Marie-Laurence Laurence Knittel

come

that plural marking in French is possible only for event nominals that display perfective aspect. This article is organised as follows. Section 2 introduces the relevant data; we distinguish two classes of event nominals based on number inflection: ENs that show variation between a singular and a plural form, and ENs that do not allow such alternation. In section 3, we consider the relationship between the lexical [START_REF] Vendler | Linguistics in Philosophy[END_REF][START_REF] Verkuyl | On the compositional nature of aspect[END_REF][START_REF] Smith | The parameter of Aspect[END_REF] or 'inner' [START_REF] Borer | Structuring sense[END_REF][START_REF] Travis | Inner Aspect : the articulation of VP[END_REF]) aspect of event nominals and number distribution. We show that both telic and atelic VPs may give rise to uninflected nominalisations, while number inflected ENs are either aspectually vague/ ambiguous, or correspond to nominalised forms of unaccusative verbs, unambiguously denoting achievements. In section 4, we provide some hypotheses to account for the facts observed in section 3. We suggest that an inner Aspect projection hosting a [±telic] feature is present in the structure of unambiguous ENs, and may co-occur with the Number projection in unaccusative nominalisations, while ambiguous ENs exhibit only a Number projection. In section 5, we examine the idea, put forward by [START_REF] Alexiadou | Après avoir marché, j'ai mal aux pieds. 'After having walked, my feet ache.' c. *Après {de la / beaucoup de} marche, j'ai mal aux pieds[END_REF] and Iordachioaia & Soare (2009), that nominalisations may display 'outer ', i.e. [±perfective] aspect. Since perfective nominalisations systematically allow both singular and plural forms, while imperfective ENs are always uninflected, we suggest that the outer aspectual value is directly linked to the distribution of Number.

[dernière phrase supprimée]

The data

French Event Nominals can be formed with several suffixes, most frequently -age (jardinage 'gardening'), -ment (bombardement 'bombing'), and -ion (utilisation 'use'). Some ENs are formed through conversion (i.e. zero affixation) of past participles (conduite 'driving') or verb bases (attaque 'attack'). In this section, we examine ENs in these morphological classes, and distinguish two patterns regarding number inflection: inflected and uninflected nominals.

Uninflected Event Nominals

This subsection is dedicated to the examination of uninflected deverbal nominals. The first class of uninflected ENs stems from the nominalisation of intransitive verbs denoting activities [START_REF] Haas | Les propriétés aspectuelles des noms d'activités[END_REF], Huyghe & Marín 2008[START_REF] Heyd | Les noms d'activité parmi les noms abstraits : propriétés aspectuelles, distributionnelles et interprétatives[END_REF] The above examples show that such nouns are introduced by the partitive determiner du / de la, which also introduces concrete mass nouns (2). These nouns do not exhibit plural marking and cannot be introduced by the singular indefinite article un(e) under normal conditions (in (3), but see (7-8) below).

( for two hours 'He has done {(some) cleaning / (some) sailing} for two hours.' Some of these ENs also have transitive counterparts, which allow them to become telic when combined with quantized arguments.

(5)a. Le nettoyage de toute la maison en deux heures m'a épuisée. the cleaning of all the house in two hours me-has exhausted 'The cleaning of the whole house in two hours has exhausted me.' b. La lingère m'a promis le repassage de cette chemise en 15 minutes the maid me-has promised the ironing of this shirt in 15 minutes 'The maid promised me the ironing of this shirt in 15 minutes.' Finally, French also displays strictly transitive atelic verbs that give rise to uninflected ENs.

(6)a. L'utilisation *(de la photocopieuse) pendant une journée complète the use *(of the copy-machine) for a day whole l'a endommagée. it-has damaged 'The use of the copy-machine for a whole day damaged it.' b. La conduite (de la voiture) pendant plus de quatre heures m'a fatiguée.

the driving (of the car) for more than four hours me-has tired 'The driving of the car for more that four hours exhausted me.' According to Knittel (2009[START_REF] Knittel | Possession vs incorporation in the nominal domain; evidence from Fench event nominal dependencies[END_REF], the presence of the definite article instead of the partitive is due to the fact that transitive ENs must occur as heads of possessive constructions. The possessive construction, which is necessary to legitimate the EN complements, is characterized in French by the presence of the definite article even at first mention. However, their mass character shows through two other properties. When modified, these nouns must be introduced by an indefinite article, like other mass nouns (7); they also allow a 'manner reading' when pluralized and modified, which parallels the 'subtype' reading displayed by plural mass nouns under the same conditions (8). ( 7 A peculiarity of EN pluralisation in French is that the described event must involve repeated occurrences with respect to the patient. This is exemplified in (12-13). In (12b), the nominal counterpart of (12a), the noun atterissage 'landing' occurs in the plural since it refers to two landing events by the same plane. Conversely, two landing events, each of which is performed by a different plane, can only be referred to by a singular EN (12d vs. e). Similar observations hold for the examples in (13), where bombardement 'bombing' must occur in the singular if it refers to one bombing event, even when several patients are involved. Consequently, example (13c) can only refer to several bombing events of each of the cities in question. the bombing of two cities has made many victims 'The bombing of two cities has made many victims.' ≠ c. Les bombardements de deux villes ont fait de nombreuses victimes the-Pl bombing-Pl of two cities have made many victims. 'The bombings of two cities have made many victims.' Evidently, this condition only applies to inflected event nominals. Thus, an event such as nettoyage 'cleaning' (14) will be denoted by an uninflected nominal, regardless of the number of repetitions or the identity of the patient. ( 14) La punition de Max a été le(*s) nettoyage(*s) de la cuisine the punishment of Max has been the(*Pl) cleaning(*Pl) of the kitchen (et de la chambre) pendant un mois. (and of the bedroom) for one month 'Max's punishment has been the cleaning of the kitchen (and of the bedroom) for one month'.

We now turn to the interaction between the aspectual properties of the nominalised events and number-inflection.

Number-inflection and (a)telicity

As exemplified in 2.1, most ENs that can be characterised as mass nouns describe atelic events. This observation is not unexpected, given Mourelatos' (1978) and [START_REF] Borer | Structuring sense[END_REF] position that only nouns referring to telic events can appear in the plural. However, if this were indeed the case, we would expect a contrast between the telic and atelic uses of the ENs under consideration, the former exhibiting number variation, but not the latter. And yet, typically atelic ENs (such as repassage 'ironing' (15) or balayage 'sweeping' ( 16 We can, therefore, conclude that the number properties of these ENs do not rely on their telic or atelic reading.

Let us now consider the aspectual properties of number-inflected ENs. for its inhabitants 'The invasion of the country in one week has been an ordeal for its inhabitants.' c. Les invasions de la Gaule au V° et au VII° siècle the-Pl invasion-Pl of the Gaul in.the 5th and in.the 7th centuries ont profondément bouleversé la société. have deeply modified the society 'The invasions of Gaul in the 5th and 7th centuries have deeply changed the society.'

The above examples illustrate the fact that ENs referring to telic events display number variation. However, as shown in examples (15-16) above, telicity does not necessarily entail number inflection. Consequently, we have to find another parameter weighing on the distribution of number. Note, however, that the ENs in (16-20) share a common property-they are all aspectually ambiguous. Consider the ENs in (21). The noun invasion 'invasion' (21a) allows aspectual modifiers introduced by pendant 'for', which should not be the case if it only admitted a telic interpretation. The same is true for lavage 'washing' (21b).

(21) a. L'invasion du pays pendant deux ans a été une épreuve the invasion of.the country for two years has been an ordeal pour ses habitants. for its inhabitants 'The invasion of the country for two years has been an ordeal for its inhabitants.' b. Le lavage de la chemise pendant une heure n'a pas donné the washing of the shirt for one hour has not given un résultat satisfaisant. a result satisfactory 'The washing of the shirt for one hour did not given a satisfactory result.'

The noun vol 'theft' can be construed either as an achievement (18b), or as an accomplishment, as shown by its compatibility with prendre X temps 'take X time' (22a); assassinat 'assassination' also allows a durative reading (22b), evident in its possible combination with pendant [START_REF] Haas | Du verbe au nom : calques et décalages aspectuels[END_REF]).

(22) a. Le vol du tableau a pris plusieurs heures. the theft of.the painting has taken several hours 'The theft of the painting has taken several hours.' b. Il a été blessé pendant l'assassinat. he has been wounded during the assassination 'He has been wounded during the assassination.' Another ambiguity is illustrated in (23); the number inflected EN bombardement 'bombing' is ambiguous between a single event (i.e. the launching of one bomb) and a multiple event activity (Smith 1991: 85) ENs referring to activities with an inceptive component, i.e. focussed on their initial point (Smith 1991, 48), constitute another class of number inflected ENs. For example, cambriolage 'burglary' involves a break-in event, followed by various events of theft. Similarly, agression 'aggression' implies an assault followed by various molestations. Such events display properties of both activities and achievements, evident in their compatibility both with PPs introduced by pendant 'for' and with those introduced by à 'at', the latter focusing on the initial point of the process.

( The last class of ENs allowing number variation is the unaccusatives. Like the verbs they are derived from, unaccusative nominals refer unambiguously to achievements, and may occur in either the singular or in the plural. Unaccusative ENs constitute a regular correlation between telicity and number inflection. Recall that the opposite pattern was noted with uninflected atelic nominals such as jardinage 'gardening' (4a). Both groups, thus, fit into the correlation between telicity and plurality put forward by [START_REF] Mourelatos | Events, Processes and States[END_REF] and [START_REF] Borer | Structuring sense[END_REF]. Conversely, the other ENs described above do not exhibit such a strict covariation between Number distribution and (a)telicity. Thus, nominals originally referring to activities do not allow pluralisation when merged with quantized objects, even though they denote telic events (15c, 16c). As for number inflected nominals, they cannot be systematically characterized as telic, but are either aspectually ambiguous or activities with an inceptive component. Our conclusion, then, is that it is not possible to establish a regular correlation between (a)telicity and the distribution of number in French.

We can now propose a syntactic account of both inflected and non-inflected French ENs.

