Abdelhafid Zitouni
email: ah_zitouni@yahoo.fr

L Seinturier
email: lionel.seinturier@lifl.fr

M Boufaida
email: boufaida@hotmail.com

Contract-based approach to analyze software components

published or not. The documents may come

Introduction

Component-based approaches have been proposed to create and deploy software systems assembled from components. The use of previously developed components should leads to faster time for complex software applications. Therefore, component-based software development is a promising solution to some of the problems that designers, developers and integrators face when building their systems [START_REF] Dong | Design component contracts[END_REF]. Software patterns are a design paradigm used to solve problems that arise when developing software within a particular context. Patterns capture the static and dynamic structure and collaboration among the components in a software design. A key promise of the pattern-based approach is that it may greatly simplify the construction of software systems, reuse experience and reduce cost. Design patterns [START_REF] Gamma | Design Patterns, Elements of Reusable Object-Oriented Software[END_REF] have been proposed to reify good design practice from conceptual design building blocks into a composable form. Formal specification and verification techniques are useful for design analysis in that the formal representations are more precise, expressive, and unambiguous than the informal ones, such as graphical and textual notations. Formal notations can be a basis for verification techniques, such as model checking [START_REF] Clarke | Model Checking[END_REF], which can be used to detect errors. A contribution of this paper is to provide a systematic approach for a software designer to model and analyze component integration during the design phase, the early planning stage of the software lifecycle. The approach includes a process of representing, instantiating and integrating design patterns which can seen as a components (called design components), and analyzing their compositions, which are captured as contracts. Compositional patterns can be formally described as connections between process components. The approach involves the modeling of design components and their composition, and a framework in which the design compositions can be analyzed. Another contribution of this paper is a proposition of a novel ADL (LOTOS-ADL) that has been designed to address specification of structural and dynamic architectures. The rest of this paper is organised as follows. Section 2 introduces an overview of our approach. In Section 3 we focus on the abstract specification of the component. Section 4 presents the concepts of LOTOS-ADL. Section 5 gives an overview of our environment of validation. Section7 illustrate a case study. Finally the last section concludes the paper and gives directions for future work.

Overview of the Approach

We present an overview of our approach and outline the general ideas in our formal models. We separate the abstract specification from its implementation.

Our main goal is to provide a systematic approach for a software designer to model and analyze component integration during the design phase, the early planning stage of the software lifecycle. The approach includes a process of representing, instantiating and integrating design components and analyzing their compositions, which are captured as contracts (figure1). This approach allows design components to be reused by making the components description available in a component library. With this approach, the designer can not only model the design component precisely, unambiguously and expressively, but also detect the interactions between components and correct design errors before implementation. As shown in figure1, our approach begins by four steps: (The analysis, the selection, the abstract specification and the instantiation steps). These steps are describes as follows:

-Analysis: the purpose of this step is to analyze the application requirements and to decide on the set of design patterns that will be used in designing the system. In [START_REF] Zitouni | Un framework pour l'utilisation des design patterns par intégration du langage de spécification LOTOS[END_REF] we are shown the specification and the description of the system configuration and its components must be put into a form amenable for analysis and design [START_REF] Zitouni | Un framework pour l'utilisation des design patterns par intégration du langage de spécification LOTOS[END_REF].

-Selection: in this step we analyze the responsibilities and the functionalities of each component and identify candidate patterns that could provide a design solution for each component. In doing so, we consider the design problem that we want to solve and match it to the solution provided by general purpose design patterns (expert pattern [START_REF] Dong | Automating the analyse of design component contracts[END_REF] is a good candidate for this task) [START_REF] Zitouni | Un framework pour l'utilisation des design patterns par intégration du langage de spécification LOTOS[END_REF].

-Abstract specification: this step (inspired from the work of Dong and al. [START_REF] Dong | Design component contracts[END_REF]) contains a formal model of design component, called design component contract. A design component contract includes structural contract, behavioural contract and interface contract.

-Instantiation: in this step, we create instances of the selected patterns and identify the relationships between these instances (This is a role of the Abstract factory pattern). Finally, we use the pattern instances and their relationships to construct the composite component. We use the LOTOS-ADL for this task. During the design or design refinement phases we could discover that a selected pattern has limitations or impacts on other design aspects. In this case, the designer would revisit this design level to choose another pattern, replace previous choices, or create a new pattern dependency or a new uses relationship.

In this article we focus on the abstract specification of the component and the ADL for describing the architecture of component-based software, which provide explicit support for specifying components. ADLs are important since they can document component-based architecture early, reason about their properties, and automate their analysis and system generation [START_REF] Gamma | Design Patterns, Elements of Reusable Object-Oriented Software[END_REF].

Abstract specification of a component

The abstract specification contains a formal model of design component, called design component contract.

A design component contract includes structural contract, behavioural contract and interface contract. Each contract defines the generic information about a design component. The instantiation operation can be used to apply a generic contract in a particular application. The integration operation formally defines how to compose two or more contracts to form a new contract.

The structural properties describe the relations of the constructs of each design component, such as connectivity of classes by inheritance or association relations in object-oriented systems. The behavioural properties are constraints such as event ordering, and action sequence of each design component. The interface contract describes the finite set of input or output ports attached to a design component and the set of messages sent to or received by a component.

Structural contracts

We define the structural aspect of a design component contract as follows: The structural aspect of a design component contract SC is a tuple SC = ( C,A,M,T,Ar,Pc,Pa,), where C is a set of classes in the design component that define the participants within each design component , A is a set of attributes defined in classes C, M is a set of methods defined in classes C, T is a set of types that are used to define the attributes and methods in classes C, and Ar is a set of access rights that the attributes and methods can have in a class of C. For example, Ar = {public, protected, private}, Pc is a set of connection predicates symbols that capture the relationships between the role each a design component. For example (Inherit, association, aggregation,..), and Pa is a set of action predicates symbols that can perform in a design component For example (invoke, new, return…) The structural aspect of a design component can be formalized using a subset of First Order Logic (FOL), because the relations between pattern participants can be easily expressed as predicates [START_REF] Lamport | The temporal logic of actions[END_REF]. The subset of FOL used to describe the structural aspect of a design component comprises variable symbols, connectives (' '), quantifiers (' '), element (є) and predicate symbols acting upon variable symbols. The variable symbols represent class, objects, while the predicate symbols represent permanent relation [START_REF] Taibi | Modeling of distributed objects computing design patterns combination[END_REF]. We define two groups of predicates, entities (Table1) and relationships (Table2).

-Entity predicates define whether a design component has a specific class (abstract or concrete), what a method (or attribute) is defined in a class….

-Relationship predicates define the relations between classes, attributes, and operations and the actions that a role can perform in a component.

Behavioural contracts

In contrast to the structural aspect of a design component contract, the behavioural contract describes the dynamic information, such as the collaboration among the objects participating in the component and the creation of new objects. The behavioural contract is essential because the structural contract only captures the static information. But components are also characterized by the interactions among the objects and operations.

