

Interrogative pronouns include mən 'who?' (mən-hu 'who is it?'); š(ə)-'what?' and its variants: -āš after a preposition; šən-in šən-hu 'what is it?'; äyy (invariable) 'which Adverbs of place: hūn, hūnāti(yyä) 'here

', vämm, vämmāti(yyä)

'there', hō w k, hō w kāti(yyä) 'over there', ilāh 'toward there', l-gäddām 'in front of', ət-taḥt 'under', əlväwg 'above'.

  Cardinals 1 and 2 agree in gender: 1 masc. wāḥəd fem. waḥdä; 2 masc. äṯnäyn fem. ṯäntäyn. The dual is still productive: kälb-äyn 'two dogs'. Certain cardinals have two forms. From 3 to 10, the long form in -a is used in the absolute state.From 11 to 19, the cardinals are used without final -əṛ in the absolute state.The number 100 is miyyä in the absolute state and mīt in the construct state. The other cardinals have an invariable form: 20 ˁəšrīn, 30 ṯlāṯīn, 40 aṛbˁīn, 50 xamsīn, 60 səttīn, 70 säbˁīn, 80 ṯmānyīn, 90 təsˁīn, 200 mītäyn, 1,000 älv.With the exception of äwwäl 'first', the ordinals have the pattern of the participle fāˁəl: ṯāni 'second'.

		absolute state	construct state
	11	aḥdaˁš	aḥdaˁšəṛ
	12	aṯnaˁš	aṯnaˁšəṛ
	13	aṯləṭṭaˁš	aṯləṭṭaˁšəṛ
	14	aṛbaˁṭaˁš	aṛbaˁṭaˁšəṛ
	15	axməsṭaˁš	axməsṭaˁšəṛ
	16	səṭṭaˁš	səṭṭaˁšəṛ
	17	äsbaˁṭaˁš	äsbaˁṭaˁšəṛ
	18	äṯmənṭaˁš	äṯmənṭaˁšəṛ
	19	ätsaˁṭaˁš	ätsaˁṭaˁšəṛ
	2.2.6 Verbs
	2.2.6.1 Patterns/stems
	2.2.6.1.1 Pattern I: 3-radicals
		absolute state	construct state
	3	(ä)ṯlāṯä	äṯlət
	4	aṛbˁa	aṛbaˁ
	5	xamsä	axməs
	6	səttä	sətt
	7	säbˁa	äsbaˁ
	8	(ä)ṯmānyä	äṯmən
	9	təsˁa	ətsaˁ
	10	ˁašṛa	əˁšəṛ

Table 6 .

 6 Derived forms

		active meaning*	reflexive meaning	passive meaning
	3-radicals	I:	VIII:	VII:
		C1C2ə/aC3	(ə)C1tC2ə/aC3	(ə)nC1C2ə/aC3
	doubling R2	II:	VI:
		C1aC2C2aC3	tC1aC2C2aC3	uC1aC2C2aC3
	lengthening V	III:	VI:
	after R1	C1āC2əC3	tC1āC2əC3	uC1āC2əC3
	prefix s-	'IV':	X:
		saC1C2aC3	staC1C2aC3	usaC1C2aC3
	lengthening V	'IX':
	after R2	(ə)C1C2āC3	-	-
	quadriradicals	C1aC2C3aC4	tC1aC2C3aC4	uC1aC2C3aC4

Table 7

 7

	. Perfect verb
		Triradicals:	Triradicals:	Quadriradicals:
		type a	type b and c
	3rd sg. masc. ktəb	vtaḥ	baˁṛaṣ
	3rd sg. fem.	kətbət	vätḥət	baˁ ə ṛṣət
	2nd sg. masc. ktəbt	vtaḥt	baˁṛaṣt
	2nd sg. fem.	kətbi	vätḥi	baˁṛaṣti
	1st sg.	ktəbt	vtaḥt	baˁṛaṣt
	3rd pl.	kətbu	vätḥu	baˁ ə ṛṣu
	2nd pl.	ktəbtu	vtaḥtu	baˁṛaṣtu
	1st pl.	ktəbnä	vtaḥnä	baˁṛaṣnä

Table 8 )

 8 The prefix vowel of the imperfect is always /ə/ for derived verbs beginning with two consonants (V, VI, VII, VIII, X and 'XI'), yətbaˁṛaṣ 'he flusters himself'. The vowel u-of the passive form is constant, yubaˁṛaṣ 'he was flustered'.

