Ré Mi Anicotte

Dominique Arrighi

Orly Cohen

Zaki Coussa

Description of the linguistic expressions of fractions

Keywords: Fractions, Numerals, Grammatical number, Partitive expressions, Divisible semantic units, Language planning ablative, ACC: accusative, ACT: active, ADJ: adjective, ART: article, CARD: cardinal, COL: collective, DAT: dative, DECL: declarative, DEF: definite, DET: determiner, DU: dual, F: feminine, FRAC: fractional, particle, PASS: passive, PERF: perfective, PL: plural, POSS

This paper proposes a typology of fractional numbers. A few fractional numbers can be expressed by suppletive (non-systematic) forms, whereas analytical (systematic) linguistic patterns of formation produce "bi-dimensional" numerical forms which refer to both the numerator and the denominator (double argument), or "mono-dimensional" forms which refer to only one of these numbers (single argument). Moreover, a fraction in a partitive expression can be an indivisible semantic unit or may, on the contrary, have a noun or a measure word inserted between its constituents.

The most inclusive source for the general history of fractions today is still Benoit, Chemla & Ritter's History of Fractions, Fractions of History, it focused on the concept of fractional numbers in ancient civilizations and only incidentally collected data on their linguistic expression. In fact, the linguistic information is often irretrievable from the symbolic notations of numbersthis means we do not know how they were read and we are not even sure they reflected spoken words (Ritter 2001)making the available written material at times inadequate for a linguistic study. To illustrate this phenomenon, let us consider our modern notation 100 for one hundred {1}{100} in English with one {1} and cent {100} in Modern French with no {1}. This gives a simple illustration of how the numerical symbols can hide the difference between various linguistic patterns 2 . A more complex situation is found with the late Egyptians, who manipulated what we conceptualize as unit-fractions 3 and wrote them in hieratic with a sign which meant part placed on top of a sign denoting the denominator, however we are not sure of the morpho-syntactic construction of the linguistic expressions corresponding to these notations 4 . Benoit et al. (1992) made us understand a probable reason why fractional numbers were seldom-considered in the study of natural languages. It is because, ultimately in this field, the specifically linguistic questions seem secondary to the conceptual frameworks. In other words, the emergence of linguistic expressions for fractions was driven much more by conceptual developments among restricted groups of technically trained peopleand of what (Haugen 1983) called the corpus planning 5 they would initiatethan by any natural or normal language evolution, be that phonetic, morphosyntactic or lexical.

Nevertheless, in this paper, we can offer a typology which enables to describe and classify the linguistic expressions for fractions. This is achieved according to way the numerical informationthe values of numerator and denominatoris expressed within fractional numbers, regardless of the intricacy of the morpho-syntactic means used to build these expressions, or the pronoun; an integer n followed by a grammatical indication related to numerals (e.g., CARD, FRAC, ORD, etc.): a numeral that expresses the integer n. These standard abbreviations are augmented with: {n}: an unanalysable (or unanalysed) form for the integer n; {n/d}: a suppletive form for the fraction n/d. 2 Following Huddleston & Pullum (2002), one needs to differentiate numerals which denote linguistic expressions (e.g., English 'one hundred' or French 'cent') versus numbers which are the numerical meaning or content (e.g., the quantity 100). Please note that 19 th century French had 'un cent' {1}{100}. 3 Egyptian calculations with fractions resulted in unit-fractions, while non-unit fractions were expressed as sums of unit-fractions (any fraction can indeed be expressed as the sum of unitary fractions.); e.g., 4/7 was conveyed as the sum of 1/2 and 1/14 both written in hieratic (Gardiner 1957: 196). The only known exceptions were the use of special signs for the two non-unit fractions 2/3 and 3/4. 4 Sethe (1916: 84-86) relied on Coptic sources and conjectured a genitive construction starting with 'r' (part) and linked by a preposition to a cardinal which expressed a denominator; e.g., 1/5 would have been expressed as 'part of five'. 5 Haugen distinguished four steps: selection of norm (which is societal and somehow exterior to the language); codification of the norm; implementation of function (includes the activities of writers and institutions); and elaboration of function (involves the production of a linguistic corpus complying with the norm).

partitive expressions including them. Actually, putting aside most of the morpho-syntactic features in the examples I gathered was essentially how I could arrive at my typology.

As should be expected, the majority of our evidence, either formal or linguistic, come from secondary literature and technical texts, e.g., mathematics, architecture, geography, hydraulics.

I tried to wider the range of languages I could account for. But I faced the lack of data in the languages of cultures with no strong mathematical traditions. These cultures did not develop linguistic means to express arithmetical matters they were not interested in, or the influence of the people who cared for these issues was not strong enough to pervade the whole society, nor the general language which is documented today. Be that as it may, these sociolinguistic aspects are outside the scope of this paper.

Producing a large-scale survey, or providing an organized language corpus, seems, at best, hardly feasible in the current state of the literature. Even the linguistic data base WALS Online6 does not consider fractions, and this is because monographies on individual languages usually do not do it either. As a matter of fact, we hope our contribution will initiate enough interest for researchers to systematically gather the relevant information on these matters.

Our typology proposes to categorize any expression of a fractional number as either a suppletive form (non-systematic) or an analytic form (systematic). The set of suppletive forms in a given language maps a finite set of fractions. The analytic form can split into monodimensional (systematic, single argument) and bi-dimensional (systematic, double argument), and they can define limited sets or general series of fractions.

The simplicity of this typology might reflect a case of what biologists call convergent evolution, that is the independent evolution of similar features in species of different periods or epochs in time, where similar forms or functions were created that were not present in the last common ancestor of those species. In the field of linguistics, we would talk of similarities in the patterns of expression which are not the result of a genetic relation nor of a process of borrowing, but emerged separately to answer the same needs.

In order to exemplify each type of expression, this paper reorganized, according to the typology, the data which initially enabled us to induce relevant descriptive criteria. One sideresult was the possibility to deduce and assess some capabilities of each type of expression.

SUPPLETIVE FORMS

Suppletive forms can be used to express a few particular fractions, they are not obviously derived from cardinal forms, at least from a synchronic view-point. In terms of their capacities in quantification or partitive phrases, Tibetan phye-ka and Thai khrueng are juxtaposed with a noun or measure word in the same manner as names of integers.

Arabic niṣf requires nouns to bear the definite article as shown in (1). English one half can make use of a preposition in (2) forming a usual partitive expression, however the juxtaposition one half the population is possible also, perhaps more colloquial.

( By the way, the two terms uncia and scripulum, besides their numerical values, were alsoor originallyused in metrological scales: uncia12 was the suppletive form {1/12}, a unit of area

(1/12 of 1 iugerium), a monetary unit and a unit of weight (1/12 of 1 as); scripulum was {1/288}

and also a unit of area (1/288 of 1 iugerium). However, according to (Maher and Makowski 2001), in example (3) the two words uncia and scripulum are to be understood as fraction names, not as submultiples within a metrological scale 13 .

Contemporary Chinese has a suppletive term bàn for 1/2 which cannot be analysed from the names è r and liǎng of the integer 2.

Classical Chinese had suppletive forms for 1/2, 1/3 and 2/3, respectively bàn (half), shǎobàn (the smaller half) and tàibàn (the larger half), both derived from bàn {1/2}; they are shown in the examples ( 4) and ( 5). These suppletive forms of fractions can be used in the same manner as integers with nouns and measure words 14 .

( Some texts 16 also contain ruòbàn (the weak half) for 1/4 and qiángbàn (the strong half) for 3/4, and the term zhōngbàn (the middle half) for 1/2. These termsbut {1/2}could be replaced by analytical forms made with 3.CARD fēn and 4.CARD fēn. Most of these suppletive forms were lost in Contemporary Chinese: only bàn {1/2} and yī bàn i.e. 1.CARD {1/2} remain 13 Another case of fluidity between fraction names and submultiples of a metrological scale can be observed in Egyptian (Ritter 2003). This lexical interchangeability can be understood in terms of the conceptual continuum between subunits and fractions, see Ritter (1992, 2001, 2003). 14 One can see Anicotte (2015 a, 2015 b, 2017, 2019 a, 2019 b) about the fraction numbers in Chinese. 15 The Suàn shù shū [A Book on the Reckonings with Counting Rods] is an excavated text dated to the beginning of the 2 nd century BCE. When we wrote this paper, two other long mathematical texts were available: Suàn shù [Reckoning Procedures] also dated to the beginning of the 2 nd century BCE, and Shù [Numbers] dated to the 3 rd century BCE. They contain hundreds of fractions expressed in natural language, with no symbolic notation. 16 The term ruòbàn for 1/4 is found in the first chapter of the Xiàhóu Yáng suàn jīng [Xiahou Yang's Mathematical Manual] (oldest version 1084). The terms qiángbàn for 3/4 and zhōngbàn for 1/2 are found in the Shù shū jiǔ zhāng [Mathematical Treatise in Nine Sections] (1247) (Libbrecht 1973: 70-71).

for 1/2. Moreover, the regular analytical form 2.CARD fēn zhī 1.CARD is also available now. The term tàibàn (or dàbàn) remains as an approximate number meaning most, no longer as an exact number.

ANALYTICAL FORMS

Using non-analytical expressions appears viable for a finite list of fractions, but a generic linguistic pattern capable of expressing any fractional number would be analytic and account for both the numerator and the denominator as we conceptualize them today.

This requirement may suggest that expressions involving two integer names should cover all situations. But this is not the case. Besides the numerical phrases which I call bidimensional because they do refer to both the denominators and the numerators, we also encounter mono-dimensional phrases referring only to their denominators or their numerators.

The bi-dimensional patterns are the most general ones, in the sense that they can potentially express any fraction; or at least any proper fractions (smaller than 1, e.g., 1/3).

Besides, we found that the expression of so-called improper fractions (greater than 1, e.g., 4/3) can rely on the same pattern or not, they can also be expressed as mixed numbers (the sum of an integer and a proper fraction, e.g., 1+1/3 instead of 4/3).

3-1 Mono-dimensional forms

Some analytical mono-dimensional patterns can express potentially infinite sets of fractions. For example, unit-fractions in Latin, Sanskrit, and in Classical Chinese, are mono-dimensional forms which mention the denominator only, while our numerator 1 is not stated 17 .

The unit-fractions (let us say 1/d with the numerator 1 and the denominator d) in Classical Chinese would usually take the form d.CARD fēn stating the cardinal d, and fēn, a term whose original meaning is part (noun and verb), forming compounds like 4.CARD fēn in (6). I call 4.CARD fēn a mono-dimensional phrase because only the numeral 4.CARD is expressed. The numerator's name 1.CARD was not compulsory and was usually omitted when the fraction name occurred as a factor in a multiplication. However, it was possible to use it, typically when communicating the result of an operation. This is the case in ( 6) with the bi-dimensional phrase 16.CARD fēn 1.CARD.

17 Benoit et al. (1992) point out that the modern concept of a fraction, that is to say an object of the form p/q where p and q are integers, is a late-comer to history, and that almost all ancient literate societies only conceived of fractions in the form now called unitary, that is, with numerator 1or more precisely, conceptualized without numerator.

