Patrick Henry
email: henry@labri.fr

Christian Bassac
email: christian.bassac@inria.fr

A toolkit for a Generative Lexicon

In this paper we describe the conception of a software toolkit designed for the construction, maintenance and collaborative use of a Generative Lexicon. In order to ease its portability and spreading use, this tool was built with free and open source products. We eventually tested the toolkit and showed it filters the adequate form of anaphoric reference to the modifier in endocentric compounds.

Introduction

Over the past ten years or so, the Generative Lexicon (from now on GL: [START_REF] Pustejovsky | The Generative Lexicon[END_REF]), has been used, among other things, as a new tool in lexical semantics. The representation language it provides has been used to investigate various linguistic phenomena, in various languages (see for instance Proceedings of The first/second/third International Workshop on Generative Approaches to the Lexicon 2001Lexicon /2003Lexicon /2005) and an increasingly unified and powerful formalism emerged (Asher and Pustejovsky forthcoming). A GL can also be considered as a promising theory which can be applied to build NLP tools (Gupta and Aha 2003), for instance with a view to validating linguistic rules on large corpora. The aim of this paper is to present an example of such a tool: basically it is a toolkit which will be used to manipulate a GL for further applications. Our presentation will go into some detail about the data structures and the software that we chose in order to build, maintain and use a GL collaboratively.

The software: objectives and constraints

First we set ourselves three constraints:

• We think that the task of building resources should be mutualized, due to the amount of work that must be put in.

Although works on the automatic acquiring of qualia structure do exist, (see for instance [START_REF] Sébillot | Apprentissage sur corpus de relations lexicales sémantiques -La linguistique et l'apprentissage au service d'applications du traitement automatique des langues[END_REF], we think that the final adjustments must be hand-made. So, right from the outset we deliberately chose a system that allows collaborative work.

• It also seems to us that this toolkit should be available and used on the usual platforms that cover most current standards, WINDOWS, UNIX/LINUX, and MAC/OS. • The kit must also be as user-friendly as possible and the building of a lexicon must be both intuitive and interactive. We also wish that the user can visualize the precise contents of the resources, consequently we think that building and maintenance should be carried out with graphic interfaces.

The tools used

Taking into account the constraints previously described, we chose the following tools:

LDAP[http://www.opendap.org/]: This protocol directly derived from X500 was initially conceived to build electronic directories. It can be observed that there are similarities between a directory and a lexicon, for instance both allow access to information from an entry. This protocol seemed to be the right thing to us as it has the following functionalities:

• It can be mutualized • Data structures can be parametrized • Its consultation is optimized for speed access • It can be used in network • Toolkits are ready to use and free • Its use is cross-platform PYTHON [http://www.python.org/]: The choice of the language is of paramount importance: in this type of application the specifications keep evolving as work goes on, and consequently an easy-handling device is needed, which also eases further work on the software. So we decided to use an interpreter. Among the functionalities we need, object-oriented programming is crucial as it allows the modification of some data structures without having to go back to the code already written. Among the functionalities that we deem relevant with PYTHON, the following are of the utmost interest:

• The interpreter is among the most efficient to-date (semi-compiled) • Object-oriented programming • The WXWINDOW graphic toolkit is available (WXPYTHON). • The LDAP toolkit is available • The language handles lists • The language has lambda-expressions JXPLORER [http://www.jxplorer.org/]: It is a simple LDAP browser-editor written in JAVA that can be used on any platform. This is the tool we used in the construction and maintenance of the lexicon. Figure 1 below shows an example of the editing of a lexical entry: the item is chosen in the left hand column, and the details of the entry are visualized in the central part, the nomenclature being inspired by the notational conventions of GL. The terminology somewhat differs from classical GL notation on three points: first the various qualia roles are not grouped together, second the key-words TRIGGER and RESULTAT (result) are used respectively for the AGENTIVE and FORMAL roles embedded in the TELIC found in various GL works, and third some variables such as OBJECTCLASS are inherited from the LDAP data structure and are not directly connected with GL. A partial view of the construction of a lexical entry is given below in figure 1. It is fairly easy to add a value by a right click on the type of the value to add. We chose to show the example of the construction and editing of this particular item because it is part of the lexicon in the theoretical study by Bassac and Bouillon (2005, 27) we later used to test our toolkit 1 . The 1 [START_REF] Bassac | Qualia structure and anaphoric reference in compounds in[END_REF] examined the data of French, English and Turkish but the validity tests here are carried out on French data only. The entry chosen final view of a lexicon entry is given in figure 2 below.

as an example here is the word pressoir [press] N (like in "cider press", see 5 below).

Figure2: View of a lexicon entry with JXPLORER Although, the tests were carried out on a limited number of lexicon entries, the extension to large scale applications should be easy as LDAP has already been optimized for quick access to data.

Functionalities available

The programs allow the following operations :

• On the LDAP server (DSA): Centralization and acceptance of requests for consultation or modification of the lexicon. The software can be configured so as to include accesscontrol. • The JXPLORER (DUA) client allows connexion to the lexicon in order to create, modify or search an entry in the lexicon. Other similar projects do exist, whose aims are the constructions of lexicons. One of these projects is the ISLE project [START_REF] Bel | Lexical Entry Tool (software) and Accompanying Documentation[END_REF]) that terminated in 2003. It is an interesting and relevant project in so far as it led to the development of tools used in the implementation of lexicons, which is our objective too. Yet, the final goals of the ISLE project were slightly different: the lexicons were multilingual, and the information content was not uniform and depended on the entry, mainly in order to perform disambiguation tasks. The whole system is a kind of meta-entry in which lexical information is not distributed according to the principles of a GL, but is layered with a view to easing the process of automatic translation. Our toolkit, here presented, was specially designed for the development of applications that conform fairly strictly to the principles of GL. More specifically, we concentrated on these aspects of lexical semantics that allow coherence checking operations and validation of linguistic hypotheses. The implementation we chose also departed from the ISLE project as it was built on data bases which can be accessed via PERL CGI scripts and a WEB browser. Our approach is more orientated towards the implementation of NLP applications such as coherence checking or generation. This is why our toolkit heavily relies on the choice of LDAP and PYTHON.

An application

We chose to test our scale model on the paper by [START_REF] Bassac | Qualia structure and anaphoric reference in compounds in[END_REF]: the reasons for our choice are that first in this paper the authors offer an explanatory account of an interesting syntactic phenomenon, and second the range of vocabulary items used is both varied and its number can be easily handled (seventy or so).

The authors show that anaphoric reference to the modifier of a compound is qualia-driven and depends both on the type of the relation R that holds between the modifier N 1 and the head N 2 , and the role it is encoded in:

• If R is a contain relation, then anaphoric reference to N 1 is possible via a definite determiner NP (DeFDetNP) but not via a possessive determiner NP (PossDetNP) nor via a demonstrative determiner NP (DemDetNP). The authors show that this accounts for the following facts:

• If R is a part_of relation,
(1) Le verre a vin est cassé, * son vin coule. The wine glass is broken, *its wine is leaking out of the crack.