	Table 8. Imperfect verb
			Triradicals:	Triradicals:	Quadriradicals
			type a and c	type b
	3rd sg. masc.	yəktəb	yävtaḥ	ibaˁṛaṣ
	3rd sg. fem.	təktəb	tävtaḥ	tbaˁṛaṣ
	2nd sg. masc.	təktəb	tävtaḥ	tbaˁṛaṣ
	2nd sg. fem.	tək ə tbi	täv ə tḥi	tbaˁ ə ṛṣi
	1st sg.		nəktəb	nävtaḥ	nbaˁṛaṣ
	3rd pl.		yək ə tbu	yäv ə tḥu	ibaˁ ə ṛṣu
	2nd pl.		tək ə tbu	täv ə tḥu	tbaˁ ə ṛṣu
	1st pl.		nək ə tbu	näv ə tḥu	nbaˁ ə ṛṣu
	2.2.6.2.3 Imperative
			Triradicals:	Triradicals:	Quadriradicals
			type a and c	type b
	2nd sg. masc.	ktəb	avtaḥ	baˁṛaṣ
	2nd sg. fem.	k ə tbi	äv ə tḥi	baˁ ə ṛṣi
	2nd pl.		k ə tbu	äv ə tḥu	baˁ ə ṛṣu
	2.2.6.3	Participles and elatives
	Table 10. Participles and elatives
			active	reflexive	passive	elative
	-		I	VIII		I and VIII
			C1āC2əC3	məC1tC2ə/aC3	mäC1C2ūC3	aC1C2aC3
	doubling R2	II	VI
			mC1aC2C2aC3	mətC1aC2C2aC3 muC1aC2C2aC3 aC1aC2C2aC3
	lengthening V	III	VI
	after R1		mC1āC2əC3	mətC1āC2əC3	muC1āC2əC3	aC1āC2əC3
	prefix s-		'IV'	X
			msaC1C2aC3	məstaC1C2aC3	musaC1C2aC3	asaC1C2aC3
	quadriradical
	s		mC1aC2aC3aC4 mətC1aC2aC3aC4 muC1aC2aC3aC4 aC1aC2aC3aC4

Table 11

 11

	. Weak verbs
		geminate	I w/y	II w/y	III w/y
	type a	2 cas:	R1=w	(rare)	R3=w/y
		đ̣ all/iđ̣ all 'to	uṣal/yäwṣal	R2=w: xāf/ixāf	(rare)
		while away the	'to arrive'	'to be scared'	nsä/yänsä
		time',		R2=y: bāt ibāt 'to	'to forget'
		tämm/itämm 'to		spend the night'
		go on'
	type b		R1=w: uzən	R2=w: gāl igūl 'to	R2=w/y
		-	yūzən 'to weigh'	tell'	(frequent)
			R1=y: ibəs	R2=y: gās igīs	šrä yəšri 'to buy'
			yībəs 'to dry'	'to head for'
	Type c	R2=R3
		baṭṭ ibəṭṭ 'to beat'
	particularities 1st, 2nd pers.	perfect 3rd pers.	1st, 2nd pers.	3rd pers. pl.
		perfect in -äy-:	f., pl. in w-: waṣlət	perfect in -ə-: gəlt	imperfect
		baṭṭäyt	wəznət / ūznət		type a: yänsāw
					type b: yəšru
	active	bāṭṭ	wāzən	R2=w > /y/:	nāsi (fem. nāsyä)
	participle			gāyəl	šāri (fem. šāryä)
	passive	mäbṭūṭ	mäwzūn	R2=w > /y/:	mänsi(yy)
	participle	(pl. mbaṭṭa)		mägyūl	məšri(yy)


François Bolley
email: bolley@ceremade.dauphine.fr

José Cañizo
email: canizo@mat.uab.es

José Antonio Carrillo
email: carrillo@mat.uab.es

JOS É A. CA ÑIZO ¸ois Bolley

Jos É A Carrillo

Mean-field limit for the stochastic Vicsek model

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

Introduction

The stochastic Vicsek model [START_REF] Vicsek | Novel type of phase transition in a system of self-driven particles[END_REF] arises in the study of collective motion of animals and it is receiving lots of attention due to the appearance of a phase transition [START_REF] Couzin | Collective memory and spatial sorting in animal groups[END_REF][START_REF] Grégoire | Onset of collective and cohesive motion[END_REF]. A continuum version and variants of this model have been proposed in the recent works [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF][START_REF] Degond | Macroscopic limits and phase transition in a system of selfpropelled particles[END_REF]. Our objective is to rigorously derive some continuum partial differential equations analysed in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF] from the stochastic Vicsek particle model. This was carried out for a family of collective behaviour models in [START_REF] Bolley | Stochastic mean-field limit: non-Lipschitz forces & swarming[END_REF] following the method of [START_REF] Sznitman | Topics in propagation of chaos[END_REF]. The present models do not fall into this analysis due to the evolution being defined on a surface as we explain next. In the models considered here, individuals are assumed to move with a fixed cruising speed trying to average their orientations with other individuals in the swarm in the presence of noise. This orientation mechanism is modelled by locally averaging in space their relative velocity to other individuals. More precisely, we are interested in the behaviour of N interacting R 2d -valued processes (X i t , V i t ) t≥0 with 1 ≤ i ≤ N with constant speed |V i t |, say unity. We define them as solutions to the coupled Stratonovich stochastic differential equations

       dX i t = V i t dt, dV i t = √ 2 P (V i t ) • dB i t -P (V i t )   1 N N j=1 K(X i t -X j t )(V i t -V j t )   dt. (1)
Here P (v) is the projection operator on the tangent space at v/|v| to the unit sphere S d-1 in R d , i.e.,

P (v) = I - v ⊗ v |v| 2 .
This stochastic system is considered with independent and commonly distributed initial data

(X i 0 , V i 0 ) ∈ R d × S d-1 with 1 ≤ i ≤ N . The (B i t ) t≥0 denote N independent standard Brownian motions in R d .
The projection operator ensures that V i t keeps constant norm, equal to 1. The second term in the evolution of V i t models the tendency of the particle i to have the same orientation as the other particles, in a way weighted by the interaction kernel K, as in the model proposed by F. Cucker and S. Smale [START_REF] Cucker | Emergent behavior in flocks[END_REF]. Let us observe that P (V i t )V i t = 0, so we can drop the corresponding term when writing [START_REF] Bolley | Stochastic mean-field limit: non-Lipschitz forces & swarming[END_REF] to recover the usual formulations as in [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF].