NumP and AspP: some hypotheses

As argued by various authors (see in particular [START_REF] Alexiadou | Après avoir marché, j'ai mal aux pieds. 'After having walked, my feet ache.' c. *Après {de la / beaucoup de} marche, j'ai mal aux pieds[END_REF], Iordachioaia & Soare 2009[START_REF] Sleeman Petra | Aspect and argument structure of deverbal nominalizations[END_REF], the functional structure of event nominals can contain nominal projections, verbal projections or a combination of both. Among the nominal projections, number (NumP) is of particular interest here, since French ENs allow number variation. Therefore, the functional projections of number inflected ENs should contain a NumP iv , as is the case for any other count noun. As for the relevant verb structure, it has been postulated by [START_REF] Borer | Structuring sense[END_REF] that the functional structure of ENs contains a projection of inner Aspect, AspP (see also [START_REF] Travis | Inner Aspect : the articulation of VP[END_REF]. As shown above (17-24), number-inflected ENs in French are frequently aspectually ambiguous. To account for this phenomenon, we assume that they lack an inner aspect projection. The presence of NumP and the absence of inner aspect suggest, in fact, that number-inflected nominals in French are closer in structure to nouns than to verbs. A similar view is presented by Iordachioaia & Soare (2009) for Romanian infinitive CENs that exhibit number-marking. According to their analysis, nominals of this class exhibit another peculiarity: they can establish anaphoric relations with the same demonstrative pronoun as nouns. Conversely, the other class of Romanian ENs, (uninflected) 'supine CENs', are referred to using the same pronoun as reference to clauses. A similar contrast is also observed in French. Anaphoric reference to inflected ENs patterns with reference to other nouns while reference to uninflected ENs patterns with reference to clauses. The following examples show that the third person personal pronouns il(s) / elle(s) are preferred as anaphoric pronouns for number-inflected ENs, compared to the neuter pronoun ça, which is used as an anaphor for clauses:

(27) a. Le bombardement de Londres a commencé en 1940. {Il These data suggest that uninflected ENs are more 'verbal' than those which allow number marking. As a consequence, we may consider their lack of number inflection as a reflex of their more verbal characteristics.

There are other arguments supporting such an analysis. Beauseroy & Knittel (to appear) show that uninflected activity nominals do not display the usual properties of nouns. They cannot be modified by qualifying adjectives ((29), see also [START_REF] Heyd | Les noms d'activité parmi les noms abstraits : propriétés aspectuelles, distributionnelles et interprétatives[END_REF], suggesting that they lack some functional projection that usually dominates NPs. Moreover, the light verb faire 'to do' can systematically be inserted to introduce these nouns (30-31).

( J'adore faire du jardinage. I love to do part.art. gardening A possible analysis, then, is that such nouns are associated with verbal projections, possibly the AspP mentioned above. Given that these nouns essentially refer to activities, we hypothesize that their structure contains an AspP projection encoding atelicity. As noted above, these nouns can combine with quantized arguments to give rise to telic DPs, similarly to verbs (15b, 16b). The absence of number inflection in such cases, also noted above, can now be seen as a result of the absence of a NumP projection in the structure of these ENs. Finally, let us turn to unaccusative ENs. Unlike other number-inflected ENs, unaccusative ENs (25-26) are not aspectually ambiguous, as they always denote achievements, which are clearly telic events. A possible analysis is that these ENs can be analysed as containing an Aspect projection, like activity ENs. But, the difference is that AspP in unaccusative ENs encodes a telic feature, rather than the atelicity found in activity nominals. Yet, unaccusative ENs systematically display number variation, suggesting that they are also dominated by a NumP. Consequently, unaccusative ENs display a mixed functional structure, containing both nominal and verbal categories. We now turn to the morphological structure of French ENs.

A note on morphology and outer aspect

In the preceding section, we argued that activity nominals are characterized by the presence of an inner aspect projection in their functional structure and lack a Number projection. Consequently, the same unmarked form is used to refer to one or several events, as exemplified (33).

(33) Le balayage de la cuisine {hier / pendant trois semaines} the sweeping of the kitchen {yesterday / for three weeks} 'The sweeping of the kitchen {yesterday / for three weeks}'

In the above example, the adverbial hier 'yesterday' entails a single event reading of the string le balayage de la cuisine 'the sweeping of the kitchen' whereas pendant trois semaines 'for three weeks' is preferably construed as iterative, denoting a series of events. This assumption leads to another question: how does this plural event reading appear, in the absence of plural marking? In the verbal domain, event plurality has been argued to be the result of a pluractional operator (see in particular [START_REF] Cusic | Verbal plurality and aspect[END_REF][START_REF] Lasersohn | Plurality, conjunction and events[END_REF][START_REF] Van Geenhoven | For-adverbials, frequentative aspect, and pluractionality[END_REF], Laca 2006 a,b). The same analysis has been applied to Romanian supine CENs that do not exhibit number variation (Iordachioaia & Soare 2009). Looking back at the examples of activity nominals provided above, one can observe that most ENs are formed with the verbal derivational suffix -age (balayage 'sweeping', nettoyage 'cleaning', jardinage 'gardening', repassage 'ironing'). According to [START_REF] Ferret | Les noms d'événements en -age et en -ée : une différenciation fondée sur l'aspect grammatical[END_REF], -age suffixation rule conveys outer aspect with imperfective value, and, consequently, the resulting nominalisations display pluractionality effects (see also [START_REF] Martin | The Semantics of Eventive Suffixes in French[END_REF][START_REF] Uth | The rivalry of French -ment and -age from a diachronic perspective[END_REF]. This analysis could be used to explain why activity nominals ending with -age do not require plural inflection; since -age suffixation already supplies a pluractional value, there is no need for any other plurality marker. However, the correlation is not complete. On the one hand, as illustrated in (34), activity nominals may be derived by various other means besides -age (suffixes, past participle conversion, verb to noun conversion, see section 2). ( 34 Note that -age nominals are not incompatible with idiomatic expressions that preclude the subdivision of events, such as en un tournemain (lit. 'in one turn of the hand') or en un clin d'oeil 'in one wink' [START_REF] Laca | Indefinites, quantifiers and pluractionals: What scope effects tell us about event pluralities[END_REF].

(36) a. L'abattage de l'arbre en un clin d'oeil 'the cutting down of the tree in one wink' b. Le démontage de l'armoire en un tournemain 'the disassembling of the cupboard in one turn of the hand' Therefore, we conclude that pluractionality with activity ENs comes from the atelic interpretation of nominals in itself, that is, from the presence of inner aspect with an atelic value. The same analysis is proposed by [START_REF] Alexiadou | Après avoir marché, j'ai mal aux pieds. 'After having walked, my feet ache.' c. *Après {de la / beaucoup de} marche, j'ai mal aux pieds[END_REF] for Polish EN -nie/-cie nominalisations. Unlike -age nominalisations, nouns ending with -ée, which are converted from past participles [START_REF] Tribout | [END_REF], are analysed as perfective by [START_REF] Ferret | Les noms d'événements en -age et en -ée : une différenciation fondée sur l'aspect grammatical[END_REF]. However, as with -age nominals, things are not so simple. First, ENs derived from past participles are not always telic (e.g., conduite 'driving', from the past participle of conduire 'to drive', which refers unambiguously to an activity, (6b)). We assume that unaccusative nominals, that are generally converted from past participles (25, 26), are the exact counterpart of activity nominals in that they are necessarily telic (see section 3) and perfective.

Recall. however, that they are dominated by NumP, as shown by their number inflection. Two analyses are possible here. On the one hand, we could assume that their outer aspect projection is dominated by a NumP, accounting for their count properties. On the other hand, it can be proposed that NumP in itself is responsible for their perfective interpretation. This analysis can be transposed to unaccusative nouns, which can always be combined with the preposition après.

(40) a. Après l'arrivée de l'acteur, les applaudissements ont commencé. 'After {the entry of the actor}, the applaudings begun' = b. Après que l'acteur est arrivé, les applaudissements ont commencé.

'After the actor came in, the applaudings begun'

Finally, let us consider additional number inflected ENs. Here we find greater morphological variation than with activity nominals. This was illustrated above with nouns ending in -ion (invasion 'invasion' (9a)), -ment (bombardement 'bombing' (9b)) and -at (assassinat 'assassination' ( 17)), as well as cases of noun-to-verb conversion (zero affixation, e.g., vol 'theft' ( 18)). Unlike uninflected nominals and unaccusatives, that display some sort of morphological regularity, number inflected ENs cannot receive a uniform analysis. In the preceding section, we have hypothesized that such nouns do not contain an inner aspect projection in their structure, in light of their aspectual ambiguity. On the other hand, we have argued that number inflection conveys perfectivity in unaccusative nouns. Extending this position to the present set of ENs, that have been defined as inflected nominals, we should expect these nominals to display perfective outer aspect. This is confirmed in (41), using the après test introduced in (38) above.

(41) a. Après {le vol / l'assassinat / le cambriolage}, un suspect a été arrête par la police.

'After {the theft / the assassination / the burglary}, a suspect has been arrested by the police.' b. La rebellion a commencé après {l'invasion / le bombardement}.

'The rebellion begun after {the invasion / the bombing}. [START_REF] Ferret | Les noms d'événements en -age et en -ée : une différenciation fondée sur l'aspect grammatical[END_REF] propose another test of perfectivity: when used as objects of filmer 'to film', perfective ENs are construed as referring to events as a whole, including their initial and final endpoints. As expected, our inflected ENs reveal perfective behaviour (42).

(42) a. J'ai filmé {le vol / l'assassinat / le cambriolage}. 'I have filmed {the theft / the assassination / the burglary}.' b. {L'invasion / le bombardement} a été filmé.

'{The invasion / the bombing} has been filmed.' Our conclusion is, therefore, that the perfective interpretation of these ENs is due to the presence of a Number Projection in their functional structure, as is the case with unaccusative nominals. Since (im)perfectivity in French is conveyed by the inflectional category of Tense in the verbal domain, the fact that it is also conveyed by Number, the only inflectional category available in the nominal domain, appears to be a welcome result.

Conclusion

In this article, we examined the behaviour of French Event Nominals with respect to number marking. We have shown that two types of nominals display number inflection: unaccusatives, that retain the telic feature of their base verbs, and a mixed class of aspectually vague or ambiguous ENs. Conversely, activity nominals never allow number inflection, although they may give rise to telic DPs when combined with quantized objects. We therefore conclude that number inflection does not arise from telicity. Turning to the examination of outer aspect, we showed that unaccusatives and vague or ambiguous ENs exhibit perfective behaviour in perfectivity tests proposed by [START_REF] Ferret | Les noms d'événements en -age et en -ée : une différenciation fondée sur l'aspect grammatical[END_REF]. Consequently, we argue that number inflection is restricted to nominals displaying a perfective interpretation, an observation with attested cross-linguistic parallels [START_REF] Alexiadou | Après avoir marché, j'ai mal aux pieds. 'After having walked, my feet ache.' c. *Après {de la / beaucoup de} marche, j'ai mal aux pieds[END_REF]. To account for this observation, we hypothesized that the projection of Number is responsible for the perfective interpretation of inflected nominals. This conclusion is further supported by a particular class of nouns derived from activity verbs, which allow both mass and count uses. We showed that a perfective interpretation of these nouns only arises when they are construed as count nouns, as predicted by the proposed analysis.

  atterrissages de l'avion se sont bien passés. the landing-Pl of.the plane Refl-be well passed 'The landings of the plane went well.' c. Les deux avions ont atterri (une fois). the two planes have landed (one time) 'The two planes have landed (once).' = d. L'atterrissage des deux avions s'est bien passé. the landing of.the two planes Refl-be well passed 'The landing of the two planes (went well.' ≠ e. Les atterrissages des deux avions se sont bien passés. the-Pl landing-Pl of.the two planes Refl-be well passed 'The landings of the two planes went well.' (13) a. L'aviation a bombardé deux villes. the air force has bombed two cities 'The air force bombed two cities.' = b. Le bombardement de deux villes a fait de nombreuses victimes.