(2) Max examina ses patins à roulettes : leurs roulettes étaient tout usées.

Max took a look at his roller skates: Their rollers were worn-out.

(3) Ce pressoir est défectueux, *ses olives restent entières. This olive press does not work properly, *its olives are not properly ground.

(4) Nous utilisons un nouveau pressoir, son cidre est excellent.

We use a new cider press, its cider is great.

In our implementation we also built and used a partial type hierarchy in order to control the unification of the types of variables that appear in the qualia structure.

The facts exemplified in (1) to ( 4) above are validated by our system as shown below for example (3) and (4):

Assessment and future work

As we have shown, this software is operational and has already been used in the construction of an NLP application. It can be readily used in the building and maintenance of a GL. One of the limits of the system is the size of the lexicon and the obligation to encode the lexicon information manually. Consequently the next steps will lead us to code wider scale mutualizable resources. For instance, we plan to use corpora or systems such as WordNet and its ontology to automatically acquire the relevant information necessary to build qualia structures (see for instance [START_REF] Sébillot | Apprentissage sur corpus de relations lexicales sémantiques -La linguistique et l'apprentissage au service d'applications du traitement automatique des langues[END_REF]. This task is fairly difficult, but it can be eased by the use of morphological indications contained in a corpus (see Namer, Bouillon, Jacquey, this volume) or by syntactic clues: for instance a sequence N 1 de N 2 in French in which N 1 is a container and N 2 is a liquid is an example of dot introduction.

The ergonomy of the system can be improved so as to make it more user-friendly. In the near future JXplorer which was used in the first steps of the development of the system should be given up for a tool which will be easier to use, so as to show only the relevant information. For large scale use we have planned to develop interfaces that allow the definition of rules rather than algorithms as well as checking procedures for the coherence of lexical entries.

Fig 1 :

 1 Fig 1: Partial view of the construction of a lexicon entry.

•o

  The entree.py (DUA) client is a python module (library) which allows the following operations : Connexion of the program to the lexicon, possibly with authentication o Saving and restoring the lexicon in LDIF or XML format o Searching a word in the lexicon o Retrieval of the features of an entry. o Retrieval of a feature value o Pretty printing of a lexicon entryThe architecture of the whole system is summed up and schematized in figure3below:

Figure 3 :

 3 Figure 3: The client-server architecture of the lexicon

  Figure 4: head/modifier relations and anaphoric reference to the modifier

Figure 5 :

 5 Figure 5: Execution on a computer


Virginie Hugouvieux

Walter Kob

Cluster Phase in enzyme-induced Gelation of Polymers

Important examples for such systems occur in plant cell walls where some polysaccharides (pectins) undergo gelation due to the presence of an enzyme (pectin methylesterase) [1].

We have developed a simple model (bead-spring polymers+soft particles) that can serve to describe such systems. Using large scale computer simulations we study how the polymer sol becomes unstable due to the increasing fraction of attractive monomers and transforms at high temperatures into a dense liquid (i.e. phase separation) or into a gel at low temperatures. We find that before the system reaches at long times one of these two phases, it forms at intermediate times a rather regular cluster phase. We rationalize this finding by a competition between two mechanisms: Nucleation of the attractive particles to form clusters and the slow relaxation dynamics of the coarsening of the polymer sol. We show that the temperature dependence of the time scale of these two processes can be understood semi-quantitatively from the bare potential between the constituent particles, i.e., monomers and catalysts, which governs the reactions. Finally we discuss how this cluster phase can be stabilized, thus giving rise to a gel with a non-trivial internal structure. [1] D.J. Cosgrove, Nat. Rev. Mol. Cell. Bio. 6, 850 (2005).

4

  th International Soft Matter Conference, 12-16 September 2016, Grenoble, France Cluster Phase in enzyme-induced Gelation of Polymers Virginie Hugouvieux a and Walter Kob b a SPO, INRA, Montpellier SupAgro, Université Montpellier, 34060 Montpellier, France b Laboratoire Charles Coulomb, CNRS, Université Montpellier, 34095 Montpellier, France * Corresponding author: virginie.hugouvieux@supagro.inra.fr Certain biopolymers transform into a gel because of a catalytic reaction in which freely moving catalysts, called enzymes, convert repulsive monomers of the polymers into attractive ones. These attractive monomers stick to each other and make that the polymer sol slowly transforms into a gel.

Figure :

 : Figure: Typical evolution of the system from (a) a homogeneous solution of enzymes (green) and repulsive (blue) homopolymers, to (b) a gel containing clusters of attractive (red) monomers (enzymes not shown), and (c) a macroscopic phase separation.


Carlos Peregrina

come

Process for treating effluents loaded with ammoniacal

nitrogen and reactor for the implementation of the process


Lucas Brivadis

Vincent Andrieu

Élodie Chabanon

Émilie Gagnière

Noureddine Lebaz

Ulysse Serres

New dynamical observer for a batch crystallization process based on solute concentration

Keywords: Observers, Crystallization, Crystal Size Distribution, Tikhonov regularization, Solute concentration

In this paper a new observer is introduced to estimate the Crystal Size Distribution (CSD) only from the measurements of the solute concentration, temperature and a model of the growth rate. No model of the nucleation rate is needed. This approach is based on the use of a Kazantzis-Kravaris/Luenberger observer which exponentially estimates functionals of the CSD. Then, the full state is estimated by means of a Tikhonov regularization procedure. Numerical simulations are provided. Our approach relies on an infinite-dimensional observer, contrarily to the usual moment based observers.

Introduction

Crystallization is one of the oldest and major processes used in industry (chemical, pharmaceutical, food, etc.) to produce, purify or separate solid compounds or products [START_REF] Biscans | Cristallisation en solution -Procédés et types d'appareils[END_REF]. This unit operation aims to produce solid crystals with well defined specifications including (among others) the Crytal Size Distribution (CSD) which is of critical importance. At the industrial scale, the CSD is neither well controlled nor monitored during the crystallization process and a grinding step is usually performed before delivering the final product. Hence, a key point is the real-time control and the "online" monitoring of the CSD during the crystallization operation in order to avoid the grinding step. Unfortunately, the CSD is nowadays not directly measurable in real-time by existing sensors.

Nevertheless, the Process Analytical Technologies (PATs) allow us to get access to real time information such as the solute concentration based on the Attenuated Total Reflectance Fourier Transform InfraRed spectroscopy (ATR-FTIR) and the Chord Length Distribution based on the Focused Beam Reflectance Measurement (FBRM ® ).

FBRM ® is commonly employed in pharmaceutical industries to detect online some process deviations. Note however that its use for online estimation of the CSD remains challenging [START_REF] Poblete | On-line monitoring of chord distributions in liquid-liquid dispersions and suspension polymerizations by using the focused beam reflectance measurement technique[END_REF][START_REF] Scheler | Ray tracing as a supportive tool for interpretation of fbrm signals from spherical particles[END_REF].