( Sanskrit 20 names for unit-fractions were also mono-dimensional compounds formed with the nouns bhāga-or aṃśa-, both meaning part, and added to either the cardinal form as in ( 10) and ( 11) or the ordinal form 21 of the denominator's name as in (12). 19 In the corpus available online or in printed editions, quindecim and nona in (8) are in fact respectively written XV and IX, but quinta is written in full letters, not as V, this is said for the sake of precision but is of no consequence for this paper. 20 Datta & Singh (1935: 185-186), Filliozat & Mazars (1987), Mazars (1992). The expression of fractions has most likely evolved during the history of Sanskrit and other Indic languages, however available instances are too scarce for us to perceive the changes. 21 Or rather a form derived from the ordinal according to the grammarian Pāṇini who commented that the term used in fraction names could show phonetic alterations with respect to the original ordinal form (Filliozat & Mazars 1987); Pāṇini is reputed to have been active around the 5 th century BCE.

( Besides names for unit-fractions (numerator 1), we wondered if there exited monodimensional names for fractions of a given denominator. We know only of one Chinese finite mono-dimensional series of terms which can express the tenths with the phrases n.CARD ché ng

(n being an integer from 1 to 9) which state only the numerator, while the denominator 10 is understood by a linguistic convention applicable to this particular pattern22 . This vernacular form occurs only in non-technical contexts (e.g., to convey the likelihood of an event).

3-2 Case of expression of the numerator with non-lexical means

One pattern in Arabic proved enlightening as to what we couldand shouldcount as an expression of the numerator.

Arabic names of denominators 3 through 10 are specific words built on the basis of a vocalic shift on the cardinal forms (Schulz et al. 2000: 214). For example the word 5.FRAC.SG for 1/5 in ( 14) may seem to be a mono-dimensional form derived from the cardinal 5.CARD given in ( 13).

(13) ḥamas (14) ḥumus 5.CARD 5.FRAC.SG '5' '1/5'

The name of 3/5 in ( 16) is obviously a bi-dimensional term made with the adjectival form of the cardinal for 3 and the plural form of 5.FRAC, that is to say the numerator 3 and the denominator 5 are expressed lexically.

(15) ḥumusain (16) ṯālṯat ʾaḥmās 5.FRAC.DU 3.ADJ 5.FRAC.PL '2/5' '3/5' Now, in the name 5.FRAC.DU of 2/5 in (15), the value 2 of the numerator can be readily retrieved from the dual form, and is therefore a bi-dimensional form. And then, we realise that in 5.FRAC.SG the value of the numerator 1 is explicitly expressed by the singular form. That is to say, this Arabic 5.FRAC.SG is also a bi-dimensional term, dissimilar to mono-dimensional expressions of unitary fractions, exposed in the previous section, and where grammatical number was not present in the expressions, or not even a feature of the language in Chinese 23 .

3-3 Bi-dimensional forms

First of all, we should be careful that not all bi-dimensional numerical phrases have to do with fractions.

When we see, or hear, the phrase 'five minus one', we retrieve two numbers in an operation, but there is no fraction involved. The situation is similar with 'five parts minus one (part)' which concerns not one fraction, but two.

Some bi-dimensional phares express proportions, e.g., 'a 40-60 split' in English or its Chinese equivalents '4 6 kāi' and '4 6 fēn ché ng'. We may consider that they underly fractions (40% and 60% for the English phrase, 4/10 and 6/10 for the Chinese one), but the denominators 100 or 10 are not expressed.

All these phrases should be kept outside in the present study which concerns the expression of the numerator and the denominator of a fraction. Now let us illustrate the morpho-syntactic variety of the bi-dimensional phrases for fractional numbers, that is to say the nature of the two numerical items, and the way they are joined together.

The denominator can be cardinals (denoted by CARD) as in Chinese, etc., ordinals (ORD) as in English, French, Latin, Sanskrit, etc., or have a special form (FRAC) as Arabic in the examples ( 14)-( 16), and also in German and Swedish which respectively possess the specific suffixes -tel and -del to produce denominator names 24 . Here is an example in Swedish:

(17) två femtedel-ar 2.CARD 5.FRAC-PL 25 '2/5' 23 Xu Dan (2012: 1-19). 24 These forms are invariable in German and can take the plural mark -ar in Swedish. Attention however to German irregular form Drittel (3.FRAC) and variant Siebtel (7.FRAC) for Siebentel (7.CARD-tel). 25 The suffix -del is used to form fraction numerals, therefore femtedelar in ( 17) could be formally construed as femte-del-ar and glossed 5.CARD-del-PL, but this would hide the specificity of the pattern.

The numerator can be in first position. This is case in English, French, German, Spanish, Swedish, Thai26 , etc. And, it was exemplified above for Arabic with the example ( 16) showing an adjectival form of the numerator's name preceding the plural form of the special numeral expressing the denominator. Below, we will see more examples with fraction in Modern Greek, Hebrew, Latin, and proper fractions in Sanskrit.

But the numerator may as well be in second position. This is exemplified, in this paper, by Chinese, Ancient Greek, Japhug, Tibetan, and improper fractions in Sanskrit. 27 A generic pattern for proper fractions in Latin can be seen in ( 18) and ( 19) cited by (Maher and Makowski 2001). It relies on the cardinal form of the numerator followed by the ordinal form of the denominator, which takes a plural mark, reflecting the pattern of quantification phrases.

( Fraction names in Modern Hebrew take their roots in Mishnaic Hebrew (1 st to 4 th centuries CE). The fraction name for 2/5 in (20) can be construed as a noun phrase with the name of the numerator and the name of the denominator, in this order. They are linked using the so-called construct pattern (based on the genitive form of the first constituent) and is to be understood as the genitival two of five:

(20) štei ḫamišiot 2.CARD.F.GEN 5.CARD.F.PL '2/5

The Modern Greek fraction name for 2/5 in ( 21) can be interpreted as a quantification phrase, with the denominator's name playing the role of the quantified item and bearing a plural marker. The Tibetan bi-dimensional pattern for fractions is explained by Goldstein et al. (1991: 200) and Wang Zhijing (1994: 108-109). The syllable cha 32 is placed after the denominator's name, and the compound "denominator's name-cha" is inserted before the numerator's name as 29 The final r in catur (a word for 1/4) could alternatively be written as its allophone ḥ forming the word catuḥpañcka. 30 The Bakhśalī manuscript is a mathematical text on pieces of birch bark which was excavated in 1881, it was arguably written around the 7 th century C.E. in a late or hybrid variety of Sanskrit (Hayashi 1995).

31 Waanders (1992) implies that very few instances of bi-dimensional expressions for fractions are known in Ancient Greek. 32 In Tibetan, fraction is said cha-grangs i.e. 'number of parts' from cha 'part' and grangs 'number'. The Tibetan-Chinese dictionary Zàng Hàn dà cí diǎn (Beijing: Mínzú chūbǎnshè [The Ethnic Publishing House], 1993, vol.1: 772) states that cha is a noun meaning part, and can also be a measure word meaning pair. It is used in the expression of fractions and also appears as a suffix in some nouns formed from other nouns.

shown in (26). A pattern which respects the order "noun + numeral" of Tibetan's quantification phrases, assuming we construe the compound suffixed with -cha as a nominal form and the numerator's name as the quantifier:

(26) lgna cha gnys 5.card cha 2.CARD '2/5' (Goldstein et al. 1991: 200) Japhug (Jacques 2008, 2021) is a Tibeto-Burman language of the rGyalrong branch (not a Tibetic language) spoken in Sichuan Province, China. The Japhug bi-dimensional expression of 1/3 in ( 27) relies on the words tɯ-tɯcɯr (part) and ŋgɯ (interior): When a measure word was involved, it was put directly after fēn as above in (29) and below in ( 31) and ( 33). The genitive morpheme zhī could optionally be inserted in the bi-dimensional construction, as can be seen in ( 30), ( 31) and ( 33 . 33 We only have a transmitted version. One can refer to Chemla & Guo (2004: 71-97) for a detailed history of the manuscript. 34 Qin Jiushao used a symbolic notation to write down numbers involved in operations; however, all the numbers were repeated in the text using the linguistic numeration. 35 The pattern "denominator's name + bun no + numerator's name" is found in Japanese (Martin 1975: 767) with bun, the Sino-Japanese reading of Chinese fēn, and no the Japanese determination linker. In However, when no measure word was involved, the pattern "denominator's name + zhī + numerator's name" (without fēn) was also found in both non-mathematical and mathematical texts 36 . Besides, there were scarce alternative bi-dimensional expressions of fractions other than the sequences with obligatory fēn and optional zhī, they were found in both excavated and transmitted texts, whether mathematical or not 37 . For instance, qǔ (take) could be found instead of zhī.

This variety of different forms at a given time means that the attempts to standardize the pattern of expression remained in the realm of technical texts written in a quasi-formal language by literati who devoted their work to mathematics, while the general corpuses would exhibit free formation.

In today's Chinese, the phrases "denominator's name + fēn zhī + numerator's name" contain the sequence fēn zhī and are indivisible: measure words are placed after these fraction names and not between their constituents "denominator's name + fēn" and "zhī + numerator's name" 38 . However, one should note that free formationor conservatismcan still be seen in Sinitic languages with, for example, zhī non-compulsory in Cantonese fraction names (Matthews & Yip 2011: 453).

We end here this accumulation of various morpho-syntactic situations, at times within one language. As we wrote at the beginning of the paper, this variety was what we had to face when we started to deal with this subject, and we used it to induce our typology of fraction expressions.

3-4 Bi-dimensional forms as divisible or indivisible semantic units

Let us now focus on one particular characteristic of bi-dimensional fraction names when they are put into a partitive expression: they can perform as indivisible semantic units or, on the contrary, the sequence of their constituents can be split by nouns or measure words.

This issue arose with the examples ( 29), ( 31) and ( 33) above where the constituents of the bi-dimensional phrases for fractional numbers in Classical Chinese could be separated, and a Korean there are "denominator's name + pun uy/ci + numerator's name" with uy the indigenous Korean genitive linker or ci its Sino-Korean counterpart (Martin 1992: 188 Examples of the use of the determination particle de are given in ( 40) and ( 41). The order "fraction name + de + noun" in (40) leaves the noun in the head position while the order "noun + de + fraction name" in (41) puts the fraction in the head position, slightly changing the focus of the noun phrase. The Contemporary Chinese indivisible bi-dimensional expressions for fraction names can be used in the same manner as integers when they occur in quantification phrases.

CONCLUSION

The words or phrases expressing fractions in one language can be indigenous or take their origin in contacts and loans from a literary language. They can be standardized or freely made up on the spot, using a great variety of morpho-syntactic means.

Some languages rely on the ordinal form or other special form of the denominator;

while cardinals and a given structure or order for the constituents can suffice in other languages.

The name of the numerator can be in first position (this is the case in Arabic, English, French, German, Modern Greek, Hebrew, Latin, Spanish, Swedish, Thai, etc.), or in second position (e.g., in Chinese, Ancient Greek, Japanese, Japhug, Korean, Tibetan, etc.), while Sanskrit had instances of both orders. The phrases expressing fractions can at times reflects the patterns of quantification phrases with adjectival or genitive constructions.

Our typology puts this diversity aside, and classifies fractional numbers according to the way the numerical information (numerator and denominator) is conveyed. We ended up with suppletive forms vs analytical forms (further divided into mono-or bi-dimensional).

In a given language, a finite list of fractions can be expressed with suppletive forms.

Analytic mono-dimensional phrases stating only the denominator or the numerator can also be observed, this scheme is restricted to a few series of fractions; for example, the infinite sets of unit fractions 1/n in Sanskrit, Latin, and Classical Chinese. Finally, the most generic patterns to express fractional numbers are analytical bi-dimensional phrases accounting, in lexical or morpho-syntactical ways, for both the numerator and the denominator; these bi-dimensional patterns of formation can potentially express any fraction of two integers.

