Catherine Taine-Cheikh

Expressiveness and evaluation in Arabic: the singular development of the diminutive in Ḥassāniyya Arabic

published or not. The documents may come

Introduction

As explained by Grandi and Körtvélyessy in their introduction to the Edinburgh Handbook of Evaluative Morphology (2015), the notion of evaluative morphology is closely linked to that of evaluative affixes which, in a language such as Italian, serve to form categories like the diminutive, augmentative, pejorative and meliorative. The label was proposed by Scalise (1984: 132-3) who noted that such affixes show specificities in their derivational and inflectional morphology, thus constituting a sort of third morphology, somewhere in between the previous two: -As in inflectional morphology, adding these suffixes changes neither the syntactic category of the base (casa 'house' -casina 'little house'), nor its subcategorization (such as gender, number, actionality...).

-As in derivational morphology, adding these suffixes changes the semantics of the basic word (e.g. ragazzo 'boy' -ragazzino 'little boy') and the composition rule can apply recurrently (fuoco -fuocherello -fuocherellino 'fire -little fire -nice little fire').

-These affixes have the particular feature, among others, of being external with respect to other derivational suffixes and internal with respect to inflectional suffixes, cf. -ucolin contrabbandierucoli 'small-time-smugglers' (= compoundword contrabband + iere AG + ucol(o) PEJ.(M.SG) + i M.PL). Evaluative morphology languages are far from presenting the same characteristics. However, if one considers Arabic, one notes that the derivation known as taṣġīr (literally 'diminution') by scholars of Classical Arabic largely corresponds to Scalise's remarks on Italian. Indeed, it is a type of double derivation which applies more or less similarly (infixation of the diphthong -ay-) to a very large number of different patterns without modifying their category or subcategory. Examples: M.SG kalb 'dog' M.PL kilāb, F.SG kalba(t) 'bitch' F.PL kalbāt; diminutives M.SG kulayb M.PL kulaybāt/kulayyib, F.SG kulayba(t) F.PL kulaybāt. Because this area of the evaluative morphology of Arabic is so widespread and systematic, it is important to explore, and that is what I will do here. However, if one considers the category of Quantity as the basic reference and examines what derivations there are, it would have been legitimate to take other domains into account, as noted by Štekauer (2015 : 43) : « [...] evaluative morphology encompasses, in addition to the 'traditional' cases of evaluative morphology, some categories of Aktionsart (frequentativeness, iterativity, intensity and distribution as well as pluractionality), attenuatives and morphologically realised forms of comparatives and superlatives: they obviously represent deviations from default values. ». Derivational morphology is applied over a vast domain in Arabic. It extends namely -in contrast to what one finds in many languages -to comparatives and superlatives. They are regularly formed on the pattern (ˀ)aR 1 R 2 aR 3 . (e.g. in literary Arabic: kabīr 'big', ˀakbaṛ 'bigger'). 1 Contrary to diminutives, they only marginally take on expressive value, in exclamatory phrases with the so-called verb of admiration: mā (ˀ)aR 1 R 2 aR 3 + N/PR! (e.g. mā ˀakbaṛ aḥmad! 'How big Aḥmad is!'). 2 The category of the highest degree and the augmentative cannot be reduced in Arabic to that of comparative-superlative. It extends to various word formations based on reduplication. In some cases this pertains to roots and one thus sees verbs with R 1 R 2 R 1 R 2 roots often having expressive, and sometimes iterative, meaning (e.g. karkara 'to laugh out loud and repeat laughter', katkata 'to laugh softly', ṭaxṭaxa 'to laugh sneeringly'). Other cases, more akin to true morphological derivation, show reduplication of the second root (pattern R 1 aR 2 R 2 aR 3 (a)). This gives verbs, among other possible values, meanings of iterativity and/or intensity (e.g. Ḥassāniyya kṣaṛ 'break' -kaṣṣaṛ 'break into pieces; to break several times'). 3 It is fundamentally these values which one finds in nouns where gemination is accompanied by lengthening of the second vowel (cf. Ḥassāniyya kaṣṣāṛ 'which breaks a lot, which has the habit of breaking'). It is on the same R 1 aR 2 R 2 āR 3 pattern that adjectives expressing qualities and various names of professions are based (e.g. xayyāt 'tailor'). The array of derived expressive forms has been well described for Classical Arabic (see for example Fleisch 1961: 365 and sq.). Most of these forms have survived in the dialects to differing degrees, but specialists of Arabic dialectology generally focus on the nominal forms, forming nouns or adjectives, of the diminutive (taṣġīr), of which almost all dialects have retained at least a trace. 4 In the first three sections of this article I will study the diminutive in the Ḥassāniyya dialect. This is a variety of Arabic spoken in the west Saharan region of West Africa (Mauritania, northern Mali, southern Morocco and Algeria) by approximately 3 1 By convention, subscript numbers are used for the root consonants of the root (R stands for any root consonant, the subscript number shows its position in the root). 2 A detailed study of the formation and uses of the superlative comparative (called elative in Arabic grammars) can be found in Taine-Cheikh 1984. 3 These values, frequent in Ḥassāniyya Arabic, are more or less so in other dialects. See for example in Syrian (Lentin 1991). 4 The augmentative forms are much more rarely mentioned (see however Singer 1984: 477-497). This confirms the non systematic and more secondary nature of the formation. million people, today almost wholly sedentary. This change in lifestyle however has had little impact on the linguistic traits of the language, the effects of the opposition « Bedouin dialects vs. sedentary dialects » are still highly visible in dialectal comparisons. My study of Ḥassāniyya began in the 1970s, and has mostly been carried out in Mauritania where the majority of inhabitants speak the language. Despite the size of the territory where it is spoken, it remains exceptionally unified. One of the numerous specificities of the dialect is its formation of the diminutive (Taine-Cheikh 1988). Following Section 2 devoted to the morphology, I will study the use of the diminutive in two distinct corpora, a corpus of tales (section 3) and a corpus of poetry (section 4). In Section 5 I will compare the uses to those of the diminutive in other Arabic dialects, so as to clarify both crosslinguistic convergence and the specificities of Arabic (in its entirety and in various dialects).

Diminutive derivation in Ḥassāniyya

Broadly speaking, formation of the diminutive is very similar in Ancient Arabic and Ḥassāniyya. The principal characteristic is infixation of the diphthong -ayfollowing the second consonant (whether it belong to the root or an affix)5 and mainly concerns nouns and adjectives. Overall, diminutive patterns show fewer distinctions than patterns for basic lexemes. They do however vary depending on the root (number of root consonants: triliteral roots R 1 R 2 R 3 or quadriliteral roots R 1 R 2 R 3 R 4 ). They also vary depending on the presence (or absence) of a consonantal prefix and/or long vowel in the basic lexeme. On the subject of gender and number, which remain unchanged in the diminutive pattern, I have two general observations. On one hand, the external plural feminine marker -āt tends to be used in all genders and with all diminutives, cf. (1d). On the other, feminines without the feminine ending -a(t) (2a) have a regularized diminutive (2b).

(1) a. The diminutives will be presented in four stages. I will begin with the regular diminutives of nouns and adjectives, first triliteral roots ( §2.1), then quadriliteral roots ( §2.2). I will continue with rarer formations, some of them innovative: those having the pattern aR 1 R 2 aR 3 ( §2.3) and those found in verb forms ( §2.4). We will then see that diminutives are not always used with the same value. To simplify presentation, I will often attribute a 'quantitative' gloss ('small', 'a little'...) or the simple abbreviation DIM.

Triliteral root nouns and adjectives

The ubiquitous triliteral roots are composed of three consonants in sequence found in all of the root's lexemes (except for particular issues such as those raised by the presence of the glide W or Y).

2.1.1. Patterns without long vowels Patterns without a long vowel -i.e. R 1 R 2 vR 3 and R 1 R 2 vR 3 (a) -make up the majority of cases. These are generally nouns, e.g. kälb 'dog' and kälbä 'bitch'. There are fewer adjectives than nouns without a long vowel, but their diminutives are formed in the same way, cf. (3a). This is also true for a few adverbs and prepositions with the same pattern, cf. (3b) and (4e). If R 3 = W/Y, the root glide is retained, cf. (4a and b). However if R 2 = W/Y, the root consonant is always represented by w, cf. (4c, d and e).

(3) a. KḤL Most have a diminutive form, beyond forms with passive meaning or bearing the long vowel ā (Cohen 1963: 141-2, Taine-Cheikh 1988: 107-9). The infix ay remains the diminutive marker, but its placement varies depending on the root and the verb pattern. Reflexive (or 'middle' voice forms) are characterized by the addition of the affix t (prefixed or infixed) which does not appear to impact placement of the infix -ay-.

Basic verbs and their reflexives: triradicals

The diminutive of basic verbs is characterized by the presence of the infix -ayfollowing R 1 , cf. (15a and b). 8 Excluding the presence of pre-root a-, the diminutive pattern is the same as that of quadriliteral verbs where R 2 = Y such as gäymär. In 'middle' voice forms bearing the infix -tfollowing R 1 , the diminutive marker -ayis placed between the -tand R 2 . (18) a. stäkḥal 'to become blackish' DIM ə staykḥal b. säkḥal 'to make blackish' DIM säykḥal 8 For each verb, the 1st form, with suffixed inflection, is a Perfective and the 2nd form, with prefixed inflection (in yəor i-), an Imperfective. I only give both forms for the first few examples. 9 Some passives in uappear to have diminutive forms, thus ugäyrmaṣ 'he was pinched a little' DIM of ugärmaṣ 'he was pinched'. 10 The infix -(a)yis not between the two R 2 as for the other adjectival-noun forms, cf. (10c). Moreover the long vowel ā is maintained.

The morphology of the diminutive in (a)y, widespread in Ḥassāniyya, thus tends to apply to almost all of the patterns used in the dialect -a feature which is particularly surprising in verb forms. Moreover this applies to almost all possible cognitive categories. The value can be that defined by Štekauer (2015: 43): « The quantityfounded deviations from the default value run across the fundamental cognitive categories of SUBSTANCE, ACTION, QUALITY and CIRCUMSTANCE ». There are however other values, as we shall demonstrate in the following corpus study.

Diminutives in Ḥassāniyya folktales

The Contes arabes de Mauritanie (Tauzin 1993) contains 28 tales of a diversity of lengths.11 The author's transcription and translation have only been marginally modified (with a few rare exceptions, which will be noted). I alone however am responsible for the morphematic analysis as well as the summaries presented here. 12 No diminutives were found in ten tales. In the other eighteen, the frequency and importance of diminutives is highly variable.