  )) can combine with a quantized argument, resulting in a telic VP/NP (b-c), without altering the properties of the EN with respect to number inflection, as shown in (d) iii : (15) a. J'ai fait du repassage pendant deux heures. I have done part.art. ironing for two hours lit. : 'I have done ironing for two hours.' b. Le repassage de la chemise en 15 minutes the ironing of the shirt in 15 minutes 'the ironing of the shirt in 15 minutes' c. J'ai repassé la chemise en 15 minutes the ironing the shirt in 15 minutes 'I ironed the shirt in 15 minutes' d. ?? Les repassage-s de la chemise (pendant 2 heures / en 15 minutes) the-Pl ironing-Pl of the shirt (for two hours / in 15 minutes) (16) a. J'ai fait du balayage pendant toute la matinée. I have done part.art. sweeping for all the morning lit. : 'I have done sweeping for the whole morning.' b. Le balayage de la cour en une demi-heure the sweeping of the courtyard in a half hour 'the sweeping of the courtyard in half an hour' c. J'ai balayé la cour en une demi-heure I have swept the courtyard in a half hour 'I swept the courtyard in half an hour' d. ?? Les balayages de la cour (pendant toute la matinée / the-Pl sweeping-Pl of the courtyard (for the whole morning / en une heure) in an hour)

  to come in / {one / indef.pl.} entry(Pl) b. L'entrée de l'acteur a été applaudie. the entry of the actor has been applauded 'The arrival of this actor has been applauded.' c. Les entrées de cet acteur sont toujours très applaudies. the-Pl entry-Pl of this actor are always much applauded 'The arrivals of this actor always are much applauded.' (26) a. arriver / {une / des} arrivée(s) to arrive / {one / indef.pl.} arrival(Pl) b. L'arrivée de Paul et Marie n'est pas passée inaperçue. the arrival of Paul and Marie be-not passed unnoticed 'The arrival of Paul and Mary did not go unnoticed.' c. Les arrivées de Paul et Marie ne sont pas passées inaperçues. the-Pl arrival-Pl of Paul and Marie be-not passed unnoticed 'The arrivals of Paul and Mary did not go unnoticed.'

(

  35) a. L'abattage de l'arbre {*pendant / en} 15 minutes 'the cutting down of the tree {*for / in} 15 minutes' b. Le démontage de l'armoire {*pendant / en} une heure 'the disassembling of the cupboard {*for / in} one hour'

  ii .

	part.art. water
	'(some) water'
	(1) a. jardiner	/ du	jardinage
	'to garden'	part.art. gardening
	b. balayer	/ du	balayage
	'to sweep'	part.art. sweeping
	c. naviguer / de la	navigation
	'to sail'		part.art. sailing
	(2) de l'	eau

  Pl theft-Pl of this painting in 1917 and in 1988 have stayed famous lit. 'The thefts of this painting in 1917 and in 1988 are still famous.'

	pour ses habitants.
				The
	following examples show that achievements (assassinat 'assassination' (17)
	and vol 'theft' (18)) and accomplishments (lavage 'washing' (19) invasion
	'invasion' (20)) give rise to nominalisations exhibiting number specification.
	(17) a. assassiner /	{un / des}	assassinat(s)
	to assassinate / {one / indef.pl.} assassination(Pl)
	b. L'assassinat	de cet opposant politique a eu lieu	à Paris.
	the assassination of this opponent political has taken place in Paris
	'The assassination of this political opponent took place in Paris.'
	c. Les assassinats	d'opposants politiques	sont fréquents
	the-Pl assassination-Pl of opponents political are frequent
	dans certains pays.
	in some countries
	'Assassinations of political opponents are frequent in some countries.'
	(18) a. voler / {un / des}	vol(s)
	to steal / {one / indef.pl.} theft(Pl)
	b. Le vol du tableau	s'est produit	à minuit.
	the theft of.the painting Refl-be occurred at midnight
	'The theft of the painting has occurred at midnight.'
	c. Les vols	de ce tableau en 1917 et en 1988	sont restés célèbres.
	the-(19) a. laver /	{un / des}	lavage(s)
	to wash / {one / indef.pl.} washing(Pl)
	b. Le lavage	de la chemise en une heure	n'a pas donné
	the washing of the shirt	in one hour	has not given
	un résultat satisfaisant.
	a result satisfactory
	'The washing of the shirt in one hour did not produce a satisfactory result.'
	c. Les lavages	répétés de la chemise ont terni	ses couleurs.
	the-Pl washing-Pl repeated of the shirt	have tarnished its shades
	lit.: 'The repeated washings of the shirt have tarnished its shades. '
	(20) a. envahir / {une / des}	invasion(s)
	to invade / {one / indef.pl.} invasion(Pl)
	b. L'invasion du pays	en une semaine a été une épreuve
	the invasion of.the country in one week	has been an ordeal

  , consisting of the launch of several bombs. (23) a. Le bombardement d'Hiroshima a eu lieu a 8h15. [single event] 'The bombing of Hiroshima took place a 8.15.' b. Le bombardement de Londres s'est poursuivi jusqu'en 1941 'The bombing of London lasted until 1941.' [multiple event activity]

  / ?? ça} s'est poursuivi jusqu'en 1941. 'The bombing of London has begun in 1940. {It/ ??that} lasted until 1941.' b. L'assassinat de cet opposant politique a eu lieu à Paris. {Il / ?? ça} a malheureusement fait d'autres victimes. 'The assassination of this political opponent took place in Paris. / le nettoyage de} cette chambre, {ça / *il} m'a épuisée. lit. 'Cleaning / the cleaning of} this room, {that / *it} has exhausted me.' c. {Conduire / la conduite de} la voiture pendant plus de quatre heures, {ça / *elle} m'a donné des maux de tête. '{Driving / the driving of} the car for more that four hours, {that / *it} has given me headache.'

	Unfortunately, {it / ??that} has made other victims.'
	c. Ils ont été traumatisés par le cambriolage de leur maison. En effet, {il / ? ça}
	a duré plusieurs heures.
	'They have been traumatised by the burglary of their house. Indeed, {it / ?
	that} lasted for several hours.'
	d. {Elles ne sont pas / ? ça n'est pas} passé(es) inaperçu(es), les arrivées de
	Paul et Marie !
	'{They / ? that} did not go unnoticed, the arrivals of Paul and Mary!'
	Uninflected ENs are referred to by the neuter pronoun ça, as are the
	corresponding VPs.
	(28) a. {Le jardinage / jardiner}, {ç' / *il} est agréable.
	lit.'{The gardening / gardening}, {that / *it} is pleasant.' (i.e. 'Gardening is
	pleasant.')
	b. {Nettoyer

  The following table summarizes the observations put forward in this section:

	(32)
	Example :	jardinage	bombardement	arrivée
	Number inflection	-	+	+
	NumP	-	+	+
	Aspectual value	-telic	ambiguous	+telic
	AspP	+	-	+

  An argument supporting the latter hypothesis is provided by nouns such as danse 'dance/dancing' or marche 'walk/walking'. As shown by various authors (Van de Velde 1995, Haas & Huyghe 2010, Haas Huyghe & Marín 2008, Heyd & Knittel 2009, Beauseroy & Knittel to appear), these nouns can be used as either mass or count nouns. After {part.art. dance / a lot of walk}, my feet ache.' Examples like (38) lead us to conclude that, in the case of danse and marche, number inflection is linked to perfectivity v (see Alexiadou et al. 2010 for a cross-linguistic generalisation).

	(37) a. une danse / de la danse
	a dance /	part.art. dance
	b. une marche / de la marche
	a walk /	part.art. walk
	As shown in (38), these nouns can be introduced by the preposition après
	'after' when used as count nouns (38a), but not when they are construed as
	mass (38c). The après test is proposed by Ferret et al. (2010) to distinguish
	perfective from imperfective nominals.

'

iii Note that PPs introduced by pendant 'for' are not excluded with quantized arguments, this combination resulting in the atelic reinterpretation of the VPs/DPs: (i) {Le repassage de la chemise / j'ai repassé la chemise} pendant 2 heures. '{The ironing of the shirt / I have ironed the shirt} for two hours'. (ii) {Le balayage de la cour / j'ai balayé la cour} pendant toute la matinée.

'The sweeping of the courtyard / I swept the courtyard} for the whole morning'. iv We assume that only count nouns are dominated by a Number Projection, and that the mass reading arises from the absence of NumP (see [START_REF] Kwon | Number from a syntactic perspective: why plural marking looks 'truer' in French than in Korean[END_REF], Knittel 2009). v Après does not influence inner aspect, since danse and marche in (37-38) are still construed as activities.


E Lima

B P Weiss

J P Cuq-Lelandais

Unraveling the simultaneous shock magnetization and demagnetization of rocks

In the natural case of an hypervelocity impact on a planetary or asteroidal surface, two competing phenomena occur: partial or complete shock demagnetization of preexisting remanence and acquisition of shock remanent magnetization (SRM). In this paper, to better understand the effects of shock on the magnetic history of rocks, we simulate this natural case through laser shock experiments in controlled magnetic field. As previously shown, SRM is strictly proportional to the ambient field at the time of impact and parallel to the ambient field. Moreover, there is no directional or intensity heterogeneity of the SRM down to the scale of ~0.2 mm 3 . We also show that the intensity of SRM is independent of the initial remanence state of the rock. Shock demagnetization and magnetization appear to be distinct phenomena that do not necessarily affect identical populations of grains. As such, shock demagnetization is not a limiting case of shock magnetization in zero field.

As a consequence, when it can be recognized in a rock, SRM must be considered as a reliable record of the direction and intensity of the ambient magnetic field at the time of impact. The natural process of hypervelocity impact where a rock carrying a remanent magnetization is shocked in the presence of an ambient field can be studied as the simple superimposition of shock demagnetization and shock magnetization. For this there are now a variety of techniques that allow experimental study of both phenomena separately or simultaneously as in this study.

These results have potential implications for the paleomagnetic study of meteorites, and lunar rocks, and for the understanding of the magnetic signature (as studied through paleomagnetism and/or magnetic anomalies) of terrestrial, lunar and Martian impact craters.