We have chosen a basic LOTOS [START_REF] Bolognesi | Introduction to the ISO specification language LOTOS[END_REF] (figure 2) for defining a formal semantic model of behavioural contracts because it represents a powerful approach to modeling of behaviour and concurrency. The choice of LOTOS is motivated by its powerful ability for describing behaviour and the availability of tools enabling formal verification and automatic generation of distributed programs. Our proposal focuses on formally describing architectures encompassing both the structural and behavioural viewpoints. We illustrate our approach with the example of a client server application

Interface contracts

We define the interface aspect of a design component contract as follow: Let a tuple IC = (P, IP,OP, IM,OM, IMI ), where P is a finite set of process names, IP is a finite set of input ports attached to a process, OP is a finite set of output ports attached to a process, IM is a finite set of input messages sent to a process and OM is a finite set of output messages sent from a process, IMI is the finite set of input messages sent from outside the design component to a process.

Interface contracts

Consider the structure (class and interaction diagram) of the observer pattern shown in figure 3 [10]: (The Observer (also called Publisher-Subscriber) regulates how a change in one object can be reflected in an unspecified number of dependant objects). description should provide a formal specification of the architecture in terms of components and connectors and how they are composed together. Further, in the case of a dynamic or mobile architecture, it must provide a specification of how its components and connectors can change or move at runtime. The behavioural viewpoint may be specified in terms of: actions a system executes or participates in, relations among actions to specify behaviours, behaviours of components and connectors, and how they interact. A large challenge for an Architecture Description Language (ADL) is the ability to describe static but also dynamic software architectures from structural and behavioural viewpoints. Indeed, for describing dynamic architectures, an ADL must be able to describe changing structures and behaviours of components and connectors (including creation/ deletion/reconfiguration/ moving) at runtime. The set of concepts that manipulated within our ADL meta model. (We are defined a meta-model for these tasks).

In our meta-model, we are mainly interested in representing static and dynamic behaviour contract using static and dynamic contract. A major benefit of separate static part from the dynamic part is that reasoning independently from any particular situations. The static contract of a component is a part that does

Validation

By verification, we mean comparison of a complex system against a set of properties characterizing the intended functioning of the system (for instance, deadlock freedom, mutual exclusion, etc.). Most of the verification algorithms are based on the labelled transition systems model, which consists of a set of states, an initial state, and a transition relation between states. This model is often generated automatically from high level descriptions of the system under study, and then compared against the system properties using various decision procedures. For the verification of our approach, we use the FOCOVE (Formal Concurrency Verification Environment) (available in www.focove.new.fr) (fig 4)

The FOCOVE environment is dedicated to the design and verification for component based software development. FOCOVE translate a LOTOS program into a Labelled Transition System (LTS for short) describing its exhaustive behaviour. This LTS can be represented either explicitly as a set of states and transitions or implicitly as a library of C functions allowing us to execute the program behaviour in a controlled way.

Case study

Consider the simple client-server system shown in Figure5. It consists of one client and one server interacting via link connector. Such a system is easy to describe in LOTOS-ADL.

A LOTOS-ADL specification describes a system through a hierarchy of components (process). A process is an entity able to realise non-observable actions, and also interact with others process through externally observable actions. The LOTOS specification at the top-level is a parallel composition of the process Client (component client), the process Server (component server) and the process connector (connector) (figure 5). The connector behaviour is defined through the temporal ordering of invocation operations in the connector interface. The connector interface is made up of four ports: invCtl to invocations from client, terCtl to returns to client, invSrv to invocations from server and terSrv to return to server

Conclusion

In this paper, we have introduced a proposition of formal model of design component based on contract and a rigorous analysis approach to software design composition based on automated verification techniques. Our approach allows us to find errors in the design composition early in the development process. This paper has illustrated how to adopt LOTOS as ADL to describe the behaviour of software architecture. This language is mathematically well-defined and expressive: it allows the description of concurrency, non-determinism, synchronous and asynchronous communications. It supports various levels of abstraction and provides several specification styles. These positive features encouraged us to adopt LOTOS as an ADL for describing both component and connector enables us to check behaviours properties. Currently, we are investigating to proposing a rulesbased transformation enabling the mapping from LOTOS specification to JAVA pseudo code.

Figure 1 .

 1 Figure 1. Overview of our approach

Figure 2 .

 2 Figure 2. LOTOS operators

For

  the verification of concurrent systems, FOCOVE allows errors the use interleaving semantics or the maximality based semantic.

Figure 4 . 1 .

 41 Figure 4.1. The FOCOVE environment

Figure 4 . 2 .

 42 Figure 4. 2. Generation of the label transition systems

Figure 5 .

 5 Figure 5. Illustration of the Client-Server specification specification Client-Server [invClt,terClt,invSrv,terSrv] : noexit:= library RESULT, SERVICES endlib behaviour Client [invClt, terClt] |[invClt, terClt]| connector [invClt, terClt, invSrv, terSrv] |[invSrv, terSrv]| Server [invSrv, terSrv] where ……… ……… Endprocess

Table 1 .

 1 Entity predicates

Table 2 .

 2 Relationship predicates

The abstract specification of structural contract is done by:

(0) Component-name is Observer where:

(

∃ abstract-class(Subject,Observer) є C;

(2) ∧ ∃ class(ConcreteObserver,ConcreteSubject)} є C;

(3) ∧ ∃ ( attach, detach, getstate, update, notify )є M;

(4) ∧ ∃ (void, datatyp)}є T;

(5) ∧ ∃ The abstract specification of interface contract is done by: (0) Component-name is Observer where: The behavioural aspect of a design component is expressed by the LOTOS specification. This specification describe the sequences of observable event:{attach,detach,getstate,setstate,update,notify,cha nge}exchanged by elements of components: {ConcreteSubject, ConcreteObserver} by means of a restricted set of operators (figure 2).

Architecture Description Language

A key aspect of the design of any software system is its architecture. An architecture description, from a runtime perspective, should provide a formal specification of the architecture in terms of components and connectors and how they are composed together.

Enabling specification of dynamic architectures is a large challenge for an Architecture Description Language (ADL) [START_REF] Oquendo | PI-ADL: An architecture description language based on the higher-order typed pi-calculus for specifying dynamic and mobile software architectures[END_REF]. This section describes LOTOS-ADL, a novel ADL that has been designed to address specification of structural and dynamic architectures.

It is a formal language based on temporal ordering of observational behaviour. While most ADLs focus on describing software architectures from a structural viewpoint, LOTOS-ADL focuses on formally describing architectures encompassing both the structural and behavioural viewpoints. The LOTOS-ADL design principles, concepts and notation are presented. An architecture description specifies architecture. Architecture can be described according to different viewpoints. From a runtime perspective, two viewpoints are frequently used in software architecture [7], [START_REF] Oquendo | PI-ADL: An architecture description language based on the higher-order typed pi-calculus for specifying dynamic and mobile software architectures[END_REF]: the structural viewpoint and the behavioural viewpoint. The structural viewpoint may be specified in terms of: components (units of computation of a system), connectors (interconnections among components for supporting their interactions), Configurations of components and connectors. Thereby, from a structural viewpoint, an architecture


F Barillet

D Boichard

Studies on dairy production of milking ewes I. -Estimates of genetic parameters for total milk composition and yield

Keywords: Dairy ewes, milk composition, milk yield, genetic parameters, selection goal. Résumé

Genetic parameters for dairy traits in first lactation (milk yield, fat and protein yields, fat and protein contents) were estimated from records of 1487 Lacaune ewe lambs born from 102 young rams undergoing progeny test and 74 proven rams. Variance and covariance components were estimated by H ENDERSON ' S methods I and III. According to the analysis, information from proven rams was totally or partially used for estimating fixed effects, or was excluded. Results appeared similar to the average literature data for dairy cows, except the correlation between fat and protein contents, which was rather high (0.8). The genetic standard deviation of fat was larger than that of protein, the ratio being about 1.3 for yields and 1.85 for contents. Accordingly, expected genetic change is likely to be smaller for protein matter than for fat matter. Whereas the genetic correlation between fat content and yield was positive, the genetic correlations between protein content and yield, or between content of one component and yield of the other, seemed to be close to zero and maybe negative. Accordingly, the selection criterion should include useful yield and content, instead of the useful yield alone. Useful yield (or content) was defined as a combination of fat and protein yields (or contents), with weighting I and 1.85 respectively.