3.1. Lexicalized diminutives Some diminutives (of nouns and adjectives) are lexicalized and more common than their corresponding basic forms. Fives tales only have this type of diminutive if one excepts, in tale 7, the ritualized opening formula: « [...] it is God alone who quenches my thirst and yours, and that of all little Muslim throats (xräyzāt -DIM de xarzāt '(internal) throats') [...] » where the use of the diminutive xräyzāt '(internal) throats') is conventional.

Tale 24 contains only one diminutive, ṭfäylä 'girlie', from ṭəvlä 'girl'. Each form corresponds to a different age range: from birth to age 8 for ṭfäylä, from 8 to 12-13 for ṭəvlä (as for ṭfäyl DIM from ṭfəl 'boy'). In tales 9 and 12 one finds bläydä, which is practically a synonym of bläd 'place; function, office'. The expected meaning 'small office' for bläydä is contradicted in tale 9 (p. 28) by the presence of the adjective kbīrä 'big'. In tale 5, a piece of charcoal (ḥmūmä F.SG) is compared to gold and characterized as šwäyn(ä) 'ugly (F)'. In this case it would not have been possible to use the basic form šäyn(ä) as it is restricted to abstract uses. Lastly there are a few adjectives, generally of the pattern R 1 R 2 īR 3 , which are habitually used more often in the diminutive form R 1 R 2 äyyəR 3 than in their basic form. This is often the case namely of ḍˤayyəv 'weak.DIM' and rgäyyəg 'sickly.DIM' which are used in tale 7 instead of ḍ(a)ˤīv 'weak' and rgīg 'sickly'. As for sġayyər (DIM de sġīr 'small' -with ġ / q depending on the region), found in other tales, the frequency of the form in -ayhas led them to lose their diminutive meaning.13 3.2. Pejorative diminutives Pejorative uses clearly dominate in seven tales. Depending on the context, pejoratives can express a broad array of emotions (disdain, pity, mockery, contempt, or even hatred or fear).

3.2.1. Tale 1: 'The ball of gum' A woman sends her children -three sons and a daughter -to harvest gum. One of the sons is 'the one with scrawny legs': bū ṣwäygāt rgäyygīn (DIM of ṣīgân rgāg 'skinny legs'). The other is 'the one with the big belly': bū gḷäyḷa (DIM of gəḷḷa 'portbelly'). The third is called 'the one with the narrow mouth': bū fwäym đ̣ wäyyəg (DIM of ṿuṃṃ đ̣ iyyəg)14 . The daughter is called 'buttery cream' (əz=zäbdä). Trying to reach the ball of gum in the tree, Scrawny-legs falls on Big-belly. One's belly bursts, the other's legs break. Narrow-mouth laughs and his small lips split. Cream, the daughter, runs to tell her mother. She is melting because the day is hot. The diminutives, which play a fundamental role in the tale, apply to the children but are pejorative. Each of the boys is characterized by a ridiculous physical feature which plays a crucial part in the sequence of catastrophic events. As in many languages, the stigmatized feature is turned into a (nick)name.

3.2.2. Tale 2: 'The three deaf people' A deaf woman is tilling her field carrying a child (ṭfäyl -DIM of ṭfəl) on her back. A man comes along, deaf as well, and asks her if she hasn't seen a small herd (gḷäyvä -DIM of gaḷve) of goats (lə=mˤîz). He finds it at the other end of her field. To thank her for the help she inadvertently gave him, he wants to give her a goat (mˤayzä -DIM of ˤanz 'goat' SG irregular) with a broken leg. Thinking he is accusing her of having broken its leg, she wants to go see the judge. The judge, also deaf, believes that the issue revolves around the paternity of the child. He orders the man to marry the woman. The three nouns in the diminutive refer to elements which play a central role in the tale. gḷäyvä and mˤayzä are markedly pejorative: the first denotes the man's poverty, the second the negligible worth of a goat with a broken leg. Their presence reinforces the globally negative tone of the tale which mocks the behavior of three deaf people.

3.2.3. Tale 4: 'Deyloul and the two-humped she-camel' To trick his enemies on the point of attacking his camp, Deyloul leads their spy to believe that he won't move that night. But he organizes the departure of his people in the night, leaving behind only an 'old she-camel' (nwäygä -DIM of nāgä) with her calf, a dog and an 'old old old servant' (xwäydəm -DIM of xâdəm + khäylä khäylä khäylä -DIM of kählä), to whom he gives a 'small mortar' with millet to grind all night. Misled by the barking, bleating and sounds of grinding, the enemies arrive too late to steal the herd. The systematic use of diminutive forms to denote the she-camel, servant and mortar corresponds to a (non stigmatizing) minorative intention as to the means used by Deyloul to beat his enemies. It is a way of ridiculing them.

3.2.4. Tale 8: 'The young man and his cousin' A young man studying the Koran is begging. He comes to his paternal cousin's place, she is sitting before her tent. He is lousy, 'dirty' (mwäyssax -DIM of mwässax) and not very handsome. She says to him: 'Yuck! You're so ugly!' (tfu mašäyyn=ak). Then she throws him a 'small kid' (ždäyy -DIM of ždi) foot that is so dry and old that there is nothing left to eat. Years later, educated and wealthy, he refuses all the potential wives his mother proposes: he wants to marry his paternal cousin. But in the middle of the wedding week, when he joins her in the tent set up for them, he shows her the dried foot of the little kid (ždäyy) that he had left at his mother's. He reminds her of her insulting behavior and leaves. 15 The diminutives used are pejorative -one applies to the young man and the other contributes to rendering his cousin's disdain for him.

3.2.5. Tale 10: 'The woman and the caravan' This tale illustrates the capacity for craftiness attributed to women. One of them presents herself to some caravaneers as a 'poor [and] weak woman' (mṛayyä wä ḍ̣ ˤayyvä), having for baggage but 'a small bottle' (bwäyš/ḅḅäyš). The use of diminutives for mṛa 'woman' and būš 'bottle' constitute self-devaluation for purposes of deception. The goal, as in the use of ḍˤayyvä, is to excite the pity of the caravaneers. She then complains that her mount is uncomfortable, and manages to work her way up the train of camels, marking on her way, with her bottle of ink, all of the loads in the caravan as being hers.

3.2.6. Tale 11: 'The woman and the gob of spit' For his future wife, a man chooses a girl who is still very young (ṭfwäylä). He decides to raise her in his home to avoid her learning women's craftiness. He locks her in the highest room where there is only a 'small window' (kwäywä -DIM of kuwwä) way up high, 'tiny' (sqayyrä -DIM of sqīrä). She thus grows up and becomes a 'young lady' (šwäybbä -DIM of šābbä). One day she leans out of the 'little window' (kwäywä), sees someone and throws him a 'small piece of clay' (ṭwäybä -DIM of ṭūbä), then a second one. She makes a clay copy of the key to her room and gives it to the man. He gets into the habit of visiting her every day when her 'father' is at the mosque. But one day the man spits and a gob sticks high on the rafters. Her 'father' sees the sputum, confirms that it belongs to a man and asks the young lady to come swear an oath at the mosque the next day. When the lover arrives she tells him: « 'Now' (đ̣ ṛäyk -DIM of đ̣ aṛk), put on a 'dirty' (mwäyssxa -DIM.F of mwässxa) 'boubou' (dṛäyṛīˤa -DIM of daṛṛāˤa), take 'pots' (gdäyḥāt -DIM.PL of gädḥāt), makes holes in them and disguise yourself as a beggar. I will be on a donkey twisting and turning and you will come up and ask for a 'little boubou' (dṛäyṛīˤa). I will pretend to fall, and you to help me up. » Everything went as planned and, upon arrival at the mosque she swore that no man had touched her since birth, except for her 'father' and the beggar, just before. The first diminutives seem principally denotative, even though they highlight the difficult captivity imposed on the girl. As for those found in the episode of the beggar, they stress social minoration (the goal being to appear as humble as possible).

3.2.7. Tale 26: 'The girl with the bracelet and the girl with the necklace' In this tale, where a young girl is victim of her step-mother's jealousy, the only diminutive used is 'the little old lady' (lə=ˤzäyyəz -DIM de lə=ˤzūz16 ). Tauzin notes that the first use of ˤzäyyəz is from the outset accompanied by the definite article lə-for her a sign that the diminutive denotes not just any little old lady, but the stereotyped figure of the mean old lady. She, like the step-mother, symbolizes the heroine's bad mother.

Caritative diminutives

Caritative uses are found in three tales, but the diminutives are rare and are only always positive in tales 20 to 22 which portray lovers and attractive feminine figures.

3.3.1. Tale 20: 'The hedgehog and the ostrich' The 'tiny' (qṣayyər) hedgehog is in love with the tall ostrich. As she is wagging her head, walking in the moonlight, he circles around her and sings: « I have seen the beautiful ostrich // The 'moon' (lə=gmäyrä -DIM of l=gämṛa 'the full moon'), that night was shining // Ah! 'walking with small steps' (yähäymsäs -DIM of ihämsäs) all night long // To meet a beautiful woman // A 'little hedgehogess'17 who 'curls up in a ball' (təddäykräš -DIM of təddäkräš) //. » These affectionate diminutives translate the hedgehog's feelings, while highlighting his tininess. But the hedgehog is mistaken in believing the ostrich the 'little hedgehogess' of his dreams. The size difference between the couple is so large indeed that the marriage cannot be consummated.

Tale 21: 'The bullock who gave birth'

This tale is exceptional in that it is given in two forms, one in prose and one in verse. It is the story of a he-jackal who has a bullock and a doe-rabbit who has a cow. The jackal takes advantage of the rabbit's absence to pretend that his bullock has given birth to a heifer. When they go to consult a judge, the latter pretends to give birth himself, thus forcing the jackal to admit his lie. The two diminutives in the prose version are found in the names given to the cattle (qšäywä -DIM of qašwä '[the one with the] white mark on the forehead', with q/ġ; (ṛwäymä -DIM of ṛāymä '[the one which] she was used to'). They are terms of endearment such as often given to cattle. In the verse rendering, there is an additional diminutive. The jackal, when called upon to return the heifer to the rabbit is addressed by the pejorative 'oh little thief' (yä ṣwäyṛəg -DIM of yä ṣāṛəg).

3.3.3. Tale 22: 'The little gazelle'18 The little gazelle (l=ūḥayšiyyä DIM of waḥšiyyä 'gazelle') was roaming with the other animals. One day they were so thirsty they decided to dig a well but the ūḥayšiyyä said that she couldn't dig with the others because her hands were 'tiny' (sqayyrīn). After that she goes to fetch water at the well at night, in secret, with her 'little canary' (gdäyrä -DIM of gədrä). She washes and sings. After several attempts, the animals manage to catch her while she is bathing. The lion asks to remain alone with her because he wants to know what she sings at the well. The ūḥayšiyye tells him: « I am 'bound' » (mušäyk ə rvä -DIM of mušäk ə rvä). Untie me and I will dance for you. Once freed, she ties on her skirt and sings. But she escapes when the lion starts to dance with his eyes closed. The hero of the story is a female animal. The use of diminutives to characterize her make her a « fragile and endearing protagonist » (Tauzin 1993: 101 n.1).