Introduction

An understanding of the effects of shock on the remanent magnetization of rocks is important for the interpretation of the magnetic anomalies of extensively cratered planetary surfaces like those on Mars (e.g., [START_REF] Hood | Distribution of crustal magnetic fields on Mars: shock effects of basin-forming impacts[END_REF][START_REF] Shahnas | Viscous and impact demagnetization of Martian crust[END_REF][START_REF] Louzada | Shock effects on pyrrhotite A c c e p t e d M a n u s c r i p t 12 magnetism and implications for Martian crustal magnetization[END_REF], and the Moon (e.g., Halekas et al., 2003;[START_REF] Hood | Antipodal effects of lunar basin-forming impacts: Initial 3D simulations and comparisons with observations[END_REF]. It is also crucial for the understanding of the paleomagnetic signal of meteorites and lunar rocks that have all been shocked (see discussion in e.g., Carporzen et al., 2010;[START_REF] Yu | Ancient stable magnetism of the Richardton H5 chondrite[END_REF], Weiss et al. 2009). To a lesser extent, the interpretation of the paleomagnetic record of terrestrial craters also requires the identification of possible shock effects on remanence (see, for instance, [START_REF] Louzada | Paleomagnetism of Lonar impact crater, India[END_REF].

The first generation of studies dealing with the effect of shocks on magnetic remanence was published in the 1970's, motivated by the return of Apollo lunar A c c e p t e d M a n u s c r i p t 2 samples [START_REF] Nagata | Introductory notes on shock remanent magnetization and shock demagnetization of igneous rocks[END_REF]Fuller et al., 1974;[START_REF] Pohl | Shock magnetization and demagnetization of basalt by transient stress up to 10 kbar[END_REF]Cisowski et al., 1976;Martelli and Newton, 1977;Fuller, 1977;Cisowski and Fuller, 1978). A second generation, partly triggered by the discovery of magnetic anomalies associated with the impact basins on Mars [START_REF] Acuña | Global Distribution of Crustal Magnetization Discovered by the Mars Global Surveyor MAG/ER Experiment[END_REF], was published during the past decade [START_REF] Dickinson | Shock magnetism in fine particle iron[END_REF][START_REF] Kletetschka | Pressure effects on martian crustal magnetization near large impact basins[END_REF][START_REF] Louzada | Effect of shock on the magnetic properties of pyrrhotite, the Martian crust and meteorites[END_REF][START_REF] Gattacceca | Investigating impact demagnetization through laser impacts and SQUID microscopy[END_REF][START_REF] Gattacceca | The effects of explosive-driven shocks on the natural remanent magnetization and the magnetic properties of rocks[END_REF][START_REF] Gattacceca | On the efficiency of shock magnetization processes[END_REF]. Most of these experimental studies dealt with shock demagnetization, i.e. the destruction of previous remanence by shock waves either in zero field (when dealing with natural remanence) or in the Earth field (equivalent to zero field when dealing with artificial isothermal saturation remanence).

Although more data are clearly needed for ferromagnetic FeNi alloys (kamacite, taenite, tetrataenite) that are the most common remanence carrier in meteorites and lunar rocks, we now have a fair understanding of shock demagnetization processes. And although definitive experimental proof is still lacking, there are indications that for pressures below the Hugoniot elastic limit (typically 1-5 GPa for silicate minerals), dynamic pressure (shock experiments) and static pressure (hydrostatic pressure experiments) give the same demagnetizing effect, so that the results of the experimental demagnetization by hydrostatic pressure (e.g. [START_REF] Rochette | High pressure magnetic transition in pyrrhotite and impact demagnetization on Mars[END_REF][START_REF] Gilder | Static stress demagnetization of single and multidomain magnetite with implications to meteorite impacts[END_REF][START_REF] Bezaeva | Pressure demagnetization of the Martian crust: ground truth from SNC meteorites[END_REF][START_REF] Bezaeva | Demagnetization of terrestrial and extraterrestrial rocks under hydrostatic pressure up to 1.2 GPa[END_REF] can be safely extended to the shock processes.

Shock magnetization, i.e. the acquisition of new remanence by shock in the presence of a magnetic field, has received much less attention [START_REF] Nagata | Introductory notes on shock remanent magnetization and shock demagnetization of igneous rocks[END_REF][START_REF] Pohl | Shock magnetization and demagnetization of basalt by transient stress up to 10 kbar[END_REF][START_REF] Pohl | The Effect of Shock on Remanence in Multi-Domain Iron Grains and Implications for Palaeointensity Measurements[END_REF][START_REF] Nagata | Shock remanent magnetization of meteorites[END_REF][START_REF] Gattacceca | On the efficiency of shock magnetization processes[END_REF][START_REF] Funaki | Acquisition of shock remanent magnetization for demagnetized samples in a weak magnetic field (7 µT) by shock pressures 5-20 GPa without plasma-induced magnetization[END_REF] in part because it is experimentally more demanding as it requires ambient field control during the shock experiments. Yet, the fundamental properties of shock remanent magnetization (SRM, defined by [START_REF] Nagata | Introductory notes on shock remanent magnetization and shock demagnetization of igneous rocks[END_REF]) are now well established [START_REF] Pohl | Shock magnetization and demagnetization of basalt by transient stress up to 10 kbar[END_REF][START_REF] Gattacceca | On the efficiency of shock magnetization processes[END_REF]: SRM is proportional to the ambient field in the low-field (< ~1 mT) limit and is strictly parallel to the ambient magnetic field for magnetically isotropic rocks; its intensity is independent of the angle between the shock wave propagation direction and the ambient field for isotropic rocks, it can have a significant intensity compared to thermoremanent magnetization (TRM) acquired in the same ambient field (e.g. up to 36 % in Gattacecca et al., 2008); and it has a coercivity spectrum shifted towards lower values compared to that of TRM.

However, in the natural case of an hypervelocity impact on a planetary or asteroidal surfaces, two competing phenomena occur: partial or complete shock demagnetization of pre-existing remanence and acquisition of SRM. In this paper, we simulate experimentally this natural case to better understand the effects of shock on the magnetic history of rocks, and, in particular, possible interactions between these two phenomena. We also incorporate new information on the basic properties of the shock magnetization process such as small-scale (mm 3 ) directional homogeneity.

Material and methods

We conducted shock experiments with a pulsed laser in controlled magnetic field using the same experimental setting as in [START_REF] Gattacceca | On the efficiency of shock magnetization processes[END_REF] which is briefly described by figure 1. Laser shock experiments were conducted at Laboratoire de Combustion et Détonique (LCD, Poitiers, France). The target rock was a basalt whose petrography and rock magnetic properties are described in length in [START_REF] Gattacceca | On the efficiency of shock magnetization processes[END_REF]. In summary, this basalt contains pseudo-single domain titanomagnetite (with Bcr/Bc = 2.29, and Mrs/Ms = 1.82, with respective standard deviation 0.20 and 0.16 as A c c e p t e d M a n u s c r i p t 3 measured on 12 samples, where Bcr is the coercivity of remanence, Bc is the coercivity, Mrs is saturation remanence, and Ms is saturation magnetization), with blocking temperatures up to 300°C. Moreover this basalt was shown to be magnetically homogeneous at least to the scale of 10 -2 cm 3 , well below the size of the samples used in this study (~cm 3 ). The mechanical properties and the behavior under shock are described in [START_REF] Berthe | Mechanical characterization of basalt using shock wave induced by laser, Application to planetary hypervelocity impact effect[END_REF]. In this work, shock wave propagation was modeled in a 2D axisymmetric configuration using the Radioss software and the Hyperworks suite from Altair ® [START_REF] Gary | Comportement des métaux à grande vitesse de déformation : modélisationTechniques de l'ingénieur[END_REF]. As discussed in [START_REF] Gattacceca | On the efficiency of shock magnetization processes[END_REF], the peak pressure reached during laser shock experiments is heterogeneous throughout the sample, as the shock wave decays rapidly away from the impact spot. Twenty samples were shocked using a laser flux of 1.8 GW•cm -2 in water-confined regime (e.g., [START_REF] Fabbro | Physical study of laser-produced plasma in confined geometry[END_REF]. This flux produces pressures ranging from 2 GPa at the impact spot, about 0.3 GPa at the bottom of the sample, and 0.1 GPa on the sides of the sample (figure 2), with a median value of about 0.3 GPa. ). An additional three samples were shocked with a higher flux of 4.0 GW•cm -2 .

In this study, we used the very same basalt cylindrical samples (10 mm high, 9.5 mm in diameter, about 2.2 g) that were used in [START_REF] Gattacceca | On the efficiency of shock magnetization processes[END_REF]. This was made possible because low-pressure laser shocks are fully nondestructive such that the same samples can be shocked repeatedly. In this former study, samples were demagnetized before the shock experiments in order to study SRM acquisition only. In the present work, samples were given a laboratory TRM (by heating up to 300°C in a controlled magnetic field in the 50 to 294 µT range, see Table 1) using a Magnetic Measurements TM MMTD oven, and successively shocked in an ambient field perpendicular to this TRM. Subsequent alternating field (AF) demagnetization allows quantification of the shock demagnetization of the TRM (by studying the component of magnetization parallel to the original TRM) and of the acquisition of SRM (by studying the component of magnetization acquired parallel to the ambient field at the time of shock). All bulk remanence measurements were performed with a 2G Superconducting Rock Magnetometer (model 755R) at CEREGE (Aix-en-Provence, France).

In detail, samples were initially demagnetized by AF, then given an initial TRM (in an ambient field of intensity between 50 and 300 µT) that was AF demagnetized in order to obtain the coercivity spectrum of the TRM. A second TRM was then imparted using the same ambient field intensity. Although there is a risk that repeated laboratory heating result in thermal alteration of magnetic minerals, TRM was chosen as the initial remanent magnetization because it is probably the most common type of crustal magnetization on planets and asteroids, and was therefore more representative of natural processes than for instance anhysteretic remanent magnetization. In order to check that the magnetic mineralogy was not modified by the heating experiments, magnetic susceptibility was measured before and after the TRM acquisitions and those few samples that showed variation greater than 10% were discarded. Similarly, two samples that showed variation between the two TRM intensities of greater than 10% were discarded. Ultimately, 20 samples were selected for the shock experiments.

Experimental results

Generalities

All results are summarized in table 1. Figure 3

Shock demagnetization of TRM

The 20 samples that were shocked with a mean laser flux of 1.85 GW•cm -2 (s.d. 0.07 GW.cm -2 ) showed an average shock demagnetization of their TRM of 27% (s.d. 5%). This value is independent of the initial TRM intensity (figure 4a). Note that the variations in initial TRM intensities among samples are linked to varying ambient magnetic field during TRM acquisition, and not to variations in their intrinsic magnetic properties. Shock demagnetization of the TRM is also independent of the strength of the magnetic field at the time of impact (figure 4b). However a possible slight tendency for increasing demagnetization with ambient field (resp. initial TRM intensity) can be observed in figure 4b (resp. 4a). The correlation in figure 4b is more robust, especially because the four samples shocked in zero field show the lowest TRM demagnetization, which doest not fit in the trend seen in figure 4a, which seem to discard a weak interaction between TRM and SRM. The possible relation between TRM demagnetization efficiency and the ambient field during shock cannot be attributed to the ambient field not being applied exactly perpendicular to the TRM during the shock experiment, because in that case the necessary misalignment angle can be estimated to ~16°, which is well above the errors on the sample positioning and the ambient field direction during the experiments. This second order effect may be caused by the fact that an increasing ambient field, together with the shock wave, participate to overcome the energy limit necessary to remagnetize a given grain. This would not be in accord with [START_REF] Néel | Théorie du trainage magnétique des ferromagnétiques en grains fins avec applications aux terres cuites[END_REF] theory for single domain grains, and attributable to the pseudo-single domain grain size of the magnetite in the studied basalt. This effect should also be visible for the TRM process (i.e. TRM of rocks with non single domain magnetic carriers may demagnetize easier in presence of a perpendicular magnetic field) but we are unaware of any study of this kind.