Obtaining an online CSD estimation from the solute concentration, temperature and a growth rate model only is an interesting problem which is the purpose of the present paper. To obtain online state estimation for dynamical systems from measured outputs, control engineers usually employ asymptotic state observers. Designing state observer for complex dynamical systems is an active research area. Some studies are devoted to nonlinear dynamics (see [START_REF] Bernard | Observer Design for Nonlinear Systems[END_REF] and references therein). Others consider infinite dimensional systems (see for instance [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF]).

Designing observers for a batch crystallization process has been addressed in recent years by several researchers (see for instance [START_REF] Mesbah | Real-time control of a semi-industrial fed-batch evaporative crystallizer using different direct optimization strategies[END_REF][START_REF] Mesbah | A comparison of nonlinear observers for output feedback model-based control of seeded batch crystallization processes[END_REF][START_REF] Motz | State estimation in batch crystallization using reduced population models[END_REF][START_REF] Nagy | Robust nonlinear model predictive control of batch processes[END_REF][START_REF] Porru | Monitoring of batch industrial crystallization with growth, nucleation, and agglomeration. part 2: Structure design for state estimation with secondary measurements[END_REF][START_REF] Uccheddu | Design of a csd observer during batch cooling crystallization dealing with uncertain nucleation parameters[END_REF][START_REF] Vissers | Model-based estimation and control methods for batch cooling crystallizers[END_REF]). These studies are based on spatial discretization of the PDE (see [START_REF] Porru | Monitoring of batch industrial crystallization with growth, nucleation, and agglomeration. part 2: Structure design for state estimation with secondary measurements[END_REF] or [START_REF] Vissers | Model-based estimation and control methods for batch cooling crystallizers[END_REF]) or more frequently on moments analysis (see survey [START_REF] Mesbah | A comparison of nonlinear observers for output feedback model-based control of seeded batch crystallization processes[END_REF]). However, the moment based approaches suffer from several drawbacks: (i) the numerical moment values exhibit a very large difference in their order of magnitude [START_REF] John | Techniques for the reconstruction of a distribution from a finite number of its moments[END_REF]. Consequently, a small numerical/experimental error has a significant impact on the moments estimation. The error will then be propagated during the computation of the moment transport equations [START_REF] Dorao | Numerical calculation of the moments of the population balance equation[END_REF]; (ii) the recovering of the CSD from a finite number of its moments is still an open area of research in mathematics and highly dependant of the moments quality [START_REF] Lebaz | Reconstruction of a distribution from a finite number and its moments : A comparitive study in the case of depolymerization process[END_REF][START_REF] Omar | Crystal Population Balance Formulation and Solution Methods : A Review[END_REF]. Furthermore, the moment based observers depend on the knowledge of the nucleation rate which may be tricky to model. All these reasons may explain the difficulty to develop an efficient algorithm following this route.

In this paper, another approach is adopted to describe the CSD without using its moments. Moreover, a key point of the proposed approach is that no information is needed on the nucleation rate.

In the first part of the paper, we recall the Kazantzis-Kravaris/Luenberger methodology for observer design which has been introduced in its original form in [START_REF] Luenberger | Observing the state of a linear system[END_REF] for linear systems and adapted for nonlinear dynamics in [START_REF] Andrieu | On the existence of a kazantzis-kravaris/luenberger observer[END_REF] and [START_REF] Kazantzis | Nonlinear observer design using lyapunov's auxiliary theorem[END_REF]. Section 3 is devoted to the modeling of the batch crystallization process. Finally some simulation results are presented in Section 4 which highlight the practical interest of the suggested methodology.

Kazantzis-Kravaris/Luenberger observer methodology

In control engineering, the algorithm which is employed to reconstruct online missing data on a partially measured dynamical process is named an observer. Observer design for nonlinear dynamical systems is a very active research area and has been the subject of numerous studies in the past 40 years (see [START_REF] Bernard | Observer Design for Nonlinear Systems[END_REF] for a recent survey on this topic).

One very efficient way to design observers for finite-dimensional linear as well as nonlinear dynamical systems is the Kazantzis-Karavis/Lunberger (KKL) observer methodology. It is an approach which follows Luenberger original idea in [START_REF] Luenberger | Observing the state of a linear system[END_REF] where asymptotic observers for linear systems were introduced for the first time. The approach of [START_REF] Luenberger | Observing the state of a linear system[END_REF] is somehow different from the way Luenberger observers are introduced nowadays (which follows Luenberger second paper on observers [START_REF] Luenberger | An introduction to observers[END_REF]). It has been recently employed by Kazantzis and Kravaris in [START_REF] Kazantzis | Nonlinear observer design using lyapunov's auxiliary theorem[END_REF] to design local observers for finite-dimensional nonlinear dynamics based on Luenberger original idea in a local version. We hope it may also be fruitful in the context of distributed parameter systems. In this section we adapt to the context of linear abstract Cauchy problem the KKL observer methodology which is a two-steps design procedure. In the first step we consider the problem of estimating a function of the state. The reconstruction of the overall state of the system is obtained in the second step by inverting this function.

Step 1: reconstruction of a function of the state

As shown in [START_REF] Andrieu | On the existence of a kazantzis-kravaris/luenberger observer[END_REF] and [START_REF] Bernard | Luenberger observers for nonautonomous nonlinear systems[END_REF], it is always possible to exponentially estimate a function of the state, even for nonlinear finite-dimensional systems, that will carry enough information about the state to estimate it in Step 2. In order to do so, it is sufficient to introduce an auxiliary dynamical system fed by the measured output such that its solutions provide an estimation of this function of the state.

Let R (resp. R + , R -and R * ) denotes the set of real numbers (resp. non-negative, non-positive and non-zero real numbers). Let X be a Hilbert space. We denote L(X ) the set of all endomorphisms of X and L(X , R) the space of linear forms from X to R. Consider the abstract Cauchy problem on X

ψ = Fψ, ψ(0) = ψ 0 , (1)
where F : D(F) ⊂ X → X is a linear operator which is the generator of a strongly continuous semi-group denoted

(T t ) t∈R+ in L(X ) and ψ 0 ∈ D(F). Let ρ(F) = {λ ∈ C | (F -λI) -1 ∈ L(X )} denote the resolvent set of F.
Moreover, consider a bounded output operator

y = Hψ, (2)
where H ∈ L(X , R) is bounded. Following the KKL methodology, we obtain the following proposition.

Proposition 2.1. For all λ in ρ(F) ∩ R * -, let T λ in L(X , R) be the operator defined as

T λ : X ψ → H(F -λI) -1 ψ ∈ R .
Then, the dynamical system

żλ = λz λ + y, (3)
is an exponential observer for T λ ψ. More precisely, for all (ψ 0 , z 0 ) in D(F) × R, it yields for all t 0

T λ (T t ψ 0 ) -z λ (t) = exp(λt) (T λ (ψ 0 ) -z 0 ) . ( 4
)
where z λ : R + → R is the solution of system (3) when y is given by (2) and initiated from z 0 .