3.4. Other cases In three tales, the dominant tonality in the diminutives is neither pejorative nor caritative. In tale 13, what rare diminutives there are appear solely denotative. In the two other tales, the more numerous diminutives concern -as in tale 2 on the three deaf people -most of the story's important elements. Diminutives thus appear to serve to single elements out, both to facilitate their identification and to stress their special role in the narrative.

3.4.1. Tale 13: 'The woman who married a demon' This tale describes relations between humans and demons, which the Moors believe in quite firmly. One of the episodes narrates the woman's departure for the demon world. Her husband had warned her never to invoke God, even if she saw strange things such as a dancing head. However, she cannot refrain herself from invoking God and causes the destruction of the demon city. The diminutives bear on the threshold between the human and demon worlds. At the first mention, only the adjective (đ̣ wäyyəg -DIM of đ̣ iyyəg 'tight') is in the diminutive. At the second, the noun (fwäym -DIM of ṿuṃṃ 'door') is too: lə=fwäym əđ̣ =đ̣ wäyyəg 'the very small, narrow door'. The narrator insists on the narrowness of the passage between the human world and that of the devils.

3.4.2. Tale 25: 'The green bird' A woman loses all her sons at birth. To ward off evil, she promises to give her first born to an ogre as soon as he is able to recite the entire Koran. In this fantasy tale, the boy escapes from the ogre and successfully overcomes a series of hurdles to obtain the hand of a princess, satisfy the desires of his mother-in-law and obtain the love of his wife. In his escape, the boy receives help from his mare: it is out of her belly that is born a 'little horse' (ḥwäyli -dim. of hawli), half silver, half gold. He arrives under a tree where there is a green bird, and when he leaves he carries away the bird's jewels. He continues until discovering a 'small mortar' (mhäyrīz -DIM of məhrāz) pounding near a 'little fire' (nwäyrä -DIM of nâṛ F.SG). Although the mare had warned him against the objects, he approaches and finds a 'tiny old woman' (ˤzäyyəz) who is adding 'tiny kindling' and pounding in a 'small mortar' (ˤwäyd -DIM of ˤûd + sqayyər). He recognizes his mother. From that day on, she churns milk for him to make little balls of cream to treat his ringworm. He is then called the 'ringwormed one' (əl=gäyṛəˤ DIM of ägṛaˤ) 'with the little balls'. Following a first ordeal, he marries the king's daughter but she refuses his advances. With the help of his little horse (ḥwäyli), he assists the other sons-in-law in bringing home the green bird that the queen wants to hear sing. But the bird demands the presence of a particular young girl, who has her own demands. The ringwormed one satisfies the first demands using 'little boubous' (qšäyšibāt -DIM of qaššābāt), 'small knives' (mwäysāt -DIM of mūsāt), 'little bits of string' (ṣräyˤāt -DIM of ṣṛəˤ 'necklaces') and a little meat. Each time he helps a son-in-law in an ordeal, he asks him for something by way of payment -an earlobe, a toe, a 'small piece' (tšäyšä -DIM of ṭaššä) of nostril -which in the end allow him to prove the role he played and to win his wife's heart. Apart from the pejorative epithet 'ringwormed' (gäyṛəˤ) the other diminutives serve above all to specify or even highlight various elements.

3.4.3. Tale 27: 'The sultan's son who married a monkey' The sultan's youngest son falls in love with an ogress's daughter. When she elopes with him, the girl takes various little things with her. These will serve to slow her mother down when she sets off in their pursuit: grains of sand which will change into sand dunes, 'little pebbles' (ḥžäyrāt -DIM of ḥažṛāt) which will transform into mountains, a 'small sip' (žqäymä -DIM of žuqmä, avec q/ġ) of water which will change into a sea, pine needles which will change into forests. But they will not prevent the ogress from catching them and transforming her daughter into a monkey. The use of diminutives is denotative and highlights the contrast between the paucity of the means and the magnitude of the results.

The frequency of diminutives varies, and, beyond lexicalized uses, they are particularly present in 'joking' tales, in poetic passages, and some fantasy tales. Pejorative uses are more frequent and varied than meliorative ones. Some diminutives are even used as (nick)names, through mockery or affection. There are times however where the use of the diminutive is more neutral, serving above all to aid the narrator in singling elements out. Distribution across the various categories is highly unequal. The best represented is that of nouns with 37 tokens (42 including those with tfäyl(ä) or bläydä). 19 Adjectives total 6 tokens (15 if the 4 lexicalized diminutives are included) -for only 4 (or 8) different adjectives. The other categories are even rarer: two verbs (both in verse), a past participle, a set expression of « admiration » and an adverb.

Diminutives in Ḥassāniyya poetry

Poetry in dialect is very dynamic in Moor society (Taine-Cheikh 2008). There are various genres of poetry, but it is in love poetry, studied in this section, that diminutives are most frequent. 20 This poetry, which is neither panegyric nor quite descriptive, is all the more appreciated as it plays on forms (puns, alliterations, assonances...) and expresses the poet's feelings indirectly. This section reproduces in large part my 1988 article where I present the major role the diminutives play in allusive expression -a role which is unfortunately highly difficult to render in translation (which is why I have maintained DIM in some non interlinearized translations). The poem references are only mentioned when they are not from my own corpus.

Declarations of affection

Courtship situations are favorable to the use of diminutives used as terms of endearment. Below is the example of a short poem where the originality resides in a to and fro between the physical charm (välāḥ -DIM vläyyiḥ) and the spiritual charm (mälāḥ DIM mläyyiḥ) of the beloved woman, as if the poet felt obliged to choose between the two attributes but couldn't. 20 With the possible exception of obscene wedding songs (Tauzin 2001: 158ff). 21 This was observed for the first time by the Lieutenant Le Borgne. I corrected the transcription but was unable to find the foot missing from the last hemistich. Moreover, a y in one of the diminutives is superscribed because it does not count in the syllable formation. The two other diminutives -kḥäylä (DIM of kaḥlä 'black.F') and especially ə ˤwäynāt (PL de ə ˤwäynä -DIM of ˤayn PL ˤaynīn 'eyes') must be understood more in their connotative rather than denotative meaning: they serve to express the poet's feelings, not to faithfully depict the woman being wooed. 22 As for the presence of the adjective 'niggardly', it is not in contradiction with the use of diminutives of endearment because b(a)xīlä 23 must be understood metaphorically in the sense 'miserly with one's time'. This reproach, frequently heard from aspiring lovers, is less the expression of a defect than an indirect compliment to the beloved whose presence is always felt to be too rare.

Ambiguous declarations

The frustration can however be real, with the poem leaving room for serious doubt as to the poet's true feelings for the woman he is wooing. This is the case for poem ( 21), where diminutives are particularly abundant: The portrait sketched by the poet is hardly flattering to the woman who is keeping out of his sight: skinniness, dull skin, frizzy hair, above all short, are all features which are considered physical shortcomings in a Moor woman. However the use of diminutives24 lends an affectionate tone to the portrait of the beloved and largely takes the sting out of the criticism. The poet is clearly disappointed, but the warmth of his feelings persists. The same could be said of poem ( 22) a ṭalˤa 'topped up' with aaabababab rhymes. The poet expresses four harmful wishes and only the presence of some diminutives (beginning with that of the desired woman's name) softens the harsh tone.

( This degree of criticism can seem very surprising in poetry, but it is linked to the fact that the Moor culture tends to avoid praise, for both aesthetic and social reasons. 26 Moreover, the antiphrastic use of disparaging terms is widespread in the Arab world to avert the 'evil eye'. Thus Moors will never say that children are healthy or plump. Instead they will mention weaknesses (real or imagined), calling them šwäyn (DIM of šäyn 'ugly'), from msäykīn ('poor little thing' -DIM of məskīn 'poor') or mġäysīl ('little rogue, little smart' -DIM of mäġsūl 'rude, oaf', littéralement 'washed') with a soft caressing tone which leaves no doubt as to the speaker's kind intentions. Intonation alone cannot transform pejorative terms into ones of endearment: it is fundamentally the diminutive form which, in a given context, steers interpretation towards affection. One should also note that the diminutive has an impact beyond the morphologically affected word: its enunciative scope tends to cover the entire context, beyond the sentence in which the diminutive is used.

Indirect declarations

In general, Moor poets tend to declare their feelings indirectly. Describing the beloved's defects rather than her qualities (especially physical ones), highlighting regrets and reproaches, all of this is ordinary. However Moor poets often go much further in their exploration of indirectness. Thus in the following poem, the declaration of love is addressed to a 'little she-kid'. The animal, singularized by the use of the diminutive, grows in the poet's tender imagination and it is clear that she indirectly represents the woman in his thoughts. Four diminutives concern the referent 'little she-kid'. Three designate her directly ( ə ždäyyä DIM of ždiyyä 'she-goat kid'; ə ždäyˤa DIM of žädˤa 'two year she-goat kid'; ə ḥwäyliyyä DIM of ḥawliyye 'young she-goat'), a fourth takes her as argument (maġaylā=k DIM of the 'admirative' phrase maġlā=k). It is the closeness, the spatial proximity between the woman and the goat in the camp, which allows the poet to express his love for the former all while pretending (at least in the six first verses) to address the latter. This 'displacement' is made easier by the enunciative function of the diminutive. Given that connotative meaning can 'float', i.e. be detached from the linguistic sign (whereas denotative meaning can only apply to the denotatum), the poet is able to express his love without making the object explicit. Only the last verse, where the poet asks the beloved for a təbrīˤa (a short love poem of two lines composed by women) implicitly poses a change in object starting around verse 7 yā mettānət wəžd=i. This poem illustrates why the diminutive attracts poets, in particular in its enunciative function, but it shows originality in the solution adopted. In general, the poet's feelings focus on the places frequented by the object of his love. Indeed, although nomads are undeniably attached to their various camps, poems which evoke a particular camp with nostalgia are often love poems calling out to the loved one in an entirely indirect manner. A very small dune (ˤläyb DIM of ˤəlb) or very small well (ḥsäyy DIM of ḥāsi) could easily play the role of the kid in the preceding poem.