1. -Paramètres génétiques de la quantité et composition totale du lait à la traite Les paramètres génétiques des caractères laitiers (quantité de lait, de matière grasse et de matière protéique, taux butyreux et protéique) sont estimés à partir d'un fichier de 1 487 agnelles de race Lacaune en 1'" lactation, issues de 102 béliers de testage et 74 mâles de service. Ils sont estimés par décomposition de la variance et de la covariance entre demi-soeurs de père, en appliquant les méthodes 1 ou III d'H ENDERSON , aux données de testage. L'information des pères de service est utilisée en totalité, partiellement, ou ignorée, pour estimer les effets fixés. Les résultats obtenus selon ces 3 analyses sont cohérents entre eux, et globalement conformes à la moyenne bibliographique connue en vache laitière, à l'exception de la corrélation génétique entre les taux butyreux et protéique qui ici apparaît plus élevée (0,8). La matière grasse est plus variable que la matière protéique, puisque le rapport des écarts-types génétiques est estimé à 1,3 pour les quantités et à 1,85 pour les taux. Les possibilités d'évolution génétique de la matière grasse sont donc plus importantes que celles de la matière protéique. Alors que la corrélation génétique entre le taux butyreux et la quantité de matière grasse est positive, les corrélations génétiques entre le taux protéique et sa matière, ou entre un taux et la quantité de l'autre matière, sont proches de zéro, et peut-être négatives. Il est conseillé en conséquence de sélectionner sur une combinaison linéaire de la quantité et du taux moyen de matière utile, plutôt que sur la matière utile seule. Les critères « quantité ou taux moyen de matière utile » sont eux-mêmes définis comme une combinai- son des quantités (ou taux) de matière grasse et protéique, avec des pondérations économiques égales respectivement à 1 et 1,85. Mots clés : Brebis laitières, composition du lait, quantité de lait, paramètres génétiques, objectif de sélection.

I. Introduction

Since the 1960's the main selection goal for dairy ewes was limited to the milk yield. Two reasons motivated this choice. On the one hand, the low level of milk production with a high concentration necessitated the fast development of a selection scheme. On the other hand, recording the milk concentration on the farm was not economically feasible within the usual type A procedure, i.e. two measurements a day once a month.

Nowadays the selection scheme applied to the whole Lacaune population is producing a continual improvement in milk yield (B ARILLET et al., 1986). The selection program may now be reexamined in order to take into account the milk composition, since sheep milk is exclusively processed into cheese. That question involves three steps : a new definition of the main selection criterion, the design of a simplified recording procedure for milk composition, suited to the species at a reasonable cost, and the integration of that procedure in the selection scheme. The genetic parameters for yields and milk composition are to be estimated first, especially since the literature on that topic is very scarce for the dairy ewes. In order to achieve that aim, a qualitative dairy recording procedure of type A (two milkings a month) has been experimentally set up on 6 798 ewes of the Lacaune selection nucleus between 1979 and 1981.

II. Material and methods

A. Definition of the variables Milk yield of dairy ewes is defined in France by the production at the milking period only, after one month of suckling. Accordingly, only the decreasing part of the lactation curve is recorded while the milk concentration is increasing throughout that period.

The following variables for this milking period were analysed : milk yield (M), fat and protein yields (F, P), fat and protein contents (F %, P %), days of milking (D), daily milk production (DM), as M divided by D, and ratio of fat to protein content (F %, P %).

Yield traits were corrected for days of milking by the multiplicative factor k of the French dairy sire evaluation scheme (PouTous et al., 1981), as follows :

Above 120 days of milking, yields do not depend any more on days of milking, while under this threshold, correlations between yields and days of milking remain highly positive. Accordingly, the within flock variability is more homogeneous and heritability of the traits is increased (P OUTOUS & MocouoT, 1975).

Useful yield (U) and content (U %) were defined as a combination of fat and protein yields or contents, with weightings of 1 and 1.85 respectively :

U and U % were the main and secondary selection criteria respectively.

B. Material

The Lacaune selection nucleus comprises 105 000 ewes in 320 herds, for which only milk yield was recorded up to 1985. However milk composition was experimentally recorded between 1979 and 1981 for 2 045 primiparous ewes distributed in 26 year x flock groups. The present analysis was restricted to the year x flock groups where at least three young unproven and two proven rams were used, in order to obtain a good connection between flocks in that sample of the selection nucleus. The data set included 1 487 first lactations distributed in 22 year x flock groups, with 763 ewe lambs born from 102 young rams undergoing progeny test, and 724 born from 74 proven rams. Table 1 summarizes the characteristics of the data set. Out of the proven rams, 27 males with 427 daughters in 22 year x flock groups were responsible for the greatest part of the connection between flocks.

In order to reduce sampling error, only the young rams tested with at least three daughters were kept in the above data set for the analysis. This threshold was rather low because progeny groups were incomplete in the qualitative recorded sample : the sires had eight daughters on average in the data set while they were tested on 30-40 daughters, for milk yield only, in the whole Lacaune selection nucleus (B ARILLET & E LSEN , 1979).

C. Methods of analysis

Genetic parameters were estimated by variance and covariance analysis of half sisters data. In order to avoid bias due to selection [START_REF] Obertson | The effect of selection on the estimation of genetic parameters[END_REF], only the 102 young rams were taken into account. However, using the information of all proven sires or of the most widely used proven rams led to a better estimation of fixed effects. The three following analyses were conducted (table 1).

Analysis 7

Henderson's method 1 [START_REF] Enderson | Estimation of variance and covariance components[END_REF] was applied to the data of young ram's daughters, with the sire effect as random. Data were previously corrected for fixed effects (age at lambing, month of lambing, year x flock) which were estimated on the whole data set with a complete model including young and proven sires effects and environmental effects. Owing to this type of correction, this method was very close to Henderson method II but the reduction in the number of degrees of freedom was not taken into account.

Analysis 2

Variance and covariance components were estimated by Henderson's method III (H ENDERSON , 1953). The model was derived from HILL et al. ( 1983), M EYER (1984) and V AN V LECK (1985). Proven sires were considered as fixed effects in order to improve connection between year x flock. The sample gathered 427 ewe lambs born from the 27 most widely used proven sires and 763 daughters of young unproven rams. The model was the following : with l i a constant, M ¡ the month of lambing effect, A j the age at lambing effect, YF, the year x flock effect, S, the fixed effect of the sires group, T, m the within group fixed effect of the proven sire or the random effect of the young ram, assumed to be normally distributed with zero expectation and variance (T , 2 .