Proof. Let ψ 0 be in D(F). Equations ( 1)-(3) yield

d dt T λ (T t ψ 0 ) -z λ (t) = T λ (FT t ψ 0 ) -λz λ (t) -HT t ψ 0 = T λ (F -λI)T t ψ 0 + λ T λ (T t ψ 0 ) -z λ (t) -HT t ψ 0 = λ(T λ (T t ψ 0 ) -z λ (t)),
where the last equality follows since T λ (F -λI) = H. Hence, (4) follows by integrating in time the former equation.

Keeping in mind that λ is negative in Proposition 2.1, (4) implies

lim t→+∞ |T λ (T t ψ 0 ) -z λ (t)| = 0. ( 5
)
This ends the proof.

Remark 1. The operator T λ is solution to the Sylvester equation :

FT λ = λT λ + H. ( 6
)
We recognize here the algebraic equation which was already given in Luenberger seminal paper [START_REF] Luenberger | Observing the state of a linear system[END_REF] and which becomes a nonlinear partial differential equation in [START_REF] Andrieu | On the existence of a kazantzis-kravaris/luenberger observer[END_REF].

2.2

Step 2: reconstruction of the entire state of the system According to step 1, we can easily estimate T λ ψ for all λ in ρ(F) ∩ R * -via the observer system (3). The idea of the KKL observer methodology is to consider the mapping T : X → R p given by ψ → (T λ1 ψ, . . . , T λp ψ) which will be exponentially estimated along the trajectory of (1) via a bench of observers of the form (3). To solve the estimation problem, the question is to solve the inverse problem

T ψ = z (7)
with the unknown ψ in X . Let ImT = {T x | x ∈ X } be the image of T . Then (7) admits a solution only if z is in ImT . The former condition is in general too restrictive due to the fact that z is only an estimation of T ψ and consequently may not be in ImT . A solution to overcome this problem is to replace the equality constraint [START_REF] Dorao | Numerical calculation of the moments of the population balance equation[END_REF] by

the minimization problem minimize T ψ -z 2 subject to ψ ∈ X . (8)
The set of solutions of ( 8) is denoted by argmin ψ∈X T ψ -z 2 . The following results can be found in [START_REF] Kern | Méthodes numériques pour les problèmes inverses[END_REF]Chapter 4]:

• If T is injective, then (8)
has at most one solution.

• If z ∈ ImT ⊕ (ImT ) ⊥ , then the set argmin ψ∈X T ψ -z 2 is closed, convex and non-empty (in particular [START_REF] Haine | Recovering the observable part of the initial data of an infinite-dimensional linear system with skew-adjoint generator[END_REF] admits at least one solution).

• If T is bijective and admits a left inverse denoted T -1 then the unique solution of (8) is ψ(t) = T -1 z(t).

For finite dimensional systems, the injectivity of T is directly linked with an observability property of the dynamical system (1). Indeed, following [START_REF] Luenberger | Observing the state of a linear system[END_REF], it can be shown that for finite dimensional systems, if the pair (F, H) is observable (in the sense that the Kalman observability matrix is full rank), then picking p pairwise distinct λ i 's with p = dim X , then the obtained mapping T is invertible. In that case, a KKL observer can simply be obtained

as ż =    λ 1 . . . λ p    z +    1 . . . 1    y , ψ(t) = T -1 z(t).
In the infinite dimensional case, there is no hope that a finite number of λ i 's can give all the information allowing to reconstruct the entire system's state.

It is an open question to know if observability properties of the infinite dimensional system allows to obtain that a certain countable set of λ i 's gives enough information. In that case, the minimization problem (8) becomes ill-posed. In other words, it may have no solution, or numerous solutions, and its solutions may depend on z in a non-continuous manner. In our case, z is an estimation of T ψ. Hence, the minimizer ψ of ( 8) may be very different of the real state ψ. A typical approach to overcome this problem is to consider a Tikhonov regularization method in which the optimization problem ( 8) is slightly modified. This new minimization problem will have solutions close to the former one, and be well-posed. We recall the next proposition on which this method is based for the convenience of the reader.

Proposition 2.2 ([12], Proposition 6.1). Given T in L(X , R p ) and δ > 0, the minimization problem

minimize T ψ -z 2 + δ ψ 2 subject to ψ ∈ X (9)
admits a unique solution and is a continuous function of z in R p .

The choice of the regularization parameter δ is of the uttermost importance. Indeed, as δ → +∞, the solution of ( 9) goes to zero. Conversely, as δ → 0, the problem (9) goes closer to the ill-posed problem [START_REF] Haine | Recovering the observable part of the initial data of an infinite-dimensional linear system with skew-adjoint generator[END_REF]. A compromise shall be made. In practice, δ is chosen experimentally, and is linked to the measure confidence: the more uncertain is the output y, the bigger is δ.

We also have the following theoretical result, that describes what happens when δ goes to zero.

Theorem 2.3 ([12, Theorem 6.1]). Let T in L(X , R p ) and z ∈ ImT . Let ψ 0 ∈ X and ψ the solution of (8) closest to ψ 0 . Let (z n ) n∈N be a sequence in R p converging to z. Let ε n = |z n -z|. Let (δ n ) n∈N be a sequence of regularization
parameters converging to zero. For any n ∈ N, let ψ n be the solution of the problem (9) associated to z n and δ n .

Then,

• |T ψ n -z n | -→ n→+∞ 0; • if ε n δ n -→ n→+∞ 0, then |T ψ n -z n | = O(ε n ) and ψ n -→ n→+∞ ψ; • if ε n δ 2 n -→ n→+∞ 0 and ψ ∈ (ImT ) * , then |T ψ n -z n | = O(ε 2 n ) and |ψ n -ψ| = O(ε n ).
To summarize, a possible observer design of a given abstract Cauchy problem in the form (1) is then given by

                 ż =    λ 1 . . . λ p    z +    1 . . . 1    y ψ(t) = argmin ψ∈X T ψ -z(t) 2 + δ ψ 2 , δ > 0 T = T λ1 , . . . , T λp , T λi = H(F -λ i I) -1 , i = 1, . . . , p, (10)
where the λ i are pairwise distinct elements of ρ(F) ∩ R * -. Note that despite the fact that this dynamical system is well-defined for all abstract Cauchy problem in the form [START_REF] Andrieu | On the existence of a kazantzis-kravaris/luenberger observer[END_REF], its convergence to the real state is not guaranteed a priori and may be linked to observability properties.

3 Application to the crystallization process

Modeling the batch crystallization process

In this subsection, a dynamical model representing the batch crystallization process and the concentration measurement is given.

Population balance equation

In a first step a batch crystallization process is modeled. We assume that the size of the crystals is described by a scalar parameter x (in m). For example, if the crystals are spherical, then x may represent their diameter.

Let us denote ψ(t, •) the Number Density Function in terms of the crystal size (NDF) at time t (in m -1 .m -3 ), so that x2 x1 ψ(t, x)dx is the total amount of crystals in the reactor at time t with size between x 1 and x 2 (in m -3 ). Let [t 0 , t 1 ] be the time window in which the crystallization process occurs.