Thus in the following poem by Mḥammäd W. Aḥmäd Yuṛa (after Ould Ahmedou Bamba 1982: 31-2), the 'little well' of the Ähl Āṃänniš appears to be the place frequented by his beloved. The circumstances of the meeting are evoked and highlighted by the use of three other diminutive forms (gṛäyn DIM of gaṛn 'tuft'; bṛäyg DIM of bṛag 'lightning' and rwäyḥa DIM of ṛīḥ F.SG 'wind'), but no mention is made of the beloved directly.

sting of the criticism and even leads to a sort of inversion of values. What these poems clearly reveal is that the use of diminutives may be dictated by pragmatics to essentially translate the poet's attitude, feelings and emotions.

Arabic diminutives from a typological perspective

In this section, I will consider the « root-and-pattern » formation process from a typological perspective, both formally and semantically. In particular, I will briefly review the other formation processes attested in Arabic with relatively diverse meanings and values.

5.1. The « root and pattern » formation process 5.1.1. A process rare in the world's languages The process for forming the diminutive described for Ḥassāniyya is also found in other dialects. Indeed, insertion of the diphthong -ay-(in a given position and for a given pattern) is, by far, the most common process in Arabic for forming diminutives. The same process is also found in other Semitic languages such as Biblical Hebrew, Aramaic, and Akkadian (Zewi 2006: 638). However, crosslinguistically speaking, this process, characteristic of « root and pattern » type morphology, is, in the works of Štekauer (2015: 53) considered limited to this language family, whereas « the most productive W[ord] F[ormation] processes are suffixation (95 per cent of the languages of the world) and compounding (90 per cent) followed by reduplication (80 per cent), prefixation (72 per cent) and conversion (63 per cent) » (ibid.: 47). While the « root and pattern » inflection process is specific to a few Semitic languages, the evaluative morphology of Arabic is in keeping on other points with universal trends, namely in that its formation of diminutives is much more common and systematic than for augmentatives: « Diminutives are the unmarked evaluative category; that is, the presence of a productive category of augmentatives in a language implies the presence of diminutives » (Merlini Barbaresi 2015: 41). In addition, as in many of the world's languages, the diminutive derivation holds a rather special place in the inflectional morphology of Arabic. Given that the diminutive pattern depends on that of the basic word (presence or not of a long vowel, prefix, or geminate), it cannot be said that the diminutive is derived directly from the root (even though the nature of the root -triliteral or quadriliteral -also plays a role). 27 5.1.2. A more or less broad domain of application There is a hierarchy in the types of words serving to form diminutives. It varies with the language, but nouns appear to be the most common category, contrary to more grammatical and less autonomous words (determiners, but also numerals and pronouns, even adverbs) which are the rarest categories (Bauer 1997: 538-540). This observation also applies to Arabic, where the main (and often only) base concerned by the diminutive derivation in -(a)yis that of nominals -nouns and adjectives. In the Maghreb, where on the whole this process for forming diminutives is the most productive, two main dialect groups are to be distinguished (cf. Taine-Cheikh 1988: 110-115). The first group (the largest and most well documented) is that of sedentary and urban dialects. In these varieties, the diminutive system has grown apart from that of Classical Arabic. The patterns R 1 R 2 ayR 3 (cf. §2.1.1) and R 1 R 2 ayR 3 īR 4 (cf. §2.2.3) have disappeared and two purely dialectal patterns (where R 2 is doubled or a w inserted) provide diminutives of adjectives (of colors and physical defects, but not only) such as kḥīḥəl. 28 In the second, more conservative, group, of dialects spoken by nomads,29 the patterns R 1 R 2 ayR 3 and R 1 R 2 ayyə/iR 3 (cf. §2.1.2) on one hand, and R 1 R 2 ayR 3 īR 4 and R 1 R 2 ayR 3 ə/iR 4 (cf. §2.2.2) on the other, are generally kept distinct. It is also in this group that one finds most of the dialects with verbal diminutives -essentially reflexive verbs bearing the prefix t-, cf. ( 25).

(25) West Algeria: among the Ulâd Bṛâhim of Saïda (Marçais 1908: 107) a. tqeibböh 'to become a little naughty' b.

tˤeiṛba 'to become a little arabized'

In the Middle-East, diminutives bearing the infix -ayare relatively rare, with the exception of particular cases such as Iraqi Arabic,30 the northern Arabic dialects and Negev Arabic. And yet it is in other dialects of the Arabian Peninsula, namely in the East and in Ṣanˤāˀ (Yemen), that one finds verbs in the diminutive form. 5.1.3. Competition across derivations Some forms show a tendency to be used on occasion to express the diminutive. This is the case of reduplicated forms, often belonging to a childish register.31 This is also the case for some diminutives bearing the suffix -a(t), generally used for expressing units and portions. 32 Masliyah gives examples in Iraqi Arabic (26a and b). He further notes diminutives bearing the suffix -ūn (26c). Some derivations are found above all in proper nouns, namely in the Middle-East and in Morocco. This often pertains to the pattern R 1 aR 2 R 2 ūR 3 (cf. 27a and 28a). It is also the case with the suffixes -ū/ō (cf. 27b) and -ūš/ōš (cf. 28b). The origins of the various suffixes are generally ascribed to the influence of a substrate or adstrate (Aramaic, Berber, or even Latin). 33 (26) Iraqi Arabic (Masliyah 1997: 72- There is thus a certain variety in Arabic diminutives, but only the root and pattern type with the infix -ayis widespread, both in terms of the number of dialects where it is productive and in terms of the number of syntactic categories affected. Diminutives of nouns and adjectives are quite common. The patterns are more varied in the so-called Bedouin dialects, and it is in this same group that one finds diminutives of verbs on the one hand, and of adjectives of color and physical defects (pattern aR 1 R 2 aR 3 ), on the other hand. There is no dialect however where derivation by the infix -ayis a developed as in Ḥassāniyya.

5.2. The polysemy of diminutives 5.2.1. The Jurafsky model Jurafsky (1996: 542) proposes a representation of the universal semantic structure of the diminutive category in the form of a radial category, where the central line is composed of the ordered notions child -> small -> female. The names of the other senses are linked by labeled arcs specifying the « mechanisms of semantic change; inference (I), metaphor (M), generalization (G), and lambda-abstraction (L) ». The central vertical line divides the structure into two parts. On one side of the line are semantic values: child -> member (M); small -> exactness (L) + partitive (L) + approximation (L, M) + small type-of (I); small type-of -> related-to (G) + imitation (G, M). On the other are values linked to pragmatics: child -> affection (I) + pets (M) + sympathy (I) + intimacy (I); small -> contempt (M) + hedges. Leaving aside the question of the source of the diminutive, which, according to Jurafsky is more likely a word meaning 'child' or 'son' rather than 'small',35 I will continue my analysis beginning with the semantic values.

5.2.2. The semantics of Arabic diminutives While defining diminutives as « [t]he quantity-founded deviations from the default value » is useful as a starting point, it does not cover all possible meanings. Indeed, in Arabic one finds diminutive forms where the meaning (or one of the meanings) corresponds not to the primary sense 'small(ness)' but to one of the derived senses observed by Jurafsky, namely imitation, approximation, partitive, exactness. a) The meaning of imitation is found in certain dialects having diminutive verb forms.

(30) West Algeria: among the Ulâd Bṛâhim of Saïda (Marçais 1908 Masliyah (1997: 77). 36 c) We saw above that the sense of partition or portion is generally expressed by suffixes such as the so called 'feminine' suffix. Thus laḥmä has the meaning 'piece of meat' in contrast to lḥam 'meat' in Ḥassāniyya. Furthermore, diminutives can serve as singulatives, not only in narratives (as in 2.4.) but also in lexical derivation. Thus Denizeau observes that in ethnonyms such as ḥwêwi 'of the fraction of the Ḥawâya' or msêˤîdi 'of the tribe of the Masāˤîd', « the function of the diminutive form, doubling that of the suffix -i, appears to be strictly singulative » (1957: 68). This is akin to the meaning of member noted by Jurafsky.

d) The last meaning is that of exactness. A few examples are to be found in the temporal domain:

(36) Ḥassāniyya a. gbaḷ 'before' DIM gbäyḷ 'a moment ago' b. đ̣ aṛk 'now' DIM đ̣ ṛäyk 'just now' (37) Southern Tunisia, in Gabès (Marçais & Farès 1932: 250, 1933: 1) a. ṣ u bâḥ 'morning' DIM ṣbi a ḥ 'daybreak' b. məɣrəb 'sunset' DIM mɣêrəb 'immediate approach of sunset' 36 C 1 C 2 ēC 3 (or fˤēl) < C 1 C 2 ayC 3 .

In the end, we will have found in Arabic almost all of the meanings established by Jurafsky. However these meanings are not very frequent whereas connotations have been noted by all authors studying Arabic diminutives. 5.2.3. The pragmatics of Arabic diminutives Distinguishing between morphosemantics (denotation and connotation) and morphopragmatics, Merlini Barbaresi (2015: 35) asserts: « word connotations, intended as stable meanings picked up from contextual uses over time, are not seen to be relevant for a pragmatic account; they are part of the complex semantics of the word ». The stability of the semantic connotations is striking in some languages such as Polish where the diminutive of each first name (especially of women: -enka, -ulka,uchna, -usia, -ạtko) has « a slightly different semantic value » (Wierzbicka 1984: 124).

There are examples of this sort in Arabic, particularly in diminutives used, as in Polish, to address someone, whether they be kinship term diminutives in ayor -u (Yassin 1978: 300; Henkin 2010: 179; Blanc 1964: 74) or diminutives of first names with a suffix -ī or -ūš (Watson 2006: 191; Colin 1999: 76). With the exception of -ūš, all of these diminutives are ordinarily caritative. Inversely, diminutives of collective nouns and verbs usually have pejorative or ironic meaning in some dialects (see, on one hand, Denizeau 1957: 68; Taine-Cheikh 1988: 90, and, on the other, Marçais 1908: 455; Dhina 1938: 331; Watson 2006: 192). The situation in sedentary dialects (namely in the Maghreb) is quite similar. In these dialects, infix derivation provides a high number of frequently used diminutive forms, but their use is restricted to specific contexts. For the main part this concerns the affectionate use women make of diminutives, especially when addressing children. 37 In Djerba, in Tunisia, a woman would never use a diminutive when addressing a man (Saada 1970: 323). The use of diminutives is very different in the Bedouin dialects (or at least in several). It is not reserved for women, nor for addressing children. Moreover, the meanings can be either caritative or pejorative, even though the latter are sometimes more frequent than the former in the speech of men. 38 The Bedouin dialects are characterized both by the high productivity of diminutives and the variety of their meanings. There is probably a link between the two, the fact that diminutives can be used pejoratively certainly multiplies the contexts where they are used. I have restricted myself here to analyzing only two corpora, but they already well illustrate the diminutive's vast morphopragmatic possibilities as described by Merlini Barbaresi (2015: 41). Further investigation of the importance of the opposition « self-oriented vs. other oriented, empathetic vs. neutral non empathetic » (Ponsonnet 2014: 35) could be useful as Arabic diminutives rarely appear to be self-oriented, although the Moor tales do provide a few examples. In any case, in Ḥassāniyya -the dialect which has clearly developed diminutives the most -, the uses of the diminutive often correspond to situations where the distance between the protagonists (or participants in a dialog) appears reduced. 39 This increase in proximity can stem from a variety of motives and pertains both to positive emotions (affection, love, commiseration, empathy...) and negative emotions or stances (derision, mockery, disdain, aggressiveness...). It is because the pragmatic dimension becomes essential that connotations may detach from lexemes in the diminutive form to transmit a more global assessment, either positive or negative depending on the context. In contrast, diminutives lacking any emotional connotation, such as those which serve to singularize elements important to a narrative, denote values pertaining to the denotatum.