E ii kl mn the residual effect assumed to be normaily distributed with zero expectation and variance a}. 2 .

Analysis 3

Variance and covariance components were estimated by Henderson's method III from the subsample of the 763 daughters of the 102 young rams. The model included the effects of year x flock, month and age at lambing as fixed, and of young ram as random.

In the three analyses approximative sampling errors were determined as described by GROSSMAN & NORTON (1974).

D. Predicted changes according to the selection criterion Predicted changes were estimated as described by R ENDEL & R OBERTSON (1950). Demographic and genetic hypotheses were derived from the actual Lacaune selection scheme (B ARILLET & E LSEN , 1979) : selection pressures on the four gene transmission pathways, sire-son, sire-daughter, dam-son and dam-daughter, were 15, 33, 10 and 70 p. 100. Generation intervals were equal to 5.5, 4.9, 4.5 and 3.5 years, respectively. Males were progeny tested on 40 daughters, and 45 p. 100 of adult ewes were mated with unproven rams. Table 8 shows the prediction of asymptotic annual genetic changes under these hypotheses according to the selection criterion.

III. Results

The estimates of heritability coefficients, genetic and phenotypic standard devia- tions and genetic correlations are shown in tables 2, 4, 5 and 7 respectively.

A. Comparison of the 3 analyses Results obtained from the three different analyses were very consistent. Thus, the structure of the data from the unproven sires could be considered as satisfactory. Indeed, the demographic constitution of the Lacaune breed was very favourable as compared to the dairy cattle (M EYER , 1985 ; V AN V LECK , 1985 ; B OICHARD & B ONA i TI , 1987). On average each year x flock group included 34 ewe lambs born from 6 young rams, with a range of 3 to 12 sires. The same pattern was observed for proven rams with 33 daughters from 8 sires on average, while 20 of them were born from some of the 27 best-represented rams in the data set. More generally speaking, the large number of ewe lambs and of young rams in each year x flock group may compensate for the possible lack of connection between sires and year x flock.

B. Heritability coefficients

Heritability of the days of milking (D) was rather low, from 0.07 to 0.09 according to the analysis (table 2), thus justifying the partial correction of the yields that are phenotypically very correlated to the days of milking. The corrected variables (CM, CF and CP) were more heritable than the original variables (M, F and P) in agreement with P OUTOUS & M OCQUOT (1975).

Heritability of milk yield (M and CM) varied from 0.27 to 0.32 according to the analysis. This result was in agreement with the average literature data for milking ewes : !.29 ( 1984). The heritabilities for fat yield (0.23 to 0.29) and protein yield (0.22 to 0.27) were similar and slightly smaller than that for milk yield. Estimated heritabilities for contents were much higher and similar, between 0.49 and 0.62 for fat content, and between 0.47 and 0.53 for protein content. These results were consistent with the average literature data for dairy cows, reviewed in 1974 by M AOALA & H ANNA (table 3) and reported by others since that time (D EB et al., 1974 ; ToNG et al., 1976 ; H ARDIE et al., 1978 ; H ARGROVE et al., 1981 1987). However, only two studies of genetic parameters for milk composition in dairy ewes are known to us. These estimates were obtained from two experimental flocks in the Sarde breed. The first one, for fat content only (BorrELtt, 1969) was similar to ours. The second study, involving both protein and fat (C ASU et al., 1975 ; F LAMANT & C ASU , 1977), reported estimates very different from ours, in particular for protein content.

C. Genetic standard deviation estimates (table 4)

Fat yield and content were more variable than protein. According to the analyses, genetic standard deviations ranged respectively from 1.24 to 1.30 kg for CF, 0.94 to 0.95 kg for CP, 4.3 to 4.9 g/l for F % and 2.4 to 2.5 g/1 for P %. The ratio of fat to protein standard deviations reached about 1.30 for yields and 1.85 for contents. Similarity of the results between species has to be pointed out. Indeed the estimates of the ratio reported by HILL et al. (1983) and B OICHARD & B ONA I TI (1987) are very close to ours. Therefore fat traits seem more likely to vary than protein ones.

D. Estimates of genetic correlations

Milk yield was more strongly correlated with protein yield (0.92 to 0.94) than with fat yield, 0.82 to 0.86 (table 5). Correlation between fat and protein yields took an intermediate position between the two previous ones (0.90 to 0.93). The average literature data for dairy cow show a very similar trend (table 6), the correlation between milk and protein yields being the highest (M AUALA & H ANNA , 1974 ; TONG et al., 1976 ; H ARGROVE et al., 1981 ; P APE et al., 1983 b ; ALPS et al., 1984 ; M EYER , 1985 ; B OICHARD & BoNAm, 1987). The estimated genetic correlation between fat and protein contents fell within a range of 0.75 to 0.81 and was higher than the usual value published for dairy cows (0.56). However, our estimates were more consistent with the two results given for the Sarde breed (C ASU et al., 1975 ; F LAMANT & C ASU , 1977). Anyway, in both species a preferential evolution of one of the contents should be easier to obtain than for one of the matter yields, since genetic correlation is lower between contents than between yields.

In our sample, genetic correlations between milk yield and concentration (— 0.34 to -0.51) were clearly negative (table 5). That opposition is not so strong in dairy cows (table 6), although more recent studies (H ARGROVE et al., 1981 ; P APE et al., 1983 & BoN A m, 1987) reported strong negative correlations. Moreover, the genetic correla- tion between milk yield and protein content (— 0.47 to — 0.51) was stronger than between milk yield and fat content (-0.34 to -0.41). Recent papers showed a similar trend in dairy cows.

Genetic correlations between fat yield and content were always positive and ranged from 0.10 to 0.24 (table 5). This observation was in agreement with results obtained in dairy cows (table 6) : indeed the average literature data is 0.30 without one negative estimate. However, the estimated correlation between protein yield and content was negative, from — 0.09 to — 0.19. Similarly, correlations between fat content and protein yield (-0.05 to -0.15) or between protein content and fat yield (-0.04 to -0.13) were slightly negative. Only the relationship between protein yield and content appeared rather atypical, as published studies show an average of 0.15 over 14 estimates with only 4 negative results (table 6). The negative cross-correlations were more usual : indeed the average over 14 estimates between protein yield and fat content is equal to — 0.08, with 12 negative results, while between protein content and fat yield it reaches 0.04 over 14 estimates with 7 negative (table 6).

IV. Discussion

The present results generally agree with already published data for dairy cows for heritability estimates, for difference between variabilities of fat and protein matter and for genetic correlations between matter yields and milk yield, between milk yield and both contents, between fat yield and content.

However, the genetic correlation between both contents seems to be higher than for dairy cows. This difference could be due to the lack of selection on milk concentration in dairy ewes, while a selection on fat content has been applied on dairy cows for a long time. It may also be explained by a difference between species, or between traits, which are not exactly the same : the average content is considered over the whole lactation for the cow but only after a month of suckling for the ewe. This difference could also be due to the low accuracy of the present estimate, obtained from a rather small data sample. Indeed, the standard error of the estimated genetic correlation between contents was close to 0.11 (table 7).