We assume that the crystals never reach a specific maximal allowable size denoted x max during the experiment.

Typically this size can simply be related to the size of the reactor in which the crystallization occurs:

∀t ∈ [t 0 , t 1 ], ψ(t, x max ) = 0. ( 11
)
We assume that all the crystals appear at the same positive size x min , and we denote u(t) the appearance rate of new crystals at size x min at time t. The function u quantifies the nucleation rate in the reactor. In [START_REF] Nagy | Recent advances in the monitoring, modelling and control of crystallization systems[END_REF], for instance, an expression for the function u is given as

u(t) = R n (t)/G(t)
, where R n is the overall rate of nucleation expressed in s -1 .m -3 and G is the growth rate of the crystals in m.s -1 . Note however that in our approach for NDF observation, we don't need to know precisely this expression. We do not use any model of u, and assume this quantity to be unknown. We have

∀t ∈ [t 0 , t 1 ], ψ(t, x min ) = u(t). (12)
Finally, let G(t, x) 0 be the growth rate of the crystals in m.s -1 , in other words the rate at which a crystal of size x grows at time t. The population balance leads to ∀t ∈ (t 0 , t 1 ), ∀x ∈ (x min , x max ), ∂ t ψ(t, x)

+ ∂ x (Gψ)(t, x) = 0. ( 13
)
By considering the McCabe assumption, we assume that G does not depend on x. In that case equation [START_REF] Lebaz | Reconstruction of a distribution from a finite number and its moments : A comparitive study in the case of depolymerization process[END_REF] becomes

∀t ∈ (t 0 , t 1 ), ∀x ∈ (x min , x max ), ∂ t ψ(t, x) + G(t)∂ x ψ(t, x) = 0. (14)
Equation ( 14) is a time-varying one-dimensional transport equation.

We also assume that at the beginning of the experiment, some seed particles are in the reactor. This yields

∀x ∈ [x min , x max ], ψ(t 0 , x) = ψ 0 (x).
To summarize, the model of the NDF in a batch crystallization process is:

     ∂ t ψ(t, x) = -G(t)∂ x ψ(t, x) ∀t ∈ (t 0 , t 1 ), ∀x ∈ (x min , x max ) ψ(t 0 , x) = ψ 0 (x) ∀x ∈ [x min , x max ] ψ(t, x min ) = u(t) ∀t ∈ [t 0 , t 1 ] ( 15
)
to which is added the other boundary condition in [START_REF] Kazantzis | Nonlinear observer design using lyapunov's auxiliary theorem[END_REF] which in our case is seen as a knowledge on the particular solution we wish to estimate.

The following theorem states that system [START_REF] Luenberger | An introduction to observers[END_REF] of this model admits weak solution in L 2 space and strong solution in

H 1 . Theorem 3.1. Let t 1 > t 0 0, x max > x min 0, ψ 0 ∈ H 1 (x min , x max ), u ∈ H 1 (t 0 , t 1 ), G ∈ C 0 ([t 0 , t 1 ]; R * + )
. Assume that u(t 0 ) = ψ 0 (x min ). Then system (15) admits a unique solution

ψ ∈ C 0 ([t 0 , t 1 ]; H 1 (x min , x max )) ∩ C 1 ([t 0 , t 1 ]; L 2 (x min , x max )).
Moreover, for all (t, x)

∈ [t 0 , t 1 ] × [x min , x max ], ψ(t, x) = ψ 0 (x -G(t)) if x -x min G(t) u • G -1 (G(t) -x + x min ) else. ( 16
)
where

G : [t 0 , t 1 ] t → t t0 G(τ )dτ .
The proof of this theorem can be found in [START_REF] Coron | Control and Nonlinearity. Mathematical surveys and monographs[END_REF]Theorem 2.4] in the case G = 1, and can be easily adapted by means of a time reparametrization. It is worth noticing that this theorem does not take into account hypothesis [START_REF] Kazantzis | Nonlinear observer design using lyapunov's auxiliary theorem[END_REF]. However, the following proposition holds.

Proposition 3.2. Assume that the hypothesis of Theorem 3.1 are satisfied. Assume that there exists x ∈ [x min , x max )

such that ψ 0 (x) = 0 for all x ∈ [x, x max ]. If x + G(t 1 ) < x max , (17)
then ψ(t, x) = 0 for all t ∈ [t 0 , t 1 ] and all x ∈ [x + G(t 1 ), x max ]. Proof. Let t ∈ [t 0 , t 1 ] and x ∈ [x + G(t 1 ), x max ]. Then
x -G(t 1 ) x x min .

Consequently, according to [START_REF] Mersmann | Crystal growth[END_REF], ψ(t, x) = ψ 0 (x -G(t)) = 0.

Hence, one must choose t 1 small enough so that the particles did not reach the size x max . Roughly speaking, this means that the observer that we are going to design must estimate the state in small time, i.e. before the particles reach the size x max . In the following, we always assume that (11) is satisfied.

Concentration sensor modeling

In the considered batch crystallization process, the measured outputs are the temperature and the solute concentration denoted C c (t). These two measurements allow to obtain online estimation of the growth rate (i.e. G) and the third moment of the NDF.

Estimation of G

The knowledge of the temperature and the solute concentration allows to obtain some approximation of the growth rate G . Indeed, following [START_REF] Uccheddu | Design of a csd observer during batch cooling crystallization dealing with uncertain nucleation parameters[END_REF], a model of G can be given for all time t ∈ [t 0 , t 1 ] by

G(t) = k g C c (t) -C * (t) C * (t) (18)
where

• k g is a known growth rate parameter (in m.s -1 ),

• C * (t) is the solubility at time t (in kg of solute per kg of solvent),

• C c (t) is the solute concentration at time t (in kg of solute per kg of solvent).

Since C * (t) depends on the temperature at time t , the growth rate G of the crystals can be estimated online with the available sensors.

Other model expressions of G are available in the literature, for more details one may refer to [START_REF] Mersmann | Crystal growth[END_REF][START_REF] Mullin | Crystallization[END_REF].

Estimation of the third moment of the NDF

It is possible to link the solute concentration with the NDF. Indeed, for each t ∈ R + , let C s (t) (in kg of solid per kg of solvent) be the solid concentration in the reactor at time t, in other words, the ratio between the total crystals mass in the reactor at time t and the solvent mass. Let ρ s (in kg.m -3 ) be the density of the solute in solid phase and M e the solvent mass (in kg). It yields:

C s (t) = ρ s M e V s (t)
where V s (t) is the volume (in m 3 ) occupied by the crystals at time t. Then the volume of a crystal with size x (in

m) is simply V = k v x 3
where k v is a volumetric shape factor (see e.g. [START_REF] Hulburt | Some problems in particle technology: A statistical mechanical formulation[END_REF][START_REF] Randolph | Theory of particulate processes[END_REF]). For example, k v = π/6 for spherical crystals. The total volume of the crystals is then

V s (t) = k v xmax xmin ψ(t, x)x 3 dx
Hence, the solid concentration in the reactor can be expressed as follows.