  Basic verbs and their reflexives: quadriradicals Quadriradicals and their reflexives bearing the prefix tform their diminutive by infixation of y between R 1 and R 2 . Thus for verbs having the root GRMS and their action nouns. (16) a. gaṛmaṣ 'to pinch' DIM gäyṛmaṣ 9 b. tgaṛmaṣ 'to pinch himself' DIM tgäyṛmaṣ c. tgaṛmīṣ 'pinch' DIM tgäyṛmīṣ 2.4.3. Patterns where the 2nd root consonant is a geminate Verbs with a geminate R 2 form their diminutives by infixation of y between R 1 and R 2 . The same holds for adjectival-noun forms in R 1 aR 2 R 2 āR 3 which derive from them, cf. (17c). 10 (17) a. bäddäl 'to change' DIM bäyddäl b. tbäddäl 'to get changed' DIM tbäyddäl c. bäddālä 'seller F' DIM bäyddālä 2.4.4. Patterns bearing a prefix in s(t)a-Derived verbs bearing the prefix sta-, as well as the few verbs bearing the prefix saexclusive to Ḥassāniyya (Taine-Cheikh 2003), form their diminutive by infixation of y before R 1 . Thus for the root KḤL.

  Many nouns and adjectives show a long vowel (ā, ū or ī) after R 2 . In the diminutive, the vowel is replaced by y(i/ə): Some lexical items (nouns and especially adjectives) are derived from a nominal base by addition of a consonantal suffix such as -ān, -āni(yy), -āwi(yy) and -i(yy). Some nouns (namely 'Berberized' lexemes) do not come under this rule. Thus āxläyxəl DIM (without ī) from axəlxāl 'anklet' (XLXL).2.3. A specific case: aR 1 R 2 aR 3 patternsFor aR 1 R 2 aR 3 patterns, the diminutive is expressed by infixation of -ayafter R 1 (and not R 2 ) -a change which can be explained by the original presence of an initial glottal stop ˀ no longer found dialectically. 7 There are two case figures.2.3.1. Adjectives of colors and bodily defects aR 1 R 2 aR 3 is the pattern of adjectives of colors and bodily defects in the masculine singular. Its diminutive follows the pattern aR 1 ayR 2 əR 3 (ˀuR 1 ayR 2 iR 3 in classical Elatives aR 1 R 2 aR 3 is also the pattern of the comparative-superlative (elative). Unmarked for gender and number, it applies to all triliteral adjective roots. Its diminutive in Ḥassāniyya follows the pattern aR 1 ayR 2 aR 3 : Verb forms and suchlike Broadly speaking, verb forms in Ḥassāniyya are either agentive, reflexive or passive.

	The diminutive is regularly formed by insertion of ay following the 2nd root
	consonant, with no change to the suffix:
	(6)	a. ʕTŠ	M.SG ˤaṭšān 'thirty'		DIM	ə ˤṭäyšān
		b. TḤT	M.SG taḥtāni 'lower'		DIM	tḥäytāni
		2.2. Quadriliteral root nouns and adjectives (or suchlike)
	Contrary to suffixal consonants, prefixal consonants count as root consonants. Thus Arabic):
	the rules that apply are the same as for quadriliteral roots.
	2.2.1. Patterns without long vowels (11) a. KḤL M.SG akḥal 'black'		DIM	äkäyḥəl 'a little black'
	Quadriliterals without a long vowel form their diminutive by insertion of -ay-after b. ṬRŠ M.SG aṭṛaš 'deaf' DIM aṭayrəš 'a little deaf'
	R 2 , cf. (7a). The same holds for triliteral root nouns bearing a consonantal prefix, cf.
	(7b). The quality of the final vowel is always ə in masculine diminutives (even when a in the base). (7) a. ʕGRB M.SG ˤagrəb 'scorpion' DIM ˤgäyrəb b. RVG M.SG maṛväg 'elbow' DIM mṛäyvəg 2.3.2. (12) a. kbīr 'big' äkbaṛ (mən) 'bigger (than)' DIM äkäybaṛ (mən)
		2.2.2. Patterns containing a long vowel following the 1st root consonant b. ṭwīl 'long' aṭwäl (mən) 'longer (than)' DIM aṭaywäl (mən)
	Triliteral root lexemes having a long vowel following R 1 show the same behavior as
	quadriliterals where R 2 = W, cf. (8a). The same is true for adjectives where the Only the final vowel (a) distinguishes this diminutive pattern from the preceding
	diphthong iy precedes R 2 = Y, cf. (8b). one. Thus adjectives for colors and bodily defects (13a) and elatives (13b) follow the
	same basic pattern, but have different diminutives (13c and d).
	(8) (13) a. aṭṛaš 1° 'deaf' a. XTM M.SG xātəm 'ring'			DIM	xwäytəm
		b. HYN b. aṭṛaš (mən) 2° (elative of 1°) 'deafer (than)' M.SG hiyyən 'easy'	DIM	hwäyyən
		c. DIM of 1° : aṭayrəš 'a little deaf'
		F.SG 2.2.3. Patterns containing a long vowel following the root final consonant ġūlä 'female ghoul' DIM ġwäylä 'little female ghoul' d. DIM of 2° : aṭayṛaš (mən) 'a little deafer (than)'
	d. XYM F.SG xaymä 'tent' e. BYN bäyn 'between' Where the quadriliteral root lexemes bear the long vowel ā or ū between R 3 and R 4 , DIM xwäymä 'little tent' it changes to ī in the diminutive (9b and c). The same holds for assimilated triliterals The elative diminutive is also used in exclamatory expressions.
	bearing a consonantal prefix (10a and b)or a geminate R 2 (10c). (14) mäk ə bṛ=u! (< mā=akbaṛ=u) 'how big he is!' DIM	mäkäybṛ=u!
	(9)	a. BHNS M.SG bähnīs 'beast' b. ḤWBR F.SG ḥawbāṛa 'debauched woman' 2.4.	DIM DIM	bhäynīs ḥwäybīṛa
	(5)	a. M.SG c. VKRN M.SG väkrūn 'turtle' kbīr 'big'	DIM	kbäyyər 'a little big' DIM vkäyrīn
	b. F.SG c. M.SG (10) a. FTḤ d. M.SG b. GRS c. BRM	kbīrä 'big' xṛūv 'lamb' M.SG məftāḥ 'key' M.SG təgrās 'freezing' lžām 'bit' F.SG bärrīmä 'crankshaft wick' DIM DIM	kbäyyrä 'a little big' DIM mfäytīḥ xṛayyəv 'little lamb' DIM tgäyrīs

F.SG kaḥlä 'black' DIM kḥaylä 'a little black' b. Ð̣ RK đ̣ aṛk 'now' DIM đ̣ ṛäyk 'just now' (4) a. ŽRW M.SG žärw 'puppy' DIM žräyw 'little puppy' b. ŽDY M.SG ždi(y) 'goat kid' DIM ždäyy 'little goat kid' c. ĠWL DIM bwäyn 'a little between' 2.1.2. Patterns containing a long vowel following the 2nd root consonant DIM lžäyyəm 'little bit' 2.1.3. Nouns and adjectives bearing a suffix DIM bräyrīmä

  I am going to tell you something you are not going to like, 'It will be for niggardly you, 'You who deprive me of your soft eyes.DIM, the gap between your teeth.DIM 'And your little black lips too black.DIM.'

		DIM.wit[of]=PR.2F.SG	NEG=PR.1SG	laught_at.PTCP.M.SG
		u vläy y ḥ=ək		mā	n-aˤṛav	kämm
		and DIM.glamor[of]=PR.2F.SG	NEG	1SG.IMPFV-know	how_much
		[u] lā	n-aˤṛav		gədd	ə mläy y ḥ=ək
		and NEG	1SG.IMPFV-know	greatness[of] DIM.wit[of]=PR.2F.SG
		u txallaṭ			əˤliy=yä	vämm
		and mix.PFV.3M.SG		on=PR.1SG	there
		ə mläy y ḥ=ək		u	vläy y ḥ=ək
		DIM.wit[of]=PR.2F.SG	and	DIM.glamor[of]=PR.2F.SG
	'I think of your charm.DIM // But do not forget your wit.DIM,
	'Your charm is incomparable // The same is true of the brilliance of your
	mind.DIM,
	'And I am getting confused here // Between your charm.DIM and the brilliance of
	your mind.DIM.'
	(20)	bäyn=i	w	hiyyä		təžlīžt=ək
		between=PR.1SG and	she		affliction[of]=PR.2F.SG
		hāḏu	l=ək			əl=baxīlä
		these_ones	for=PR.2F.SG		DEF=niggardly.F.SG
		bə=ˤwäynāt=ək	u		vläyžt=ək
		with=DIM.eyes[of]=PR.2F.SG and	DIM.diastema[of]=PR.2F.SG
		u kmäyyəmt=ək			lə=khäylä
		and DIM.black_lips[of]=PR.2F.SG		DEF=DIM.black.F.SG

In this ṭalˤa (a poem of 6 verses of aaabab rhymes), as in the gâv which follows (poem of four verses of abab rhymes), the diminutives have clearly affectionate overtones. This adds to the denotative effect corresponding to an attribute of the beloved. It is often a physical or moral quality. Thus in (20) of the 'diastema (space between the teeth)' (välžä) and the 'fact of having the area around the lips of a very dark color' (kmāmä) which, in Moor aesthetics, are considered desirable attributes. '

  Oh how I love you, little goat // How dear.DIM you are to my heart! 'If my tenderness continues to grow // Until you become a big kid.DIM, 'Then that you become a young goat.DIM // Then she will kill me with sadness. 'Oh source of my torment // Accept this little poem 'And accept this he-kid // But recite me a few verses of love.'