The 3 samples shocked with a higher flux (4.0 GW.cm -2 , s.d. 0.5 GW.cm -2 ) were demagnetized by 58% (s.d. 5%). This indicates that the TRM shock demagnetization process has not reached its limit at the relatively low pressures (~0.5 GPa median pressure for the 1.8 GW.cm -2 laser flux) used in this study. For all experiments, shock demagnetization affects preferentially low coercivity grains as discussed in more details in §4.

Pressure demagnetization of TRM for the same basalt was studied in Bezaeva (2010) through hydrostratic pressure experiments with typical hold time at peak pressure of 10 2 -10 3 s. The demagnetization was of about 10% for 0.2 GPa, 30% for 0.5 GPa, and 45% for 1 GPa. These values are in general agreement with the 26% bulk demagnetization that was measured in this study for samples shocked with 1.8 GW•cm -2 fluxes [most of whose volumes reached pressures in the 0.2 to 1 GPa range (figure 2)] and typical hold time at peak pressure of 25-30 ns.

A c c e p t e d M a n u s c r i p t 5

Shock magnetization

In addition to our previous work [START_REF] Gattacceca | On the efficiency of shock magnetization processes[END_REF], we provide here additional information about SRM directional homogeneity. Indeed, there has been some debate about whether small-scale heterogeneities could exists, due to, for instance, to local transient magnetic fields during shock wave propagation or other hypothesized effects (Crawford andSchultz, 1988, 1999;[START_REF] Soloviev | Generation of electric and magnetic field during detonation of high explosive charges in boreholes[END_REF][START_REF] Funaki | Acquisition of shock remanent magnetization for demagnetized samples in a weak magnetic field (7 µT) by shock pressures 5-20 GPa without plasma-induced magnetization[END_REF]. To study the homogeneity of SRM acquisition, an additional basalt sample was AF demagnetized and imparted with an SRM using a 1.9 GW•cm -2 laser flux. Eleven mutually oriented ~10 mm 3 sub-samples were then cut with a wire saw and AF demagnetized. The SRM directions obtained for the 11 sub-samples show a remarkable clustering with a Fisher (1953) precision parameter k=259. The associated semi-angle of the 95% confidence cone (95=2.8°) is well within the orientation uncertainties during sub-sampling and magnetic measurements, so that SRM exhibits directional homogeneity at least down to the scale of a few mm 3 .

Previous SRM acquisition experiments on the same basalt with the exact same shock protocol showed that the intensity of SRM was surprisingly almost homogeneous in the sample despite heterogeneous peak pressure values [START_REF] Gattacceca | On the efficiency of shock magnetization processes[END_REF]. In order to verify that this property holds at the microscale and to check for possible variations in SRM intensity throughout the sample, we prepared a 500 m -thick polished thin section from the central vertical section of the shocked cylinder. The sawing direction was such that the SRM was perpendicular to the slice (figure 5a). The vertical (out-of-the-plane) component of the thin section's magnetic field was then mapped using the Superconducting Quantum Interference Device (SQUID) microscope [START_REF] Fong | High-resolution room-temperature sample scanning superconducting quantum interference device microscope configurable for geological and biomagnetic applications[END_REF][START_REF] Weiss | Paleomagnetic analysis using SQUID microscopy[END_REF] in the MIT Paleomagnetism Laboratory (sensor-to-sample distance of ~600 m). Figure 5b shows the measured map of the vertical component of the field, and figures 5c and 5d show the maps of the horizontal components estimated from the measured data using the technique described in [START_REF] Lima | Obtaining vector magnetic field maps from single-component measurements of geological samples[END_REF].

Based on the field maps, the magnetization distribution within the sample was recovered by means of an inverse problem. We used an inversion algorithm presented in [START_REF] Weiss | Paleomagnetic analysis using SQUID microscopy[END_REF], incorporating the non negligible vertical sample thickness. In essence, three parallel layers of unidirectional (vertically oriented) dipolar sources were evenly distributed within the sample, and the intensity of each dipole was obtained. While this multilayer model yields better accuracy, the variations in magnetization intensity along the sample's thickness are not meaningful. Therefore, we combined all three layers for visualization purposes by adding them together. This two-dimensional magnetization distribution can be viewed as an "average" distribution, resulting from integrating along the thickness of the sample. We provide an image of the SRM distribution acquired by the basalt sample (figure 5e). SRM intensity appears homogeneous throughout the sample. For comparison and normalization purposes, we provide as well an image of the anhysteretic remanent magnetization (ARM) distribution of the same sample (figure 5f). The two images are qualitatively similar, which indicates that SRM is indeed acquired with roughly the same intensity throughout the whole sample as evidenced at the scale of a few mm 3 in [START_REF] Gattacceca | On the efficiency of shock magnetization processes[END_REF]. This implies that the SRM intensity does not depend strongly on pressure in the 0.2 to 1 GPa range for this particular basalt.

To further illustrate this point, Figure 5g shows the average magnetization calculated along the top-bottom direction. Only sources with magnitudes greater than 10% of the A c c e p t e d M a n u s c r i p t 6 maximum magnetization of the corresponding line were used in the averaging process, so as to minimize artifacts due to non-magnetic minerals and irregular sample shape. In addition, each curve was normalized by the corresponding overall average moment. It is clear that there is no major difference between the SRM and ARM magnetization profiles. However, the profiles suggest that the SRM may be slightly relatively concentrated in the center of the sample, which would be consistent with the location of highest pressure. As a consequence, even though the peak pressure is not homogeneous within the sample, we show here that the SRM is mostly homogeneous in intensity throughout the sample. In addition, the small residuals obtained in our inversions using a unidirectional model (root-mean-square (RMS) residuals of ~5% of RMS field) indicate that the magnetization direction is also constant throughout the sample, with the possible exception of a few isolated spots located at the edges of the specimen. These artifacts-which we attribute to sample preparation-locally increased the residuals and had a negative impact on the RMS value. However, within the sample region, residuals were 4 to 5 times smaller than the RMS value, thereby showing an excellent agreement between model and data. Thus, the observed homogeneity in magnetization direction and intensity down to the microscale justifies working on the bulk samples instead of studying separate sub-samples.

As already shown in previous studies [START_REF] Pohl | Shock magnetization and demagnetization of basalt by transient stress up to 10 kbar[END_REF][START_REF] Gattacceca | On the efficiency of shock magnetization processes[END_REF] the intensity of the SRM acquired during the shock experiments is proportional to the ambient field at the time of impact and is estimated here to be 1.51×10 -5 Am 2 /(kg•µT) (s.d. = 0.29×10 -5 Am 2 /(kg•µT), n=16). From the SRM intensities measured by [START_REF] Gattacceca | On the efficiency of shock magnetization processes[END_REF] on previously demagnetized samples, we compute a SRM of 1.26×10 -5 Am 2 /(kg•µT) (s.d. = 0.15×10 -5 Am 2 /(kg•µT), n=28). These two indistinguishable values show that SRM acquisition is the same whether the sample is demagnetized or carries a remanence before the shock. This is exemplified by figure 6 which displays the SRM acquired by the basalt carrying a TRM before shock (this study, SRM1 in Table 1) and the SRM intensity acquired by the same basalt sample demagnetized before shock (from Gattacceca et al., 2008, SRM0 in Table 1) (all shock experiments were performed under the same laser and ambient field conditions: laser flux of 1.8-1.9 GW×cm -2 ).

Discussion

These results demonstrate the two competing magnetic effects that occur when a shock wave passes through a rock: demagnetization of previous remanence and acquisition of shock remanence. It is noteworthy that the intensity of the SRM is independent of the initial state of the rock before the shock (figure 6). Similarly, the amount of shock demagnetization of the original NRM does not depend on the intensity of the ambient magnetic field at the time of impact (figure 4b). The two competing phenomena are thus independent even when they occur simultaneously. The exact same would be valid for TRM acquisition and the associated thermal demagnetization of the initial remanence.

Given that the pressure is not homogeneous within our shocked samples, it is not possible to interpret the curved shape of the demagnetization plots (figure 3c-d) as a proof that SRM and the remaining post-shock TRM have overlapping coercivity spectra, because we are measuring a bulk sample whose different regions have been shocked at different pressures. However, as shown in previous studies of demagnetization by hydrostatic pressure (e.g., [START_REF] Bezaeva | Demagnetization of terrestrial and extraterrestrial rocks under hydrostatic pressure up to 1.2 GPa[END_REF], there is no equivalence between A c c e p t e d M a n u s c r i p t 7 pressure demagnetization at a given pressure and a given alternating demagnetazation field, so that pressure demagnetization affects a large fraction of the coercivity spectrum with preferential demagnetization of low coercivity grains. It is therefore very probable that the curved shape of the demagnetization plots in figure 3c-d is due to overlapping coercivity spectra of SRM and post-shock TRM, i.e. incomplete TRM demagnetization and incomplete SRM acquisition over a large part of the coercivity spectrum.

As we know from this study (see §3) and from [START_REF] Gattacceca | On the efficiency of shock magnetization processes[END_REF] that SRM acquisition in theses samples is almost homogeneous in the studied samples despite inhomogeneous pressure, it is reasonable to study the bulk behavior of the sample and compare the coercivity spectra of SRM and post-shock TRM (noted TRM2 in the following). We use as reference the coercivity spectrum of pre-shock TRM (noted TRM1 in the following). Over every coercivity window defined by the AF demagnetization steps, we have computed the ratio ∆TRM2/∆TRM1 to evaluate which fraction of the original TRM was erased by the shock, as well as the ratio ∆SRM/∆TRM1 to evaluate the efficiency of SRM acquisition with respect to TRM acquisition. This was done for a set of 13 samples that were shocked in the exact same conditions. Because there is only a small variation among the 13 samples, we provide the average values (figure 7). Shock demagnetization affects preferentially the low coercivity fraction of TRM as already discovered in a number of previous studies [START_REF] Louzada | Effect of shock on the magnetic properties of pyrrhotite, the Martian crust and meteorites[END_REF][START_REF] Bezaeva | Pressure demagnetization of the Martian crust: ground truth from SNC meteorites[END_REF][START_REF] Bezaeva | Demagnetization of terrestrial and extraterrestrial rocks under hydrostatic pressure up to 1.2 GPa[END_REF], which is sometimes referred as shock "hardening" of the initial remanence (e.g. [START_REF] Brecher | The magnetic effects of brecciation and shock in meteorites: II. The ureilites and evidence for strong nebular magentic fields[END_REF]. Figure 7a shows that about 70% of the TRM carried by grains with coercivities below 5 mT is demagnetized by shock whereas grains with coercivity higher than 25 mT are not affected. Figure 7a also shows that SRM efficiency with respect to TRM in the same field intensity decreases with coercivity. Indeed in the coercivity window below 3 mT SRM is almost as intense as the original TRM whereas it is less than 10% above 10 mT. The bulk efficiency is 18% (s.d. = 5%, n=20), in agreement with the value of 14% obtained in [START_REF] Gattacceca | On the efficiency of shock magnetization processes[END_REF]. Another interesting aspect of figure 7a is that the two curves are significantly distinct. This difference is magnified in figure 7b, which displays the ratio ∆SRM/∆TRM2 as a function of coercivity. The fact that the curve in figure 7b is not a horizontal line shows that although the two phenomena (shock magnetization and shock demagnetization) preferentially affect low coercivity grains, they are distinct phenomena that affect different grains with different intensities.