The cross-linguistic variety of the partitive expressions with fractions is due to the cross-linguistic diversity of the partitive expressions themselves, but actually this was not relevant to this paper, except for the fact it is only when they become part of a partitive expression that it becomes apparent whether bi-dimensional phrases for fractional numbers are indivisible semantic units or not. Parenthetically, it should be noted that mono-dimensional expressions, as well as suppletive forms, do not work in the same manner as numerals in quantitative phrases in all languages. This is why, whenever enough data is accessible, not only the details of the formation of fractional numbers, but also their syntactic insertion in partitive expressions or quantification phrases deserve to accounted for.

  are not limited to proper fractions (smaller than 1). They can also be improper fractions (greater than 1) like 5/4 in (24), showing the name for 5/4 with the numerator put in final position, formed with the suffix -ka added to the cardinal 5literally: 'a fivesome of fourths' according to[Filliozat & Mazars 1987]) (Bakhśalī manuscript 30 ) Other examples of expressions with the denominator in first position are found in Ancient Greek 31 in (25) which shows a noun phrase expressing 2/5. It relied on the juxtaposition of the genitive form 5.CARD.GEN of the denominator's name followed by the noun moira (part), itself quantified by the numerator's name 2.CARD. -GEN.PL 5.CARD ART.DEF-ACC.F.PL 2.CARD part-ACC.F.PL of the five the two parts nem-ontai occupy-PRS.IND.MID.3PL occupy (Thucydides [5 th c. BCE], The History of the Peloponnesian War, 1-1-10)

  expressed as 'one part within three parts') (Jacques [personal communication by e-mail, March 2014]) In Classical Chinese, the bi-dimensional pattern stated the denominator and the numerator in this order; see examples (28)zhū' (zhū is a unit of weight) (Suàn shù shū, strip 28)

  In the Jiǔ zhāng suàn shù [Nine Chapters on the Mathematical Art] 33 , the item zhī is used systematically in bi-dimensional fraction names. This form was also used in the Shù shū jiǔ zhāng [Mathematical Treatise in Nine Sections] 34 (1247) and it was transmitted to Korea and Japan 35

	(30)	gè	shòu sān shí	fēn	zhī	è r shí sān
		each get	30.CARD	part GEN 23.CARD
		'each gets 23/30' (about the sharing of a sum of money)
		(Suàn shù shū, strip 26)
	(31)	jīn	qī	fēn zhū zhī sān
		gold 7.CARD part zhū GEN 3.CARD
		'3/7 zhū of gold' (zhū is a unit of weigh)
		(Suàn shù shū, strip 30)
	No improper fraction would be mentioned. However, there were mixed numbers (sums of an
	integer and a proper fraction); as in (32), the sum of 12 and 11/72 with no measure word
	expressed by juxtaposition, and in (33) with a measure word. Sometimes, a conjunction yòu can
	be found between the integer and the proper fraction.
	(32)	shí è r		qī shí èr fēn shí yī
		12.CARD 72.CARD part 11.CARD
		'12 11/72'
		(Suàn shù shū, strip 36)
	(33)	zòng	yī	bù liù	fēn bù zhī yī
		length 1.CARD bù 6.CARD part bù GEN 1.CARD
		'a length of 1 bù 1/6 bù' (bù is a unit of length)
		(Suàn shù shū, strip 121)
						):

  ).36 Notably by Li Zhizao in his Tóng wé n suàn zhǐ [Arithmetic Guidance of the Combined Learning] (1613) and Wu Jiashan in his Suàn xué è r shí yī zhǒng [Twenty-one Books of Arithmetic] (1863) only to cite modern mathematical texts. 37 They were listed, in Chinese,by Hu Changqing (1996), Zhang Shoujun (1997), Fang Wenyi (1995, 2009),Da Zhengyue & Yin Shunmin (2008). One can seeAnicotte (2015 b), in English, for a study on this issue.38 It is hard to assess the detailed process of the change which occurred in the context of the abandonment of Classical Chinese, and of educational reforms. Exploring the evolution of fraction names would be in itself an object of research, the part played by institutions of standardization also.On the contrary, the English partitive expressions in (34) and the Modern Greek one in (35) both involve fraction names which are indivisible units connected to the quantified item with a genitive construction (a preposition in English and the genitive case in Greek).

	(39)	wǔ	fēn zhī èr		rénkǒu
		5.CARD fēn zhī 2.CARD population
		'2/5 [of the] population'
	(34)	two	fifth-s	of a	litre of water
		2.CARD 5.ORD-PL of ART-SG litre of water
	(35)	t-a	dyo	pempt-a	t-on	Ellen-on
		ART.DEF-NOM.PL 2.CARD 5.CARD-NOM.PL ART.DEF-GEN.PL Greek-GEN.PL
		'2/5 of the Greeks'
		Modern Standard Arabic partitive expressions in (36) and (37) also involve indivisible
	fraction names; an article is required on the noun.
	(36)	ḥumusay	aš-šaʿb
		5.FRAC.DU.ADJ ART.DEF-people
		'2/5 of the people'
	(37)	ṯālṯat	ʾaḥmās	aš-šaʿb
		3.CARD.ADJ 5.FRAC.PL ART.DEF-people
		'3/5 of the people'
		In Contemporary Chinese, the sequences "denominator's name + fēn zhī + numerator's
	name" are also indivisible semantic units. They can occur before or after what they quantify,
	directly juxtaposed or linked with the determination particle de. Examples of juxtaposition are
	given in (38) with a measure word and in (39) with a noun:
	(38)	wǔ	fēn zhī èr		shēng
		5.CARD fēn zhī 2.CARD litre
		'2/5 [of a] litre'

Dryer, Matthew S. & Haspelmath, Martin (eds.) 2013. The World Atlas of Language Structures Online. Leipzig: Max Planck Institute for Evolutionary Anthropology: http://wals.info [accessed on 2022.03.23].

The Latin word uncia, through two different processes of phonological changes, has given the English length unit inch (1/12 of a foot) and the mass unit ounce (it has at times been defined as 1/16 or 1/12 of a pound).

Second entry for ché ng in the Xiandai Hanyu guifan cidian [Dictionary of Contemporary Chinese], 2010, Beijing: Waiyu jiaoxue yu yanjiu chubanshe, p.163.

Smyth (2002: 177-199): The bi-dimensional expressions are "numerator's name + nai + denominator's name" built with the preposition nai (in, of).

This order is also found in Japanese(Martin 1975: 767) and Korean(Martin 1992: 188).

nd century BCE; dates are tentative because the transmission of the corpus was only oral for centuries.

measure word would be placed just after the sequence "denominator's name + fēn", and therefore before the numerator's name.


Keywords: AMS Classification: 54C70, 35L45, 35Q72, 74H20, 74H25 Cauchy's problem, non-linear transport equations, non-local transport equations, system of hyperbolic equations, Riesz transform, Hlder-Zygmund space, dynamics of dislocation densities

Short time existence and uniqueness in Hölder spaces

for the 2D dynamics of dislocation densities

Introduction 1.Physical motivation and presentation of the model

Real crystals show certain defects in the organization of their crystalline structure, called dislocations. These defects were introduced in the Thirties by Taylor [START_REF] Taylor | The mechanism of plastic deformation of crystals[END_REF], Orowan [START_REF] Orowan | Zur kristallplastizitat i-iii[END_REF] and Polanyi [START_REF] Polanyi | Uber eine art gitterstorung, die einem kristall plastisch machen konnte[END_REF] as the principal explanation of plastic deformation of materials at the microscopic scale. Dislocations can move under the action of exterior stresses applied to the material.

Groma and Balogh in [START_REF] Groma | Investigation of dislocation pattern formation in a twodimensional self-consistent field approximation[END_REF] considered the particular case where these defects are parallel lines in the three-dimensional space, that can be viewed as points in a plane considering 1 Université d'Orléans, Laboratoire MAPMO, Route de Chartres, 45000 Orlans cedex 2, France.

1 their cross-section.

In this model we consider two types of "edge dislocations" in the plane (x 1 , x 2 ). Typically, for a given velocity field, those dislocations of type (+) propagate in the direction + e 1 where e 1 = (1, 0) is the Burgers vector, while those of type (-) propagate in the directione 1 . We refer the reader to the book of Hith and Lothe [START_REF] Hirth | Theory of dislocations, Second Edition[END_REF], for a detailled description of the classical notion in physics of edge dislocations and of the Burgers vector associated to these dislocations.

In [START_REF] Groma | Investigation of dislocation pattern formation in a twodimensional self-consistent field approximation[END_REF] Groma and Balogh have considered the case of densities of dislocations. More precisely, this 2-D system is given by the following coupled non-local and non-linear transport equations (see Cannone et al. [4, Section 2] for more modeling details):

                 ∂ρ + ∂t (x, t) + u ∂ρ + ∂x 1 (x, t) = 0 on R 2 × (0, T ), ∂ρ - ∂t (x, t) -u ∂ρ - ∂x 1 (x, t) = 0 on R 2 × (0, T ), u = R 2 1 R 2 2 (ρ + -ρ -).
(1.1)

The unknowns of this system are the two scalar functions ρ + and ρ -at the time t and the position x = (x 1 , x 2 ), that we denote for simplification by ρ ± . This term correspond to the plastic deformations in a crystal. Its derivative in the x 1 -direction ∂ρ ± ∂x 1 represents the dislocation densities of type (±). Physically, these quantities are non-negative. The function u is the velocity vector field which is equal to the shear stress in the material, solving the equations of elasticity. The operators R 1 (resp. R 2 ) are the 2D Riesz transform associated to x 1 (resp. x 2 ). More precisely, the Fourier transform of these 2D Riesz transforms R 1 and R 2 are given by

R k f (ξ) = ξ k |ξ| f (ξ) for ξ ∈ R 2 , k = 1, 2.
The goal of this work is to establish local existence and uniqueness result of the solution of (1.1) when the initial datum

ρ ± (x 1 , x 2 , t = 0) = ρ ± 0 (x 1 , x 2 ) = ρ± 0 (x 1 , x 2 ) + Lx 1 , L ∈ R (1.2) with ρ± 0 ∈ C r (R 2 ) ∩ L p (R 2 ), for r > 1, p ∈ (1, +∞), where C r (R 2
) is the Hlder-Zygmund space defined in Section 2. The choice L > 0 guarantee the possibility to choose ρ± 0 ∈ L p (R 2 ) such that the assumption is compatible with the non-negativity of ∂ρ ± 0 ∂x 1

. In a particular case where the initial datum is increasing, the global existence of a solution was proved by Cannone et al. [START_REF] Cannone | Global existence for a system of non-linear and non-local transport equations describing the dynamics of dislocation densities[END_REF], using especially an entropy inequality satisfies by the dislocation densities. The fundamental issue of uniqueness for global solutions remains open.