∀t

∈ [t 0 , t 1 ], C s (t) = ρ s k v M e xmax xmin ψ(t, x)x 3 dx. (19)
Assume moreover that ρ s is a known parameter. This implies that we can associate to system (15) the measurement y defined as

∀t ∈ [t 0 , t 1 ], y(t) = xmax xmin ψ(t, x)x 3 dx. ( 20
)
From there, the observation problem we intend to solve is the following. From the knowledge of the output function y(t) and the growth rate, give an online estimation of the NDF.

The purpose of the Section 3.4 is to propose a novel algorithm to solve this problem based on the KKL methodology which was described in Section 2. Note however that before implementing the observer an observability analysis can be carried out.

About the observability of the crystallization model

In this section, we study how the third moment may help us to estimate the NDF. First, we have the following result.

Proposition 3.3. Let τ ∈ (t 0 , t 1 ]. Assume that there exists µ > 0 such that G(t) µ for all t ∈ [t 0 , t 1 ]. Then for all y ∈ C 0 (t 0 , τ ), there exists at most one function u ∈ H 4 (t 0 , τ ) such that the solution ψ of (15) given by u and

ψ 0 = 0 satisfies y(t) = xmax xmin ψ(t, x)x 3 dx, ∀t ∈ [t 0 , τ ].
In other words, Proposition 3.3 states that the map u → y is injective, where y denotes the third moment of the solution of ( 15) with null initial condition. Its proof can be found in appendix. Hence, one can hope that our method may reconstruct ψ from y, at least when the initial condition is zero (i.e. there is no crystals at the beginning of the process).

However, one can wonder what happens if the initial condition is not zero. Can we still reconstruct the state from the measurement of its third moment and the knowledge of its dynamics? In other words, is the map ψ → y injective? If yes, then one can hope that our algorithm is robust, so that the estimation of the state converges to the actual NDF. Unfortunately, the answer is no. Indeed, we have the following proposition which is a slight modification of [30, Theorem 3.2.3] that we state in our own context only. For the convenience of the reader, its proof is given in appendix.

Proposition 3.4. Let τ ∈ (t 0 , t 1 ]. Assume that there exists µ > 0 such that G(t) µ for all t ∈ [t 0 , t 1 ]. There exist infinitely many solutions of (15) with different initial conditions and boundary conditions that have the same third moment y ∈ C 0 (t 0 , τ ).

We shall say that system [START_REF] Luenberger | An introduction to observers[END_REF] with measurement ( 20) is not observable. Thus, we cannot guarantee that our estimation of the NDF converges to the actual NDF. Despite this fact, our methodology should be able to reconstruct partially the actual NDF. Indeed, the linear function that maps the NDF to its third moment has rank 4. The image of this state-output mapping is sometimes called the observable part of the system (see e.g. [START_REF] Haine | Recovering the observable part of the initial data of an infinite-dimensional linear system with skew-adjoint generator[END_REF]), due to the fact that an observer shall estimate at least the projection of the actual state on this subspace. See the proof in appendix for more details.

A dynamical observer from the concentration measurement

Eventhough it was shown in the former section that the system is not observable, in this subsection, we show how the KKL observer approach can be employed on the considered model. Following the procedure given in Section 2, we consider λ a negative real number and the dynamical system

ż = λz + y . ( 21
)
We must find a mapping T λ which is estimated by this dynamical equation. We have the following proposition.

Proposition 3.5. Let T λ : C 1 ([t 0 , t 1 ]; L 2 (x min , x max )) → C 1 ([t 0 , t 1 ]
) be the functional defined as

T λ (ψ) : t → xmax xmin a(t, x)ψ(t, x)dx ( 22
)
where

     ∂ t a(t, x) + G(t)∂ x a(t, x) = λa(t, x) + x 3 ∀t ∈ (t 0 , t 1 ), ∀x ∈ (x min , x max ) a(t 0 , x) = 0 ∀x ∈ [x min , x max ] a(t, x min ) = 0 ∀t ∈ [t 0 , t 1 ], (23)
then, along the solution of (21) which satisfies [START_REF] Kazantzis | Nonlinear observer design using lyapunov's auxiliary theorem[END_REF] where y is given in [START_REF] Mullin | Crystallization[END_REF], it yields for all t in [t 0 , t 1 ]

T λ (ψ)(t) -z(t) = exp(λt)(T λ (ψ)(0) -z 0 ). ( 24
)
Proof. Using ( 15) and an integration by parts yields

d dt (T λ (ψ)(t) -z(t)) = xmax xmin ∂ t a(t, x)ψ(t, x)dx - xmax xmin G(t)a(t, x)∂ x ψ(t, x)dx -λz(t) - xmax xmin x 3 ψ(t, x)dx = xmax xmin ∂ t a(t, x)ψ(t, x)dx + xmax xmin G(t)∂ x a(t, x)ψ(t, x)dx -G(t) a(t, x)ψ(t, x) xmax xmin -λz(t) - xmax xmin x 3 ψ(t, x)dx.
Hence, with [START_REF] Omar | Crystal Population Balance Formulation and Solution Methods : A Review[END_REF] and also the boundary condition in ( 15) and ( 11), this implies

d dt (T λ (ψ)(t) -z(t)) = λ(T λ (ψ)(t) -z(t)).
By integrating in time the former equation, we obtain [START_REF] Poblete | On-line monitoring of chord distributions in liquid-liquid dispersions and suspension polymerizations by using the focused beam reflectance measurement technique[END_REF].

Consequently, for each negative λ we exponentially estimate the functional T λ ψ(t). It is interesting to remark that no information on the nucleation rate is needed to obtain this estimation. The state observer is given as [START_REF] John | Techniques for the reconstruction of a distribution from a finite number of its moments[END_REF], after a choice of the regularization parameter δ.

Numerical simulations

In this subsection numerical simulations are carried out. Let (x j ) 1 j Nx be a uniform discretization of the space interval (x min , x max ) with space step ∆x and (t k ) 1 k Nt be a uniform discretization of the time interval (t 0 , t 1 ) with time step ∆t. We fix N x = N t = 100. Let (λ j ) 1 i p be the considered negative values of λ. An approximation of (T λi ψ)(t k ) is given by ∆x Nx j=1 a i,j,k ψ j,k where a i,j,k is an approximation of a λi (t k , x j ) (solution of ( 23)) and ψ j,k an approximation of ψ(t k , x j ). The transport equation which describes the crystallization process is simulated via the method of characteristics. We consider the system [START_REF] Luenberger | An introduction to observers[END_REF] with G as in [START_REF] Mesbah | A comparison of nonlinear observers for output feedback model-based control of seeded batch crystallization processes[END_REF] with a null initial condition ψ 0 = 0 and a boundary condition similar to a truncated normal distribution reaching its maximum at t = 3 s and with a compact support [0, 6] (see Fig. 1a). The unique solution of this system is drawn in Fig. 1c (solid line), and the corresponding growth rate is drawn in Fig. 1b.