Conclusion

We have performed laser shock experiments on TRM-bearing basalt samples in a controlled magnetic field and studied the two competing phenomena that affect the magnetic remanence of these samples: shock demagnetization of the original remanence and shock magnetization in the ambient field at the time of impact.

As previously evidenced, SRM is strictly proportional to the ambient field at the time of impact and parallel to the ambient field. Moreover, we show that there is no directional or intensity heterogeneity of the SRM down to the scale of ~0.2 mm 3 . We also show that the intensity of SRM is independent of the initial remanence state of the rock. As a consequence, when it can be recognized in a rock, SRM must be considered as a reliable record of the direction and intensity of the ambient magnetic field at the time of impact. This has potential implications for the paleomagnetic study of meteorites, and lunar rocks, and for the understanding of the magnetic signature (as studied through A c c e p t e d M a n u s c r i p t 8 paleomagnetism and/or magnetic anomalies) of terrestrial, lunar and Martian impact craters.

Comparison of the coercivity spectra of SRM and shock demagnetized TRM show that the two phenomena (shock demagnetization and shock magnetization) preferentially affect low coercivity grains but not with the same intensity. As such, shock demagnetization cannot be considered as a limiting case of shock magnetization in zero field. The natural process of hypervelocity impact where a rock carrying a remanent magnetization is shocked in the presence of an ambient field can be studied as the simple superimposition of shock demagnetization and shock magnetization. For this there is now a variety of techniques that allow experimental study of both phenomena separately [START_REF] Gilder | Static stress demagnetization of single and multidomain magnetite with implications to meteorite impacts[END_REF][START_REF] Louzada | Effect of shock on the magnetic properties of pyrrhotite, the Martian crust and meteorites[END_REF][START_REF] Bezaeva | Demagnetization of terrestrial and extraterrestrial rocks under hydrostatic pressure up to 1.2 GPa[END_REF][START_REF] Gattacceca | On the efficiency of shock magnetization processes[END_REF], or simultaneously as in this study. Simultaneous analysis of the two phenomena, although more experimentally demanding, permits reproducibility of both pressure and modification of rock magnetic and mechanical properties for both processes. In both cases, the magnetization is out-of-page. The spatial scale is the same for figures a to f. g) Average magnetization calculated along the topbottom direction for both maps (solid line = SRM; dashed line = ARM). Each curve is normalized by the corresponding overall average moment.

figure 6: SRM intensity acquired by basalt samples carrying a TRM (this study) vs SRM intensity acquired by the same samples demagnetized before shock [START_REF] Gattacceca | On the efficiency of shock magnetization processes[END_REF]. In all cases, the shock experiment was performed with a laser flux of 1.8-1.9 GW•cm -2 . The ambient field varies from 5 to 294 µT. Each circle is for the same sample shocked in the same ambient field. Note the logarithmic scale on both axes.

figure 7: a) SRM efficiency vs TRM (blue boxes) and TRM demagnetization (red circles) as a function of coercivity. The AF value is the middle of the coercivity window of interest. SRM efficiency is defined as the ratio of SRM to preshock TRM (TRM1) acquired by grains with coercivities in the specified range. TRM demagnetization is defined as the ratio of TRM that was shock demagnetized (TRM2) to TRM1 for a given coercivity range. (b) Ratio of SRM acquired to TRM demagnetized as a function of coercivity (i.e. ratio of the two curves of figure 7a). The AF value is the middle of the coercivity window of interest. Above 20 mT, this ratio is dominated by noise.

A c c e p t e d M a n u s c r i p t 1 / 1 l i n e SRM intensity (from demagnetized state) in Am 2 .kg -1 SRM intensity (from TRM state) in Am 2 .kg -1 (µT) GW.cm -2 (Am 2 .kg -1 ) (µT) (Am 2 .kg -1 ) (GW.cm-2 ) (Am 2 .kg

  shows typical demagnetization diagrams obtained during demagnetization of the samples. A first batch of 4 samples A c c e p t e d M a n u s c r i p t 4 was shocked in zero field (< 100 nT) in order to independently study shock demagnetization of the TRM. These samples display a single component of magnetization which is a partially shock-demagnetized TRM (fig 3a). The pre-shock TRM demagnetization plot for the same sample is shown for comparison (fig 3b). The remaining 16 samples were shocked in an ambient field equal to the field that was used to impart TRM. Their AF demagnetization diagrams display two components of magnetization: the partially shock-demagnetized TRM and the newly acquired SRM (fig 3c-d).

  Figure captions figure 1 : Diagram of the experimental setup used in this work (modified from Gattacceca et al. (2008).

  Figure 01 Click here to download Figure(s): Fig01.eps

  Figure 02 Click here to download Figure(s): fig02.eps

  Figure 03 Click here to download Figure(s): fig03.eps

  Figure 05 Click here to download Figure(s): fig05.eps

  Figure 06 Click here to download Figure(s): fig06.eps

  Figure 07 Click here to download Figure(s): fig07.eps

Table 1 :

 1 Results of shock and thermoremanent magnetization experiments SRM0 is the shock remanent magnetization acquired by initially demagnetized samples (from[START_REF] Gattacceca | On the efficiency of shock magnetization processes[END_REF]. TRM1 is a thermoremanent magnetization acquired by heating up to 300°C. SRM1 is the shock remanent magnetization acquired by the samples initially carrying TRM1 ( † indicates that the shock was performed in null ambient field). The TRM demagnetization percentage indicates the decrease of TRM1 after shock.

	10

Table 01

 01

A c c e p t e d M a n u s c r i p t


Keywords: Valeurs propres, résolvante, déterminant, logarithme complexe

Dénombrement de valeurs propres dans le plan complexe Résumé : Nous proposons une approche pour compter le nombre de valeurs propres d'une matrice, situées dans un domaine borné du plan complexe. L'approche est fondée sur l'application du théorème des résidus. L'intégration nous ramène à l'évaluation de l'argument principal du logarithme d'une fonction. Nous proposons une stratégie pour le choix du pas qui permette de rester sur la même branche du logarithme pendant l'intégration. Des résultats numériques sont obtenus à partir de tests conduits sur des matrices tirées d'ensembles classiques de matrices.

Introduction

The localization of eigenvalues of a given matrix A in a domain of the complex plane is of interest in scientific applications. When the matrix is real symmetric or complex hermitian, a procedure based on computations of Sturm sequences allows to safely apply bisections on real intervals to localize the eigenvalues. The problem is much harder for non symmetric or non hermitian matrices and especially for non normal ones. This last case is the main concern of this work. Proceeding by trying to compute the eigenvalues of the matrix may not always be appropriate for two reasons.

First most of the iterative methods frequently used to calculate eigenvalues of large and sparse matrices may loose some of them, since only a part of the spectrum is computed, and as such there is no guarantee to localize all the eigenvalues of the selected domain. When a shift-and-invert transformation is used, the eigenvalues are obtained in an order more or less dictated by their distance from the shift, and if one eigenvalue is skipped, there is no easy strategy that allows to recover it.

Second the entries of the matrix may be given with some errors and then the eigenvalues can only be localized in domains of C.

Many authors have defined regions in the complex plane that include the eigenvalues of a given matrix. One of the main tool is the Gershgorin theorem. Since a straight application of the theorem often leads to large disks, some authors extended the family of inequalities for obtaining smaller regions by intersections which include eigenvalues (see e.g., [START_REF] Brualdi | Regions in the complex plane containing the eigenvalues of a matrix[END_REF][START_REF] Huang | Regions containing eigenvalues of a matrix[END_REF]). Other techniques consist to consider bounds involving the singular values (see e.g., [START_REF] Beattie | Inclusion regions for matrix eigenvalues[END_REF]), the eigenvalues of the hermitian part and the skew-hermitian part of the matrix (see e.g., [START_REF] Adam | An eigenvalue inequality and spectrum localization for complex matrices[END_REF]), or the field of values of inverses of the shifted matrices (see e.g., [START_REF] Hochstenbach | Eigenvalue inlusion regions from inverses and of shifted matrices[END_REF]).

For taking into account, possible perturbations of the matrix, Godunov [START_REF] Godunov | Spectral portraits of matrices and criteria of spectrum dichotomy[END_REF] and Trefethen [START_REF] Trefethen | Pseudospectra of matrices[END_REF] have separately defined the notion of the of ǫ-spectrum or pseudospectrum of a matrix to address the problem. The problem can then be reformulated as that of determining level curves of the 2-norm of the resolvent R(z) = (zI -A) -1 of the matrix A.

The previous approaches determine a priori enclosures of the eigenvalues. A dual approach can be considered: given some curve (Γ) in the complex plane, count the number of eigenvalues of the matrix A that are surrounded by (Γ). This problem was considered in [START_REF] Bertrand | Counting the eigenvalues surrounded by a closed curve[END_REF] where several procedures were proposed. In this paper, we make some progress with respect to the work in [START_REF] Bertrand | Counting the eigenvalues surrounded by a closed curve[END_REF]. Our procedure is based on the application of the residual theorem: the integration process leads to the evaluation of the principal argument of the logarithm of the function g(z) = det((z + h)I -A)/det(zI -A). This function is also considered in [START_REF] Bindel | Error bounds and error estimates for nonlinear eigenvalue problems[END_REF] to count the eigenvalues when a nonlinear eigenvalue problem is perturbed.

This work is mainly concerned with the control of the integration path so as to stay on the same branch along an interval when evaluating the principal argument of a logarithm.

In section 2, we present the mathematical tools. In section 3, we present the basis of our strategy for following a branch of the logarithm function and conditions for controlling the path length. Section 4 deals with the implementation of our method: we show how to safely compute the determinant and how to include new points along the boundary. In section 5 we present numerical test results carried out on some test matrices and in section 6, we conclude with some few remarks and future works.

Mathematical tool and previous works

In this section we present the Cauchy's argument principle and some previous works on counting eigenvalues in regions of the complex field.