In a particular sub-case of model (1.1) where the dislocation densities depend on a single variable x 1 + x 2 , the existence and uniqueness of a Lipschitz solution was proved by El Hajj et al. in [START_REF] Hajj | A convergent scheme for a non-local coupled system modelling dislocations densities dynamics[END_REF] in the framework of viscosity solutions. Also the existence and uniqueness of a strong solution in W 1,2 loc (R × [0, +∞)) was proved by El Hajj [START_REF] Hajj | Well-posedness theory for a nonconservative Burgers-type system arising in dislocation dynamics[END_REF] in the framework of Sobolev spaces. For a similar model describing moreover boundary layer effects (see Groma, Csikor, Zaiser [START_REF] Groma | Spatial correlations and higher-order gradient terms in a continuum description of dislocation dynamics[END_REF]), we refer the reader to Ibrahim [START_REF] Ibrahim | Existence and uniqueness for a non-linear parabolic/Hamilton-Jacobi system describing the dynamics of dislocation densities, to appear[END_REF] where a result of existence and uniqueness is established, using the framework of viscosity solutions and also entropy solution for nonlinear hyperbolic equations.

Our study of the dynamics of dislocation densities in a special geometry is related to the more general dynamics of dislocation lines. We refer the interested reader to the work of Alvarez et al. [START_REF] Alvarez | Dislocation dynamics: short-time existence and uniqueness of the solution[END_REF], for a local existence and uniqueness of some non-local Hamilton-Jacobi equation. We also refer to Barles et al. [START_REF] Barles | Global existence results and uniqueness for dislocation equations[END_REF] for some long-time existence results.

Main results

We shall show that the system (1.1) possesses a unique local-in-time solution for any initial datum satisfy (1.2) such that ρ±

0 ∈ C r (R 2 ) ∩ L p (R 2
), for r > 1 and for p ∈ (1, +∞). This functional setting allows us to control the velocity field u in terms of ρ +ρ -(see the third line of (1.1)). As we wrote it before, the velocity u is related to ρ

+ -ρ -through the two-dimensional Riesz transforms R 1 , R 2 . Riesz transforms do not map C r (R 2 ) into C r (R 2
), but they are bounded on C r (R 2 ) ∩ L p (R 2 ), for r ∈ [0, +∞) and for p ∈ (1, +∞), as we will see later.

For notational convenience, we define the space Y r,p , for r ∈ [0, +∞) and p ≥ 1 as follows

Y r,p = {f = (f 1 , f 2 ) such that f k ∈ C r (R 2 ) ∩ L p (R 2 ), for k = 1, 2 } ,
where C r (R 2 ) is the inhomogeneous Hlder-Zygmund space (see Section 2, for more precise definition). This space is a Banach space endowed with the following norm: for

f = (f 1 , f 2 ) f r,p = max k=1,2 ( f k C r ) + max k=1,2 ( f k L p ) .
In order to avoid technical difficulties, we first consider (see Theorem 1.1) the case L = 0. Then (see Theorem 1.2) we treat the general case L ∈ R.

ρ 0 = (ρ + 0 , ρ - 0 ) ∈ Y r,p . (1.3) If r > 1 and p ∈ (1, +∞), then (1.1) has a unique solution ρ = (ρ + , ρ -) ∈ L ∞ ([0, T ]; Y r,p )
, where the time T > 0 depends only on ρ 0 r,p . Moreover, the solution ρ satisfies

ρ ∈ Lip([0, T ]; Y r-1,p ).
In order to prove this theorem, we strongly use the fact that the Riesz transforms are continuous on C r (R 2 ) ∩ L p (R 2 ) for r ∈ [0, +∞), p ∈ (1, +∞). This result ensures that the velocity vector field remains bounded on C r (R 2 ) ∩ L p (R 2 ). Using this property and some commutator estimates, we can prove that there exists some T > 0 such that the solution ρ n of an approached system of (

= (ρ + , ρ -) ∈ L ∞ ([0, T ]; Y r,p )
, where the time T > 0 depends only on L and ρ0 r,p . Moreover,

ρ ± (x 1 , x 2 , t) = ρ± (x 1 , x 2 , t) + Lx 1 , where ρ = (ρ + , ρ-) ∈ Lip([0, T ]; Y r-1,p
≥ 0 for all (x, t) ∈ R × [0, T ].
Related to our analysis in the present paper, we get the following theorem as a byproduct.

Theorem 1.4 (Global existence and uniqueness for linear transport equations)

Take g 0 ∈ C r (R 2 ) ∩ L p (R 2 ) and v = (v 1 , v 2 ) ∈ L ∞ ([0, T ); Y r,p
) for all T > 0, r > 1 and 1 < p < +∞. Then, there exists a unique solution

g ∈ L ∞ ([0, T ); C r (R 2 ) ∩ L p (R 2 )) ∩ Lip([0, T ); C r-1 (R 2 ) ∩ L p (R 2 ))
of the linear transport equation

     ∂g ∂t + v • ∇g = 0 on R 2 × (0, T ) g(x, 0) = g 0 (x) on R 2 .
(1.4)

Organization of the paper

This paper is organized as follows. In Section 2, we recall the characterization of Hlder spaces and gather several important estimates. In particular, the boundedness of Riesz transforms on 2 Some results on Hlder-Zygmund spaces This is a preparatory section in which we recall some results on Hlder-Zygmund spaces, and gather several estimates that will be used in the subsequent sections. A major part of the following results can be found in Meyer [START_REF] Meyer | Ondelettes et opérateurs. I, II, Actualités Mathématiques[END_REF] and Meyer, Coifman [START_REF] Meyer | Ondelettes et opérateurs. III, Actualités Mathématiques[END_REF].

C r (R 2 ) ∩ L p (R
We start with a dyadic decomposition of R d , where d > 0 is an integer. To this end, we take an arbitrary radial function

χ ∈ C ∞ 0 (R d ), such that supp χ ⊂ ξ : |ξ| ≤ 4 3 , χ ≡ 1 for |ξ| ≤ 3 4 , χ L 1 = 1. It is a classical result that, for φ(ξ) = χ( ξ 2 ) -χ(ξ), we have φ ∈ C ∞ 0 (R d ) and supp φ ⊂ ξ : 3 4 ≤ |ξ| ≤ 8 3 , χ(ξ) + j≥0 φ(2 -j ξ) = 1, for all ξ ∈ R d .
For the purpose of isolating different Fourier frequencies, define the operators ∆ i for i ∈ Z as follows:

∆ i f =                  0 if i ≤ -2, χ(D)f = χ(y)f (x -y)dy if i = -1, φ(2 -i D)f = 2 id φ(2 i y)f (x -y)dy if i ≥ 0, (2.5)
where χ and φ are the inverse Fourier transforms of χ and φ, respectively.

For i ∈ Z, S i is the sum of ∆ j with j ≤ i -1, i.e.

S i f = ∆ -1 f + ∆ 0 f + ∆ 1 f + • • • + ∆ i-1 f = 2 id χ(2 i y)f (x -y)dy.
It can be shown for any tempered distribution f that S i f → f in the distributional sense, as i → ∞.

For any r ∈ R and p, q ∈ [1, ∞], the inhomogeneous Besov space B r p,q (R d ) consists of all tempered distributions f such that the sequence {2 jr ∆ j f L p } j∈Z belongs to l q (Z). When both p and q are equal to ∞, the Besov space B r p,q (R d ) reduces to the inhomogeneous Hlder-Zygmund space

C r (R d ), i.e. B r ∞,∞ (R d ) = C r (R d ). More explicitly, C r (R d ) with r ∈ R contains any function f satisfying f C r = sup j∈Z 2 jr ∆ j f L ∞ < ∞. (2.6)
It is easy to check that C r (R d ) endowed with the norm defined in (2.6) is a Banach space.

For r ≥ 0, C r (R d ) is closely related to the classical Hlder space Cr (R d ) equipped with the norm

f Cr = |β|≤[r] ∂ β f L ∞ + sup x =y ∂ [r] f (x) -∂ [r] f (y) |x -y| r-[r] . (2.7)
In fact, if r is not an integer, then the norm (2.6) and (2.7) are equivalent, and C r (R d ) = Cr (R d ). The proof for this equivalence is classical and can be found in Chemin [START_REF] Chemin | Oxford Lecture Series in Mathematics and its Applications[END_REF]. When r is an integer, say r = k, Ck (R d ) is the space of bounded functions with bounded j-th derivatives for any j ≤ k. In particular, C1 (R d ) contains the usual Lipschitz functions and is sometimes denoted by Lip(R d ). As a consequence of Bernstein's Lemma (stated below), Cr (R d ) is a subspace of C r (R d ). Explicit examples can be constructed to show that such an inclusion is genuine. In addition, according to Proposition 2.2, Cr (R d ) includes C r+ε (R d ) for any ε > 0. In summary, for any integer k ≥ 0 and ε > 0,

C k+ε (R d ) ⊂ Ck (R d ) ⊂ C k (R d ).
Proposition 2.1 (Bernstein's Lemma, Meyer [START_REF] Meyer | Ondelettes et opérateurs. I, II, Actualités Mathématiques[END_REF] ) Let d > 0 be an integer and

α 1 > α 2 > 0 be two real numbers. 1) If 1 ≤ p ≤ q ≤ ∞ and supp f ⊂ ξ ∈ R d : |ξ| ≤ α 1 2 j , then max |α|=k ∂ α f L q ≤ C2 jk+d( 1 p -1 q ) f L p ,
where C > 0 is a constant depending only on k and α 1 .

2

) If 1 ≤ p ≤ ∞ and supp f ⊂ ξ ∈ R d : α 1 2 j ≤ |ξ| ≤ α 2 2 j , then C -1 2 jk f L p ≤ max |α|=k ∂ α f L p ≤ C2 jk f L p ,
where C > 0 is a constant depending only on k, α 1 and α 2 .

∈ C ε (R d ), we have f L ∞ ≤ C ε f C 0 log e + f C ε f C 0 ≤ C ε f C ε .
(2.8)

In the system (1.1), the velocity field u is determined by ρ +ρ -through the 2D Riesz transforms. These Riesz transforms do not map a C r (R d ) Hlder-Zygmund space to itself, but their action on

C r (R d ) is indeed bounded in C r (R d ) ∩ L p (R d ) for p ∈ (1, +∞) (see Proposition 2.4
). We first recall a general result concerning the boudedness of Fourier multiplier operators on Hlder spaces.

Proposition 2.3 (Fourier multiplier operators on Hlder spaces, Meyer [START_REF] Meyer | Ondelettes et opérateurs. I, II, Actualités Mathématiques[END_REF]) Let d > 0 be an integer and F be an infinitely differentiable function on R d . Assume that for some R > 0 and m ∈ R, we have

F (λξ) = λ m F (ξ)
for any ξ ∈ R d with |ξ| > R and λ ≤ 1. Then the Fourier multiplier operator F (D)

maps continuously C r (R d ) into C r-m (R d ) for any r ∈ R.
R k f C r ≤ C f C r ∩L p
where k = 1, 2.