We will work with stochastic processes defined on R 2d instead of R d × S d-1 . We will check later on that solutions of (1) with initial data in R d ×S d-1 remain there for all times. We have written [START_REF] Bolley | Stochastic mean-field limit: non-Lipschitz forces & swarming[END_REF] in the Stratonovich sense, since the term involving noise corresponds to Brownian motion on the sphere S d-1 as in [START_REF] Hsu | Stochastic analysis on manifolds[END_REF]Section 1.4] and [START_REF] Rogers | Diffusions, Markov processes, and martingales[END_REF]Section V.31].

By symmetry of the initial configuration and of the evolution, all particles have the same distribution. Even though they are initially independent, correlation builds up in time due to the interaction term. Nevertheless, this interaction term is of order 1/N, and thus, it seems reasonable that two of these interacting particles (or a fixed number k of them) become less and less correlated as N gets large (propagation of chaos).

Following [START_REF] Sznitman | Topics in propagation of chaos[END_REF] we shall show that the N interacting processes (X i t , V i t ) t≥0 respectively behave as N → ∞ like the auxiliary processes (

X i t , V i t ) t≥0 , solutions to      dX i t = V i t dt, dV i t = √ 2 P (V i t ) • dB i t -P (V i t )(H * f t )(X i t , V i t ) dt, (X i 0 , V i 0 ) = (X i 0 , V i 0 ), f t = law(X i t , V i t ) (2)
in the Stratonovich sense. Here the Brownian motions (B i t ) t≥0 are those governing the evolution of the (X i t , V i t ) t≥0 and

(H * f )(x, v) = R 2d K(x -x ′ ) (v -v ′ ) f (x ′ , v ′ ) dx ′ dv ′ , x, v ∈ R d .
Note that (2) consists of N equations which can be solved independently of each other. Each of them involves the condition that f t is the distribution of (X i t , V i t ), thus making it nonlinear. The processes (X i t , V i t ) t≥0 with i ≥ 1 are independent since the initial conditions and driving Brownian motions are independent.

We will show that these processes defined on R 2d are identically distributed, take values in R d × S d-1 if initially so, and their common law f t at time t, as a measure on R d × S d-1 , evolves according to

∂ t f t + ω • ∇ x f t = ∆ ω f t + ∇ ω • f t (I -ω ⊗ ω)(H * f t ) , t > 0, x ∈ R d , ω ∈ S d-1 . (3) Now the convolution H * f is over R d × S d-1 : (H * f )(x, ω) = R d ×S d-1 K(x -x ′ ) (ω -ω ′ ) f (x ′ , ω ′ ) dx ′ dω ′ , x ∈ R d , ω ∈ S d-1 .
Moreover, ∇ x stands for the gradient with respect to the position variable x ∈ R d whereas ∇ ω , ∇ ω • and ∆ ω respectively stand for the gradient, divergence and Laplace-Beltrami operators with respect to the velocity variable ω ∈ S d-1 . This equation is proposed in [START_REF] Degond | Macroscopic limits and phase transition in a system of selfpropelled particles[END_REF] as a continuous version of the original Vicsek model [START_REF] Vicsek | Novel type of phase transition in a system of self-driven particles[END_REF], and one of our purposes is to make this derivation rigorous. The asymptotic behavior and the appearance of a phase transition in the space-homogeneous version of (3) (i.e., without the space variable) has been recently studied in [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF]. It is also known as the Doi-Onsager equation, introduced by Doi in [START_REF] Doi | Molecular dynamics and rheological properties of concentrated solutions of rodlike polymers in isotropic and liquid crystalline phases[END_REF] as a model for the non-equilibrium Statistical Mechanics of a suspension of polymers in which their spatial orientation (given by the parameter ω ∈ S d-1 ) is taken into account.

The main result of this paper can be summarized as: Theorem 1. Let f 0 be a probability measure on R d × S d-1 with finite second moment in x ∈ R d and let (X i 0 , V i 0 ) for 1 ≤ i ≤ N be N independent variables with law f 0 . Let also K be a Lipschitz and bounded map on R d . Then, i) There exists a pathwise unique global solution to the SDE system (1) with initial data (X i 0 , V i 0 ) for 1 ≤ i ≤ N ; moreover, the solution is such that all V i t have norm 1. ii) There exists a pathwise unique global solution to the nonlinear SDE (2) with initial datum (X i 0 , V i 0 ); moreover, the solution is such that V i t has norm 1. iii) There exists a unique global weak solution to the nonlinear PDE (3) with initial datum f 0 . Moreover, it is the law of the solution to (2).

Solutions to general SDE's can be built in submanifolds of R d by means of the Brownian motion of the ambient space as in [START_REF] Rogers | Diffusions, Markov processes, and martingales[END_REF]Theorem V.34.86] for instance; then one can interpret the generator in terms of the corresponding Laplace-Beltrami operator. For example, the Brownian motion on a submanifold Σ of R d is the solution to the SDE

dW t = P Σ (W t ) • dB t
on R d and with P Σ (w) being the orthogonal projection of R d onto the tangent space at w to Σ. Here, we give the full construction and derivation of the evolution of the law as it can be done explicitly in the case of the sphere S d-1 . Let us also emphasize that we have only partial diffusion since it is a kinetic model.

We observe that existence of L 2 and classical solutions for the space-homogeneous version of (3) has also been considered in [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF].