Use of the argument principle

The localization of the eigenvalues of matrix A involves the calculation of determinants. Indeed let (Γ) be a closed piecewise regular Jordan curve (piecewise C 1 and of winding number 1) in the complex plane which does not include eigenvalues of A. The number N Γ of eigenvalues surrounded by (Γ) can be expressed by the Cauchy formula (see e.g., [START_REF] Rudin | Real and complex analysis[END_REF][START_REF] Silverman | Introductory Complex Analysis[END_REF]):

N Γ = 1 2iπ Γ f ′ (z) f (z) dz, (1)
where

f (z) = det(zI -A) is the characteristic polynomial of A.
If γ(t) 0≤t≤1 is a parametrization of Γ the equation ( 1) can be rewritten as

N Γ = 1 2iπ 1 0 f ′ (γ(t)) f (γ(t)) γ ′ (t)dt. (2)
The primitive ϕ defined by

ϕ(u) = u 0 f ′ (γ(t)) f (γ(t)) γ ′ (t)dt, u ∈ [0, 1],
is a continuous function which is a determination of log(f • γ) (e.g. see [START_REF] Silverman | Introductory Complex Analysis[END_REF]):

log f (γ(t)) = log |f (γ(t))| + i arg(f (γ(t))), t ∈ [0, 1].
It then follows that

N Γ = 1 2π ϕ I (1),
where ϕ I (1) is the imaginary part of ϕ(1) since its real part vanishes.

Counting the eigenvalues in a region surrounded by a closed curve

In [START_REF] Bertrand | Counting the eigenvalues surrounded by a closed curve[END_REF], two procedures were proposed for counting the eigenvalues in a domain surrounded by a closed curve. The first method is based on the series expansion of log(I + hR(z)), where R(z) = (zI -A) -1 , combined with a path following technique. The method uses a predictor -corrector scheme with constant step size satisfying the constraint

|ϕ I (z + ∆z) -ϕ I (z)| < π, RR n°7770
for a discrete list of points z. The implementation of the algorithm requires the computation of a few of the smallest singular values and the corresponding left and right singular vectors of (zI -A); they are used to follow the tangent to the level curve of the smallest singular value of (zI -A).

In the second procedure, the domain is surrounded by a parameterized userdefined curve z = γ(t) and thus

N Γ = 1 2iπ γ(1) γ(0) d dt det(γ(t)I -A) det(γ(t)I -A) dt (3)
Since γ(0) = γ(1), the function γ(t) defined on [0, 1], can be extended onto R by γ ext (t) = γ(t mod 1).

By subdividing the interval [γ(0), γ(1)] into subintervals of equal length, and by assuming that γ ext ∈ C ∞ , they make use of a fundamental result from quadrature of periodic function to prove an exponential convergence of the integral.

The method is compared to other integrators with adaptive step sizes.

Each of these methods makes use of the computation of

u(t) = det(γ(t)I -A) |det(γ(t)I -A)| ,
which is efficiently computed through a LU factorization of the matrix (γ(t)I -A) with partial pivoting. In order to avoid underflow or overflow, the quantity is computed by

det(γ(t)I -A) |det(γ(t)I -A)| = n i=1 u ii |u ii |
where u ii is the i-th diagonal element of U in the LU factorization. The product is computed using the procedure that will be described later on in section 4.

Our work, which can be viewed as an improvement of [START_REF] Bertrand | Counting the eigenvalues surrounded by a closed curve[END_REF], mostly deals with the control of the integration so as to stay on the same branch along an interval, during the evaluation of the principal argument of the logarithm of the function g(z) defined in the introduction.

Integrating along a curve

In this section, we describe strategies for the integration of the function g

(z) = f ′ (z) f (z)
, where f (z) = det(zI -A), along the boundary of a domain limited by a user-defined curve (Γ) that does not include eigenvalues of A.

Following a branch of log(f(z)) along the curve

To simplify the presentation and without loss of generalization, let us assume that Γ =

N -1 i=0 [z i , z i+1 ] is a polygonal curve. Let Arg(z) ∈ (-π, π]
denote the principal determination of the argument of a complex number z, and arg(z) ≡ Arg(z) (2π), be any determination of the argument of z. In this section, we are concerned with the problem of following a branch of log(f (z)) when z runs along (Γ). The branch (i.e. a determination RR n°7770 arg 0 of the argument), which is to be followed along the integrating process, is fixed by selecting an origin z 0 ∈ (Γ) and by insuring

arg 0 (f (z 0 )) = Arg(f (z 0 )). (4)
Let z and z + h two points of (Γ). Since

(z + h)I -A = (zI -A) + hI = (zI -A)(I + hR(z)),
where R(z) = (zI -A) -1 , it then follows that

f (z + h) = f (z) det(I + hR(z)). ( 5
) Let Φ z (h) = det(I + hR(z)), then z+h z f ′ (z) f (z) dz = log(f (z + h)) -log(f (z)) = log f (z + h) f (z) = log(Φ z (h)) = log |Φ z (h)| + i arg(Φ z (h)).
In the previous approach [START_REF] Bertrand | Counting the eigenvalues surrounded by a closed curve[END_REF], given z, the step h is chosen such that condition

|arg(Φ z (h))| < π, (6)
is satisfied. In [START_REF] Bertrand | Counting the eigenvalues surrounded by a closed curve[END_REF] condition [START_REF] Bindel | Error bounds and error estimates for nonlinear eigenvalue problems[END_REF] is only checked at point z + h but we want the condition to be satisfied at all the points s ∈ [z, z + h], so as to guarantee that we stay on the same branch along the interval [z, z + h]. We need a more restrictive condition which is mathematically expressed by the following lemma:

Lemma 3.1 (Condition (A)) Let z and h be such that [z, z + h] ⊂ (Γ). If |Arg(Φ z (s))| < π, ∀s ∈ [0, h], (7)
then,

arg 0 (f (z + h)) = arg 0 (f (z)) + Arg(Φ z (h)), ( 8
)
where arg 0 is the determination of the argument determined as in ( 4) by an a priori given z 0 ∈ (Γ).

Proof. We prove it by contradiction. Let us assume that there exists k ∈ Z \ {0} such that

arg 0 (f (z + h)) = arg 0 (f (z)) + Arg(Φ z (h)) + 2kπ.
By continuity of the branch, there exists s ∈ [0, h] such that |Arg(Φ z (s))| = π, which contradicts [START_REF] Brualdi | Regions in the complex plane containing the eigenvalues of a matrix[END_REF]. ⋄ Condition (7) is called Condition (A).

RR n°7770

Step size control

In our approach, given z, the step h is chosen such that condition of Lemma (3.1) is satisfied. Condition (A) is equivalent to

Φ z (s) / ∈ (-∞, 0], ∀s ∈ [0, h].
In order to find a practical criterion to insure it, we look for a more severe condition by requiring that Φ z (s) ∈ Ω, where Ω is an open convex set, neighborhood of 1, and included in Ω ⊂ C \ (-∞, 0]. Possible options for Ω are the positive real half-plane, or any disk included in it and centered in 1.

Since Φ z (0) = 1, let Φ z (s) = 1 + δ, with δ = ρe iθ .
A sufficient condition for (7) be to satisfied is ρ < 1, i.e.

|Φ z (s) -1| < 1, ∀s ∈ [0, h] (9)
This condition will be referred to as Condition (B), and, when only verified

at z + h, i.e. |Φ z (h) -1| < 1, (10)
it will be referred to as Condition (B'). This last condition is the condition used in [START_REF] Bertrand | Counting the eigenvalues surrounded by a closed curve[END_REF]. It is clear that Condition (B) implies Condition (A) whereas this is not the case for Condition (B').

Since it is very difficult to check [START_REF] Golub | Matrices, Moments and Quadrature with Applications[END_REF], we apply the condition on the linear approximation Ψ z (s) = 1 + sΦ ′ z (0) of Φ z (s) at 0. Replacing function Φ z by its tangent Ψ z in (9), leads to

|Ψ z (s) -1| < 1, ∀s ∈ [0, h], (11)
which is equivalent to the following condition, referred as Condition (C):

|h| < 1 |Φ ′ z (0)| . ( 12
) Example 3.1 (First illustration) Let A = 0 0 0 1 . It then follows that f (z) = z(z -1), Φ z (h) = (1 + h z )(1 + h z -1 ), Φ ′ z (0) = 1 z + 1 z -1 .
Let us assume that we are willing to integrate along the segment from z = 2 to z = 1 + i. In order to see if intermediate points are needed to insure that the branch of the logarithm is correctly followed, we consider the previously introduced conditions on h = t(-1 + i) where t ∈ [0, 1].

Condition (A): Φ 2 (h) = 1 + 3h 2 + h 2 2 is
f (z) = (z -λ) n , Φ z (h) = 1 + h z -λ n , Φ ′ z (0) = n z -λ .
Let us assume that we are willing to integrate from z = λ + 1 to z + h = λ + e iθ . We consider the previously introduced conditions on h.

Condition (A): |θ| < π n . Condition (B): |θ| < π 3n . Condition (B'): cos nθ > 1
2 which is satisfied for values that violate Condition (A).

Condition (C): | θ 2 | < arcsin 1 2n
, which is more severe than |θ| < 1 n and therefore guaranties Condition (A).

In this example, if (Γ) is the circle with center λ and radius 1, the step size must be reduced in such a way that more than 2n intervals are considered to satisfy Condition (A), or even 6n and 2πn intervals with Condition (B) and Condition (C) respectively.

Practically, we consider that Condition (C) implies Condition (A), as long as the linear approximation is valid. Problems may occur when Φ ′ z vanishes. The following example illustrates such a situation.

Implementation

In this section, we describe the numerical implementation of our method. Strategies for including new points and a procedure for safely computing the determinants are given.

Avoiding overflows and underflows

The implementation of our method requires the computation of

Φ z (h) = det((z + h) -A) det(zI -A) .
In order to avoid underflow or overflow, we proceed as follows:

For any non singular matrix M ∈ C n×n , let us consider its LU factorization P M = LU where P is a permutation matrix of signature σ. Then det(M ) = σ n i=1 (u ii ) where u ii ∈ C are the diagonal entries of U . If the matrix M is not correctly scaled, the product n i=1 (u ii ) may generate an overflow or an underflow. To avoid this, the determinant is characterized by the triplet (ρ, K, n) so that det(A) = ρK n

where:

ρ = σ n i=1 u ii |u ii |
, (ρ ∈ C with |ρ| = 1), and

K = n n i=1 |u ii | (K > 0).
The quantity K is computed through its logarithm:

log(K) = 1 n n i=1 log(|u ii |).
By this way, the exact value of the determinant is not computed, as long as the scaling of the matrix is not adequate.