Proof of Proposition 2.4:

Using the operator ∆ -1 defined in (2.5), we divide R k f into two parts,

R k f = ∆ -1 R k f + (1 -∆ -1 )R k f. (2.9) Since supp χ(ξ) ∩ supp φ(2 -j ξ) = ∅ for j ≥ 1, the operator ∆ j ∆ -1 = 0 when j ≥ 1.
Thus, according to (2.6),

∆ -1 R k f C r = sup j∈Z 2 jr ∆ j ∆ -1 R k f L ∞ = max [2 -r ∆ -1 ∆ -1 R k f L ∞ , ∆ 0 ∆ -1 R k f L ∞ ] ≤ max [1, 2 -r ] ∆ -1 R k f L ∞ .
Let q be the conjugate of p, namely 1 p + 1 q = 1. It then follow, since Riesz transforms are bounded on L p (R d ), that for all p ∈ (1, +∞):

∆ -1 R k f C r ≤ max [1, 2 -r ] χ * R k f L ∞ ≤ max [1, 2 -r ] χ L q R k f L p = C f L p
where C is a constant depending only on r and p. To estimate the second part in (2.9), we apply Proposition 2.3 with

F (ξ) = (1 -χ(ξ))
ξ k |ξ| and m = 0, and hence we conclude

that it maps C r (R d ) into C r (R d ).
This gives the proof of Proposition 2.4. 2

Finally, we recall the notion of Bony's paraproduct (see Bony [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF]). The usual product uv of two functions u and v can be decomposed into three parts. More precisely, using

v = j∈Z ∆ j v, u = j∈Z ∆ j u and ∆ j ∆ k v = 0 if |j -k| ≥ 1, ∆ j (S k-1 v∆ k v) = 0 if |j -k| ≥ 5,
we can write

uv = T u v + T v u + R(u, v),
where

T u v = j≥1 S j-1 (u)∆ j v, R(u, v) = |i-j|≤1 ∆ i u∆ j v.
We remark that the previous decomposition allows one to distinguish different types of terms in the product of uv. The Fourier frequencies of u and v in T u v and T v u are separated from each other while those of the terms in R(u, v) are close to each other.

Using this decomposition, one can show that

uv C s ≤ u C s v L ∞ + u L ∞ v C s for s > 0. (2.10)
For the prove of (2.10) see Chen [8, Prop. 5.1].

Two commutator estimates

Two major commutator estimates are stated and proved in this section. We remark that this commutator estimates was often used to resolve the Navier-Stokes equations (see for instance Cannone et al. [START_REF] Cannone | Littlewood-Paley decomposition and Navier-Stokes equations[END_REF], [START_REF] Cannone | On the regularity of the bilinear term for solutions to the incompressible Navier-Stokes equations[END_REF]). Here, we apply these techniques on our system (1.1).

Lemma 3.1 (L ∞ commutator estimates) Let j ≥ -1 be an integer and r > 0. Then, for some absolute constant C, we have 1)

u ∂ ∂x α , ∆ j f L ∞ ≤ C2 -jr ∂f ∂x α L ∞ u C r + ∇u L ∞ f C r , for α = 1, 2, 2) u ∂ ∂x α , ∆ j f L ∞ ≤ C2 -jr ( f L ∞ u C r+1 + ∇u L ∞ f C r ) , for α = 1, 2,
where the bracket [ , ] represents the commutator, namely

u ∂ ∂x α , ∆ j f = u ∂ ∂x α (∆ j f ) -∆ j u ∂f ∂x α , for α = 1, 2. (3.11)
Proof of Lemma 3.1:

Proof of 1): Using the paraproduct notations T and R, we decompose u ∂ ∂x α , ∆ j f,

for α = 1, 2, into five parts, u ∂ ∂x α , ∆ j ρ = I 1 + I 2 + I 3 + I 4 + I 5 ,
where

I 1 = T u ∂ ∂x α , ∆ j f = T u ∂ ∂x α (∆ j f ) -∆ j T u ∂f ∂x α , I 2 = -∆ j T ∂f ∂xα u, I 3 = T∂(∆ j f ) ∂xα u, I 4 = R u, ∂(∆ j f ) ∂x α , I 5 = -∆ j R u, ∂f ∂x α .
Back to the definition of T , we can write

I 1 = k≥1 S k-1 (u)∆ j ∂(∆ k f ) ∂x α -∆ j k≥1 S k-1 (u)∆ k ∂f ∂x α = k≥1 S k-1 (u)∆ j ∂(∆ k f ) ∂x α -∆ j S k-1 (u) ∂(∆ k f ) ∂x α .
(3.12)

Since ∆ j ∆ k = 0 for |j -k| > 1 and

supp S k-1 (u) ∂(∆ k f ) ∂x α ⊂ ξ : 1 3 2 k-2 ≤ |ξ| ≤ 5 3 2 k+1 ,
the sum in (3.12) only involves those terms with k satisfying |j -k| ≤ 4. We only take j ≥ 0 since the case j = -1 can be handled similarly. Applying the definition of ∆ j in (2.5), we obtain

I 1 = |j-k|≤4 2 jd φ(2 j (x -y)) [S k-1 (u(x)) -S k-1 (u(y))] ∂(∆ k f ) ∂x α (y)dy = |j-k|≤4 φ(y) S k-1 (u(x)) -S k-1 (u(x -2 -j y)) ∂(∆ k f ) ∂x α (x -2 -j y)dy.
Using the fact that φ ∈ S(R d ) and S j are continuous from L ∞ onto itself, we get for r ∈ R and an absolute constant C:

I 1 L ∞ ≤ C2 -j ∇u L ∞ ∂(∆ j f ) ∂x α L ∞ ≤ C ∇u L ∞ ∆ j f L ∞ ≤ C2 -jr ∇u L ∞ f C r , (3.13)
where we have used Proposition 2.1 in the second inequality. To estimate I 2 and I 3 , we first write them as

I 2 = - |j-k|≤4 ∆ j S k-1 ∂f ∂x α ∆ k u , I 3 = |j-k|≤4 S k-1 ∂(∆ j f ) ∂x α ∆ k u.
Similarly, only terms with k satisfying |j -k| ≤ 4 are considered in the above sums. Thus, since ∆ j and S j are continuous from L ∞ onto itself, we have for r ∈ R:

I 2 L ∞ ≤ C ∆ j u L ∞ S j ∂f ∂x α L ∞ ≤ C2 -jr u C r ∂f ∂x α L ∞ , (3.14)
I 3 L ∞ ≤ C ∆ j u L ∞ S j ∆ j ∂f ∂x α L ∞ ≤ C2 -jr u C r ∂f ∂x α L ∞ , (3.15)
where the C ′ s in the above inequalities are absolute constants. From the definition of R, we have

I 4 = |k 1 -k 2 |≤1 ∆ k 1 (u)∆ k 2 ∂(∆ j f ) ∂x α .
Obviously, only a finite number of terms involved in the above sums are non-zeros. Then,

I 4 L ∞ ≤ C ∆ j u L ∞ ∆ j ∂f ∂x α L ∞ ≤ C2 -jr u C r ∂f ∂x α L ∞ . (3.16)
Note that from the definition of R and ∆ j , j ≥ -1, we can write I 5 as

I 5 = - k≥j-3 k+1 k 1 =k-1 ∆ j ∆ k (u)∆ k 1 ∂f ∂x α .
Therefore, for an absolute constant C, we have:

I 5 L ∞ ≤ C k≥j-3 k+1 k 1 =k-1 ∆ k u L ∞ ∆ k 1 ∂f ∂x α L ∞ ≤ C ∂f ∂x α L ∞ u C r k≥j-3 2 -kr ≤ C2 -jr ∂f ∂x α L ∞ u C r .
(3.17)

Gathering the estimates in (3.13)-(3.17), we establish the desired inequality in 1).

Proof of 2): As in the proof of 1), we decompose u ∂ ∂x α , ∆ j f as the sum of I 1 , I 2 , I 3 , I 4 and I 5 . The estimate on I 1 remains untouched, while different bounds are needed for I 2 , I 3 , I 4 and I 5 . Indeed, for j ≥ 1:

I 2 L ∞ ≤ C ∆ j u L ∞ ∂S j-1 f ∂x α L ∞ ≤ C2 j ∆ j u L ∞ S j-1 f L ∞ ≤ C2 -jr u C r+1 f L ∞ , (3.18)
where we have used Proposition 2.1 in the second inequality. I 3 and I 4 can be similarly estimated as I 2 :

I 3 L ∞ ≤ C ∂∆ j f ∂x α L ∞ ∆ j u L ∞ ≤ C2 j ∆ j u L ∞ ∆ j f L ∞ ≤ C2 -jr u C r+1 f L ∞ , (3.19)
I 4 L ∞ ≤ C ∆ j u L ∞ ∂∆ j f ∂x α L ∞ ≤ C2 j ∆ j u L ∞ ∆ j f L ∞ ≤ C2 -jr u C r+1 f L ∞ . (3.20)
Finally, we have

I 5 L ∞ ≤ C k≥j-3 k+1 k 1 =k-1 ∆ k u L ∞ ∂∆ k 1 f ∂x α L ∞ ≤ C k≥j-3 k+1 k 1 =k-1 2 k 1 ∆ k u L ∞ ∆ k 1 f L ∞ ≤ C f L ∞ k≥j-3 2 k ∆ k u L ∞ ≤ C f L ∞ u C r+1 k≥j-3 2 -kr ≤ C2 -jr f L ∞ u C r+1 . ( 3

Local existence and uniqueness results

This section is devoted to the proofs of Theorems 1.1 and 1.2. For the sake of a clear presentation, we divide it into three subsections. In the first subsection, we show a basic a priori estimate and we prove Theorem 1.4. With the aid of this estimate, we prove Theorems 1.1 and 1.2 in the next subsections.

An a priori estimate

Proposition 4.1 (A priori estimate) Let r > 1 and p > 1. For all T > 0,

ρ 0 = (ρ + 0 , ρ - 0 ) ∈ Y r,q and u ∈ L ∞ ([0, T ); C r (R 2 ) ∩ L p (R 2 ))
, there exists a unique solution ρ = (ρ + , ρ -) ∈ L ∞ ([0, T ); Y r,p ) of the following system of linear transport equations

∂ρ ± ∂t ± u ∂ρ ± ∂x 1 = 0 (4.22)
Moreover, for all t ∈ [0, T ], we have

ρ(•, t) r,p ≤ ρ 0 r,p exp C t 0 u(•, τ ) C r ∩L p dτ
where C > 0 is a constant depending only on r and p.

Proof of Proposition 4.1:

From the fact that u(•, t) ∈ C r (R 2 ) ∩ L p (R 2 ), for t ∈ [0, T ], we can define the flow map

X ± (•, t) satisfying      ∂X ± (x, t) ∂t = ±ū(X(x, t), t)
, where ū = (u, 0),

X ± (x, 0) = x. (4.23)
By the characteristics method, we know that, if (X ± ) -1 is the inverse function of X ± with respect to x, then ρ ± (x, t) = ρ ± 0 ((X ± ) -1 (x, t)) is the unique solution of system (4.22) (see Serre [START_REF] Serre | Systems of conservation laws. I[END_REF] for more details).