Step 1: reconstruction of a function of the state

Following the methodology developed in Section 2, we first try to estimate the function T λ (ψ) of the state via the dynamical system (21) for some fixed negative values of λ. All along the simulation of ( 15), we compute y and estimate the solution of ( 23) via the method of characteristics. We integrate the solution of ( 21) with the first order Euler's method. Then we plot the evolution of the relative error between z and T λ (ψ) in Fig. 2 for some values of λ. One can check that the error goes to zero as t → +∞. Moreover, the bigger is |λ|, the faster is the convergence. This is due to the exponential convergence of z -T λ (ψ) to zero given by (4). Hence, we are able to approximate any function T λ (ψ) of the state. Now, we can move to the second part of the methodology of Section 2.

4.2

Step 2: reconstruction of the entire state of the system

Following Step 1, we estimate simultaneously numerous functions T λi (ψ) which correspond to different values λ i < 0. These estimations are denoted z λi . The aim of this section is to estimate the state ψ from the knowledge of (z λi ) 1 i p .

Then, we choose a regularization parameter δ > 0 and solve the discrete version of the quadratic minimization problem (9) at each time step, that is for each time t k , find ψ j,k 1 j Nx that minimize ∆x a i,j,k

1 i p,1 j Nx • ψ j,k 1 j Nx -z λi (t k ) 1 i p 2 + δ ψ j,k 1 j Nx 2 . ( 25
)
This is a quadratic minimization problem, which we solve via an interior-point method (see e.g. [START_REF] Boyd | Convex Optimization[END_REF]Chapter III.11]. We need to fix an initial condition to apply this algorithm. Following a continuation method, we choose as an initial condition at time t k the minimum value obtained at time t k-1 , transported during a time ∆t at speed

G(t k-1 ).
The choice of parameters p, λ 1 , . . . , λ p and δ and their influence are investigated in the paragraphs below.

• Choice of the p and (λ i ). Note that the matrix a i,j,k 1 i p,1 j Nx may be injective only if p N x , that is if the discretization in λ is thinner than in x. Therefore, we fix p = 2N x = 200. Moreover, even if the matrix (a i,j,k ) i,j is injective, a regularization method is needed to left-inverse it. Indeed, for all t ∈ (t 0 , t 1 ), the operator

L 2 (x min , x max ) ψ → λ → xmax xmin a λ (t, x)ψ(x)dx ∈ L 2 (λ min , λ max )
is compact (as an integral operator). Hence, even if it is injective, its inverse is not continuous. The matrix (a i,j,k ) i,j is a discretization of this operator. Then, the more the discretization is thinner, the more it is ill-conditioned. This emphasizes the necessity of using a regularization method.

In Fig. 3, we plot the estimation of the NDF for different values of (λ i ). For large values of |λ|, z converges quickly to T ψ. However, it appears that functions a λ carry less information for large values of |λ|, so that the map T is more difficult to inverse. This explains Fig. 3b, on which the estimation ψ is worst than on Fig. 3c.

On the contrary, for small values of |λ|, it seems that functions a λ carry more information, since the estimation ψ is similar on Fig. 3a and Fig. 3c at t = 10 s. However, we also see a peaking phenomenon (for t 5 s on Fig. 3a), due to the fact that z is slower to converge to T ψ than for large values of |λ|.

Thus, one must find a compromise for the choice of (λ i ): take large values for fast convergence and avoiding peaking, and small values for efficient estimation.

• Choice of the regularization parameter δ.

The regularization parameter δ must be chosen numerically, in order to find a compromise between the minimization of the norm of the state, and the minimization of the gap T ψ -z. This compromise can be interpreted as a measurement reliability. Indeed, if the measurement has a small uncertainty, then we choose a small δ. On the contrary, if the measurement is highly uncertain, then we fix a large value of δ in order to regularize the solution.

In Fig. 4, we plot the actual NDF ψ and its estimation ψ at different times, for different values of δ, and with or without measurement noise. Measurement noise is fixed at 2% of the maximal value of the output on the time interval. For small values of δ and/or with measurement noise, we see that a peaking phenomenon appear: this is due to a lack of regularization of the solution. On the contrary, if δ is too large, then the minimization of the norm of the state takes too much importance in the minimization problem, and ψ is too attenuated.

Conclusion

In this paper a new observer has been introduced to estimate the Crystal Size Distribution from the measurements of the solute concentration, temperature and a model of the growth rate. No model of the nucleation rate is needed. This approach is based on the use of Kazantzis-Kravaris/Luenberger observer and a Tikhonov regularization procedure. The numerical results obtained are promising. Even though, the knowledge of the solute concentration alone does not allow to accurately reconstruct the full CSD as shown by our observability analysis, we believe that this tool could be used in addition with supplementary information given by other sensors (for instance FBRM ® ).

This approach will be evaluated on experiments in a future research project.

Appendix: proofs of Propositions 3.3 and 3.4.

Assume that G µ > 0. Let τ ∈ (t 0 , t 1 ] and u ∈ H 4 (t 0 , τ ). Let ψ be the solution of ( 15) with initial condition ψ 0 and boundary condition u. We introduce a time reparametrization t = t t0 G(s)ds, which is well defined since G µ. Let ψ, ũ and ỹ be such that ψ( t) = ψ(t), ũ( t) = u(t) and ỹ( t) = y(t) for all t ∈ [t 0 , t 1 ]. Then      ∂ t ψ( t, x) -∂ x ψ( t, x) ψ(t 0 , x) = ψ 0 (x) ψ( t, x min ) = ũ( t) [START_REF] Randolph | Theory of particulate processes[END_REF] and ỹ( t) = xmax xmin ψ( t, x)x 3 dx. Since the observability properties are not affected by the time reparametrization, one can investigate observability properties of the system (26) instead of [START_REF] Luenberger | An introduction to observers[END_REF]. Therefore, one can assume without loss of generality that G = 1 in the rest of the proof. Since u ∈ H 4 (t 0 , τ ), we have y ∈ C 4 (t 0 , τ ). Equation [START_REF] Kazantzis | Nonlinear observer design using lyapunov's auxiliary theorem[END_REF] = 6u + 6x min u + 3x 2 min u (2) + x 3 min u (3) .

End of the proof of Proposition 3.3. By hypothesis, ψ 0 = 0. Consequently, Equations ( 27)-( 29) yield      y (t 0 ) = x 3 min u(t 0 ) y (2) (t 0 ) = 3x 2 min u(t 0 ) + x 3 min u (t 0 ) y (3) (t 0 ) = 6x min u(t 0 ) + 3x 2 min u (t 0 ) + x 3 min u (2) (t 0 ), which is a triangular system with non vanishing diagonal since x min > 0. Hence u(t 0 ), u (t 0 ) and u (2) (t 0 ) are determined by y. Moreover, on [t 0 , τ ], u satisfies Equation [START_REF] Vissers | Model-based estimation and control methods for batch cooling crystallizers[END_REF] which is a 3rd order ordinary differential equation. Hence, according to the Cauchy-Lipschitz theorem, there exits a unique solution u to this problem. Thus y determines u uniquely, that u → y is injective.