As a direct consequence of the classical Sznitman's theory, we get the following mean-field limit result: Theorem 2. With the assumptions of Theorem 1 and for the respective solutions (X i t , V i t ) t≥0 and (X i t , V i t ) t≥0 of (1) and (2), for all T > 0 there exists a constant C such that

E |X i t -X i t | 2 + |V i t -V i t | 2 ≤ C N for all 0 ≤ t ≤ T , N ≥ 1 and 1 ≤ i ≤ N .
This estimate classically ensures quantitative estimates on (see [START_REF] Sznitman | Topics in propagation of chaos[END_REF][START_REF] Bolley | Stochastic mean-field limit: non-Lipschitz forces & swarming[END_REF] for details) i) the convergence in N of the law at time t of any (by symmetry) of the processes (X i t , V i t ) towards f t , ii) the propagation of chaos for the particle system through the convergence of the law at time t of any k particles towards the tensor product f ⊗k t (for k fixed or k = o(N )), iii) the convergence of the empirical measure at time t of the particle system towards f t .

Of course, the same techniques lead to a corresponding mean-field limit result for the spacehomogeneous particle system instead of (1), obtaining the corresponding space-homogeneous PDE.

Proofs

Using the standard Itô-Stratonovich calculus, see [8, p. 99] for instance, equations ( 1) and ( 2) are respectively equivalent to the Itô stochastic differential equations

       dX i t = V i t dt, dV i t = √ 2 P (V i t )dB i t -P (V i t )   1 N N j=1 K(X i t -X j t )(V i t -V j t )   dt -(d -1) V i t |V i t | 2 dt . (4)
and

           dX i t = V i t dt, dV i t = √ 2 P (V i t )dB i t -P (V i t )(H * f t )(X i t , V i t ) dt -(d -1) V i t |V i t | 2 dt, (X i 0 , V i 0 ) = (X i 0 , V i 0 ), f t = law(X i t , V i t ) (5)
which we now consider.

We start with the proof of Theorem 1. We use a regularization of the diffusion and drift coefficients. We let σ 1 be a d × d matrix valued map on R d with bounded derivatives of all orders such that σ 1 (v) = P (v) for all v with |v| ≥ 1/2, and σ 2 and σ 3 be maps on R d , again with bounded derivatives of all orders, such that

σ 2 (v) = v/|v| 2 if |v| ≥ 1/2 and σ 3 (v) = v if |v| ≤ 2.
Existence and uniqueness for the particle system (4). Given such σ 1 , σ 2 , the system of equations

       dX i t = V i t dt, dV i t = √ 2 σ 1 (V i t ) dB i t -σ 1 (V i t )   1 N N j=1 K(X i t -X j t )(V i t -V j t )   dt -(d -1)σ 2 (V i t )dt (6)
starting from (X i 0 , V i 0 ) ∈ R d × S d-1 for 1 ≤ i ≤ N has locally Lipschitz coefficients. Moreover, by the Itô formula and as long as

|V i t | ≥ 1/2, d|V i | 2 = 2 √ 2 V i • P (V i )dB i -2 V i • P (V i )   1 N N j=1 K(X i -X j )(V i -V j )   dt -2 (d -1)dt + 2 d k,l=1 δ kl d B i k - d p=1 V i k V i p |V i | 2 B i p , B i l - d q=1 V i l V i q |V i | 2 B i q = -2 (d -1)dt + 2 d k=1   1 -2 (V i k ) 2 |V i | 2 + d p=1 (V i p ) 2 (V i k ) 2 |V i | 4   dt = 0.
Here we dropped the time dependence, wrote y = (y 1 , . . . y d ) ∈ R d and used the fact that V i • P (V i )y = 0 for all vectors y ∈ R d . Hence |V i t | = 1 up to explosion time. Since moreover dX i t = V i t dt, this ensures that the explosion time is infinite, hence global existence and pathwise uniqueness for [START_REF] Doi | Molecular dynamics and rheological properties of concentrated solutions of rodlike polymers in isotropic and liquid crystalline phases[END_REF]. Now the solution to (6) for given σ 1 , σ 2 is a solution to (4) since all velocities have norm 1, which provides global existence of solutions to [START_REF] Degond | Macroscopic limits and phase transition in a system of selfpropelled particles[END_REF]. If now we consider two solutions to [START_REF] Degond | Macroscopic limits and phase transition in a system of selfpropelled particles[END_REF] for the same initial data and Brownian motions, then they have velocities equal to 1, so that are solutions to (4) for any σ 1 , σ 2 , for which pathwise uniqueness holds: hence they are equal. This proves the first part in Theorem 1.

Existence and uniqueness for the artificial processes [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF]. Let σ 1 , σ 2 , σ 3 be any maps as above and let

H σ 3 [f ](x) = R 2d K(x -y) σ 3 (v -w) f (y, w) dy dw.
Then, given a distribution f 0 on R d × S d-1 with finite second moment in x ∈ R d and (X 0 , V 0 ) with law f 0 , the nonlinear equation

     dX t = V t dt, dV t = √ 2 σ 1 (V t )dB t -σ 1 (V t )(H σ 3 * f t )(X t , V t )dt -(d -1)σ 2 (V t )dt, f t = law(X t , V t ) (7)
has bounded and Lipschitz coefficients on R 2d , so admits a pathwise unique global solution according to [START_REF] Sznitman | Topics in propagation of chaos[END_REF]Theorem 1.1]. Moreover, as long as |V t | ≥ 1/2, then we can repeat the argument above to prove that d|V t | 2 = 0, so that |V t | = 1 for all time. In particular the obtained solution (X t , V t ) t≥0 is a global solution to the genuine nonlinear equation [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF]. Pathwise uniqueness of solutions to (5) can be obtained as for [START_REF] Degond | Macroscopic limits and phase transition in a system of selfpropelled particles[END_REF].