Let j ≥ -1. Applying the operator ∆ j to both sides of the system (1.1) yields

∂∆ j ρ ± ∂t ±u ∂∆ j ρ ± ∂x 1 = ± u ∂ ∂x 1 , ∆ j ρ ± ,
where u ∂ ∂x 1 , ∆ j ρ ± is defined in (3.11). This equation can be rewritten in the following form

∆ j ρ ± (x, t) = ∆ j ρ ± 0 ((X ± ) -1 (x, t))± t 0 u ∂ ∂x 1 , ∆ j ρ ± (X ± ((X ± ) -1 (x, t), s), s)ds.
Taking the L ∞ -norm of both sides of this equality, we get:

∆ j ρ ± (•, t) L ∞ ≤ ∆ j ρ ± 0 L ∞ + t 0 u ∂ ∂x 1 , ∆ j ρ ± (•, s) L ∞ ds.
Applying Lemma 3.1 (1), we obtain

ρ ± (•, t) C r ≤ ρ ± 0 C r +C t 0 ∂ρ ± ∂x 1 (•, s) L ∞ u(•, s) C r + ∇u(•, s) L ∞ ρ ± (•, s) C r ds.
According to (2.8), we know that for r > 1 and a constant C = C(r) > 0, we have:

∂ρ ± ∂x 1 L ∞ ≤ C ρ ± C 1 log e + ρ ± C r ρ ± C 1 ≤ C ρ ± C r .
In a similar way, we can obtain ∇u

L ∞ ≤ C u C r . Therefore, for C = C(r) > 0, ρ ± (•, t) C r ≤ ρ ± 0 C r + C t 0 u(•, s) C r ρ ± (•, s) C r ds ≤ max ± ρ ± 0 C r + C t 0 u(•, s) C r ρ(•, s) r,p ds.
Moreover, integrating in time the system (1.1), we get the following L p estimate:

ρ ± (•, t) L p ≤ ρ ± 0 L p + t 0 u(•, s) L p ∂ρ ± ∂x 1 (•, s) L ∞ ds ≤ ρ ± 0 L p + C t 0 u(•, s) L p ρ(•, s) r,p ds,
where we have used Hlder inequality in the first line and (2.8) in the second line. Now, adding the two previous inequalities, we obtain

ρ(•, t) r,p ≤ ρ 0 r,p + C t 0 u(•, s) C r ∩L p ρ(•, s) r,p ds,
where The proof of Theorem 1.4 is a consequence of the proof of Proposition 4.1. Indeed, just consider the characteristic equation

• C r ∩L p = • C r + • L p .
     ∂X(x, t) ∂t = v(X(x, t), t), X(x, 0) = x. (4.25)
Then, as in the proof of Proposition 4.1, we use the commutator estimates proved in Lemma 3.1 (1), to show the following estimate:

g(•, t) C r ∩L p ≤ g 0 C r ∩L p exp C t 0 v(•, s) r,p ds , (4.26)
which proves the result. 2

Proof of Theorem 1.1

The proof starts with the construction of a successive approximation sequence

{ρ n = (ρ +,n , ρ -,n )} n≥1 satisfying                        ρ 1 = (ρ + 0 , ρ - 0 ) = ρ 0 , ∂ρ ±,n+1 ∂t ± u n ∂ρ ±,n+1 ∂x 1 = 0, on R 2 × (0, T ) u n = R 2 1 R 2 2 (ρ +,n -ρ -,n ) , ρ ±,n+1 (x, 0) = ρ ± 0 . (4.27)
First of all, according to Proposition 2.4,

ρ 1 ∈ Y r,p implies that u 1 ∈ C r (R 2 ) ∩ L p (R 2 ).
Thus, applying Proposition 4.1, we can prove that, for all T > 0, there exists a unique solution ρ 2 ∈ L ∞ ([0, T ); Y r,p ) for (4.27) with n = 2. Arguing in a similar manner we can show that this approached problem (4.27) has a unique solution ρ n for all n ≥ 1.

The rest of the proof can be divided into two major steps. The first step establishes the existence of T 1 > 0 such that {ρ n = (ρ +,n , ρ -,n )} n≥1 is uniformly bounded in Y r,p for any t ∈ [0, T 1 ]. The second step shows that for some T 2 ∈ [0, T 1 ], we have

{ρ n = (ρ +,n , ρ -,n )} n≥1 is a Cauchy sequence in C([0, T 2 ], Y r-1,p ).
Step 1 (A uniform bound): Using similar arguments as in the proof of Proposition 4.1, estimate (4.24) yields, by Proposition 2.4, the following bound on {ρ n = (ρ +,n , ρ -,n )} n≥1 : Then ρ n (•, t) r,p ≤ M for all n ≥ 1 and t ∈ [0, T 1 ]. Since, ρ 1 r,p ≤ ρ 0 r,p < M and ρ k (•, t) r,p < M, we obtain

ρ n+1 (•, t) r,p ≤ ρ 0 r,p exp(C 0 T 1 M) ≤ M.
(4.28) Furthermore, since r > 1, we use (2.10) and Proposition 2.4 to get:

∂ρ ±,n ∂t C r-1 ≤ u n ∂ρ ±,n+1 ∂x 1 C r-1 ≤ u n C r-1 ∂ρ ±,n+1 ∂x 1 L ∞ + u n L ∞ ∂ρ ±,n+1 ∂x 1 C r-1 ≤ C u n C r-1 ρ ±,n+1 C r ≤ CM 2 ,
where C = C(r). We can also check that the following L p estimate on ρ ±,n is valid:

∂ρ ±,n ∂t L p ≤ u n ∂ρ ±,n+1 ∂x 1 L p ≤ u n L p ∂ρ ±,n+1 ∂x 1 L ∞ ≤ C u n L p ρ ±,n+1 C r ≤ CM 2
, where we have used Hlder inequality in the first line, then (2.8) and Proposition 2.4 in the second line. Adding the two previous inequalities, we deduce that

∂ρ ±,n ∂t C r-1 + ∂ρ ±,n ∂t L p ≤ CM 2 , (4.29)
where C = C(r). Thus, by (4.28)-(4.29), we obtain that

ρ n ∈ L ∞ ([0, T 1 ]; Y r,p ) ∩ Lip([0, T 1 ]; Y r-1,p )
is uniformly bounded.

Step 2 (Cauchy sequence): To show that {ρ n = (ρ +,n , ρ -,n )} n≥1 is a Cauchy sequence in Y r-1,q , we consider the difference η ±,n = ρ ±,nρ ±,n-1 . Rigorously speaking, we should consider the more general difference η ±,m,n = ρ ±,mρ ±,n , but the analysis for η m,n = (η +,m,n , η -,m,n ) is parallel to what we shall present for η n = (η +,n , η -,n ) and thus we consider η n for the sake of a concise presentation. It follows from (4.27) that

{η n = (η +,n , η -,n )} n≥1 satisfies                        η ±,1 = ρ ± 0 , ∂η ±,n+1 ∂t ± u n ∂η ±,n+1 ∂x 1 = ∓w n ∂ρ ±,n ∂x 1 , w n = R 2 1 R 2 2 (η +,n -η -,n ), η ±,n+1 (x, 0) = η ±,n+1 0 (x) = 0. (4.30)
Proceeding as in the proof of Proposition 4.1, we obtain for any integer j ≥ -1,

∆ j η ±,n+1 (•, t) L ∞ ≤ t 0 u n ∂ ∂x 1 , ∆ j η ±,n+1 (•, s) L ∞ ds K 1 + t 0 ∆ j w n ∂ρ ±,n ∂x 1 (•, s) L ∞ ds. K 2
Estimating K 1 by Lemma 3.1 (2), and K 2 by (2.10), we get:

η ±,n+1 (•, t) C r-1 ≤ C t 0 ∇u n (•, s) L ∞ η ±,n+1 (•, s) C r-1 + u n (•, s) C r η ±,n+1 (•, s) L ∞ ds +C t 0 w n (•, s) L ∞ ∂ρ ±,n ∂x 1 (•, s) C r-1 + w n (•, s) C r-1 ∂ρ ±,n ∂x 1 (•, s) L ∞ ds. Since r > 1, Proposition 2.2 implies, ∇u n L ∞ ≤ C u n C r , η ±,n+1 L ∞ ≤ C η ±,n+1 C r-1 , ∂ρ ±,n ∂x 1 L ∞ ≤ C ρ ±,n C r and w n L ∞ ≤ C w n C r-1 .
Therefore, for a constant C depending only on r,

η ±,n+1 (•, t) C r-1 ≤ C t 0 u n (•, s) C r η ±,n+1 (•, s) C r-1 ds +C t 0 w n (•, s) C r-1 ρ ±,n (•, s) C r ds.
However, it follows from a basic L p estimate that

η ±,n+1 (•, t) L p ≤ C t 0 ∇u n (•, s) L ∞ η ±,n+1 (•, s) L p ds +C t 0 w n (•, s) L p ∂ρ ±,n ∂x 1 (•, s) L ∞ ds ≤ C t 0 u n (•, s) C r η ±,n+1 (•, s) L p ds +C t 0 w n (•, s) L p ρ ±,n (•, s) C r ds.
Adding the last two inequalities, yields

η n+1 (•, t) r-1,p ≤ C t 0 u n (•, s) C r ∩L p η n+1 (•, s) r-1,p ds +C t 0 w n (•, s) C r-1 ∩L p ρ n (•, s) r,p ds.
The components of w n are the Riesz transforms of η n and thus, according to Proposition 2.4:

w n C r-1 ∩L p ≤ C η n r-1,p .
We thus have reached an iterative relationship between η n r-1,p and η n+1 r-1,p : We conclude from Steps 1 and 2 that there exists ρ = (ρ + , ρ -) satisfying First of all, we note that for all L ∈ R, if ρ ± are solutions of (1.1) then ρ± (x 1 , x 2 , t) = ρ ± (x 1 , x 2 , t) -Lx 1 solves the following system:

η n+1 (•, t) r-1,p ≤ C 1 t 0 ρ n (•, s) r,p η n+1 (•, s) r-1,p ds +C 1 t 0 η n (•, s) r-1,p ρ n (•, s)
       ∂ ρ± ∂t (x, t) ± u ∂ ρ± ∂x 1 (x, t) = ∓Lu on R 2 × (0, T ), u = R 2 1 R 2 2 (ρ + -ρ-), (4.32)
and respects the following initial data: ρ± 0 (x 1 , x 2 , t) = ρ ± 0 (x 1 , x 2 ) -Lx 1 . Now, to prove Theorem 1.2, it suffices to show that, for all initial data ρ± 0 ∈ Y r,p , the system (4.32) has a unique local solution ρ± ∈ L ∞ ([0, T ); Y r,p ) for r > 1 and p ∈ (1, +∞).

In order to do this, we proceed as in the proof of Theorem 1.1. We consider the following

u n = R 2 1 R 2 2 (ρ +,n -ρ-,n ) ,
ρ±,n+1 (x, 0) = ρ0 ± .

(4.33)

We remark that, the only change that appears here, compared to the approached system (4.27) is the right-hand side Lu n of the second equation of (4.33). However, by Proposition 2.4, we know that this term remains bounded in L ∞ ([0, T ); C r (R 2 ) ∩ L p (R 2 )) for r > 1 and p ∈ (1, +∞). Which permits us to easily follow the same steps of the proof of Theorem 1.1. This finally proves that, for some small T > 0, we have on the one hand: the sequence ρn = (ρ +,n , ρ-,n ) is uniformly bounded in L ∞ ([0, T ); Y r,p ), and on the other hand, this sequence is a Cauchy sequence in L ∞ ([0, T ); Y r-1,p ). This terminate the proof. 2

Theorem 1 . 1 (

 11 Local existence and uniqueness, case L = 0) Consider the initial data

Proposition 2 . 2 (

 22 Logarithmic Sobolev inequalities in Hlder-Zygmund space, Kozono et al. [15, Th 2.1] ) Let d > 0 be an integer. There exists a constant C = C(d) such that for any ε > 0 and f

Proposition 2 . 4 (

 24 Boundedness of Riesz transforms on C r (R d ) ∩ L p (R d )) Let r ∈ R and p ∈ (1, +∞). Then there exists a positive constant C depending only on r and p such that

ρ

  n+1 (•, t) r,p ≤ ρ 0 r,p exp C 0 t 0 u(•, s) C r ∩L p ds , ρ 0 r,p exp C 0 t 0 ρ n (•, s) r,p ds , where r > 1, p ∈ (1, +∞) and C 0 = C 0 (r, p). Choose T 1 and M satisfying M = 2 ρ 0 r,p and exp(C 0 T 1 M) ≤ 2 or T 1 = ln(2) 2C 0 ρ 0 r,p .