End of the proof of Proposition 3.4. Substituting the boundary condition in equation [START_REF] Vissers | Model-based estimation and control methods for batch cooling crystallizers[END_REF] with u = 0 yields y (4) = 0 identically on [t 0 , τ ]. Hence y is a polynomial function of degree less or equal than 3. Thus the linear function that maps any solution of [START_REF] Luenberger | An introduction to observers[END_REF] with null boundary condition to its third moment has rank 4. Since ψ lies in an infinite dimensional vector space, we get by the rank-nullity theorem that its kernel is non-trivial, i.e. the state-output map ψ → y is not injective, and the system has a 4-dimensional observable part.

Note that Proposition 3.4 relies deeply on Hypothesis [START_REF] Kazantzis | Nonlinear observer design using lyapunov's auxiliary theorem[END_REF]. Hence the non-injectivity of the measurement is due to the fact that the system is observed on a too small time interval. If the system was observed on [t 0 , +∞), then one could show with similar argues an injectivity result.

  Simulation of the NDF ψ

Figure 1 :

 1 Figure1: Numerical simulation of the process with (x min , x max ) = (0, 10), (t 0 , t 1 ) = (0, 10) and N x = N t = 100.

Figure 2 :

 2 Figure 2: Convergence of T λ (ψ) -z to zero for different values of λ. We choose z(0) = 0 arbitrarily. The bigger is |λ|, the faster is the convergence. By means of a linear regression, one can estimate the convergence rate of the relative error to zero: O(e -7.4t ) if λ = -0, 1, O(e -8.2t ) if λ = -1, O(e -14.2t ) if λ = -10, O(e -32.9t ) if λ = -100.

  (λi) ⊂ [-100, -1]

Figure 3 :

 3 Figure 3: Influence of (λ i ) 1 i p on the reconstruction of the CSD.

2 (Figure 4 :

 24 Figure 4: Influence of the regularization parameter δ and measurement noise α on the reconstruction of the CSD

  (•, x)dx -x 3 ψ(•, x) , x)dx + 3 x 2 ψ(•, x) x)dx + 6x min u + 3x 2 min u + x 3 min u (2) ,(29)y (4) = -6 [ψ(•, x)] xmax xmin + 6x min u + 3x min u (2) (t) + x 3 min u(3)


stabilized by a lecithin containing lyso and anionic phospholipids, exhibited a major population of very mobile negatively charged oil droplets with average diameter around 240 nm together with small particles around 40 nm. It exhibited a good physical stability but an increase of 1 H NMR fatty acid molecular diffusion coefficient. The TAG-DHA formulation, stabilized by a mixture of soybean phosphatidylcholine and Tween-80 that exhibited repulsive interactions at the interface as shown by monolayer surface pressure-area isotherms, presented an average droplet diameter of 203 nm and contained small particles (20 nm). It was characterized by less negative interface layer and lower supramolecular diffusion coefficient and was less physically stable than PL-DHA formulation. Both systems oxidized slowly at 37°C in the dark although a progressive disappearance of tocopherols was observed with an higher rate in PL-DHA formulations.

These formulations could be used as delivery systems with FA compositions equilibrated according to nutritional recommendations. This work was carried out within the project AGEcaninox, ANR-08-ALIA-002

12th

  Euro Fed Lipid Congress, Montpellier, FRANCE Session: Physical Chemistry, Lipid Biophysics and Fomulations Structure and dynamics and stability of lipid-based delivery systems containing DHA esterified in triacylglycerols or phospholipids C. Genot, T.H. Kabri, C. Gaillard, L. Foucat, A. Meynier, INRA UR1268 Biopolymères Interactions Assemblages, Nantes, France Our objective was to formulate food delivery systems providing docosahexahenoic acid (DHA) DHA as phospholipids (PL-DHA) or triacylglycerols (TAG-DHA). Two formulations based on food grade oils and lecithins and mineral water (70 wt%) with similar and with equilibrated fatty acid compositions were formulated and characterized by laser diffraction, dynamic light scattering, transmission electron microscopy, coherent anti-stokes Raman spectroscopy (CARS) and proton NMR. The PL-DHA formulation,


est destinée au dépôt

Assessment of the synchrony between incubation and latency periods of peach trees infected by Plum pox virus

Loup Rimbaud, Agnès Delaunay, Sylvie Dallot, Sonia Borron, Samuel Soubeyrand, Gael Thébaud, Emmanuel Jacquot

To cite this version: The relative durations of incubation (the time between inoculation and symptom expression) and latency (the time between inoculation and infectiousness of the host) are poorly documented for plant diseases. However, the extent of asynchrony between the ends of these two periods (i.e. their mismatch) is a key determinant of the epidemic dynamics of many diseases and consequently is of primary interest in the design of disease management strategies. In order to assess this mismatch, we developed an estimation approach based on experiments and we applied it on sharka, a severe disease caused by Plum pox virus (PPV, genus Potyvirus) on trees belonging to the Prunus genus. Leaves of infected young peach trees were used individually as viral sources in aphid-mediated transmission experiments carried out at different time points post inoculation to bracket symptom onset. By fitting a binomial generalized linear model to the obtained transmission rates, we demonstrated that the first symptoms appear on a leaf 1 day before it rapidly becomes infectious. In addition, among the symptomatic leaves, we found a positive correlation between symptom intensity and transmission rate. The latter result strengthens the conclusion that, under our experimental conditions, latency and incubation of PPV infection are almost synchronized.

Key words: infectious potential, latent period, Myzus persicae, Potyviridae, symptom severity

01602151 Talk: Assessment of the synchrony between incubation and latency periods of peach trees infected by Plum pox virus

	Loup Rimbaud 1 , Agnès Delaunay 2 , Sylvie Dallot 2 , Sonia Borron 2 , Samuel
	Soubeyrand 3 , Gaël Thébaud 2 , Emmanuel Jacquot 2
	1 Montpellier SupAgro, UMR BGPI, Cirad TA A-54/K, F-34398 Montpellier Cedex 5
	2 INRA, UMR BGPI, Cirad TA A-54/K, F-34398 Montpellier Cedex 5
	3 INRA, UR 546 Biostatistics and Spatial Processes, F-84914 Avignon Cedex 9
	loup.rimbaud@supagro.inra.fr

Loup Rimbaud, Agnès Delaunay, Sylvie Dallot, Sonia Borron, Samuel Soubeyrand, et al.. Assessment of the synchrony between incubation and latency periods of peach trees infected by Plum pox virus. 15. Rencontres de Virologie Végétale (RVV), Centre de Coopération Internationale en Recherche Agronomique pour le Développement (CIRAD). Montpellier, FRA. Institut de Recherche pour le Développement (IRD), FRA., Jan 2015, Aussois, France. hal-