		here_is		this_one	DIM.poème
		u	rāˤi	ḏāk	ə ždi
		and	here_is	that_one	kid.M
		gūlī=l=i		təbrīˤa
		say.IMPER.F.SG=to= PR.1SG	distich.
	(23)	yowgi			bī=k	ə ždäyyä
		how_much_love	with=PR.2SG	DIM.she-goat_kid
		maġaylā=k		əˤliy=yä
		DIM.how_beloved=PR.2SG	for=PR.1SG
		w īlä		kəbrət	vī=yä
		and if		grow_up.PFV.3F.SG	in=PR.1SG
		läyn		ə tˤūd	ə ždäyˤa
		until		3F.SG.IMPFV.become	DIM.two_year_she-goat_kid
		wə		tˤūd	ə ḥwäyliyyä
		and		3F.SG.IMPFV.become	DIM.young_she-goat
		təktəl=ni		bə	l=läyˤa
		3F.SG.IMPFV.kill=PR.1SG	with	DEF.pain
		[u]	yā	mättānət	wəžd=i
		[and] oh	[the]strengthener.F.SG[of]	loving_sadness[of]= PR.1SG
		rāˤi		ḏīk	ə ṭläyˤa

'

  With plural diminutives, attenuation can bear on the number.Denizeau (1957: 68) notes that in this same Southern Tunisian dialect, « tmērāt means 'a few dates, rare dates', 'less numerous' and not 'smaller or worse' which would be the case of the affirmative plural tämṛât ». Lastly, to finish with cases of attenuation, let us further note in Iraqi Arabic: « a number of wild animals, reptiles, and harmful plants occur in fˤēl pattern probably to demean or lessen the fear that such creatures may cause to children. Examples: xmayyis 'lion', abu glēgil 'wolf' [...] » (

	(33) Ṣanˤānī Arabic (Watson 2006: 192)
		tjaymal		'to make oneself (o.s.) appear beautiful'
	b) The meaning of approximation or attenuation appears the most widespread. Very
	common in the diminutives of adjectives, it is also to be found in some verbal
	diminutives such as in (34) and in participial diminutives as in (35) which indicate an
	incomplete change in color.
	(34) West Algeria: among the Ulâd Bṛâhim of Saïda (Marçais 1908: 107)
	tfêiqqöh	'to have a poor knowledge of the law'
	(35) Southern Tunisia: among the Marāzîg (Boris 1958)
	a.	mətḥêmər	'reddish'
	b.	məššêheb	'greyish'
				: 108)
	a.	tbêiles		'to play the devil'
	b.	tmêilöḥ		'to play the saint'
	(31) Southern Algeria: among the ˀArbâˤ nomads (Dhina 1938: 331)
	a.	tˤe i raf		'to play the expert'
	b.	tme i las		'to play the sweetish'
	c.	e šše i ṭar (< tše i ṭar) 'to swank'
	d.	tne i fax		'to play the little conceited'
	(32) Arabic dialects of eastern Arabia (Holes 2004: 109)
	a.	tṣēmax		'to pretend to be deaf'
	b.	thēbal		'to act as if stupid'

In Ḥassāniyya, the diphthong -aycan be reduced to the long vowel ē but never to ī as in many Maghreb dialects.

Some nouns can however form their diminutive plural directly on the base, e.g. ṛžäyžlä DIM of ṛažžālä 'men' (rather than a plural formed on the DIM.SG rwäyžəl).

This prefix a-< ˀais not governed by the same rules as the prefix a-/āof nominals such as axəlxāl, whether they have been borrowed from Berber or 'Berberized'. See above, §2.2.3.

Tauzin classifies them as follows: to laugh (1 and 2); on wisdom (3 to 9); on the cunning of women (10 and 11); on demons (12 and 13); on blacksmiths, slaves (14 to 17); for children (18 to 28).

The events as well as their unfolding are not always easy to understand, Tauzin stayed as close as possible to the original versions provided by local speakers. One must therefore not be surprised if some events remain somewhat hazy.

The same is true for gläyyil and gräyyib -a phenomenon which is found namely in Negev Arabic: « Lexemes that have superseded their original forms include ṣġayyir 'small', gḷayyil 'little', gṛayyib 'near', gṣayyir 'short' [...] »(Henkin 2008: 366).

Fwaym is the specific diminutive form of an originally biliteral root noun (cf. classical Arabic fam 'mouth').

Leaving the young bride without consummating the marriage is a way for the young man to avenge his honor. He had only kept the little kid's foot in order to remind her of her past misdeeds.

ˤzäyyəz is an exception to the rule: it does not bear the ending -a, like the feminine noun ˤzūz.

In fact the feminine form for hedgehog is a neologism, only the masculine gänvūd 'hedgehog' is used normally.

Tauzin translated this by The little wild animal -a translation which does not make it possible to render the feminine gender of uḥayšiyyä and which is contested by my informant (the hassāniyya meaning is different from that of Classical Arabic).

'Tell me, this one who comes to woo you // What may he hope? 'You are certainly not // A buxom woman. 'It is so difficult to see you! // You are a bit dull and ugly 'And your sparse frizzy hair // Barely covers your ear.DIM, 'And in your mouth.DIM // In disarray are your little teeth.22 Small eyes are not a quality for the Moors. Mentioning them is therefore equivalent to criticizing the physique of the desired woman, which makes no sense in the context.23 It is the meter which imposes the classicized form baxīla (for bxîla).

Two adjectives (dxäynä DIM of daxnä and šwäynä DIM of šäynä) and four nouns (ṛwäyṣ DIM of ṛāṣ; uḏäynä DIM of ūḏən F.SG; f̣ f̣ w äym [irregular] DIM of ṿuṃṃ and snäynä DIM of sənn).

This expression is difficult to understand, although it is clearly insulting because wähd is not part of the dialect's ordinary vocabulary.

Praise, which implies a certain reverence on the part of the praiser for the recipient -and often inferior status -is especially practiced in poetry by griots. Even for them however, praise is not devoid of risk as it is easy to offend one person by praising another (Taine-Cheikh 2004).

The diminutives are not the only double-derived forms in Arabic. It is a well known phenomenon which, as for derivation through prefixes and suffixes, does not contradict, in my view, the general rule of root and pattern crossing established byCantineau (1950). For a discussion of generativist theses, see[START_REF] Watson | Arabic morphology: diminutive verbs and diminutive nouns in San'ani Arabic[END_REF]

In Classical Arabic diminutive forms have been observed, not only as in Ḥassāniyya, for prepositions, adverbs and admirative phrases, as well as for demonstratives and pronouns.

This could be a dialect spoken by former Bedouins (as is Ḥassāniyya) or a 'Bedouinized' dialect.

Iraqi Arabic has retained the pattern specific to diminutives of adjectives of color and deformities. It has even extended it to the feminine: ixēris F ixērisa -DIM of axras F xarsa 'mute'(Masliyah 1997: 77).

A trend observed both in Arabic[START_REF] Colin | Arabe marocain. Inédits de Georges S. Colin édité par Dominique Caubet & Zakia Iraqui-Sinaceur[END_REF]: 114-8, Caubet 1991[START_REF] Lentin | À propos de la valeur "intensive" de la IIème forme verbale en arabe syrien : modalité et expressivité. Vers un renouvellement du système verbal[END_REF]) and crosslinguistically[START_REF] Morgenstern | La réduplication : universaux iconiques et valeurs en systèmes[END_REF] Ponsonnet, this volume).

 32 Use of the so-called 'feminine' marker for the diminutive is particularly clear in Berber Zenaga (Taine-Cheikh 2002), even though it can also serve to express the augmentative in other Berber languages[START_REF] Vycichl | Diminutiv und Augmentativ im Berberischen[END_REF][START_REF] Kossmann | Derivational gender in Moroccan Berber: examples from Ayt Seghruschen[END_REF]).

See also the suffixes -eika and -ča in Uzbekistan Arabic and the suffixes -īn(a) and -ett(a) of Romance origin in Maltese(Zimmermann 209: 616; Mifsud 2008: 153).

In Moroccan Arabic, it is often the pattern R 1 aR 2 R 2 ūR 3 where š replaces R 3 (whence the geminate R 2 ).

I can say however that in the case of Zenaga Berber, the prefixal diminutive morpheme (M ag-) does in fact stem from the meaning 'son' (see[START_REF] Vycichl | Diminutiv und Augmentativ im Berberischen[END_REF], Taine-Cheikh 2002).

In a dialect such as that of Dellys in Algeria, diminutives are practically mandatory when addressing young children(Souag, pers. comm. 03/2016).

In Negev Arabic, empathetic diminutives are mostly used by women and older men whereas « [t]he pejorative function of diminutives is less gender-based »(Henkin (2010: 179).

On « enunciative distance », seeKerbrat-Orecchioni,1980: 32; Taine-Cheikh 1988: 99-100.

XXIX-XXX (1984-1986), 73-106.


Torbjørn Cunis

Laurent Burlion

Jean-Philippe Condomines
email: torbjoern.cunis|jean-philippe.condomines@recherche.enac.fr

Non-linear Analysis and Control Proposal for In-flight Loss-of-control

Keywords: Aerospace, UAVs, Navigation, Guidance, and Control, Upset recovery, Application of nonlinear analysis and design, Stability of nonlinear systems, Control of constrained systems, Control of bifurcation and chaos

In-flight loss-of-control (LOC-I) still poses a severe threat to today's commercial aviation. Hence, we review the literature for non-linear analysis and control methods of LOC-I and upset recovery. Using state-of-the-art methods such as continuation theory and reachability estimation, we sketch an analysis of an aircraft's flight envelope in terms of its trim conditions and propose control approaches both within and outside the envelope.

INTRODUCTION

Over the past three decades, in-flight loss-of-control events (LOC-I) have remained the foremost cause of fatal accidents [START_REF] Boeing | Statistical Summary of Commercial Jet Airplane Accidents[END_REF][START_REF] Boeing | Statistical Summary of Commercial Jet Airplane Accidents[END_REF][START_REF] Boeing | Statistical Summary of Commercial Jet Airplane Accidents[END_REF]. With a contribution of almost 50 % of fatalities in civil aviation while representing less than a tenth of the total accidents 1 , the International Air Transport Association (IATA, 2015a) lists LOC-I as "highest risk to aviation safety." In response, aircraft manufacturers, commercial airlines, and national and international authorities and association have provided procedures and trainings for flight crews in order to tackle-or even avoid-events of LOC-I (Carbaugh et al., 2008;IATA, 2015b).