Neither was the correlation between protein yield and both contents accurately estimated. These relationships were low but their sign could not be clearly established. However, the same question remains without answer for dairy cows. The number of estimates has to be pointed out, being half the corresponding number of estimates for fat yield (table 6), because of the lack of systematic recording of protein content in all countries.

In France, the sheep milk is exclusively processed into a specific cheese known for its high ratio of fat to dry content. For this reason, useful matter yield (U) and content (U %), defined as above, were chosen as main and secondary selection criteria [START_REF] Arillet | Amelioration g!nitique de la composition du lait des brebis : 1'exemple de la race Lacaune[END_REF]. Since each content is weighted with the reciprocal of its standard deviation, U % gives the same economic value to an increase of one genetic standard deviation in fat as well as in protein content. So the genetic correlation of U % was the same with F % as with P %, 0.94 and 0.93 respectively (table 7). The main selection criterion U can be considered as a cheese output, i.e. a dry matter yield (F + P), by the ewe. In the same way, the secondary criterion represents on one hand the cheese yield best predicted by the dry matter content (F % + P %), and on the other hand the fat to dry matter ratio in the cheese, corresponding to the ratio of fat to protein contents in the milk (table 8). It can be pointed out that the ratio F %/P % is to be monitored in the milking ewe, while the reciprocal ratio P %/F % is of major interest in dairy cattle (IVI AI7ALA & H ANNA , 1974 ; B OICHARD & B ONA I TI , 1987). The genetic correlation between U and U % was null (table 7). Such a result was already known in dairy cattle : using different weightings (1 to 1.20) of protein relative to fat, B ONA I TI & MOCQUOT (1982), ALPS & A VERDUNK (1984) and B OICHARD & B ONA I TI (1987) obtained null or slightly positive estimates of genetic correlation between useful yield and content.

Although selecting exclusively on U should maintain dry matter content at a stable genetic level, it should not be advised in dairy ewes. According to the sign of the genetic correlation between protein yield and both contents, this selection leads to a possible decrease in protein content and to an almost certain increase in fat content. These results are clearly shown by the expected genetic changes (table 8) based on the demographic hypotheses of B ARILLET & E LSEN (1979). Of course, selection on U gives the largest yearly response in dry matter yield (F + P), 0.391 kg, while maintaining dry matter content. But fat content increases by 0.05 g/i each year while protein content decreases by 0.04 g/I. A regular decrease in P % may have, in the long term, negative technological consequences. Accordingly, it seems preferable to use as selection crite- rion a combination of U and U %, or U and P %. As shown in table 8, the first option, with weighting 1 for U expressed in kg and 0.1 for U % expressed in g/1, is rather satisfactory. The response in the main trait (F + P) is almost the same as with selection on U only, while positive yearly gains are expected for secondary traits : + 0.30 g/l for fat content, + 0.10 g/I for protein content, + 0.003 for F %/ / P %.

V. Conclusion

This study was the first on-farm estimation of genetic parameters for dairy traits (milk yield and composition) in milking ewes. The present estimates were generally consistent with those obtained in dairy cows for 25 years. Therefore in the Lacaune breed, selection on milk yield must be discarded and replaced by the selection on useful yield. Because of the genetic relationship between protein content and protein and fat yields, the selection criterion must include both useful yield and content instead of useful yield alone.

Since 1985, fat and protein contents were recorded with a simpler procedure in the Lacaune breed (B ARILLET et al., 1986) and, since 1986, an experimental A type recording procedure was set up in some flocks of the selection nucleus in the Manech and Basco-Bearnaise breeds. In few years we will be able to obtain more accurate estimates of genetic parameters for the main French breeds from larger data samples.

Received November 25, 1986. Accepted April 1, 1987.

  b ; ALPS et al., 1984 ; S CHNEEBERGER & H AGGER , 1984 ; M EYER , 1985 ; B OICHARD

  D ASSAT & MASON, 1954 ; F INCI , 1957 ; D ASSAT & S ARTORE , 1962 ; B OY -

	AZOGLU B t l ll., 1965 ; S OLLER B l al., 1966 ; C ALCED O O RD ONE Z , 1968 ; H INKOVSKI , 1968 ;
	BoNELLI, 1969 ; H ORAK , 1969 ; T ECZA , 1969 ; M INEV et al., 1971 ; RoMER et al., 1971 ;
	Y ARKIN & T UNCEL , 1972 ; O JEDA S AHAGUN , 1974 ; C ASU et C ll., 1975 ; F LAMANT &
	C ASU , 1977 ; C ARRIEDO & S AN P RIMITIVO , 1982 ; M AVROGENIS , 1982 ; B OICHARD et C ll.,


Roger Lewandowski
email: roger.lewandowski@univ-rennes1.fr

Yves Preaux
email: preaux@free.fr

Attractors for a deconvolution model of turbulence

Keywords: MCS Classification : 76D05, 35Q30, 76F65, 76D03 Navier-Stokes equations, Large eddy simulation, Deconvolution models

We consider a deconvolution model for 3D periodic flows. We show the existence of a global attractor for the model.

Introduction

This note is concerned by the deconvolution model of order N introduced in [START_REF] Layton | A high accuracy leray-deconvolution model of turbulence and its limiting behavior[END_REF] (model (2.7) below) for 3D periodic flows. This model takes inspiration in the class of the so called α-models (see in [START_REF] Cheskidov | On a leray-α model of turbulence[END_REF] and [START_REF] Foias | The Navier-Stokes-alpha model of fluid turbulence[END_REF] and references inside) and also in the class of ADM models (see in [START_REF] Stolz | An approximate deconvolution model for large-eddy simulation with application to incompressible wall-bounded flows[END_REF]). We are interested by the question of the existence of a global attractor for this model. The question of attractors has already been considered for the alpha model (see [START_REF] Chepyzhov | On the convergence of the leray-alpha model to the trajectory attractor of the 3D Navier-Stokes system[END_REF]), corresponding to the case N = 0. We prove in this work the existence of an attractor for each N (see Theorem 3.1). In order to make the paper self contained, we describe carrefully how is constructed the deconvolution model. Next, we recall basic notions on the attractors, notions that can be founded in the book of R. Temam (see [START_REF] Temam | Infinite Dimensional Dynamical Systems in Mechanics and Physics[END_REF]). Finally we prove the existence of the attractor. The question of its dimension is under progress.

H s = w = k we ik•x , ∇ • w = 0, w(0) = 0, k |k| 2s | w(k, t)| 2 < ∞ . (2.2) ||w|| 2 s = k |k| 2s | w(k, t)| 2 ,
where of course ||w|| 2 0 = ||w|| 2 . It can be shown that when s is an integer, ||w|| 2 s = ||∇ s w|| 2 (see [START_REF] Doering | Applied analysis of the Navier-Stokes equations[END_REF]).

We denote by P L The Helmholtz-Leray orthogonal projection of (L 2 ) 3 onto H 0 and by A the Stokes operator defined by A = -P L △ on D(A) = H 0 ∩ (H 2 ) 3 . We note that in the space-periodic case Aw = -△w for all w ∈ D(A). The operator A -1 is a sef-adjoint positive definite compact operator from H s onto H s , for s = 1 and s = 2 (see [START_REF] Foias | Navier-Stokes Equations and Turbulence[END_REF]). We denote λ 1 the smallest eigenvalue of A. We introduce the trilinear form b defined by

(2.3) b(u, v, w) = i,j Ω u i ∂ i v j w j dx.
wherever the integrals make sense. Note that b(u, w, w) = 0 when ∇ • u = 0.