ρ 2 4. 3

 23 ∈ L ∞ ([0, T 2 ]; Y r,p ) ∩ Lip([0, T 2 ]; Y r-1,p ) such that ρ n converges to ρ in C([0, T 2 ]; Y r-1,p ).The proof of uniqueness follows directly from Step 2. This completes the proof of Theorem 1.1. Proof of Theorem 1.2It is worth mentioning that the ideas of the proof of Theorem 1.2 are already contained in the proof of Theorem 1.1.

± u n ∂ ρ±,n+1 ∂x 1 =

 1 ∓Lu n , on R 2 × (0, T )

  ).

	Remark 1.3 If at the initial time we have ∂ρ ± ∂ρ ± ∂x 1 then this remains true for 0 ≤ t ≤ T , i.e., ∂x 1	(•, •, t = 0) ≥ 0 two positive quantities,

  2 ) is established. Section 3 presents two key commutator estimates (Lemma 3.1). Finally in Section 4, we prove Theorem 1.4 and a basic a priori estimate. Then, thanks to this a priori estimate, we give in Subsections 4.2 and 4.3 the proofs of Theorems 1.1 and 1.2 respectively.

  By Gronwall's Lemma and Proposition 2.4, ρ(•, t) r,p ≤ ρ 0 r,p exp C

	t
	0	u(•, s) C r ∩L p ds .	(4.24)
	Which completes the proof of Proposition 4.1.
			2
	Proof of Theorem 1.4:

  r,p ds,(4.31) where the constants are labeled as C 1 for the purpose of defining T 2 . It has been shown in Step 1 that for t ≤ T 1 ,ρ n r,p ≤ M. Now, choose T 2 > 0 satisfying T 2 ≤ T 1 , C 1 MT 2 ≤ 1 4 ,we can show that {ρ n (•, t)} n≥1 is a Cauchy sequence in Y r-1,p for t ≤ T 2 . Indeed, for any given ε > 0, if η n r-1,p ≤ ε for t ≤ T 2 , then (4.31) implies that:η n+1 r-1,p ≤ C 1 εMT 2 + C 1 Mis valid for any t ≤ T 2 . It then follows from Gronwall's inequality that η n+1 r-1,p ≤ ε, for any t ≤ T 2 which completes Step 2.

t 0 η n+1 (•, s) r-1,p ds,

Acknowledgements

The author would like to thank M. Cannone and R. Monneau for fruitful remarks that helped in the preparation of the paper. This work was partially supported by the program "PPF, programme pluri-formations mathmatiques financires et EDP", (2006-2010), Universit Paris-Est.


Théo Charnay

Véronique Blanck

Mathieu Cerino

Marc Bartoli

Florence Riccardi

Nathalie Bonello-Palot

Christophe Pécheux

Karine Nguyen

Nicolas Lévy

Svetlana Gorokhova
email: svetlana.gorokhova@univ-amu.fr

Martin Krahn
email: martin.krahn@univ-amu.fr

Retrospective analysis and reclassification of DYSF variants in a large French series of dysferlinopathy patients

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

Dysferlinopathies are a group of autosomal recessive muscular dystrophies caused by mutations in the gene encoding dysferlin (DYSF; MIM 603009, 2p13, NM_003494.4). [START_REF] Liu | Dysferlin, a novel skeletal muscle gene, is mutated in Miyoshi myopathy and limb girdle muscular dystrophy[END_REF][START_REF] Bashir | A gene to Caenorhabditis elegans spermatogenesis factor fer-1 is mutated in limb-girdle muscular dystrophy type 2B[END_REF] The two most common clinical presentations of dysferlinopathies are autosomal recessive limb-girdle muscular dystrophy 2 (LGMDR2; previously LGMD2B, MIM 253601) and Miyoshi muscular dystrophy 1 (MM; MIM 254130). [START_REF] Aoki | Dysferlinopathy[END_REF] Since description of the first pathogenic DYSF variants in 1998, [START_REF] Liu | Dysferlin, a novel skeletal muscle gene, is mutated in Miyoshi myopathy and limb girdle muscular dystrophy[END_REF][START_REF] Bashir | A gene to Caenorhabditis elegans spermatogenesis factor fer-1 is mutated in limb-girdle muscular dystrophy type 2B[END_REF] more than 600 unique pathogenic or likely pathogenic variants have been reported (664 in the Leiden Open Variation Database [LOVD] [accessed 20 November 2020] and 318 variants in ClinVar [version 7 July 2020]). A significant number of these variants were identified in our diagnostic center in the Medical Genetics Department at the Timone Hospital (Marseille, France), since our laboratory was one of the first centers in Europe to offer diagnostic sequencing of all 55 exons of the DYSF gene. [START_REF] Nguyen | Dysferlin mutations in LGMD2B, Miyoshi myopathy, and atypical dysferlinopathies[END_REF][START_REF] Nguyen | Phenotypic study in 40 patients with dysferlin gene mutations: high frequency of atypical phenotypes[END_REF][START_REF] Krahn | Analysis of the DYSF mutational spectrum in a large cohort of patients[END_REF] Recent evolution of sequencing technologies and bioinformatic tools has allowed accumulation of an unprecedented amount of genetic data, leading to development of large databases with variants from the general population as well as rapid expansion of databases with clinically relevant variants identified in patients with genetic diseases. These advances profoundly changed the ways variant pathogenicity is established and led to reclassification of multiple variants identified during prior diagnostic sequencing. For example, a number of variants that were previously considered pathogenic were subsequently identified in the general population at frequencies that were not compatible with pathogenicity, leading to reclassification of these variants as benign. [START_REF] Piton | XLID-causing mutations and associated genes challenged in light of data from large-scale human exome sequencing[END_REF][START_REF] Bell | Carrier testing for severe childhood recessive diseases by nextgeneration sequencing[END_REF][START_REF] Whiffin | Using high-resolution variant frequencies to empower clinical genome interpretation[END_REF][START_REF] Di Fruscio | Are all the previously reported genetic variants in limb girdle muscular dystrophy genes pathogenic?[END_REF] The dramatic expansion of clinical sequencing also created a critical need for standardization of variant pathogenicity interpretation in diagnostic settings, leading to the development of a new variant classification system by the American College of Medical Genetics and Genomics and the Association for Molecular Pathology (ACMG/AMP). [START_REF] Richards | Standards and guidelines for the interpretation of sequence variants: a joint consensus recommendation of the American College of Medical Genetics and Genomics and the Association for Molecular Pathology[END_REF] The evolution of the variant interpretation standards led to many discrepancies between the pathogenicity attributed to newly identified variants and the pathogenicity classifications of "legacy" variants identified in the earlier days of genetic sequencing. Presence of these misclassified variants in the literature and the variant databases greatly interferes with variant interpretation in the current diagnostic setting. To solve this problem for the DYSF gene, we have retrospectively evaluated the pathogenicity of 176 unique variants identified in dysferlinopathy patients referred to our center for diagnostic sequencing and updated the classification of these variants in the public database LOVD. Our results highlight the importance of variant reclassification and join the first steps of the international effort to review and reanalyze the pathogenicity classifications of variants present in the literature and public databases.

MATERIALS AND METHODS

Selection of DYSF variants for reclassification

The variants described in this study were identified in patients that were referred for sequencing of the DYSF gene in our diagnostic laboratory at the Timone Children's Hospital (Marseille, France) since 2001. The referred patients had clinical presentations highly suggestive of a primary dysferlinopathy, often with dysferlin protein deficiency identified on muscle biopsy samples (by immunohistochemistry and/or western blotting). The diagnostic approaches used during these past 20 years included single-strand conformation polymorphism (SSCP) and denaturing high performance liquid chromatography (DHPLC) with subsequent direct sequencing of identified abnormal fragments, multiplex ligationdependent probe amplification (MLPA), quantitative polymerase chain reaction (Q-PCR), and recently a gene panel. The variants identified in dysferlinopathy patients diagnosed at our center were described in several reports [START_REF] Nguyen | Dysferlin mutations in LGMD2B, Miyoshi myopathy, and atypical dysferlinopathies[END_REF][START_REF] Nguyen | Phenotypic study in 40 patients with dysferlin gene mutations: high frequency of atypical phenotypes[END_REF][START_REF] Krahn | Analysis of the DYSF mutational spectrum in a large cohort of patients[END_REF] and were the basis for the UMD-DYSF database. [START_REF] Blandin | UMD-DYSF, a novel locus specific database for the compilation and interactive analysis of mutations in the dysferlin gene[END_REF] Since 2001, 572 patients and their family members have undergone diagnostic sequencing of DYSF gene in our laboratory (Fig. 1a). Of 464 suspected dysferlinopathy patients and family members referred by French neurologists, 242 patients (191 probands) carried two DYSF variants that were considered pathogenic at the moment of diagnosis. As more information about pathogenicity of DYSF variants became available over the years, classifications of certain variants have been modified. In this study, we analyzed all 176 unique DYSF variants, including those that were at one point returned as pathogenic but were then considered as benign in subsequent diagnostic returns.

Variant classification and data analysis are described in the Supplementary Methods.

RESULTS

To address the critical issue of outdated pathogenicity classifications in public databases, we performed a retrospective analysis of 176 DYSF variants identified in the large number of patients referred to our center for diagnostic sequencing of the DYSF gene since 2001. Our laboratory was one of the first diagnostic centers in Europe to offer analysis of all 55 exons of the DYSF gene, providing this diagnostic test to 572 suspected dysferlinopathy patients and their family members over the last 20 years. We focused on families referred for sequencing by French neurologists and curated 176 unique DYSF variants identified in 242 patients (191 probands) carrying two DYSF variants that were considered pathogenic at the moment of diagnosis (Fig. 1a). It is important to note that certain variants were initially reported as pathogenic, but later were reconsidered as benign when more information about population frequencies or in trans/cis occurrences became available. We decided to include this type of variant in our analysis, since certain of them are still listed as pathogenic in public databases and could potentially influence clinical interpretations of these variants in the future. The variants curated in this study are listed in Supplementary Table S1, showing each variant's location, type and the final pathogenicity group as well as the concordance of our classification with classifications in ClinVar and LOVD at the time of the study. Supplementary Table S2 provides the detailed information about the lines of evidence and the ACMG/AMP codes used to classify each variant.

The vast majority of variants in our study were returned before the five-tier ACMG/AMP classification was adopted in our laboratory in 2016. Figure 1b shows the number of variants reclassified into five ACMG/AMP pathogenicity classes, grouped by type of the variant (35 nonsense variants, 53 indels, 28 variants affecting splicing, and 60 missense variants). Nonsense and indel variants had the largest proportion of variants assigned to the pathogenic group (34/35 and 46/53 respectively). The missense variant group had the largest proportion of variants with downgraded pathogenicity (16 of 60). Of 176 variants, 17 variants were reclassified as variants of uncertain significance (VUS), likely benign, or benign (overall concordance with original classification of 90%). LOVD entries for all variants with changes in classification were updated. Supplementary Table S3 provides detailed information used for assigning these variants into the pathogenicity classes and lists the types of evidence that were missing to classify a given variant to a pathogenic or likely pathogenic category. The major reasons for assigning variants to VUS/likely benign/benign categories were lack of other patients carrying the analyzed variant (7/17), absence of the second pathogenic/likely pathogenic variant (6/17), or association with two pathogenic/ likely pathogenic variants (4/17). As a consequence, the PM3 code was not attributed or attributed at a "supporting" level for 12 and 5 variants respectively and the PP4 code was not attributed for 12 variants. Seven variants in this group had elevated allele frequencies (BA1, BS1, or lack of PM2) and six variants were not predicted as deleterious by in silico algorithms (no PP3). Finally, three variants in this group were associated in cis with a pathogenic or a likely pathogenic variant (BP2).