The Federal Aviation Administration (FAA, 2016) defines LOC-I flight events as deviation from the desired flight condition, "often" leading to upsets characterized by unstable, highly non-linear behaviour of the aircraft aerodynamic system, such as stall, spin, and post-stall rotations [START_REF] Chambers | Aerodynamic Characteristics of Airplanes at High Angles of Attack[END_REF]. Control approaches for upset recovery include throttle-only control [START_REF] Burcham | Development and Flight Test of an Emergency Flight Control System Using Only Engine Thrust on an MD-11 Transport Airplane[END_REF][START_REF] Burcham | Manual Throttles-Only Control Effectiveness for Emergency Flight Control of Transport Aircraft[END_REF][START_REF] Urnes Sr | Flight Control for Multi-engine UAV Aircraft using Propulsion Control[END_REF] in case of hydraulic failures of the control surfaces, linear-optimal control [START_REF] Chang | Aircraft Trim Recovery from Highly Nonlinear Upset Conditions[END_REF], L 1 adaptive control [START_REF] Xargay | L 1 adaptive controller for multi-input multi-output systems in the presence of nonlinear unmatched uncertainties[END_REF], state-based switching control [START_REF] Engelbrecht | A Multi-mode Upset Recovery Flight Control System for Large Transport Aircraft[END_REF], non-linear dynamic inversion [START_REF] Stepanyan | Stall Recovery Guidance Algorithms Based on Constrained Control Approaches[END_REF], and Lyapunov-based control [START_REF] Engelbrecht | Automatic Flight Envelope Recovery for Large Transport Aircraft[END_REF].

Several LOC-I prevention and upset recovery systems were designed [START_REF] Engelbrecht | A Multi-mode Upset Recovery Flight Control System for Large Transport Aircraft[END_REF][START_REF] Engelbrecht | Automatic Flight Envelope Recovery for Large Transport Aircraft[END_REF]Stepanyan et al., 2016a,b;[START_REF] Tekles | Design of a Flight Envelope Protection System for NASA's Transport Class Model[END_REF] for the NASA generic transport model (GTM; [START_REF] Jordan | AirSTAR: A UAV Platform for Flight Dynamics and Control System Testing[END_REF] and evaluated in pilot-in-the-loop simulations (Cunning-1 Counting both fatal and non-fatal accidents. ham et al., 2011;[START_REF] Crespo | Analysis of Control Strategies for Aircraft Flight Upset Recovery[END_REF][START_REF] Richards | Vehicle Upset Detection and Recovery for Onboard Guidance and Control[END_REF] and in-flight tests [START_REF] Gregory | Flight Test of L 1 Adaptive Control Law: Offset Landings and Large Flight Envelope Modeling Work[END_REF]. The GTM, a down-scaled model of a typical transport aircraft has been studied exhaustively [START_REF] Foster | Dynamics Modeling and Simulation of Large Transport Airplanes in Upset Conditions[END_REF][START_REF] Frink | Computational Aerodynamic Modeling Tools for Aircraft Loss of Control[END_REF] and provides an open-source six-degree-of-freedom model for MATLAB/Simulink (NASA, 2016).

For the analysis of non-linear regimes, two disparate methods have recently been applied: bifurcation and reachability analysis. The first has been developed from the mathematical continuation and bifurcation theory to a stateof-the-art analysis tool for trim conditions and periodical orbits of the non-linear aircraft dynamics (cf. [START_REF] Caroll | Bifurcation Analysis of Nonlinear Aircraft Dynamics[END_REF][START_REF] Jahnke | Application of Dynamical Systems Theory to Nonlinear Aircraft Dynamics[END_REF][START_REF] Goman | Application of bifurcation methods to nonlinear flight dynamics problems[END_REF][START_REF] Kwatny | Nonlinear Analysis of Aircraft Loss of Control[END_REF][START_REF] Engelbrecht | Automatic Flight Envelope Recovery for Large Transport Aircraft[END_REF] for more than thirty years now. The second, on the other hand, is a relatively new technique based on hybrid system theory, where sub-sets of the state space are evolved over time, determining possible violations of predefined constraints (e.g., [START_REF] Lombaerts | Safe maneuvering envelope estimation based on a physical approach[END_REF][START_REF] Mcdonough | Recoverable sets of initial conditions and their use for aircraft flight planning after a loss of control event[END_REF][START_REF] Mcdonough | Fast Computable Recoverable Sets and Their Use for Aircraft Loss-of-Control Handling[END_REF]. A particular form of reachability analysis is the computation of control-invariant sub-sets, or safe sets [START_REF] Lygeros | On reachability and minimum cost optimal control[END_REF][START_REF] Tedrake | LQR-trees: Feedback Motion Planning via Sums-of-Squares Verification[END_REF][START_REF] Chakraborty | Nonlinear region of attraction analysis for flight control verification and validation[END_REF].

The CONVEX thesis 2 aims to contribute to LOC-I handling by design, implementation, and flight-test evaluation of non-linear upset recovery for micro air vehicle (MAV) while benefiting from the experimental gestalt of an MAV. In this paper, we present and propose first steps towards non-linear upset recovery control, including but not limited to formal definitions and a brief introduction to continuation and bifurcation theory; a non-linear analysis of trim conditions within, but undesired equilibria and periodic orbits beyond the flight envelope; and control strategies for both stable flight and upset recovery.

DEFINITIONS

The flight dynamics of an aircraft are commonly given by a system of first-order differential equations d dt

X = f (X, U ) (1)
of the states X and inputs U : the state vector at time t is then given by X = [ V, γ, χ, θ, φ, ψ, p, q, r, x g , y g , z g ] , T

(2) where V , γ, χ representing the aircraft's velocity (flightpath), θ, φ, ψ the aircraft's attitude with respect to the normal earth-fixed axes, p, q, r the aircraft's angular rates with respect to the body-fixed axes, and x g , y g , z g the aircraft's position in the normal earth-fixed reference system; the control inputs to the aircraft are further given by the input vector

U = [ η, ξ, ζ, T ] , T (3)
with elevator, aileron, and rudder deflections η, ξ, ζ and thrust T .

The state space is the subset of all possible states, X ⊆ R n , (4) and we define the flight envelope as set of desired states X E ⊂ X as well as the set of viable control inputs, U ⊂ R m .

Finally, we have the (controlled) flow of the system (1) as

φ(X 0 , u(•) , t) = x(t) (5) for u : t → U and x(•) is solution to the initial value problem ẋ(t) = f (x(t) , u(t)) with x(0) = X 0 .
The safe set of X is then the largest control-invariant set, i.e.

X safe = {X ∈ X |∃u(•) ∈ U . ∀t ≥ 0. φ u (X, t) ∈ X E } . (6)
Given an initial state X 0 outside the flight envelope, i.e. X 0 ∈ X -X E , upset recovery is formally given as the task to find a control law u : R → U such that for a t R > 0, φ(X 0 , u(•) , t) ∈ X E for all t > t R . Candidate upset recovery approaches can be evaluated by the time of recovery t R , the initial set of states X 0 ⊆ X -X E which can be recovered by the control law u(•, •) in time t R ≤ tR , and the undesired region of the state-space intersected by the controlled flow, that is φ(X

0 , u(•) , [0, t R ]) ⊆ X -X E .
3. CONTINUATION AND BIFURCATION [START_REF] Crawford | Introduction to bifurcation theory[END_REF] relates a bifurcation point to a significant change in the dynamics of a system. Here, given a dynamic system similar to (1) Ẋ = f (X, µ) , (7) where X denotes the state vector again and λ the continuation parameters, which may include state variables, control inputs, system parameters, and external influences [START_REF] Kwatny | Nonlinear Analysis of Aircraft Loss of Control[END_REF]. Recalling that any point (X * , µ * ) is an equilibrium if and only if

f (X * , µ * ) = 0, (8)
it is furthermore a bifurcation point if at least one real eigenvalue λ-or complex-conjugated pair-crosses the imaginary axis, i.e. λ(X * , µ * ) = 0. By continuation of the parameters µ, bifurcation analysis discusses creation, vanishing, and changes of stability of the branches of equilibria of (7) as function of µ * .

TRIM CONDITION ANALYSIS

In [START_REF] Kwatny | Nonlinear Analysis of Aircraft Loss of Control[END_REF], the longitudinal trim conditions of the GTM have been analyzed. By assuming a considerably damped pitch motion, that is q = 0, the system dynamics of f are restricted to speed V and flight-path angle γ as states, elevator η and thrust T as inputs, and the angle of attack α as output. A trim condition is given by

(V * , γ * , η * , T * ) if and only if f V,γ (V * , γ * , η * , T * ) = 0.
(9) As obtained from Fig. 1, for speeds greater than a certain speed V there are two trim conditions at low and high angle of attack, respectively. While there are no trim conditions for V < V , at V = V the trim conditions diminish to a single one. In other words, for flights slower than V there are no conditions, and thus no angle of attack, to maintain trimmed flight. Recall that is just the definition of stall, i.e. V ≡ V Stall , and the stall speed varies with the flight-path angle γ. As for V > V Stall (γ) there are two branches of trim conditions, the condition at V Stall (γ) is a bifurcation point and maneuverability of the system is lowered [START_REF] Berg | Unfolding the zero structure of a linear control system[END_REF][START_REF] Kwatny | Nonlinear Analysis of Aircraft Loss of Control[END_REF].

While the limits of elevator deflection and thrust obviously restrict the achievable trim conditions, we define without loss of generality the flight envelope around the set of (viable) trim conditions,

X trim = (V, γ) ∈ X ∃ (η, T ) ∈ U . f V,γ (V, γ, η, T ) = 0 .
(10) Hence, the stall trim conditions constitute a boundary of the flight envelope.

LQR SAFE SET ANALYSIS

The system can be linearized at a reasonable large number of trim conditions. Thus, one can easily derive a set of linear controllers for stable flight in the flight envelope.

Let K i be a linear-optimal regulator (LQR) and S i the corresponding solution to the algebraic Riccati equation for a linearization of f V,γ around a trim condition (X * i , U * i ). We can employ S i for a quadratic Lyapunov-candidate function [START_REF] Tedrake | LQR-trees: Feedback Motion Planning via Sums-of-Squares Verification[END_REF]) [START_REF] Slotine | Applied Nonlinear Control[END_REF]. Thus, X stable i is safe in the sense of (6). ).