The Filter and the deconvolution process

Let w ∈ H 0 and w ∈ H 1 be the unique solution to the following Stokes problem with periodic boundary conditions:

(2.4)

-δ 2 △w + w + ∇r = w in R 3 , ∇ • w = 0, Ω w = 0.
We denote the filtering operation by G so that w = Gw. Writing w(x, t) = k w(k, t)e -ik•x , it is easily seen that ∇r = 0 and w(x, t)

= k w(k, t) 1 + δ 2 |k| 2 e -ik•x .
Then writing w = G(w), we see that in the corresponding spaces of the type H s , the transfer function of G, denoted by G, is the function

G(k) = 1 1+δ 2 |k| 2 ,
and we also can write on the H s type spaces

(2.5) -δ 2 △w + w = w in R 3 , ∇ • w = 0, Ω w = 0.
The procedure of deconvolution by the Van Citter approximation is described in [START_REF] Layton | A high accuracy leray-deconvolution model of turbulence and its limiting behavior[END_REF]. This yields the operator

D N w = N n=0 (I -G) n w. Definition 2.1 The truncation operator H N : H s → H s is defined by H N w := D N w = (D N • G)w.
Note that, for any s 0 we have the following proprieties (see [START_REF] Layton | A high accuracy leray-deconvolution model of turbulence and its limiting behavior[END_REF]) :

(2.6)

H N w s w s , H N w s+2 C(δ, N) w s .

The model

Let u 0 ∈ H 0 , f ∈ H -1 . For δ > 0, let the averaging be defined by (2.4). The problem we consider is the following: for a fixed T > 0, find (w, q)

(2.7)

       w ∈ L 2 ([0, T ], H 1 ) ∩ L ∞ ([0, T ], H 0 ), ∂ t w ∈ L 2 ([0, T ], H -1 ) q ∈ L 2 ([0, T ], L 2 per,0 ), ∂ t w + (H N (w) • ∇) w -ν△w + ∇q = H N (f) in D ′ ([0, T ] × IR 3 ), w(x, 0) = H N (u 0 ) = w 0 .
where L 2 per,0 denotes the scalar fields in L 2 loc (IR 3 ), 2π-periodic with zero mean value. We prove in [START_REF] Layton | A high accuracy leray-deconvolution model of turbulence and its limiting behavior[END_REF] the following result.

Theorem 2.1 Problem (2.7) admits a unique solution (w, q), w ∈ L ∞ ([0, T ], H 1 ) ∩ L 2 ([0, T ], H 2 ), and the following energy equality holds:

(2.8) 1 2 w(t) 2 + ν t 0 Ω |∇w| 2 dxdt ′ = 1 2 H N (u 0 ) 2 + t 0 Ω H N (f).w dxdt ′ .
3 Main result

Recall of basic notions about attractors

We denote by w(t, •) = S(t)(w 0 ) the (unique) solution of system (2.7) at time t. We recall the definitions of a global attractor and an absorbing set (see in [START_REF] Temam | Infinite Dimensional Dynamical Systems in Mechanics and Physics[END_REF]).

S(t)(A).

Proposition 3.1 Assume that there exists an absorbing bounded set A and that the semi group S(t) is uniformly compact, then A = ω(A) is the global attractor for the dynamical system defined by S(t).

see the proof in [START_REF] Temam | Infinite Dimensional Dynamical Systems in Mechanics and Physics[END_REF]).

Existence of a global attractor

We are now in order to state and prove the main result of this note.

Theorem 3.1 The system (2.7) has a global attractor.

Proof. Thanks to Proposition 3.1, it remains to prove that system (2.7) has an absorbing set and that S(t) is uniformly compact, in the sense of definition 3.2. Both things are derived from basic estimates that we detail in the following.

Absorbing set in H 0 : We take the inner product of the first quation of system (2.7) with w to obtain

(3.1) 1 2 d dt w 2 + b(H N (w), w, w) + ν||w|| 2 1 = (H N (f), w).
Observing that b(H N (w), w, w) = 0 due to ∇ • H N (w) = 0, applying Young inequality, Poincare inequality ||w|| λ

-1 2
1 ||w|| 1 and using (2.6) there remains

(3.2) d dt w 2 + νλ 1 ||w|| 2 1 νλ 1 f 2 .
So, noting ρ 0 = 1 νλ 1 f and applying Gronwall lema we obtain

(3.3) w 2 w 0 2 e -νλ 1 t + ρ 2 0 (1 -e -νλ 1 t ).
Considering w 0 included in a ball B(0, R) and choosing ρ ′ 0 > ρ 0 , the previous inequality implies that, for t > T 0 , (3.4)

w(t) 2 < ρ ′ 0 2 , with T 0 = 1 νλ 1 ln R 2 ρ ′ 0 2 -ρ 0 2 .
Since each bounded set of H 0 is included in a ball B(0, R), one deduces that B(0, ρ ′ 0 ) is an absorbing set in H 0 . More, as an alternative of (3.2) we may obtain

(3.5) d dt w 2 + ν||w|| 2 1 1 νλ 1 f 2 .
Integrating between t and t + r, we observe than, for u 0 ∈ B(0, R), ρ ′ 0 > ρ 0 and t > T 0 with T 0 = 1 νλ 1 ln R 2

3 .

 3 We denote by ω(A) the set ω(A) = s≥0 t≥s

2 d dt w 2 1 +

 21 Absorbing set in H 1 : We take now the inner product of the first equation of system (2.7) with Aw to obtain(3.7) 1 b(H N (w), w, Aw) + ν||Aw|| 2 = (H N (f), Aw),

  Definition 3.2 1. A set A ⊂ H 0 is an absorbing set if and only if for every bounded subset B ⊂ H 0 there exists t 1 > 0 such that for all t ≥ t 1 one has S(t)(B) ⊂ A. 2. We say that the semi group S(t) is uniformly compact if and only if for every bounded subset B ⊂ H 0 there exists t 2 = t 2 (B) such that

	S(t)(B) is compact.
	t≥t 2

Definition 3.1 We say that A ⊂ H 0 is a global attractor for the dynamical system (2.7) if and only if (P1) A is compact in the space H 0 , (P2) ∀ t ∈ IR, S(t)(A) ⊂ A, (P3) For every bounded subset B ⊂ H 0 , ρ(S(t)(B), A) goes to zero when t goes to infinity, where ρ(S(t)(B), A) = sup v∈B inf u∈A ||u -v||.

The trilinear form b satisfies the folowing inequality (see in [START_REF] Layton | A high accuracy leray-deconvolution model of turbulence and its limiting behavior[END_REF]) :

Therefore, one has .