In section 3, it was indicated that our algorithm will heavily be based on the computation of Φ z (h) = det(I + hR(z)). For h of moderate modulus, the determinant does not overflow. This can be verified since

Φ z (h) = det((z + h)I -A) det(zI -A) = K n 2 ρ 2 K n 1 ρ 1 ,
where det(zI -A) and det((z + h)I -A) are respectively represented by the triplets (ρ 1 , K 1 , n) and (ρ 2 , K 2 , n). Before raising to power n, to protect from under-or overflow, the ratio K 2 /K 1 must be in the interval [

1 n √ M f l , n M f l ]
where M f l is the largest floating point number. When this situation is violated, intermediate points must be inserted between z and z + h.

RR n°7770

Estimating the derivative

An easy computation shows that the derivative Φ ′ z (0) can be expressed by :

Φ ′ z (0) = trace(R(z)). ( 14
)
The evaluation of this simple expression involves many operations, as we show it now. By using the LU-factorization P (zI -A) = LU which is available at z, and by using ( 14), we may compute

Φ ′ z (0) = n i=1 u * i l i
, where l i = L -1 e i and u i = (U * ) -1 e i , with e i being the i-th column of the identity matrix. When A is a sparse matrix, the factors L and U are sparse but not the vectors u i and l i . Therefore, the whole computation involves 2n sparse triangular systems. Experiments showed that they involve more operations than the LU factorization of the matrix. Approximations of the trace of the inverse of a matrix have been investigated. They involve less operations than use of the LU factorization but they are only valid for symmetric or hermitian matrices [START_REF] Bai | Bounds for the trace of the inverse and the determinant of symmetric positive definite matrices[END_REF][START_REF] Golub | Matrices, Moments and Quadrature with Applications[END_REF].

If the derivative is approximated by its first order approximation, sparsity helps. More specifically, given z and z + h, the derivative of Φ z at 0 is estimated by

Φ ′ z (0) ≈ Φ z (s) -1 s ,
where s = αh with α = min(10 -6 µ/|h|, 1), and µ = max z∈Γ |z|. Therefore, the computation imposes an additional LU factorization for evaluating the quantity Φ z (s). It is known that, for a sparse matrix, the sparse LU factorization involves much less operations that its dense counterpart.

Test for including new points

In this subsection, we describe a heuristic procedure for including new points in the interval [z, z + h]. In section 3, we introduced Condition (B) which is more severe than Condition (A) but might be easier to verify, and we proposed to test its linear approximation called Condition (C). Unfortunately Example 3.3 has exhibited that Condition (C) may be satisfied while Condition (B') and therefore Condition (B) is violated. To increase our confidence in accepting the point z + h, we simultaneously check Condition (C) and Condition (B').

When Condition (C) is violated, we insert M regularly spaced points between z and z + h where

M = min |h| |Φ ′ z (0)| , M max , (15)
with M max being some user defined parameter.

In addition, we insist that Condition (C) is satisfied at each bound of the segment [z, z + h]. Therefore, on exit, the condition |h| < 1 |Φ ′ z+h (0)| must also be guarantied. When it is violated, we insert the point z + h/2 in the list.

The following example illustrates the effect of this step size control. The polygonal line (Γ) is determined by 10 points regularly spaced on the circle of center 0 and radius 1.3. In Figure 1, are displayed the eigenvalues of A, the line (Γ) and the points that are automatically inserted by the procedure. The figure illustrates that, when the line gets closer to some eigenvalue, the segment length becomes smaller.

Global algorithm

The algorithm is sketched in Table 1. From a first list Z of points, it extends the list Z in order to determine a safe split of the integral [START_REF]Matrix market[END_REF]. The complexity of the algorithm is based on the number of computed determinants. For each z ∈ Z, the numbers det(zI -A) and Φ ′ z (0) are computed; they involve two evaluations of the determinant. Therefore, for N final points in Z, the complexity can be expressed by:

C = 2L LU N,
where L LU is the number of operations involved in the complex LU factorization of zI -A.

When the matrix A is real and, assuming that the polygonal line (Γ) is symmetric w.r.t. the real axis and intersects it only in two points, half of the computation can be saved since

N Γ = 1 π I Γ+ f ′ (z) f (z) dz ,
where (Γ + ) is the upper part of (Γ) when split by the real axis, and I(Z) denotes the imaginary part of Z.

Numerical tests

The tests are run on a laptop Dell (Processor Intel Core i7-2620M CPU, clock: 2.70 GHz, RAM: 4 GB). The program eigencnt is coded in Matlab.

In the following tests, we describe the performances of the algorithm for three real matrices chosen from the set Matrix Market [START_REF]Matrix market[END_REF]. The maximum inserted points in an interval is M max = 10. When (Γ) is symmetric w.r.t. the real axis, only half of the integration is performed. The storage of the matrices is kept sparse (except for computing the spectra of the matrices of the two first examples). This matrix is of small order and its spectrum is displayed in Figure 2.

The first experiment consists to focusing on the right part of the spectrum by defining a regular polygon of 10 vertices; the polygon is centered in the origin, symmetric w.r.t. with the real axis as shown in figure 3 (only its upper part is drawn), and of radius R = 10 -3 . Five eigenvalues were correctly found as surrounded by the polygon. Some statistics are displayed in the first line of Table 2.

The second experiment focuses on the bifurcation between real and complex eigenvalues in the neighborhood of -3.5. In the box [-4, -3.4]×[-10 -4 i, 10 -4 i], 89 eigenvalues are counted (see the statistics in the second line of Table 2). The aspect ratio of the box is large. The refining process proceeds in 16 steps to produce 1519 intervals from the initial four. If the integral is computed by the RR n°7770 ments the ARPACK code) returns the eigenvalues {8.3713 ± 64.653i, 8.8025 ± 64.876i, 16.203, 20.166}. By increasing the number p of requested eigenvalues, only a few of them converged: for instance for p = 20, only the two rightmost were found. Increasing even further up to p = 100, 14 eigenvalues were given back, including the already computed two rightmost and 12 additional ones with real parts belonging to the interval [12.2,12.9]. Therefore, the user is inclined to ask for the exact situation in this region. Defining the rectangle Γ = Γ + ∩ Γ - where Γ + = (14, 14 + 2i, 12 + 2i, 12) and where Γ -is the symmetric of Γ + w.r. Actually, the right number of eigenvalues was already given before the last refining step with 3994 intervals. Taking into account the result of the experiment, after several tries of shifts in eigs, all the 116 eigenvalues surrounded by (Γ) were obtained by requesting p = 200 eigenvalues in the neighborhood of the shift σ = 13.5 (elapsed time: 10.2s).

Conclusion

In this paper, we have developed a reliable method for counting the eigenvalues in a region surrounded by a user-defined polygonal line. The main difficulty to tackle lies in the step control which must be used during the complex integration along the line. The method is reliable but it involves a high level of computation. This is the price to pay for the reliability. In forthcoming works, a parallel version of the method will be developed and implemented. The code involves a high potential for parallelism since most of the determinant computations are independent. A second level of parallelism is also investigated within the computation of a determinant for matrices arising in domain decompositions.

Example 3 . 3 (

 33 Critical situation) Let us consider the matrix of Example 3.1. For z = 1/2, Φ 1/2 (h) = 1 -4h 2 , and Φ ′ 1/2 (0) = 0, and the conditions become Condition (A): h / ∈ R or |h| < 1/2, Condition (B): |h| < 1/2, Condition (B'): |h| < 1/2, Condition (C): is satisfied for all h ∈ C. RR n°7770

Example 4 . 1

 41 Let A be the random matrix : -0.31 -0.81 0.44 -0.94 0.16 0.93 0.75 -0.09 -0.91 -0.83 -0.70 -0.83 -0.92 0.03 -0.58 -0.87 -0.26 -0.93 -0.60 -0.92 -0.36

Figure 1 :

 1 Figure 1: Example 4.1. The eigenvalues are indicated by the stars. The polygonal line is defined by the 10 points with circles; the other points of the line are automatically introduced to insure the conditions as specified in section 4.3 (M max = 1 in (15)).

3 Figure 2 :

 32 Figure 2: Spectrum of the matrix of Example 5.1.

Example 5 . 1 (

 51 Matrix ODEP400A) This matrix is a model eigenvalue problem coming from an ODE.

3 Figure 3 :

 33 Figure 3: Example 5.1: First experiment on the right end of the spectrum.

Exper. 1 :

 1 Box=[-20, 0] × [75i, 125i] Exper. 2: Box=[-20, 0] × [-500i, 500i]

Figure 4 :

 4 Figure 4: Example 5.2: Spectrum of the matrix (up) and zooms on the two regions of experiments.

  In the second example, we illustrate the lack of reliability of Condition (B').Example 3.2 (Second illustration) Let A = λI n , where λ ∈ R and I n is the identity matrix of order n. It then follows that

	Condition (B): this condition is equivalent to |h||3 + h| < 2. By studying the function φ(t) = |h||3 + h| = √ 2t|3 -t + it|, the parameter t must remain smaller than α ≈ 0.566.
	Condition (B'): in this example, this condition is equivalent to the previous
	one, since the function φ(t) is increasing with t.
	Condition (C): since Φ interval to |h| < 2 3 or equivalently t < ′ 2 (h) = 3 2 + h, this condition limits the extent of the √ 2 3 ≈ 0.471 .

a non positive real number if and only

if h ∈ [-2, -1] (-3 2 + iR).

From that, it can easily be seen that the segment [0, -1 + i] does not intersect the forbidden region. Therefore no intermediate points are needed.

Table 2 :

 2 Statistics for Example 5.1. Nb. of eigenvalues in (Γ) nb. of intervals elapsed time Two experiments consider the right part of the spectrum. A first box [-20, 0] × [75i, 125i] is not symmetric w.r.t. the real axis. Therefore, the integration is not reduced. Eight eigenvalues are decounted. The second box [-20, 0] × [-500i, 500i] is symmetric w.r.t. the real axis but it includes 542 eigenvalues. The statistics are reported in Table3. In Figure4, the spectrum and two zooms on it are displayed.

	Exper. 1	5		25	5.6e-2 s
	Exper. 2	89	1519	1.1 s
	relation (8) at each step (hence even if the necessary conditions for correctness
	are not satisfied), it would only have been correct at the fifth step and after ;
	this corresponds to 825 intervals. This illustrates the loss in efficiency which is
	imposed by the constraint for a safe computation.
	Example 5.2 (Matrix TOLS2000) This matrix comes from a stability anal-
	ysis of a model of an airplane in flight.
	n	A 1	Spectral radius Spectrum included in
	2000 5.96 ×10 6	r= 2.44 ×10 3	[-750,0]x[-r,+r]
		Nb. of eigenvalues in (Γ) nb. of intervals elapsed time
	Exper. 1	8		2611	11.0 s
	Exper. 2	542	15669	57.7 s

Table 3 :

 3 Statistics for Example 5.2.

Table 4 :

 4 t. the real axis, the procedure eigencnt returns Statistics for Example 5.3.

	Number of eigenvalues in (Γ) number of intervals elapsed time
	116	7986	54 h 42 mn

RR n°7770