V i = 1 2 XT S i X > 0, X = 0, (11)
X = X -X * i , to have X stable i = X ∈ X V i X ≤ ρ i with ρ i > 0 being a stable neighbourhood of X * i if and only if d dt V i = XT S i f X * i + X, U * i -K i X < 0 (12) for all X ∈ X stable i -{X * i } (Slotine
Introducing a polynomial, positive semi-definite Lagrange multiplier h ∈ K [X] Tedrake et al. reduce ( 12) to a sumof-squares problem [START_REF] Parillo | Semidefinite programming relaxations for semialgebraic problems[END_REF]:

XT S i f + h X ρ i -V i X ≤ -X 2 2 , ( 13
)
where • 2 denotes the L 2 -norm and > 0. Here, ρ i -V i (•) equals the signed distance to ∂X stable i and with h(•) ≥ 0 for all X = 0 compensates for non-negative derivatives of V i outside the stable neighbourhood. Hence finding a Langrange multiplier proofs the stability of X * i in X stable i by the linear controller K i . Outside the flight envelope, and beyond the stall speed in particular, control of the aircraft may be restricted by limited control effectiveness [START_REF] Kwatny | Nonlinear Analysis of Aircraft Loss of Control[END_REF] and non-linear modes like periodic orbits. In order to tackle these, we propose a further, non-linear analysis; eventually, we will develop a selecting approach of suitable trim conditions on the boundary of the flight envelope and a flight control law for recovery from an upset condition to the respective trim condition.

Except for [START_REF] Xargay | L 1 adaptive controller for multi-input multi-output systems in the presence of nonlinear unmatched uncertainties[END_REF], uncertainties of the underlying aerodynamic model or in the outputs are not considered in the literature reviewed. Hence a first step towards an upset recovery law is to estimate the effects of uncertainties to the flight envelope as defined in (10).

CONCLUSION

In this paper, we have reviewed recent LOC-I prevention and upset recovery approaches, various linear and nonlinear control methods, and analysis techniques such as bifurcation and reachability. We have then formally defined an aircraft dynamic system, its state space, and the flight envelope. By considering the non-linear analysis of the generic transport model by [START_REF] Kwatny | Nonlinear Analysis of Aircraft Loss of Control[END_REF] we have exemplary shown the results of bifurcation theory and continuation and discussed the outcomes.

Along the branch of level flight trim conditions, we have derived linear-quadratic optimal regulators (LQR) around selected trim conditions and computed the safe sets as stable neighbourhoods of the trim conditions for the linear controlled system. We thus have proposed a control approach based on the bifurcation analysis and reachability.

As argued, further analysis of the non-linear dynamics are required; in particular, uncertainties need to be taken into account before any control law can be designed and implemented to an MAV. We also expect further analysis to give insights on suitable recovery approaches which are able to control the aircraft outside the flight envelope.

  Fig. 1. Trim conditions obtained for level flight and varying speed, with respect to angle of attack, elevator deflection, and thrust. At stall speed, a saddle-node bifurcation occurs ( ) which depends on the respective flight-path angle ().

Fig. 2 .

 2 Fig. 2. Safe sets of LQR control at selected trim conditions ( ) at level flight and low angle of attack.

Command based On Non-linear piloting to bring an aerial Vehicle back in its flight Envelope X E ; PhD thesis of the first author.

Appendix A. PHUGOID DYNAMICS

In this paper, we were discussing the phugoid dynamics of the GTM adapted from [START_REF] Kwatny | Nonlinear Analysis of Aircraft Loss of Control[END_REF],

where S and c a are wing area and aerodynamic mean chord; C D , C L , C X , C Z , C m are the aerodynamic coefficients of drag, lift, force body x-axis, force body z-axis, and moment body y-axis, respectively, as functions of angle of attack, elevator deflection, and pitch rate; x ref cg , z ref cg , x cg , z cg are the reference and actual position of the center of gravity with respect to x and z; and l t is the engine's displacement along the z-axis.

A.1 Restricted longitudinal model

To reduce the number of states, we consider the phugoid motion to be damped-either a priori or by a suitable damping system in inner-loop-and q ≡ 0 ( [START_REF] Kwatny | Nonlinear Analysis of Aircraft Loss of Control[END_REF]. For the trim condition we then get M = I y q = 0 (A.2) in addition to V = γ = 0 at (V * , γ * , η * , T * , α * ).

A.2 Linear control approach

For design and analysis of linear control, we can assume an inner controller of the angle of attack and neglect the effect of the elevator to the lift and drag coefficients. We thus get the phugoid dynamics around a trim condition

where T , α are inputs to the inner system.

A.3 Aerodynamic coefficients

Using the MATLAB Curve fitting toolbox, the aerodynamic coefficients of the generic transport model in the body axis system has been fitted to the polynomials C X (α, η) = -0.0186 + 0.2413α -0.0135η + 1.4957α 2 + 0.1849αη -0.0941η 2 -7.4482α 3 -0.2617α 2 η + 0.2723αη 2 + 4.5867α 4 + 0.0628α 3 η -0.2583α 2 η 2 (A.4) C Z (α, η) = -0.0418 -5.2246α -0.4420η + 3.6670α 2 + 0.0866αη -0.2135η 2 + 8.7973α 3 + 0.5947α 2 η -0.0499αη 2 -6.8839α 4 -0.3686α 3 η + 0.3358α 2 η 2 (A.5) C m (α, η) = + 0.1866 -1.5743α -1.7199η + 2.1987α 2 + 0.0662αη -0.6995η 2 -4.4762α 3 + 3.1578α 2 η + 0.1736αη 2 + 2.2467α 4 -1.7551α 3 η + 0.7337α 2 η 2 (A.6) and the lift and drag coefficients can be calculated by rotation, C L (α, η) = -C Z (α, η) cos α + C X (α, η) sin α, (A.7) C D (α, η) = -C Z (α, η) sin α -C X (α, η) cos α, (A.8) from body to air-path axis system.


Amelia Carolina Sparavigna

The group of the Fermat Numbers

Keywords: generalized sum, groups, Abelian groups, transcendental functions, logarithmic and exponential functions, Fermat numbers

In this work we are discussing the group that we can obtain if we consider the Fermat numbers with a generalized sum.

Turin, 24 May 2018. DOI: 10.5281/zenodo.1252422

In [START_REF] Weisstein | Fermat Number[END_REF] we find that there are two definitions of the Fermat numbers. We have a less common definition giving a Fermat number as F n =2 n + 1 , which is obtained by setting x=1 in a Fermat polynomial of x, and the commonly encounter definition F n =2 2 n +1 , which is a subset of the previous assembly of numbers. Here we will consider numbers F n =2 n + 1 and -as we have recently proposed in [START_REF] Sparavigna | The q-integers and the Mersenne numbers[END_REF] for qintegers and Mersenne numbers -investigate the set of them to find its generalized sum which defines the operation of the group.

Let us remember that a group is a set A having an operation • which is combining the elements of A.

That is, the operation combines any two elements a,b to form another element of the group denoted a•b. To qualify (A,•) as a group, the set and operation must satisfy the following requirements.

F m ⊕F n =2-F m -F n + F m F n =(1-F m )+(1-F n )+ F m F n (1)
To have [START_REF] Weisstein | Fermat Number[END_REF], let us evaluate:

F m +n =2 m +n +1=F m ⊕F n =2-F m -F n + F m F n =2-(2 m +1)-(2 n +1)+(2 m +1)(2 n +1) 2 m+ n +1=2-2 m -2 n -2+2 m 2 n + 2 n + 2 m +1
This gives also the closure of the group.

We can provide a recurrence relation as:

F n +1 =2 n+1 +1=F n ⊕F 1
From (1), we can see that the neutral element is not 0. We have to use as a neutral element the integer 2, which is F 0 =2 0 + 1=2 and then an element of the group. We have:

F n ⊕F 0 =2-F n -F 0 + F n F 0 =F n
The opposite element is defined by F n ⊕Opposite(F n )=2 . We have:

Opposite( F n )= F n F n -1 =1+ 2 -n =F -n (2)
Then, to have a group we need to add numbers (2) to the set of the Fermat numbers.

Therefore, we consider 2 as the neutral element , and the opposite element as given by ( 2).

Let us consider three Fermat numbers F n , F m , F l ; to have a group we need the associativity of the generalized sum, so that (

F m ⊕F n )⊕F l =F m ⊕( F n ⊕F l ) .
Let us call x=F n , y =F m , z=F l and evaluate:

(x⊕ y )⊕z=2-( x⊕ y )-z+(x⊕ y ) z=2-2+ x+ y-xy-z +2 z -xz-yz + xyz (x⊕ y )⊕z=x+ y+ z-xy-xz-yz + xyz (3)
And:

x⊕( y ⊕z)=2-x-( y ⊕z)+ x ( y⊕z )=2-x-(2-y-z + yz)+ x( 2-y-z + yz ) x⊕( y ⊕z)=x + y + z-xy-xz -yz+ xyz (4)
From ( 3) and ( 4), we have the associativity. The commutativity is evident.

We have already considered the generalized sum (1) in a recent work [START_REF] Sparavigna | Generalized Sums Based on Transcendental Functions[END_REF].

In [START_REF] Sparavigna | Generalized Sums Based on Transcendental Functions[END_REF], we consider some functions G(x) , having inverses so that G -1

(G( x))=x , which are generators of group law [START_REF] Sicuro | Groups, information theory, and Einstein's likelihood principle[END_REF][START_REF] Curado | A new entropy based on a group-theoretical structure[END_REF][6]:

Φ( x , y)=G(G -1 ( x)+G -1 ( y ))
The group law is giving the generalized sum of the group x⊕ y=G(G

-1 (x )+G -1 ( y)) .
In [START_REF] Sparavigna | Generalized Sums Based on Transcendental Functions[END_REF] we considered the following generator and inverse:

G(x)=e -2 x ( e 2 x +1) G -1 ( x)=ln( 1 √x-1 ) (5)
and investigate a possible group from them. The group law Φ(x , y) gives the generalized sum: Let us also note that, if we use (5), we need x >1. And this is a condition satisfied by the Fermat numbers and their opposites (2).

x⊕ y=G(G -1 (x )+G -1 ( y))=G( ln( 1 √x-1 )+ ln( 1 √ y -1 ))=G( ln( 1 √x-1 1 √ y-1 ))=G( ln 1 Z ) G(ln 1 Z )=e

Closure:

  For all a,b in A, the result of the operation a•b is also in A. Associativity: For all a,b and c in A, it holds (a•b)•c = a•(b•c). Identity element: An element e exists in A, such that for all elements a in A, it is e•a = a•e = a. Inverse element: For each a in A, there exists an element b in A such that a•b = b•a = e, where e is the identity (the notation is inherited from the multiplicative operation). A further requirement, is the commutativity: For all a,b in A, a•b = b•a. In this case, the group is an Abelian group. For an Abelian group, one may choose to denote the operation by + , the identity element becomes the neutral element and the inverse element the opposite element. In this case, the group is called an additive group. The generalized sum for the Fermat numbers F n =2 n + 1 is:

  x-y+ xy=(1-x)+(1-y)+ xy (6)And (6) is the generalized sum (1) proposed for the Fermat numbers.