By Young inequality we obtain

We now use a Gronwall type proposition (see the proof in [START_REF] Temam | Infinite Dimensional Dynamical Systems in Mechanics and Physics[END_REF]): We can now finish the proof. Thanks to (3.4) and (3.6), using this lemma with

Thus, after a time T 1 = T 1 ( w 0 , f , ν), w is included in a ball or radius R = R ( f , ν, δ, N). One deduces that there exists an absorbing set in H 1 . Let B be a bounded set in H 1 . Estimate (3.14) implies that t≥T 0 +r S(t)B is a bounded set in H 1 wich is compactly imbeded in H 0 , so S(t) is uniformly compact. Estimate (3.14) also implies the existence of an absorbing bonded set since k 1 , k 2 and k 3 are independant of w 0 . Thanks to (3.1), this achieves the proof of the theorem.


Jérémy Jacob
email: jeremy.jacob@univ-orleans.fr

Jean-Robert Disnar

Gérard Bardoux

Carbon isotope evidence for sedimentary miliacin as a tracer of Panicum miliaceum (broomcorn millet) in the sediments of Lake le Bourget (French Alps)

published or not. The documents may come

Carbon isotope evidence for sedimentary miliacin as a tracer of Panicum miliaceum (broomcorn millet) in the sediments of Lake le Bourget (French Alps).

Introduction

The development of novel tracers permitting unravelling possible interactions between past climate and human activities is of direct relevance within the context of future Global Change [START_REF] Dearing | Climate-human-environment interactions: resolving our past[END_REF]. We recently proposed that miliacin (olean-18-en-3β-ol methyl ether or germanicol methyl ether, Figure 1) detected in the sediments of Lake le Bourget (French Alps) could be a specific tracer for the former cultivation of Panicum miliaceum (broomcorn millet) in the catchment [START_REF] Jacob | Millet cultivation history in the French Alps as evidenced by a sedimentary molecule[END_REF]. This assumption was based on the fact that miliacin is the sole pentacyclic triterpene methyl ether (PTME) found in Lake le Bourget sediments. Because of the peculiar resistance of PTMEs to diagenesis [START_REF] Jacob | Pentacyclic triterpene methyl ethers in recent lacustrine sediments (Lagoa do Caçó, Brazil)[END_REF], this implies that miliacin was certainly the sole PTME produced in large amounts in the catchment. Few plants are reputed to synthesize miliacin as the sole PTME. These consist of Microstegium vimineum, Syntherisma sanguinalis, Glyceria acutiflora, Panicum sp. and P. miliaceum [START_REF] Jacob | Pentacyclic triterpene methyl ethers in recent lacustrine sediments (Lagoa do Caçó, Brazil)[END_REF]. By comparing this list with an archaeobotanical inventory of plants that developed around the lake during the Bronze Age [START_REF] Bouby | Economie agraire à la fin de l'âge du Bronze sur les bords du Lac du Bourget (Savoie, France)[END_REF], we conclude that P. miliaceum is the most probable biological source of miliacin in the sediments. Therefore, miliacin in the sediments of Lake Le Bourget attest to the large development of P. miliaceum in the catchment. Since this plant is not indigenous to the study area, this implies that this plant was intentionally introduced (for agriculture) and that sedimentary miliacin attests to the cultivation of P. miliaceum in the watershed. As a matter of fact, the first appearance of miliacin in the sediments around 1700 BC slightly precedes the expected introduction of P. miliaceum in the region [START_REF] Marinval | Données carpologiques françaises sur les Millets (Panicum miliaceum L. et Setaria italica L. Beauv.) de la Protohistoire au Moyen Age[END_REF]. In addition, the drastic decrease of miliacin concentrations around 800 BC coincides with several evidences of human abandonment of lake shore dwellings due to a phase of climatic deterioration [START_REF] Magny | Holocene climate variability as reflected by mid-European lake-level fluctuations and its probable impact on prehistoric settlements[END_REF][START_REF] Billaud | Habitats lacustres du Néolithique et de l'âge du Bronze dans les lacs alpins français[END_REF]. Despite these arguments, the certification of P. miliaceum as the source of miliacin in the sediments of Lake le Bourget requires additional investigation. P. miliaceum is a C4 plant imported from Asia whereas plants growing naturally around in the Alps have a C3 metabolism. By comparing the δ 13 C of miliacin extracted form these sediments with that of miliacin extracted from P. miliaceum and Chionochloa sp., we aim to confirm the use of sedimentary miliacin as a tracer of P. miliaceum, and by extension, of human cropping.

Samples and Methods

Isolation of miliacin

Sediment extraction was described previously [START_REF] Jacob | Millet cultivation history in the French Alps as evidenced by a sedimentary molecule[END_REF]. The resultant neutral lipid fractions of eleven 1 cm-thick sediment samples (22 g in total) from Lake le Bourget core LDB04 (at depths corresponding to the 1225 -770 BC time interval; [START_REF] Jacob | Millet cultivation history in the French Alps as evidenced by a sedimentary molecule[END_REF] were combined in order to obtain enough miliacin for δ 13 C measurements. This extract was dissolved in 0.5 ml of hexane and then fractionated by flash chromatography on silica first activated at 110 °C for 48 h and then deactivated with 5% water. After the successive elution of aliphatic hydrocarbons (fraction F1 obtained with 2 ml hexane) and aromatics (fraction F2 obtained with 1 ml hexane and 2 ml hexane:toluene 3:1), miliacin was found in fractions F3 and F4 obtained with 2 ml hexane:toluene ( 1:1) and 2 ml hexane:ethyl acetate (19:1,), respectively. Fractions F3 and F4 were combined and dried under nitrogen. 4.3 g of P. miliaceum seeds and 2.4 g of P. miliaceum bran were extracted with the same protocol as described in [START_REF] Jacob | Millet cultivation history in the French Alps as evidenced by a sedimentary molecule[END_REF]. The total extracts were dissolved in 2 ml of hexane, and 0.5 ml of each extract was fractionated as above. Purified miliacin was obtained from Pr. R. Smith (Loughborough University, UK) and Pr. H.E. Connor (Christchurch University, NZ). This compound was purified from Chionochloa sp. extracts [START_REF] Russell | Triterpene methyl ethers of Chinochloa (Gramineae)[END_REF][START_REF] Connor | Triterpene methyl ether differentiation in Chionochloa (Gramineae)[END_REF][START_REF] Connor | Flora of New Zealand -Gramineae. Supplement I: Danthonioideae[END_REF].

GC-MS analyses

In order to verify the purity of miliacin in the combined F3+F4 fraction of P. miliaceum seeds and bran as well as in F3+F4 fraction of LDB04 sediment extracts, these fractions were analysed by GC-MS on a TRACE-PolarisGCQ. The gas chromatograph was fitted with a Rtx-5MS capillary column (30 m, 0.25 mm i.d., 0.25 µm film thickness) with 5 m of guard column. The GC operating conditions were: temperature held at 40 °C for 1 min, then increased from 40 to 120 °C at 30 °C min -1 , 120 to 300 °C at 3 °C min -1 , with a final isothermal hold at 300 °C over 30 min. The sample was injected splitless, with the injector temperature set at 280 °C. Helium was the carrier gas. The mass spectrometer was operated in the electron ionisation (EI) mode at 70 eV ionization energy and scanned from 50 to 600 Da.

Concentrations of miliacin were estimated by using pure miliacin isolated from Chionochloa sp. as external standard.