Existence and uniqueness for the PDE (3). Let f 0 be a distribution on R d × S d-1 with finite second moment in x ∈ R d , (X 0 , V 0 ) with law f 0 , and let (X t , V t ) t≥0 be the solution to [START_REF] Degond | Continuum limit of self-driven particles with orientation interaction[END_REF] with initial datum (X 0 , V 0 ). Its law f t , as a measure on R 2d , satisfies

d dt R 2d ϕ df t = R 2d v •∇ x ϕ+ Hess v ϕ : (I -v ⊗ v)+ ∇ v ϕ•(I -v ⊗ v)(H * f t )-(d-1)v •∇ v ϕ df t
for all smooth ϕ on R 2d by the Itô formula; here ∇ v and ∆ v are respectively the gradient and Laplace operators with respect to v ∈ R d , and Hess v ϕ : M is the term by term product of the Hessian with respect to v matrix of ϕ with a matrix M . We have observed that |V t | = 1 a.s., so f t is concentrated on R d × S d-1 . We now define the restriction

F t of f t on R d × S d-1 by R d ×S d-1 Φ dF t = R 2d ϕ df t for all continuous maps Φ on R d × S d-1
, where ϕ is any continuous and bounded map on

R 2d equal to Φ on R d × S d-1 . Let now Φ be a C ∞ c map on R d × S d-1 and ϕ be a C ∞ c map on R 2d such that ϕ(x, v) = Φ(x, v/|v|) for all 1/2 ≤ |v| ≤ 2. Then ϕ is 0-homogeneous in v in the annulus 1/2 ≤ |v| ≤ 2, so that v • ∇ v ϕ = 0 for all (x, v) in the support of f t . In particular d dt R d ×S d-1 Φ dF t = d dt R 2d ϕ df t = R 2d v • ∇ x ϕ + ∆ v ϕ + ∇ v ϕ • (I -v ⊗ v)(H * f t ) df t .
Then the maps v

• ∇ x Φ and v • ∇ x ϕ are equal on R d × S d-1 since Φ and ϕ have the same x-dependence. Moreover, ∇ ω Φ = ∇ v ϕ and ∆ ω Φ = ∆ v ϕ for (x, ω) ∈ R d × S d-1
. This last point can be checked by direct computations. Hence

d dt R d ×S d-1 Φ dF t = R d ×S d-1 (ω • ∇ x Φ + ∆ ω Φ + ∇ ω Φ • (I -ω ⊗ ω)(H * F t )) dF t .
This ensures that F t is a weak solution to [START_REF] Cucker | Emergent behavior in flocks[END_REF].

We now turn to uniqueness of solutions to [START_REF] Cucker | Emergent behavior in flocks[END_REF]. For that purpose we let f 1 and f 2 be two solutions with the same initial datum f 0 , and at each time t we view them as measures on R 2d concentrated on the surface R d × S d-1 . We let (X 1 t , V 1 t ) t≥0 and (X 2 t , V 2 t ) t≥0 be the solutions to [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF] with drift given by H σ 3 * f 1 t and H σ 3 * f 2 t respectively, and common initial datum (X 0 , V 0 ) with law f 0 . Then their respective laws g 1 t and g 2 t , as measures on R 2d , are solutions to the linear PDE

∂ t g i t + v • ∇ x g i t = d k,l=1 ∂ 2 ∂v k ∂v l (σ 1 σ * 1 ) kl g i t + ∇ v • g i t σ 1 (H σ 3 * f i t ) + (d -1)σ 2 .
Since f i t is also a measure solution to this linear PDE on R 2d with bounded and regular coefficients, for which uniqueness classically holds, it follows that g i t = f i t (i = 1, 2). Consequently, the (X i t , V i t ) t≥0 are solutions to the nonlinear SDE [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF], for which we have already proved uniqueness. Hence (X [START_REF] Frouvelle | Dynamics in a kinetic model of oriented particles with phase transition[END_REF]. Hence we may apply the estimates in [START_REF] Sznitman | Topics in propagation of chaos[END_REF]Theorem 1.4] to obtain Theorem 2.

1 t , V 1

Date:

  February 1, 2011.

  ) t≥0 are solutions of the corresponding equations with bounded and Lipschitz diffusion and drifts coefficients as in

	f 1 t (= g 1 t ) = (g 2 t =)f 2 t .	t ) t≥0 and (X	2 t , V	2 t ) t≥0 are equal, and in particular
	Proof of Theorem 2. Since |V i t | = |V (X i i t , V t	i t | = 1 for all i and t, the processes (X i t , V i t ) t≥0 and

Acknowledgments. The authors wish to thank Pierre Degond and Vlad Panferov for drawing their attention to this question and for discussion on it. This note was written while the authors were visiting the Isaac Newton Institute, Cambridge; it is a pleasure for them to thank this institution for its kind hospitality.