Of the variants with confirmed pathogenicity, 22 had at least one entry listed as uncertain in LOVD or ClinVar at the time of the study (Table S1). In most cases, additional previously unreported patient or family segregation data allowed to establish firmer pathogenicity, thus decreasing the possibility of misdiagnosis for future patients with these variants. Variant locations along the DYSF protein and its functional domains are shown in Fig. 2. Fortythree clinically relevant (pathogenic and likely pathogenic) missense variants identified in patients referred to the Marseille center for DYSF sequencing are shown in the top panel (gray). One hundred fifteen clinically relevant indel, nonsense, and splice variants are shown in the lower panel (purple). Loss-of-function variants (purple) are distributed homogeneously along the protein length, while missense variants (gray) are more often found in C2 and Dysf domains.

Forty variants identified in dysferlinopathy patients in our center were novel, as they were not present in LOVD or in ClinVar. We submitted all variants that have not been previously reported by our group and updated the previously reported variants [START_REF] Nguyen | Dysferlin mutations in LGMD2B, Miyoshi myopathy, and atypical dysferlinopathies[END_REF][START_REF] Nguyen | Phenotypic study in 40 patients with dysferlin gene mutations: high frequency of atypical phenotypes[END_REF][START_REF] Krahn | Analysis of the DYSF mutational spectrum in a large cohort of patients[END_REF] in LOVD.

DISCUSSION

Since the ACMG/AMP classification system became widely accepted, a certain number of studies have described retrospective reanalysis of variants in various genetic conditions. Reinterpretation of variants can be approached from two perspectives: variant-based and patient-based. In the first approach, variants are reinterpreted based on publicly available data, such as population frequency or recent publications. Largescale studies of this type can be undertaken by any group without need to access patient-related information. [START_REF] Niestroj | Guideline-based and bioinformatic reassessment of lesionassociated gene and variant pathogenicity in focal human epilepsies[END_REF][START_REF] Harrison | Clinical laboratories collaborate to resolve differences in variant interpretations submitted to ClinVar[END_REF] In comparison, patient-based variant reclassification takes into account patient and family related data, such as segregation and phenotypic data, in addition to the updated variant frequency and literature information. This reclassification has to be done by the same group that published the original cohort, thus making this type of study more difficult to perform and less frequent. [START_REF] Cerino | Refining NGS diagnosis of muscular disorders[END_REF][START_REF] Krenn | Genotype-guided diagnostic reassessment after exome sequencing in neuromuscular disorders: experiences with a two-step approach[END_REF][START_REF] Tsai | Outcomes of 92 patient-driven family studies for reclassification of variants of uncertain significance[END_REF] In our study, we retrospectively analyzed variants from a large series of dysferlinopathy patients, allowing reclassification of 17 variants.

Of 176 variants analyzed in this study, 159 retained pathogenic or likely pathogenic classification after the re-evaluation. Several aspects of our study explain this relatively low concordance (90%). First, we included in the analysis certain variants that were identified as pathogenic at the very beginning of diagnostic sequencing of DYSF when the knowledge about this gene and its variants was still very limited. Even though these variants were later reinterpreted as benign or uncertain and were no longer returned to patients as pathogenic, we thought that it was important to formally reclassify these variants and to correct the "pathogenic" classification in public databases such as LOVD. Second, earlier pathogenicity classifications were attributing much higher weight to predictions by in silico algorithms, in particular for the interpretation of missense variants. In the current ACMG/ AMP classifications, pathogenicity predictions are used as only a supporting line of evidence (evidence code PP3). [START_REF] Richards | Standards and guidelines for the interpretation of sequence variants: a joint consensus recommendation of the American College of Medical Genetics and Genomics and the Association for Molecular Pathology[END_REF] Third, dysferlinopathies are much rarer than many other inherited disorders, making it difficult to interpret the pathogenicity of a variant found in only one or several patients. Indeed, a recent study reinterpreting hereditary cancer variants in such frequently mutated genes as BRCA1 had a higher concordance for pathogenic variants (99.3%). [START_REF] Mersch | Prevalence of variant reclassification following hereditary cancer genetic testing[END_REF] Fourth, interpretation of loss-offunction variants is much more straightforward than that of missense variants. As seen from Fig. 1b andS1, most of the discordance came from missense variants. Indeed, no protein truncating variants had downgraded classifications, while the classifications of 16 missense variants and one small in-frame indel were changed to VUS, likely benign, or benign (see Supplementary Table S3 for more details). Since missense variants in the DYSF gene are a common cause of dysferlinopathy, the concordance between reinterpretations of DYSF variant pathogenicity is expected to be lower compared with the concordance between reinterpretations of pathogenic variants in a gene where only lossof-function variants cause disease.

Despite recent advances aimed at standardizing the classification of variants identified by diagnostic sequencing, there are still many examples of erroneous pathogenicity classification in the scientific literature and public databases of pathogenic variants. [START_REF] Piton | XLID-causing mutations and associated genes challenged in light of data from large-scale human exome sequencing[END_REF][START_REF] Bell | Carrier testing for severe childhood recessive diseases by nextgeneration sequencing[END_REF][START_REF] Di Fruscio | Are all the previously reported genetic variants in limb girdle muscular dystrophy genes pathogenic?[END_REF] The variant interpretation in skeletal muscle disorders could be even more difficult due to reduced penetrance of certain variants as well as possible digenic or multigenic modes of inheritance. Ongoing international curation efforts, such as the ones led by ClinGen, aim to establish gene-specific interpretation guidelines and to curate a certain number of variants. [START_REF] Rivera-Muñoz | ClinGen Variant Curation Expert Panel experiences and standardized processes for disease and gene-level specification of the ACMG/ AMP guidelines for sequence variant interpretation[END_REF] However, the outdated variant classifications that are currently found in the [START_REF] Sula | Crystal structures of the human Dysferlin inner DysF domain[END_REF] The variants analyzed in this study are listed in the Supplementary Table S1. Detailed information about the lines of evidence used for variant classification is provided in the Supplementary Table S2. S1. Detailed information about the lines of evidence used for variant classification is provided in the Supplementary Tables S2 andS3.

Fig. 2

 2 Fig. 2 Distribution of pathogenic and likely pathogenic variants along the dysferlin protein. Forty-three clinically relevant (pathogenic and likely pathogenic) missense variants identified in dysferlinopathy patients referred for sequencing to the Department of Medical Genetics in Marseille are shown in the top panel (gray). One hundred fifteen clinically relevant indel, nonsense, splice variants identified in this set of samples are shown in the lower panel (purple). Exon-intron junctions of the DYSF gene as well as the functional domains of dysferlin are visualized. The dysferlin domain nomenclature is shown according to Sula et al.[START_REF] Sula | Crystal structures of the human Dysferlin inner DysF domain[END_REF] The variants analyzed in this study are listed in the Supplementary TableS1. Detailed information about the lines of evidence used for variant classification is provided in the Supplementary TableS2.

Fig. 1

 1 Fig. 1 Selection and reclassification of 176 unique DYSF variants identified in dysferlinopathy patients referred for sequencing to the Medical Genetics Department, Marseille. (a) Five hundred seventy-two patients with a clinical presentation highly suggestive of a primary dysferlinopathy were referred for sequencing of the DYSF gene in our diagnostic laboratory at the Timone Children's Hospital (Marseille, France) since 2001. Only patients and family members referred by French neurologists were included in this study (464 individuals). Of these, 242 patients (191 probands) carried two DYSF variants that were considered pathogenic at the time of diagnosis. The set of analyzed variants consisted of 176 unique DYSF variants, including those that were at one point returned as pathogenic but were then considered as benign in subsequent diagnostic returns. (b) The vast majority of 176 variants were returned before the five-tier American College of Medical Genetics and Genomics and the Association for Molecular Pathology (ACMG/AMP) classification system was adopted in our laboratory. Left side of the panel shows the four types of variants analyzed: 35 nonsense variants, 53 indels, 28 variants affecting splicing, and 60 missense variants. Right side of the panel shows the numbers of variants reclassified into five ACMG/AMP pathogenic classes (pathogenic, likely pathogenic, variant of uncertain significance [VUS], likely benign, and benign), grouped by variant type. Of 176 variants, pathogenicity classification of 17 variants was downgraded to VUS, likely benign, or benign (overall concordance with original classification of 90%). Missense variant group had the largest proportion of variants with downgraded pathogenicity (16 of 60). One in-frame indel was also downgraded to VUS. The variants analyzed in this study are listed in the Supplementary TableS1. Detailed information about the lines of evidence used for variant classification is provided in the Supplementary TablesS2 and S3.

Genetics in Medicine_#####################_

ACKNOWLEDGEMENTS

We thank the neurologists in France and beyond for referring patients to our diagnostic center. We also thank the entire team of the Laboratory of Molecular Genetics at the Timone Hospital, Marseille, for their assistance in establishing the diagnostic sequencing of the DYSF gene as well as for their help with this analysis over the years. This study was supported by Genetics Institute for Patients, Therapies Innovation & Science (GIPTIS), AFM-TELETHON, and Assistance Publique-Hôpitaux de Marseille (AP-HM).

DATA AVAILABILITY

The authors confirm that all data underlying the findings are fully available without restriction. All relevant data are within the paper and its Supporting Information files. Forty novel variants were submitted to LOVD-DYSF database (www.lovd.nl/DYSF). Overall, 46 new entries were made to LOVD-DYSF and 130 previous entries were updated.

variant databases and the literature are not likely to be completely corrected by these efforts. Our study retrospectively analyzes DYSF variants identified, published, and submitted to variant databases during 20 years of diagnostic sequencing at the Medical Genetics Department in Marseille, France. [START_REF] Nguyen | Dysferlin mutations in LGMD2B, Miyoshi myopathy, and atypical dysferlinopathies[END_REF][START_REF] Nguyen | Phenotypic study in 40 patients with dysferlin gene mutations: high frequency of atypical phenotypes[END_REF][START_REF] Krahn | Analysis of the DYSF mutational spectrum in a large cohort of patients[END_REF] Besides direct benefit for diagnostics of dysferlinopathies, our study is an example of much needed effort to re-evaluate variant interpretations in the previously published cohorts and to work with curators of variant databases to update the information for the erroneously classified variants.

AUTHOR CONTRIBUTIONS

T.C. had a major role in the analysis of data and participated in writing of the manuscript; V.B. and C.P. participated in data acquisition; M.B., M.C., F.R., N.B.-P., K.N., and N.L. revised the manuscript for content; M.K. and S.G. had major roles in the conception of the study, data interpretation, and manuscript writing.

COMPETING INTERESTS

The authors declare no competing interests.

ETHICS DECLARATION

The data on DYSF variants do not include any identifying information. These data correspond to retrospective analysis that does not require ethics committee approval at our institution. The appropriate consent was collected from each patient at the moment of variant identification as part of the standard procedure during a diagnostic genetic testing in our laboratory.

ADDITIONAL INFORMATION Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41436-021-01164-3.

Correspondence and requests for materials should be addressed to S.G. or M.K.

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