GC-C-IRMS analyses

Miliacin standard, miliacin extracted from P. miliaceum bran and the miliacincontaining fraction from Lake le Bourget sediments were analysed by GC-C-IRMS, in a continuous flow mode. δ 13 C values were determined with a HP5860 gas chromatograph coupled to a Micromass Optima (GVI) isotope ratio mass spectrometer via an Isochron III combustion interface heated at 850 °C. The gas chromatograph was fitted with a BPX 5 column (60m, 0.32 mm i.d., 0.25 μm film thickness). The GC operating conditions were: temperature held at 50 °C for 2 min, then increased from 40 to 200°C at 30 °C min -1 , 120 to 350 °C at 5 °C min -1 , with final isothermal hold at 350 °C over 20 min. The samples were injected in splitless mode, with the injector temperature set at 280 °C. Helium was the carrier gas. Ultra high purity carbon dioxide gas with known δ 13 C value was pulsed three times at the beginning and end of each analysis. The accuracy of carbon isotope measurements was evaluated using a standard of three n-alkanes and dodecanoic methyl ester with offline measured δ 13 C values (VGmix from Micromass). All δ 13 C values are reported in standard ‰ notation relative to VPDB.

TC/EA-IRMS analyses

Standard miliacin crystals purified from Chinochloa sp. extracts were analysed by Thermal Conversion/Elemental Analyzer-Isotope Ratio Mass Spectrometry (TC/EA-IRMS) in dual inlet mode. δ 13 C values were determined with a CARLO ERBA NA 1500 elemental analyser coupled to a Micromass SIRA 10, via a triple trap purification interface.

Results and discussion

Partial Total Ion Chromatograms (TIC) of the F3+F4 fraction obtained from Lake le Bourget sediments, P. miliaceum seeds and bran extracts and pure miliacin extracted from Chionochloa sp. are displayed in Figure 2. All TIC traces show relatively pure miliacin that elutes at 40.75 min in our temperature programme. The absence of any other fragment than those typical for miliacin mass spectra in the peaks eluting at 40.75 min with our temperature programme allowed us to exclude any significant coelution in all analyses (Figure 3).

Concentrations in miliacin of the different samples are reported in Table 1. P. miliaceum seeds and bran contain 0.3 and 0.16 mg.g -1 (plant dry mass) of miliacin, respectively. The average concentration of miliacin in the selected samples from Lake le Bourget sediments reaches 0.49 µg.g -1 sediment.

The first record of miliacin in the literature was provided by [START_REF] Itô | The chemical investigation of some gramineae oils[END_REF] in a paper dealing with the chemical composition of some gramineae oils. This author found miliacin in P. miliaceum and Syntherisma sanguinalis (hairy crabgrass). Miliacin was also found in several species of Chionochloa, sometimes associated with other PTMEs such as arundoin, lupeol methyl ether, β-amyrin methyl ether, parkeol and cycloartenol methyl ethers [START_REF] Russell | Triterpene methyl ethers of Chinochloa (Gramineae)[END_REF][START_REF] Connor | Triterpene methyl ether differentiation in Chionochloa (Gramineae)[END_REF][START_REF] Connor | Flora of New Zealand -Gramineae. Supplement I: Danthonioideae[END_REF]. Miliacin is the sole PTME detected in Glyceria acutiflora. It also occurs in Eragrostis ferruginea where it is associated with βamyrin methyl ether (see [START_REF] Ohmoto | Triterpenoids of the Gramineae[END_REF]. As a conclusion, miliacin is found principally in gramineae with either a C3 or C4 metabolism.

Offline and online measurements of purified miliacin extracted from Chionochloa sp. are in rather good agreement (-31.2 and -33 ‰, respectively) and are consistent with a C3 metabolism. The difference between these two values might arise from a compound present in low amounts in the "purified" extract.

Miliacin extracted from P. miliaceum seeds and bran have δ 13 C values of -23.5 ‰ and -23 ‰, respectively, i.e. typical for a lipid derived from C4 plants. The δ 13 C of miliacin extracted from Lake le Bourget sediments (-21.5 ‰) is close to that of miliacin extracted from P. miliaceum, as opposed to the δ 13 C of miliacin extracted from Chionochloa sp. (-33 ‰).

This indicates that miliacin detected in the Lake le Bourget sediments was produced by a C4 plant. Since most of plants growing naturally in the Alps are C3 plants, miliacin extracted from Lake le Bourget sediments was probably produced by a C4 producer plant foreign to the study area, such as P. miliaceum. This is a further support for miliacin as a tracer specifically for P. miliaceum around Lake le Bourget.

These results not only provide further evidence on the origin of sedimentary miliacin but also allow the formulation of new assumptions on the transportation of this molecule from the plant to the sediment. We previously assumed that miliacin is present in plant epicuticular waxes [START_REF] Jacob | Millet cultivation history in the French Alps as evidenced by a sedimentary molecule[END_REF], as is the case for other pentacyclic triterpenes. High concentrations of miliacin in P. miliaceum bran suggest that this part of the plant could also be an important contributor of sedimentary miliacin, for example by wind transport during winnowing.

Conclusion

The carbon isotopic composition of miliacin detected in the sediments of Lake le Bourget provides further evidence that P. miliaceum, a C4 plant, was the biological source of this molecule in this context. This result strengthens the confidence with which miliacin can be used to unravel the history of millet cultivation in sedimentary archives. In addition, the unique source of miliacin makes this biomarker a suitable target to unravel the evolution of environmental parameters from its carbon and hydrogen isotopic compositions. In a larger way, these findings also illustrate the possibilities of the molecular biomarker approach to better understand past interactions between climate, environment and human societies.

Figure captions

Figure 1 :

 1 Figure 1: Structure of miliacin (olean-18-en-3βol methyl ether or germanicol methyl ether).

Figure 2 :

 2 Figure 2: Partial GC-MS TIC traces of: (i) Chionochloa sp. purified miliacin; (ii) F3+F4

Figure 3 :

 3 Figure 3: Mass spectra of (a) miliacin extracted from Lake le Bourget sediments, (b) miliacin

Table captions Table 1 :

 captions1 Concentrations and δ 13 C values of miliacin extracted from P. miliaceum seeds and bran, Chionochloa sp. and the sediments of Lake le Bourget. Uncertainties are standard errors of the mean for 2 or 3 measurements. Where no uncertainty is given, values represent a single measurement.

	Sample	Weight extracted Miliacin concentration	Miliacin δ 13 C (‰ VPDB) EA-IRMS GC-IRMS
	P. miliaceum seeds	4.3 g	0.3 mg/g		-23.5±0.3
	P. miliaceum bran	2.4 g	0.16 mg/g		-23.0±0.3
	LDB04 sediment samples	22 g	0.49 µg/g		-21.5±0.3
	Chionochloa sp.	Purified compound	-31.2	-33.0±0.3

Acknowledgments

This work is part of the APHRODYTE project, supported by the French CNRS (Programme ECLIPSE 2005-2006). E. Gaillard from HITEX is kindly acknowledged for providing bran and seeds of P. miliaceum. Nicolas Péchot is acknowledged for performing EA-IRMS measurements. The authors wish to thank Pr. H.E. Connor (Christchurch University, New Zealand) and Pr. R. Smith (Loughborough University, United Kingdom) for providing purified miliacin.