The last two authors acknowledge support from the project MTM2008-06349-C03-03 DGI-MCI (Spain), the 2009-SGR-345 from AGAUR-Generalitat de Catalunya and the French-Spanish acciones integradas program FR2009-0019. All authors were partially supported by the ANR-08-BLAN-0333-01 Projet CBDif-Fr.


H Than Quoc

T Pham Huu

T Le Van

J M Cormier

A Khacef
email: ahmed.khacef@univ-orleans.fr

Application of Atmospheric Non Thermal Plasma-Catalysis Hybrid System for Air Pollution Control: Toluene removal

Keywords: Non thermal Plasma, Catalysis, Toluene, oxidation

The combination of heterogeneous catalysts with non-thermal plasma (DBD operated in sinusoidal mode) was investigated for the removal of toluene from air at room temperature. The DBD reactor gives the possibility to introduce the heterogeneous catalyst directly into the discharge zone (in-plasma catalysis, IPC) or to place the catalyst downstream the plasma zone (post-plasma catalysis, PPC). Various catalytic materials based on Ag, Au, Cu, Co, Mn, La, and Nb, supported on Al 2 O 3 and CeO 2 in powder form has been tested to select the catalysts that present the highest activity in the presence of plasma.

Without plasma, the reaction of toluene conversion over perovskite, Al 2 O 3 , and OMS-2 based catalysts are promoted at high temperature (200-400°C). In the absence of catalysts, the toluene removal efficiencies were in the 55-60% range.

When the plasma was combined to catalyst, the toluene removal efficiency strongly increased when the catalysts are in IPC position. Best results, up to 96% toluene conversion, were obtained when using 1wt%Au/Al 2 O 3 and Nb 2 O 5 catalysts.

Introduction

Volatile organic compounds (VOCs) are an important category of air pollutants and therefore, become a serious problem for damaging the human health and the environment in general [1,2]. Well established technologies for VOC abatement are thermal and catalytic oxidation, adsorption, condensation, biofiltration, membrane separation or ultra-violet oxidation [3,4]. The thermal and catalytic methods, which are widely used, are very efficient to oxidize VOCs. However, these technologies which require a substantial supply of thermal energy (in the range of 700-800°C for thermal incineration and 200-600°C for catalytic oxidation) are not well adapted and energetically expensive in the case of moderate gas flow rates containing low VOC concentrations. In addition, the catalyst poisoning followed by deactivation represents a serious problem for these technologies.

As an alternative to conventional VOCs abatement techniques, atmospheric non-thermal plasma (NTP) technology received increasing interest during the last decade [5][6][7][8][9][10][11][12] for the removal of dilute VOCs from many sources. The main advantage of these non equilibrium plasmas consists in the ability to generate high-energy electrons, while keeping the background gas close to room temperature. Thus, a highly reactive environment is created without spending energy on gas heating as in thermal processes. The energetic electrons excite, dissociate and ionize the gas molecules producing chemically active species. VOCs removal is generally attributed to radical reactions initiated by O or OH radicals [13,14].

However, depending on the reaction conditions, other radical and ionic reactions also result in VOC conversion to CO 2 , H 2 O and other degradation products [15,16].

Atmospheric pressure NTPs for environmental applications may be produced by a variety of electrical discharge devices (corona, surface, and dielectric-barrier discharges) [17][18][19][20]. For some industrial and domestic applications, pulsed discharges are much more suited because of their high selectivity, capability of removing various pollutants simultaneously, moderate operating conditions, and relatively low maintenance requirements resulting in relatively low energy costs of the pollutant treatment. However, NTP technology for indoor air treatment has the disadvantage to produce undesirable byproducts such as ozone and NOx [21][22][23]. To overcome the by-products formation and increasing energy efficiency, non thermal plasma could take advantage of its synergetic effect through combination with heterogeneous catalysts [9]. This combination can be either single-stage or two-stage. In the two-stage system, the catalyst materials are usually placed downstream from NTP reactor. In the single stage system, the catalysts are placed directly in the NTP reactor.

In this work, various catalytic materials were combined with a dielectric barrier discharge (DBD) for VOC abatement at room temperature. Reported results concern the catalyst effect, the plasma effect and the plasma-catalyst effect on the conversion efficiency of toluene used as a representative VOC. A systematic investigation is carried out in order to select the optimal positioning of the catalyst regarding the plasma discharge (inside the plasma or in post-plasma).

Experimental set-up

Plasma reactor

The plasma reactor we used is a dielectric-barrier discharge (DBD) reactor in a cylindrical configuration. A schematic view of DBD reactor is shown in figure 1. It consist consisted of a 1.5 mm-diameter tungsten wire centered in a 20 mm long dielectric tube, and fixed with two ceramic rings. The outer surface of tube was coated with a brass wire to form the outer electrode. The length of the discharge zone can be adjusted and then determine the active volume of the plasma reactor. That configuration gives the possibility to combine heterogeneous catalyst with plasma reactor in two different ways: by introducing the catalyst in the discharge zone (in-plasma catalysis, IPC configuration) or by placing the catalyst downstream the discharge zone (post-plasma catalysis, PPC configuration) (Figure 1a andb). The plasma reactor was powered by a high voltage AC generator delivering HV amplitude up to 18 kV at a frequency of 50 Hz.

Gas containing mixtures of O 2 , N 2 , and toluene was prepared in a gas handling system and their composition was controlled with calibrated flow meters. Experiments were conducted at total flow rate of 3 L/h and toluene concentration of about 800 ppm. The experiment was carried out at room temperature and atmospheric pressure.

Materials

Various catalytic materials based on Ag, Au, Cu, Co, Mn, La, and Nb, supported on Al 2 O 3 and CeO 2 in powder form, have been tested to select the catalysts presenting the highest activity in the presence of non-thermal plasma operating at low temperature (room temperature). These catalysts include: OMS-2 (cryptomelane type manganese octahedral molecular sieve), 5-15wt%CuO/OMS- The Au/support catalysts were prepared by the deposition-precipitation method with urea [24].

The OMS-2 catalysts were prepared by the hydrothermal method as described in [25]. Two solutions (x(g)KMnO 4 in 100 ml H 2 O, and y(g) MnSO 4 .H 2 O in 300 ml H 2 O with adding HNO 3 to maintain the pH at 1.2) were heated to 100 0 C during 24h. The precipitate was recovered by filtration, washed and dried in air (120 °C during 12 hours) and calcined in static air using a range of times and temperatures.

Finally, the LaMnO 3 and La 0.7 Ag 0.3 MnO 3 catalysts were prepared by the citrate method. All the catalysts were also characterized after reaction by XRD, IR Spectroscopy, and FESEM.

The catalytic tests were made in continuous flow system at a total flow rate of about 4 L/h at room temperature (30°C). The weight of the catalysts in powder form was about 200 mg.

Catalysts characterization

The catalysts characterization was performed by XRD, FE-SEM, BET, and IR. The X-ray diffraction (XRD) measurements were performed on the catalysts with a Siemens D500 diffractometer using the K  line of Cu at =1.5418 Å. The 2 range was recorded between 5 and 80° at a rate of 0.02 s -1 . The infrared spectroscopy (IR) was carried out with Brucker Tensor 37. The BET (Brunauer, Emmett and Teller) surface area, S BET , of the catalysts was measured by nitrogen adsorption-desorption method at 77 K using NOVA 1200 surface area analyzer (Quantachrome).

Oxidation experiments

The activity measurements were performed in a continuous flow fixed-bed reactor. The total flow rate through the catalyst bed was kept constant at 3 L/h (oxygen: 0.8 L/h, nitrogen: 2.2 L/h) leading a weight hourly space velocity (WHSV) of about 20 000 h -1 ). The toluene concentration was fixed at about 800 ppm. Under standard reaction conditions, this gas mixture was fed over 200 mg catalysts introduced in the plasma reactor in post-plasma or in-plasma configuration.

The reactants and reaction products were analyzed by gas chromatography (INTERSMAT IGC 120FL) equipped with FID and SE 30 10% Column (1.5 m length, interior diameter 0.18 mm) and by gas chromatography coupled with mass spectrometer (GC-MS).

The analyses of the products were performed by GC HP 6890 Series II equipped with electron capture detector (ECD), phenomenex ZB5 column (30 m x 0.25 mm, i.d. of 0.25 m) and GC-MS Agilent Technologies 5973 equipped with flame ionization detector (FID), HP5 column (30 m x 0.18 mm, i.d. of 0.32 m).

Results

Systematic experiments of toluene conversion were carried out in three systems: only heterogeneous catalysts, only NTP, and NTP-catalyst system with two combinations: in-plasma catalysis (IPC) and postplasma catalysis (PPC).

Toluene removal by heterogeneous catalysis

The catalytic activity is expressed in terms of the temperature (T e ) required for 95% conversion of toluene. The temperature of catalytic ignition (T i ) corresponds to the beginning of the catalytic reaction.

Toluene removal by NTP without catalyst

In the absence of catalyst material, the plasma plays a role of "oxidative media". A significant fraction of input plasma energy is dissipated in the dissociation of O 2 which becomes the dominant process because O 2 dissociation is much more efficient than N 2 dissociation in the electron energy range produced. This dominant oxidation process leads to the formation of the main products of the plasma such as O 3 , CO, CO 2 , and NOx (NO, NO 2 ). In our experimental conditions, toluene conversions in the range 55-60% were obtained. As expected, the plasma partially oxidizes the toluene leading to the formation of byproducts such as formic acid (CH 2 O 2 ), acetic acid (C 2 H 4 O 2 ), and benzene (C 6 H 6 ).

Toluene removal efficiency by NTP/Catalyst hybrid system

Systematic experiments were carried out for the two catalyst positions in regard to plasma reactor. When the IPC is operated, short lifetime reactive species, in the range of tens of nanoseconds, as radicals and excited states (O( 3 P), O( 1 D), OH,) created by the NTP can be used efficiently by the catalyst. On the contrary, when the catalyst is placed post-plasma (PPC), long lifetime reactive species, in the range of milliseconds, such as O 3 , H 2 O 2 , NO, and NO 2 could be used.

The NTP-catalyst hybrid system is operated at room temperature. The toluene removal efficiency significantly increases when the catalyst was coupled to the plasma reactor whatever the nature of catalyst used. The toluene conversion measured over 0-15wt%Cu/OMS-2 catalysts is reported Figure 3. By increasing the loading of Cu from 0 to 15 wt%, the toluene conversion increases from 70 to 79% for IPC and from 65 to 69% for PPC. Similar results were obtained when Co was substituted to Cu at the same contents.

Figure 3: Toluene conversion over OMS-2 supported Cu catalysts.

Figure 4 shows toluene conversions over alumina based catalysts. Results clearly show that the highest toluene conversion (up to 96%) was obtained when 1wt%Au/Al 2 O 3 was used into the plasma discharge. However, when these catalysts were positioned in the post-discharge region, lower toluene removal (up to 80%) was measured. One can notice that, in the absence of catalyst, the toluene removal efficiency was only 60%. The products of toluene oxidation measured at the outlet of the plasma-catalyst system are CO 2 , CO, and H 2 O. In contrast to plasma experiments without catalyst, ozone and NOx are not measured as reaction products. It means that the toluene reaction conversion occurs at room temperature by consumption of O 3 and NOx produced by the plasma discharge. Deposit accumulation was observed on the wall of the plasma reactor. This deposit was identified as benzyl alcohol, benzoic acid, and 5-Methyl-2-Nitrophenol.

Conclusion

Atmospheric dielectric barrier discharge was combined to catalysts in two configurations (in-plasma (IPC) and post-plasma (PPC) catalysis) for the abatement of toluene at room temperature. Different catalysts were prepared and characterized and toluene conversions were measured.

Without plasma, the reaction of toluene conversion over different catalysts (perovskite, Al 2 O 3 , and OMS-2 based catalysts) is promoted at high temperature (higher than 200°C, depending on the nature of the catalyst).

In the absence of catalyst, the toluene removal conversions were in the 55-60% range and the reactions products were O 3 , CO, CO 2 , and NOx.

When the plasma was combined to catalyst, the toluene removal efficiencies strongly increased when the catalysts were in IPC position. Best toluene removal conversions, up to 96%, were obtained when using 1wt%Au/Al 2 O 3 and Nb 2 O 5 catalysts. The products of the C 7 H 8 oxidation were CO and CO 2 .

Figure 1 :

 1 Figure 1: Schematic overview of the two plasma-catalyst hybrid configurations: (a) Post-Plasma Configuration (PPC), (b) In-Plasma Configuration (IPC).

Figure 2

 2 shows the data giving T i and T e for toluene conversion as a function of the nature of catalyst. The reaction of toluene conversion over perovskite, Al 2 O 3 , and OMS-2 based catalysts are promoted at high temperature. As example, these data indicates that T i and T e are in the range of 300-480°C in the case of Al 2 O 3 , and 200-400°C when OMS-2 was used. In the presence of catalyst modified with Cu, Co, Ag, or Au, the ignition temperature decreases drastically to reach 90°C in the case of 1wt%Au/Al 2 O 3 .The supported metals on OMS-2 need a high loading (10-15wt%) of Cu or Co to reach a high activity. In comparison to OMS-2 alone, the temperature of the end of reaction over 15wt%Cu/OMS-2 decreases from 400°C to 250°C. Obviously, adding Co, Ag, or Cu creates new active sites which change the activity of the catalysts. Similar behavior was observed in the case of perovskite catalyst.

Figure 2 :

 2 Figure 2: Toluene oxidation temperatures: Ignition reaction (a) and end reaction (b) as a function of the nature of catalysts.

Figure 4 :

 4 Figure 4: Toluene conversion over γ-Al 2 O 3 metal supported catalysts.

  2, 5-10wt%Co 2 O 3 /OMS-2, CeO 2 , 1wt%Au/Ce 2 O 3 , Al 2 O 3 , 5-15wt%Co 2 O 3 /Al 2 O 3 , 1-2wt%Ag/Al 2 O 3 , 1wt%Au/Al 2 O 3 , LaMnO 3 , LaMnO 3 /Al 2 O 3 , La 0.7 Ag 0.3 MnO 3 , La 0.7 Ag 0.3 MnO 3 /Al 2 O 3 , Nb 2 O 5 . Al 2 O 3 , and Nb 2 O 5 catalysts, all the other catalysts were prepared by different methods. The supported metal catalysts were prepared by the wetness impregnation method. The aqueous solutions of nitrate salt ((Cu(NO 3 ) 2. 3H 2 O, Co(NO 3 ) 2 6H 2 O, AgNO 3 ,) were slowly heated at 70°C until all the precursor was completely dissolved. A calculated amount of the support (OMS-2, Al 2 O 3 , Nb 2 O 5 , and CeO 2 ) was added to the solution. The slurry was heated to approximately 80-100°C and stirred until the total evaporation of water. The samples thus obtained were dried at 120°C during 5 hours. The powder was crushed using a mortar and pestle and subsequently calcined at 500°C in air for 4 hours.

	2.2.1. Catalysts preparation
	Except CeO 2 ,

Table 1

 1

summarizes the activity of CeO 2 , Nb 2 O 5 and perovskite catalysts. As shown in previous data, the in-plasma catalysis configuration seems better suited for toluene conversion. When the IPC is operated, the conversion of toluene over various catalysts follows the order: CeO 2 <LaMnO 3 /Al 2 O 3 <LaMnO 3 <La 0.7 Ag 0.3 MnO 3 /Al 2 O 3 <1wt%Au/CeO 2 <La 0.7 Ag 0.3 MnO 3 ; Nb 2 O 5 .

Table 1 :

 1 Toluene conversion over CeO 2 and Al 2 O 3 based catalysts, niobium oxide, and perovskite.

Acknowledgement:

The authors would thank the CNRS (French Scientific Research National Center), the VAST (Vietnam Academy of Science and Technology), the GDRI (International Group of Research) "Catalysis for Environment: Depollution, Renewable Energy and Clean Fuels" for their supports, and Dr. J. A. Rodrigues (INPE-Brazil) for providing the Nb 2 O 5 catalyst.